% Complex Analytic and Differential Geometry, Chapter I % J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France \magnification=1200 \input ./cpmono01.ini %%\vsize=17cm \voffset=0.3cm \hoffset=-0.3cm \vsize=20.4cm \hsize=13.5cm %%\hsize=11cm \def\ii{{\rm i}} \def\bu{{\scriptstyle\bullet}} \let\wh=\widehat \let\wt=\widetilde \let\ol=\overline \let\ssm=\smallsetminus \def\soul{{\raise-4.6pt\hbox{$-$}\kern-7pt}} \def\compact{\subset\!\subset} \def\ort{\hbox{~\vrule width4.0pt height0.4pt depth0pt \vrule width0.4pt height6.0pt depth0pt~$\,$}} \def\lhra{\lhook\joinrel\longrightarrow} \def\srelbar{\mathrel{\smash{\hbox{\rm-}}}} \def\merto{\srelbar\,\srelbar\,\rightarrow} \def\larex#1 {\hbox to #1 pt{\rightarrowfill}} \def\mapstex#1 {\hbox to #1 pt{$\mapstochar$\rightarrowfill}} \def\buildo#1\over#2{\mathrel{\mathop{\null#2}\limits^{#1}}} \def\buildu#1\under#2{\mathrel{\mathop{\null#2}\limits_{#1}}} \catcode`@=11 \def\cmalign#1{\null\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$&&$##$\hfil\crcr 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\def\cW{{\Cal W}} \def\cX{{\Cal X}} \def\cY{{\Cal Y}} \def\cZ{{\Cal Z}} \special{header=/home/demailly/psinputs/mathdraw/grlib.ps} \newdimen\figmm \figmm=0.694444mm \long\def\InsertFig#1 #2 #3 #4\EndFig{\par \hbox{\hskip #1\figmm$\vbox to #2\figmm{\vfil\special{" 0.694444 dup scale #3}}#4$}} \long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1\figmm\raise#2\figmm\hbox{#3}}} \def\rahmen#1{\vskip#1truecm} \def\ga{{\frak a}} \def\gm{{\frak m}} \def\stimes{\mathop{\kern0.7pt \vrule height 0.4pt depth 0pt width 5pt\kern-5pt \vrule height 5.4pt depth -5pt width 5pt\kern-5pt \vrule height 5.4pt depth 0pt width 0.4pt\kern4.6pt \vrule height 5.4pt depth 0pt width 0.4pt\kern-6.5pt \raise0.3pt\hbox{$\times$}\kern-0.7pt}\nolimits} \def\hotimes{\mathrel{\widehat\otimes}} \catcode`@=11 \def\rightarrowfil{\m@th\mathord-\mkern-6mu% \cleaders\hbox{$\mkern-2mu\mathord-\mkern-2mu$}\hfill \mkern-6mu\mathord\rightarrow} \catcode`@=12 \def\gge{{\scriptscriptstyle\ge}} \def\Ll{\langle\!\langle} \def\Gg{\rangle\!\rangle} \def\lraww{\mathrel{\rlap{$\longrightarrow$}\kern-1pt\longrightarrow}} \def\hookdown{\big\downarrow\kern-25pt \raise6.3pt\hbox{$\scriptscriptstyle\cap$}} \def\begarrow{\smash{\kern-0.5pt\vrule height-1.6pt depth2pt width4pt}} \def\Ker{\mathop{\rm Ker}\nolimits} \def\Id{\mathop{\rm Id}\nolimits} \def\Vol{\mathop{\rm Vol}\nolimits} \def\rk{\mathop{\rm rk}\nolimits} \def\Tr{\mathop{\rm Tr}\nolimits} \def\deg{\mathop{\rm deg}\nolimits} \def\Re{\mathop{\rm Re}\nolimits} \def\Im{\mathop{\rm Im}\nolimits} \def\mod{\mathop{\rm mod}\nolimits} \def\codim{\mathop{\rm codim}\nolimits} \def\Supp{\mathop{\rm Supp}\nolimits} \def\Coker{\mathop{\rm Coker}\nolimits} \def\Hom{\mathop{\rm Hom}\nolimits} \def\End{\mathop{\rm End}\nolimits} \def\Aut{\mathop{\rm Aut}\nolimits} \def\Ext{\mathop{\rm Ext}\nolimits} \def\aff{{\rm aff}} \def\BM{{\rm BM}} \def\Diag{{\rm Diag}} \def\Spm{\mathop{\rm Spm}\nolimits} \def\Proj{\mathop{\rm Proj}\nolimits} \def\Cycl{\mathop{\rm Cycl}\nolimits} \def\nn{{\rm n\hbox{\sevenrm-}n}} \def\norm{{\rm norm}} \def\alg{{\rm alg}} \def\red{{\rm red}} \def\cHom{{\cal H}{\it om}} \def\GL{{\rm GL}} \def\Coker{\mathop{\rm Coker}\nolimits} \def\stred{{\rm st\hbox{\sevenrm-}red}} \def\topdim{{\rm topdim}\,} \def\Tor{{\rm Tor}} \def\Ext{{\rm Ext}} \def\pr{{\rm pr}} \def\bu{\bullet} \def\Gl{{\rm Gl}} \def\Herm{{\rm Herm}} \def\BC{{\rm BC}} \def\DR{{\rm DR}} \def\Gr{{\rm Gr}} \def\prim{{\rm prim}} \def\rank{\mathop{\rm rank}\nolimits} \def\Jac{\mathop{\rm Jac}\nolimits} \def\Alb{\mathop{\rm Alb}\nolimits} \def\Prim{\mathop{\rm Prim}\nolimits} \def\Nak{{\rm Nak}} \def\Grif{{\rm Grif}} \def\Dom{\hbox{\rm Dom}\,} \def\Ric{{\rm Ricci}} \def\wtP{\smash{\wt P}} \def\omcirc{({\ol\Omega})^\circ} \def\gm{{\frak m}} \def\red{{\rm red}} \def\toor{\hbox{\rm T\^or}} \def\Coker{{\rm Coker}} \def\loc{{\rm loc}} \def\ord{{\rm ord}} \def\reg{{\rm reg}} \def\sing{{\rm sing}} \def\Sh{{\rm Sh}} \def\Psh{{\rm Psh}} \def\div{{\rm div}} \def\Div{{\rm Div}} \def\DR{{\rm DR}} \def\pr{{\rm pr}} \def\Vect{{\rm Vect}} \let\bC=\bbbc \let\bH=\bbbh \let\bK=\bbbk \let\bN=\bbbn \let\bP=\bbbp \let\bQ=\bbbq \let\bR=\bbbr \let\bZ=\bbbz \let\bOne=\bbbone $${}$$ \vskip80pt \headline={\hfill} \centerline{\headfont Complex Analytic and}\medskip \centerline{\headfont Differential Geometry}\bigskip\bigskip\bigskip \centerline{\tenrmmfour Jean-Pierre Demailly}\bigskip\bigskip\bigskip\bigskip \centerline{\tenrmmthree Universit\'e de Grenoble I}\medskip \centerline{\tenrmmthree Institut Fourier, UMR 5582 du CNRS}\medskip \centerline{\tenrmmthree 38402 Saint-Martin d'H\`eres, France} \bigskip\bigskip\bigskip\bigskip \centerline{Typeset on Monday September 1, 2008} \vfill\eject $\strut$ \vfill\eject \headline={\petit\def\newline{ }\def\fonote#1{}% \ifodd\pageno \hfil \botmark\unskip \kern\headlineindent \llap{\folio}% \else \leftheadline \fi}% %%\input agtoc.tex \titlea{Chapter I.}{\newline Complex Differential Calculus and Pseudoconvexity} \begpet This introductive chapter is mainly a review of the basic tools and concepts which will be employed in the rest of the book: differential forms, currents, holomorphic and plurisubharmonic functions, holomorphic convexity and pseudoconvexity. Our study of holomorphic convexity is principally concentrated here on the case of domains in $\bC^n$. The more powerful machinery needed for the study of general complex varieties (sheaves, positive currents, hermitian differential geometry) will be introduced in Chapters II to~V. Although our exposition pretends to be almost self-contained, the reader is assumed to have at least a vague familiarity with a few basic topics, such as differential calculus, measure theory and distributions, holomorphic functions of one complex variable, $\ldots\,$. Most of the necessary background can be found in the books of (Rudin, 1966) and (Warner, 1971); the basics of distribution theory can be found in Chapter~I of (H\"ormander 1963). On the other hand, the reader who has already some knowledge of complex analysis in several variables should probably bypass this chapter. \endpet \titleb{\S 1.}{Differential Calculus on Manifolds} \titlec{\S 1.A.}{Differentiable Manifolds} The notion of manifold is a natural extension of the notion of submanifold defined by a set of equations in $\bR^n$. However, as already observed by Riemann during the 19th century, it is important to define the notion of a manifold in a flexible way, without necessarily requiring that the underlying topological space is embedded in an affine space. The precise formal definition was first introduced by H.~Weyl in (Weyl, 1913). Let $m\in\bN$ and $k\in\bN\cup\{\infty,\omega\}$. We denote by $C^k$ the class of functions which are $k$-times differentiable with continuous derivatives if $k\ne\omega$, and by $C^\omega$ the class of real analytic functions. A {\it differentiable manifold} $M$ of real dimension $m$ and of class $C^k$ is a topological space (which we shall always assume Hausdorff and separable, i.e.\ possessing a countable basis of the topology), equipped with an atlas of class $C^k$ with values in $\bR^m$. An {\it atlas} of class $C^k$ is a collection of homeomorphisms $\tau_\alpha:U_\alpha\longrightarrow V_\alpha$, $\alpha\in I$, called {\it differentiable charts}, such that $(U_\alpha)_{\alpha\in I}$ is an open covering of $M$ and $V_\alpha$ an open subset of $\bR^m$, and such that for all $\alpha,\beta\in I$ the {\it transition map} $$\tau_{\alpha\beta}=\tau_\alpha\circ\tau_\beta^{-1}: \tau_\beta(U_\alpha\cap U_\beta)\longrightarrow\tau_\alpha(U_\alpha\cap U_\beta) \leqno(1.1)$$ is a $C^k$ diffeomorphism from an open subset of $V_\beta$ onto an open subset of $V_\alpha$ (see Fig.~1). Then the components $\tau_\alpha(x)=(x^\alpha_1,\ldots,x^\alpha_m)$ are called the {\it local coordinates} on $U_\alpha$ defined by the chart $\tau_\alpha$~; they are related by the transition relation $x^\alpha=\tau_{\alpha\beta} (x^\beta)$. \input epsfiles/fig_1_1.tex \vskip8mm \centerline{{\bf Fig.~I-1} Charts and transition maps} \vskip6mm If $\Omega\subset M$ is open and $s\in\bN\cup\{\infty,\omega\}$, $0\le s\le k$, we denote by $C^s(\Omega,\bR)$ the set of functions $f$ of class $C^s$ on $\Omega$, i.e.\ such that $f\circ \tau_\alpha^{-1}$ is of class $C^s$ on $\tau_\alpha(U_\alpha\cap\Omega)$ for each $\alpha$~; if $\Omega$ is not open, $C^s(\Omega,\bR)$ is the set of functions which have a $C^s$ extension to some neighborhood of~$\Omega$. A {\it tangent vector} $\xi$ at a point $a\in M$ is by definition a differential operator acting on functions, of the type $$C^1(\Omega,\bR)\ni f\longmapsto \xi\cdot f=\sum_{1\le j\le m}\xi_j\,{\partial f\over\partial x_j}(a)$$ in any local coordinate system $(x_1,\ldots,x_m)$ on an open set $\Omega\ni a$. We then simply write $\xi=\sum\xi_j\,\partial/\partial x_j$. For every $a\in\Omega$, the $n$-tuple $(\partial/\partial x_j)_{1\le j\le m}$ is therefore a basis of the {\it tangent space} to $M$ at $a$, which we denote by $T_{M,a}$. The {\it differential} of a function $f$ at $a$ is the linear form on $T_{M,a}$ defined by $$df_a(\xi)=\xi\cdot f=\sum\xi_j\,\partial f/\partial x_j(a),\qquad \forall\xi\in T_{M,a}.$$ In particular $dx_j(\xi)=\xi_j$ and we may write $df=\sum(\partial f/\partial x_j)dx_j$. Therefore $(dx_1,\ldots,dx_m)$ is the dual basis of $(\partial/\partial x_1,\ldots, \partial/\partial x_m)$ in the cotangent space $T^\star_{M,a}$. The disjoint unions $T_M=\bigcup_{x\in M}T_{M,x}$ and $T^\star_M= \bigcup_{x\in M}T^\star_{M,x}$ are called the {\it tangent} and {\it cotangent bundles} of~$M$. If $\xi$ is a vector field of class $C^s$ over $\Omega$, that is, a map $x\mapsto\xi(x)\in T_{M,x}$ such that $\xi(x)=\sum\xi_j(x)\,\partial/ \partial x_j$ has $C^s$ coefficients, and if $\eta$ is another vector field of class $C^s$ with $s\ge 1$, the {\it Lie bracket} $[\xi,\eta]$ is the vector field such that $$[\xi,\eta]\cdot f=\xi\cdot (\eta\cdot f)-\eta\cdot(\xi\cdot f).\leqno(1.2)$$ In coordinates, it is easy to check that $$[\xi,\eta]=\sum_{1\le j,k\le m}\Big(\xi_j{\partial\eta_k\over \partial x_j}-\eta_j{\partial\xi_k\over\partial x_j}\Big)\, {\partial\over\partial x_k}.\leqno(1.3)$$ \titlec{\S 1.B.}{Differential Forms} A differential form $u$ of degree $p$, or briefly a $p$-form over $M$, is a map $u$ on $M$ with values $u(x)\in\Lambda^pT^\star_{M,x}$. In a coordinate open set $\Omega\subset M$, a differential $p$-form can be written $$u(x)=\sum_{|I|=p}u_I(x)\,dx_I,$$ where $I=(i_1,\ldots,i_p)$ is a multi-index with integer components, \hbox{$i_1<\ldots \varepsilon\}$, converges weakly to $T$ as $\varepsilon$ tends to $0$. Indeed, $\langle T\star\rho_\varepsilon,f\rangle=\langle T, \rho_\varepsilon\star f\rangle$ and $\rho_\varepsilon\star f$ converges to $f$ in $\cD^p(\Omega)$ with respect to all seminorms $p^s_K$. \titled{\S 2.D.4. Poincar\'e Lemma for Currents.} Let $T\in{}^s\cD^{\prime\,q}(\Omega)$ be a closed current on an open set $\Omega\subset\bR^m$. We first show that $T$ is cohomologous to a smooth form. In fact, let $\psi\in C^\infty(\bR^m)$ be a cut-off function such that $\Supp\psi\subset\ol\Omega$, $0<\psi\le 1$ and $|d\psi|\le 1$ on $\Omega$. For any vector $v\in B(0,1)$ we set $$F_v(x)=x+\psi(x)v.$$ Since $x\mapsto\psi(x)v$ is a contraction, $F_v$ is a diffeomorphism of $\bR^m$ which leaves $\complement\Omega$ invariant pointwise, so $F_v(\Omega)=\Omega$. This diffeomorphism is homotopic to the identity through the homotopy $H_v(t,x)=F_{tv}(x):[0,1]\times\Omega \longrightarrow\Omega$ which is proper for every $v$. Formula (2.22) implies $$(F_v)_\star T-T=d\big((H_v)_\star([0,1]\otimes T)\big).$$ After averaging with a smoothing kernel $\rho_\varepsilon(v)$ we get $\Theta-T=dS$ where $$\Theta=\int_{B(0,\varepsilon)}(F_v)_\star T\,\rho_\varepsilon(v)\,dv, ~~~~S=\int_{B(0,\varepsilon)}(H_v)_\star([0,1]\otimes T) \,\rho_\varepsilon(v)\,dv.$$ Then $S$ is a current of the same order $s$ as $T$ and $\Theta$ is smooth. Indeed, for $u\in\cD^p(\Omega)$ we have $$\langle\Theta,u\rangle=\langle T,u_\varepsilon\rangle~~~~\hbox{\rm where}~~ u_\varepsilon(x)=\int_{B(0,\varepsilon)}F_v^\star u(x)\, \rho_\varepsilon(v)\,dv~;$$ we can make a change of variable $z=F_v(x)\Leftrightarrow v=\psi(x)^{-1}(z-x)$ in the last integral and perform derivatives on $\rho_\varepsilon$ to see that each seminorm $p^t_K(u_\varepsilon)$ is controlled by the sup norm of $u$. Thus $\Theta$ and all its derivatives are currents of order $0$, so $\Theta$ is smooth. Now we have $d\Theta=0$ and by the usual Poincar\'e lemma (1.22) applied to $\Theta$ we obtain \begstat{(2.24) Theorem} Let $\Omega\subset\bR^m$ be a starshaped open subset and $T\in{}^s\cD^{\prime\,q}(\Omega)$ a current of degree $q\ge 1$ and order $s$ such that $dT=0$. There exists a current $S\in{}^s\cD^{\prime\,q-1} (\Omega)$ of degree $q-1$ and order $\le s$ such that $dS=T$ on $\Omega$.\qed \endstat \titleb{\S 3.}{Holomorphic Functions and Complex Manifolds} \titlec{\S 3.A.}{Cauchy Formula in One Variable} We start by recalling a few elementary facts in one complex variable theory. Let $\Omega\subset\bC$ be an open set and let $z=x+\ii y$ be the complex variable, where $x,y\in\bR$. If $f$ is a function of class $C^1$ on $\Omega$, we have $$df={\partial f\over\partial x}\,dx+{\partial f\over\partial y}\,dy ={\partial f\over\partial z}\,dz+{\partial f\over\partial\ol z}\,d\ol z$$ with the usual notations $${\partial\over\partial z}={1\over 2}\Big({\partial\over\partial x} -\ii{\partial\over\partial y}\Big),~~~~ {\partial\over\partial\ol z}={1\over 2}\Big({\partial\over\partial x} +\ii{\partial\over\partial y}\Big).\leqno(3.1)$$ The function $f$ is holomorphic on $\Omega$ if $df$ is $\bC$-linear, that is, $\partial f/\partial\ol z=0$. \begstat{(3.2) Cauchy formula} Let $K\subset\bC$ be a compact set with piecewise $C^1$ boun\-dary $\partial K$. Then for every $f\in C^1(K,\bC)$ $$f(w)={1\over 2\pi\ii}\int_{\partial K}{f(z)\over z-w}\,dz- \int_K{1\over\pi(z-w)}\,{\partial f\over\partial\ol z}\,d\lambda(z),~~~~ w\in K^\circ$$ where $d\lambda(z)={\ii\over 2}dz\wedge d\ol z=dx\wedge dy$ is the Lebesgue measure on $\bC$. \endstat \begproof{} Assume for simplicity $w=0$. As the function $z\mapsto 1/z$ is locally integrable at $z=0$, we get $$\eqalign{ \int_K{1\over\pi z}\,{\partial f\over\partial\ol z}\,d\lambda(z) &=\lim_{\varepsilon\to 0}\int_{K\ssm D(0,\varepsilon)} {1\over\pi z}\,{\partial f\over\partial\ol z}\,{\ii\over 2}dz\wedge d\ol z\cr &=\lim_{\varepsilon\to 0}\int_{K\ssm D(0,\varepsilon)} d\Big[{1\over 2\pi\ii}\,f(z)\,{dz\over z}\Big]\cr &={1\over 2\pi\ii}\int_{\partial K}f(z)\,{dz\over z}- \lim_{\varepsilon\to 0}{1\over 2\pi\ii}\int_{\partial D(0,\varepsilon)} f(z)\,{dz\over z}\cr}$$ by Stokes' formula. The last integral is equal to ${1\over 2\pi}\int_0^{2\pi}f(\varepsilon e^{\ii\theta})\,d\theta$ and converges to $f(0)$ as $\varepsilon$ tends to $0$.\qed \endproof When $f$ is holomorphic on $\Omega$, we get the usual Cauchy formula $$f(w)={1\over 2\pi\ii}\int_{\partial K}{f(z)\over z-w}\,dz,~~~~ w\in K^\circ,\leqno(3.3)$$ from which many basic properties of holomorphic functions can be derived: power and Laurent series expansions, Cauchy residue formula, $\ldots$ Another interesting consequence is: \begstat{(3.4) Corollary} The $L^1_\loc$ function $E(z)=1/\pi z$ is a fundamental solution of the operator $\partial/\partial\ol z$ on $\bC$, i.e.\ $\partial E/\partial\ol z=\delta_0$ $($Dirac measure at $0)$. As a consequence, if $v$ is a distribution with compact support in $\bC$, then the convolution $u=(1/\pi z)\star v$ is a solution of the equation $\partial u/\partial\ol z=v$. \endstat \begproof{} Apply (3.2) with $w=0$, $f\in\cD(\bC)$ and $K\supset{\rm Supp}\,f$, so that $f=0$ on the boundary $\partial K$ and $f(0)=\langle 1/\pi z,-\partial f/\partial\ol z\rangle$.\qed \endproof \begstat{(3.5) Remark} \rm It should be observed that this formula cannot be used to solve the equation $\partial u/\partial\ol z=v$ when $\Supp v$ is not compact; moreover, if $\Supp v$ is compact, a solution $u$ with compact support need not always exist. Indeed, we have a necessary condition $$\langle v,z^n\rangle=-\langle u,\partial z^n/\partial\ol z\rangle=0$$ for all integers $n\ge 0$. Conversely, when the necessary condition $\langle v,z^n\rangle=0$ is satisfied, the canonical solution $u=(1/\pi z)\star v$ has compact support: this is easily seen by means of the power series expansion $(w-z)^{-1}=\sum z^nw^{-n-1}$, if we suppose that $\Supp v$ is contained in the disk $|z|R$. \endstat \titlec{\S 3.B.}{Holomorphic Functions of Several Variables} Let $\Omega\subset\bC^n$ be an open set. A function $f:\Omega\to\bC$ is said to be holomorphic if $f$ is continuous and separately holomorphic with respect to each variable, i.e.\ $z_j\mapsto f(\ldots,z_j,\ldots)$ is holomorphic when $z_1,\ldots,z_{j-1}$, $z_{j+1},\ldots,z_n$ are fixed. The set of holomorphic functions on $\Omega$ is a ring and will be denoted $\cO(\Omega)$. We first extend the Cauchy formula to the case of polydisks. The open polydisk $D(z_0,R)$ of center $(z_{0,1},\ldots,z_{0,n})$ and (multi)radius $R=(R_1,,\ldots,R_n)$ is defined as the product of the disks of center $z_{0,j}$ and radius $R_j>0$ in each factor~$\bC$~: $$D(z_0,R)=D(z_{0,1},R_1)\times\ldots\times D(z_{0,n},R_n)\subset\bC^n. \leqno(3.6)$$ The {\it distinguished boundary} of $D(z_0,R)$ is by definition the product of the boundary circles $$\Gamma(z_0,R)=\Gamma(z_{0,1},R_1)\times\ldots\times\Gamma(z_{0,n},R_n). \leqno(3.7)$$ It is important to observe that the distinguished boundary is smaller than the topological boundary $\partial D(z_0,R)= \bigcup_j\{z\in\ol D(z_0,R)\,;\,|z_j-z_{0,j}|=R_j\}$ when $n\ge 2$. By induction on $n$, we easily get the \begstat{(3.8) Cauchy formula on polydisks} If $\ol D(z_0,R)$ is a closed polydisk contained in $\Omega$ and $f\in\cO(\Omega)$, then for all $w\in D(z_0,R)$ we have $$f(w)={1\over(2\pi\ii)^n}\int_{\Gamma(z_0,R)} {f(z_1,\ldots,z_n)\over(z_1-w_1)\ldots(z_n-w_n)}\,dz_1\ldots dz_n.\eqno\qed$$ \endstat The expansion $(z_j-w_j)^{-1}=\sum (w_j-z_{0,j})^{\alpha_j} (z_j-z_{0,j})^{-\alpha_j-1}$, $\alpha_j\in\bN$, $1\le j\le n$, shows that $f$ can be expanded as a convergent power series $f(w)=\sum_{\alpha\in\bN^n}a_\alpha(w-z_0)^\alpha$ over the polydisk $D(z_0,R)$, with the standard notations $z^\alpha=z_1^{\alpha_1}\ldots z_n^{\alpha_n}$, $\alpha!=\alpha_1!\ldots\alpha_n!$ and with $$a_\alpha= {1\over(2\pi\ii)^n}\int_{\Gamma(z_0,R)}{f(z_1,\ldots,z_n)\,dz_1\ldots dz_n \over(z_1-z_{0,1})^{\alpha_1+1}\ldots(z_n-z_{0,n})^{\alpha_n+1}} ={f^{(\alpha)}(z_0)\over\alpha!}.\leqno(3.9)$$ As a consequence, $f$ is holomorphic over $\Omega$ if and only if $f$ is $\bC$-analytic. Arguments similar to the one variable case easily yield the \begstat{(3.10) Analytic continuation theorem} If $\Omega$ is connected and if there exists a point $z_0\in\Omega$ such that $f^{(\alpha)}(z_0)=0$ for all $\alpha\in\bN^n$, then $f=0$ on~$\Omega$.\qed \endstat Another consequence of (3.9) is the {\it Cauchy inequality} $$|f^{(\alpha)}(z_0)|\le{\alpha!\over R^\alpha}\sup_{\Gamma(z_0,R)}|f|,~~~~ \ol D(z_0,R)\subset\Omega,\leqno(3.11)$$ From this, it follows that every bounded holomorphic function on $\bC^n$ is constant (Liouville's theorem), and more generally, every holomorphic function $F$ on $\bC^n$ such that $|F(z)|\le A(1+|z|)^B$ with suitable constants $A,B\ge 0$ is in fact a polynomial of total degree $\le B$. We endow $\cO(\Omega)$ with the topology of uniform convergence on compact sets $K\compact\Omega$, that is, the topology induced by $C^0(\Omega,\bC)$. Then $\cO(\Omega)$ is closed in $C^0(\Omega,\bC)$. The Cauchy inequalities (3.11) show that all derivations $D^\alpha$ are continuous operators on $\cO(\Omega)$ and that any sequence $f_j\in\cO(\Omega)$ that is uniformly bounded on all compact sets $K\compact\Omega$ is locally equicontinuous. By Ascoli's theorem, we obtain \begstat{(3.12) Montel's theorem} Every locally uniformly bounded sequence $(f_j)$ in $\cO(\Omega)$ has a convergent subsequence $(f_{j(\nu)})$. \endstat In other words, bounded subsets of the Fr\'echet space $\cO(\Omega)$ are relatively compact (a Fr\'echet space possessing this property is called a Montel space). \titlec{\S 3.C.}{Differential Calculus on Complex Analytic Manifolds} A {\it complex analytic manifold} $X$ of dimension $\dim_\bC X=n$ is a differentiable manifold equipped with a holomorphic atlas $(\tau_\alpha)$ with values in $\bC^n$~; this means by definition that the transition maps $\tau_{\alpha\beta}$ are holomorphic. The tangent spaces $T_{X,x}$ then have a natural complex vector space structure, given by the coordinate isomorphisms $$d\tau_\alpha(x):T_{X,x}\longrightarrow\bC^n,\qquad U_\alpha\ni x\,;$$ the induced complex structure on $T_{X,x}$ is indeed independent of $\alpha$ since the differentials $d\tau_{\alpha\beta}$ are $\bC$-linear isomorphisms. We denote by $T^\bR_X$ the underlying real tangent space and by $J\in{\rm End}(T^\bR_X)$ the {\it almost complex structure}, i.e.\ the operator of multiplication by $\ii=\sqrt{-1}$. If $(z_1,\ldots,z_n)$ are complex analytic coordinates on an open subset $\Omega\subset X$ and $z_k=x_k+\ii y_k$, then $(x_1,y_1,\ldots,x_n,y_n)$ define real coordinates on $\Omega$, and $T^\bR_{X\restriction\Omega}$ admits $(\partial/\partial x_1$, $\partial/\partial y_1$, $\ldots$, $\partial/\partial x_n$, $\partial/\partial y_n)$ as a basis$\,$; the almost complex structure is given by $J(\partial/\partial x_k)=\partial/\partial y_k$, $J(\partial/\partial y_k)=-\partial/\partial x_k$. The complexified tangent space $\bC\otimes T_X=\bC\otimes_\bR T^\bR_X= T^\bR_X\oplus\ii T^\bR_X$ splits into conjugate complex subspaces which are the eigenspaces of the complexified endomorphism ${\rm Id}\otimes J$ associated to the eigenvalues $\ii$ and $-\ii$. These subspaces have respective bases $$~{\partial\over\partial z_k}={1\over 2}\Big({\partial\over\partial x_k}- \ii{\partial\over\partial y_k}\Big),~~~~ {\partial\over\partial\ol z_k}={1\over 2}\Big({\partial\over\partial x_k}+ \ii{\partial\over\partial y_k}\Big),~~~~1\le k\le n\leqno(3.13)$$ and are denoted $T^{1,0}X$ ({\it holomorphic vectors} or {\it vectors of type} $(1,0)$) and $T^{0,1}X$ ({\it antiholomorphic vectors} or {\it vectors of type} $(0,1)$). The subspaces $T^{1,0}X$ and $T^{0,1}X$ are canonically isomorphic to the complex tangent space $T_X$ (with complex structure $J$) and its conjugate $\ol{T_X}$ (with conjugate complex structure $-J$), via the $\bC$-linear embeddings $$\cmalign{T_X&\longrightarrow T^{1,0}_X\subset\bC\otimes T_X, \hfill\ol{T_X}&\longrightarrow T^{0,1}_X\subset\bC\otimes T_X\cr \hfill\xi&\longmapsto{1\over2}(\xi-\ii J\xi),~~~~~ \xi&\longmapsto{1\over2}(\xi+\ii J\xi).\cr}$$ We thus have a canonical decomposition $\bC\otimes T_X=T^{1,0}_X\oplus T^{0,1}_X\simeq T_X\oplus\ol{T_X}$, and by duality a decomposition $$\Hom_\bR(T^\bR_X;\bC)\simeq\Hom_\bC(\bC\otimes T_X;\bC)\simeq T^\star_X\oplus\ol{T^\star_X}$$ where $T^\star_X$ is the space of $\bC$-linear forms and $\ol{T^\star_X}$ the space of conjugate $\bC$-linear forms. With these notations, $(dx_k,dy_k)$ is a basis of $\Hom_\bR(T_\bR X,\bC)$, $(dz_j)$ a basis of $T^\star_X$, $(d\ol z_j)$ a basis of $\ol{T^\star_X}$, and the differential of a function $f\in C^1(\Omega,\bC)$ can be written $$df=\sum_{k=1}^n {\partial f\over\partial x_k}\,dx_k+{\partial f\over\partial y_k}\,dy_k =\sum_{k=1}^n {\partial f\over\partial z_k}\,dz_k+{\partial f\over\partial\ol z_k}\, d\ol z_k.\leqno(3.14)$$ The function $f$ is holomorphic on $\Omega$ if and only if $df$ is $\bC$-linear, i.e.\ if and only if $f$ satisfies the {\it Cauchy-Riemann equations} $\partial f/\partial\ol z_k=0$ on $\Omega$, $1\le k\le n$. We still denote here by $\cO(X)$ the algebra of holomorphic functions on $X$. Now, we study the basic rules of complex differential calculus. The complexified exterior algebra $\bC\otimes_\bR\Lambda^\bu_\bR(T_X^\bR)^\star =\Lambda^\bu_\bC(\bC\otimes T_X)^\star$ is given by $$\Lambda^k(\bC\otimes T_X)^\star=\Lambda^k\big(T_X\oplus\ol{T_X}\big)^\star =\bigoplus_{p+q=k}\Lambda^{p,q}T^\star_X,~~~~0\le k\le 2n$$ where the exterior products are taken over $\bC$, and where the components $\Lambda^{p,q}T^\star_X$ are defined by $$\Lambda^{p,q}T^\star_X=\Lambda^p T^\star_X\otimes \Lambda^q\ol{T^\star_X}.\leqno(3.15)$$ A complex differential form $u$ on $X$ is said to be of {\it bidegree} or {\it type} $(p,q)$ if its value at every point lies in the component $\Lambda^{p,q}T^\star_X$~; we shall denote by $C^s(\Omega,\Lambda^{p,q} T_X^\star)$ the space of differential forms of bidegree $(p,q)$ and class $C^s$ on any open subset $\Omega$ of $X$. If $\Omega$ is a coordinate open set, such a form can be written $$u(z)=\sum_{|I|=p,|J|=q}u_{I,J}(z)\,dz_I\wedge d\ol z_J,~~~~ u_{I,J}\in C^s(\Omega,\bC).$$ This writing is usually much more convenient than the expression in terms of the real basis $(dx_I\wedge dy_J)_{|I|+|J|=k}$ which is not compatible with the splitting of $\Lambda^k T^\star_\bC X$ in its $(p,q)$ components. Formula (3.14) shows that the exterior derivative $d$ splits into $d=d'+d''$, where $$\leqalignno{ d'&:C^\infty(X,\Lambda^{p,q}T^\star_X)\longrightarrow C^\infty(X,\Lambda^{p+1,q}T^\star_X),\cr d''&:C^\infty(X,\Lambda^{p,q}T^\star_X)\longrightarrow C^\infty(X,\Lambda^{p,q+1}T^\star_X),\cr d'u&=\sum_{I,J}\sum_{1\le k\le n}{\partial u_{I,J}\over\partial z_k}\,dz_k \wedge dz_I\wedge d\ol z_J,&(3.16')\cr d''u&=\sum_{I,J}\sum_{1\le k\le n.}{\partial u_{I,J}\over\partial\ol z_k} \,d\ol z_k\wedge dz_I\wedge d\ol z_J.&(3.16'')\cr}$$ The identity $d^2=(d'+d'')^2=0$ is equivalent to $$d^{\prime 2} = 0,~~~~d'd''+d''d' = 0,~~~~d^{\prime\prime 2}=0, \leqno(3.17)$$ since these three operators send $(p,q)$-forms in $(p+2,q)$, $(p+1,q+1)$ and $(p,q+2)$-forms, respectively. In particular, the operator $d''$ defines for each $p=0,1,\ldots,n$ a complex, called the {\it Dolbeault complex} $$C^\infty(X,\Lambda^{p,0}T^\star_X)\buildo d''\over\longrightarrow\cdots\longrightarrow C^\infty(X,\Lambda^{p,q}T^\star_X)\buildo d''\over\longrightarrow C^\infty(X,\Lambda^{p,q+1}T^\star_X)$$ and corresponding {\it Dolbeault cohomology groups} $$H^{p,q}(X,\bC)={\Ker d^{\prime\prime\,p,q}\over \Im d^{\prime\prime\,p,q-1}},\leqno(3.18)$$ with the convention that the image of $d''$ is zero for $q=0$. The coho\-mo\-logy group $H^{p,0}(X,\bC)$ consists of $(p,0)$-forms $u=\sum_{|I|=p}u_I(z)\,dz_I$ such that $\partial u_I/\partial\ol z_k=0$ for all $I,k$, i.e.\ such that all coefficients $u_I$ are holomorphic. Such a form is called a {\it holomorphic $p$-form} on $X$. Let $F:X_1\longrightarrow X_2$ be a holomorphic map between complex manifolds. The pull-back $F^\star u$ of a $(p,q)$-form $u$ of bidegree $(p,q)$ on $X_2$ is again homogeneous of bidegree $(p,q)$, because the components $F_k$ of $F$ in any coordinate chart are holomorphic, hence $F^\star dz_k=dF_k$ is \hbox{$\bC$-linear.} In particular, the equality $dF^\star u=F^\star du$ implies $$d'F^\star u=F^\star d'u,~~~~d''F^\star u=F^\star d''u. \leqno(3.19)$$ Note that these commutation relations are no longer true for a non holomorphic change of variable. As in the case of the De Rham cohomology groups, we get a pull-back morphism $$F^\star: H^{p,q}(X_2,\bC)\longrightarrow H^{p,q}(X_1,\bC).$$ The rules of complex differential calculus can be easily extended to currents. We use the following notation. \begstat{(3.20) Definition} There are decompositions $$\cD^k(X,\bC)=\bigoplus_{p+q=k}\cD^{p,q}(X,\bC),~~~~ \cD'_k(X,\bC)=\bigoplus_{p+q=k}\cD'_{p,q}(X,\bC).$$ The space $\cD'_{p,q}(X,\bC)$ is called the space of currents of bidimension $(p,q)$ and bidegree $(n-p,n-q)$ on~$X$, and is also denoted $\cD^{\prime\,n-p,n-q}(X,\bC)$. \endstat \titlec{\S 3.D.}{Newton and Bochner-Martinelli Kernels} The {\it Newton kernel} is the elementary solution of the usual Laplace operator $\Delta=\sum\partial^2/\partial x_j^2$ in~$\bR^m$. We first recall a construction of the Newton kernel. Let $d\lambda=dx_1\ldots dx_m$ be the Lebesgue measure on $\bR^m$. We denote by $B(a,r)$ the euclidean open ball of center $a$ and radius $r$ in $\bR^m$ and by $S(a,r)=\partial B(a,r)$ the corresponding sphere. Finally, we set $\alpha_m=\Vol\big(B(0,1)\big)$ and $\sigma_{m-1}=m\alpha_m$ so that $$\Vol\big(B(a,r)\big)=\alpha_mr^m,~~~~{\rm Area}\big(S(a,r)\big)= \sigma_{m-1}r^{m-1}.\leqno(3.21)$$ The second equality follows from the first by derivation. An explicit computation of the integral $\int_{\bR^m}e^{-|x|^2}d\lambda(x)$ in polar coordinates shows that $\alpha_m=\pi^{m/2}/(m/2)!$ where $x!=\Gamma(x+1)$ is the Euler Gamma function. The {\it Newton kernel} is then given by: $$\cases{ N(x)=\displaystyle{1\over 2\pi}\,\log|x|&if~~$m=2$,\cr N(x)=\displaystyle-{1\over(m-2)\sigma_{m-1}}\,|x|^{2-m} &if~~$m\ne 2$.\cr}\leqno(3.22)$$ The function $N(x)$ is locally integrable on $\bR^m$ and satisfies $\Delta N=\delta_0$. When $m=2$, this follows from Cor.~3.4 and the fact that $\Delta=4\partial^2/\partial z\partial\ol z$. When $m\ne 2$, this can be checked by computing the weak limit $$\eqalign{ \lim_{\varepsilon\to 0}~\Delta(|x|^2+\varepsilon^2)^{1-m/2}&= \lim_{\varepsilon\to 0}~m(2-m)\varepsilon^2(|x|^2+\varepsilon^2)^{-1-m/2}\cr &=m(2-m)\,I_m\,\delta_0\cr}$$ with~ $I_m=\int_{\bR^m}(|x|^2+1)^{-1-m/2}\,d\lambda(x)$. The last equality is easily seen by performing the change of variable $y=\varepsilon x$ in the integral $$\int_{\bR^m}\varepsilon^2(|x|^2+\varepsilon^2)^{-1-m/2} \,f(x)\,d\lambda(x)=\int_{\bR^m}(|y|^2+1)^{-1-m/2} \,f(\varepsilon y)\,d\lambda(y),$$ where $f$ is an arbitrary test function. Using polar coordinates, we find that $I_m=\sigma_{m-1}/m$ and our formula follows. The {\it Bochner-Martinelli kernel} is the $(n,n-1)$-differential form on $\bC^n$ with $L^1_\loc$ coefficients defined by $$\leqalignno{ k_\BM(z)&=c_n\sum_{1\le j\le n}(-1)^j {\ol z_j\,dz_1\wedge\ldots dz_n\wedge d\ol z_1\wedge\ldots \widehat{d\ol z_j}\ldots\wedge d\ol z_n\over|z|^{2n}},&(3.23)\cr c_n&=(-1)^{n(n-1)/2}{(n-1)!\over(2\pi\ii)^n}.\cr}$$ \begstat{(3.24) Lemma} $d''k_\BM=\delta_0$ on $\bC^n$. \endstat \begproof{} Since the Lebesgue measure on $\bC^n$ is $$d\lambda(z)=\bigwedge_{1\le j\le n}{\ii\over 2}dz_j\wedge d\ol z_j =\Big({\ii\over 2}\Big)^n(-1)^{\textstyle{n(n-1)\over 2}} dz_1\wedge\ldots dz_n\wedge d\ol z_1\wedge\ldots d\ol z_n,$$ we find $$\eqalignno{ d''k_\BM&=-{(n-1)!\over\pi^n}\sum_{1\le j\le n}{\partial\over\partial \ol z_j}\Big({\ol z_j\over |z|^{2n}}\Big)d\lambda(z)\cr &=-{1\over n(n-1)\alpha_{2n}}\sum_{1\le j\le n}{\partial^2\over\partial z_j\partial\ol z_j}\Big({1\over |z|^{2n-2}}\Big)d\lambda(z)\cr &=\Delta N(z)d\lambda(z)=\delta_0.&\square\cr}$$ \endproof We let $K_\BM(z,\zeta)$ be the pull-back of $k_\BM$ by the map $\pi:\bC^n\times\bC^n\to\bC^n$, $(z,\zeta)\longmapsto z-\zeta$. Then Formula (2.19) implies $$d''K_\BM=\pi^\star\delta_0=[\Delta],\leqno(3.25)$$ where $[\Delta]$ denotes the current of integration on the diagonal $\Delta\subset\bC^n\times\bC^n$. \begstat{(3.26) Koppelman formula} Let $\Omega\subset\bC^n$ be a bounded open set with piecewise $C^1$ boundary. Then for every $(p,q)$-form $v$ of class $C^1$ on $\ol\Omega$ we have $$\eqalign{ v(z)=\int_{\partial\Omega}&K_\BM^{p,q}(z,\zeta)\wedge v(\zeta)\cr &\quad{}+d''_z\int_\Omega K_\BM^{p,q-1}(z,\zeta)\wedge v(\zeta)+ \int_\Omega K_\BM^{p,q}(z,\zeta)\wedge d''v(\zeta)\cr}$$ on $\Omega$, where $K_\BM^{p,q}(z,\zeta)$ denotes the component of $K_\BM(z,\zeta)$ of type $(p,q)$ in~$z$ and $(n-p,n-q-1)$ in~$\zeta$. \endstat \begproof{} Given $w\in\cD^{n-p,n-q}(\Omega)$, we consider the integral $$\int_{\partial\Omega\times\Omega}K_\BM(z,\zeta)\wedge v(\zeta)\wedge w(z).$$ It is well defined since $K_\BM$ has no singularities on $\partial\Omega \times\Supp v\compact\partial\Omega\times\Omega$. Since $w(z)$ vanishes on $\partial\Omega$ the integral can be extended as well to $\partial(\Omega\times \Omega)$. As $K_\BM(z,\zeta)\wedge v(\zeta)\wedge w(z)$ is of total bidegree $(2n,2n-1)$, its differential $d'$ vanishes. Hence Stokes' formula yields $$\eqalign{ \int_{\partial\Omega\times\Omega}&K_\BM(z,\zeta)\wedge v(\zeta)\wedge w(z)=\int_{\Omega\times\Omega}d''\big(K_\BM(z,\zeta)\wedge v(\zeta)\wedge w(z)\big)\cr &=\int_{\Omega\times\Omega}d''K_\BM(z,\zeta)\wedge v(\zeta)\wedge w(z) -K^{p,q}_\BM(z,\zeta)\wedge d''v(\zeta)\wedge w(z)\cr &\qquad-(-1)^{p+q}\int_{\Omega\times\Omega}K^{p,q-1}_\BM(z,\zeta)\wedge v(\zeta)\wedge d''w(z).\cr}$$ By (3.25) we have $$\int_{\Omega\times\Omega}\!\!\!d''K_\BM(z,\zeta)\wedge v(\zeta)\wedge w(z)=\int_{\Omega\times\Omega}\![\Delta]\wedge v(\zeta)\wedge w(z)=\int_\Omega v(z)\wedge w(z)$$ Denoting $\langle~,~\rangle$ the pairing between currents and test forms on $\Omega$, the above equality is thus equivalent to $$\eqalign{ \langle\int_{\partial\Omega}K_\BM(z,\zeta)\wedge v(\zeta),w(z)\rangle &=\langle v(z)-\int_\Omega K^{p,q}_\BM(z,\zeta)\wedge d''v(\zeta),w(z) \rangle\cr &{}-(-1)^{p+q}\langle\int_\Omega K^{p,q-1}_\BM(z,\zeta)\wedge v(\zeta),d''w(z)\rangle,\cr}$$ which is itself equivalent to the Koppelman formula by integrating $d''v$ by parts.\qed \endproof \begstat{(3.27) Corollary} Let $v\in{}^s\cD^{p,q}(\bC^n)$ be a form of class $C^s$ with compact support such that $d''v=0$, $q\ge 1$. Then the $(p,q-1)$-form $$u(z)=\int_{\bC^n}K_\BM^{p,q-1}(z,\zeta)\wedge v(\zeta)$$ is a $C^s$ solution of the equation $d''u=v$. Moreover, if $(p,q)=(0,1)$ and $n\ge 2$ then $u$ has compact support, thus the Dolbeault cohomology group with compact support $H^{0,1}_c(\bC^n,\bC)$ vanishes for $n\ge 2$. \endstat \begproof{} Apply the Koppelman formula on a sufficiently large ball $\ol\Omega=\ol B(0,R)$ containing $\Supp v$. Then the formula immediately gives $d''u=v$. Observe that the coefficients of $K_\BM(z,\zeta)$ are $O(|z-\zeta|^{-(2n-1)})$, hence $|u(z)|=O(|z|^{-(2n-1)})$ at infinity. If $q=1$, then $u$ is holomorphic on $\bC^n\ssm\ol B(0,R)$. Now, this complement is a union of complex lines when $n\ge 2$, hence $u=0$ on $\bC^n\ssm\ol B(0,R)$ by Liouville's theorem.\qed \endproof \begstat{(3.28) Hartogs extension theorem} Let $\Omega$ be an open set in $\bC^n$, $n\ge 2$, and let $K\subset\Omega$ be a compact subset such that $\Omega\ssm K$ is connected. Then every holomorphic function $f\in\cO(\Omega\ssm K)$ extends into a function $\wt f\in\cO(\Omega)$. \endstat \begproof{} Let $\psi\in\cD(\Omega)$ be a cut-off function equal to $1$ on a neighborhood of~$K$. Set $f_0=(1-\psi)f\in C^\infty(\Omega)$, defined as $0$ on~$K$. Then $v=d''f_0=-fd''\psi$ can be extended by $0$ outside $\Omega$, and can thus be seen as a smooth $(0,1)$-form with compact support in~$\bC^n$, such that $d''v=0$. By Cor.~3.27, there is a smooth function $u$ with compact support in $\bC^n$ such that $d''u=v$. Then $\wt f=f_0-u\in\cO(\Omega)$. Now $u$ is holomorphic outside $\Supp\psi$, so $u$ vanishes on the unbounded component $G$ of $\bC^n\ssm\Supp\psi$. The boundary $\partial G$ is contained in $\partial\Supp\psi\subset \Omega\ssm K$, so $\wt f=(1-\psi)f-u$ coincides with $f$ on the non empty open set $\Omega\cap G\subset\Omega\ssm K$. Therefore $\wt f=f$ on the connected open set $\Omega\ssm K$.\qed \endproof A refined version of the Hartogs extension theorem due to Bochner will be given in Exercise~8.13. It shows that $f$ need only be given as a $C^1$ function on $\partial\Omega$, satisfying the tangential Cauchy-Riemann equations (a so-called {\it CR-function}). Then $f$ extends as a holomorphic function $\wt f\in\cO(\Omega)\cap C^0(\ol\Omega)$, provided that $\partial\Omega$ is connected. \titlec{\S 3.E.}{The Dolbeault-Grothendieck Lemma} We are now in a position to prove the Dolbeault-Grothendieck lemma (Dolbeault 1953), which is the analogue for $d''$ of the Poincar\'e lemma. The proof given below makes use of the Bochner-Martinelli kernel. Many other proofs can be given, e.g.\ by using a reduction to the one dimensional case in combination with the Cauchy formula~(3.2), see Exercise~8.5 or (H\"ormander 1966). \begstat{(3.29) Dolbeault-Grothendieck lemma} Let $\Omega$ be a neighborhood of $0$ in $\bC^n$ and $v\in{}^s\cE^{p,q}(\Omega,\bC)$, $[$resp.\ $v\in{}^s\cD^{\prime\,p,q}(\Omega,\bC)]$, such that $d''v=0$, where $1\le s\le\infty$. \medskip \item{\rm a)} If $q=0$, then $v(z)=\sum_{|I|=p}v_I(z)\,dz_I$ is a holomorphic $p$-form, i.e.\ a form whose coefficients are holomorphic functions. \smallskip \item{\rm b)} If $q\ge 1$, there exists a neighborhood $\omega\subset \Omega$ of $0$ and a form $u$ in ${}^s\cE^{p,q-1}(\omega,\bC)$ $[$resp.\ a current $u\in{}^s\cD^{\prime\,p,q-1}(\smash{\omega},\bC)]$ such that $d''u=v$ on~$\omega$. \vskip0pt \endstat \begproof{} We assume that $\Omega$ is a ball $B(0,r)\subset\bC^n$ and take for simplicity $r>1$ (possibly after a dilation of coordinates). We then set $\omega=B(0,1)$. Let $\psi\in\cD(\Omega)$ be a cut-off function equal to $1$ on $\omega$. The Koppelman formula (3.26) applied to the form $\psi v$ on $\Omega$ gives $$\psi(z)v(z)=d''_z\!\int_\Omega\! K_\BM^{p,q-1}(z,\zeta)\wedge \psi(\zeta)v(\zeta)+{}\!\int_\Omega\! K_\BM^{p,q}(z,\zeta)\wedge d''\psi(\zeta)\wedge v(\zeta).$$ This formula is valid even when $v$ is a current, because we may regularize $v$ as $v\star\rho_\varepsilon$ and take the limit. We introduce on $\bC^n\times\bC^n\times\bC^n$ the kernel $$K(z,w,\zeta)=c_n\sum_{j=1}^n{(-1)^j(w_j-\ol\zeta_j) \over((z-\zeta)\cdot(w-\ol\zeta))^n} \bigwedge\limits_k(dz_k-d\zeta_k)\wedge \bigwedge\limits_{k\ne j}(dw_k-d\ol\zeta_k).$$ By construction, $K_\BM(z,\zeta)$ is the result of the substitution $w=\ol z$ in $K(z,w,\zeta)$, i.e. $K_\BM=h^\star K$ where $h(z,\zeta)=(z,\ol z,\zeta)$. We denote by $K^{p,q}$ the component of $K$ of bidegree $(p,0)$ in $z$, $(q,0)$ in $w$ and $(n-p,n-q-1)$ in~$\zeta$. Then $K_\BM^{p,q}=h^\star K^{p,q}$ and we find $$v=d''u_0+g^\star v_1~~~~\hbox{\rm on $\omega$,}$$ where $g(z)=(z,\ol z)$ and $$\eqalign{ u_0(z)&=\int_\Omega K_\BM^{p,q-1}(z,\zeta)\wedge\psi(\zeta)v(\zeta),\cr v_1(z,w)&=\int_\Omega K^{p,q}(z,w,\zeta)\wedge d''\psi(\zeta)\wedge v(\zeta). \cr}$$ By definition of $K^{p,q}(z,w,\zeta)$, $v_1$ is holomorphic on the open set $$U=\big\{(z,w)\in\omega\times\omega\,;\,\forall\zeta\notin\omega,~ \Re(z-\zeta)\cdot(w-\ol\zeta)>0\big\},$$ which contains the ``conjugate-diagonal'' points $(z,\ol z)$ as well as the points $(z,0)$ and $(0,w)$ in $\omega\times\omega$. Moreover $U$ clearly has convex slices $(\{z\}\times\bC^n)\cap U$ and $(\bC^n\times\{w\})\cap U$. In particular $U$ is starshaped with respect to $w$, i.e. $$(z,w)\in U\Longrightarrow (z,tw)\in U,~~~~\forall t\in[0,1].$$ As $u_1$ is of type $(p,0)$ in $z$ and $(q,0)$ in $w$, we get $d''_z(g^\star v_1)=g^\star d_wv_1=0$, hence $d_wv_1=0$. For~$q=0$ we have $K_\BM^{p,q-1}=0$, thus $u_0=0$, and $v_1$ does not depend on $w$, thus $v$ is holomorphic on~$\omega$. For $q\ge 1$, we can use the homotopy formula (1.23) with respect to $w$ (considering $z$ as a parameter) to get a holomorphic form $u_1(z,w)$ of type $(p,0)$ in $z$ and $(q-1,0)$ in~$w$, such that $d_wu_1(z,w)=v_1(z,w)$. Then we get $d''g^\star u_1= g^\star d_wu_1=g^\star v_1$, hence $$v=d''(u_0+g^\star u_1)~~~~\hbox{\rm on $\omega$}.$$ Finally, the coefficients of $u_0$ are obtained as linear combinations of convolutions of the coefficients of $\psi v$ with $L^1_\loc$ functions of the form $\ol\zeta_j|\zeta|^{-2n}$. Hence $u_0$ is of class $C^s$ (resp.\ is a current of order $s$), if $v$ is.\qed \endproof \begstat{(3.30) Corollary} The operator $d''$ is hypoelliptic in bidegree $(p,0)$, i.e.~if a current $f\in\cD^{\prime\,p,0}(X,\bC)$ satisfies $d''f\in\cE^{p,1}(X,\bC)$, then $f\in\cE^{p,0}(X,\bC)$. \endstat \begproof{} The result is local, so we may assume that $X=\Omega$ is a neighborhood of $0$ in $\bC^n$. The $(p,1)$-form $v=d''f\in \cE^{p,1}(X,\bC)$ satisfies $d''v=0$, hence there exists $u\in\cE^{p,0}(\smash{\wt\Omega},\bC)$ such that $d''u=d''f$. Then $f-u$ is holomorphic and $f=(f-u)+u\in\cE^{p,0} (\smash{\wt\Omega},\bC)$.\qed \endproof \titleb{\S 4.}{Subharmonic Functions} A {\it harmonic} (resp.\ {\it subharmonic}) function on an open subset of $\bR^m$ is essentially a function (or distribution) $u$ such that $\Delta u=0$ (resp.\ $\Delta u\ge 0$). A fundamental example of subharmonic function is given by the Newton kernel~$N$, which is actually harmonic on $\bR^m\ssm\{0\}$. Subharmonic functions are an essential tool of harmonic analysis and potential theory. Before giving their precise definition and properties, we derive a basic integral formula involving the Green kernel of the Laplace operator on the ball. \titlec{\S 4.A.}{Construction of the Green Kernel} The {\it Green kernel} $G_\Omega(x,y)$ of a smoothly bounded domain $\Omega\compact\bR^m$ is the solution of the following {\it Dirichlet boundary problem} for the Laplace operator $\Delta$ on~$\Omega\,$: \begstat{(4.1) Definition} The Green kernel of a smoothly bounded domain $\Omega\compact\bR^m$ is a function $G_\Omega(x,y):\ol\Omega\times \ol\Omega\to[-\infty,0]$ with the following properties: \medskip \item{\rm a)} $G_\Omega(x,y)$ is $C^\infty$ on $\ol\Omega\times\ol\Omega \ssm\Diag_\Omega$~~ $(\Diag_\Omega={}$ diagonal$\,)~;$ \smallskip \item{\rm b)} $G_\Omega(x,y)=G_\Omega(y,x)~;$ \smallskip \item{\rm c)} $G_\Omega(x,y)<0$ on $\Omega\times\Omega$ and $G_\Omega(x,y)=0$ on $\partial\Omega\times\Omega\,;$ \smallskip \item{\rm d)} $\Delta_x G_\Omega(x,y)=\delta_y$~ on $\Omega$ for every fixed $y\in\Omega$. \endstat It can be shown that $G_\Omega$ always exists and is unique. The uniqueness is an easy consequence of the maximum principle (see Th.~4.14 below). In the case where $\Omega=B(0,r)$ is a ball (the only case we are going to deal with), the existence can be shown through explicit calculations. In fact the Green kernel $G_r(x,y)$ of $B(0,r)$ is $$G_r(x,y)=N(x-y)-N\Big({|y|\over r}\Big(x-{r^2\over|y|^2}\,y\Big)\Big),~~~~ x,y\in\ol B(0,r).\leqno(4.2)$$ A substitution of the explicit value of $N(x)$ yields: $$\eqalign{ &G_r(x,y)={1\over 4\pi}\log{|x-y|^2\over r^2-2\langle x,y\rangle+{1\over r^2} |x|^2\,|y|^2}~~~~\hbox{\rm if}~~m=2,~~~\hbox{\rm otherwise}\cr &G_r(x,y)={-1\over(m-2)\sigma_{m-1}}\Big(|x-y|^{2-m}-\big(r^2- 2\langle x,y\rangle+{1\over r^2}|x|^2\,|y|^2\big)^{1-m/2}\Big).\cr}$$ \begstat{(4.3) Theorem} The above defined function $G_r$ satisfies all four properties $(4.1\,\hbox{\rm a--d})$ on $\Omega=B(0,r)$, thus $G_r$ is the Green kernel of $B(0,r)$. \endstat \begproof{} The first three properties are immediately verified on the formulas, because $$r^2-2\langle x,y\rangle+{1\over r^2}|x|^2\,|y|^2=|x-y|^2+{1\over r^2} \big(r^2-|x|^2\big)\big(r^2-|y|^2\big).$$ For property d), observe that $r^2y/|y|^2\notin\ol B(0,r)$ whenever $y\in B(0,r)\ssm\{0\}$. The second Newton kernel in the right hand side of (4.1) is thus harmonic in $x$ on $B(0,r)$, and $$\Delta_xG_r(x,y)=\Delta_x N(x-y)=\delta_y~~~\hbox{\rm on}~~B(0,r).\eqno\qed$$ \endproof \titlec{\S 4.B.}{Green-Riesz Representation Formula and Dirichlet Problem} \titled{\S 4.B.1. Green-Riesz Formula.} For all smooth functions $u,v$ on a smoothly bounded domain $\Omega\compact \bR^m$, we have $$\int_\Omega (u\,\Delta v-v\,\Delta u)\,d\lambda=\int_{\partial\Omega} \Big(u\,{\partial v\over\partial\nu}-v\,{\partial u\over\partial\nu}\Big)\, d\sigma\leqno(4.4)$$ where $\partial/\partial\nu$ is the derivative along the outward normal unit vector $\nu$ of $\partial\Omega$ and $d\sigma$ the euclidean area measure. Indeed $$(-1)^{j-1}\,dx_1\wedge\ldots\wedge\wh{dx_j}\wedge\ldots \wedge dx_{m\,\restriction\partial\Omega}=\nu_j\,d\sigma,$$ for the wedge product of $\langle\nu,dx\rangle$ with the left hand side is $\nu_j\,d\lambda$. Therefore $${\partial v\over\partial\nu}\,d\sigma =\sum_{j=1}^m~{\partial v\over\partial x_j}\,\nu_j\,d\sigma =\sum_{j=1}^m~(-1)^{j-1}{\partial v \over\partial x_j}\,dx_1\wedge\ldots\wedge\wh{dx_j}\wedge\ldots\wedge dx_m.$$ Formula (4.4) is then an easy consequence of Stokes' theorem. Observe that (4.4) is still valid if $v$ is a distribution with singular support relatively compact in $\Omega$. For $\Omega=B(0,r)$, $u\in C^2\big(\ol B(0,r),\bR\big)$ and $v(y)=G_r(x,y)$, we get the {\it Green-Riesz representation formula}\/: $$u(x)=\int_{B(0,r)}\Delta u(y)\,G_r(x,y)\,d\lambda(y)+\int_{S(0,r)}u(y)\,P_r(x,y) \,d\sigma(y)\leqno(4.5)$$ where $P_r(x,y)=\partial G_r(x,y)/\partial\nu(y)$, $(x,y)\in B(0,r)\times S(0,r)$. The function $P_r(x,y)$ is called the {\it Poisson kernel}. It is smooth and satisfies $\Delta_xP_r(x,y)=0$ on $B(0,r)$ by (4.1~d). A simple computation left to the reader yields: $$P_r(x,y)={1\over\sigma_{m-1}r}\,{r^2-|x|^2\over|x-y|^m}.\leqno(4.6)$$ Formula (4.5) for $u\equiv 1$ shows that $\smash{\int_{S(0,r)}}P_r(x,y)\, d\sigma(y)=1$. When $x$ in $B(0,r)$ tends to $x_0\in S(0,r)$, we see that $P_r(x,y)$ converges uniformly to $0$ on every compact subset of $S(0,r)\ssm\{x_0\}$~; it follows that the measure $P_r(x,y)\,d\sigma(y)$ converges weakly to $\delta_{x_0}$ on $S(0,r)$. \titled{\S 4.B.2. Solution of the Dirichlet Problem.} For any bounded measurable function $v$ on $S(a,r)$ we define $$P_{a,r}[v](x)=\int_{S(a,r)}v(y)\,P_r(x-a,y-a)\,d\sigma(y),~~~~ x\in B(a,r).\leqno(4.7)$$ If $u\in C^0\big(\ol B(a,r),\bR\big)\cap C^2\big(B(a,r),\bR\big)$ is harmonic, i.e.\ $\Delta u=0$ on $B(a,r)$, then (4.5) gives $u=P_{a,r}[u]$ on $B(a,r)$, i.e.\ the Poisson kernel reproduces harmonic functions. Suppose now that $v\in C^0\big(S(a,r),\bR\big)$ is given. Then $P_r(x-a,y-a)\,d\sigma(y)$ converges weakly to $\delta_{x_0}$ when $x$ tends to $x_0\in S(a,r)$, so $P_{a,r}[v](x)$ converges to $v(x_0)$. It follows that the function $u$ defined by $$\cases{u=P_{a,r}[v]&on~~$B(a,r)$,\cr u=v &on~~$S(a,r)$\cr}$$ is continuous on $\ol B(a,r)$ and harmonic on $B(a,r)$~; thus $u$ is the solution of the Dirichlet problem with boundary values $v$. \titlec{\S 4.C.}{Definition and Basic Properties of Subharmonic Functions} \titled{\S 4.C.1. Definition. Mean Value Inequalities.} If $u$ is a Borel function on $\ol B(a,r)$ which is bounded above or below, we consider the mean values of $u$ over the ball or sphere: $$\leqalignno{ \mu_B(u\,;a,r)&={1\over\alpha_m r^m}\int_{B(a,r)}u(x)\,d\lambda(x),&(4.8)\cr \mu_S(u\,;a,r)&={1\over\sigma_{m-1} r^{m-1}}\int_{S(a,r)}u(x)\,d\sigma(x). &(4.8')\cr}$$ As $d\lambda=dr\,d\sigma$ these mean values are related by $$\leqalignno{ \mu_B(u\,;a,r)&={1\over\alpha_mr^m}\int_0^r\sigma_{m-1}t^{m-1}\, \mu_S(u\,;a,t)\,dt&(4.9)\cr &=m\int_0^1 t^{m-1}\,\mu_S(u\,;a,rt)\,dt.\cr}$$ Now, apply formula (4.5) with $x=0$. We get $P_r(0,y)=1/\sigma_{m-1}r^{m-1}$ and $G_r(0,y)=(|y|^{2-m}-r^{2-m})/(2-m)\sigma_{m-1}= -(1/\sigma_{m-1})\int^r_{|y|}t^{1-m}dt$, thus $$\eqalign{ \int_{B(0,r)}\Delta u(y)\,G_r(0,y)\,d\lambda(y)&=-{1\over\sigma_{m-1}}\int_0^r {dt\over t^{m-1}}\int_{|y|0$} (resp.\ $\Delta u(a)<0$), Formula (4.10) shows that $\mu_S(u\,;a,r)>u(a)$ (resp.\ $\mu_S(u\,;a,r)-\infty$ almost everywhere on $W$. If $x\in\ol W$, one can choose $a\in W$ such that $|a-x|-\infty$. Then $B(a,r)$ is a neighborhood of $x$, $\smash{\ol B}(a,r)\subset\Omega$ and $\mu_B(u\,;a,r)\ge u(a)>-\infty$. Therefore $x\in W$, $W$ is also closed. We must have $W=\Omega$ or $W=\emptyset$~; in the last case $u\equiv-\infty$ by the mean value inequality.\qed \endproof \begstat{(4.18) Theorem} Let $u\in\Sh(\Omega)$ be such that $u\not\equiv-\infty$ on each connected component of $\Omega$. Then \medskip \item{\rm a)} $r\longmapsto\mu_S(u\,;a,r)$, $r\longmapsto\mu_B(u\,;a,r)$ are non decreasing functions in the interval $]0,d(a,\complement\Omega)[\,$, and $\mu_B(u\,;a,r)\le\mu_S(u\,;a,r)$. \smallskip \item{\rm b)} For any family $(\rho_\varepsilon)$ of smoothing kernels, $u\star\rho_\varepsilon\in\Sh(\Omega_\varepsilon)\cap C^\infty(\Omega_\varepsilon,\bR)$, the family $(u\star\rho_\varepsilon)$ is non decreasing in $\varepsilon$ and $\lim_{\varepsilon\to 0}u\star \rho_\varepsilon=u$.\vskip0pt \endstat \begproof{} We first verify statements a) and b) when $u\in C^2(\Omega,\bR)$. Then $\Delta u\ge 0$ and $\mu_S(u\,;a,r)$ is non decreasing in virtue of (4.10). By (4.9), we find that $\mu_B(u\,;a,r)$ is also non decreasing and that $\mu_B(u\,;a,r)\le \mu_S(u\,;a,r)$. Furthermore, Formula (4.11) shows that $\varepsilon\longmapsto u\star\rho_\varepsilon(a)$ is non decreasing (provided that $\rho_\varepsilon$ is radially symmetric). In the general case, we first observe that property 4.12~c) is equivalent to the inequality $$u\le u\star\mu_r~~~~\hbox{\rm on}~~\Omega_r,~~~~\forall r>0,$$ where $\mu_r$ is the probability measure of uniform density on $B(0,r)$. This inequality implies $u\star\rho_\varepsilon\le u\star\rho_\varepsilon\star \mu_r$ on $(\Omega_r)_\varepsilon=\Omega_{r+\varepsilon}$, thus $u\star\rho_\varepsilon\in C^\infty(\Omega_\varepsilon,\bR)$ is subharmonic on $\Omega_\varepsilon$. It follows that $u\star\rho_\varepsilon\star\rho_\eta$ is non decreasing in $\eta$~; by symmetry, it is also non decreasing in $\varepsilon$, and so is $u\star\rho_\varepsilon=\lim_{\eta\to 0}u\star\rho_\varepsilon\star\rho_\eta$. We have $u\star\rho_\varepsilon\ge u$ by (4.19) and $\limsup_{\varepsilon\to 0}u\star\rho_\varepsilon\le u$ by the upper semicontinuity. Hence $\lim_{\varepsilon\to 0}u\star\rho_\varepsilon=u$. Property~a) for $u$ follows now from its validity for $u\star\rho_\varepsilon$ and from the monotone convergence theorem.\qed \endproof \begstat{(4.19) Corollary} If $u\in\Sh(\Omega)$ is such that $u\not\equiv-\infty$ on each connected component of $\Omega$, then $\Delta u$ computed in the sense of distribution theory is a positive measure. \endstat Indeed $\Delta(u\star\rho_\varepsilon)\ge 0$ as a function, and $\Delta(u\star\rho_\varepsilon)$ converges weakly to $\Delta u$ in $\cD'(\Omega)$. Corollary~4.19 has a converse, but the correct statement is slightly more involved than for the direct property: \begstat{(4.20) Theorem} If $v\in\cD'(\Omega)$ is such that $\Delta v$ is a positive measure, there exists a unique function $u\in\Sh(\Omega)$ locally integrable such that $v$ is the distribution associated to $u$. \endstat We must point out that $u$ need not coincide everywhere with $v$, even when $v$ is a locally integrable upper semicontinuous function: for example, if $v$ is the characteristic function of a compact subset $K\subset\Omega$ of measure $0$, the subharmonic representant of $v$ is $u=0$. \begproof{} Set $v_\varepsilon=v\star\rho_\varepsilon\in C^\infty(\Omega_\varepsilon,\bR)$. Then $\Delta v_\varepsilon=(\Delta v) \star\rho_\varepsilon\ge 0$, thus $v_\varepsilon\in\Sh(\Omega_\varepsilon)$. Arguments similar to those in the proof of Th.~4.18 show that $(v_\varepsilon)$ is non decreasing in~$\varepsilon$. Then $u:=\lim_{\varepsilon\to 0}~v_\varepsilon\in\Sh(\Omega)$ by Th.~4.15. Since $v_\varepsilon$ converges weakly to $v$, the monotone convergence theorem shows that $$\langle v,f\rangle=\lim_{\varepsilon\to 0}\int_\Omega v_\varepsilon\,f\, d\lambda=\int_\Omega u\,f\,d\lambda,~~~~\forall f\in\cD(\Omega),~~~f\ge 0,$$ which concludes the existence part. The uniqueness of $u$ is clear from the fact that $u$ must satisfy $u=\lim u\star\rho_\varepsilon=\lim v\star\rho_\varepsilon$.\qed \endproof The most natural topology on the space $\Sh(\Omega)$ of subharmonic functions is the topology induced by the vector space topology of $L^1_\loc(\Omega)$ (Fr\'echet topology of convergence in $L^1$ norm on every compact subset of $\Omega$). \begstat{(4.21) Proposition} The convex cone $\Sh(\Omega)\cap L^1_\loc(\Omega)$ is closed in $L^1_\loc(\Omega)$, and it has the property that every bounded subset is relatively compact. \endstat \begproof{} Let $(u_j)$ be a sequence in $\Sh(\Omega)\cap L^1_\loc(\Omega)$. If $u_j\to u$ in $L^1_\loc(\Omega)$ then $\Delta u_j\to \Delta u$ in the weak topology of distributions, hence $\Delta u\ge 0$ and $u$ can be represented by a subharmonic function thanks to Th.~4.20. Now, suppose that $\|u_j\|_{L^1(K)}$ is uniformly bounded for every compact subset $K$ of~$\Omega$. Let $\mu_j=\Delta u_j\ge 0$. If $\psi\in\cD(\Omega)$ is a test function equal to $1$ on a neighborhood $\omega$ of~$K$ and such that $0\le\psi\le 1$ on $\Omega$, we find $$\mu_j(K)\le\int_\Omega \psi\,\Delta u_j\,d\lambda= \int_\Omega \Delta\psi\,u_j\,d\lambda\le C\|u_j\|_{L^1(K')},$$ where $K'=\Supp\psi$, hence the sequence of measures $(\mu_j)$ is uniformly bounded in mass on every compact subset of~$\Omega$. By weak compactness, there is a subsequence $(\mu_{j_\nu})$ which converges weakly to a positive measure $\mu$ on~$\Omega$. We claim that $f\star (\psi\mu_{j_\nu})$ converges to $f\star(\psi\mu)$ in $L^1_\loc(\bR^m)$ for every function $f\in L^1_\loc(\bR^m)$. In fact, this is clear if $f\in C^\infty(\bR^m)$, and in general we use an approximation of $f$ by a smooth function $g$ together with the estimate $$\|(f-g)\star(\psi\mu_{j_\nu})\|_{L^1(A)}\le\|(f-g)\|_{L^1(A+K')} \mu_{j_\nu}(K'),\qquad\forall A\compact\bR^m$$ to get the conclusion. We apply this when $f=N$ is the Newton kernel. Then $h_j=u_j-N\star(\psi\mu_j)$ is harmonic on $\omega$ and bounded in $L^1(\omega)$. As $h_j=h_j\star\rho_\varepsilon$ for any smoothing kernel $\rho_\varepsilon$, we see that all derivatives $D^\alpha h_j=h_j\star (D^\alpha\rho_\varepsilon)$ are in fact uniformly locally bounded in~$\omega$. Hence, after extracting a new subsequence, we may suppose that $h_{j_\nu}$ converges uniformly to a limit $h$ on~$\omega$. Then $u_{j_\nu}=h_{j_\nu}+N\star(\psi\mu_{j_\nu})$ converges to $u=h+N\star(\psi\mu)$ in $L^1_\loc(\omega)$, as desired.\qed \endproof We conclude this subsection by stating a generalized version of the Green-Riesz formula. \begstat{(4.22) Proposition} Let $u\in\Sh(\Omega)\cap L^1_\loc(\Omega)$ and $\ol B(0,r)\subset\Omega$. \medskip \item{\rm a)} The Green-Riesz formula still holds true for such an $u$, namely, for every $x\in B(0,r)$ $$u(x)=\int_{B(0,r)}\Delta u(y)\,G_r(x,y)\,d\lambda(y)+\int_{S(0,r)} u(y)\,P_r(x,y)\,d\sigma(y).$$ \item{\rm b)} {\rm (Harnack inequality)}\hfill\break If $u\ge 0$ on $\ol B(0,r)$, then for all $x\in B(0,r)$ $$0\le u(x)\le\int_{S(0,r)}u(y)\,P_r(x,y)\,d\sigma(y)\le {r^{m-2}(r+|x|)\over(r-|x|)^{m-1}}\,\mu_S(u\,;0,r).$$ If $u\le 0$ on $\ol B(0,r)$, then for all $x\in B(0,r)$ $$u(x)\le\int_{S(0,r)}u(y)\,P_r(x,y)\,d\sigma(y)\le {r^{m-2}(r-|x|)\over(r+|x|)^{m-1}}\,\mu_S(u\,;0,r)\le 0.$$ \vskip0pt \endstat \begproof{} We know that a) holds true if $u$ is of class $C^2$. In general, we replace $u$ by $u\star\rho_\varepsilon$ and take the limit. We only have to check that $$\int_{B(0,r)}\mu\star\rho_\varepsilon(y)\,G_r(x,y)\,d\lambda(y)= \lim_{\varepsilon\to 0}\int_{B(0,r)}\mu(y)\,G_r(x,y)\,d\lambda(y)$$ for the positive measure $\mu=\Delta u$. Let us denote by $\wt G_x(y)$ the function such that $$\wt G_x(y)=\cases{ G_r(x,y)&if $x\in B(0,r)$\cr 0&if $x\notin B(0,r)$.\cr}$$ Then $$\eqalign{ \int_{B(0,r)}\mu\star\rho_\varepsilon(y)\,G_r(x,y)\,d\lambda(y) &=\int_{\bR^m}\mu\star\rho_\varepsilon(y)\,\wt G_x(y)\,d\lambda(y)\cr &=\int_{\bR^m}\mu(y)\,\wt G_x\star\rho_\varepsilon(y)\,d\lambda(y).\cr}$$ However $\wt G_x$ is continuous on $\bR^m\ssm\{x\}$ and subharmonic in a neighborhood of~$x$, hence $\wt G_x\star\rho_\varepsilon$ converges uniformly to $\wt G_x$ on every compact subset of $\bR^m\ssm\{x\}$, and converges pointwise monotonically in a neighborhood of~$x$. The desired equality follows by the monotone convergence theorem. Finally, b) is a consequence of a), for the integral involving $\Delta u$ is nonpositive and $${1\over\sigma_{m-1}r^{m-1}}{r^{m-2}(r-|x|)\over (r+|x|)^{m-1}}\le P_r(x,y)\le {1\over\sigma_{m-1}r^{m-1}}{r^{m-2}(r+|x|)\over(r-|x|)^{m-1}}$$ by (4.6) combined with the obvious inequality $(r-|x|)^m\le |x-y|^m\le (r+|x|)^m$.\qed \endproof \titled{\S 4.C.3. Upper Envelopes and Choquet's Lemma.} Let $\Omega\subset\bR^n$ and let $(u_\alpha)_{\alpha\in I}$ be a family of upper semicontinuous functions $\Omega\longrightarrow[-\infty,+\infty[$. We assume that $(u_\alpha)$ is locally uniformly bounded above. Then the upper envelope $$u=\sup u_\alpha$$ need not be upper semicontinuous, so we consider its {\it upper semicontinuous regularization}\/: $$u^\star(z)=\lim_{\varepsilon\to 0}\sup_{B(z,\varepsilon)}u\ge u(z).$$ It is easy to check that $u^\star$ is the smallest upper semicontinuous function which is${}\ge u$. Our goal is to show that $u^\star$ can be computed with a countable subfamily of $(u_\alpha)$. Let $B(z_j,\varepsilon_j)$ be a countable basis of the topology of~$\Omega$. For each $j$, let $(z_{jk})$ be a sequence of points in $B(z_j,\varepsilon_j)$ such that $$\sup_k u(z_{jk})=\sup_{B(z_j,\varepsilon_j)}u,$$ and for each pair $(j,k)$, let $\alpha(j,k,l)$ be a sequence of indices $\alpha\in I$ such that $u(z_{jk})=\sup_l u_{\alpha(j,k,l)}(z_{jk})$. Set $$v=\sup_{j,k,l}u_{\alpha(j,k,l)}.$$ Then $v\le u$ and $v^\star\le u^\star$. On the other hand $$\sup_{B(z_j,\varepsilon_j)}v\ge\sup_k v(z_{jk})\ge\sup_{k,l} u_{\alpha(j,k,l)}(z_{jk})=\sup_k u(z_{jk})=\sup_{B(z_j,\varepsilon_j)}u.$$ As every ball $B(z,\varepsilon)$ is a union of balls $B(z_j,\varepsilon_j)$, we easily conclude that $v^\star\ge u^\star$, hence $v^\star=u^\star$. Therefore: \begstat{(4.23) Choquet's lemma} Every family $(u_\alpha)$ has a countable subfamily $(v_j)=(u_{\alpha(j)})$ such that its upper envelope $v$ satisfies $v\le u\le u^\star=v^\star$.\qed \endstat \begstat{(4.24) Proposition} If all $u_\alpha$ are subharmonic, the upper regularization $u^\star$ is subharmonic and equal almost everywhere to $u$. \endstat \begproof{} By Choquet's lemma we may assume that $(u_\alpha)$ is countable. Then $u=\sup u_\alpha$ is a Borel function. As each $u_\alpha$ satisfies the mean value inequality on every ball $\ol B(z,r)\subset\Omega$, we get $$u(z)=\sup u_\alpha(z)\le\sup\mu_B(u_\alpha\,;\,z,r)\le\mu_B(u\,;\,z,r).$$ The right-hand side is a continuous function of $z$, so we infer $$u^\star(z)\le\mu_B(u\,;\,z,r)\le\mu_B(u^\star\,;\,z,r)$$ and $u^\star$ is subharmonic. By the upper semicontinuity of $u^\star$ and the above inequality we find $u^\star(z)=\lim_{r\to 0}\mu_B(u\,;\,z,r)$, thus $u^\star=u$ almost everywhere by Lebesgue's lemma.\qed \endproof \titleb{\S 5.}{Plurisubharmonic Functions} \titlec{\S 5.A.}{Definition and Basic Properties} Plurisubharmonic functions have been introduced independently by (Lelong 1942) and (Oka 1942) for the study of holomorphic convexity. They are the complex counterparts of subharmonic functions. \begstat{(5.1) Definition} A function $u:\Omega\longrightarrow[-\infty,+\infty[$ defined on an open subset $\Omega\subset\bC^n$ is said to be plurisubharmonic if \medskip \item{\rm a)} $u$ is upper semicontinuous~$;$ \smallskip \item{\rm b)} for every complex line $L\subset\bC^n$, $u_{\restriction\Omega\cap L}$ is subharmonic on $\Omega\cap L$. \smallskip \noindent The set of plurisubharmonic functions on $\Omega$ is denoted by $\Psh(\Omega)$. \vskip0pt \endstat An equivalent way of stating property b) is: for all $a\in\Omega$, $\xi\in\bC^n$, $|\xi|0$. All functions $u'_j(z)=u_j(z)-\gamma_{z_0}(z-z_0)+\varepsilon|z-z_0|^2$ are plurisubharmonic near $z_0$. It follows that $$M_\eta(u'_1,\ldots,u'_p)=u-\gamma_{z_0}(z-z_0)+\varepsilon|z-z_0|^2$$ is also plurisubharmonic near $z_0$. Since $\varepsilon>0$ was arbitrary, e) follows.\qed \endproof \begstat{(5.19) Corollary} Let $u_\alpha\in C^\infty(\ol\Omega_\alpha)\cap \Psh(\Omega_\alpha)$ where $\Omega_\alpha\compact X$ is a locally finite open covering of $X$. Assume that $u_\beta(z)<\max\{u_\alpha(z)\}$ at every point $z\in\partial\Omega_\beta$, when $\alpha$ runs over the indices such that $\Omega_\alpha\ni z$. Choose a family $(\eta_\alpha)$ of positive numbers so small that $u_\beta(z)+\eta_\beta\le\max_{\Omega_\alpha\ni z} \{u_\alpha(z)-\eta_\alpha\}$ for all $\beta$ and $z\in\partial\Omega_\beta$. Then the function defined by $$\wt u(z)=M_{(\eta_\alpha)}\big(u_\alpha(z)\big)~~~~ {\rm for~\alpha~such~that}~~\Omega_\alpha\ni z$$ is smooth and plurisubharmonic on $X$.\qed \endstat \begstat{(5.20) Definition} A function $u\in\Psh(X)$ is said to be strictly plurisubharmonic if $u\in L^1_\loc(X)$ and if for every point $x_0\in X$ there exists a neighborhood $\Omega$ of $x_0$ and $c>0$ such that $u(z)-c|z|^2$ is plurisubharmonic on $\Omega$, i.e.\ $\sum~({\partial^2u/\partial z_j\partial\ol z_k})\xi_j\ol\xi_k\ge c|\xi|^2$ $($as distributions on $\Omega)$ for all $\xi\in\bC^n$. \endstat \begstat{(5.21) Theorem {\rm(Richberg 1968)}} Let $u\in\Psh(X)$ be a continuous function which is strictly plurisubharmonic on an open subset $\Omega\subset X$, with $Hu\ge\gamma$ for some continuous positive hermitian form $\gamma$ on $\Omega$. For any continuous function $\lambda\in C^0(\Omega)$, $\lambda>0$, there exists a plurisubharmonic function $\wt u$ in $C^0(X)\cap C^\infty(\Omega)$ such that $u\le\wt u\le u+\lambda$ on $\Omega$ and $\wt u=u$ on $X\ssm\Omega$, which is strictly plurisubharmonic on $\Omega$ and satisfies $H\wt u\ge(1-\lambda)\gamma$. In particular, $\wt u$ can be chosen strictly plurisubharmonic on $X$ if $u$ has the same property. \endstat \begproof{} Let $(\Omega_\alpha)$ be a locally finite open covering of $\Omega$ by relatively compact open balls contained in coordinate patches of $X$. Choose concentric balls $\Omega''_\alpha\subset\Omega'_\alpha\subset\Omega_\alpha$ of respective radii $r''_\alpha\eta_\alpha$ on $\ol\Omega''_\alpha$, we have $u_\alphau+\eta_\alpha$ on $\ol\Omega''_\alpha$, so that the condition required in Corollary 5.19 is satisfied. We define $$\wt u=\cases{u&on~~$X\ssm\Omega$,\cr M_{(\eta_\alpha)}(u_\alpha)&on~~$\Omega$.\cr}$$ By construction, $\wt u$ is smooth on $\Omega$ and satisfies $u\le\wt u\le u+\lambda$, $Hu\ge(1-\lambda)\gamma$ thanks to 5.18~(b,e). In order to see that $\wt u$ is plurisubharmonic on $X$, observe that $\wt u$ is the uniform limit of $\wt u_\alpha$ with $$\wt u_\alpha=\max\big\{u\,,\,M_{(\eta_1\ldots\eta_\alpha)} (u_1\ldots u_\alpha)\big\}~~~\hbox{\rm on}~~\bigcup_{1\le\beta\le\alpha} \Omega_\beta$$ and $\wt u_\alpha=u$ on the complement.\qed \endproof \titlec{\S 5.F.}{Polar and Pluripolar Sets.} Polar and pluripolar sets are sets of $-\infty$ poles of subharmonic and plurisubharmonic functions. Although these functions possess a large amount of flexi\-bility, pluripolar sets have some properties which remind their loose relationship with holomorphic functions. \begstat{(5.22) Definition} A set $A\subset\Omega\subset\bR^m$ $($resp.\ $A\subset X,$ ${\rm dim}_\bC X=n)$ is said to be polar $($resp.\ pluripolar$)$ if for every point $x\in\Omega$ there exist a connected neighborhood $W$ of $x$ and $u\in\Sh(W)$ $($resp.\ $u\in\Psh(W))$, $u\not\equiv-\infty$, such that~ $A\cap W\subset\{x\in W~;~u(x)=-\infty\}$. \endstat Theorem~4.17 implies that a polar or pluripolar set is of zero Lebesgue measure. Now, we prove a simple extension theorem. \begstat{(5.23) Theorem} Let $A\subset\Omega$ be a closed polar set and $v\in\Sh(\Omega\ssm A)$ such that $v$ is bounded above in a neighborhood of every point of $A$. Then $v$ has a unique extension $\wt v\in\Sh(\Omega)$. \endstat \begproof{} The uniqueness is clear because $A$ has zero Lebesgue measure. On the other hand, every point of $A$ has a neighborhood $W$ such that $$A\cap W\subset\{x\in W~;~u(x)=-\infty\},~~~~u\in\Sh(W),~~~ u\not\equiv-\infty.$$ After shrinking $W$ and subtracting a constant to $u$, we may assume $u\le 0$. Then for every $\varepsilon>0$ the function $v_\varepsilon=v+\varepsilon u \in\Sh(W\ssm A)$ can be extended as an upper semicontinuous on $W$ by setting $v_\varepsilon=-\infty$ on $A\cap W$. Moreover, $v_\varepsilon$ satisfies the mean value inequality $v_\varepsilon(a)\le\mu_S (v_\varepsilon\,;a,r)$ if $a\in W\ssm A$, $r1$. Then $|g_{\nu,a}(z)|>1$ in a neighborhood of $a$~; by the Borel-Lebesgue lemma, one can find finitely many functions $(g_{\nu,a})_{a\in I_\nu}$ such that $$\max_{a\in I_\nu}\big\{|g_{\nu,a}(z)|\big\}>1~~\hbox{\rm for}~~z\in L_\nu,~~~~\max_{a\in I_\nu}\big\{|g_{\nu,a}(z)|\big\}<1 ~~\hbox{\rm for}~~z\in K_\nu.$$ For a sufficiently large exponent $p(\nu)$ we get $$\sum_{a\in I_\nu}|g_{\nu,a}|^{2p(\nu)}\ge\nu~~\hbox{\rm on}~~L_\nu,~~~~ \sum_{a\in I_\nu}|g_{\nu,a}|^{2p(\nu)}\le 2^{-\nu}~~\hbox{\rm on}~~K_\nu.$$ It follows that the series $$\psi(z)=\sum_{\nu\in\bN}\sum_{a\in I_\nu}|g_{\nu,a}(z)|^{2p(\nu)}$$ converges uniformly to a real analytic function $\psi\in\Psh(X)$ (see Exercise~8.11). By construction $\psi(z)\ge\nu$ for $z\in L_\nu$, hence $\psi$ is an exhaustion.\qed \endproof \begstat{(6.15) Example} \rm The converse to Theorem~6.14 does not hold. In fact let $X=\bC^2/\Gamma$ be the quotient of $\bC^2$ by the free abelian group of rank $2$ generated by the affine automorphisms $$g_1(z,w)=(z+1,e^{\ii\theta_1}w),~~~~g_2(z,w)=(z+\ii,e^{\ii\theta_2}w),~~~~ \theta_1,\,\theta_2\in\bR.$$ Since $\Gamma$ acts properly discontinuously on $\bC^2$, the quotient has a structure of a complex (non compact) $2$-dimensional manifold. The function $w\mapsto |w|^2$ is $\Gamma$-invariant, hence it induces a function $\psi((z,w)^\sim)=|w|^2$ on~$X$ which is in fact a plurisubharmonic exhaustion function. Therefore $X$ is weakly pseudoconvex. On the other hand, any holomorphic function $f\in\cO(X)$ corresponds to a $\Gamma$-invariant holomorphic function $\smash{\wt f}(z,w)$ on~$\bC^2$. Then $z\mapsto\wt f(z,w)$ is bounded for $w$ fixed, because $\wt f(z,w)$ lies in the image of the compact set $K\times\ol D(0,|w|)$, $K={}$ unit square in $\bC$. By Liouville's theorem, $\wt f(z,w)$ does not depend on $z$. Hence functions $f\in\cO(X)$ are in one-to-one correspondence with holomorphic functions $\wt f(w)$ on $\bC$ such that $\smash{\wt f}(e^{\ii\theta_j}w)= \smash{\wt f}(w)$. By looking at the Taylor expansion at the origin, we conclude that $\smash{\wt f}$ must be a constant if $\theta_1\notin\bQ$ or $\theta_1\notin\bQ$ (if $\theta_1,\theta_2\in\bQ$ and $m$ is the least common denominator of $\theta_1,\theta_2$, then ${\wt f}$ is a power series of the form $\sum\alpha_kw^{mk}$). From this, it follows easily that $X$ is holomorphically convex if and only if $\theta_1,\theta_2\in\bQ$. \endstat \titlec{\S 6.C.}{Stein Manifolds} The class of holomorphically convex manifolds contains two types of manifolds of a rather different nature: \smallskip\noindent $\bu$ domains of holomorphy $X=\Omega\subset\bC^n\,;$ \smallskip\noindent $\bu$ compact complex manifolds. \smallskip\noindent In the first case we have a lot of holomorphic functions, in fact the functions in $\cO(\Omega)$ separate any pair of points of~$\Omega$. On the other hand, if $X$ is compact and connected, the sets $\Psh(X)$ and $\cO(X)$ consist of constant functions merely (by the maximum principle). It is therefore desirable to introduce a clear distinction between these two subclasses. For this purpose, (Stein 1951) introduced the class of manifolds which are now called Stein manifolds. \begstat{(6.16) Definition} A complex manifold $X$ is said to be a Stein manifold if \medskip \item{\rm a)} $X$ is holomorphically convex\/$;$ \smallskip \item{\rm b)} $\cO(X)$ locally separates points in $X$, i.e.\ every point $x\in X$ has a neighborhood $V$ such that for any $y\in V\ssm\{x\}$ there exists $f\in\cO(X)$ with $f(y)\ne f(x)$. \vskip0pt \endstat The second condition is automatic if $X=\Omega$ is an open subset of $\bC^n$. Hence an open set $\Omega\subset\bC^n$ is Stein if and only if $\Omega$ is a domain of holomorphy. \begstat{(6.17) Lemma} If a complex manifold $X$ satisfies the axiom $(6.16~{\rm b})$ of local separation, there exists a smooth nonnegative strictly plurisubharmonic function $u\in\Psh(X)$. \endstat \begproof{} Fix $x_0\in X$. We first show that there exists a smooth nonnegative function $u_0\in\Psh(X)$ which is strictly plurisubharmonic on a neighborhood of~$x_0$. Let $(z_1,\ldots,z_n)$ be local analytic coordinates centered at $x_0$, and if necessary, replace $z_j$ by $\lambda z_j$ so that the closed unit ball $\ol B=\{\sum|z_j|^2\le 1\}$ is contained in the neighborhood $V\ni x_0$ on which (6.16~b) holds. Then, for every point $y\in\partial B$, there exists a holomorphic function $f\in\cO(X)$ such that $f(y)\ne f(x_0)$. Replacing $f$ with $\lambda(f-f(x_0))$, we can achieve \hbox{$f(x_0)=0$} and $|f(y)|>1$. By compactness of $\partial B$, we find finitely many functions $f_1,\ldots,f_N\in\cO(X)$ such that $v_0=\sum|f_j|^2$ satisfies $v_0(x_0)=0$, while $v_0\ge 1$ on~$\partial B$. Now, we set $$u_0(z)=\cases{ v_0(z)&on $X\ssm B$,\cr M_\varepsilon\{v_0(z),(|z|^2+1)/3\}&on $B$.\cr}$$ where $M_\varepsilon$ are the regularized max functions defined in~5.18. Then $u_0$ is smooth and plurisubharmonic, coincides with $v_0$ near $\partial B$ and with $(|z|^2+1)/3$ on a neighborhood of~$x_0$. We can cover $X$ by countably many neighborhoods~$(V_j)_{j\ge 1}$, for which we have a smooth plurisubharmonic functions $u_j\in\Psh(X)$ such that $u_j$ is strictly plurisubharmonic on~$V_j$. Then select a sequence $\varepsilon_j>0$ converging to $0$ so fast that $u=\sum\varepsilon_ju_j\in C^\infty(X)$. The function $u$ is nonnegative and strictly plurisubharmonic everywhere on~$X$.\qed \endproof \begstat{(6.18) Theorem} Every Stein manifold is strongly pseudoconvex. \endstat \begproof{} By Th.~6.14, there is a smooth exhaustion function $\psi\in\Psh(X)$. If \hbox{$u\ge 0$} is strictly plurisubharmonic, then $\psi'=\psi+u$ is a strictly plurisubharmonic exhaustion.\qed \endproof The converse problem to know whether every strongly pseudoconvex manifold is actually a Stein manifold is known as the {\it Levi problem}, and was raised by (Levi 1910) in the case of domains $\Omega\subset\bC^n$. In that case, the problem has been solved in the affirmative independently by (Oka 1953), (Norguet 1954) and (Bremermann 1954). The general solution of the Levi problem has been obtained by (Grauert 1958). Our proof will rely on the theory of $L^2$ estimates for $d''$, which will be available only in Chapter~VIII. \input epsfiles/fig_1_4.tex \vskip6mm \centerline{{\bf Fig.~I-4} Hartogs figure with excrescence} \vskip6mm \begstat{(6.19) Remark} \rm It will be shown later that Stein manifolds always have enough holomorphic functions to separate finitely many points, and one can even interpolate given values of a function and its derivatives of some fixed order at any discrete set of points. In particular, we might have replaced condition (6.16~b) by the stronger requirement that $\cO(X)$ separates any pair of points. On the other hand, there are examples of manifolds satisfying the local separation condition (6.16~b), but not global separation. A simple example is obtained by attaching an excrescence inside a Hartogs figure, in such a way that the resulting map $\pi:X\to D=D(0,1)^2$ is not one-to-one (see Figure~I-4 above); then $\cO(X)$ coincides with $\pi^\star\cO(D)$. \endstat \titlec{\S 6.D.}{Heredity Properties} Holomorphic convexity and pseudoconvexity are preserved under quite a number of natural constructions. The main heredity properties can be summarized in the following Proposition. \begstat{(6.20) Proposition} Let $\cC$ denote the class of holomorphically convex $($resp.\ of Stein, or weakly pseudoconvex, strongly pseudoconvex manifolds$)$. \medskip \item{\rm a)} If $X,Y\in\cC$, then $X\times Y\in\cC$. \smallskip \item{\rm b)} If $X\in\cC$ and $S$ is a closed complex submanifold of $X$, then $S\in\cC$. \smallskip \item{\rm c)} If $(S_j)_{1\le j\le N}$ is a collection of $($not necessarily closed$)$ submanifolds of a complex manifold~$X$ such that $S=\bigcap S_j$ is a submanifold of~$X$, and if $S_j\in\cC$ for all~$j$, then $S\in\cC$. \smallskip \item{\rm d)} If $F:X\to Y$ is a holomorphic map and $S\subset X$, $S'\subset Y$ are $($not necessarily closed$)$ submanifolds in the class $\cC$, then $S\cap F^{-1}(S')$ is in~$\cC$, as long as it is a submanifold of~$X$. \smallskip \item{\rm e)} If $X$ is a weakly $($resp.\ strongly$)$ pseudoconvex manifold and $u$ is a smooth plurisubharmonic function on~$X$, then the open set $\Omega=u^{-1}(]-\infty,c[$ is weakly $($resp.\ strongly$)$ pseudoconvex. In particular the sublevel sets $$X_c=\psi^{-1}(]-\infty,c[)$$ of a $($strictly$)$ plurisubharmonic exhaustion function are weakly $($resp.\ strongly$)$ pseudoconvex. \vskip0pt \endstat \begproof{} All properties are more or less immediate to check, so we only give the main facts. \medskip\noindent a) For $K\subset X$, $L\subset Y$ compact, we have $(K\times L)^{\wedge}_{\cO(X\times Y)}=\hbox{$\wh K_{\cO(X)}\times \wh K_{\cO(Y)}$}$, and if $\varphi$, $\psi$ are plurisubharmonic exhaustions of $X$, $Y$, then \hbox{$\varphi(x)+\psi(y)$} is a plurisubharmonic exhaustion of~$X\times Y$. \medskip\noindent b) For a compact set $K\subset S$, we have $\wh K_{\cO(S)}\subset \wh K_{\cO(X)}\cap S$, and if $\psi\in\Psh(X)$ is an exhaustion, then $\psi{\restriction S}\in\Psh(S)$ is an exhaustion (since $S$ is closed). \medskip\noindent c) $\bigcap S_j$ is a closed submanifold in $\prod S_j$ (equal to its intersection with the diagonal of $X^N$). \medskip\noindent d) For a compact set $K\subset S\cap F^{-1}(S')$, we have $$\wh K_{\cO(S\cap F^{-1}(S'))}\subset\wh K_{\cO(S)}\cap F^{-1}(\wh{F(K)}_{\cO(S')}),$$ and if $\varphi$, $\psi$ are plurisubharmonic exhaustions of $S$, $S'$, then $\varphi+\psi\circ F$ is a plurisubharmonic exhaustion of $S\cap F^{-1}(S')$. \medskip\noindent e) $\varphi(z):=\psi(z)+1/(c-u(z))$ is a (strictly) plurisubharmonic exhaustion function on~$\Omega$.\qed \endproof \titleb{\S 7.}{Pseudoconvex Open Sets in $\bC^n$} \titlec{\S 7.A.}{Geometric Characterizations of Pseudoconvex Open Sets} We first discuss some characterizations of pseudoconvex open sets in~$\bC^n$. We will need the following elementary criterion for plurisubharmonicity. \begstat{(7.1) Criterion} Let $v:\Omega\longrightarrow[-\infty,+\infty[$ be an upper semicontinuous function. Then $v$ is plurisubharmonic if and only if for every closed disk $\ol\Delta=z_0+\ol D(1)\eta\subset\Omega$ and every polynomial $P\in\bC[t]$ such that $v(z_0+t\eta)\le\Re P(t)$ for $|t|=1$, then $v(z_0)\le\Re P(0)$. \endstat \begproof{} The condition is necessary because $t\longmapsto v(z_0+t\eta)-\Re P(t)$ is subharmonic in a neighborhood of $\ol D(1)$, so it satisfies the maximum principle on $D(1)$ by Th.~4.14. Let us prove now the sufficiency. The upper semicontinuity of $v$ implies $v=\lim_{\nu\to+\infty} v_\nu$ on $\partial\Delta$ where $(v_\nu)$ is a strictly decreasing sequence of continuous functions on $\partial\Delta$. As trigonometric polynomials are dense in $C^0(S^1,\bR)$, we may assume $v_\nu(z_0+e^{\ii\theta}\eta)=\Re P_\nu (e^{\ii\theta})$, $P_\nu\in\bC[t]$. Then $v(z_0+t\eta)\le\Re P_\nu(t)$ for $|t|=1$, and the hypothesis implies $$v(z_0)\le\Re P_\nu(0)={1\over 2\pi}\int_0^{2\pi}\Re P_\nu(e^{\ii\theta})\, d\theta={1\over 2\pi}\int_0^{2\pi}v_\nu(z_0+e^{\ii\theta}\eta)\,d\theta.$$ Taking the limit when $\nu$ tends to $+\infty$ shows that $v$ satisfies the mean value inequality (5.2).\qed \endproof For any $z\in\Omega$ and $\xi\in\bC^n$, we denote by $$\delta_\Omega(z,\xi)=\sup\big\{r>0~;~z+D(r)\,\xi\subset\Omega\big\}$$ the distance from $z$ to $\partial\Omega$ in the complex direction $\xi$. \begstat{(7.2) Theorem} Let $\Omega\subset\bC^n$ be an open subset. The following properties are equivalent: \medskip \item{\rm a)} $\Omega$ is strongly pseudoconvex $($according to Def.~6.13~{\rm b)}$;$ \smallskip \item{\rm b)} $\Omega$ is weakly pseudoconvex\/$;$ \smallskip \item{\rm c)} $\Omega$ has a plurisubharmonic exhaustion function $\psi$. \smallskip \item{\rm d)} $-\log\delta_\Omega(z,\xi)$ is plurisubharmonic on $\Omega\times\bC^n~;$ \smallskip \item{\rm e)} $-\log d(z,\complement\Omega)$ is plurisubharmonic on $\Omega$. \medskip\noindent If one of these properties hold, $\Omega$ is said to be a pseudoconvex open set. \endstat \begproof{} The implications a) $\Longrightarrow$ b) $\Longrightarrow$ c) are obvious. For the implication c) $\Longrightarrow$ d), we use Criterion 7.1. Consider a disk $\ol\Delta=(z_0,\xi_0)+\ol D(1)\,(\eta,\alpha)$ in $\Omega\times\bC^n$ and a polynomial $P\in\bC[t]$ such that $$-\log\delta_\Omega(z_0+t\eta,\xi_0+t\alpha)\le\Re P(t)~~~ \hbox{\rm for}~~|t|=1.$$ We have to verify that the inequality also holds when $|t|<1$. Consider the holomorphic mapping $h:\bC^2\longrightarrow\bC^n$ defined by $$h(t,w)=z_0+t\eta+we^{-P(t)}(\xi_0+t\alpha).$$ By hypothesis $$\eqalign{ &h\big(\ol D(1)\times\{0\}\big)={\rm pr}_1(\ol\Delta)\subset\Omega,\cr &h\big(\partial D(1)\times D(1)\big)\subset\Omega~~~\hbox{\rm(since}~ |e^{-P}|\le\delta_\Omega~~\hbox{\rm on}~\partial\Delta),\cr}$$ and the desired conclusion is that $h\big(\ol D(1)\times D(1)\big) \subset\Omega$. Let $J$ be the set of radii $r\ge 0$ such that $h\big(\ol D(1)\times\ol D(r)\big)\subset\Omega$. Then $J$ is an open interval $[0,R[$, $R>0$. If $R<1$, we get a contradiction as follows. Let $\psi\in\Psh(\Omega)$ be an exhaustion function and $$K=h\big(\partial D(1)\times\ol D(R)\big)\compact\Omega,~~~~ c=\sup_K\psi.$$ As $\psi\circ h$ is plurisubharmonic on a neighborhood of $\ol D(1)\times D(R)$, the maximum principle applied with respect to $t$ implies $$\psi\circ h(t,w)\le c~~~\hbox{\rm on}~~\ol D(1)\times D(R),$$ hence $h\big(\ol D(1)\times D(R)\big)\subset\Omega_c\compact\Omega$ and $h\big(\ol D(1)\times\ol D(R+\varepsilon)\big)\subset\Omega$ for some $\varepsilon>0$, a contradiction. \medskip \noindent{d)} $\Longrightarrow$ e). The function~ $-\log d(z,\complement \Omega)$ is continuous on $\Omega$ and satisfies the mean value inequality because $$-\log d(z,\complement\Omega)=\sup_{\xi\in\ol B}\big(-\log\delta_\Omega(z,\xi) \big).$$ \medskip \noindent{e)} $\Longrightarrow$ a). It is clear that $$u(z)=|z|^2+\max\{\log d(z,\complement\Omega)^{-1},0\}$$ is a continuous strictly plurisubharmonic exhaustion function. Richberg's theorem 5.21 implies that there exists $\psi\in C^\infty(\Omega)$ strictly plurisubharmonic such that $u\le\psi\le u+1$. Then $\psi$ is the required exhaustion function.\qed \endproof \begstat{(7.3) Proposition} \smallskip \item{\rm a)} Let $\Omega\subset\bC^n$ and $\Omega'\subset\bC^p$ be pseudoconvex. Then $\Omega\times\Omega'$ is pseudoconvex. For every holomorphic map $F:\Omega\to\bC^p$ the inverse image $F^{-1}(\Omega')$ is pseudoconvex. \smallskip \item{\rm b)} If $(\Omega_\alpha)_{\alpha\in I}$ is a family of pseudoconvex open subsets of $\bC^n$, the interior of the intersection $\Omega=\big(\bigcap_{\alpha\in I} \Omega_\alpha\big)^\circ$ is pseudoconvex. \smallskip \item{\rm c)} If $(\Omega_j)_{j\in\bN}$ is a non decreasing sequence of pseudoconvex open subsets of $\bC^n$, then $\Omega=\bigcup_{j\in\bN}\Omega_j$ is pseudoconvex. \vskip0pt \endstat \begproof{} a) Let $\varphi,\psi$ be smooth plurisubharmonic exhaustions of $\Omega,\Omega'$. Then $(z,w)\longmapsto\varphi(z)+\psi(w)$ is an exhaustion of $\Omega\times\Omega'$ and $z\longmapsto\varphi(z)+ \psi(F(z))$ is an exhaustion of $F^{-1}(\Omega')$. \medskip \noindent{b)} We have $-\log d(z,\complement\Omega)=\sup_{\alpha\in I} -\log d(z,\complement\Omega_\alpha)$, so this function is pluri\-sub\-harmonic. \medskip \noindent{c)} The limit $-\log d(z,\complement\Omega)={{\lim}{\downarrow}\,}_{j\to+\infty} -\log d(z,\complement\Omega_j)$ is plurisubharmonic, hence $\Omega$ is pseudoconvex. This result cannot be generalized to strongly pseudoconvex manifolds: J.E.~Fornaess in (Fornaess 1977) has constructed an increasing sequence of $2$-dimensional Stein (even affine algebraic) manifolds $X_\nu$ whose union is not Stein; see Exercise~8.16.\qed \endproof \begstat{(7.4) Examples} \smallskip\rm \noindent{a)} An {\it analytic polyhedron} in $\bC^n$ is an open subset of the form $$P=\{z\in\bC^n\,;\,|f_j(z)|<1,~1\le j\le N\}$$ where $(f_j)_{1\le j\le N}$ is a family of analytic functions on $\bC^n$. By 7.3~a), every analytic polyhedron is pseudoconvex. \medskip \noindent{b)} Let $\omega\subset\bC^{n-1}$ be pseudoconvex and let $u:\omega\longrightarrow[-\infty,+\infty[$ be an upper semicontinuous function. Then the {\it Hartogs domain} $$\Omega=\big\{(z_1,z')\in\bC\times\omega\,;\,\log|z_1|+u(z')<0\big\}$$ is pseudoconvex if and and only if $u$ is plurisubharmonic. To see that the plurisubharmonicity of $u$ is necessary, observe that $$u(z')=-\log\delta_\Omega\big((0,z'),(1,0)\big).$$ Conversely, assume that $u$ is plurisubharmonic and continuous. If $\psi$ is a plurisubharmonic exhaustion of $\omega$, then $$\psi(z')+\big|\log|z_1|+u(z')\big|^{-1}$$ is an exhaustion of $\Omega$. This is no longer true if $u$ is not continuous, but in this case we may apply Property 7.3~c) to conclude that $$\Omega_\varepsilon=\big\{(z_1,z')\,;\,d(z',\complement\omega)> \varepsilon,~\log|z_1|+u\star\rho_\varepsilon(z')<0\big\},~~~~ \Omega=\bigcup\Omega_\varepsilon$$ are pseudoconvex. \medskip \noindent{c)} An open set $\Omega\subset\bC^n$ is called a {\it tube} of base $\omega$ if $\Omega=\omega+\ii\bR^n$ for some open subset $\omega\subset\bR^n$. Then of course $-\log d(z,\complement\Omega)=-\log(x,\complement\omega)$ depends only on the real part $x=\Re z$. By Th.~5.13, this function is plurisubharmonic if and only if it is locally convex in $x$. Therefore $\Omega$ if pseudoconvex if and only if every connected component of $\omega$ is convex. \medskip \noindent{d)} An open set $\Omega\subset\bC^n$ is called a {\it Reinhardt domain} if $(e^{\ii\theta_1}z_1,\ldots,e^{\ii\theta_n}z_n)$ is in $\Omega$ for every $z=(z_1,\ldots,z_n)\in\Omega$ and $\theta_1,\ldots,\theta_n\in\bR^n$. For such a domain, we consider the {\it logarithmic indicatrix} $$\omega^\star=\Omega^\star\cap\bR^n~~~\hbox{\rm with}~~~ \Omega^\star=\{\zeta\in\bC^n\,;\,(e^{\zeta_1},\ldots,e^{\zeta_n})\in\Omega\}.$$ It is clear that $\Omega^\star$ is a tube of base $\omega^\star$. Therefore every connected component of $\omega^\star$ must be convex if $\Omega$ is pseudoconvex. The converse is not true: $\Omega=\bC^n\ssm\{0\}$ is not pseudoconvex for $n\ge 2$ although $\omega^\star=\bR^n$ is convex. However, the Reinhardt open set $$\Omega^\bullet=\big\{(z_1,\ldots,z_n)\in(\bC\ssm\{0\})^n\,;\, (\log|z_1|,\ldots,\log|z_n|)\in\omega^\star\big\}\subset\Omega$$ is easily seen to be pseudoconvex if $\omega^\star$ is convex: if $\chi$ is a convex exhaustion of $\omega^\star$, then $\psi(z)=\chi(\log|z_1|,\ldots,\log|z_n|)$ is a plurisubharmonic exhaustion of $\Omega^\bullet$. Similarly, if $\omega^\star$ is convex and such that $x\in\omega^\star \Longrightarrow y\in\omega^\star$ for $y_j\le x_j$, we can take $\chi$ increasing in all variables and tending to $+\infty$ on $\partial\omega^\star$, hence the set $$\wt\Omega=\big\{(z_1,\ldots,z_n)\in\bC^n\,;\,|z_j|\le e^{x_j}~~ \hbox{\rm for~some~~}x\in\omega^\star\big\}$$ is a pseudoconvex Reinhardt open set containing $0$.\qed \endstat \titlec{\S 7.B.}{Kiselman's Minimum Principle} We already know that a maximum of plurisubharmonic functions is plurisubharmonic. However, if $v$ is a plurisubharmonic function on $X\times\bC^n$, the partial minimum function on $X$ defined by $u(\zeta)=\inf_{z\in\Omega}v(\zeta,z)$ need not be plurisubharmonic. A simple counterexample in $\bC\times\bC$ is given by $$v(\zeta,z)=|z|^2+2\Re(z\zeta)=|z+\ol\zeta|^2-|\zeta|^2,~~~~ u(\zeta)=-|\zeta|^2.$$ It follows that the image $F(\Omega)$ of a pseudoconvex open set $\Omega$ by a holomorphic map $F$ need not be pseudoconvex. In fact, if $$\Omega=\{(t,\zeta,z)\in\bC^3\,;\,\log|t|+v(\zeta,z)<0\}$$ and if $\Omega'\subset\bC^2$ is the image of $\Omega$ by the projection map $(t,\zeta,z)\longmapsto(t,\zeta)$, then $\Omega'=\{(t,\zeta)\in\bC^2\,;\,\log|t|+u(\zeta)<0\}$ is not pseudoconvex. However, the minimum property holds true when $v(\zeta,z)$ depends only on $\Re z$~: \begstat{(7.5) Theorem {\rm (Kiselman 1978)}} Let $\Omega\subset\bC^p\times\bC^n$ be a pseudoconvex open set such that each slice $$\Omega_\zeta=\{z\in\bC^n\,;\,(\zeta,z)\in\Omega\},~~~~\zeta\in\bC^p,$$ is a convex tube $\omega_\zeta+\ii\bR^n$, $\omega_\zeta\subset\bR^n$. For every plurisubharmonic function $v(\zeta,z)$ on $\Omega$ that does not depend on $\Im z$, the function $$u(\zeta)=\inf_{z\in\Omega_\zeta}~v(\zeta,z)$$ is plurisubharmonic or locally $\equiv-\infty$ on $\Omega'= {\rm pr}_{\bC^n}(\Omega)$. \endstat \begproof{} The hypothesis implies that $v(\zeta,z)$ is convex in $x=\Re z$. In addition, we first assume that $v$ is smooth, plurisubharmonic in $(\zeta,z)$, strictly convex in $x$ and $\lim_{x\to\{\infty\}\cup\partial\omega_\zeta}v(\zeta,x)=+\infty$ for every $\zeta\in\Omega'$. Then $x\longmapsto v(\zeta,x)$ has a unique minimum point $x=g(\zeta)$, solution of the equations $\partial v/\partial x_j(x,\zeta)=0$. As the matrix $(\partial^2 v/\partial x_j\partial x_k)$ is positive definite, the implicit function theorem shows that $g$ is smooth. Now, if $\bC\ni w\longmapsto\zeta_0+wa$, $a\in\bC^n$, $|w|\le 1$ is a complex disk $\Delta$ contained in $\Omega$, there exists a holomorphic function $f$ on the unit disk, smooth up to the boundary, whose real part solves the Dirichlet problem $$\Re f(e^{\ii\theta})=g(\zeta_0+e^{\ii\theta}a).$$ Since $v(\zeta_0+wa,f(w))$ is subharmonic in $w$, we get the mean value inequality $$v(\zeta_0,f(0))\le{1\over 2\pi}\int_0^{2\pi}v\big(\zeta_0+e^{\ii\theta}a, f(e^{\ii\theta})\big)d\theta= {1\over 2\pi}\int_{\partial\Delta}v(\zeta,g(\zeta))d\theta.$$ The last equality holds because $\Re f=g$ on $\partial\Delta$ and $v(\zeta,z)=v(\zeta,\Re z)$ by hypothesis. As $u(\zeta_0)\le v(\zeta_0,f(0))$ and $u(\zeta)=v(\zeta,g(\zeta))$, we see that $u$ satisfies the mean value inequality, thus $u$ is plurisubharmonic. Now, this result can be extended to arbitrary functions $v$ as follows: let $\psi(\zeta,z)\ge 0$ be a continuous plurisubharmonic function on $\Omega$ which is independent of $\Im z$ and is an exhaustion of $\Omega\cap(\bC^p\times\bR^n)$, e.g. $$\psi(\zeta,z)=\max\{|\zeta|^2+|\Re z|^2,-\log\delta_\Omega(\zeta,z)\}.$$ There is slowly increasing sequence $C_j\to+\infty$ such that each function\break $\psi_j=(C_j-\psi\star\rho_{1/j})^{-1}$ is an ``exhaustion" of a pseudoconvex open set $\Omega_j\compact\Omega$ whose slices are convex tubes and such that $d(\Omega_j,\complement\Omega)>2/j$. Then $$v_j(\zeta,z)=v\star\rho_{1/j}(\zeta,z)+{1\over j}|\Re z|^2+\psi_j(\zeta,z)$$ is a decreasing sequence of plurisubharmonic functions on $\Omega_j$ satisfying our previous conditions. As $v=\lim v_j$, we see that $u=\lim u_j$ is plurisubharmonic.\qed \endproof \begstat{(7.6) Corollary} Let $\Omega\subset\bC^p\times\bC^n$ be a pseudoconvex open set such that all slices $\Omega_\zeta$, $\zeta\in\bC^p$, are convex tubes in $\bC^n$. Then the projection $\Omega'$ of $\Omega$ on $\bC^p$ is pseudoconvex. \endstat \begproof{} Take $v\in\Psh(\Omega)$ equal to the function $\psi$ defined in the proof of Th.~7.5. Then $u$ is a plurisubharmonic exhaustion of $\Omega'$.\qed \endproof \titlec{\S 7.C.}{Levi Form of the Boundary} For an arbitrary domain in $\bC^n$, we first show that pseudoconvexity is a local property of the boundary. \begstat{(7.7) Theorem} Let $\Omega\subset\bC^n$ be an open subset such that every point $z_0\in\partial\Omega$ has a neighborhood $V$ such that $\Omega\cap V$ is pseudoconvex. Then $\Omega$ is pseudoconvex. \endstat \begproof{} As $d(z,\complement\Omega)$ coincides with $d\big(z,\complement (\Omega\cap V)\big)$ in a neighborhood of $z_0$, we see that there exists a neighborhood $U$ of $\partial\Omega$ such that $-\log d(z,\complement\Omega)$ is plurisubharmonic on $\Omega\cap U$. Choose a convex increasing function $\chi$ such that $$\chi(r)>\sup_{(\Omega\ssm U)\cap\ol B(0,r)}-\log d(z,\complement \Omega),~~~~\forall r\ge 0.$$ Then the function $$\psi(z)=\max\big\{\chi(|z|),-\log d(z,\complement\Omega)\big\}$$ coincides with $\chi(|z|)$ in a neighborhood of $\Omega\ssm U$. Therefore $\psi\in\Psh(\Omega)$, and $\psi$ is clearly an exhaustion.\qed \endproof Now, we give a geometric characterization of the pseudoconvexity property when $\partial\Omega$ is of class $C^2$. Let $\rho\in C^2(\ol\Omega)$ be a {\it defining function} of $\Omega$, i.e.\ a function such that $$\rho<0~~\hbox{\rm on}~~\Omega,~~~~\rho=0~~\hbox{\rm and}~~d\rho\ne 0~~ \hbox{\rm on}~~\partial\Omega.\leqno(7.9)$$ The {\it holomorphic tangent space} to $\partial\Omega$ is by definition the largest complex subspace which is contained in the tangent space $T_{\partial\Omega}$ to the boundary: $${}^hT_{\partial\Omega}=T_{\partial\Omega}\cap JT_{\partial\Omega}. \leqno(7.9)$$ It is easy to see that ${}^hT_{\partial\Omega,z}$ is the complex hyperplane of vectors $\xi\in\bC^n$ such that $$d'\rho(z)\cdot\xi= \sum_{1\le j\le n}{\partial\rho\over\partial z_j}\,\xi_j=0.$$ The {\it Levi form} on ${}^hT_{\partial\Omega}$ is defined at every point $z\in\partial\Omega$ by $$L_{\partial\Omega,z}(\xi)={1\over|\nabla\rho(z)|}\sum_{j,k} {\partial^2\rho\over\partial z_j\partial\ol z_k}\,\xi_j\ol\xi_k,~~~~ \xi\in {}^hT_{\partial\Omega,z}.\leqno(7.10)$$ The Levi form does not depend on the particular choice of $\rho$, as can be seen from the following intrinsic computation of $L_{\partial\Omega}$ (we still denote by $L_{\partial\Omega}$ the associated sesquilinear form). \begstat{(7.11) Lemma} Let $\xi,\eta$ be $C^1$ vector fields on $\partial\Omega$ with values in ${}^hT_{\partial\Omega}$. Then $$\langle[\xi,\eta],J\nu\rangle=4\Im L_{\partial\Omega}(\xi,\eta)$$ where $\nu$ is the outward normal unit vector to $\partial\Omega$, $[~,~]$ the Lie bracket of vector fields and $\langle~,~\rangle$ the hermitian inner product. \endstat \begproof{} Extend first $\xi,\eta$ as vector fields in a neighborhood of $\partial\Omega$ and set $$\xi'=\sum\xi_j{\partial\over\partial z_j}={1\over 2}(\xi-\ii J\xi),~~~~ \eta''=\sum\ol\eta_k{\partial\over\partial\ol z_k}={1\over 2}(\eta+\ii J\eta).$$ As $\xi,J\xi,\eta,J\eta$ are tangent to $\partial\Omega$, we get on $\partial\Omega$~: $$0=\xi'.(\eta''.\rho)+\eta''.(\xi'.\rho)= \sum_{1\le j,k\le n}2{\partial^2\rho\over\partial z_j\partial\ol z_k}\, \xi_j\ol\eta_k+\xi_j{\partial\ol\eta_k\over\partial z_j} {\partial\rho\over\partial\ol z_k}+\ol\eta_k{\partial\xi_j\over\partial\ol z_k} {\partial\rho\over\partial z_j}.$$ Since $[\xi,\eta]$ is also tangent to $\partial\Omega$, we have $\Re\langle[\xi,\eta],\nu\rangle=0$, hence $\langle J[\xi,\eta],\nu\rangle$ is real and $$\langle[\xi,\eta],J\nu\rangle=-\langle J[\xi,\eta],\nu\rangle= -{1\over|\nabla\rho|}\big(J[\xi,\eta].\rho\big)= -{2\over|\nabla\rho|}\Re\big(J[\xi',\eta''].\rho\big)$$ because $J[\xi',\eta']=i[\xi',\eta']$ and its conjugate $J[\xi'',\eta'']$ are tangent to $\partial\Omega$. We find now $$\eqalignno{ J[\xi',\eta'']&=-\ii\sum\xi_j{\partial\ol\eta_k\over\partial z_j} {\partial\over\partial\ol z_k}+\ol\eta_k{\partial\xi_j\over\partial\ol z_k} {\partial\over\partial\ol z_j},\cr \Re\big(J[\xi',\eta''].\rho\big)&= \Im\sum\xi_j{\partial\ol\eta_k\over\partial z_j} {\partial\rho\over\partial\ol z_k}+\ol\eta_k {\partial\xi_j\over\partial\ol z_k}{\partial\rho\over\partial z_j}=-2\Im\sum {\partial^2\rho\over\partial z_j\partial\ol z_k}\,\xi_j\ol\eta_k,\cr \langle[\xi,\eta],J\nu\rangle&={4\over|\nabla\rho|}\Im\sum {\partial^2\rho\over\partial z_j\partial\ol z_k}\,\xi_j\ol\eta_k= 4\Im L_{\partial\Omega}(\xi,\eta).&\square\cr}$$ \endproof \begstat{(7.12) Theorem} An open subset $\Omega\subset\bC^n$ with $C^2$ boundary is pseudoconvex if and only if the Levi form $L_{\partial\Omega}$ is semipositive at every point of $\partial\Omega$. \endstat \begproof{} Set $\delta(z)=d(z,\complement\Omega)$, $z\in\ol\Omega$. Then $\rho=-\delta$ is $C^2$ near $\partial\Omega$ and satisfies (7.9). If $\Omega$ is pseudoconvex, the plurisubharmonicity of $-\log(-\rho)$ means that for all $z\in\Omega$ near $\partial\Omega$ and all $\xi\in\bC^n$ one has $$\sum_{1\le j,k\le n}\Big({1\over|\rho|}\,{\partial^2\rho\over\partial z_j \partial\ol z_k}+{1\over\rho^2}\,{\partial\rho\over\partial z_j}\, {\partial\rho\over\partial\ol z_k}\Big)\xi_j\ol\xi_k\ge 0.$$ Hence $\sum~(\partial^2\rho/\partial z_j\partial\ol z_k)\xi_j\ol\xi_k\ge 0$ if $\sum~(\partial\rho/\partial z_j)\xi_j=0$, and an easy argument shows that this is also true at the limit on $\partial\Omega$. Conversely, if $\Omega$ is not pseudoconvex, Th.~7.2 and 7.7 show that $-\log\delta$ is not plurisubharmonic in any neighborhood of $\partial\Omega$. Hence there exists $\xi\in\bC^n$ such that $$c=\Big({\partial^2\over\partial t\partial\ol t}\,\log\delta(z+t\xi) \Big)_{|t=0}>0$$ for some $z$ in the neighborhood of $\partial\Omega$ where $\delta\in C^2$. By Taylor's formula, we have $$\log\delta(z+t\xi)=\log\delta(z)+\Re(at+bt^2)+c|t|^2+{\rm o}(|t|^2)$$ with $a,b\in\bC$. Now, choose $z_0\in\partial\Omega$ such that $\delta(z)=|z-z_0|$ and set $$h(t)=z+t\xi+e^{at+bt^2}(z_0-z),~~~~t\in\bC.$$ Then we get $h(0)=z_0$ and $$\eqalign{ \delta(h(t))&\ge\delta(z+t\xi)-\delta(z)\,\big|e^{at+bt^2}\big|\cr &\ge\delta(z)\,\big|e^{at+bt^2}\big|\,\big(e^{c|t|^2/2}-1\big) \ge\delta(z)\,c|t|^2/3\cr}$$ when $|t|$ is sufficiently small. Since $\delta(h(0))=\delta(z_0)=0$, we obtain at $t=0$~: $${\partial\over\partial t}\,\delta(h(t))=\sum {\partial\delta\over\partial z_j}(z_0)\,h'_j(0)=0,$$ $${\partial^2\over\partial t\partial\ol t}\,\delta(h(t))=\sum {\partial^2\delta\over\partial z_j\partial\ol z_k}(z_0)\,h'_j(0)\ol{h'_k(0)}>0 ,$$ hence $h'(0)\in {}^hT_{\partial\Omega,z_0}$ and $L_{\partial\Omega,z_0} (h'(0))<0$.\qed \endproof \begstat{(7.13) Definition} The boundary $\partial\Omega$ is said to be weakly $($resp.\ strongly$)$ pseudoconvex if $L_{\partial\Omega}$ is semipositive $($resp.\ positive definite$)$ on~$\partial\Omega$. The boundary is said to be Levi flat if $L_{\partial\Omega}\equiv 0$. \endstat \begstat{(7.14) Remark} \rm Lemma 7.11 shows that $\partial\Omega$ is Levi flat if and only if the subbundle ${}^hT_{\partial\Omega}\subset T_{\partial\Omega}$ is integrable (i.e.\ stable under the Lie bracket). Assume that $\partial\Omega$ is of class $C^k$, $k\ge 2$. Then ${}^hT_{\partial\Omega}$ is of class $C^{k-1}$. By Frobenius' theorem, the integrability condition implies that ${}^hT_{\partial\Omega}$ is the tangent bundle to a $C^k$ foliation of $\partial\Omega$ whose leaves have real dimension $2n-2$. But the leaves themselves must be complex analytic since ${}^hT_{\partial\Omega}$ is a complex vector space (cf.\ Lemma~7.15 below). Therefore $\partial\Omega$ is Levi flat if and only if it is foliated by complex analytic hypersurfaces. \endstat \begstat{(7.15) Lemma} Let $Y$ be a $C^1$-submanifold of a complex analytic manifold $X$. If the tangent space $T_{Y,x}$ is a complex subspace of $T_{X,x}$ at every point $x\in Y$, then $Y$ is complex analytic. \endstat \begproof{} Let $x_0\in Y$. Select holomorphic coordinates $(z_1,\ldots,z_n)$ on $X$ centered at $x_0$ such that $T_{Y,x_0}$ is spanned by $\partial/\partial z_1,\ldots,\partial/\partial z_p$. Then there exists a neighborhood $U=U'\times U''$ of $x_0$ such that $Y\cap U$ is a graph $$z''=h(z'),~~~~z'=(z_1,\ldots,z_p)\in U',~~z''=(z_{p+1},\ldots,z_n)$$ with $h\in C^1(U')$ and $dh(0)=0$. The differential of $h$ at $z'$ is the composite of the projection of $\bC^p\times\{0\}$ on $T_{Y,(z',h(z'))}$ along $\{0\}\times\bC^{n-p}$ and of the second projection $\bC^n\to\bC^{n-p}$. Hence $dh(z')$ is $\bC$-linear at every point and $h$ is holomorphic.\qed \endproof \titleb{\S 8.}{Exercises}\begpet \titled{8.1.} Let $\Omega\subset\bC^n$ be an open set such that $$z\in\Omega,~~\lambda\in\bC,~~|\lambda|\le 1\Longrightarrow \lambda z\in\Omega.$$ Show that $\Omega$ is a union of polydisks of center $0$ (with arbitrary linear changes of coordinates) and infer that the space of polynomials $\bC[z_1,\ldots,z_n]$ is dense in $\cO(\Omega)$ for the topology of uniform convergence on compact subsets and in $\cO(\Omega)\cap C^0(\ol\Omega)$ for the topology of uniform convergence on~$\ol\Omega$.\newline {\it Hint}\/: consider the Taylor expansion of a function $f\in\cO(\Omega)$ at the origin, writing it as a series of homogeneous polynomials. To deal with the case of $\ol\Omega$, first apply a dilation to~$f$. \titled{8.2.}Let $B\subset\bC^n$ be the unit euclidean ball, $S=\partial B$ and $f\in\cO(B)\cap C^0(\ol B)$. Our goal is to check the following Cauchy formula: $$f(w)={1\over\sigma_{2n-1}}\int_S{f(z)\over(1-\langle w,z\rangle)^n}\, d\sigma(z).$$ \item{a)} By means of a unitary transformation and Exercise 8.1, reduce the question to the case when $w=(w_1,0,\ldots,0)$ and $f(z)$ is a monomial $z^\alpha$. \smallskip \item{b)} Show that the integral $\int_B z^\alpha\ol z_1^k\,d\lambda(z)$ vanishes unless $\alpha=(k,0,\ldots,0)$. Compute the value of the remaining integral by the Fubini theorem, as well as the integrals $\int_S z^\alpha \ol z_1^k\,d\sigma(z)$. \smallskip \item{c)} Prove the formula by a suitable power series expansion. \titled{8.3.} A current $T\in\cD'_p(M)$ is said to be {\it normal} if both $T$ and $dT$ are of order zero, i.e.\ have measure coefficients. \smallskip \item{a)} If $T$ is normal and has support contained in a $C^1$ submanifold $Y\subset M$, show that there exists a normal current $\Theta$ on $Y$ such that $T=j_\star\Theta$, where $j:Y\longrightarrow M$ is the inclusion.\newline {\it Hint}\/: if $x_1=\ldots=x_q=0$ are equations of $Y$ in a coordinate system $(x_1,\ldots,x_n)$, observe that $x_jT=x_jdT=0$ for $1\le j\le q$ and infer that $dx_1\wedge\ldots\wedge dx_q$ can be factorized in all terms of $T$. \smallskip \item{b)} What happens if $p>\dim Y$~? \smallskip \item{c)} Are a) and b) valid when the normality assumption is dropped~? \titled{8.4.} Let $T=\sum_{1\le j\le n}T_jd\ol z_j$ be a closed current of bidegree $(0,1)$ with compact support in $\bC^n$ such that $d''T=0$. \smallskip \item{a)} Show that the partial convolution $S=(1/\pi z_1)~\star_1~T_1$ is a solution of the equation $d''S=T$. \smallskip \item{b)} Let $K={\rm Supp}\,T$. If $n\ge 2$, show that $S$ has support in the compact set $\wt K$ equal to the union of $K$ and of all bounded components of $\bC^n\ssm K$.\newline {\it Hint}\/: observe that $S$ is holomorphic on $\bC^n\ssm K$ and that $S$ vanishes for \hbox{$|z_2|+\ldots+|z_n|$} large. \titled{8.5.} Alternative proof of the Dolbeault-Grothendieck lemma. Let $v=\sum_{|J|=q}\!\!v_Jd\ol z_J$, $q\ge 1$, be a smooth form of bidegree $(0,q)$ on a polydisk $\Omega=D(0,R)\subset\bC^n$, such that $d''v=0$, and let $\omega=D(0,r)\compact\omega$. Let $k$ be the smallest integer such that the monomials $d\ol z_J$ appearing in $v$ only involve $d\ol z_1$, $\ldots$, $d\ol z_k$. Prove by induction on $k$ that the equation $d''u=v$ can be solved on $\omega$.\newline {\it Hint}\/: set $v=f\wedge d\ol z_k+g$ where $f$, $g$ only involve $d\ol z_1$, $\ldots$, $d\ol z_{k-1}$. Then consider $v-d''F$ where $$F=\sum_{|J|=q-1}F_Jd\ol z_J,~~~~F_J(z)=(\psi(z_k)f_J)\star_k \Big({1\over \pi z_k}\Big),$$ where $\star_k$ denotes the partial convolution with respect to~$z_k$, $\psi(z_k)$ is a cut-off function equal to 1 on $D(0,r_k+\varepsilon)$ and $f=\sum_{|J|=q-1}f_Jd\ol z_J$. \titled{8.6.} Construct locally bounded non continuous subharmonic functions on $\bC$.\newline {\it Hint}\/: consider $e^u$ where $u(z)=\sum_{j\ge 1}2^{-j}\log|z-1/j|$. \titled{8.7.} Let $\omega$ be an open subset of $\bR^n$, $n\ge 2$, and $u$ a subharmonic function which is not locally $-\infty$. \smallskip \item{a)} For every open set $\omega\compact\Omega$, show that there is a positive measure $\mu$ with support in $\ol\omega$ and a harmonic function $h$ on $\omega$ such that $u=N\star\mu+h$ on $\omega$. \smallskip \item{b)} Use this representation to prove the following properties: $u\in L^p_\loc$ for all $p0$ small, show that $\Omega$ contains the Hartogs figure $$\eqalign{ \{\varepsilon/2<|w_1|<\varepsilon\}&\times\{|w_j|<\varepsilon^2\}_{10$ there is a constant $C_\varepsilon>0$ such that $${-H\delta_z(\xi)\over\delta(z)}+\varepsilon{|d'\delta_z.\xi|^2\over |\delta(z)|^2}+C_\varepsilon|\xi|^2\ge0$$ for $\xi\in\bC^n$ and $z$ near $\partial\Omega$. \smallskip \item{b)} Set $\psi(z)=-\log\delta(z)+K|z|^2$. Show that for $K$ large and $\alpha$ small the function $$\rho(z)=-\exp\big(-\alpha\psi(z)\big)=-\big(e^{-K|z|^2}\delta(z) \big)^\alpha$$ is plurisubharmonic. \smallskip \item{c)} Prove the existence of a plurisubharmonic exhaustion function $u:\Omega\to[-1,0[$ of class $C^2$ such that $|u(z)|$ has the same order of magnitude as $\delta(z)^\alpha$ when $z$ tends to $\partial\Omega$.\newline {\it Hint}\/: consult (Diederich-Fornaess 1976). \titled{8.15.} Let $\Omega=\omega+\ii\bR^n$ be a connected tube in $\bC^n$ of base $\omega$. \smallskip \item{a)} Assume first that $n=2$. Let $T\subset\bR^2$ be the triangle $x_1\ge 0$, $x_2\ge 0$, $x_1+x_2\le 1$, and assume that the two edges $[0,1]\times\{0\}$ and $\{0\}\times[0,1]$ are contained in $\omega$. Show that every holomorphic function $f\in\cO(\Omega)$ extends to a neighborhood of $T+\ii\bR^2$.\newline {\it Hint}\/: let $\pi:\bC^2\longrightarrow\bR^2$ be the projection on the real part and $M_\varepsilon$ the intersection of $\pi^{-1}((1+\varepsilon)T)$ with the Riemann surface $z_1+z_2-{\varepsilon\over 2}(z_1^2+z_2^2)=1$ (a non degenerate affine conic). Show that $M_\varepsilon$ is compact and that $$\eqalign{ &\pi(\partial M_\varepsilon)\subset([0,1+\varepsilon]\times\{0\})\cup (\{0\}\times[0,1+\varepsilon])\subset\omega,\cr &\pi([0,1]\cdot M_\varepsilon)\supset T\cr}$$ for $\varepsilon$ small. Use the Cauchy formula along $\partial M_\varepsilon$ (in some parametrization of the conic) to obtain an extension of $f$ to $[0,1]\cdot M_\varepsilon+\ii\bR^n$. \smallskip \item{b)} In general, show that every $f\in\cO(\Omega)$ extends to the convex hull $\wh\Omega$.\newline {\it Hint}\/: given $a,b\in\omega$, consider a polygonal line joining $a$ and $b$ and apply a) inductively to obtain an extension along $[a,b]+\ii\bR^n$. \titled{8.16.} For each integer $\nu\ge 1$, consider the algebraic variety $$X_\nu=\big\{(z,w,t)\in\bC^3\,;\,wt=p_\nu(z)\big\},~~~~ p_\nu(z)=\prod_{1\le k\le\nu}(z-1/k),$$ and the map $j_\nu:X_\nu\to X_{\nu+1}$ such that $$j_\nu(z,w,t)=\Big(z,\,w,\,t\Big(z-{1\over\nu+1}\Big)\Big).$$ \item{a)} Show that $X_\nu$ is a Stein manifold, and that $j_\nu$ is an embedding of $X_\nu$ onto an open subset of $X_{\nu+1}$. \item{b)} Define $X=\lim (X_\nu,j_\nu)$, and let $\pi_\nu:X_\nu\to \bC^2$ be the projection to the first two coordinates. Since $\pi_{\nu+1}\circ j_\nu=\pi_\nu$, there exists a holomorphic map $\pi:X\to\bC^2$, $\pi=\lim\pi_\nu$. Show that $$\bC^2\ssm\pi(X)=\big\{(z,0)\in\bC^2\,;\,z\ne 1/\nu,~\forall\nu\in\bN,~ \nu\ge 1\big\},$$ and especially, that $(0,0)\notin\pi(X)$. \item{c)} Consider the compact set $$K=\pi^{-1}\big(\{(z,w)\in\bC^2\,;\,|z|\le 1,~|w|=1\}\big).$$ By looking at points of the forms $(1/\nu,w,0)$, $|w|=1$, show that $\pi^{-1}(1/\nu,1/\nu)\in\wh K_{\cO(X)}$. Conclude from this that $X$ is not holomorphically convex (this example is due to Fornaess~1977). \titled{8.17.} Let $X$ be a complex manifold, and let $\pi:\wt X\to X$ be a holomorphic unramified covering of~$X$ ($X$ and $\wt X$ are assumed to be connected). \item{a)} Let $g$ be a complete riemannian metric on~$X$, and let $\wt d$ be the geodesic distance on $\wt X$ associated to $\wt g=\pi^\star g$ (see VIII-2.3 for definitions). Show that $\wt g$ is complete and that $\delta_0(x):=\wt d(x,x_0)$ is a continuous exhaustion function on~$\wt X$, for any given point $x_0\in\wt X$. \item{b)} Let $(U_\alpha)$ be a locally finite covering of $X$ by open balls contained in coordinate open sets, such that all intersections $U_\alpha\cap U_\beta$ are diffeomorphic to convex open sets (see Lemma IV-6.9). Let $\theta_\alpha$ be a partition of unity subordinate to the covering $(U_\alpha)$, and let $\delta_{\varepsilon_\alpha}$ be the convolution of $\delta_0$ with a regularizing kernel $\rho_{\varepsilon_\alpha}$ on each piece of $\pi^{-1}(U_\alpha)$ which is mapped biholomorphically onto $U_\alpha$. Finally, set $\delta=\sum(\theta_\alpha\circ\pi)\delta_{\varepsilon_\alpha}$. Show that if $(\varepsilon_\alpha)$ is a collection of sufficiently small positive numbers, then $\delta$ is a smooth exhaustion function on~$\wt X$. \item{c)} Using the fact that $\delta_0$ is $1$-Lipschitz with respect to $\wt d$, show that derivatives $\partial^{|\nu|}\delta(x)/\partial x^\nu$ of a given order with respect to coordinates in $U_\alpha$ are uniformly bounded in all components of $\pi^{-1}(U_\alpha)$, at least when $x$ lies in the compact subset $\Supp\theta_\alpha$. Conclude from this that there exists a positive hermitian form with continuous coefficients on $X$ such that $H\delta\ge-\pi^\star\gamma$ on $\wt X$. \item{d)} If $X$ is strongly pseudoconvex, show that $\wt X$ is also strongly pseudoconvex.\newline {\it Hint}\/: let $\psi$ be a smooth strictly plurisubharmonic exhaustion function on~$X$. Show that there exists a smooth convex increasing function $\chi:\bR\to\bR$ such that $\delta+\chi\circ\psi$ is strictly plurisubharmonic. \endpet \titlea{Chapter II.}{\newline Coherent Sheaves and Analytic Spaces} \begpet The chapter starts with rather general and abstract concepts concerning sheaves and ringed spaces. Introduced in the decade 1950-1960 by Leray, Cartan, Serre and Grothendieck, sheaves and ringed spaces have since been recognized as the adequate tools to handle algebraic varieties and analytic spaces in a unified framework. We then concentrate ourselves on the theory of complex analytic functions. The second section is devoted to a proof of the Weierstrass preparation theorem, which is nothing but a division algorithm for holomorphic functions. It is used to derive algebraic properties of the ring $\cO_n$ of germs of holomorphic functions in~$\bC^n$. Coherent analytic sheaves are then introduced and the fundamental coherence theorem of Oka is proved. Basic properties of analytic sets are investigated in detail: local parametrization theorem, Hilbert's Nullstellensatz, coherence of the ideal sheaf of an analytic set, analyticity of the singular set. The formalism of complex spaces is then developed and gives a natural setting for the proof of more global properties (decomposition into global irreducible components, maximum principle). After a few definitions concerning cycles, divisors and meromorphic functions, we investigate the important notion of normal space and establish the Oka normalization theorem. Next, the Remmert-Stein extension theorem and the Remmert proper mapping theorem on images of analytic sets are proved by means of semi-continuity results on the rank of morphisms. As an application, we give a proof of Chow's theorem asserting that every analytic subset of $\bP^n$ is algebraic. Finally, the concept of analytic scheme with nilpotent elements is introduced as a generalization of complex spaces, and we discuss the concepts of bimeromorphic maps, modifications and blowing-up. \endpet \titleb{\S 1.}{Presheaves and Sheaves} \titlec{\S 1.A.}{Main Definitions} Sheaves have become a very important tool in analytic or algebraic geometry as well as in algebraic topology. They are especially useful when one wants to relate global properties of an object to its local properties (the latter being usually easier to establish). We first introduce the axioms of presheaves and sheaves in full generality and give some basic examples. \begstat{(1.1) Definition} Let $X$ be a topological space. A presheaf $\cA$ on $X$ consists of the following data: \medskip \item{\rm a)} a collection of non empty sets $\cA(U)$ associated with every open set $U\subset X$, \smallskip \item{\rm b)} a collection of maps $\rho_{U,V}:\cA(V)\longrightarrow\cA(U)$ defined whenever $U\subset V$ and satisfying the transitivity property \smallskip \item{\rm c)} $\rho_{U,V}\circ\rho_{V,W}=\rho_{U,W}$~~~for~~$U\subset V\subset W,\qquad\rho_{U,U}=\Id_U$~~~for every~$U$. \medskip\noindent The set $\cA(U)$ is called the set of sections of the presheaf $\cA$ over~$U$. \endstat Most often, the presheaf $\cA$ is supposed to carry an additional algebraic structure. For instance: \begstat{(1.2) Definition} A presheaf $\cA$ is said to be a presheaf of abelian groups $($resp.\ rings, $R$-modules, algebras$)$ if all sets $\cA(U)$ are abelian groups $($resp.\ rings, $R$-modules, algebras$)$ and if the maps $\rho_{U,V}$ are morphisms of these algebraic structures. In this case, we always assume that $\cA(\emptyset)=\{0\}$. \endstat \begstat{(1.3) Example} \rm If we assign to each open set $U\subset X$ the set $\cC(U)$ of all real valued continuous functions on~$U$ and let $\rho_{U,V}$ be the obvious restriction morphism $\cC(V)\to\cC(U)$, then $\cC$ is a presheaf of rings on~$X$. Similarly if $X$ is a differentiable (resp.\ complex analytic) manifold, there are well defined presheaves of rings $\cC^k$ of functions of class $C^k$ (resp.\ $\cO$) of holomorphic functions) on~$X$. Because of these examples, the maps $\rho_{U,V}$ in Def.~1.1 are often viewed intuitively as ``restriction homomorphisms'', although the sets $\cA(U)$ are not necessarily sets of functions defined over~$U$. For the simplicity of notation we often just write $\rho_{U,V}(f)= f_{\restriction U}$ whenever $f\in\cA(V)$, $V\supset U$.\qed \endstat For the above presheaves $\cC$, $\cC^k$, $\cO$, the properties of functions under consideration are purely local. As a consequence, these presheaves satisfy the following additional {\it gluing axioms}, where $(U_\alpha)$ and $U=\bigcup U_\alpha$ are arbitrary open subsets of $X\,$: $$\leqalignno{ &\hbox{\rm If $F_\alpha\in\cA(U_\alpha)$ are such that $\rho_{U_\alpha\cap U_\beta,U_\alpha}(F_\alpha)= \rho_{U_\alpha\cap U_\beta,U_\beta}(F_\beta)$}&(1.4')\cr &\hbox{\rm for all $\alpha,\beta$, there exists $F\in\cA(U)$ such that $\rho_{U_\alpha,U}(F)=F_\alpha\,$;}\cr}$$ $$\hbox{\rm If $F,G\in\cA(U)$ and $\rho_{U_\alpha,U}(F)= \rho_{U_\alpha,U}(G)$ for all $\alpha$, then $F=G\,$;}\leqno(1.4'')$$ in other words, local sections over the sets $U_\alpha$ can be glued together if they coincide in the intersections and the resulting section on $U$ is uniquely defined. Not all presheaves satisfy $(1.4')$ and $(1.4'')$: \begstat{(1.5) Example} \rm Let $E$ be an arbitrary set with a distinguished element~$0$ (e.g.\ an abelian group, a $R$-module,~$\ldots$). The {\it constant presheaf} $E_X$ on $X$ is defined to be $E_X(U)=E$ for all~$\emptyset\ne U\subset X$ and $E_X(\emptyset)=\{0\}$, with restriction maps $\rho_{U,V}=\Id_E$ if $\emptyset\ne U\subset V$ and $\rho_{U,V}=0$ if~$U=\emptyset$. Then axiom $(1.4')$ is not satisfied if $U$ is the union of two disjoint open sets $U_1$, $U_2$ and $E$ contains a non zero element. \endstat \begstat{(1.6) Definition} A presheaf $\cA$ is said to be a sheaf if it satisfies the gluing axioms $(1.4')$ and $(1.4'')$. \endstat If $\cA$, $\cB$ are presheaves of abelian groups (or of some other algebraic structure) on the same space~$X$, a presheaf morphism $\varphi:\cA\to\cB$ is a collection of morphisms $\varphi_U:\cA(U)\to\cB(U)$ commuting with the restriction morphisms, i.e.\ such that for each pair $U\subset V$ there is a commutative diagram $$\eqalign{ &~~\cA(V)\buildo{\displaystyle\varphi_V}\over\longrightarrow\cB(V)\cr &\rho^\cB_{U,V}\big\downarrow~~\phantom{\longrightarrow}~~\big\downarrow\rho^\cA_{U,V}\cr &~~\cA(U)\buildo{\displaystyle\varphi_U}\over\longrightarrow\cB(U).\cr}$$ We say that $\cA$ is a subpresheaf of $\cB$ in the case where $\varphi_U:\cA(U)\subset\cB(U)$ is the inclusion morphism; the commutation property then means that $\rho^\cB_{U,V}(\cA(V))\subset\cA(U)$ for all $U$, $V$, and that $\rho^\cA_{U,V}$ coincides with $\rho^\cB_{U,V}$ on~$\cA(V)$. If $\cA$ is a subpresheaf of a presheaf $\cB$ of abelian groups, there is a presheaf quotient $\cC=\cB/\cA$ defined by $\cC(U)=\cB(U)/\cA(U)$. In a similar way, one defines the presheaf kernel (resp.\ presheaf image, presheaf cokernel) of a presheaf morphism $\varphi:\cA\to\cB$ to be the presheaves $$U\mapsto\Ker\varphi_U,\qquad U\mapsto\Im\varphi_U,\qquad U\mapsto\Coker\varphi_U.$$ The direct sum $\cA\oplus\cB$ of presheaves of abelian groups $\cA$, $\cB$ is the presheaf $U\mapsto\cA(U)\oplus\cB(U)$, the tensor product $\cA\otimes\cB$ of presheaves of $R$-modules is $U\mapsto\cA(U)\otimes_R\cB(U)$, etc $\ldots$ \begstat{(1.7) Remark} \rm The reader should take care of the fact that the presheaf quotient of a sheaf by a subsheaf is not necessarily a sheaf. To give a specific example, let $X=S^1$ be the unit circle in~$\bR^2$, let $\cC$ be the sheaf of continuous complex valued functions and $\cZ$ the subsheaf of integral valued continuous functions (i.e.\ locally constant functions to $\bZ$). The exponential map $$\varphi=\exp(2\pi\ii\bu):\cC\longrightarrow\cC^\star$$ is a morphism from $\cC$ to the sheaf $\cC^\star$ of invertible continuous functions, and the kernel of $\varphi$ is precisely~$\cZ$. However $\varphi_U$ is surjective for all $U\ne X$ but maps $\cC(X)$ onto the multiplicative subgroup of continuous functions of $\cC^\star(X)$ of degree~$0$. Therefore the quotient presheaf $\cC/\cZ$ is not isomorphic with $\cC^\star$, although their groups of sections are the same for all $U\ne X$. Since $\cC^\star$ is a sheaf, we see that $\cC/\cZ$ does not satisfy property~$(1.4')$.\qed \endstat In order to overcome the difficulty appearing in Example~1.7, it is necessary to introduce a suitable process by which we can produce a sheaf from a presheaf. For this, it is convenient to introduce a slightly modified viewpoint for sheaves. \begstat{(1.8) Definition} If $\cA$ is a presheaf, we define the set $\wt\cA_x$ of germs of $\cA$ at a point $x\in X$ to be the abstract inductive limit $$\wt\cA_x=\lim_{\displaystyle\longrightarrow\atop\scriptstyle U\ni x} \big(\cA(U),\rho_{U,V}\big).$$ More explicitely, $\wt\cA_x$ is the set of equivalence classes of elements in the disjoint union $\coprod_{U\ni x}\cA(U)$ taken over all open neighborhoods $U$ of~$x$, with two elements $F_1\in\cA(U_1)$, $F_2\in\cA(U_2)$ being equivalent, $F_1\sim F_2$, if and only if there is a neighborhood $V\subset U_1,\,U_2$ such that $F_{1\restriction V}= F_{2\restriction V}$, i.e., $\rho_{VU_1}(F_1)=\rho_{VU_2}(F_2)$. The germ of an element $F\in\cA(U)$ at a point $x\in U$ will be denoted by~$F_x$. \endstat Let $\cA$ be an arbitrary presheaf. The disjoint union $\wt\cA=\coprod_{x\in X}\wt\cA_x$ can be equipped with a natural topology as follows: for every $F\in\cA(U)$, we set $$\Omega_{F,U}=\big\{F_x~;~x\in U\big\}$$ and choose the $\Omega_{F,U}$ to be a basis of the topology of~$\wt\cA\,$; note that this family is stable by intersection: $\Omega_{F,U}\cap\Omega_{G,V}=\Omega_{H,W}$ where $W$ is the (open) set of points $x\in U\cap V$ at which $F_x=G_x$ and $H=\rho_{W,U}(F)$. The obvious projection map $\pi:\wt\cA\to X$ which sends $\wt\cA_x$ to $\{x\}$ is then a local homeomorphism (it is actually a homeomorphism from $\Omega_{F,U}$ onto~$U$). This leads in a natural way to the following definition: \begstat{(1.9) Definition} Let $X$ and $\cS$ be topological spaces (not necessarily Hausdorff), and let $\pi:\cS\longrightarrow X$ be a mapping such that \smallskip \item{\rm a)} $\pi$ maps $\cS$ onto $X~;$ \smallskip \item{\rm b)} $\pi$ is a local homeomorphism, that is, every point in $\cS$ has an open neigh\-borhood which is mapped homeomorphically by $\pi$ onto an open subset of~$X$. \smallskip \noindent Then $\cS$ is called a sheaf-space on $X$ and $\pi$ is called the projection of $\cS$ on~$X$. If $x\in X$, then $\cS_x=\pi^{-1}(x)$ is called the stalk of $\cS$ at $x$. \endstat If $Y$ is a subset of $X$, we denote by $\Gamma(Y,\cS)$ the set of sections of $\cS$ on $Y$, i.e.\ the set of continuous functions $F:Y\to\cS$ such that $\pi\circ F=\Id_Y$. It is clear that the presheaf defined by the collection of sets $\cS'(U):=\Gamma(U,\cS)$ for all open sets $U\subset X$ together with the restriction maps $\rho_{U,V}$ satisfies axioms $(1.4')$ and $(1.4'')$, hence $\cS'$ is a sheaf. The set of germs of $\cS'$ at $x$ is in one-to-one correspondence with the stalk $\cS_x=\pi^{-1}(x)$, thanks to the local homeomorphism assumption 1.9~b). This shows that one can associate in a natural way a sheaf $\cS'$ to every sheaf-space~$\cS$, and that the sheaf-space $(\cS')^\sim$ can be considered to be identical to the original sheaf-space $\cS$. Since the assignment $\cS\mapsto\cS'$ from sheaf-spaces to sheaves is an equivalence of categories, we will usually omit the prime sign in the notation of $\cS'$ and thus use the same symbols for a sheaf-space and its associated sheaf of sections; in a corresponding way, we write $\Gamma(U,\cS)=\cS(U)$ when $U$ is an open set. Conversely, given a presheaf $\cA$ on $X$, we have an associated sheaf-space $\wt\cA$ and an obvious presheaf morphism $$\cA(U)\longrightarrow\wt\cA'(U)=\Gamma(U,\wt\cA),~~~~ F\longmapsto \wt F=(U\ni x\mapsto F_x).\leqno(1.10)$$ This morphism is clearly injective if and only if $\cA$ satisfies axiom $(1.4'')$, and it is not difficult to see that $(1.4')$ and $(1.4'')$ together imply surjectivity. Therefore $\cA\to\wt\cA'$ is an isomorphism if and only if $\cA$ is a sheaf. According to the equivalence of categories between sheaves and sheaf-spaces mentioned above, we will use from now on the same symbol $\wt\cA$ for the sheaf-space and its associated sheaf $\wt\cA'$; one says that $\wt\cA$ is the {\it sheaf associated with the presheaf}~$\cA$. If $\cA$ itself is a sheaf, we will again identify $\wt\cA$ and $\cA$, but we will of course keep the notational difference for a presheaf $\cA$ which is not a sheaf. \begstat{(1.11) Example} \rm The sheaf associated to the constant presheaf of stalk $E$ over $X$ is the sheaf of locally constant functions~$X\to E$. This sheaf will be denoted merely by $E_X$ or $E$ if there is no risk of confusion with the corresponding presheaf. In Example~1.7, we have $\cZ=\bZ_X$ and the sheaf $(\cC/\bZ_X)^\sim$ associated with the quotient presheaf $\cC/\bZ_X$ is isomorphic to $\cC^\star$ via the exponential map.\qed \endstat In the sequel, we usually work in the category of sheaves rather than in the category of presheaves themselves. For instance, the quotient $\cB/\cA$ of a sheaf $\cB$ by a subsheaf $\cA$ generally refers to the sheaf associated with the quotient presheaf: its stalks are equal to $\cB_x/\cA_x$, but a section $G$ of $\cB/\cA$ over an open set $U$ need not necessarily come from a global section of $\cB(U)\,$; what can be only said is that there is a covering $(U_\alpha)$ of $U$ and local sections $F_\alpha\in\cB(U_\alpha)$ representing $G_{\restriction U_\alpha}$ such that $(F_\beta-F_\alpha)_{\restriction U_\alpha\cap U_\beta}$ belongs to $\cA(U_\alpha\cap U_\beta)$. A sheaf morphism $\varphi:\cA\to\cB$ is said to be injective (resp.\ surjective) if the germ morphism $\varphi_x:\cA_x\to\cB_x$ is injective (resp.\ surjective) for every $x\in X$. Let us note again that a surjective sheaf morphism $\varphi$ does not necessarily give rise to surjective morphisms $\varphi_U:\cA(U)\to\cB(U)$. \titlec{\S 1.B.}{Direct and Inverse Images of Sheaves} Let $X$, $Y$ be topological spaces and let $f:X\to Y$ be a continuous map. If $\cA$ is a presheaf on $X$, the {\it direct image} $f_\star\cA$ is the presheaf on $Y$ defined by $$f_\star\cA(U)=\cA\big(f^{-1}(U)\big)\leqno(1.12)$$ for all open sets $U\subset Y$. When $\cA$ is a sheaf, it is clear that $f_\star\cA$ also satisfies axioms $(1.4')$ and $(1.4'')$, thus $f_\star\cA$ is a sheaf. Its stalks are given by $$(f_\star\cA)_y=\lim_{\displaystyle\longrightarrow\atop\scriptstyle V\ni y} \cA\big(f^{-1}(V)\big)\leqno(1.13)$$ where $V$ runs over all open neighborhoods of $y\in Y$. Now, let $\cB$ be a sheaf on~$Y$, viewed as a sheaf-space with projection map $\pi:\cB\to Y$. We define the {\it inverse image} $f^{-1}\cB$ by $$f^{-1}\cB=\cB\times_Y X=\big\{(s,x)\in\cB\times X\,;~\pi(s)=f(x)\big\} \leqno(1.14)$$ with the topology induced by the product topology on $\cB\times X$. It is then easy to see that the projection $\pi'={\rm pr}_2:f^{-1}\cB\to X$ is a local homeomorphism, therefore $f^{-1}\cB$ is a sheaf on~$X$. By construction, the stalks of $f^{-1}\cB$ are $$(f^{-1}\cB)_x=\cB_{f(x)},\leqno(1.15)$$ and the sections $\sigma\in f^{-1}\cB(U)$ can be considered as continuous mappings \hbox{$s:U\to\cB$} such that $\pi\circ\sigma=f$. In particular, any section $s\in\cB(V)$ on an open set $V\subset Y$ has a {\it pull-back} $$f^\star s=s\circ f~~\in~~f^{-1}\cB\big(f^{-1}(V)\big).\leqno(1.16)$$ There are always natural sheaf morphisms $$f^{-1}f_\star\cA\longrightarrow\cA,~~~~\cB\longrightarrow f_\star f^{-1}\cB\leqno(1.17)$$ defined as follows. A germ in $(f^{-1}f_\star\cA)_x=(f_\star\cA)_{f(x)}$ is defined by a local section $s\in (f_\star\cA)(V)=\cA(f^{-1}(V))$ for some neighborhood $V$ of $f(x)\,$; this section can be mapped to the germ $s_x\in\cA_x$. In the opposite direction, the pull-back $f^\star s$ of a section $s\in\cB(V)$ can be seen by (1.16) as a section of $f_\star f^{-1}\cB(V)$. It is not difficult to see that these natural morphisms are not isomorphisms in general. For instance, if $f$ is a finite covering map with $q$ sheets and if we take $\cA=E_X$, $\cB=E_Y$ to be constant sheaves, then $f_\star E_X\simeq E^q_Y$ and $f^{-1}E_Y=E_X$, thus $f^{-1}f_\star E_X\simeq E^q_X$ and $f_\star f^{-1}E_Y\simeq E_Y^q$. \titlec{\S 1.C.}{Ringed Spaces} Many natural geometric structures considered in analytic or algebraic geometry can be described in a convenient way as topological spaces equipped with a suitable ``structure sheaf'' which, most often, is a sheaf of commutative rings. For instance, a lot of properties of $C^k$ differentiable (resp.\ real analytic, complex analytic) manifolds can be described in terms of their sheaf of rings $\cC^k_X$ of differentiable functions (resp.\ $\cC^\omega_X$ of real analytic functions, $\cO_X$ of holomorphic functions). We first recall a few standard definitions concerning rings, referring to textbooks on algebra for more details (see e.g.\ Lang 1965). \begstat{(1.18) Some definitions and conventions about rings} All our rings $R$ are supposed implicitly to have a unit element $1_R$ $($if $R=\{0\}$, we agree that \hbox{$1_R=0_R\,)$}, and a ring morphism $R\to R'$ is supposed to map $1_R$ to $1_{R'}$. In the subsequent definitions, we assume that all rings under consideration are commutative. \smallskip \item{\rm a)} An ideal $I\subset R$ is said to be prime if $xy\in I$ implies $x\in I$ or $y\in I$, i.e., if the quotient ring $R/I$ is entire. \smallskip \item{\rm b)} An ideal $I\subset R$ is said to be maximal if $I\ne R$ and there are no ideals $J$ such that $I\subsetneq J\subsetneq R$ $($equivalently, if the quotient ring $R/I$ is a field$)$. \smallskip \item{\rm c)} The ring $R$ is said to be a local ring if $R$ has a unique maximal ideal $\gm$ $($equivalently, if $R$ has an ideal $\gm$ such that all elements of $R\ssm\gm$ are invertible$)$. Its residual field is defined to be the quotient field~$R/\gm$. \smallskip \item{\rm d)} The ring $R$ is said to be Noetherian if every ideal $I\subset R$ is finitely gene\-rated $($equivalently, if every increasing sequence of ideals $I_1\subset I_2\subset\ldots$ is~stationary$)$. \smallskip \item{\rm e)} The radical $\sqrt{I}$ of an ideal $I$ is the set of all elements $x\in R$ such that some power $x^m$, $m\in\bN^\star$, lies in in~$I$. Then $\sqrt{I}$ is again an ideal of~$R$. \smallskip \item{\rm f)} The nilradical $N(R)=\sqrt{\{0\}}$ is the ideal of nilpotent elements of~$R$. The ring $R$ is said to be reduced if $N(R)=\{0\}$. Otherwise, its reduction is defined to be the reduced ring~$R/N(R)$. \vskip0pt \endstat We now introduce the general notion of a ringed space. \begstat{(1.19) Definition} A ringed space is a pair $(X,\cR_X)$ consisting of a topolo\-gical space $X$ and of a sheaf of rings $\cR_X$ on $X$, called the structure sheaf. A~morphism $$F:(X,\cR_X)\to(Y,\cR_Y)$$ of ringed spaces is a pair $(f,F^\star)$ where $f:X\to Y$ is a continuous map and $$F^\star~:~~f^{-1}\cR_Y\to\cR_X,~~~~F^\star_x~:~~\cR_{Y,f(x)} \to\cR_{X,x}$$ a homomorphism of sheaves of rings on $X$, called the comorphism of $F$. \endstat If $F:(X,\cR_X)\to(Y,\cR_Y)$ and $G:(Y,\cR_Y)\to(Z,\cR_Z)$ are morphisms of ringed spaces, the composite $G\circ F$ is the pair consisting of the map $g\circ f:X\to Z$ and of the comorphism $(G\circ F)^\star=F^\star\circ f^{-1}G^\star\,$: $$\cmalign{ \hfill&F^\star\circ f^{-1}G^\star~:~~&f^{-1}g^{-1}\cR_Z~\,&\kern-1.9mm \buildo f^{-1}G^\star\over{\relbar\mkern-4mu\relbar\mkern-4mu\longrightarrow}f^{-1}\cR_Y &\kern-1.9mm\buildo F^\star\over{\relbar\mkern-4mu\longrightarrow}\cR_X,\cr &F^\star_x\circ G^\star_{f(x)}~:~~~&\cR_{Z,g\circ f(x)}~\,&{\relbar\mkern-4mu \relbar\mkern-4mu\longrightarrow}~\cR_{Y,f(x)}~&{\relbar\mkern-4mu\longrightarrow}~\cR_{X,x}.\cr} \leqno(1.20)$$ We say of course that $F$ is an isomorphism of ringed spaces if there exists $G$ such that $G\circ F=\Id_X$ and $F\circ G=\Id_Y$. If $(X,\cR_X)$ is a ringed space, the nilradical of $\cR_X$ defines an ideal subsheaf $\cN_X$ of $\cR_X$, and the identity map $\Id_X:X\to X$ together with the ring homomorphism $\cR_X\to\cR_X/\cN_X$ defines a ringed space morphism $$(X,\cR_X/\cN_X)\to (X,\cR_X)\leqno(1.21)$$ called the {\it reduction morphism}. Quite often, the letter $X$ by itself is used to denote the ringed space $(X,\cR_X)\,$; we then denote by $X_\red=(X,\cR_X/\cN_X)$ its reduction. The ringed space $X$ is said to be {\it reduced} if $\cN_X=0$, in which case the reduction morphism $X_\red\to X$ is an isomorphism. In all examples considered later on in this book, the structure sheaf $\cR_X$ will be a sheaf of {\it local rings} over some field~$k$. The relevant definition is as follows. \begstat{(1.22) Definition} \smallskip \item{\rm a)} A local ringed space is a ringed space $(X,\cR_X)$ such that all stalks $\cR_{X,x}$ are local rings. The maximal ideal of~$\cR_{X,x}$ will be denoted by $\gm_{X,x}$. A morphism $F=(f,F^\star):(X,\cR_X)\to(Y,\cR_Y)$ of local ringed spaces is a morphism of ringed spaces such that $F^\star_x(\gm_{Y,f(x)}) \subset\gm_{X,x}$ at any point $x\in X$ $($i.e., $F^\star_x$ is a ``local'' homomorphism of rings$)$. \smallskip \item{\rm b)} A local ringed space over a field $k$ is a local ringed space $(X,\cR_X)$ such that all rings $\cR_{X,x}$ are local $k$-algebras with residual field $\cR_{X,x}/\gm_{X,x}\simeq k$. A morphism $F$ between such spaces is supposed to have its comorphism defined by local $k$-homomorphisms $F^\star_x:\cR_{Y,f(x)}\to\cR_{X,x}$. \vskip0pt \endstat If $(X,\cR_X)$ is a local ringed space over~$k$, we can associate to each section $s\in\cR_X(U)$ a function $$\ol s:U\to k,\qquad x\mapsto\ol s(x)\in k=\cR_{X,x}/\gm_{X,x},$$ and we get a sheaf morphism $\cR_X\to\ol\cR_X$ onto a subsheaf of rings $\ol\cR_X$ of the sheaf of functions from $X$ to $k$. We clearly have a factorization $$\cR_X\to\cR_X/\cN_X\to\ol\cR_X,$$ and thus a corresponding factorization of ringed space morphisms (with $\Id_X$ as the underlying set theoretic map) $$X_\stred\to X_\red\to X$$ where $X_\stred=(X,\ol R_X)$ is called the {\it strong reduction} of $(X,\cR_X)$. It is easy to see that $X_\stred$ is actually a reduced local ringed space over~$k$. We say that $X$ is strongly reduced if $\cR_X\to\ol\cR_X$ is an isomorphism, that is, if $\cR_X$ can be identified with a subsheaf of the sheaf of functions $X\to k$ (in our applications to the theory of algebraic or analytic schemes, the concepts of reduction and strong reduction will actually be the same$\,$; in general, these notions differ, see Exercise ??.??). It is important to observe that reduction (resp.\ strong reduction) is a fonctorial process:\newline if $F=(f,F^\star):(X,\cR_X)\to(Y,\cR_Y)$ is a morphism of ringed spaces (resp.\ of local ringed spaces over~$k$), there are natural reductions $$\eqalign{ F_\red&=(f,F^\star_\red):X_\red\to Y_\red,\quad F^\star_\red:\cR_{Y,f(x)}/\cN_{Y,f(x)}\to\cR_{X,x}/\cN_{X,x},\cr F_\stred&=(f,f^\star):X_\stred\to Y_\stred,\quad f^\star:\ol\cR_{Y,f(x)}\to\ol\cR_{X,x},\quad\ol s\mapsto \ol s\circ f\cr}$$ where $f^\star$ is the usual pull-back comorphism associated with~$f$. Therefore, if $(X,\cR_X)$ and $(Y,\cR_Y)$ are strongly reduced, the morphism $F$ is completely determined by the underlying set-theoretic map~$f$. Our first basic examples of (strongly reduced) ringed spaces are the various types of mani\-folds already defined in Chapter~I. The language of ringed spaces provides an equivalent but more elegant and more intrinsic definition. \begstat{(1.23) Definition} Let $X$ be a Hausdorff separable topological space. One can define the category of $C^k$, $k\in\bN\cup\{\infty,\omega\}$, differentiable manifolds $($resp.\ complex analytic manifolds$)$ to be the category of reduced local ringed spaces $(X,\cR_X)$ over~$\bR$ $($resp.\ over~$\bC)$, such that every point $x\in X$ has a neighborhood $U$ on which the restriction $(U,\cR_{X\restriction U})$ is isomorphic to a ringed space $(\Omega,\cC^k_\Omega)$ where $\Omega\subset\bR^n$ is an open set and $\cC^k_\Omega$ is the sheaf of $C^k$ differentiable functions $($resp.\ $(\Omega,\cO_\Omega)$, where $\Omega\subset\bC^n$ is an open subset, and $\cO_\Omega$ is the sheaf of holomorphic functions on~$\Omega)$. \endstat We say that the ringed spaces $(\Omega,\cC^k_\Omega)$ and $(\Omega,\cO_\Omega)$ are the {\it models} of the category of differentiable (resp.\ complex analytic) manifolds, and that a general object $(X,\cR_X)$ in the category is {\it locally isomorphic} to one of the given model spaces. It is easy to see that the corresponding ringed spaces morphisms are nothing but the usual concepts of differentiable and holomorphic maps. \titlec{\S 1.D.}{Algebraic Varieties over a Field} As a second illustration of the notion of ringed space, we present here a brief introduction to the formalism of algebraic varieties, referring to (Hartshorne 1977) or (EGA 1967) for a much more detailed exposition. Our hope is that the reader who already has some background of analytic or algebraic geometry will find some hints of the strong interconnections between both theories. Beginners are invited to skip this section and proceed directly to the theory of complex analytic sheaves in~\S$,$2. All rings or algebras occurring in this section are supposed to be commutative rings with unit. \titled{\S 1.D.1. Affine Algebraic Sets.} Let $k$ be an algebraically closed field of any characteristic. An {\it affine algebraic set} is a subset $X\subset k^N$ of the affine space $k^N$ defined by an arbitrary collection $S\subset k[T_1,\ldots,T_N]$ of polynomials, that is, $$X=V(S)=\big\{(z_1,\ldots,z_N)\in k^N\,;\,P(z_1,\ldots,z_N)=0,\, \forall P\in S\big\}.$$ Of course, if $J\subset k[T_1,\ldots,T_N]$ is the ideal generated by $S$, then $V(S)=V(J)$. As $k[T_1,\ldots,T_N]$ is Noetherian, $J$ is generated by finitely many elements $(P_1,\ldots,P_m)$, thus $X=V(\{P_1,\ldots,P_m\})$ is always defined by finitely many equations. Conversely, for any subset $Y\subset k^N$, we consider the ideal $I(Y)$ of $k[T_1,\ldots,T_N]$, defined by $$I(Y)=\big\{P\in k[T_1,\ldots,T_N]\,;\,P(z)=0,~\forall z\in Y\big\}.$$ Of course, if $Y\subset k^N$ is an algebraic set, we have $V(I(Y))=Y$. In the opposite direction, we have the following fundamental result. \begstat{(1.24) Hilbert's Nullstellensatz {\rm (see Lang 1965)}} If $J\subset k[T_1,\ldots,T_N]$ is an ideal, then $I(V(J))=\sqrt{J}$. \endstat If $X=V(J)\subset k^N$ is an affine algebraic set, we define the (reduced) ring $\cO(X)$ of algebraic functions on $X$ to be the set of all functions $X\to k$ which are restrictions of polynomials, i.e., $$\cO(X)=k[T_1,\ldots,T_N]/I(X)=k[T_1,\ldots,T_N]/\sqrt{J}.\leqno(1.25)$$ This is clearly a reduced $k$-algebra. An (algebraic) morphism of affine algebraic sets $X=V(J)\subset k^N$, $Y=V(J')\subset k^{N'}$ is a map $f:Y\to X$ which is the restriction of a polynomial map $k^{N'}to k^N$. We then get a $k$-algebra homomomorphism $$f^\star:\cO(X)\to\cO(Y),\qquad s\mapsto s\circ f,$$ called the {\it comorphism} of~$f$. In this way, we have defined a contravariant fonctor $$X\mapsto\cO(X),\qquad f\mapsto f^\star\leqno(1.26)$$ from the category of affine algebraic sets to the category of finitely generated reduced $k$-algebras. We are going to show the existence of a natural fonctor going in the opposite direction. In fact, let us start with an arbitrary finitely generated algebra~$A$ (not necessarily reduced at this moment). For any choice of gene\-rators $(g_1,\ldots,g_N)$ of $A$ we get a surjective morphism of the polynomial ring $k[T_1,\ldots,T_N]$ onto~$A$, $$k[T_1,\ldots,T_N]\to A,\qquad T_j\mapsto g_j,$$ and thus $A\simeq k[T_1,\ldots,T_N]/J$ with the ideal $J$ being the kernel of this morphism. It is well-known that every maximal ideal $\gm$ of $A$ has codimension~$1$ in $A$ (see Lang 1965), so that $\gm$ gives rise to a $k$-algebra homomorphism $A\to A/\gm=k$. We thus get a bijection $$\Hom_\alg(A,k)\to\Spm(A),\qquad u\mapsto\Ker u$$ between the set of $k$-algebra homomorphisms and the set $\Spm(A)$ of maximal ideals of~$A$. In fact, if $A=k[T_1,\ldots,T_N]/J$, an element $\varphi\in\Hom_\alg(A,k)$ is completely determined by the values $z_j=\varphi(T_j\mod J)$, and the corresponding algebra homomorphism $k[T_1,\ldots,T_N]\to k$, $P\mapsto P(z_1,\ldots,z_N)$ can be factorized mod $J$ if and only if $z=(z_1,\ldots,z_N)\in k^N$ satisfies the equations $$P(z_1,\ldots,z_N)=0,\qquad\forall P\in J.$$ We infer from this that $$\Spm(A)\simeq V(J)=\big\{(z_1,\ldots,z_N)\in k^N\,;\,P(z_1,\ldots,z_N)=0,\, \forall P\in J\big\}$$ can be identified with the {\it affine algebraic set} $V(J)\subset k^N$. If we are given an algebra homomorphism $\Phi:A\to B$ of finitely generated $k$-algebras we get a corresponding map $\Spm(\Phi):\Spm(B)\to\Spm(A)$ described either as $$\eqalign{ &\Spm(B)\to\Spm(A),\quad\gm\mapsto\Phi^{-1}(\gm)\quad\hbox{or}\cr &\Hom_\alg(B,k)\to\Hom_\alg(A,k),\quad v\mapsto v\circ\Phi.\cr}$$ If $B=k[T'_1,\ldots,T'_{N'}]/J'$ and $\Spm(B)=V(J')\subset k^{N'}$, it is easy to see that $\Spm(\Phi):\Spm(B)\to\Spm(A)$ is the restriction of the polynomial map $$f:k^{N'}\to k^N,\qquad w\mapsto f(w)=(P_1(w),\ldots,P_N(w)),$$ where $P_j\in k[T'_1,\ldots,T'_{N'}]$ are polynomials such that $P_j=\Phi(T_j)\mod J'$ in~$B$. We have in this way defined a contravariant fonctor $$A\mapsto\Spm(A),\qquad \Phi\mapsto\Spm(\Phi)\leqno(1.27)$$ from the category of finitely generated $k$-algebras to the category of affine algebraic sets. Since $A=k[T_1,\ldots,T_N]/J$ and its reduction $A/N(A)=k[T_1,\ldots,T_N]/\sqrt{J}$ give rise to the same algebraic set $$V(J)=\Spm(A)=\Spm(A/N(A))=V(\sqrt{J}),$$ we see that the category of affine algebraic sets is actually equivalent to the subcategory of {\it reduced} finitely generated $k$-algebras. \begstat{(1.28) Example} \rm The simplest example of an affine algebraic set is the affine space $$k^N=\Spm(k[T_1,\ldots,T_N]),$$ in particular $\Spm(k)=k^0$ is just one point. We agree that $\Spm(\{0\})=\emptyset$ (observe that $V(J)=\emptyset$ when $J$ is the unit ideal in $k[T_1,\ldots,T_N])$. \endstat \titled{\S 1.D.2. Zariski Topology and Affine Algebraic Schemes.} Let $A$ be a finitely generated algebra and $X=\Spm(A)$. To each ideal $\ga\subset A$ we associate the zero variety $V(\ga)\subset X$ which consists of all elements $\gm\in X=\Spm(A)$ such that $\gm\supset \ga\,$; if $$A\simeq k[T_1,\ldots,T_N]/J\quad\hbox{and}\quad X\simeq V(J)\subset k^N,$$ then $V(\ga)$ can be identified with the zero variety $V(J_\ga)\subset X$ of the inverse image $J_\ga$ of $\ga$ in $k[T_1,\ldots,T_N]$. For any family $(\ga_\alpha)$ of ideals in $A$ we have $$V(\sum\ga_\alpha)=\bigcap V(\ga_\alpha),\qquad V(\ga_1)\cup V(\ga_2)=V(\ga_1\ga_2),$$ hence there exists a unique topology on $X$ such that the closed sets consist precisely of all algebraic subsets $(V(\ga))_{\ga\subset A}$ of~$X$. This topology is called the Zariski topology. The Zariski topology is almost never Hausdorff (for example, if $X=k$ is the affine line, the open sets are $\emptyset$ and complements of finite sets, thus any two nonempty open sets have nonempty intersection). However, $X$ is a {\it Noetherian space}, that is, a topological space in which every decreasing sequence of closed sets is stationary; an equivalent definition is to require that every open set is quasi-compact (from any open covering of an open set, one can extract a finite covering). We now come to the concept of affine open subsets. For $s\in A$, the open set $D(s)=X\ssm V(s)$ can be given the structure of an affine algebraic variety. In fact, if $A=k[T_1,\ldots,T_N]/J$ and $s$ is represented by a polynomial in $k[T_1,\ldots,T_N]$, the localized ring $A[1/s]$ can be written as $A[1/s]=k[T_1,\ldots,T_N,T_{N+1}]/J_s$ where $J_s=J[T_{N+1}]+(sT_{N+1}-1)$, thus $$V(J_s)=\{(z,w)\in V(J)\times k\,;\, s(z)\,w=1\}\simeq V(I)\ssm s^{-1}(0)$$ and $D(s)$ can be identified with $\Spm(A[1/s])$. We have $D(s_1)\cap D(s_2)=D(s_1s_2)$, and the sets $(D(s))_{s\in A}$ are easily seen to be a basis of the Zariski topology on~$X$. The open sets $D(s)$ are called {\it affine open sets}. Since the open sets $D(s)$ containing a given point $x\in X$ form a basis of neighborhoods, one can define a sheaf space $\cO_X$ such that the ring of germs $\cO_{X,x}$ is the inductive limit $$\cO_{X,x}=\lim_{\displaystyle\longrightarrow\atop\scriptstyle D(s)\ni x}A[1/s]= \{\hbox{fractions}~p/q\,;\,p,q\in A,\,q(x)\ne 0\}.$$ This is a local ring with maximal ideal $$\gm_{X,x}=\{p/q\,;\,p,q\in A,\,p(x)=0,\,q(x)\ne 0\},$$ and residual field $\cO_{X,x}/\gm_{X,x}=k$. In this way, we get a ringed space $(X,\cO_X)$ over~$k$. It is easy to see that $\Gamma(X,\cO_X)$ coincides with the finitely generated $k$-algebra~$A$. In fact, from the definition of~$\cO_X$, a global section is obtained by gluing together local sections $p_j/s_j$ on affine open sets $D(s_j)$ with \hbox{$\bigcup D(s_j)=X$}, $1\le j\le m$. This means that the ideal $\ga=(s_1,\ldots,s_m)\subset A$ has an empty zero variety $V(\ga)$, thus $\ga=A$ and there are elements $u_j\in A$ with $\sum u_js_j=1$. The compatibility condition $p_j/s_j=p_k/s_k$ implies that these elements are induced by $$\sum u_jp_j/\sum u_js_j=\sum u_jp_j\in A,$$ as desired. More generally, since the open sets $D(s)$ are affine, we get $$\Gamma(D(s),\cO_X)=A[1/s].$$ It is easy to see that the ringed space $(X,\cO_X)$ is reduced if and only if $A$ itself is reduced; in this case, $X$ is even strongly reduced as Hilbert's Nullstellensatz shows. Otherwise, the reduction $X_\red$ can obtained from the reduced algebra $A_\red=A/N(A)$. Ringed spaces $(X,\cO_X)$ as above are called {\it affine algebraic schemes} over~$k$ (although substantially different from the usual definition, our definition can be shown to be equivalent in this special situation; compare with (Hartshorne 1977); see also Exercise ??.??). The category of affine algebraic schemes is equivalent to the category of finitely generated $k$-algebras (with the arrows reversed). \titled{1.D.3. Algebraic Schemes.} Algebraic schemes over $k$ are defined to be ringed spaces over $k$ which are locally isomorphic to affine algebraic schemes, modulo an ad hoc separation condition. \begstat{(1.29) Definition} An algebraic scheme over $k$ is a local ringed space $(X,\cO_X)$ over $k$ such that \smallskip \item{\rm a)} $X$ has a finite covering by open sets $U_\alpha$ such that $(U_\alpha,\cO_{X\restriction U_\alpha})$ is isomorphic as a ringed space to an affine algebraic scheme $(\Spm(A_\alpha),\cO_{\Spm(A_\alpha)})$. \smallskip \item{\rm b)} $X$ satisfies the algebraic separation axiom, namely the diagonal $\Delta_X$ of $X\times X$ is closed for the Zariski topology. \medskip\noindent A morphism of algebraic schemes is just a morphism of the underlying local ringed spaces. An $($abstract$)$ algebraic variety is the same as a reduced algebraic scheme. \vskip0pt \endstat In the above definition, some words of explanation are needed for b), since the product $X\times Y$ of algebraic schemes over $k$ {\it is not} the ringed space theoretic product, i.e., the product topological space equipped with the structure sheaf $\pr_1^\star\cO_X\otimes_k\pr_2^\star\cO_Y$. Instead, we define the product of two affine algebraic schemes $X=\Spm(A)$ and $Y=\Spm(B)$ to be $X\times Y=\Spm(A\otimes_k B)$, equipped with the Zariski topology and the structural sheaf associated with~$A\otimes_k B$. Notice that the Zariski topology on $X\times Y$ {\it is not the product topology} of the Zariski topologies on $X$,~$Y$, as the example $k^2=k\times k$ shows; also, the rational function $1/(1-z_1-z_2)\in \cO_{k^2,(0,0)}$ is not in $\cO_{k,0}\otimes_k\cO_{k,0}$. In general, if $X$, $Y$ are written as $X=\bigcup U_\alpha$ and $Y=\bigcup V_\beta$ with affine open sets $U_\alpha$, $V_\beta$, we define $X\times Y$ to be the union of all open affine charts $U_\alpha\times V_\beta$ with their associated structure sheaves of affine algebraic varieties, the open sets of $X\times Y$ being all unions of open sets in the various charts $U_\alpha\times V_\beta$. The separation axiom b) is introduced for the sake of excluding pathological examples such as an affine line $k\amalg\{0'\}$ with the origin changed into a double point. \titled{1.D.4. Subschemes.} If $(X,\cO_X)$ is an affine algebraic scheme and $A=\Gamma(X,\cO_X)$ is the associated algebra, we say that $(Y,\cO_Y)$ is a {\it subscheme} of $(X,\cO_X)$ if there is an ideal $\ga$ of $A$ such that $Y\hookrightarrow X$ is the morphism defined by the algebra morphism $A\to A/\ga$ as its comorphism. As $\Spm(A/\ga)\to\Spm(A)$ has for image the set $V(\ga)$ of maximal ideals $\gm$ of $A$ containing $\ga$, we see that $Y=V(\ga)$ as a set; let us introduce the ideal subsheaf $\cJ=\ga\cO_X\subset\cO_X$. Since the structural sheaf $\cO_Y$ is obtained by taken localizations $A/\ga[1/s]$, it is easy to see that $\cO_Y$ coincides with the quotient sheaf $\cO_X/\cJ$ restricted to~$Y$. Since $\ga$ has finitely many generators, the ideal sheaf $\cJ$ is locally finitely generated (see \S~2 below). This leads to the following definition. \begstat{(1.30) Definition} If $(X,\cO_X)$ is an algebraic scheme, a $($closed$)$ subscheme is an algebraic scheme $(Y,\cO_Y)$ such that $Y$ is a Zariski closed subset of~$X$, and there is a locally finitely generated ideal subsheaf $\cJ\subset\cO_X$ such that $Y=V(\cJ)$ and $\cO_Y=(\cO_X/\cJ)_{\restriction Y}$. \endstat If $(Y,\cO_Y)$, $(Z,\cO_Z)$ are subschemes of $(X,\cO_X)$ defined by ideal subsheaves $\cJ,\,\cJ'\subset\cO_X$, there are corresponding subschemes $Y\cap Z$ and $Y\cup Z$ defined as ringed spaces $$(Y\cap Z,\cO_X/(\cJ+\cJ')),\qquad(Y\cup Z,\cO_X/\cJ\cJ').$$ \titled{\S 1.D.5. Projective Algebraic Varieties.} A very important subcategory of the category of algebraic varieties is provided by {\it projective algebraic varieties}. Let $\bP^N_k$ be the projective $N$-space, that is, the set $k^{N+1}\ssm\{0\}/k^\star$ of equivalence classes of $(N+1)$-tuples $(z_0,\ldots,z_N)\in k^{N+1}\ssm\{0\}$ under the equivalence relation given by $(z_0,\ldots,z_N)\sim\lambda (z_0,\ldots,z_N)$, $\lambda\in k^\star$. The corresponding element of $\bP^N_k$ will be denoted $[z_0:z_1:\ldots:z_N]$. It is clear that $\bP_N^k$ can be covered by the $(N+1)$ affine charts $U_\alpha$, $0\le\alpha\le N$, such that $$U_\alpha=\big\{[z_0:z_1:\ldots:z_N]\in\bP^N_k\,\;\,z_\alpha\ne 0\big\}.$$ The set $U_\alpha$ can be identified with the affine $N$-space $k^N$ by the map $$U_\alpha\to k^N,\qquad[z_0:z_1:\ldots:z_N]\mapsto \Big({z_0\over z_\alpha},\,{z_1\over z_\alpha},\ldots, {z_{\alpha-1}\over z_\alpha},\,{z_{\alpha+1}\over z_\alpha},\ldots, {z_N\over z_\alpha}\Big).$$ With this identification, $\cO(U_\alpha)$ is the algebra of homogeneous rational functions of degree $0$ in $z_0,\ldots,z_N$ which have just a power of $z_\alpha$ in their denominator. It is easy to see that the structure sheaves $\cO_{U_\alpha}$ and $\cO_{U_\beta}$ coincide in the intersections $U_\alpha\cap U_\beta\,$; they can be glued together to define an algebraic variety structure $(\bP^N_k,\cO_{\bP^N})$, such that $\cO_{\bP^N,[z]}$ consists of all homogeneous rational functions $p/q$ of degree $0$ (i.e., $\deg p=\deg q$), such that $q(z)\ne 0$. \begstat{(1.30) Definition} An algebraic scheme or variety $(X,\cO_X)$ is said to be projective if it is isomorphic to a closed subscheme of some projective space $(\bP^N_k,\cO_{\bP^N})$. \endstat We now indicate a standard way of constructing projective schemes. Let $S$ be a collection of homogeneous polynomials $P_j\in k[z_0,\ldots,z_N]$, of degree $d_j\in\bN$. We define an associated {\it projective algebraic set} $$\wt V(S)=\big\{[z_0:\ldots:z_N]\in\bP^N_k\,;\,P(z)=0,~\forall P\in S\big\}.$$ Let $J$ be the {\it homogeneous ideal} of $k[z_0,\ldots,z_N]$ generated by $S$ (recall that an ideal $J$ is said to be homogeneous if $J=\bigoplus J_m$ is the direct sum of its homogeneous components, or equivalently, if $J$ is generated by homogeneous elements). We have an associated graded algebra $$B=k[z_0,\ldots,z_N]/J=\bigoplus B_m,\qquad B_m=k[z_0,\ldots,z_N]_m/J_m$$ such that $B$ is generated by $B_1$ and $B_m$ is a finite dimensional vector space over $k$ for each~$k$. This is enough to construct the desired scheme structure on $\wt V(J):=\bigcap\wt V(J_m)$, as we see in the next subsection. \titled{1.D.6. Projective Scheme Associated with a Graded Algebra.} Let us start with a reduced graded $k$-algebra $$B=\bigoplus_{m\in\bN}B_m$$ such that $B$ is generated by $B_0$ and $B_1$ as an algebra, and $B_0$, $B_1$ are finite dimensional vector spaces over $k$ (it then follows that $B$ is finitely generated and that all $B_m$ are finite dimensional vector spaces). Given $s\in B_m$, $m>0$, we define a $k$-algebra $A_s$ to be the ring of all fractions of homogeneous degree~$0$ with a power of $s$ as their denominator, i.e., $$A_s=\big\{p/s^d\,;\,p\in B_{dm},\,d\in\bN\big\}.\leqno(1.31)$$ Since $A_s$ is generated by ${1\over s}B_1^m$ over $B_0$, $A_s$ is a finitely generated algebra. We define $U_s=\Spm(A_s)$ to be the associated affine algebraic variety. For $s\in B_m$ and $s'\in B_{m'}$, we clearly have algebra homomorphisms $$A_s\to A_{ss'},\qquad A_{s'}\to A_{ss'},$$ since $A_{ss'}$ is the algebra of all $0$-homogeneous fractions with powers of $s$ and $s'$ in the denominator. As $A_{ss'}$ is the same as the localized ring $A_s[s^{m'}/s^{\prime m}]$, we see that $U_{ss'}$ can be identified with an affine open set in $U_s$, and we thus get canonical injections $$U_{ss'}\hookrightarrow U_s,\qquad U_{ss'}\hookrightarrow U_{s'}.$$ \begstat{(1.32) Definition} If $B=\bigoplus_{m\in\bN}B_m$ is a reduced graded algebra generated by its finite dimensional vector subspaces $B_0$ and $B_1$, we associate an algebraic scheme $(X,\cO_X)=\Proj(B)$ as follows. To each finitely generated algebra $A_s=\big\{p/s^d\,;\,p\in B_{dm},\,d\in\bN\big\}$ we associate an affine algebraic variety $U_s=\Spm(A_s)$. We let $X$ be the union of all open charts $U_s$ with the identifications $U_s\cap U_{s'}=U_{ss'}\,$; then the collection $(U_s)$ is a basis of the topology of $X$, and $\cO_X$ is the unique sheaf of local $k$-algebras such that $\Gamma(U_s,\cO_X)=A_s$ for each~$U_s$. \endstat The following proposition shows that only finitely many open charts are actually needed to describe~$X$ (as required in Def.~1.29~a)). \begstat{(1.33) Lemma} If $s_0,\ldots,s_N$ is a basis of $B_1$, then $\Proj(B)=\bigcup\limits_{0\le j\le N}U_{s_j}$. \endstat \begproof{} In fact, if $x\in X$ is contained in a chart $U_s$ for some $s\in B_m$, then $U_s=\Spm(A_s)\ne\emptyset$, and therefore $A_s\ne\{0\}$. As $A_s$ is generated by ${1\over s}B_1^m$, we can find a fraction $f=s_{j_1}\ldots s_{j_m}/s$ representing an element $f\in\cO(U_s)$ such that $f(x)\ne 0$. Then $x\in U_s\ssm f^{-1}(0)$, and $U_s\ssm f^{-1}(0)=\Spm(A_s[1/f])=U_s\cap U_{s_{j_1}}\cap\ldots\cap U_{s_{j_m}}$. In particular $x\in U_{s_{j_1}}$.\qed \endproof \begstat{(1.34) Example} \rm One can consider the {\it projective space} $\bP^N_k$ to be the algebraic scheme $$\bP^N_k=\Proj(k[T_0,\ldots,T_N]).$$ \endstat The Proj construction is fonctorial in the following sense: if we have a graded homomorphism $\Phi:B\to B'$ (i.e.\ an algebra homomorphism such that $\Phi(B_m)\subset B'_m$, then there are corresponding morphisms $A_s\to A'_{\Phi(s)}$, $U'_{\Phi(s)}\to U_s$, and we thus find a scheme morphism $$F:\Proj(B')\to\Proj(B).$$ Also, since $p/s^d=ps^l/s^{d+l}$, the algebras $A_s$ depend only on components $B_m$ of large degree, and we have $A_s=A_{s^l}$. It follows easily that there is a canonical isomorphism $$\Proj(B)\simeq\Proj\Big(\bigoplus_m B_{lm}\Big).$$ Similarly, we may if we wish change a finite number of components $B_m$ without affecting $\Proj(B)$. In particular, we may alway assume that $B_0=k\,1_B$. By selecting finitely many generators $g_0,\ldots,g_N$ in $B_1$, we then find a surjective graded homomorphism $k[T_0,\ldots,T_N]\to B$, thus $B\simeq k[T_0,\ldots,T_N]/J$ for some graded ideal $J\subset B$. The algebra homomorphism $k[T_0,\ldots,T_N]\to B$ therefore yields a scheme embedding $\Proj(B)\to\bP^N$ onto $V(J)$. We will not pursue further the study of algebraic varieties from this point of view$\,$; in fact we are mostly interested in the case $k=\bC$, and algebraic varieties over $\bC$ are a special case of the more general concept of complex analytic space. \titleb{\S 2.}{The Local Ring of Germs of Analytic Functions} \titlec{\S 2.A.}{The Weierstrass Preparation Theorem} Our first goal is to establish a basic factorization and division theorem for analytic functions of several variables, which is essentially due to Weierstrass. We follow here a simple proof given by C.L. Siegel, based on a clever use of the Cauchy formula. Let $g$ be a holomorphic function defined on a neighborhood of $0$ in $\bC^n$, $g\not\equiv 0$. There exists a dense set of vectors $v\in\bC^n\ssm\{0\}$ such that the function $\bC\ni t\longmapsto g(tv)$ is not identically zero. In fact the Taylor series of $g$ at the origin can be written $$g(tv)=\sum_{k=0}^{+\infty}{1\over k!}\,t^k\,g^{(k)}(v)$$ where $g^{(k)}$ is a homogeneous polynomial of degree $k$ on $\bC^n$ and $g^{(k_0)}\not\equiv 0$ for some index $k_0$. Thus it suffices to select $v$ such that $g^{(k_0)}(v)\ne 0$. After a change of coordinates, we may assume that $v=(0,\ldots,0,1)$. Let $s$ be the vanishing order of $z_n\longmapsto g(0,\ldots,0,z_n)$ at $z_n=0$. There exists $r_n>0$ such that $g(0,\ldots,0,z_n)\ne 0$ when $0<|z_n|\le r_n$. By continuity of $g$ and compactness of the circle $|z_n|=r_n$, there exists $r'>0$ and $\varepsilon>0$ such that $$g(z',z_n)\ne 0~~~~\hbox{\rm for}~z'\in\bC^{n-1}, ~~~|z'|\le r',~~~r_n-\varepsilon\le|z_n|\le r_n+\varepsilon.$$ For every integer $k\in\bN$, let us consider the integral $$S_k(z')={1\over 2\pi\ii}\int_{|z_n|=r_n}{1\over g(z',z_n)}{\partial g\over \partial z_n}(z',z_n)\,z_n^k\,dz_n.$$ Then $S_k$ is holomorphic in a neighborhood of $|z'|\le r'$. Rouch\'e's theorem shows that $S_0(z')$ is the number of roots $z_n$ of $g(z',z_n)=0$ in the disk $|z_n|0$ for all~$j$. Then $$P(0,z_n)=z_n^s=P_1(0,z_n)\ldots P_N(0,z_n),$$ hence $P_j(0,z_n)=z_n^{s_j}$ and therefore $P_j$ is a Weierstrass polynomial. \smallskip \noindent b) Set $s=\deg\,P$ and $P(0,z_n)=z_n^s$. Assume that $P$ is reducible in $\cO_n$, with $P(z',z_n)=g_1(z)g_2(z)$ for non invertible elements $g_1,g_2\in\cO_n$. Then $g_1(0,z_n)$ and $g_2(0,z_n)$ have vanishing orders $s_1,s_2>0$ with $s_1+s_2=s$, and $$g_j=u_jP_j,~~~~\deg\,P_j=s_j,~~~j=1,2,$$ where $P_j$ is a Weierstrass polynomial and $u_j\in\cO_n$ is invertible. Therefore $P_1P_2=uP$ for an invertible germ $u\in\cO_n$. Lemma~2.8 shows that $P$ divides $P_1P_2$ in $\cO_{n-1}[z_n]\,$; since $P_1$, $P_2$ are unitary and $s=s_1+s_2$, we get $P=P_1P_2$, hence $P$ is reducible in $\cO_{n-1}[z_n]$. The converse implication is obvious from~a). \qed \endproof \begstat{(2.10) Theorem} $\cO_n$ is a factorial ring, i.e.\ $\cO_n$ is entire and: \smallskip \item{\rm a)} every non zero germ $f\in\cO_n$ admits a factorization $f=f_1\ldots f_N$ in irreducible elements$\,;$ \smallskip \item{\rm b)} the factorization is unique up to invertible elements. \endstat \begproof{} The existence part a) follows from Lemma~2.9 if we take $f$ to be a Weierstrass polynomial and $f=f_1\ldots f_N$ be a decomposition of maximal length $N$ into polynomials of positive degree. In order to prove the uniqueness, it is sufficient to verify the following statement: \smallskip \item{\rlap{\hbox{${\rm b}')$}}} {\it If $g$ is an irreducible element that divides a product $f_1f_2$, then $g$ divides either $f_1$ or $f_2$.} \smallskip \noindent By Th.~2.1, we may assume that $f_1$, $f_2$, $g$ are Weierstrass polynomials in $z_n$. Then $g$ is irreducible and divides $f_1f_2$ in $\cO_{n-1}[z_n]$ thanks to Lemmas 2.8 and 2.9~b). By induction on $n$, we may assume that $\cO_{n-1}$ is factorial. The standard Gauss lemma (Lang 1965, Chapter V) says that the polynomial ring $A[T]$ is factorial if the ring $A$ is factorial. Hence $\cO_{n-1}[z_n]$ is factorial by induction and thus $g$ must divide $f_1$ or $f_2$ in $\cO_{n-1}[z_n]$.\qed \endproof \begstat{(2.11) Theorem} If $f,g\in\cO_n$ are relatively prime, then the germs $f_z,~g_z$ at every point $z\in\bC^n$ near $0$ are again relatively prime. \endstat \begproof{} One may assume that $f=P,~g=Q$ are Weierstrass polynomials. Let us recall that unitary polynomials $P,Q\in {\cal A}[X]$ (${\cal A}={\rm a}$ factorial ring) are relatively prime if and only if their resultant $R\in{\cal A}$ is non zero. Then the resultant $R(z')\in\cO_{n-1}$ of $P(z',z_n)$ and $Q(z',z_n)$ has a non zero germ at $0$. Therefore the germ $R_{z'}$ at points $z'\in\bC^{n-1}$ near $0$ is also non zero.\qed \endproof \titleb{\S 3.}{Coherent Sheaves} \titlec{\S 3.1.}{Locally Free Sheaves and Vector Bundles} Section~9 will greatly develope this philosophy. Before introducing the more general notion of a coherent sheaf, we discuss the notion of locally free sheaves over a sheaf a ring. All rings occurring in the sequel are supposed to be commutative with unit (the non commutative case is also of considerable interest, e.g.\ in view of the theory of $\cD$-modules, but this subject is beyond the scope of the present book). \begstat{(3.1) Definition} Let $\cA$ be a sheaf of rings on a topological space $X$ and let $\cS$ a sheaf of modules over $\cA$ (or briefly a $\cA$-module). Then $\cS$ is said to be locally free of rank $r$ over $\cA$, if $\cS$ is locally isomorphic to $\cA^{\oplus r}$ on a neighborhood of every point, i.e.\ for every $x_0\in X$ one can find a neighborhood $\Omega$ and sections $F_1,\ldots,F_r\in\cS(\Omega)$ such that the sheaf homomorphism $$F:\cA_{\restriction\Omega}^{\oplus r}\longrightarrow\cS_{\restriction\Omega},~~~~ \cA_x^{\oplus r}\ni(w_1,\ldots,w_r)\longmapsto \sum_{1\le j\le r}w_jF_{j,x}\in\cS_x$$ is an isomorphism. \endstat By definition, if $\cS$ is locally free, there is a covering $(U_\alpha)_{\alpha\in I}$ by open sets on which $\cS$ admits free generators $F_\alpha^1,\ldots,F_\alpha^r\in\cS(U_\alpha)$. Because the generators can be uniquely expressed in terms of any other system of independent generators, there is for each pair $(\alpha,\beta)$ a $r\times r$ matrix $$G_{\alpha\beta}=(G_{\alpha\beta}^{jk})_{1\le j,k\le r},\qquad G_{\alpha\beta}^{jk}\in\cA(U_\alpha\cap U_\beta),$$ such that $$F_\beta^k=\sum_{1\le j\le r}F_\alpha^jG^{jk}_{\alpha\beta}\qquad \hbox{\rm on}\quad U_\alpha\cap U_\beta.$$ In other words, we have a commutative diagram $$\eqalign{ &\cA^{\oplus r}_{\restriction U_\alpha\cap U_\beta} \buildo{\displaystyle F_\alpha}\over\longrightarrow \cS_{\restriction U_\alpha\cap U_\beta}\cr &G_{\alpha\beta}\Big\uparrow\quad\phantom{\longrightarrow}\qquad \Big|~\Big|\cr &\cA^{\oplus r}_{\restriction U_\alpha\cap U_\beta} \buildu{\displaystyle F_\beta}\under\longrightarrow \cS_{\restriction U_\alpha\cap U_\beta}\cr}$$ It follows easily from the equality $G_{\alpha\beta}=F_\alpha^{-1}\circ F_\beta$ that the {\it transition matrices} $G_{\alpha\beta}$ are invertible matrices satisfying the transition relation $$G_{\alpha\gamma}=G_{\alpha\beta}G_{\beta\gamma}\qquad\hbox{\rm on}\quad U_\alpha\cap U_\beta\cap U_\gamma\leqno(3.2)$$ for all indices $\alpha,\beta,\gamma\in I$. In particular $G_{\alpha\alpha}= \Id$ on $U_\alpha$ and $G_{\alpha\beta}^{-1}=G_{\beta\alpha}$ on $U_\alpha\cap U_\beta$. Conversely, if we are given a system of invertible $r\times r$ matrices $G_{\alpha\beta}$ with coefficients in $\cA(U_\alpha\cap U_\beta)$ satisfying the transition relation~(3.2), we can define a locally free sheaf $\cS$ of rank $r$ over $\cA$ by taking $\cS\simeq\cA^{\oplus r}$ over each $U_\alpha$, the identification over $U_\alpha\cap U_\beta$ being given by the isomorphism $G_{\alpha\beta}$. A~section $H$ of $\cS$ over an open set $\Omega\subset X$ can just be seen as a collection of sections $H_\alpha=(H_\alpha^1,\ldots,H_\alpha^r)$ of $\cA^{\oplus r}(\Omega\cap U_\alpha)$ satisfying the transition relations $H_\alpha=G_{\alpha\beta} H_\beta$ over $\Omega\cap U_\alpha\cap U_\beta$. The notion of locally free sheaf is closely related to another essential notion of differential geometry, namely the notion of vector bundle (resp.\ topological, differentiable, holomorphic $\ldots$, vector bundle). To describe the relation between these notions, we assume that the sheaf of rings $\cA$ is a subsheaf of the sheaf $\cC_\bK$ of continous functions on $X$ with values in the field $\bK=\bR$ or $\bK=\bC$, containing the sheaf of locally constant functions $X\to\bK$. Then, for each $x\in X$, there is an evaluation map $$\cA_x\to\bK,\qquad w\mapsto w(x)$$ whose kernel is a maximal ideal $\gm_x$ of $\cA_x$, and $\cA_x/\gm_x=\bK$. Let $\cS$ be a locally free sheaf of rank $r$ over~$\cA$. To each $x\in X$, we can associate a $\bK$-vector space $E_x=\cS_x/\gm_x\cS_x$: since $\cS_x\simeq\cA_x^{\oplus r}$, we have $E_x\simeq(\cA_x/\gm_x)^{\oplus r}=\bK^r$. The set $E=\coprod_{x\in X} E_x$ is equipped with a natural projection $$\pi:E\to X,\qquad \xi\in E_x\mapsto \pi(\xi):=x,$$ and the fibers $E_x=\pi^{-1}(x)$ have a structure of $r$-dimensional $\bK$-vector space: such a structure $E$ is called a {\it $\bK$-vector bundle of rank $r$} over~$X$. Every section $s\in \cS(U)$ gives rise to a {\it section} of $E$ over $U$ by setting $s(x)=s_x$~mod~$\gm_x$. We obtain a function (still denoted by the same symbol) $s:U\to E$ such that $s(x)\in E_x$ for every $x\in U$, i.e.\ $\pi\circ s=\Id_U$. It is clear that $\cS(U)$ can be considered as a $\cA(U)$-submodule of the $\bK$-vector space of functions $U\to E$ mapping a point $x\in U$ to an element in the fiber~$E_x$. Thus we get a subsheaf of the sheaf of $E$-valued sections, which is in a natural way a $\cA$-module isomorphic to~$\cS$. This subsheaf will be denoted by $\cA(E)$ and will be called the {\it sheaf of $\cA$-sections} of~$E$. If we are given a $\bK$-vector bundle $E$ over $X$ and a subsheaf $\cS=\cA(E)$ of the sheaf of all sections of $E$ which is in a natural way a locally free $\cA$-module of rank~$r$, we say that $E$ (or more precisely the pair $(E,\cA(E))$) is a $\cA$-vector bundle of rank~$r$ over~$X$. \begstat{(3.3) Example} \rm In case $\cA=\cC_{X,\bK}$ is the sheaf of all $\bK$-valued continuous functions on~$X$, we say that $E$ is a {\it topological} vector bundle over~$X$. When $X$ is a manifold and $\cA=\cC^p_{X,\bK}$, we say that $E$ is a {\it $C^p$-differentiable} vector bundle; finally, when $X$ is complex analytic and $\cA=\cO_X$, we say that $E$ is a {\it holomorphic} vector bundle. \endstat Let us introduce still a little more notation. Since $\cA(E)$ is a locally free sheaf of rank $r$ over any open set $U_\alpha$ in a suitable covering of $X$, a choice of generators $(F_\alpha^1,\ldots,F_\alpha^r)$ for $\cA(E)_{\restriction U_\alpha}$ yields corresponding generators $(e_\alpha^1(x),\ldots,e_\alpha^r(x))$ of the fibers~$E_x$ over~$\bK$. Such a system of gene\-rators is called a {\it $\cA$-admissible frame} of $E$ over~$U_\alpha$. There is a corresponding isomorphism $$\theta_\alpha:E_{\restriction U_\alpha}:=\pi^{-1}(U_\alpha)\longrightarrow U_\alpha\times\bK^r\leqno(3.4)$$ which to each $\xi\in E_x$ associates the pair $(x,(\xi_\alpha^1,\ldots,\xi_\alpha^r))\in U_\alpha\times\bK^r$ composed of $x$ and of the components $(\xi_\alpha^j)_{1\le j\le r}$ of $\xi$ in the basis $(e_\alpha^1(x),\ldots,e_\alpha^r(x))$ of~$E_x$. The bundle $E$ is said to be {\it trivial} if it is of the form $X\times\bK^r$, which is the same as saying that $\cA(E)=\cA^{\oplus r}$. For this reason, the isomorphisms $\theta_\alpha$ are called {\it trivializations} of $E$ over~$U_\alpha$. The corresponding {\it transition automorphisms} are $$\eqalign{ &\theta_{\alpha\beta}:=\theta_\alpha\circ\theta_\beta^{-1}: (U_\alpha\cap U_\beta)\times \bK^r\longrightarrow (U_\alpha\cap U_\beta)\times \bK^r,\cr &\theta_{\alpha\beta}(x,\xi)=(x,g_{\alpha\beta}(x)\cdot \xi), \qquad (x,\xi)\in (U_\alpha\cap U_\beta)\times \bK^r,\cr}$$ where $(g_{\alpha\beta})\in\GL_r(\cA)(U_\alpha\cap U_\beta)$ are the transition matrices already described (except that they are just seen as matrices with coefficients in $\bK$ rather than with coefficients in a sheaf). Conversely, if we are given a collection of matrices $g_{\alpha\beta}=(g_{\alpha\beta}^{jk})\in\GL_r(\cA)(U_\alpha\cap U_\beta)$ satisfying the transition relation $$g_{\alpha\gamma}=g_{\alpha\beta}g_{\beta\gamma}\qquad\hbox{\rm on}\quad U_\alpha\cap U_\beta\cap U_\gamma,$$ we can define a $\cA$-vector bundle $$E=\big(\coprod_{\alpha\in I}U_\alpha\times\bK^r\big)/\sim$$ by gluing the charts $U_\alpha\times\bK^r$ via the identification $(x_\alpha,\xi_\alpha)\sim(x_\beta,\xi_\beta)$ if and only if $x_\alpha=x_\beta=x\in U_\alpha\cap U_\beta$ and $\xi_\alpha=g_{\alpha\beta}(x)\cdot\xi_\beta$. \begstat{(3.5) Example} \rm When $X$ is a real differentiable manifold, an interesting example of real vector bundle is the {\it tangent bundle} $T_X$~; if $\tau_\alpha:U_\alpha\to\bR^n$ is a collection of coordinate charts on $X$, then \hbox{$\theta_\alpha=\pi\times d\tau_\alpha:T_{X\restriction U_\alpha}\to U_\alpha\times\bR^m$} define trivializations of $T_X$ and the transition matrices are given by $g_{\alpha\beta}(x)= d\tau_{\alpha\beta}(x^\beta)$ where $\tau_{\alpha\beta}= \tau_\alpha\circ\smash{\tau_\beta^{-1}}$ and $x^\beta=\tau_\beta(x)$. The dual $T^\star_X$ of $T_X$ is called the {\it cotangent bundle} of~$X$. If $X$ is complex analytic, then $T_X$ has the structure of a holomorphic vector bundle. \endstat We now briefly discuss the concept of sheaf and bundle morphisms. If $\cS$ and $\cS'$ are sheaves of $\cA$-modules over a topological space~$X$, then by a morphism $\varphi:\cS\to\cS'$ we just mean a $\cA$-linear sheaf morphism. If $\cS=\cA(E)$ and $\cS'=\cA(E')$ are locally free sheaves, this is the same as a $\cA$-linear bundle morphism, that is, a fiber preserving $\bK$-linear morphism $\varphi(x):E_x\to E'_x$ such that the matrix representing $\varphi$ in any local $\cA$-admissible frames of $E$ and $E'$ has coefficients in~$\cA$. \begstat{(3.6) Proposition} Suppose that $\cA$ is a sheaf of local rings, i.e.\ that a section of $\cA$ is invertible in $\cA$ if and only if it never takes the zero value in~$\bK$. Let $\varphi:\cS\to \cS'$ be a $\cA$-morphism of locally free $\cA$-modules of rank $r$,~$r'$. If the rank of the $r'\times r$ matrix $\varphi(x)\in M_{r'r}(\bK)$ is constant for all $x\in X$, then $\Ker\varphi$ and $\Im\varphi$ are locally free subsheaves of $\cS$, $\cS'$ respectively, and $\Coker\varphi=\cS'/\Im\varphi$ is locally free. \endstat \begproof{} This is just a consequence of elementary linear algebra, once we know that non zero determinants with coefficients in $\cA$ can be inverted. \qed \endproof Note that all three sheaves $\cC_{X,\bK}$, $\cC^p_{X,\bK}$, $\cO_X$ are sheaves of local rings, so Prop.~3.6 applies to these cases. However, even if we work in the holomorphic category ($\cA=\cO_X$), a difficulty immediately appears that the kernel or cokernel of an arbitrary morphism of locally free sheaves is in general not locally free. \begstat{(3.7) Examples} {\rm\smallskip \item{a)} Take $X=\bC$, let $\cS=\cS'=\cO$ be the trivial sheaf, and let $\varphi:\cO\to\cO$ be the morphism $u(z)\mapsto z\,u(z)$. It is immediately seen that $\varphi$ is injective as a sheaf morphism ($\cO$ being an entire ring), and that $\Coker\varphi$ is the {\it skyscraper sheaf} $\bC_0$ of stalk $\bC$ at $z=0$, having zero stalks at all other points $z\ne 0$. Thus $\Coker\varphi$ is not a locally free sheaf, although $\varphi$ is everywhere injective (note however that the corresponding morphism \hbox{ $\varphi:E\to E'$}, $(z,\xi)\mapsto (z,z\xi)$ of trivial rank~$1$ vector bundles $E=E'=\bC\times\bC$ is {\it not injective} on the zero fiber $E_0$). \smallskip \item{b)} Take $X=\bC^3$, $\cS=\cO^{\oplus 3}$, $\cS'=\cO$ and $$\varphi:\cO^{\oplus 3}\to\cO,\qquad (u_1,u_2,u_3)\mapsto \sum_{1\le j\le 3}z_ju_j(z_1,z_2,z_3).$$ Since $\varphi$ yields a surjective bundle morphism on $\bC^3\ssm\{0\}$, one easily sees that $\Ker\varphi$ is locally free of rank~$2$ over $\bC^3\ssm\{0\}$. However, by looking at the Taylor expansion of the $u_j$'s at $0$, it is not difficult to check that $\Ker\varphi$ is the $\cO$-submodule of $\cO^{\oplus 3}$ generated by the three sections $(-z_2,z_1,0)$, $(-z_3,0,z_1)$ and $(0,z_3,-z_2)$, and that any two of these three sections cannot generate the $0$-stalk $(\Ker\varphi)_0$. Hence $\Ker\varphi$ is not locally free.\smallskip} \endstat Since the category of locally free $\cO$-modules is not stable by taking kernels or cokernels, one is led to introduce a more general category which will be stable under these operations. This leads to the notion of {\it coherent sheaves}. \titlec{\S 3.2.}{Notion of Coherence} The notion of coherence again deals with sheaves of modules over a sheaf of rings. It is a semi-local property which says roughly that the sheaf of modules locally has a finite presentation in terms of generators and relations. We describe here some general properties of this notion, before concentrating ourselves on the case of coherent $\cO_X$-modules. \begstat{(3.8) Definition} Let $\cA$ be a sheaf of rings on a topological space $X$ and $\cS$ a sheaf of modules over $\cA$ (or briefly a $\cA$-module). Then $\cS$ is said to be locally finitely generated if for every point $x_0\in X$ one can find a neighborhood $\Omega$ and sections $F_1,\ldots,F_q\in\cS(\Omega)$ such that for every $x\in\Omega$ the stalk $\cS_x$ is generated by the germs $F_{1,x},\ldots, F_{q,x}$ as an $\cA_x$-module. \endstat \begstat{(3.9) Lemma} Let $\cS$ be a locally finitely generated sheaf of $\cA$-modules on $X$ and $G_1,\ldots,G_N$ sections in $\cS(U)$ such that $G_{1,x_0},\ldots,G_{N,x_0}$ generate $\cS_{x_0}$ at $x_0\in U$. Then $G_{1,x},\ldots,G_{N,x}$ generate $\cS_x$ for $x$ near $x_0$. \endstat \begproof{} Take $F_1,\ldots,F_q$ as in Def.~3.8. As $G_1,\ldots,G_N$ generate $\cS_{x_0}$, one can find a neighborhood $\Omega'\subset\Omega$ of $x_0$ and $H_{jk}\in\cA(\Omega')$ such that $F_j=\sum H_{jk}G_k$ on $\Omega'$. Thus $G_{1,x},\ldots,G_{N,x}$ generate $\cS_x$ for all $x\in\Omega'$.\qed \endproof \titled{\S 3.2.1.}{Definition of Coherent Sheaves.} If $U$ is an open subset of $X$, we denote by $\cS_{\restriction U}$ the restriction of $\cS$ to $U$, i.e.\ the union of all stalks $\cS_x$ for $x\in U$. If $F_1,\ldots,F_q\in\cS(U)$, the kernel of the sheaf homomorphism $F:\cA_{\restriction U}^{\oplus q}\longrightarrow\cS_{\restriction U}$ defined by $$\cA_x^{\oplus q}\ni(g^1,\ldots,g^q)\longmapsto\sum_{1\le j\le q}g^jF_{j,x} \in\cS_x,~~~~x\in U\leqno(3.10)$$ is a subsheaf $\cR(F_1,\ldots,F_q)$ of $\cA_{\restriction U}^{\oplus q}$, called the {\it sheaf of relations} between $F_1,\ldots,F_q$. \begstat{(3.11) Definition} A sheaf $\cS$ of $\cA$-modules on $X$ is said to be coherent if: \smallskip \item{\rm a)} $\cS$ is locally finitely generated~$;$ \smallskip \item{\rm b)} for any open subset $U$ of $X$ and any $F_1,\ldots,F_q\in\cS(U)$, the sheaf of relations $\cR(F_1,\ldots,F_q)$ is locally finitely generated.\smallskip \endstat Assumption a) means that every point $x\in X$ has a neighborhood $\Omega$ such that there is a surjective sheaf morphism $F:\cA_{\restriction\Omega}^{\oplus q}\longrightarrow \cS_{\restriction \Omega}$, and assumption b) implies that the kernel of $F$ is locally finitely generated. Thus, after shrinking $\Omega$, we see that $\cS$ admits over $\Omega$ a finite presentation under the form of an exact sequence $$\cA_{\restriction\Omega}^{\oplus p}\buildo G\over\longrightarrow\cA_{\restriction\Omega}^{\oplus q} \buildo F\over\longrightarrow\cS_{\restriction\Omega}\longrightarrow 0,\leqno(3.12)$$ where $G$ is given by a $q\times p$ matrix $(G_{jk})$ of sections of $\cA(\Omega)$ whose columns $(G_{j1}),\ldots,(G_{jp})$ are generators of $\cR(F_1,\ldots,F_q)$. It is clear that every locally finitely generated subsheaf of a coherent sheaf is coherent. From this we easily infer: \begstat{(3.13) Theorem} Let $\varphi:\cF\longrightarrow\cG$ be a $\cA$-morphism of coherent sheaves. Then $\Im\varphi$ and $\ker\varphi$ are coherent. \endstat \begproof{} Clearly $\Im\varphi$ is a locally finitely generated subsheaf of $\cG$, so it is coherent. Let $x_0\in X$, let $F_1,\ldots,F_q\in\cF(\Omega)$ be generators of $\cF$ on a neighborhood $\Omega$ of $x_0$, and $G_1,\ldots,G_r\in \cA(\Omega')^{\oplus q}$ be generators of $\cR\big(\varphi(F_1),\ldots,\varphi(F_q)\big)$ on a neighborhood $\Omega'\subset\Omega$ of $x_0$. Then $\ker\varphi$ is generated over $\Omega'$ by the sections $$H_j=\sum_{k=1}^q G_j^kF_k\in\cF(\Omega'),~~~~1\le j\le r.\eqno\square$$ \endproof \begstat{(3.14) Theorem} Let $0\longrightarrow\cF\longrightarrow\cS\longrightarrow\cG\longrightarrow 0$ be an exact sequence of $\cA$-modules. If two of the sheaves $\cF,\cS,\cG$ are coherent, then all three are coherent. \endstat \begproof{} If $\cS$ and $\cG$ are coherent, then $\cF=\ker(\cS\to\cG)$ is coherent by Th.~3.13. If $\cS$ and $\cF$ are coherent, then $\cG$ is locally finitely generated; to prove the coherence, let $G_1,\ldots,G_q\in\cG(U)$ and $x_0\in U$. Then there is a neighborhood $\Omega$ of $x_0$ and sections $\tilde G_1,\ldots,\tilde G_q\in\cS(\Omega)$ which are mapped to $G_1,\ldots,G_q$ on $\Omega$. After shrinking $\Omega$, we may assume also that $\cF_{\restriction\Omega}$ is generated by sections $F_1,\ldots,F_p\in\cF(\Omega)$. Then $\cR(G_1,\ldots,G_q)$ is the projection on the last $q$-components of $\cR(F_1,\ldots,F_p,\tilde G_1,\ldots,\tilde G_q)\subset\cA^{p+q}$, which is finitely generated near $x_0$ by the coherence of $\cS$. Hence $\cR(G_1,\ldots,G_q)$ is finitely generated near $x_0$ and $\cG$ is coherent. Finally, assume that $\cF$ and $\cG$ are coherent. Let $x_0\in X$ be any point, let $F_1,\ldots,F_p\in\cF(\Omega)$ and $G_1,\ldots,G_q\in\cG(\Omega)$ be generators of $\cF$, $\cG$ on a neighborhood $\Omega$ of $x_0$. There is a neighborhood $\Omega'$ of $x_0$ such that $G_1,\ldots,G_q$ admit liftings $\tilde G_1,\ldots,\tilde G_q\in\cS(\Omega')$. Then $(F_1,\ldots,F_q,\tilde G_1,\ldots,\tilde G_q)$ generate $\cS_{\restriction\Omega'}$, so $\cS$ is locally finitely generated. Now, let $S_1,\ldots,S_q$ be arbitrary sections in $\cS(U)$ and $\ol S_1,\ldots,\ol S_q$ their images in $\cG(U)$. For any $x_0\in U$, the sheaf of relations $\cR(\ol S_1,\ldots,\ol S_q)$ is generated by sections $P_1,\ldots,P_s\in\cA(\Omega)^{\oplus q}$ on a small neighborhood $\Omega$ of $x_0$. Set $P_j=(P_j^k)_{1\le k\le q}$. Then $H_j=P_j^1S_1+\ldots+P_j^qS_q$, $1\le j\le s$, are mapped to $0$ in $\cG$ so they can be seen as sections of $\cF$. The coherence of $\cF$ shows that $\cR(H_1,\ldots,H_s)$ has generators $Q_1,\ldots,Q_t\in\cA(\Omega')^s$ on a small neighborhood $\Omega'\subset\Omega$ of $x_0$. Then $\cR(S_1,\ldots,S_q)$ is generated over $\Omega'$ by $R_j=\sum Q_j^kP_k\in\cA(\Omega')$, $1\le j\le t$, and $\cS$ is coherent.\qed \endproof \begstat{(3.15) Corollary} If $\cF$ and $\cG$ are coherent subsheaves of a coherent analytic sheaf $\cS$, the intersection $\cF\cap\cG$ is a coherent sheaf. \endstat \begproof{} Indeed, the intersection sheaf $\cF\cap\cG$ is the kernel of the composite morphism $\cF\lhra\cS\longrightarrow\cS/\cG,$ and $\cS/\cG$ is coherent.\qed \endproof \titled{\S 3.2.2. Coherent Sheaf of Rings.} A sheaf of rings $\cA$ is said to be coherent if it is coherent as a module over itself. By Def.~3.11, this means that for any open set $U\subset X$ and any sections $F_j\in\cA(U)$, the sheaf of relations $\cR(F_1,\ldots,F_q)$ is finitely generated. The above results then imply that all free modules $\cA^{\oplus p}$ are coherent. As a consequence: \begstat{(3.16) Theorem} If $\cA$ is a coherent sheaf of rings, any locally finitely gene\-rated subsheaf of $\cA^{\oplus p}$ is coherent. In particular, if $\cS$ is a coherent \hbox{\it$\cA$-module} and $F_1,\ldots,F_q\in\cS(U)$, the sheaf of relations $\cR(F_1,\ldots,F_q)\subset\cA^{\oplus q}$ is also coherent. \endstat Let $\cS$ be a coherent sheaf of modules over a coherent sheaf of ring $\cA$. By an iteration of construction (3.12), we see that for every integer $m\ge 0$ and every point $x\in X$ there is a neighborhood $\Omega$ of $x$ on which there is an exact sequence of sheaves $$\cA_{\restriction\Omega}^{\oplus p_m}\buildo F_m\over\longrightarrow \cA_{\restriction\Omega}^{\oplus p_{m-1}}\longrightarrow\cdots\longrightarrow \cA_{\restriction\Omega}^{\oplus p_1}\buildo F_1\over\longrightarrow \cA_{\restriction\Omega}^{\oplus p_0}\buildo F_0\over\longrightarrow \cS_{\restriction\Omega}\longrightarrow 0,\leqno(3.17)$$ where $F_j$ is given by a $p_{j-1}\times p_j$ matrix of sections in $\cA(\Omega)$. \titlec{\S 3.3.}{Analytic Sheaves and the Oka Theorem} Many properties of holomorphic functions which will be considered in this book can be expressed in terms of sheaves. Among them, analytic sheaves play a central role. The Oka theorem (Oka 1950) asserting the coherence of the sheaf of holomorphic functions can be seen as a far-reaching deepening of the noetherian property seen in Sect.~1. The theory of analytic sheaves could not be presented without it. \begstat{(3.18) Definition} Let $M$ be a $n$-dimensional complex analytic manifold and let $\cO_M$ be the sheaf of germs of analytic functions on~$M$. An analytic sheaf over $M$ is by definition a sheaf $\cS$ of modules over~$\cO_M$. \endstat \begstat{(3.19) Coherence theorem of Oka} The sheaf of rings $\cO_M$ is coherent for any complex manifold~$M$. \endstat Let $F_1,\ldots,F_q\in\cO(U)$. Since $\cO_{M,x}$ is Noetherian, we already know that every stalk $\cR(F_1,\ldots,F_q)_x\subset\cO_{M,x}^{\oplus q}$ is finitely generated, but the important new fact expressed by the theorem is that the sheaf of relations is locally finitely generated, namely that the ``same'' generators can be chosen to generate each stalk in a neighborhood of a given point. \begproof{} By induction on $n=\dim_\bC M$. For $n=0$, the stalks $\cO_{M,x}$ are equal to $\bC$ and the result is trivial. Assume now that $n\ge 1$ and that the result has already been proved in dimension $n-1$. Let $U$ be an open set of $M$ and $F_1,\ldots,F_q\in\cO_M(U)$. To show that $\cR(F_1,\ldots,F_q)$ is locally finitely generated, we may assume that $U=\Delta=\Delta'\times\Delta_n$ is a polydisk in $\bC^n$ centered at $x_0=0$~; after a change of coordinates and multiplication of $F_1,\ldots,F_q$ by invertible functions, we may also suppose that $F_1,\ldots,F_q$ are Weierstrass polynomials in $z_n$ with coefficients in $\cO(\Delta')$. We need a lemma. \endproof \begstat{(3.20) Lemma} If $x=(x',x_n)\in\Delta$, the $\cO_{\Delta,x}$-module $\cR(F_1,\ldots,F_q)_x$ is generated by those of its elements whose components are germs of analytic polynomials in $\cO_{\Delta',x'}[z_n]$ with a degree in $z_n$ at most equal to $\mu$, the maximum of the degrees of $F_1,\ldots,F_q$. \endstat \begproof{} Assume for example that $F_q$ is of the maximum degree $\mu$. By the Weierstrass preparation Th.~1.1 and Lemma~1.9 applied at $x$, we can write $F_{q,x}=f'f''$ where $f',f''\in\cO_{\Delta',x'} [z_n]$, $f'$ is a Weierstrass polynomial in $z_n-x_n$ and $f''(x)\ne 0$. Let $\mu'$ and $\mu''$ denote the degrees of $f'$ and $f''$ with respect to $z_n$, so $\mu'+\mu''=\mu$. Now, take $(g^1,\ldots,g^q)\in \cR(F_1,\ldots,F_q)_x$. The Weierstrass division theorem gives $$g^j=F_{q,x}t^j+r^j,~~~~j=1,\ldots,q-1,$$ where $t^j\in\cO_{\Delta,x}$ and $r^j\in\cO_{\Delta',x'}[z_n]$ is a polynomial of degree $<\mu'$. For $j=q$, define $r^q=g^q+\sum_{1\le j\le q-1} t^jF_{j,x}$. We can write $$(g^1,\ldots,g^q)=\sum_{1\le j\le q}t^j(0,\ldots,F_q,\ldots,0,-F_j)_x+(r^1,\ldots,r^q) \leqno(3.21)$$ where $F_q$ is in the $j$-th position in the $q$-tuples of the summation. Since these\break $q$-tuples are in $\cR(F_1,\ldots,F_q)_x$, we have $(r^1,\ldots,r^q)\in \cR(F_1,\ldots,F_q)_x$, thus $$\sum_{1\le j\le q-1}F_{j,x}r^j+f'f''r^q=0.$$ As the sum is a polynomial in $z_n$ of degree $<\mu+\mu'$, it follows from Lemma~1.9 that $f''r^q$ is a polynomial in $z_n$ of degree $<\mu$. Now we have $$(r^1,\ldots,r^q)=1/f''(f''r^1,\ldots,f''r^q)$$ where $f''r^j$ is of degree $<\mu'+\mu''=\mu$. In combination with (3.21) this proves the lemma.\qed \endproof \begproof{of Theorem 3.19 (end)} If $g=(g^1,\ldots,g^q)$ is one of the polynomials of $\cR(F_1,\ldots,F_q)_x$ described in Lemma~3.20, we can write $$g^j=\sum_{0\le k\le\mu}u^{jk}z_n^k,~~~~u^{jk}\in\cO_{\Delta',x'}.$$ The condition for $(g^1,\ldots,g^q)$ to belong to $\cR(F_1,\ldots,F_q)_x$ therefore consists of $2\mu+1$ linear conditions for the germ $u=(u^{jk})$ with coefficients in $\cO(\Delta')$. By the induction hypothesis, $\cO_{\Delta'}$ is coherent and Th.~3.16 shows that the corresponding modules of relations are generated over $\cO_{\Delta',x'}$, for $x'$ in a neighborhood $\Omega'$ of $0$, by finitely many $(q\times\mu)$-tuples $U_1,\ldots,U_N\in\cO(\Omega')^{q\mu}$. By Lemma~3.20, $\cR(F_1,\ldots,F_q)_x$ is generated at every point $x\in\Omega=\Omega'\times\Delta_n$ by the germs of the corresponding polynomials $$G_l(z)=\Big(\sum_{0\le k\le\mu}U^{jk}_l(z')z_n^k\Big)_{1\le j\le q}, ~~~~z\in\Omega,~~~1\le l\le N.\eqno\square$$ \endproof \begstat{(3.22) Strong Noetherian property} Let $\cF$ be a coherent analytic sheaf on a complex manifold $M$ and let $\cF_1\subset\cF_2\subset \ldots$ be an increasing sequence of coherent subsheaves of~$\cF$. Then the sequence $(\cF_k)$ is stationary on every compact subset of~$M$. \endstat \begproof{} Since $\cF$ is locally a quotient of a free module $\cO_M^{\oplus q}$, we can pull back the sequence to $\cO_M^{\oplus q}$ and thus suppose $\cF=\cO_M$ (by easy reductions similar to those in the proof of Th.~3.14). Suppose $M$ connected and $\cF_{k_0}\ne\{0\}$ for some index $k_0$ (otherwise, there is nothing to prove). By the analytic continuation theorem, we easily see that $\cF_{k_0,x}\ne\{0\}$ for every $x\in M$. We can thus find a non zero Weierstrass polynomial $P\in\cF_{k_0}(V)$, ${\rm deg}_{z_n}P(z',z_n)=\mu$, in a coordinate neighborhood $V=\Delta'\times\Delta_n$ of any point $x\in M$. A division by $P$ shows that for $k\ge k_0$ and $x\in V$, all stalks $\cF_{k,x}$ are generated by $P_x$ and by polynomials of degree $<\mu$ in $z_n$ with coefficients in $\cO_{\Delta',x'}$. Therefore, we can apply induction on $n$ to the coherent $\cO_{\Delta'}$-module $$\cF'=\cF\cap\big\{Q\in\cO_{\Delta'}[z_n]\,;\,{\rm deg}\,Q \le\mu\big\}$$ and its increasing sequence of coherent subsheaves $\cF'_k= \cF_k\cap\cF'$.\qed \endproof \titleb{\S 4.}{Complex Analytic Sets. Local Properties} \titlec{\S 4.1.}{Definition. Irreducible Components} A complex analytic set is a set which can be defined locally by finitely many holomorphic equations; such a set has in general singular points, because no assumption is made on the differentials of the equations. We are interested both in the description of the singularities and in the study of algebraic properties of holomorphic functions on analytic sets. For a more detailed study than ours, we refer to H.~Cartan's seminar (Cartan 1950), to the books of (Gunning-Rossi 1965), (Narasimhan 1966) or the recent book by (Grauert-Remmert 1984). \begstat{(4.1) Definition} Let $M$ be a complex analytic manifold. A subset $A\subset M$ is said to be an analytic subset of $M$ if $A$ is closed and if for every point $x_0\in A$ there exist a neighborhood $U$ of $x_0$ and holomorphic functions $g_1,\ldots,g_n$ in $\cO(U)$ such that $$A\cap U=\{z\in U~;~g_1(z)=\ldots=g_N(z)=0\}.$$ Then $g_1,\ldots,g_N$ are said to be $($local$)$ equations of $A$ in $U$. \endstat It is easy to see that a finite union or intersection of analytic sets is analytic: if $(g'_j)$, $(g''_k)$ are equations of $A'$, $A''$ in the open set $U$, then the family of all products $(g'_jg''_k)$ and the family $(g'_j)\cup(g''_k)$ define equations of $A'\cup A''$ and $A'\cap A''$ respectively. \begstat{(4.2) Remark} \rm Assume that $M$ is connected. The analytic continuation theorem shows that either $A=M$ or $A$ has no interior point. In the latter case, each piece $A\cap U=g^{-1}(0)$ is the set of points where the function $\log|g|^2=\log(|g_1|^2+\cdots+|g_N|^2)\in\Psh(U)$ takes the value $-\infty$, hence $A$ is pluripolar. In particular $M\ssm A$ is connected and every function $f\in\cO(M\ssm A)$ that is locally bounded near $A$ can be extended to a function $\tilde f\in\cO(M)$.\qed \endstat We focus now our attention on local properties of analytic sets. By definition, a germ of set at a point $x\in M$ is an equivalence class of elements in the power set ${\cal P}(M)$, with $A\sim B$ if there is an open neighborhood $V$ of $x$ such that $A\cap V=B\cap V$. The germ of a subset $A\subset M$ at $x$ will be denoted by $(A,x)$. We most often consider the case when $A\subset M$ is a analytic set in a neighborhood $U$ of~$x$, and in this case we denote by $\cI_{A,x}$ the ideal of germs $f\in\cO_{M,x}$ which vanish on $(A,x)$. Conversely, if $\cJ=(g_1,\ldots,g_N)$ is an ideal of $\cO_{M,x}$, we denote by $\big(V(\cJ),x\big)$ the germ at $x$ of the zero variety $V(\cJ)=\{z\in U~;~g_1(z)=\ldots=g_N(z)=0\}$, where $U$ is a neighborhood of $x$ such that $g_j\in\cO(U)$. It is easy to check that the germ $(V(cJ),x)$ does not depend on the choice of generators. Moreover, it is clear that $$\leqalignno{ &\hbox{\rm for every ideal $\cJ$ in the ring $\cO_{M,x}$,}\qquad~~\, \cI_{V(\cJ),x}\supset\cJ,&(4.3')\cr &\hbox{\rm for every germ of analytic set $(A,x)$,}\qquad \big(V(\cI_{A,x}),x\big)=(A,x).&(4.3'')\cr}$$ \begstat{(4.4) Definition} A germ $(A,x)$ is said to be irreducible if it has no decom\-po\-sition $(A,x)=(A_1,x)\cup(A_2,x)$ with analytic sets \hbox{$(A_j,x)\ne(A,x)$, $j=1,2$}. \endstat \begstat{(4.5) Proposition} A germ $(A,x)$ is irreducible if and only if $\cI_{A,x}$ is a prime ideal of the ring $\cO_{M,x}$. \endstat \begproof{} Let us recall that an ideal $\cJ$ is said to be {\it prime} if $fg\in\cJ$ implies $f\in\cJ$ or $g\in\cJ$. Assume that $(A,x)$ is irreducible and that $fg\in\cI_{A,x}$. As we can write $(A,x)=(A_1,x)\cup(A_2,x)$ with $A_1=A\cap f^{-1}(0)$ and $A_2=A\cap g^{-1}(0)$, we must have for example $(A_1,x)=(A,x)$~; thus $f\in\cI_{A,x}$ and $\cI_{A,x}$ is prime. Conversely, if $(A,x)=(A_1,x)\cup(A_2,x)$ with $(A_j,x)\ne(A,x)$, there exist $f\in\cI_{A_1,x}$, $g\in\cI_{A_2,x}$ such that $f,g\notin \cI_{A,x}$. However $fg\in\cI_{A,x}$, thus $\cI_{A,x}$ is not prime.\qed \endproof \begstat{(4.6) Theorem} Every decreasing sequence of germs of analytic sets $(A_k,x)$ is stationary. \endstat \begproof{} In fact, the corresponding sequence of ideals $\cJ_k=\cI_{A_k,x}$ is increa\-sing, thus $\cJ_k=\cJ_{k_0}$ for $k\ge k_0$ large enough by the Noetherian property of~$\cO_{M,x}$. Hence $(A_k,x)=\big(V(\cJ_k),x\big)$ is constant for $k\ge k_0$. This result has the following straightforward consequence:\qed \endproof \begstat{(4.7) Theorem} Every analytic germ $(A,x)$ has a finite decomposition $$(A,x)=\bigcup_{1\le k\le N}(A_k,x)$$ where the germs $(A_j,x)$ are irreducible and $(A_j,x)\not\subset(A_k,x)$ for $j\ne k$. The decomposition is unique apart from the ordering. \endstat \begproof{} If $(A,x)$ can be split in several components, we split repeatedly each component as long as one of them is reducible. The process must stop by Th.~4.6, whence the existence. For the uniqueness, assume that\break $(A,x)=\bigcup(A'_l,x)$, $1\le l\le N'$, is another decomposition. Since $(A_k,x)=\bigcup_l(A_k\cap A'_l,x)$, we must have $(A_k,x)=(A_k\cap A'_l,x)$ for some $l=l(k)$, i.e.\ $(A_k,x)\subset (A'_{l(k)},x)$, and likewise $(A'_{l(k)},x)\subset(A_j,x)$ for some~$j$. Hence $j=k$ and $(A'_{l(k)},x)=(A_k,x)$.\qed \endproof \titlec{\S 4.2.}{Local Structure of a Germ of Analytic Set} We are going to describe the local structure of a germ, both from the holomorphic and topological points of view. By the above decomposition theo\-rem, we may restrict ourselves to the case of irreducible germs Let~$\cJ$ be a prime ideal of $\cO_n=\cO_{\bC^n,0}$ and let $A=V(\cJ)$ be its zero variety. We set $\cJ_k=\cJ\cap\bC\{z_1,\ldots,z_k\}$ for each $k=0,1,\ldots,n$. \begstat{(4.8) Proposition} There exist an integer $d$, a basis $(e_1,\ldots,e_n)$ of $\bC^n$ and associated coordinates $(z_1,\ldots,z_n)$ with the following properties: $\cJ_d=\{0\}$ and for every integer $k=d+1,\ldots,n$ there is a Weierstrass polynomial $P_k\in\cJ_k$ of the form $$P_k(z',z_k)=z_k^{s_k}+\sum_{1\le j\le s_k}a_{j,k}(z')\,z_k^{s_k-j},~~~~ a_{j,k}(z')\in\cO_{k-1},\leqno(4.9)$$ where $a_{j,k}(z')=O(|z'|^j)$. Moreover, the basis $(e_1,\ldots,e_n)$ can be chosen arbitrarily close to any preassigned basis $(e^0_1,\ldots,e^0_n)$. \endstat \begproof{} By induction on $n$. If $\cJ=\cJ_n=\{0\}$, then $d=n$ and there is nothing to prove. Otherwise, select a non zero element $g_n\in\cJ$ and a vector $e_n$ such that $\bC\ni w\longmapsto g_n(we_n)$ has minimum vanishing order $s_n$. This choice excludes at most the algebraic set $g_n^{(s_n)}(v)=0$, so $e_n$ can be taken arbitrarily close to $e^0_n$. Let $(\tilde z_1,\ldots,\tilde z_{n-1},z_n)$ be the coordinates associated to the basis $(e^0_1,\ldots,e^0_{n-1},e_n)$. After multiplication by an invertible element, we may assume that $g_n$ is a Weierstrass polynomial $$P_n(\tilde z,z_n)=z_n^{s_n}+\sum_{1\le j\le s_n}a_{j,n}(\tilde z)\,z_n^{s_n-j}, ~~~~a_{j,n}\in\cO_{n-1},$$ and $a_{j,n}(\tilde z)=O(|\tilde z|^j)$ by Remark~2.2. If $\cJ_{n-1}=\cJ\cap\bC\{\tilde z\}=\{0\}$ then $d=n-1$ and the construction is finished. Otherwise we apply the induction hypothesis to the ideal $\cJ_{n-1}\subset\cO_{n-1}$ in order to find a new basis $(e_1,\ldots,e_{n-1})$ of $\Vect(e^0_1,\ldots,e^0_{n-1})$, associated coordinates $(z_1,\ldots,z_{n-1})$ and Weierstrass polynomials $P_k\in\cJ_k$, $d+1\le k\le n-1$, as stated in the lemma.\qed \endproof \begstat{(4.10) Lemma} If $w\in\bC$ is a root of $w^d+a_1w^{d-1}+\cdots+a_d=0$, $a_j\in\bC$, then $|w|\le 2\max|a_j|^{1/j}$. \endstat \begproof{} Otherwise $|w|>2|a_j|^{1/j}$ for all $j=1,\ldots,d$ and the given equation $-1=a_1/w+\cdots+a_d/w^d$ implies $1\le 2^{-1}+\cdots+2^{-d}$, a contradiction.\qed \endproof \begstat{(4.11) Corollary} Set $z'=(z_1,\ldots,z_d)$, $z''=(z_{d+1},\ldots,z_n)$, and let $\Delta'$ in $\bC^d$, $\Delta''$ in $\bC^{n-d}$ be polydisks of center $0$ and radii $r',r''>0$. Then the germ $(A,0)$ is contained in a cone $|z''|\le C|z'|$, $C=\hbox{\it constant}$, and the restriction of the projection map $\bC^n\longrightarrow\bC^d$, $(z',z'')\longmapsto z'~:$ $$\pi:A\cap(\Delta'\times\Delta'')\longrightarrow\Delta'$$ is proper if $r''$ is small enough and $r'\le r''/C$. \endstat \begproof{} The polynomials $P_k(z_1,\ldots,z_{k-1}\,;\,z_k)$ vanish on $(A,0)$. By Lemma~4.10 and (4.9), every point $z\in A$ sufficiently close to $0$ satisfies $$|z_k|\le C_k(|z_1|+\cdots+|z_{k-1}|),~~~~d+1\le k\le n,$$ thus $|z''|\le C|z'|$ and the Corollary follows.\qed \endproof Since $\cJ_d=\{0\}$, we have an injective ring morphism $$\cO_d=\bC\{z_1,\ldots,z_d\}\lhra\cO_n/\cJ.\leqno(4.12)$$ \begstat{(4.13) Proposition} $\cO_n/\cJ$ is a finite integral extension of $\cO_d$. \endstat \begproof{} Let $f\in\cO_n$. A division by $P_n$ yields $f=P_nq_n+R_n$ with a remainder $R_n\in\cO_{n-1}[z_n]$, $\deg_{z_n}R_nk$. Hence $$f=R_d+\sum_{d+1\le k\le n}P_kq_k=R_d~~~\hbox{\rm mod}~~(P_{d+1},\ldots,P_n) \subset\cJ\leqno(4.14)$$ and $\cO_n/\cJ$ is finitely generated as an $\cO_d$-module by the family of monomials $z_{d+1}^{\alpha_{d+1}}\ldots z_n^{\alpha_n}$ with $\alpha_jd}c_kz_k$. If $\Delta',\Delta''$ are polydisks of sufficiently small radii $r',r''$ and if $r'\le r''/C$ with $C$ large, the projection map $\pi:A\cap(\Delta'\times\Delta'')\longrightarrow\Delta'$ is a ramified covering with $q$ sheets, whose ramification locus is contained in $S=\{z'\in\Delta';\delta(z')=0\}$. This means that: \smallskip \item{\rm a)} the open subset $A_S=A\cap\big((\Delta'\ssm S) \times\Delta''\big)$ is a smooth $d$-dimensional manifold, dense in $A\cap(\Delta'\times\Delta'')~;$ \smallskip \item{\rm b)} $\pi:A_S\longrightarrow\Delta'\ssm S$ is a covering~$;$ \smallskip \item{\rm c)} $\,$the fibers $\pi^{-1}(z')$ have exactly $q$ elements if $z'\notin S$ and at most $q$ if $z'\in S.$ \smallskip \noindent Moreover, $A_S$ is a connected covering of $\Delta'\ssm S$, and $A\cap(\Delta'\times\Delta'')$ is contained in a cone $|z''|\le C|z'|$~ $($see Fig. 1$)$. \endstat \input epsfiles/fig_2_1.tex \vskip0mm \centerline{{\bf Fig.~II-1} Ramified covering from $A$ to $\Delta'\subset\bC^p$.} \vskip6mm \begproof{} After a linear change in the coordinates $z_{d+1},\ldots,z_n$, we may assume $u(z'')=z_{d+1}$, so $W_u=W_{d+1}$ and $B_{d+1}(z'\,;T)=\delta(z')T$. By Lemma~4.18, we have $$\cG=\big(W_{d+1}(z',z_{d+1})\,,\,\delta(z')z_k-B_k(z',z_{d+1})\big)_{k\ge d+2} \subset\cJ,~~~~\delta^m\cJ\subset\cG.$$ We can thus find a polydisk $\Delta=\Delta'\times\Delta''$ of sufficiently small radii $r',r''$ such that $V(\cJ)\subset V(\cG)\subset V(\delta^m\cJ)$ in~$\Delta$. As $V(\cJ)=A$ and $V(\delta)\cap\Delta=S\times \Delta''$, this implies $$A\cap\Delta\subset V(\cG)\cap\Delta\subset(A\cap\Delta)\cup (S\times\Delta'').$$ In particular, the set $A_S=A\cap\big((\Delta'\ssm S)\times\Delta''\big)$ lying above $\Delta'\ssm S$ coincides with $V(\cG)\cap\big((\Delta'\ssm S)\times\Delta''\big)$, which is the set of points $z\in\Delta$ parametrized by the equations $$\cases{ \delta(z')\ne 0,~~~W_{d+1}(z',z_{d+1})=0,\cr z_k=B_k(z',z_{d+1})/\delta(z'),~~d+2\le k\le n.\cr}\leqno(4.20)$$ As $\delta(z')$ is the resultant of $W_{d+1}$ and $\partial W_{d+1}/\partial T$, we have $$\partial W_{d+1}/\partial T(z',z_{d+1})\ne 0~~~~\hbox{\rm on}~~A_S.$$ The implicit function theorem shows that $z_{d+1}$ is locally a holomorphic function of $z'$ on $A_S$, and the same is true for $z_k=B_k(z',z_{d+1})/\delta(z')$, $k\ge d+2$. Hence $A_S$ is a smooth manifold, and for $r'\le r''/C$ small, the projection map $\pi:A_S\longrightarrow\Delta'\ssm S$ is a proper local diffeomorphism; by (4.20) the fibers $\pi^{-1}(z')$ have at most $q$ points corresponding to some of the $q$ roots $w$ of $W_{d+1}(z'\,;w)=0$. Since $\Delta'\ssm S$ is connected (Remark~4.2), either $A_S=\emptyset$ or the map $\pi$ is a covering of constant sheet number $q'\le q$. However, if $w$ is a root of $W_{d+1}(z',w)=0$ with $z'\in\Delta'\ssm S$ and if we set $z_{d+1}=w$, $z_k=B_k(z',w)/\delta(z')$, $k\ge d+2$, relation $(4.17)$ shows that $W_k(z',z_k)=0$, in particular $|z_k|=O(|z'|^{1/q})$ by Lemma~4.10. For $z'$ small enough, the $q$ points $z=(z',z'')$ defined in this way lie in $\Delta$, thus $q'=q$. Property b) and the first parts of a) and c) follow. Now, we need the following lemma. \endproof \begstat{(4.21) Lemma} If $\cJ\subset\cO_n$ is prime and $A=V(\cJ)$, then $\cI_{A,0}=\cJ$. \endstat \begproof It is obvious that $\cI_{A,0}\supset\cJ$. Now, for any $f\in\cI_{A,0}$, Prop.~4.13 implies that $\tilde f$ satisfies in $\cO_n/\cI$ an irreducible equation $$f^r+b_1(z')\,f^{r-1}+\cdots+b_r(z')=0~~~~\hbox{\rm (mod}~\cJ).$$ Then $b_r(z')$ vanishes on $(A,0)$ and the first part of c) gives $b_r=0$ on $\Delta'\ssm S$. Hence $\tilde b_r=0$ and the irreducibility of the equation of~$\tilde f$ implies $r=1$, so \hbox{$f\in\cJ$,} as desired.\qed \endproof \begproof{of Theorem 4.19 (end).} It only remains to prove that $A_S$ is connected and dense in $A\cap\Delta$ and that the fibers $\pi^{-1}(z')$, $z'\in S$, have at most $q$ elements. Let $A_{S,1},\ldots,A_{S,N}$ be the connected components of $A_S$. Then $\pi:A_{S,j}\longrightarrow\Delta'\ssm S$ is a covering with $q_j$ sheets, $q_1+\cdots+q_N=q$. For every point $\zeta'\in\Delta'\ssm S$, there exists a neighborhood $U$ of $\zeta'$ such that $A_{S,j}\cap\pi^{-1}(U)$ is a disjoint union of graphs $z''=g_{j,k}(z')$ of analytic functions $g_{j,k}\in \cO(U)$, $1\le k\le q_j$. If $\lambda(z'')$ is an arbitrary linear form in $z_{d+1},\ldots,z_n$ and $z'\in\Delta'\ssm S$, we set $$P_{\lambda,j}(z'\,;T)=\prod_{\{z''\,;\,(z',z'')\in A_{S,j}\}}\big(T-\lambda (z'')\big)=\prod_{1\le k\le k_j}\big(T-\lambda\circ g_{j,k}(z')\big).$$ This defines a polynomial in $T$ with bounded analytic coefficients on $\Delta'\ssm S$. These coefficients have analytic extensions to $\Delta'$ (Remark~4.2), thus $P_{\lambda,j}\in\cO(\Delta')[T]$. By construction, $P_{\lambda,j}\big(z'\,;\lambda(z'')\big)$ vanishes identically on $A_{S,j}$. Set $$P_\lambda=\prod_{1\le j\le N}P_{\lambda,j},~~~~ f(z)=\delta(z')\,P_\lambda\big(z'\,;\lambda(z'')\big)~;$$ $f$ vanishes on $A_{S,1}\cup\ldots\cup A_{S,N}\cup(S\times\Delta'')\supset A\cap\Delta$. Lemma~4.21 shows that $\cI_{A,0}$ is prime; as $\delta\notin\cI_{A,0}$, we get $P_{\lambda,j}\big(z'\,;\lambda(z'')\big)\in\cI_{A,0}$ for some~$j$. This is a contradiction if $N\ge 2$ and if $\lambda$ is chosen in such a way that $\lambda$ separates the $q$ points $z''_\nu$ in each fiber $\pi^{-1}(z'_\nu)$, for a sequence $z'_\nu\to 0$ in $\Delta'\ssm S$. Hence $N=1$, $A_S$ is connected, and for every $\lambda\in(\bC^{n-d})^\star$ we have $P_\lambda\big(z',\lambda(z'')\in\cI_{(A,0)}$. By construction $P_\lambda\big(z',\lambda(z'')\big)$ vanishes on $A_S$, so it vanishes on $\ol A_S$~; hence, for every $z'\in S$, the fiber $\ol A_S\cap\pi^{-1}(z')$ has at most $q$ elements, otherwise selecting $\lambda$ which separates $q+1$ of these points would yield $q+1$ roots $\lambda(z'')$ of $P_\lambda(z'\,;T)$, a contradiction. Assume now that $A_S$ is not dense in $A\cap\Delta$ for arbitrarily small polydisks $\Delta$. Then there exists a sequence $A\ni z_\nu=(z'_\nu,z''_\nu)\to 0$ such that $z'_\nu\in S$ and $z''_\nu$ is not in $F_\nu:=\hbox{\rm pr}''\big(\ol A_S\cap \pi^{-1}(z'_\nu)\big)$. The continuity of the roots of the polynomial $P_\lambda(z'\,;T)$ as $\Delta'\ssm S\ni z'\to z'_\nu$ implies that the set of roots of $P_\lambda(z'_\nu\,;T)$ is $\lambda(F_\nu)$. Select $\lambda$ such that $\lambda(z''_\nu)\notin\lambda(F_\nu)$ for all $\nu$. Then $P_\lambda\big(z'_\nu\,;\lambda(z''_\nu)\big)\ne 0$ for every $\nu$ and $P_\lambda\big(z'\,;\lambda(z'')\big)\notin\cI_{A,0}$, a contradiction.\qed \endproof At this point, it should be observed that many of the above statements completely fail in the case of real analytic sets. In $\bR^2$, for example, the prime ideal $\cJ=(x^5+y^4)$ defines an irreducible germ of curve $(A,0)$ and there is an injective integral extension of rings $\bR\{x\}\lhra\bR\{x,y\}/\cJ$ of degree 4; however, the projection of $(A,0)$ on the first factor, $(x,y)\mapsto x$, has not a constant sheet number near $0$, and this number is not related to the degree of the extension. Also, the prime ideal $\cJ=(x^2+y^2)$ has an associated variety $V(\cJ)$ reduced to $\{0\}$, hence $\cI_{A,0}=(x,y)$ is strictly larger than $\cJ$, in contrast with Lemma~4.21. Let us return to the complex situation, which is much better behaved. The result obtained in Lemma~4.21 can then be extended to non prime ideals and we get the following important result: \begstat{(4.22) Hilbert's Nullstellensatz} For every ideal $\cJ\subset\cO_n$ $$\cI_{V(\cJ),0}=\sqrt{\cJ},$$ where $\sqrt{\cJ}$ is the radical of $\cJ$, i.e.\ the set of germs $f\in\cO_n$ such that some power $f^k$ lies in $\cJ$. \endstat \begproof{} Set $B=V(\cJ)$. If $f^k\in\cJ$, then $f^k$ vanishes on $(B,0)$ and $f\in\cI_{B,0}$. Thus $\sqrt{\cJ}\subset\cI_{B,0}$. Conversely, it is well known that $\sqrt{\cJ}$ is the intersection of all prime ideals $\cP\supset\cJ$ (Lang~1965, Chapter~VI). For such an ideal $(B,0)=\big(V(\cJ),0)\supset\big(V(\cP),0\big)$, thus $\cI_{B,0}\subset\cI_{V(\cP),0}=\cP$ in view of Lemma~4.21. Therefore $\cI_{B,0}\subset\bigcap_{\cP\supset\cJ}\cP=\sqrt{\cJ}$ and the Theorem is proved.\qed \endproof In other words, if a germ $(B,0)$ is defined by an arbitrary ideal $\cJ\subset\cO_n$ and if $f\in\cO_n$ vanishes on $(B,0)$, then some power $f^k$ lies in $\cJ$. \titlec{\S 4.3.}{Regular and Singular Points. Dimension} The above powerful results enable us to investigate the structure of singularities of an analytic set. We first give a few definitions. \begstat{(4.23) Definition} Let $A\subset M$ be an analytic set and $x\in A$. We say that $x\in A$ is a regular point of $A$ if $A\cap\Omega$ is a $\bC$-analytic submanifold of $\Omega$ for some neighborhood $\Omega$ of $x$. Otherwise $x$ is said to be singular. The corresponding subsets of $A$ will be denoted respectively $A_\reg$ and $A_\sing$. \endstat It is clear from the definition that $A_\reg$ is an open subset of $A$ (thus $A_\sing$ is closed), and that the connected components of $A_\reg$ are $\bC$-analytic submanifolds of $M$ (non necessarily closed). \begstat{(4.24) Proposition} If $(A,x)$ is irreducible, there exist arbitrarily small neighborhoods $\Omega$ of $x$ such that $A_\reg\cap\Omega$ is dense and connected in $A\cap\Omega$. \endstat \begproof{} Take $\Omega=\Delta$ as in Th.~4.19. Then $A_S\subset A_\reg\cap\Omega\subset A\cap\Omega$, where $A_S$ is connected and dense in $A\cap\Omega$~; hence $A_\reg\cap\Omega$ has the same properties.\qed \endproof \begstat{(4.25) Definition} The dimension of an irreducible germ of analytic set $(A,x)$ is defined by $\dim(A,x)=\dim(A_\reg,x)$. If $(A,x)$ has several irreducible components $(A_l,x)$, we set $$\dim (A,x)=\max\{\dim(A_l,x)\},~~~~\codim(A,x)=n-\dim(A,x).$$ \endstat \begstat{(4.26) Proposition} Let $(B,x)\subset(A,x)$ be germs of analytic sets. If $(A,x)$ is irreducible and $(B,x)\ne(A,x)$, then $\dim(B,x)<\dim(A,x)$ and $B\cap\Omega$ has empty interior in $A\cap\Omega$ for all sufficiently small neighborhoods $\Omega$ of $x$. \endstat \begproof{} We may assume $x=0$, $(A,0)\subset(\bC^n,0)$ and $(B,0)$ irreducible. Then $\cI_{A,0} \subset\cI_{B,0}$ are prime ideals. When we choose suitable coordinates for the ramified coverings, we may at each step select vectors $e_n,e_{n-1},\ldots$ that work simultaneously for $A$ and $B$. If $\dim B=\dim A$, the process stops for both at the same time, i.e.\ we get ramified coverings $$\pi:A\cap(\Delta'\times\Delta'')\longrightarrow\Delta',~~~~ \pi:B\cap(\Delta'\times\Delta'')\longrightarrow\Delta'$$ with ramification loci $S_A,S_B$. Then $B\cap\big((\Delta'\ssm (S_A\cup S_B))\times\Delta''\big)$ is an open subset of the manifold $A_S=A\cap\big((\Delta'\ssm S_A)\times\Delta''\big)$, therefore $B\cap A_S$ is an analytic subset of $A_S$ with non empty interior. The same conclusion would hold if $B\cap\Delta$ had non empty interior in $A\cap\Delta$. As $A_S$ is connected, we get $B\cap A_S=A_S$, and as $B\cap\Delta$ is closed in $\Delta$ we infer $B\cap\Delta\supset\ol A_S=A\cap\Delta$, hence $(B,0)=(A,0)$, in contradiction with the hypothesis.\qed \endproof \begstat{(4.27) Example: parametrization of curves} {\rm Suppose that $(A,0)$ is an irreducible germ of curve ($\dim(A,0)=1$). If the disk $\Delta'\subset\bC$ is chosen so small that $S=\{0\}$, then $A_S$ is a connected covering of $\Delta'\ssm\{0\}$ with $q$ sheets. Hence, there exists a covering isomorphism between $\pi$ and the standard covering $$\bC\supset\Delta(r)\ssm\{0\}\longrightarrow \Delta(r^q)\ssm\{0\},~~~~ t\longmapsto t^q,~~~~r^q=\hbox{\rm radius of~~}\Delta',$$ i.e.\ a map $\gamma:\Delta(r)\ssm\{0\}\longrightarrow A_S$ such that $\pi\circ\gamma(t)=t^q$. This map extends into a bijective holomorphic map $\gamma:\Delta(r)\longrightarrow A\cap\Delta$ with $\gamma(0)=0$. This means that every irreducible germ of curve can be parametrized by a bijective holomorphic map defined on a disk in $\bC$ (see also Exercise~10.8).} \endstat \titlec{\S 4.4.}{Coherence of Ideal Sheaves} Let $A$ be an analytic set in a complex manifold $M$. The {\it sheaf of ideals} $\cI_A$ is the subsheaf of $\cO_M$ consisting of germs of holomorphic functions on $M$ which vanish on $A$. Its stalks are the ideals $\cI_{A,x}$ already considered; note that $\cI_{A,x}=\cO_{M,x}$ if $x\notin A$. If $x\in A$, we let $\cO_{A,x}$ be the ring of germs of functions on $(A,x)$ which can be extended as germs of holomorphic functions on $(M,x)$. By definition, there is a surjective morphism $\cO_{M,x}\longrightarrow\cO_{A,x}$ whose kernel is $\cI_{A,x}$, thus $$\cO_{A,x}=\cO_{M,x}/\cI_{A,x},~~~~\forall x\in A,\leqno(4.28)$$ i.e.\ $\cO_A=(\cO_M/\cI_A)_{\restriction A}$. Since $\cI_{A,x}=\cO_{M,x}$ for $x\notin A$, the quotient sheaf $\cO_M/\cI_A$ is zero on $M\ssm A$. \begstat{(4.29) Theorem {\rm (Cartan 1950)}} For any analytic set $A\subset M$, the sheaf of ideals $\cI_A$ is a coherent analytic sheaf. \endstat \begproof{} It is sufficient to prove the result when $A$ is an analytic subset in a neighborhood of $0$ in $\bC^n$. If $(A,0)$ is not irreducible, there exists a neighborhood $\Omega$ such that $A\cap\Omega=A_1\cup\ldots\cup A_N$ where $A_k$ are analytic sets in $\Omega$ and $(A_k,0)$ is irreducible. We have $\cI_{A\cap\Omega}=\bigcap\cI_{A_k}$, so by Cor.~3.15 we may assume that $(A,0)$ is irreducible. Then we can choose coordinates \hbox{$z'$, $z''$}, polydisks $\Delta',\Delta''$ and a primitive element $u(z'')=c_{d+1} z_{d+1}+\cdots+c_nz_n$ such that Th.~4.19 is valid. Since $\delta(z')= \prod_{jp$ for all $x\in Z$. Then the closure $\ol Z$ of $Z$ in $X$ is an analytic subset. \endstat \begstat{(8.8) Remmert's proper mapping theorem} Let $F:X\longrightarrow Y$ be a proper holomorphic map. Then $F(X)$ is an analytic subset of $Y$. \endstat \begproof{} We let $(8.7_m)$ denote statement (8.7) for $\dim Z\le m$ and $(8.8_m)$ denote statement (8.8) for $\dim X\le m$. We proceed by induction on $m$ in two steps:\medskip \noindent{\it Step 1.} $(8.7_m)$ and $(8.8_{m-1})$ imply $(8.8_m)$.\newline \noindent{\it Step 2.} $(8.8_{m-1})$ implies $(8.7_m)$. \medskip \noindent As $(8.8_m)$ is obvious for $m=0$, our statements will then be valid for all $m$, i.e.\ for all complex spaces of bounded dimension. However, Th.~8.7 is local on $X$ and Th.~8.8 is local on $Y$, so the general case is immediately reduced to the finite dimensional case. \medskip \noindent{\it Proof of step 1.} The analyticity of $F(X)$ is a local question in $Y$. Since $F:F^{-1}(U)\longrightarrow U$ is proper for any open set $U\subset Y$ and $F^{-1}(U)\compact X$ if $U\compact Y$, we may suppose that $Y$ is embedded in an open set $\Omega\subset\bC^n$ and that $X$ only has finitely many irreducible components $X_\alpha$. Then we have $F(X)=\bigcup F(X_\alpha)$ and we are reduced to the case when $X$ is irreducible, $\dim X=m$ and $Y=\Omega$. First assume that $X$ is a manifold and that the rank of $dF$ is constant. The constant rank theorem implies that every point in $X$ has a neighborhood $V$ such that $F(V)$ is a closed submanifold in a neighborhood $W$ of $F(x)$ in $Y$. For any point $y\in Y$, the fiber $F^{-1}(y)$ can be covered by finitely many neighborhoods $V_j$ of points $x_j\in F^{-1}(y)$ such that $F(V_j)$ is a closed submanifold in a neighborhood $W_j$ of $y$. Then there is a neighborhood of $y$ $W\subset\bigcap W_j$ such that $F^{-1}(W)\subset\bigcup V_j$, so $F(X)\cap W=\bigcup F(V_j)\cap W$ is a finite union of closed submanifolds in $W$ and $F(X)$ is analytic in $Y$. Now suppose that $X$ is a manifold, set $r=\ol\rho(F)$ and let $Z\subset X$ be the analytic subset of points $x$ where the rank of $dF_x$ is $s}\lambda_jz_j$ be a linear form on $\bC^n$ in the coordinates of index $j>s$. For $z'\in B'\ssm(A'\cup S'_1\cup S'_2)$, we let $\sigma_j(z')$ be the elementary symmetric functions in the $q$ complex numbers $\lambda(z)$ corresponding to $z\in\pi^{-1}(z')$. Then these functions can be extended as bounded holomorphic functions on $B'$ and we get a polynomial $P_\lambda(z'\,;\,T)$ such that $P_\lambda\big(z'\,;\,\lambda(z'')\big)$ vanishes identically on $Z\ssm\pi^{-1}(A'\cup S'_1\cup S'_2)$. Since $\pi$ is finite, $Z\cap\pi^{-1}(A'\cup S'_1\cup S'_2)$ is a union of three (non necessarily closed) analytic subsets of dimension $\le s-1$, thus has empty interior in $Z$. It follows that the closure $\ol Z\cap(B'\times B'')$ is contained in the analytic set $W\subset B'\times B''$ equal to the common zero set of all functions $P_\lambda\big(z'\,;\,\lambda(z'')\big)$. Moreover, by construction, $$Z\ssm\pi^{-1}(A'\cup S'_1\cup S'_2)= W\ssm\pi^{-1}(A'\cup S'_1\cup S'_2).$$ As in the proof of Cor.~5.4, we easily conclude that $\ol Z\cap(B'\times B'')$ is equal to the union of all irreducible components of $W$ that are not contained in $\pi^{-1}(A'\cup S'_1\cup S'_2)$. Hence $\ol Z$ is analytic.\qed\medskip Finally, we give two interesting applications of the Remmert-Stein theorem. We assume here that the reader knows what is the complex projective space $\bP^n$. For more details, see Sect. 5.15. \begstat{(8.10) Chow's theorem {\rm(Chow 1949)}} Let $A$ be an analytic subset of the complex projective space $\bP^n$. Then $A$ is algebraic, i.e.\ $A$ is the common zero set of finitely many homogeneous polynomials $P_j(z_0,\ldots,z_n)$, $1\le j\le N$. \endstat \begproof{} Let $\pi:\bC^{n+1}\ssm\{0\}\longrightarrow\bP^n$ be the natural projection and $Z=\pi^{-1}(A)$. Then $Z$ is an analytic subset of $\bC^{n+1}\ssm\{0\}$ which is invariant by homotheties and $\dim Z=\dim A+1\ge 1$. The Remmert-Stein theorem implies that $\ol Z=Z\cup\{0\}$ is an analytic subset of $\bC^{n+1}$. Let $f_1,\ldots,f_N$ be holomorphic functions on a small polydisk $\Delta\subset\bC^{n+1}$ of center $0$ such that $\ol Z\cap\Delta=\bigcap f_j^{-1}(0)$. The Taylor series at $0$ gives an expansion $f_j=\sum_{k=0}^{+\infty}P_{j,k}$ where $P_{j,k}$ is a homogeneous polynomial of degree $k$. We claim that $\ol Z$ coincides with the common zero $W$ set of the polynomials $P_{j,k}$. In fact, we clearly have $W\cap\Delta \subset\bigcap f_j^{-1}(0)=\ol Z\cap\Delta$. Conversely, for $z\in\ol Z\cap\Delta$, the invariance of $Z$ by homotheties shows that $f_j(tz)=\sum P_{j,k}(z)t^k$ vanishes for every complex number $t$ of modulus $<1$, so all coefficients $P_{j,k}(z)$ vanish and $z\in W\cap\Delta$. By homogeneity $\ol Z=W$~; since $\bC[z_0,\ldots,z_n]$ is Noetherian, $W$ can be defined by finitely many polynomial equations.\qed \endproof \begstat{(8.11) E.E.\ Levi's continuation theorem} Let $X$ be a normal complex space and $A$ an analytic subset such that $\dim(A,x)\le\dim(X,x)-2$ for all $x\in A$. Then every meromorphic function on $X\ssm A$ has a meromorphic extension to $X$. \endstat \begproof{} We may suppose $X$ irreducible, $\dim X=n$. Let $f$ be a meromorphic function on $X\ssm A$. By Th.~6.13, the pole set $P_f$ has pure dimension $(n-1)$, so the Remmert-Stein theorem implies that $\ol P_f$ is analytic in $X$. Fix a point $x\in A$. There is a connected neighborhood $V$ of $x$ and a non zero holomorphic function $h\in\cO_X(V)$ such that $\ol P_f\cap V$ has finitely many irreducible components $\ol P_{f,j}$ and $\ol P_f\cap V\subset h^{-1}(0)$. Select a point $x_j$ in $\ol P_{f,j}\ssm(X_\sing\cup(\ol P_f)_\sing\cup A)$. As $x_j$ is a regular point on $X$ and on $\ol P_f$, there is a local coordinate $z_{1,j}$ at $x_j$ defining an equation of $\ol P_{f,j}$, such that $z_{1,j}^{m_j}f\in\cO_{X,x_j}$ for some integer $m_j$. Since $h$ vanishes along $P_f$, we have $h^{m_j}f\in\cO_{X,x}$. Thus, for $m=\max\{m_j\}$, the pole set $P_g$ of $g=h^m f$ in $V\ssm A$ does not contain $x_j$. As $P_g$ is $(n-1)$-dimensional and contained in $P_f\cap V$, it is a union of irreducible components $\ol P_{f,j}\ssm A$. Hence $P_g$ must be empty and $g$ is holomorphic on $V\ssm A$. By Cor.~7.8, $g$ has an extension to a holomorphic function $\tilde g$ on $V$. Then $\tilde g/h^m$ is the required meromorphic extension of $f$ on~$V$.\qed \endproof \titleb{\S 9.}{Complex Analytic Schemes} Our goal is to introduce a generalization of the notion of complex space given in Def.~5.2. A complex space is a space locally isomorphic to an analytic set $A$ in an open subset $\Omega\subset\bC^n$, together with the sheaf of rings $\cO_A=(\cO_\Omega/\cI_A)_{\restriction A}$. Our desire is to enrich the structure sheaf $\cO_A$ by replacing $\cI_A$ with a possibly smaller ideal $\cJ$ defining the same zero variety $V(\cJ)=A$. In this way holomorphic functions are described not merely by their values on $A$, but also possibly by some ``transversal derivatives'' along $A$. \titlec{\S 9.1.}{Ringed Spaces} We start by an abstract notion of ringed space on an arbitrary topological space. \begstat{(9.1) Definition} A ringed space is a pair $(X,\cR_X)$ consisting of a topolo\-gical space $X$ and of a sheaf of rings $\cR_X$ on $X$, called the structure sheaf. A~morphism $$F:(X,\cR_X)\longrightarrow(Y,\cR_Y)$$ of ringed spaces is a pair $(f,F^\star)$ where $f:X\longrightarrow Y$ is a continuous map and $$F^\star~:~~f^{-1}\cR_Y\longrightarrow\cR_X,~~~~F^\star_x~:~~(\cR_Y)_{f(x)} \longrightarrow(\cR_X)_x$$ a homomorphism of sheaves of rings on $X$, called the comorphism of $F$. \endstat If $F:(X,\cR_X)\longrightarrow(Y,\cR_Y)$ and $G:(Y,\cR_Y)\longrightarrow(Z,\cR_Z)$ are morphisms of ringed spaces, the composite $G\circ F$ is the pair consisting of the map $g\circ f:X\longrightarrow Z$ and of the comorphism $(G\circ F)^\star=F^\star\circ f^{-1}G^\star\,$: $$\cmalign{ \hfill&F^\star\circ f^{-1}G^\star~:~~&f^{-1}g^{-1}\cR_Z~\,&\kern-1.9mm \buildo f^{-1}G^\star\over{\relbar\mkern-4mu\relbar\mkern-4mu\longrightarrow}f^{-1}\cR_Y &\kern-1.9mm\buildo F^\star\over{\relbar\mkern-4mu\longrightarrow}\cR_X,\cr &F^\star_x\circ G^\star_{f(x)}~:~~~&(\cR_Z)_{g\circ f(x)}~\,&{\relbar\mkern-4mu \relbar\mkern-4mu\longrightarrow}~(\cR_Y)_{f(x)}~&{\relbar\mkern-4mu\longrightarrow}~(\cR_X)_x.\cr} \leqno(9.2)$$ \titlec{\S 9.2.}{Definition of Complex Analytic Schemes} We begin by a description of what will be the local model of an analytic scheme. Let $\Omega\subset\bC^n$ be an open subset, $\cJ\subset\cO_\Omega$ a coherent sheaf of ideals and $A=V(\cJ)$ the analytic set in $\Omega$ defined locally as the zero set of a system of generators of $\cJ$. By Hilbert's Nullstellensatz~4.22 we have $\cI_A=\sqrt\cJ$, but $\cI_A$ differs in general from $\cJ$. The sheaf of rings $\cO_\Omega/\cJ$ is supported on $A$, i.e.\ $(\cO_\Omega/\cJ)_x=0$ if $x\notin A$. Ringed spaces of the type $(A,\cO_\Omega/\cJ)$ will be used as the local models of analytic schemes. \begstat{(9.3) Definition} A morphism $$F=(f,F^\star):(A,\cO_\Omega/\cJ_{\restriction A})\longrightarrow(A',\cO_{\Omega'}/\cJ' _{\restriction A'})$$ is said to be analytic if for every point $x\in A$ there exists a neighborhood $W_x$ of $x$ in $\Omega$ and a holomorphic function $\Phi:W_x\longrightarrow\Omega'$ such that $f_{\restriction A\cap W_x}= \Phi_{\restriction A\cap W_x}$ and such that the comorphism $$F^\star_x:(\cO_{\Omega'}/\cJ')_{f(x)}\longrightarrow(\cO_\Omega/\cJ)_x$$ is induced by $\Phi^\star:\cO_{\Omega',f(x)}\ni u\longmapsto u\circ\Phi \in\cO_{\Omega,x}$ with $\Phi^\star\cJ'\subset\cJ$. \endstat \begstat{(9.4) Example} \rm Take $\Omega=\bC^n$ and $\cJ=(z_n^2)$. Then $A$ is the hyperplane $\bC^{n-1}\times\{0\}$, and the sheaf $\cO_{\bC^n}/\cJ$ can be identified with the sheaf of rings of functions $u+z_nu'$, $u,u'\in\cO_{\bC^{n-1}}$, with the relation $z_n^2=0$. In particular, $z_n$ is a nilpotent element of $\cO_{\bC^n}/\cJ$. A morphism $F$ of $(A,\cO_{\bC^n}/\cJ)$ into itself is induced (at least locally) by a holomorphic map $\Phi=(\wt\Phi,\Phi_n)$ defined on a neighborhood of $A$ in $\bC^n$ with values in $\bC^n$, such that $\Phi(A)\subset A$, i.e.\ $\Phi_{n\restriction A}=0$. We see that $F$ is completely determined by the data $$\cmalign{ \hfill f(z_1,\ldots,z_{n-1})&=\wt\Phi(z_1,\ldots,z_{n-1},0),~~~~\hfill f~&:~~\bC^{n-1}\longrightarrow\bC^{n-1},\cr f'(z_1,\ldots,z_{n-1})&=\displaystyle{\partial\Phi\over\partial z_n} (z_1,\ldots,z_{n-1},0),~~~~f'~&:~~\bC^{n-1}\longrightarrow\bC^n,\cr}$$ which can be chosen arbitrarily. \endstat \begstat{(9.5) Definition} A complex analytic scheme is a ringed space $(X,\cO_X)$ over a separable Hausdorff topological space $X$, satisfying the following property: there exist an open covering $(U_\lambda)$ of $X$ and isomorphisms of ringed spaces $$G_\lambda:(U_\lambda,\cO_{X\restriction U_\lambda}) \longrightarrow(A_\lambda,\cO_{\Omega_\lambda}/\cJ_{\lambda\,\restriction A_\lambda})$$ where $A_\lambda$ is the zero set of a coherent sheaf of ideals $\cJ_\lambda$ on an open subset $\Omega_\lambda\subset\bC^{N_\lambda}$, such that every transition morphism $G_\lambda\circ G_\mu^{-1}$ is a holomorphic isomorphism from $g_\mu(U_\lambda\cap U_\mu)\subset A_\mu$ onto $g_\lambda(U_\lambda\cap U_\mu)\subset A_\lambda$, equipped with the respective structure sheaves $\cO_{\Omega_\mu}/\cJ_{\mu\,\restriction A_\mu}$, $\cO_{\Omega_\lambda}/\cJ_{\lambda\,\restriction A_\lambda}$. \endstat We shall often consider the maps $G_\lambda$ as identifications and write simply $U_\lambda=A_\lambda$. A morphism $F:(X,\cO_X)\longrightarrow(Y,\cO_Y)$ of analytic schemes obtained by gluing patches $(A_\lambda,\cO_{\Omega_\lambda}/\cJ_{\lambda\, \restriction A_\lambda})$ and $(A'_\mu,\cO_{\Omega'_\mu}/\cJ'_{\mu\,A'_\mu})$, respectively, is a morphism $F$ of ringed spaces such that for each pair $(\lambda,\mu)$, the restriction of $F$ from $A_\lambda\cap f^{-1}(A'_\mu)\subset X$ to $A'_\mu\subset Y$ is holomorphic in the sense of Def.~9.3. \titlec{\S 9.3.}{Nilpotent Elements and Reduced Schemes} Let $(X,\cO_X)$ be an analytic scheme. The set of {\it nilpotent elements} is the sheaf of ideals of $\cO_X$ defined by $$\cN_X=\{u\in\cO_X\,;\,u^k=0~~\hbox{\rm for~some}~\,k\in\bN\}.\leqno(9.6)$$ Locally, we have $\cO_{X\restriction A_\lambda}= (\cO_{\Omega_\lambda}/\cJ_\lambda)_{\restriction A_\lambda}$, thus $$\leqalignno{ &\cN_{X\restriction A_\lambda}=(\sqrt{\cJ_\lambda}/\cJ_\lambda )_{\restriction A_\lambda},&(9.7)\cr &(\cO_X/\cN_X)_{\restriction A_\lambda}\simeq(\cO_{\Omega_\lambda}/ \sqrt{\cJ_\lambda})_{\restriction A_\lambda}= (\cO_{\Omega_\lambda}/\cI_{A_\lambda})_{\restriction A_\lambda}= \cO_{A_\lambda}.&(9.8)\cr}$$ The scheme $(X,\cO_X)$ is said to be {\it reduced} if $\cN_X=0$. The associated ringed space $(X,\cO_X/\cN_X)$ is reduced by construction; it is called the {\it reduced scheme} of $(X,\cO_X)$. We shall often denote the original scheme by the letter $X$ merely, the associated reduced scheme by $X_\red$, and let $\cO_{X,\red}=\cO_X/\cN_X$. There is a canonical morphism $X_\red\to X$ whose comorphism is the reduction morphism $$\cO_X(U)\longrightarrow\cO_{X,\red}(U)=(\cO_X/\cN_X)(U),\qquad \hbox{$\forall U$ open set in $X$}.\leqno(9.9)$$ By~(9.8), the notion of reduced scheme is equivalent to the notion of complex space introduced in Def.~5.2. It is easy to see that a morphism $F$ of reduced schemes $X,Y$ is completely determined by the set-theoretic map $f:X\longrightarrow Y$. \titlec{\S 9.4.}{Coherent Sheaves on Analytic Schemes} If $(X,\cO_X)$ is an analytic scheme, a sheaf $\cS$ of $\cO_X$-modules is said to be {\it coherent} if it satisfies the same properties as those already stated when $X$ is a manifold: \smallskip \noindent{$(9.10)$}~~$\cS$ is locally finitely generated over $\cO_X\,$; \hfill\break \noindent{$(9.10')$}~for any open set $U\subset X$ and any sections $G_1,\ldots,G_q\in\cS(U)$, the\break \phantom{$(9.10')$~}relation sheaf $\cR(G_1,\ldots,G_q)\subset \cO_{X\restriction U}^{\oplus q}$ is locally finitely generated. \smallskip \noindent Locally, we have $\cO_{X\restriction A_\lambda}=\cO_{\Omega_\lambda}/\cJ_\lambda$, so if $i_\lambda:A_\lambda\to\Omega_\lambda$ is the injection, the direct image $\cS_\lambda=(i_\lambda)_\star(\cS_{\restriction A_\lambda})$ is a module over $\cO_{\Omega_\lambda}$ such that $\cJ_\lambda.\cS_\lambda=0$. It is clear that $\cS_{\restriction\Omega_\lambda}$ is coherent if and only if $\cS_\lambda$ is coherent as a module over $\cO_{\Omega_\lambda}$. It follows immediately that the Oka theorem and its consequences 3.16--20 are still valid over analytic schemes. \titlec{\S 9.5.}{Subschemes} Let $X$ be an analytic scheme and $\cG$ a coherent sheaf of ideals in $\cO_X$. The image of $\cG$ in $\cO_{X,\red}$ is a coherent sheaf of ideals, and its zero set $Y$ is clearly an analytic subset of $X_\red$. We can make $Y$ into a scheme by introducing the structure sheaf $$\cO_Y=(\cO_X/\cG)_{\restriction Y},\leqno(9.11)$$ and we have a scheme morphism $F:(Y,\cO_Y)\longrightarrow(X,\cO_X)$ such that $f$ is the inclusion and $F^\star:f^{-1}\cO_X\longrightarrow\cO_Y$ the obvious map of $\cO_{X\restriction Y}$ onto its quotient $\cO_Y$. The scheme $(Y,\cO_Y)$ will be denoted $V(\cG)$. When the analytic set $Y$ is given, the structure sheaf of $V(\cG)$ depends of course on the choice of the equations of $Y$ in the ideal $\cG\,$; in general $\cO_Y$ has nilpotent elements. \titlec{\S 9.6.}{Inverse Images of Coherent Sheaves} Let $F:(X,\cO_X)\longrightarrow(Y,\cO_Y)$ be a morphism of analytic schemes and $\cS$ a coherent sheaf over~$Y$. The sheaf theoretic inverse image $f^{-1}\cS$, whose stalks are $(f^{-1}\cS)_x=\cS_{f(x)}$, is a sheaf of modules over $f^{-1}\cO_Y$. We define the {\it analytic inverse image} $F^\star\cS$ by $$F^\star\cS=\cO_X\otimes_{f^{-1}\cO_Y}f^{-1}\cS,~~~~ (F^\star\cS)_x=\cO_{X,x}\otimes_{\cO_{Y,f(x)}}\cS_{f(x)}.\leqno(9.12)$$ Here the tensor product is taken with respect to the comorphism $F^\star:f^{-1}\cO_Y\to\cO_X$, which yields a ring morphism $\cO_{Y,f(x)}\to\cO_{X,x}$. If $\cS$ is given over $U\subset Y$ by a local presentation $$\cO_{Y\restriction U}^{\oplus p}\buildo A\over\longrightarrow\cO_{Y\restriction U}^{\oplus q}\longrightarrow \cS_{\restriction U}\longrightarrow 0$$ where $A$ is a $(q\times p)$-matrix with coefficients in $\cO_Y(U)$, our definition shows that $F^\star\cS$ is a coherent sheaf over $\cO_X$, given over $f^{-1}(U)$ by the local presentation $$\cO_{X\restriction f^{-1}(U)}^{\oplus p}\buildo{F^\star A}\over{\larex 24 } \cO_{X\restriction f^{-1}(U)}^{\oplus q}\longrightarrow F^\star\cS_{\restriction f^{-1}(U)} \longrightarrow 0.\leqno(9.13)$$ \titlec{\S 9.7.}{Products of Analytic Schemes} Let $(X,\cO_X)$ and $(Y,\cO_Y)$ be analytic schemes, and let $(A_\lambda,\cO_{\Omega_\lambda}/\cJ_\lambda)$,\break $(B_\mu,\cO_{\Omega'_\mu}/\cJ'_\mu)$ be local models of $X$, $Y$, respectively. The {\it product scheme} $(X\times Y,\cO_{X\times Y})$ is obtained by gluing the open patches $$\Big(A_\lambda\times B_\mu~,~\cO_{\Omega_\lambda\times\Omega'_\mu}\big/ \big({\rm pr}_1^{-1}\cJ_\lambda+{\rm pr}_2^{-1}\cJ'_\mu\big) \cO_{\Omega_\lambda\times\Omega'_\mu}\Big).\leqno(9.14)$$ In other words, if $A_\lambda$, $B_\mu$ are the subschemes of $\Omega_\lambda$, $\Omega'_\mu$ defined by the equations $g_{\lambda,j}(x)=0$, $g'_{\mu,k}(y)=0$, where $(g_{\lambda,j})$ and $(g'_{\mu,k})$ are generators of $\cJ_\lambda$ and $\cJ'_\mu$ respectively, then $A_\lambda\times B_\mu$ is equipped with the structure sheaf $\cO_{\Omega_\lambda\times\Omega'_\mu}\big/\big( g_{\lambda,j}(x),g'_{\mu,k}(y)\big)$. Now, let $\cS$ be a coherent sheaf over $\cO_X$ and let $\cS'$ be a coherent sheaf over $\cO_Y$. The (analytic) {\it external tensor product} $\cS\stimes\cS'$ is defined to be $$\cS\stimes\cS'={\rm pr}_1^\star\cS\otimes_{\cO_{X\times Y}} {\rm pr}_2^\star\cS'.\leqno(9.15)$$ If we go back to the definition of the inverse image, we see that the stalks of $\cS\stimes\cS'$ are given by $$(\cS\stimes\cS')_{(x,y)}=\cO_{X\times Y,(x,y)}\otimes_ {\cO_{X,x}\otimes\cO_{Y,y}}(\cS_x\otimes_\bC\cS'_y)~,\leqno(9.15')$$ in particular $(\cS\stimes\cS')_{(x,y)}$ {\it does not coincide} with the sheaf theoretic tensor product $\cS_x\otimes\cS'_y$ which is merely a module over $\cO_{X,x}\otimes\cO_{Y,y}$. If $\cS$ and $\cS'$ are given by local presentations $$\cO_{X\restriction U}^{\oplus p}\buildo A\over\longrightarrow\cO_{X\restriction U}^{\oplus q}\longrightarrow \cS_{\restriction U}\longrightarrow 0,~~~~ \cO_{Y\restriction U'}^{p'}\buildo B\over\longrightarrow\cO_{Y\restriction U'}^{q'} \longrightarrow\cS'_{\restriction U'}\longrightarrow 0,$$ then $\cS\stimes\cS'$ is the coherent sheaf given by $$\cO_{X\times Y\restriction U\times U'}^{pq'\oplus qp'} \buildo{(A(x)\otimes\Id,\Id\otimes B(y))}\over{\larex 84 } \cO_{X\times Y\restriction U\times U'}^{qq'}\longrightarrow (\cS\stimes\cS')_{\restriction U\times U'}\longrightarrow 0.$$ \titlec{\S 9.8.}{Zariski Embedding Dimension} If $x$ is a point of an analytic scheme $(X,\cO_X)$, the {\it Zariski embedding dimension} of the germ $(X,x)$ is the smallest integer $N$ such that $(X,x)$ can be embedded in $\bC^N$, i.e.\ such that there exists a patch of $X$ near $x$ isomorphic to $(A,\cO_\Omega/\cJ)$ where $\Omega$ is an open subset of $\bC^N$. This dimension is denoted $$\hbox{\rm embdim}(X,x)=~\hbox{\rm smallest~such}~~N.\leqno(9.16)$$ Consider the maximal ideal $\gm_{X,x}\subset\cO_{X,x}$ of functions which vanish at point~$x$. If $(X,x)$ is embedded in $(\Omega,x)=(\bC^N,0)$, then $\gm_{X,x}/\gm_{X,x}^2$ is generated by $z_1,\ldots,z_N$, so $d=\dim\gm_{X,x}/\gm_{X,x}^2\le N$. Let $s_1,\ldots,s_d$ be germs in $\gm_{\Omega,x}$ which yield a basis of $\gm_{X,x}/\gm_{X,x}^2\simeq\gm_{\Omega,x}/(\gm^2_{\Omega,x}+\cJ_x)$. We can write $$z_j=\sum_{1\le k\le d}c_{jk}s_k+u_j+f_j,~~~c_{jk}\in\bC,~~u_j\in \gm^2_{\Omega,x},~~f_j\in\cJ_x,~~1\le j\le n.$$ Then we find $dz_j=\sum c_{jk}\,ds_k(x)+df_j(x)$, so that the rank of the system of diffe\-rentials $\big(df_j(x)\big)$ is at least $N-d$. Assume for example that $df_1(x),\ldots,df_{N-d}(x)$ are linearly independant$\,$. By the implicit function theorem, the equations $f_1=\ldots=f_{N-d}=0$ define a germ of smooth subvariety $(Z,x)\subset(\Omega,x)$ of dimension $d$ which contains $(X,x)$. We have $\cO_Z=\cO_\Omega/(f_1,\ldots,f_{N-d})$ in a neighborhood of $x$, thus $$\cO_X=\cO_\Omega/\cJ\simeq\cO_Z/\cJ'~~~~\hbox{\rm where}~~ \cJ'=\cJ/(f_1,\ldots,f_{N-d}).$$ This shows that $(X,x)$ can be imbedded in $\bC^d$, and we get $$\hbox{\rm embdim}(X,x)=\dim\gm_{X,x}/\gm_{X,x}^2.\leqno(9.17)$$ \begstat{(9.18) Remark} \rm For a given dimension $n=\dim(X,x)$, the embedding dimension $d$ can be arbitrarily large. Consider for example the curve $\Gamma\subset\bC^N$ parametrized by $\bC\ni t\longmapsto(t^N,t^{N+1},\ldots, t^{2N-1})$. Then $\cO_{\Gamma,0}$ is the ring of convergent series in $\bC\{t\}$ which have no terms $t,t^2,\ldots,t^{N-1}$, and $\gm_{\Gamma,0}/\gm_{\Gamma,0}^2$ admits precisely $z_1=\smash{t^N} ,\ldots,z_N=\smash{t^{2N-1}}$ as a basis. Therefore $n=1$ but $d=N$ can be as large as we want. \endstat \titleb{\S 10.}{Bimeromorphic maps, Modifications and Blow-ups} It is a very frequent situation in analytic or algebraic geometry that two complex spaces have isomorphic dense open subsets but are nevertheless different along some analytic subset. These ideas are made precise by the notions of modification and bimeromorphic map. This will also lead us to generalize the notion of meromorphic function to maps between analytic schemes. If $(X,\cO_X)$ is an analytic scheme, $\cM_X$ denotes the sheaf of meromorphic functions on~$X$, defined at the beginning of \S~6.2. \begstat{(10.1) Definition} Let $(X,\cO_X)$, $(Y,\cO_Y)$ be analytic schemes. An analytic morphism $F:X\to Y$ is said to be a modification if $F$ is proper and if there exists a nowhere dense closed analytic subset $B\subset Y$ such that the restriction $F:X\ssm F^{-1}(B)\to Y\ssm B$ is an isomorphism. \endstat \begstat{(10.2) Definition} If $F:X\to Y$ is a modification, then the comorphism $F^\star:f^\star\cO_Y\to\cO_X$ induces an isomorphism $F^\star:f^\star\cM_Y\to\cM_X$ for the sheaves of meromorphic functions on $X$ and~$Y$. \endstat \begproof{} Let $v=g/h$ be a section of $\cM_Y$ on a small open set $\Omega$ where $u$ is actually given as a quotient of functions $g,h\in\cO_Y(\Omega)$. Then $F^\star u=(g\circ F)/(h\circ F)$ is a section of $\cM_X$ on $F^{-1}(\Omega)$, for $h\circ F$ cannot vanish identically on any open subset $W$ of~$F^{-1}(\Omega)$ (otherwise $h$ would vanish on the open subset $F(W\ssm F^{-1}(B))$ of $\Omega\ssm B$). Thus the extension of the comorphism to sheaves of meromorphic functions is well defined. Our claim is that this is an isomorphism. The injectivity of $F^\star$ is clear: $F^\star u=0$ implies $g\circ F=0$, which implies $g=0$ on $\Omega\ssm B$ and thus $g=0$ on $\Omega$ because $B$ is nowhere dense. In order to prove surjectivity, we need only show that every section $u\in\cO_X(F^{-1}(\Omega))$ is in the image of $\cM_Y(\Omega)$ by~$F^\star$. For this, we may shrink $\Omega$ into a relatively compact subset $\Omega'\compact\Omega$ and thus assume that $u$ is bounded (here we use the properness of $F$ through the fact that $F^{-1}(\Omega')$ is relatively compact in $F^{-1}(\Omega))$. Then $v=u\circ F^{-1}$ defines a bounded holomorphic function on~$\Omega\ssm B$. By Th.~7.2, it follows that $v$ is weakly holomorphic for the reduced structure of~$Y$. Our claim now follows from the following Lemma.\qed \endproof \begstat{(10.3) Lemma} If $(X,\cO_X)$ is an analytic scheme, then every holomorphic function $v$ in the complement of a nowhere dense analytic subset $B\subset Y$ which is weakly holomorphic on~$X_\red$ is meromorphic on~$X$. \endstat \begproof{} It is enough to argue with the germ of $v$ at any point $x\in Y$, and thus we may suppose that $(Y,\cO_Y)=(A,\cO_\Omega/\cI)$ is embedded in~$\bC^N$. Because $v$ is weakly holomorphic, we can write $v=g/h$ in $Y_\red$, for some germs of holomorphic functions $g,h$. Let $\wt g$ and $\wt h$ be extensions of $g$, $h$ to $\cO_{\Omega,x}$. Then there is a neighborhood $U$ of $x$ such that $\wt g-v\wt h$ is a nilpotent section of $cO_\Omega(U\ssm B)$ which is in $\cI$ on \endproof \begstat{(10.4) Definition} A meromorphic map $F:X\merto Y$ is a scheme morphism $F:X\ssm A\to Y$ defined in the complement of a nowhere dense analytic subset $A\subset X$, such that the closure of the graph of $F$ in $X\times Y$ is an analytic subset $($for the reduced complex space structure of $X\times Y)$. \endstat \titleb{\S 11.}{Exercises}\begpet \titled{11.1.} Let $\cA$ be a sheaf on a topological space~$X$. If the sheaf space $\wt\cA$ is Hausdorff, show that $\cA$ satisfies the following {\it unique continuation principle}: any two sections $s,s'\in\cA(U)$ on a connected open set $U$ which coincide on some non empty open subset~$V\subset U$ must coincide identically on~$U$. Show that the converse holds if $X$ is Hausdorff and locally connected. \titled{11.2.} Let $\cA$ be a sheaf of abelian groups on~$X$ and let $s\in\cA(X)$. The support of~$s$, denoted $\Supp\,s$, is defined to be $\{x\in X\,;\,s(x)\ne 0\}$. Show that $\Supp\,s$ is a closed subset of~$X$. The support of $\cA$ is defined to be $\Supp\,\cA=\{x\in X\,;\,\cA_x\ne 0\}$. Show that $\Supp\,\cA$ is not necessarily closed: if $\Omega$ is an open set in $X$, consider the sheaf $\cA$ such that $\cA(U)$ is the set of continuous functions $f\in\cC(U)$ which vanish on a neighborhood of $U\cap(X\ssm\Omega)$. \titled{11.3.} Let $\cA$ be a sheaf of rings on a topological space~$X$ and let $\cF$, $\cG$ be sheaves of $\cA$-modules. We define a presheaf $\cH=\cHom_\cA(\cF,\cG)$ such that $\cH(U)$ is the module of all sheaf-homomorphisms $\cF_{\restriction U}\to\cG_{\restriction U}$ which are $\cA$-linear.\smallskip \item{a)} Show that $\cHom_\cA(\cF,\cG)$ is a sheaf and that there is a canonical homomorphism $\varphi_x:\cHom_\cA(\cF,\cG)_x\longrightarrow \hom_{\cA_x}(\cF_x,\cG_x)$ for every $x\in X$. \smallskip \item{b)} If $\cF$ is locally finitely generated, then $\varphi_x$ is injective, and if $\cF$ has local finite presentations as in (3.12), then $\varphi_x$ is bijective. \smallskip \item{c)} Suppose that $\cA$ is a coherent sheaf of rings and that $\cF$, $\cG$ are coherent modules over~$\cA$. Then $\cHom_\cA(\cF,\cG)$ is a coherent $\cA$-module.\newline {\it Hint}\/: observe that the result is true if $\cF=\cA^{\oplus p}$ and use a local presentation of $\cF$ to get the conclusion. \titled{11.4.} Let $f:X\to Y$ be a continuous map of topological spaces. Given sheaves of abelian groups $\cA$ on $X$ and $\cB$ on~$Y$, show that there is a natural isomorphism $$\hom_X(f^{-1}\cB,\cA)=\hom_Y(\cB,f_\star\cA).$$ {\it Hint}\/: use the natural morphisms (2.17). \titled{11.5.} Show that the sheaf of polynomials over $\bC^n$ is a coherent sheaf of rings (with either the ordinary topology or the Zariski topology on $\bC^n$). Extend this result to the case of regular algebraic functions on an algebraic variety.\newline {\it Hint}\/: check that the proof of the Oka theorem still applies. \titled{11.6.} Let $P$ be a non zero polynomial on~$\bC^n$. If $P$ is irreducible in $\bC[z_1,\ldots,z_n]$, show that the hypersurface $H=\smash{P^{-1}(0)}$ is globally irreducible as an analytic set. In general, show that the irreducible components of $H$ are in a one-to-one correspondence with the irreducible factors of~$P$.\newline {\it Hint}\/: for the first part, take coordinates such that $P(0,\ldots,0,z_n)$ has degree equal to $P\,$; if $H$ splits in two components $H_1$, $H_2$, then $P$ can be written as a product $P_1P_2$ where the roots of $P_j(z',z_n)$ correspond to points in~$H_j$. \titled{11.7.} Prove the following facts:\smallskip \item{a)} For every algebraic variety $A$ of pure dimension $p$ in~$\bC^n$, there are coordinates $z'=(z_1,\ldots,z_p)$, $z''=(z_{p+1},\ldots,z_n)$ such that $\pi:A\to\smash{\bC^p}$, $z\mapsto z''$ is proper with finite fibers, and such that $A$ is entirely contained in a cone $$|z''|\le C(|z'|+1).$$ {\it Hint}\/: imitate the proof of Cor.~4.11. \smallskip \item{b)} Conversely if an analytic set $A$ of pure dimension $p$ in $\bC^n$ is contained in a cone $|z''|\le C(|z'|+1)$, then $A$ is algebraic.\newline {\it Hint}\/: first apply $(5.9)$ to conclude that the projection $\pi: A\to\bC^p$ is finite. Then repeat the arguments used in the final part of the proof of Th.~4.19. \item{c)} Deduce from a), b) that an algebraic set in $\bC^n$ is irreducible if and only if it is irreducible as an analytic set. \titled{11.8.} Let $\Gamma:f(x,y)=0$ be a germ of analytic curve in $\bC^2$ through~$(0,0)$ and let $(\Gamma_j,0)$ be the irreducible components of $(\Gamma,0)$. Suppose that $f(0,y)\not\equiv 0$. Show that the roots $y$ of $f(x,y)=0$ corresponding to points of $\Gamma$ near $0$ are given by {\it Puiseux expansions} of the form $y=g_j(x^{1/q_j})$, where $g_j\in\cO_{\bC,0}$ and where $q_j$ is the sheet number of the projection $\Gamma_j\to\bC$, $(x,y)\mapsto x$.\newline {\it Hint}\/: special case of the parametrization obtained in~(4.27). \titled{11.9.} The goal of this exercise is to prove the existence and the analyticity of the {\it tangent cone} to an arbitrary analytic germ $(A,0)$ in~$\bC^n$. Suppose that $A$ is defined by holomorphic equations $f_1=\ldots=f_N=0$ in a ball $\Omega=B(0,r)$. Then the (set theoretic) tangent cone to $A$ at $0$ is the set $C(A,0)$ of all limits of sequences $t_\nu^{-1}z_\nu$ with $z_\nu\in A$ and $\bC^\star\ni t_\nu$ converging to~$0$.\smallskip \item{a)} Let $E$ be the set of points $(z,t)\in\Omega\times\bC^\star$ such that $z\in t^{-1}A$. Show that the closure $\smash{\ol E}$ in $\Omega\times\bC$ is analytic.\newline {\it Hint}\/: observe that $E=A\ssm(\Omega\times\{0\})$ where $A=\{f_j(tz)=0\}$ and apply Cor.~5.4.\smallskip \item{b)} Show that $C(A,0)$ is a conic set and that $\ol E\cap(\Omega\times \{0\})=C(A,0)\times\{0\}$ and conclude. Infer from this that $C(A,0)$ is an algebraic subset of~$\bC^n$. \titled{11.10.} Give a new proof of Theorem~5.5 based on the coherence of ideal sheaves and on the strong noetherian property. \titled{11.11.} Let $X$ be an analytic space and let $A$, $B$ be analytic subsets of pure dimensions. Show that $\codim_X(A\cap B)\le\codim_X A+\codim_X B$ if $A$ or $B$ is a local complete intersection, but that the equality does not necessarily hold in general.\newline {\it Hint}\/: see Remark~(6.5). \titled{11.12.} Let $\Gamma$ be the curve in $\bC^3$ parametrized by $\bC\ni t\longmapsto(x,y,z)=(t^3,t^4,t^5)$. Show that the ideal sheaf $\cI_\Gamma$ is generated by the polynomials $xz-y^2$, $x^3-yz$ and $x^2y-z^2$, and that the germ $(\Gamma,0)$ is not a (sheaf theoretic) complete intersection.\newline {\it Hint}\/: $\Gamma$ is smooth except at the origin. Let $f(x,y,z)= \sum a_{\alpha\beta\gamma}x^\alpha y^\beta z^\gamma$ be a convergent power series near~$0$. Show that $f\in\cI_{\Gamma,0}$ if and only if all weighted homogeneous components $f_k=\sum_{3\alpha+4\beta+5\gamma=k} a_{\alpha\beta\gamma}x^\alpha y^\beta z^\gamma$ are in~$\cI_{\Gamma,0}$. By means of suitable substitutions, reduce the proof to the case when $f=f_k$ is homogeneous with all non zero monomials satisfying $\alpha\le 2$, $\beta\le 1$, $\gamma\le 1$; then check that there is at most one such monomial in each weighted degree${}\le 15$ the product of a power of $x$ by a homogeneous polynomial of weighted degree~$\le 8$. \endpet \titlea{Chapter III}{\newline Positive Currents and Lelong Numbers} \begpet In 1957, P. Lelong introduced natural positivity concepts for currents of pure bidimension $(p,p)$ on complex manifolds. With every analytic subset is associated a current of integration over its set of regular points and all such currents are positive and closed. The important closedness property is proved here via the Skoda-El Mir extension theorem. Positive currents have become an important tool for the study of global geometric problems as well as for questions related to local algebra and intersection theory. We develope here a differential geometric approach to intersection theory through a detailed study of wedge products of closed positive currents (Monge-Amp\`ere operators). The Lelong-Poincar\'e equation and the Jensen-Lelong formula are basic in this context, providing a useful tool for studying the location and multiplicities of zeroes of entire functions on $\bC^n$ or on a manifold, in relation with the growth at infinity. Lelong numbers of closed positive currents are then introduced; these numbers can be seen as a generalization to currents of the notion of multiplicity of a germ of analytic set at a singular point. We prove various properties of Lelong numbers (e.g. comparison theorems, semi-continuity theorem of Siu, transformation under holomorphic maps). As an application to Number Theory, we prove a general Schwarz lemma in $\bC^n$ and derive from it Bombieri's theorem on algebraic values of meromorphic maps and the famous theorems of Gelfond-Schneider and Baker on the transcendence of exponentials and logarithms of algebraic numbers. \endpet \titleb{1.}{Basic Concepts of Positivity} \titlec{1.A.}{Positive and Strongly Positive Forms} Let $V$ be a complex vector space of dimension $n$ and $(z_1,\ldots,z_n)$ coordinates on $V$. We denote by $(\partial/\partial z_1,\ldots, \partial/\partial z_n)$ the corresponding basis of $V$, by $(dz_1,\ldots,dz_n)$ its dual basis in $V^\star$ and consider the exterior algebra $$\Lambda V^\star_\bC=\bigoplus\Lambda^{p,q}V^\star,~~~~ \Lambda^{p,q}V^\star=\Lambda^pV^\star\otimes\Lambda^q\ol{V^\star}.$$ We are of course especially interested in the case where $V=T_xX$ is the tangent space to a complex manifold $X$, but we want to emphasize here that our considerations only involve linear algebra. Let us first observe that $V$ has a canonical orientation, given by the $(n,n)$-form $$\tau(z)=\ii dz_1\wedge d\ol z_1\wedge\ldots\wedge\ii dz_n\wedge d\ol z_n= 2^n\,dx_1\wedge dy_1\wedge\ldots\wedge dx_n\wedge dy_n$$ where $z_j=x_j+iy_j$. In fact, if $(w_1,\ldots,w_n)$ are other coordinates, we find $$\eqalign{ &dw_1\wedge\ldots\wedge dw_n=\det(\partial w_j/\partial z_k)\,dz_1\wedge\ldots \wedge dz_n,\cr &\tau(w)=\big|\det(\partial w_j/\partial z_k)\big|^2\,\tau(z).\cr}$$ In particular, a complex manifold always has a canonical orientation. More generally, natural positivity concepts for $(p,p)$-forms can be defined. \begstat{(1.1) Definition} A $(p,p)$-form $u\in\Lambda^{p,p}V^\star$ is said to be positive if for all $\alpha_j\in V^\star$, $1\le j\le q=n-p$, then $$u\wedge \ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge \ii\alpha_q\wedge \ol\alpha_q$$ is a positive $(n,n)$-form. A $(q,q)$-form $v\in\Lambda^{q,q}V^\star$ is said to be strongly positive if $v$ is a convex combination $$v=\sum\gamma_s~\ii\alpha_{s,1}\wedge\ol\alpha_{s,1} \wedge\ldots\wedge \ii\alpha_{s,q}\wedge\ol\alpha_{s,q}$$ where $\alpha_{s,j}\in V^\star$ and $\gamma_s\ge 0$. \endstat \begstat{(1.2) Example} \rm Since $\ii^p(-1)^{p(p-1)/2}=\ii^{p^2}$, we have the commutation rules $$\eqalign{ &\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge \ii\alpha_p\wedge\ol\alpha_p= i^{p^2}\alpha\wedge\ol\alpha,~~~~\forall\alpha=\alpha_1\wedge\ldots\wedge\alpha_p \in\Lambda^{p,0}V^\star,\cr &i^{p^2}\beta\wedge\ol \beta\wedge i^{m^2}\gamma\wedge\ol \gamma= i^{(p+m)^2}\beta\wedge \gamma\wedge\ol{\beta\wedge \gamma},~~~~ \forall \beta\in\Lambda^{p,0}V^\star,~\forall \gamma\in\Lambda^{m,0}V^\star.\cr}$$ Take $m=q$ to be the complementary degree of $p$. Then $\beta\wedge \gamma=\lambda dz_1\wedge\ldots\wedge dz_n$ for some $\lambda\in\bC$ and $i^{n^2}\beta\wedge\gamma\wedge\ol{\beta\wedge \gamma}= |\lambda|^2\tau(z)$. If we set $\gamma=\alpha_1\wedge\ldots\wedge\alpha_q$, we find that $i^{p^2}\beta\wedge\ol \beta$ is a positive $(p,p)$-form for every $\beta\in\Lambda^{p,0}V^\star\,$; in particular, strongly positive forms are positive.\qed \endstat The sets of positive and strongly positive forms are closed convex cones, i.e. closed and stable under convex combinations. By definition, the positive cone is dual to the strongly positive cone via the pairing $$\cmalign{ &\Lambda^{p,p}V^\star\times&\Lambda^{q,q}V^\star&\longrightarrow\bC\cr &\hfill(u,&v)&\longmapsto u\wedge v/\tau,\cr}\leqno(1.3)$$ that is, $u\in\Lambda^{p,p}V^\star$ is positive if and only if $u\wedge v\ge 0$ for all strongly positive forms $v\in\Lambda^{q,q}V^\star$. Since the bidual of an arbitrary convex cone $\Gamma$ is equal to its closure $\ol\Gamma$, we also obtain that $v$ is strongly positive if and only if $v\wedge u=u\wedge v$ is $\ge 0$ for all positive forms $u$. Later on, we will need the following elementary lemma. \begstat{(1.4) Lemma} Let $(z_1,\ldots,z_n)$ be arbitrary coordinates on $V$. Then $\Lambda^{p,p}V^\star$ admits a basis consisting of strongly positive forms $$\beta_s=\ii\beta_{s,1}\wedge\ol\beta_{s,1}\wedge\ldots \wedge\ii\beta_{s,p}\wedge\ol\beta_{s,p},~~~~1\le s\le{n\choose p}^2$$ where each $\beta_{s,l}$ is of the type $dz_j\pm dz_k$ or $dz_j\pm \ii dz_k$, $1\le j,k\le n$. \endstat \begproof{} Since one can always extract a basis from a set of generators, it is sufficient to see that the family of forms of the above type generates $\Lambda^{p,p}V^\star$. This follows from the identities $$\cmalign{ &4dz_j\wedge d\ol z_k=~~~&(dz_j+dz_k)&\wedge&(\ol{dz_j+dz_k})&- &(dz_j-dz_k)&\wedge&(\ol{dz_j-dz_k})\cr &\hfill+\ii&(dz_j+\ii dz_k)&\wedge&(\ol{dz_j+\ii dz_k})&-\ii&(dz_j-\ii dz_k)&\wedge&(\ol{dz_j-\ii dz_k}),\cr}$$ $$dz_{j_1}\wedge\ldots\wedge dz_{j_p}\wedge d\ol z_{k_1}\wedge \ldots\wedge d\ol z_{k_p}=\pm\bigwedge_{1\le s\le p}dz_{j_s}\wedge d\ol z_{k_s}.\eqno{\square}$$ \endproof \begstat{(1.5) Corollary} All positive forms $u$ are real, i.e. satisfy $\ol u=u$. In terms of coordinates, if $u=i^{p^2}\sum_{|I|=|J|=p}u_{I,J}\, dz_I\wedge d\ol z_J$, then the coefficients satisfy the hermitian symmetry relation $\ol{u_{I,J}}=u_{J,\,I}$. \endstat \begproof{} Clearly, every strongly positive $(q,q)$-form is real. By Lemma 1.4, these forms generate over $\bR$ the real elements of $\Lambda^{q,q}V^\star$, so we conclude by duality that positive $(p,p)$-forms are also real.\qed \endproof \begstat{(1.6) Criterion} A form $u\in\Lambda^{p,p}V^\star$ is positive if and only if its restriction $u_{\restriction S}$ to every $p$-dimensional subspace $S\subset V$ is a positive volume form on $S$. \endstat \begproof{} If $S$ is an arbitrary $p$-dimensional subspace of $V$ we can find coordinates $(z_1,\ldots,z_n)$ on $V$ such that $S=\{z_{p+1}=\ldots=z_n=0\}$. Then $$u_{\restriction S}=\lambda_S\,\ii dz_1\wedge d\ol z_1\wedge\ldots\wedge \ii dz_p\wedge d\ol z_p$$ where $\lambda_S$ is given by $$u\wedge\ii dz_{p+1}\wedge d\ol z_{p+1}\wedge\ldots\wedge \ii dz_n\wedge d\ol z_n=\lambda_S\,\tau(z).$$ If $u$ is positive then $\lambda_S\ge 0$ so $u_{\restriction S}$ is positive for every $S$. The converse is true because the $(n-p,n-p)$-forms $\bigwedge_{j>p}\ii dz_j\wedge d\ol z_j$ generate all strongly positive forms when $S$ runs over all $p$-dimensional subspaces.\qed \endproof \begstat{(1.7) Corollary} A form $u=\ii\sum_{j,k}u_{jk}\,dz_j\wedge d\ol z_k$ of bidegree $(1,1)$ is positive if and only if $\xi\mapsto\sum u_{jk}\xi_j\ol\xi_k$ is a semi-positive hermitian form on $\bC^n$. \endstat \begproof{} If $S$ is the complex line generated by $\xi$ and $t\mapsto t\xi$ the parametrization of $S$, then $u_{\restriction S}= \big(\sum u_{jk}\xi_j\ol\xi_k\big)\,\ii dt\wedge d\ol t$.\qed \endproof Observe that there is a canonical one-to-one correspondence between hermitian forms and real $(1,1)$-forms on $V$. The correspondence is given by $$h=\sum_{1\le j,k\le n}h_{jk}(z)\,dz_j\otimes d\ol z_k\longmapsto u=\ii\sum_{1\le j,k\le n}h_{jk}(z)\,dz_j\wedge d\ol z_k\leqno(1.8)$$ and does not depend on the choice of coordinates: indeed, as $\ol h_{jk}= h_{kj}$, one finds immediately $$u(\xi,\eta)=\ii\sum h_{jk}(z)(\xi_j\ol\eta_k-\eta_j\ol\xi_k)= -2\Im h(\xi,\eta),~~~~\forall\xi,\eta\in TX.$$ Moreover, $h$ is $\ge 0$ as a hermitian form if and only if $u\ge 0$ as a $(1,1)$-form. A diagonalization of $h$ shows that every positive $(1,1)$-form $u\in\Lambda^{1,1}V^\star$ can be written $$u=\sum_{1\le j\le r}\ii\gamma_j\wedge\ol\gamma_j~,~~~~\gamma\in V^\star, ~~r={\rm rank~of~~}u,$$ in particular, every positive $(1,1)$-form is strongly positive. By duality, this is also true for $(n-1,n-1)$-forms. \begstat{(1.9) Corollary} The notions of positive and strongly positive $(p,p)$-forms coincide for $p=0,1,n-1,n$.\qed \endstat \begstat{(1.10) Remark} \rm It is not difficult to see, however, that positivity and strong positivity differ in all bidegrees $(p,p)$ such that $2\le p\le n-2$. Indeed, a positive form $i^{p^2}\beta\wedge\ol\beta$ with $\beta\in\Lambda^{p,0}V^\star$ is strongly positive if and only if $\beta$ is decomposable as a product $\beta_1\wedge\ldots\wedge\beta_p$. To see this, suppose that $$i^{p^2}\beta\wedge\ol\beta=\sum_{1\le j\le N}i^{p^2}\gamma_j\wedge\ol\gamma_j$$ where all $\gamma_j\in\Lambda^{p,0}V^\star$ are decomposable. Take $N$ minimal. The equality can be also written as an equality of hermitian forms $|\beta|^2=\sum|\gamma_j|^2$ if $\beta,\gamma_j$ are seen as linear forms on $\Lambda^pV$. The hermitian form $|\beta|^2$ has rank one, so we must have $N=1$ and $\beta= \lambda\gamma_j$, as desired. Note that there are many non decomposable $p$-forms in all degrees $p$ such that $2\le p\le n-2$, e.g. $(dz_1\wedge dz_2+dz_3\wedge dz_4)\wedge\ldots\wedge dz_{p+2}\,$: if a $p$-form is decomposable, the vector space of its contractions by elements of $\bigwedge^{p-1}V$ is a $p$-dimensional subspace of $V^\star$; in the above example the dimension is $p+2$. \endstat \begstat{(1.11) Proposition} If $u_1,\ldots,u_s$ are positive forms, all of them strongly positive $($resp. all except perhaps one$)$, then $u_1\wedge\ldots\wedge u_s$ is strongly positive $($resp. positive$)$. \endstat \begproof{} Immediate consequence of Def.~1.1. Observe however that the wedge product of two positive forms is not positive in general (otherwise we would infer that positivity coincides with strong positivity).\qed \endproof \begstat{(1.12) Proposition} If $\Phi:W\longrightarrow V$ is a complex linear map and $u\in\Lambda^{p,p}V^\star$ is $($strongly$)$ positive, then $\Phi^\star u\in\Lambda^{p,p}W^\star$ is $($strongly$)$ positive. \endstat \begproof{} This is clear for strong positivity, since $$\Phi^\star(\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge \ii\alpha_p\wedge\ol\alpha_p)=\ii\beta_1\wedge\ol\beta_1\wedge\ldots\wedge \ii\beta_p\wedge \ol\beta_p$$ with $\beta_j=\Phi^\star\alpha_j\in W^\star$, for all $\alpha_j\in V^\star$. For $u$ positive, we may apply Criterion~1.6: if $S$ is a $p$-dimensional subspace of $W$, then $u_{\restriction\Phi(S)}$ and $(\Phi^\star u)_{\restriction S}=(\Phi_\restriction S)^\star u_{\restriction\Phi(S)}$ are positive when $\Phi_{\restriction S}: S\longrightarrow\Phi(S)$ is an isomorphism; otherwise we get $(\Phi^\star u)_{\restriction S}=0$.\qed \endproof \titlec{1.B.}{Positive Currents} The duality between the positive and strongly positive cones of forms can be used to define corresponding positivity notions for currents. \begstat{(1.13) Definition} A current $T\in\cD'_{p,p}(X)$ is said to be positive $($resp. strongly positive$)$ if $\langle T,u\rangle\ge 0$ for all test forms $u\in\cD_{p,p}(X)$ that are strongly positive $($resp. positive$)$ at each point. The set of positive $($resp. strongly positive$)$ currents of bidimension $(p,p)$ will be denoted $$\cD^{\prime+}_{p,p}(X),~~~~\hbox{\rm resp.}~~\cD^{\prime\oplus}_{p,p}(X).$$ \endstat It is clear that (strong) positivity is a local property and that the sets $\cD^{\prime\oplus}_{p,p}(X)\subset\cD^{\prime+}_{p,p}(X)$ are closed convex cones with respect to the weak topology. Another way of stating Def.~1.13 is: \medskip \noindent{\it $T$ is positive $($strongly positive$)$ if and only if $T\wedge u\in\cD'_{0,0}(X)$ is a positive measure for all strongly positive $($positive$)$ forms $u\in C^\infty_{p,p}(X)$.} \medskip \noindent This is so because a distribution $S\in\cD'(X)$ such that $S(f)\ge 0$ for every non-negative function $f\in\cD(X)$ is a positive measure. \begstat{(1.14) Proposition} Every positive current $T=i^{(n-p)^2}\sum T_{I,J}\,dz_I \wedge d\ol z_J$ in $\cD^{\prime +}_{p,p}(X)$ is real and of order $0$, i.e. its coefficients $T_{I,J}$ are complex measures and satisfy $\ol{T_{I,J}}=T_{J,\,I}$ for all multi-indices $|I|=|J|=n-p$. Moreover $T_{I,I}\ge 0$, and the absolute values $|T_{I,J}|$ of the measures $T_{I,J}$ satisfy the inequality $$\lambda_I\lambda_J\,|T_{I,J}|\le 2^p\sum_M~\lambda_M^2\,T_{M,M},~~~~ I\cap J\subset M\subset I\cup J$$ where $\lambda_k\ge 0$ are arbitrary coefficients and $\lambda_I=\prod_{k\in I}\lambda_k$. \endstat \begproof{} Since positive forms are real, positive currents have to be real by duality. Let us denote by $K=\complement I$ and $L=\complement J$ the ordered complementary multi-indices of $I,J$ in $\{1,2,\ldots,n\}$. The distribution $T_{I,I}$ is a positive measure because $$T_{I,I}~\tau=T\wedge i^{p^2}dz_K\wedge d\ol z_K\ge 0.$$ On the other hand, the proof of Lemma~1.4 yields $$\eqalign{ T_{I,J}~\tau&=\pm\,T\wedge i^{p^2}dz_K\wedge d\ol z_L= \sum_{a\in(\bZ/4\bZ)^p}\varepsilon_a\,T\wedge\gamma_a ~~~~{\rm where}\cr \gamma_a&=\bigwedge_{1\le s\le p}{\ii\over 4}(dz_{k_s}+ i^{a_s}dz_{l_s})\wedge(\ol{dz_{k_s}+i^{a_s}dz_{l_s}}),~~~~ \varepsilon_a=\pm 1,\pm i.\cr}$$ Now, each $T\wedge\gamma_a$ is a positive measure, hence $T_{I,J}$ is a complex measure and $$\eqalign{ |T_{I,J}|\,\tau&\le\sum_a T\wedge\gamma_a= T\wedge\sum_a\gamma_a\cr &=T\wedge\bigwedge_{1\le s\le p}\Big(\sum_{a_s\in\bZ/4\bZ}{\ii\over 4} (dz_{k_s}+i^{a_s}dz_{l_s})\wedge(\ol{dz_{k_s}+i^{a_s}dz_{l_s}}) \Big)\cr &=T\wedge\bigwedge_{1\le s\le p}\big(\ii dz_{k_s}\wedge d\ol z_{k_s}+ \ii dz_{l_s}\wedge d\ol z_{l_s}\big).\cr}$$ The last wedge product is a sum of at most $2^p$ terms, each of which is of the type $i^{p^2}dz_M\wedge d\ol z_M$ with $|M|=p$ and $M\subset K\cup L$. Since $T\wedge i^{p^2}dz_M\wedge d\ol z_M=T_{\complement M,\complement M}\,\tau$ and $\complement M\supset\complement K\cap\complement L=I\cap J$, we find $$|T_{I,J}|\le 2^p\sum_{M\supset I\cap J}T_{M,M}.$$ Now, consider a change of coordinates $(z_1,\ldots,z_n)=\Lambda w= (\lambda_1 w_1,\ldots,\lambda_nw_n)$ with $\lambda_1,\ldots,\lambda_n>0$. In the new coordinates, the current $T$ becomes $\Lambda^\star T$ and its coefficients become $\lambda_I\lambda_J\,T_{I,J}(\Lambda w)$. Hence, the above inequality implies $$\lambda_I\lambda_J\,|T_{I,J}|\le 2^p\sum_{M\supset I\cap J} \lambda_M^2\,T_{M,M}.$$ This inequality is still true for $\lambda_k\ge 0$ by passing to the limit. The inequality of Prop.~1.14 follows when all coefficients $\lambda_k$, $k\notin I\cup J$, are replaced by $0$, so that $\lambda_M=0$ for $M\not\subset I\cup J$.\qed \endproof \begstat{(1.15) Remark} \rm If $T$ is of order $0$, we define the {\it mass measure} of $T$ by $\|T\|=\sum|T_{I,J}|$ (of course $\|T\|$ depends on the choice of coordinates). By the Radon-Nikodym theorem, we can write $T_{I,J}=f_{I,J}\|T\|$ with a Borel function $f_{I,J}$ such that $\sum|f_{I,J}|=1$. Hence $$T=\|T\|\,f,~~~~{\rm where}~~f=i^{(n-p)^2}\sum f_{I,J}\,dz_I\wedge d\ol z_J.$$ Then $T$ is (strongly) positive if and only if the form $f(x)\in \Lambda^{n-p,n-p}T^\star_xX$ is (strongly) positive at $\|T\|$-almost all points $x\in X$. Indeed, this condition is clearly sufficient. On the other hand, if $T$ is (strongly) positive and $u_j$ is a sequence of forms with constant coefficients in $\Lambda^{p,p}T^\star X$ which is dense in the set of strongly positive (positive) forms, then $T\wedge u_j=||T||\,f\wedge u_j$, so $f(x)\wedge u_j$ has to be a positive $(n,n)$-form except perhaps for $x$ in a set $N(u_j)$ of $\|T\|$-measure~$0$. By a simple density argument, we see that $f(x)$ is (strongly) positive outside the $\|T\|$-negligible set $N=\bigcup N(u_j)$. As a consequence of this proof, $T$ is positive (strongly positive) if and only if $T\wedge u$ is a positive measure for all strongly positive (positive) forms $u$ of bidegree $(p,p)$ with {\it constant coefficients} in the given coordinates $(z_1,\ldots,z_n)$. It follows that if $T$ is (strongly) positive in a coordinate patch $\Omega$, then the convolution $T\star\rho_\varepsilon$ is (strongly) positive in $\Omega_\varepsilon= \{x\in\Omega\,;\,d(x,\partial\Omega)>\varepsilon\}$.\qed \endstat \begstat{(1.16) Corollary} If $T\in\cD'_{p,p}(X)$ and $v\in C^0_{s,s}(X)$ are positive, one of them $($resp. both of them$)$ strongly positive, then the wedge product $T\wedge v$ is a positive $($resp. strongly positive$)$ current. \endstat This follows immediately from Remark~1.15 and Prop.~1.11 for forms. Similarly, Prop.~1.12 on pull-backs of positive forms easily shows that positivity properties of currents are preserved under direct or inverse images by holomorphic maps. \begstat{(1.17) Proposition} Let $\Phi:X\longrightarrow Y$ be a holomorphic map between complex analytic manifolds. \medskip \item{\rm a)} If $T\in\cD^{\prime+}_{p,p}(X)$ and $\Phi_{\restriction {\rm Supp}\,T}$ is proper, then $\Phi_\star T\in\cD^{\prime+}_{p,p}(Y)$. \medskip \item{\rm b)} If $T\in\cD^{\prime+}_{p,p}(Y)$ and if $\Phi$ is a submersion with $m$-dimensional fibers, then $\,\Phi^\star T\in\cD^{\prime+}_{p+m,p+m}(X)$. \medskip \noindent Similar properties hold for strongly positive currents.\qed \endstat \titlec{1.C.}{Basic Examples of Positive Currents} We present here two fundamental examples which will be of interest in many circumstances. \titled{(1.18) Current Associated to a Plurisubharmonic Function} Let $X$ be a complex manifold and $u\in\Psh(X)\cap L^1_\loc(X)$ a plurisubharmonic function. Then $$T=\ii d'd''u=\ii\sum_{1\le j,k\le n}{\partial^2u\over\partial z_j\partial\ol z_k}\,dz_j\wedge d\ol z_k$$ is a positive current of bidegree $(1,1)$. Moreover $T$ is closed (we always mean here $d$-closed, that is, $dT=0$). Assume conversely that $\Theta$ is a closed real $(1,1)$-current on $X$. Poincar\'e's lemma implies that every point $x_0\in X$ has a neighborhood $\Omega_0$ such that $\Theta=dS$ with $S\in\cD'_1(\Omega_0,\bR)$. Write $S=S^{1,0}+S^{0,1}$, where $S^{0,1}=\ol{S^{1,0}}$. Then $d''S=\Theta^{0,2}=0$, and the Dolbeault-Grothendieck lemma shows that $S^{0,1}=d''v$ on some neighborhood $\Omega\subset\Omega_0$, with $v\in\cD'(\Omega,\bC)$. Thus $$\eqalign{ S&=\ol{d''v}+d''v=d'\ol v+d''v,\cr \Theta&=dS=d'd''(v-\ol v)=\ii d'd''u,\cr}$$ where $u=2\Re v\in\cD'(\Omega,\bR)$. If $\Theta\in C^\infty_{1,1}(X)$, the hypoellipticity of $d''$ in bidegree $(p,0)$ shows that $d'u$ is of class $C^\infty$, so $u\in C^\infty(\Omega)$. When $\Theta$ is positive, the distribution $u$ is a plurisubharmonic function (Th.~I.3.31). We have thus proved: \begstat{(1.19) Proposition} If $\Theta\in\cD^{\prime+}_{n-1,n-1}(X)$ is a closed positive current of bidegree $(1,1)$, then for every point $x_0\in X$ there exists a neighborhood $\Omega$ of $x_0$ and $u\in\Psh(\Omega)$ such that $\Theta=\ii d'd''u$.\qed \endstat \titled{(1.20) Current of Integration on a Complex Submanifold} Let $Z\subset X$ be a closed $p$-dimensional complex submanifold with its canonical orientation and $T=[Z]$. Then $T\in\cD^{\prime\oplus}_{p,p}(X)$. Indeed, every $(r,s)$-form of total degree $r+s=2p$ has zero restriction to $Z$ unless $(r,s)=(p,p)$, therefore we have $[Z]\in\cD'_{p,p}(X)$. Now, if $u\in\cD_{p,p}(X)$ is a positive test form, then $u_{\restriction Z}$ is a positive volume form on $Z$ by Criterion~1.6, therefore $$\langle[Z],u\rangle=\int_Z u_{\restriction Z}\ge 0.$$ In this example the current $[Z]$ is also closed, because $d[Z]=\pm [\partial Z]=0$ by Stokes' theorem.\qed \titlec{1.D.}{Trace Measure and Wirtinger's Inequality} We discuss now some questions related to the concept of area on complex submanifolds. Assume that $X$ is equipped with a hermitian metric $h$, i.e. a positive definite hermitian form $h=\sum h_{jk}dz_j\otimes d\ol z_k$ of class $C^\infty$~; we denote by $\omega=\ii\sum h_{jk}dz_j\wedge d\ol z_k \in C^\infty_{1,1}(X)$ the associated positive $(1,1)$-form. \begstat{(1.21) Definition} For every $T\in\cD^{\prime+}_{p,p}(X)$, the trace measure of $T$ with respect to $\omega$ is the positive measure $$\sigma_T={1\over 2^pp!}\,T\wedge\omega^p.$$ \endstat If $(\zeta_1,\ldots,\zeta_n)$ is an orthonormal frame of $T^\star X$ with respect to $h$ on an open subset $U\subset X$, we may write $$\eqalign{ &\omega=\ii\sum_{1\le j\le n}\zeta_j\wedge\ol\zeta_j,~~~~ \omega^p=i^{p^2}p!\,\sum_{|K|=p}\zeta_K\wedge\ol\zeta_K,\cr &T=i^{(n-p)^2}\sum_{|I|=|J|=n-p}T_{I,J}\,\zeta_I\wedge\ol\zeta_J,~~~~ T_{I,J}\in\cD'(U),\cr}$$ where $\zeta_I=\zeta_{i_1}\wedge\ldots\wedge\zeta_{i_{n-p}}$. An easy computation yields $$\sigma_T=2^{-p}\Big(\sum_{|I|=n-p}T_{I,I}\Big)\,\ii\zeta_1\wedge\ol\zeta_1 \wedge\ldots\wedge \ii\zeta_n\wedge\ol\zeta_n.\leqno(1.22)$$ For $X=\bC^n$ with the standard hermitian metric $h=\sum dz_j\otimes d\ol z_j$, we get in particular $$\sigma_T=2^{-p}\Big(\sum_{|I|=n-p}T_{I,I}\Big)\, \ii dz_1\wedge d\ol z_1\wedge\ldots\wedge \ii dz_n\wedge d\ol z_n. \leqno(1.22')$$ Proposition 1.14 shows that the mass measure $||T||=\sum|T_{I,J}|$ of a positive current $T$ is always dominated by $C\sigma_T$ where $C>0$ is a constant. It follows easily that the weak topology of $\cD_p'(X)$ and of $\cD_p^{0\,\prime}(X)$ coincide on $\cD_p^{\prime+}(X)$, which is moreover a metrizable subspace: its weak topology is in fact defined by the collection of semi-norms $T\longmapsto |\langle T,f_\nu\rangle|$ where $(f_\nu)$ is an arbitrary dense sequence in $\cD_p(X)$. By the Banach-Alaoglu theorem, the unit ball in the dual of a Banach space is weakly compact, thus: \begstat{(1.23) Proposition} Let $\delta$ be a positive continuous function on $X$. Then the set of currents $T\in\cD_p^{\prime+}(X)$ such that $\int_X \delta\,T\wedge\omega^p\le 1$ is weakly compact. \endstat \begproof{} Note that our set is weakly closed, since a weak limit of positive currents is positive and $\int_X \delta\,T\wedge\omega^p= \sup\langle T,\theta\delta\omega^p\rangle$ when $\theta$ runs over all elements of $\cD(X)$ such that $0\le\theta\le 1$.\qed \endproof Now, let $Z$ be a $p$-dimensional complex analytic submanifold of $X$. We claim that $$\sigma_{[Z]}={1\over 2^pp!}[Z]\wedge\omega^p={\rm Riemannian~volume~measure ~on~~}Z.\leqno(1.24)$$ This result is in fact a special case of the following important inequality. \begstat{(1.25) Wirtinger's inequality} Let $Y$ be an oriented real submanifold of class $C^1$ and dimension $2p$ in $X$, and let $dV_Y$ be the Riemannian volume form on $Y$ associated with the metric $h_{\restriction Y}$. Set $${1\over 2^pp!}\omega^p_{\restriction Y}=\alpha\,dV_Y,~~~~\alpha\in C^0(Y).$$ Then $|\alpha|\le 1$ and the equality holds if and only if $Y$ is a complex analytic submanifold of $X$. In that case $\alpha=1$ if the orientation of $Y$ is the canonical one, $\alpha=-1$ otherwise. \endstat \begproof{} The restriction $\omega_{\restriction Y}$ is a real antisymmetric $2$-form on $TY$. At any point $z\in Y$, we can thus find an oriented orthonormal $\bR$-basis $(e_1,e_2,\ldots,e_{2p})$ of $T_zY$ such that $$\eqalign{ {1\over 2}\omega&=\sum_{1\le k\le p}\alpha_k\,e^\star_{2k-1}\wedge e^\star_{2k}~~~{\rm on}~~T_zY,~~~~{\rm where}\cr \alpha_k&={1\over 2}\omega(e_{2k-1},e_{2k})=-\Im h(e_{2k-1},e_{2k}).\cr}$$ We have $dV_Y=e^\star_1\wedge\ldots\wedge e^\star_{2p}$ by definition of the Riemannian volume form. By taking the $p$-th power of $\omega$, we get $${1\over 2^pp!}\omega^p_{\restriction T_zY}=\alpha_1\ldots\alpha_p\, e^\star_1\wedge\ldots\wedge e^\star_{2p}=\alpha_1\ldots\alpha_p\,dV_Y.$$ Since $(e_k)$ is an orthonormal $\bR$-basis, we have $\Re h(e_{2k-1},e_{2k})=0$, thus $h(e_{2k-1},e_{2k})=-\ii\alpha_k$. As $|e_{2k-1}|=|e_{2k}|=1$, we get $$0\le|e_{2k}\pm Je_{2k-1}|^2=2\big(1\pm\Re h(Je_{2k-1},e_{2k})\big) =2(1\pm\alpha_k).$$ Therefore $$|\alpha_k|\le 1,~~~~|\alpha|=|\alpha_1\ldots\alpha_p|\le 1,$$ with equality if and only if $\alpha_k=\pm1$ for all $k$, i.e. $e_{2k}=\pm Je_{2k-1}$. In this case $T_zY\subset T_zX$ is a complex vector subspace at every point $z\in Y$, thus $Y$ is complex analytic by Lemma~I.4.23. Conversely, assume that $Y$ is a $\bC$-analytic submanifold and that $(e_1,e_3,\ldots,e_{2p-1})$ is an orthonormal complex basis of $T_zY$. If $e_{2k}:=Je_{2k-1}$, then $(e_1,\ldots,e_{2p})$ is an orthonormal $\bR$-basis corresponding to the canonical orientation and $${1\over 2}\omega_{\restriction Y}= \sum_{1\le k\le p}e^\star_{2k-1}\wedge e^\star_{2k},~~~~ {1\over 2^pp!}\omega^p_{\restriction Y}=e^\star_1\wedge\ldots\wedge e^\star_{2p} =dV_Y.\eqno\square$$ \endproof Note that in the case of the standard hermitian metric $\omega$ on $X=\bC^n$, the form $\omega=\ii\sum dz_j\wedge d\ol z_j= d\big(\ii\sum z_j\,d\ol z_j\big)$ is globally exact. Under this hypothesis, we are going to see that $\bC$-analytic submanifolds are always \hbox{\it minimal surfaces} for the Plateau problem, which consists in finding a compact subvariety $Y$ of minimal area with prescribed boundary~$\partial Y$. \begstat{(1.26) Theorem} Assume that the $(1,1)$-form $\omega$ is exact, say $\omega=d\gamma$ with $\gamma\in C^\infty_1(X,\bR)$, and let $Y,Z\subset X$ be $(2p)$-dimensional oriented compact real submanifolds of class $C^1$ with boundary. If $\partial Y=\partial Z$ and $Z$ is complex analytic, then $${\rm Vol}(Y)\ge{\rm Vol(Z)}.$$ \endstat \begproof{} Write $\omega=d\gamma$. Wirtinger's inequality and Stokes' theorem imply $$\eqalignno{ {\rm Vol}(Y)&\ge{1\over 2^pp!}\Big|\int_Y\omega^p\Big| ={1\over 2^pp!}\Big|\int_Y d(\omega^{p-1}\wedge\gamma)\Big| ={1\over 2^pp!}\Big|\int_{\partial Y}\omega^{p-1}\wedge\gamma\Big|,\cr {\rm Vol}(Z)&={1\over 2^pp!}\int_Z\omega^p={1\over 2^pp!}\int_{\partial Z} \omega^{p-1}\wedge\gamma=\pm{1\over 2^pp!}\int_{\partial Y} \omega^{p-1}\wedge\gamma.&\square\cr}$$ \endproof \titleb{2.}{Closed Positive Currents} \titlec{2.A.}{The Skoda-El Mir Extension Theorem} We first prove the Skoda-El Mir extension theorem (Skoda 1982, El Mir 1984), which shows in particular that a closed positive current defined in the complement of an analytic set $E$ can be extended through~$E$ if (and only~if) the mass of the current is locally finite near~$E$. El Mir simplified Skoda's argument and showed that it is enough to assume $E$ complete pluripolar. We follow here the exposition of Sibony's survey article (Sibony 1985). \begstat{(2.1) Definition} A subset $E\subset X$ is said to be complete pluripolar in $X$ if for every point $x_0\in X$ there exist a neighborhood $\Omega$ of $x_0$ and a function $u\in\Psh(\Omega)\cap L^1_\loc(\Omega)$ such that $E\cap\Omega=\{z\in\Omega~;~u(z)=-\infty\}$. \endstat Note that any closed analytic subset $A\subset X$ is complete pluripolar: if $g_1=\ldots=g_N=0$ are holomorphic equations of $A$ on an open set $\Omega\subset X$, we can take $u=\log(|g_1|^2+\ldots+|g_N|^2)$. \begstat{(2.2) Lemma} Let $E\subset X$ be a closed complete pluripolar set. If $x_0\in X$ and $\Omega$ is a sufficiently small neighborhood of $x_0$, there exists: \medskip \item{\rm a)} a function $v\in\Psh(\Omega)\cap C^\infty(\Omega\ssm E)$ such that $v=-\infty$ on $E\cap\Omega~;$ \medskip \item{\rm b)} an increasing sequence $v_k\in\Psh(\Omega)\cap C^\infty(\Omega)$, $0\le v_k\le 1$, converging uniformly to $1$ on every compact subset of $\Omega\ssm E$, such that $v_k=0$ on a neighborhood of $E\cap\Omega$. \smallskip \endstat \begproof{} Assume that $\Omega_0\compact X$ is a coordinate patch of $X$ containing $x_0$ and that $E\cap\Omega_0=\{z\in\Omega_0~;~u(z)=-\infty\}$, $u\in\Psh(\Omega_0)$. In addition, we can achieve $u\le 0$ by shrinking $\Omega_0$ and subtracting a constant to $u$. Select a convex increasing function $\chi\in C^\infty([0,1],\bR)$ such that $\chi(t)=0$ on $[0,1/2]$ and $\chi(1)=1$. We set $$u_k=\chi\big(\exp(u/k)\big).$$ Then $0\le u_k\le1$, $u_k$ is plurisubharmonic on $\Omega_0$, $u_k=0$ in a neighborhood $\omega_k$ of $E\cap\Omega_0$ and $\lim u_k=1$ on $\Omega_0\ssm E$. Let $\Omega\compact\Omega_0$ be a neighborhood of $x_0$, let $\delta_0=d(\Omega,\complement\Omega_0)$ and $\varepsilon_k\in{}]0,\delta_0[$ be such that $\varepsilon_kn-p$ at every point of $M\cap\Omega$.\break After a change of the ordering, we may assume for example that $\zeta_1=d'g_1$, $\zeta_2=d'g_2$, $\ldots$, $\zeta_{n-p+1}=d'g_{n-p+1}$ are linearly independent on $\Omega$ (shrink $\Omega$ if necessary). Complete $(\zeta_1,\ldots,\zeta_{n-p+1})$ into a continuous frame $(\zeta_1,\ldots,\zeta_n)$ of $T^\star X_{\restriction\Omega}$ and set $$\Theta=\sum_{|I|=|J|=n-p}\Theta_{I,J}\,\zeta_I\wedge\ol\zeta_J~~~{\rm on}~~ \Omega.$$ As $\Theta$ and $d'\Theta$ have measure coefficients supported on $M$ and $g_k=0$ on $M$, we get $g_k\Theta=g_kd'\Theta=0$, thus $$d'g_k\wedge\Theta=d'(g_k\Theta)-g_kd'\Theta=0,~~~~1\le k\le m,$$ in particular $\zeta_k\wedge\Theta=0$ for all $1\le k\le n-p+1$. When $|I|=n-p$, the multi-index $\complement I$ contains at least one of the elements $1,\ldots,n-p+1$, hence $\Theta\wedge\zeta_{\complement I}\wedge\ol \zeta_{\complement J}=0$ and $\Theta_{I,J}=0$.\qed \endproof \begstat{(2.11) Corollary} Let $\Theta\in\cD'_{p,p}(X)$ be a normal current. If ${\rm Supp}\,\Theta$ is contained in an analytic subset $A$ of dimension $0$ such that $$C_1||\Theta||_K\le\int_K\Theta\wedge\beta^p\le C_2||\Theta||_K.$$ We denote by $L^1(K)$, resp. by $L^\infty(K)$, the space of integrable (resp. bounded measurable) functions on $K$ with respect to any smooth positive density on~$X$. \begstat{(3.3) Chern-Levine-Nirenberg inequalities {\rm(1969)}} For all compact subsets $K,L$ of $X$ with $L\subset K^\circ$, there exists a constant $C_{K,L}\ge0$ such that $$||dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T||_L\le C_{K,L}~ ||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}\,||T||_K.$$ \endstat \begproof{} By induction, it is sufficient to prove the result for $q=1$ and $u_1=u$. There is a covering of $L$ by a family of balls $B'_j\compact B_j\subset K$ contained in coordinate patches of $X$. Let $\chi\in\cD(B_j)$ be equal to $1$ on $\smash{\ol B'_j}$. Then $$||dd^cu\wedge T||_{L\cap\ol B'_j}\le C\int_{\ol B'_j}dd^c u\wedge T \wedge\beta^{p-1}\le C\int_{B_j}\chi\,dd^c u\wedge T\wedge\beta^{p-1}.$$ As $T$ and $\beta$ are closed, an integration by parts yields $$||dd^cu\wedge T||_{L\cap\ol B'_j}\le C\int_{B_j}u\,T\wedge dd^c\chi \wedge\beta^{p-1}\le C'||u||_{L^\infty(K)}||T||_K$$ where $C'$ is equal to $C$ multiplied by a bound for the coefficients of the smooth form $dd^c\chi\wedge\beta^{p-1}$.\qed \endproof \begstat{(3.4) Remark} \rm With the same notations as above, any plurisubharmonic function $V$ on $X$ satisfies inequalities of the type \medskip \item{\rm a)} $\quad||dd^cV||_L\le C_{K,L}\,||V||_{L^1(K)}$. \medskip \item{\rm b)} $\quad{\displaystyle\sup_L}\,V_+\le C_{K,L}\,||V||_{L^1(K)}$. \medskip \noindent In fact the inequality $$\int_{L\cap\ol B'_j}dd^cV\wedge\beta^{n-1}\le\int_{B_j}\chi dd^cV\wedge \beta^{n-1}=\int_{B_j}Vdd^c\chi\wedge\beta^{n-1}$$ implies a), and b) follows from the mean value inequality. \endstat \begstat{(3.5) Remark} \rm Products of the form $\Theta=\gamma_1\wedge\ldots\wedge \gamma_q\wedge T$ with mixed $(1,1)$-forms $\gamma_j=dd^cu_j$ or $\gamma_j=dv_j\wedge d^cw_j+dw_j\wedge d^cv_j$ are also well defined whenever $u_j$, $v_j$, $w_j$ are locally bounded plurisubharmonic functions. Moreover, for $L\subset K^\circ$, we have $$||\Theta||_L\le C_{K,L}||T||_K\prod||u_j||_{L^\infty(K)} \prod||v_j||_{L^\infty(K)}\prod||w_j||_{L^\infty(K)}.$$ To check this, we may suppose $v_j,w_j\ge 0$ and $||v_j||=||w_j||=1$ in~$L^\infty(K)$. Then the inequality follows from (3.3) by the polarization identity $$2(dv_j\wedge d^cw_j+dw_j\wedge d^cv_j)=dd^c(v_j+w_j)^2 -dd^cv_j^2-dd^cw_j^2-v_jdd^cw_j-w_jdd^cv_j$$ in which all $dd^c$ operators act on plurisubharmonic functions. \endstat \begstat{(3.6) Corollary} Let $u_1,\ldots,u_q$ be continuous $($finite$)$ plurisubharmonic functions and let $u_1^k,\ldots,u_q^k$ be sequences of plurisubharmonic functions converging locally uniformly to $u_1,\ldots,u_q$. If $T_k$ is a sequence of closed positive currents converging weakly to $T$, then \smallskip \item{\rm a)} $u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k \wedge T_k\longrightarrow u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$. \smallskip \item{\rm b)} $dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T_k\longrightarrow dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$. \endstat \begproof{} We observe that b) is an immediate consequence of a) by the weak continuity of $dd^c$. By using induction on $q$, it is enough to prove result a) when~$q=1$. If $(u^k)$ converges locally uniformly to a finite continuous plurisubharmonic function $u$, we introduce local regularizations $u_\varepsilon=u\star\rho_\varepsilon$ and write $$u^kT_k-uT=(u^k-u)T_k+(u-u_\varepsilon)T_k+u_\varepsilon(T_k-T).$$ As the sequence $T_k$ is weakly convergent, it is locally uniformly bounded in mass, thus $||(u^k-u)T_k||_K\le ||u^k-u||_{L^\infty(K)}||T_k||_K$ converges to $0$ on every compact set~$K$. The same argument shows that $||(u-u_\varepsilon)T_k||_K$ can be made arbitrarily small by choosing $\varepsilon$ small enough. Finally $u_\varepsilon$ is smooth, so $u_\varepsilon(T_k-T)$ converges weakly to~$0$.\qed \endproof Now, we prove a deeper monotone continuity theorem due to (Bedford-Taylor 1982) according to which the continuity and uniform convergence assumptions can be dropped if each sequence $(u_j^k)$ is decreasing and $T_k$ is a constant sequence. \begstat{(3.7) Theorem} Let $u_1,\ldots,u_q$ be locally bounded plurisubharmonic functions and let $u_1^k,\ldots,u_q^k$ be decreasing sequences of plurisubharmonic functions converging pointwise to $u_1,\ldots,u_q$. Then \medskip \item{\rm a)} $u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k \wedge T\longrightarrow u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$. \medskip \item{\rm b)} $dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T\longrightarrow dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\it weakly}$. \endstat \begproof{} Again by induction, observing that a)~$\Longrightarrow$~b) and that a) is obvious for $q=1$ thanks to Lebesgue's bounded convergence theorem. To proceed with the induction step, we first have to make some slight modifications of our functions $u_j$ and $u_j^k$. As the sequence $\smash{(u_j^k)}$ is decreasing and as $u_j$ is locally bounded, the family $\smash{(u_j^k)_{k\in\bN}}$ is locally uniformly bounded. The results are local, so we can work on a Stein open set $\Omega\compact X$ with strongly pseudoconvex boundary. We use the following notations: \medskip \itemitem{\kern-\parindent\rlap{\hbox{(3.8)}}} let $\psi$ be a strongly plurisubharmonic function of class $C^\infty$ near $\smash{\ol\Omega}$ with $\psi<0$ on $\Omega$ and $\psi=0$, $d\psi\ne 0$ on~$\partial\Omega\,;$ \smallskip \itemitem{\kern-\parindent\rlap{\hbox{$(3.8')$}}} we set $\Omega_\delta=\{z\in\Omega\,;\,\psi(z)<-\delta\}$ for all $\delta>0$. \input epsfiles/fig_3_1.tex \vskip6mm \centerline{{\bf III-1} Construction of $v_j^k$} \vskip6mm \noindent After addition of a constant we can assume that $-M\le\smash{u_j^k}\le-1$ near~$\smash{\ol\Omega}$. Let us denote by $\smash{(u_j^{k,\varepsilon})}$, $\varepsilon\in{}]0,\varepsilon_0]$, an increasing family of regularizations converging to $\smash{u_j^k}$ as $\varepsilon\to 0$ and such that $-M\le\smash{u^{k,\varepsilon}_j}\le-1$ on $\smash{\ol\Omega}$. Set $A=M/\delta$ with $\delta>0$ small and replace $\smash{u_j^k}$ by $\smash{v_j^k=\max\{A\psi,u_j^k\}}$ and $\smash{u_j^{k,\varepsilon}}$ by $\smash{v_j^{k,\varepsilon}= \max_\varepsilon\{A\psi,u_j^{k,\varepsilon}\}}$ where $\max_\varepsilon=\max~\star~\rho_\varepsilon$ is a regularized max function. Then $\smash{v_j^k}$ coincides with $\smash{u_j^k}$ on $\Omega_\delta$ since $A\psi<-A\delta=-M$ on $\Omega_\delta$, and $\smash{v_j^k}$ is equal to $A\psi$ on the corona $\Omega\setminus\Omega_{\delta/M}$. Without loss of generality, we can therefore assume that all $\smash{u_j^k}$ (and similarly all $\smash{u_j^{k,\varepsilon}})$ coincide with $A\psi$ on a fixed neighborhood of $\partial\Omega$. We need a lemma. \endproof \begstat{(3.9) Lemma} Let $f_k$ be a decreasing sequence of upper semi-continuous functions conver\-ging to $f$ on some separable locally compact space $X$ and $\mu_k$ a sequence of positive measures converging weakly to $\mu$ on $X$. Then every weak limit $\nu$ of $f_k\mu_k$ satisfies $\nu\le f\mu$. \endstat Indeed if $(g_p)$ is a decreasing sequence of continuous functions converging to $f_{k_0}$ for some $k_0$, then $f_k\mu_k\le f_{k_0}\mu_k \le g_p\mu_k$ for $k\ge k_0$, thus $\nu\le g_p\mu$ as $k\to+\infty$. The monotone convergence theorem then gives $\nu\le f_{k_0}\mu$ as $p\to+\infty$ and $\nu\le f\mu$ as $k_0\to+\infty$.\qed \begproof{of Theorem 3.7 (end).} Assume that a) has been proved for $q-1$. Then $$S^k=dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k\wedge T\longrightarrow S=dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T.$$ By 3.3 the sequence $(u_1^k S^k)$ has locally bounded mass, hence is relatively compact for the weak topology. In order to prove a), we only have to show that every weak limit $\Theta$ of $u_1^k S^k$ is equal to $u_1 S$. Let $(m,m)$ be the bidimension of $S$ and let $\gamma$ be an arbitrary smooth and strongly positive~form of bidegree $(m,m)$. Then the positive measures $S^k\wedge\gamma$ converge weakly to $S\wedge\gamma$ and Lemma~3.9 shows that $\Theta\wedge\gamma\le u_1S\wedge\gamma$, hence $\Theta\le u_1S$. To get~the equality, we set $\beta=dd^c\psi>0$ and show that $\int_\Omega u_1 S\wedge\beta^m\le\int_\Omega\Theta\wedge\beta^m$, i.e. $$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\le \liminf_{k\to+\infty}\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k \wedge T\wedge\beta^m.$$ As $u_1\le u_1^k\le u_1^{k,\varepsilon_1}$ for every $\varepsilon_1>0$, we get $$\eqalign{\int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr &\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2 \wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr &=\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge u_2dd^cu_3 \wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr}$$ after an integration by parts (there is no boundary term because $\smash{u_1^{k,\varepsilon_1}}$ and $u_2$ both vanish on $\partial\Omega$). Repeating this argument with $u_2,\ldots,u_q$, we obtain $$\eqalign{ \int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr &\le\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge\ldots\wedge dd^c u_{q-1}^{k,\varepsilon_{q-1}}\wedge u_q T\wedge\beta^m\cr &\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2^{k,\varepsilon_2} \wedge\ldots\wedge dd^c u_q^{k,\varepsilon_q}\wedge T\wedge\beta^m.\cr}$$ Now let $\varepsilon_q\to 0,\ldots,\varepsilon_1\to 0$ in this order. We have weak convergence at each step and $\smash{u_1^{k,\varepsilon_1}}=0$ on the boundary; therefore the integral in the last line converges and we get the desired inequality $$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\le\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k\wedge T\wedge\beta^m.\eqno{\square}$$ \endproof \begstat{(3.10) Corollary} The product $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ is symmetric with respect to $u_1,\ldots,u_q$. \endstat \begproof{} Observe that the definition was unsymmetric. The result is true when $u_1,\ldots,u_q$ are smooth and follows in general from Th.~3.7 applied to the sequences $u_j^k=u_j\star\rho_{1/k}$, $1\le j\le q$.\qed \endproof \begstat{(3.11) Proposition} Let $K,L$ be compact subsets of $X$ such that $L\subset K^\circ$. For any plurisubharmonic functions $V,u_1,\ldots,u_q$ on $X$ such that $u_1,\ldots,u_q$ are locally bounded, there is an inequality $$||Vdd^cu_1\wedge\ldots\wedge dd^cu_q||_L\le C_{K,L}\,||V||_{L^1(K)}||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}.$$ \endstat \begproof{} We may assume that $L$ is contained in a strongly pseudoconvex open set $\Omega=\{\psi<0\}\subset K$ (otherwise we cover $L$ by small balls contained in $K$). A suitable normalization gives $-2\le u_j\le -1$ on $K\,$; then we can modify $u_j$ on $\Omega\setminus L$ so that $u_j=A\psi$ on $\Omega\setminus\Omega_\delta$ with a fixed constant $A$ and $\delta>0$ such that $L\subset\Omega_\delta$. Let $\chi\ge 0$ be a smooth function equal to $-\psi$ on $\Omega_\delta$ with compact support in $\Omega$. If we take $||V||_{L^1(K)}=1$, we see that $V_+$ is uniformly bounded on $\Omega_\delta$ by 3.4~b); after subtraction of a fixed constant we can assume $V\le 0$ on $\Omega_\delta$. First suppose $q\le n-1$. As $u_j=A\psi$ on $\Omega\setminus\Omega_\delta$, we find $$\eqalign{ &\int_{\Omega_\delta}-V\,dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge\beta^{n-q}\cr &~\,{}=\int_\Omega V\,dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge\beta^{n-q-1}\wedge dd^c\chi-A^q\int_{\Omega\setminus \Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi\cr &~\,{}=\int_\Omega\chi\,dd^cV\wedge dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge\beta^{n-q-1}-A^q\int_{\Omega\setminus \Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi.\cr}$$ The first integral of the last line is uniformly bounded thanks to 3.3 and 3.4~a), and the second one is bounded by $||V||_{L^1(\Omega)}\le$ constant. Inequality 3.11 follows for $q\le n-1$. If $q=n$, we can work instead on $X\times\bC$ and consider $V,u_1,\ldots,u_q$ as functions on $X\times\bC$ independent of the extra variable in~$\bC$.\qed \endproof{} \titleb{4.}{Case of Unbounded Plurisubharmonic Functions} We would like to define $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ also in some cases when $u_1,\ldots,u_q$ are not bounded below everywhere, especially when the $u_j$ have logarithmic poles. Consider first the case $q=1$ and let $u$ be a plurisubharmonic function on $X$. The {\it pole set} of $u$ is by definition $P(u)=u^{-1}(-\infty)$. We define the {\it unbounded locus} $L(u)$ to be the set of points $x\in X$ such that $u$ is unbounded in every neighborhood of~$x$. Clearly $L(u)$ is closed and we have $L(u)\supset\smash{\ol{P(u)}}$ but in general these sets are different: in fact, $u(z)=\sum k^{-2}\log(|z-1/k|+e^{-k^3})$ is everywhere finite in $\bC$ but $L(u)=\{0\}$. \begstat{(4.1) Proposition} We make two additional assumptions: \smallskip \item{\rm a)} $T$ has non zero bidimension $(p,p)$ (i.e. degree of $T<2n$). \smallskip \item{\rm b)} $X$ is covered by a family of Stein open sets $\Omega\compact X$ whose boundaries $\partial\Omega$ do not intersect $L(u)\cap\Supp\,T$. \smallskip \noindent Then the current $uT$ has locally finite mass in X. \endstat For any current $T$, hypothesis 4.1~b) is clearly satisfied when $u$ has a discrete unbounded locus $L(u)$; an interesting example is $u=\log|F|$ where $F=(F_1,\ldots,F_N)$ are holomorphic functions having a discrete set of common zeros. Observe that the current $uT$ need not have locally finite mass when $T$ has degree $2n$ (i.e. $T$ is a measure); example: $T=\delta_0$ and $u(z)=\log|z|$\break in $\bC^n$. The result also fails when the sets $\Omega$ are not assumed to be Stein; example: $X={}$ blow-up of $\bC^n$ at $0$, $T=[E]={}$ current of integration on the exceptional divisor and $u(z)=\log|z|$ (see \S~7.12 for the definition of blow-ups). \begproof{} By shrinking $\Omega$ slightly, we may assume that $\Omega$ has a smooth strongly pseudoconvex boundary. Let $\psi$ be a defining function of $\Omega$ as in (3.8). By subtracting a constant to $u$, we may assume $u\le-\varepsilon$ on $\ol\Omega$. We fix $\delta$ so small that $\ol\Omega\ssm\Omega_\delta$ does not intersect $L(u)\cap\Supp\,T$ and we select a neighborhood $\omega$ of $(\ol\Omega\ssm\Omega_\delta)\cap \Supp\,T$ such that $\ol\omega\cap L(u)=\emptyset$. Then we define $$u_s(z)=\cases{ \max\{u(z),A\psi(z)\}&on $\omega$,\cr \max\{u(z),s\}&on $\Omega_\delta=\{\psi<-\delta\}$.\cr}$$ By construction $u\ge -M$ on $\omega$ for some constant $M>0$. We fix $A\ge M/\delta$ and take $s\le -M$, so $$\max\{u(z),A\psi(z)\}=\max\{u(z),s\}=u(z)~~~~\hbox{\rm on}~~ \omega\cap\Omega_\delta$$ and our definition of $u_s$ is coherent. Observe that $u_s$ is defined on $\omega\cup\Omega_\delta$, which is a neighborhood of $\ol\Omega\cap\Supp\,T$. Now, $u_s=A\psi$ on $\omega\cap(\Omega\ssm\Omega_{\varepsilon/A})$, hence Stokes' theorem implies $$\eqalign{ \int_\Omega dd^cu_s\wedge T\wedge(dd^c\psi)^{p-1}&{}- \int_\Omega Add^c\psi\wedge T\wedge(dd^c\psi)^{p-1}\cr &=\int_\Omega dd^c\big[(u_s-A\psi)T\wedge(dd^c\psi)^{p-1}\big]=0\cr}$$ because the current $[\ldots]$ has a compact support contained in $\ol\Omega_{\varepsilon/A}$. Since $u_s$ and $\psi$ both vanish on $\partial\Omega$, an integration by parts gives $$\eqalign{ \int_\Omega u_sT\wedge(dd^c\psi)^p &=\int_\Omega\psi dd^cu_s\wedge T\wedge(dd^c\psi)^{p-1}\cr &\ge-||\psi||_{L^\infty(\Omega)}\int_\Omega T\wedge dd^cu_s\wedge (dd^c\psi)^{p-1}\cr &=-||\psi||_{L^\infty(\Omega)}A\int_\Omega T\wedge(dd^c\psi)^p.\cr}$$ Finally, take $A=M/\delta$, let $s$ tend to $-\infty$ and use the inequality $u\ge-M$ on~$\omega$. We obtain $$\eqalign{ \int_\Omega u\,T\wedge(dd^c\psi)^p &\ge -M\int_\omega T\wedge(dd^c\psi)^p+\lim_{s\to-\infty} \int_{\Omega_\delta}u_sT\wedge(dd^c\psi)^p\cr &\ge -\big(M+||\psi||_{L^\infty(\Omega)}M/\delta\big) \int_\Omega T\wedge(dd^c\psi)^p.\cr}$$ The last integral is finite. This concludes the proof.\qed \endproof \begstat{(4.2) Remark} \rm If $\Omega$ is smooth and strongly pseudoconvex, the above proof shows in fact that $$||uT||_{\ol\Omega}\le{C\over\delta}||u||_{L^\infty( (\ol\Omega\ssm\Omega_\delta)\cap\Supp\,T)}||T||_{\ol\Omega}$$ when $L(u)\cap\Supp\,T\subset\Omega_\delta$. In fact, if $u$ is continuous and if $\omega$ is chosen sufficiently small, the constant $M$ can be taken arbitrarily close to $||u||_{L^\infty( (\ol\Omega\ssm\Omega_\delta)\cap\Supp\,T)}$. Moreover, the maximum principle implies $$||u_+||_{L^\infty(\ol\Omega\cap\Supp\,T)}= ||u_+||_{L^\infty(\partial\Omega\cap\Supp\,T)},$$ so we can achieve that $u<-\varepsilon$ on a neighborhood of $\ol\Omega \cap\Supp\,T$ by subtracting $||u||_{L^\infty((\ol\Omega\ssm\Omega_\delta) \cap\Supp\,T)}+2\varepsilon$ $[$Proof of maximum principle: if $u(z_0)>0$ at $z_0\in\Omega\cap\Supp\,T$ and $u\le 0$ near $\partial\Omega\cap\Supp\,T$, then $$\int_\Omega u_+T\wedge(dd^c\psi)^p=\int_\Omega\psi dd^cu_+\wedge T \wedge(dd^c\psi)^{p-1}\le 0,$$ a contradiction$]$.\qed \endstat \begstat{(4.3) Corollary} Let $u_1,\ldots,u_q$ be plurisubharmonic functions on $X$ such that $X$ is covered by Stein open sets $\Omega$ with $\partial\Omega\cap L(u_j)\cap\Supp\,T=\emptyset$. We use again induction to define $$dd^c u_1\wedge dd^cu_2\wedge\ldots\wedge dd^c u_q\wedge T= dd^c(u_1dd^c u_2\ldots\wedge dd^c u_q\wedge T).$$ Then, if $u_1^k,\ldots,u_q^k$ are decreasing sequences of plurisubharmonic functions converging pointwise to $u_1,\ldots,u_q$, $q\le p$, properties~$(3.7\,{\rm a,b})$ hold. \endstat \input epsfiles/fig_3_2.tex \vskip6mm \centerline{{\bf III-2} Modified construction of $v_j^k$} \vskip6mm \begproof{} Same proof as for Th.~3.7, with the following minor modification: the max procedure $v_j^k:=\max\{u_j^k,A\psi\}$ is applied only on a neighborhood $\omega$ of $\Supp\,T\cap(\ol\Omega\ssm\Omega_\delta)$ with $\delta>0$ small, and $u_j^k$ is left unchanged near \hbox{$\Supp\,T\cap\ol\Omega_\delta$.} Observe that the integration by part process requires the functions $u_j^k$ and $\smash{u_j^{k,\varepsilon}}$ to be defined only near $\ol\Omega\cap\Supp\,T$.\qed \endproof \begstat{(4.4) Proposition} Let $\Omega\compact X$ be a Stein open subset. If $V$ is a plurisubharmonic function on $X$ and $u_1,\ldots,u_q$, $1\le q\le n-1$, are plurisubharmonic functions such that $\partial\Omega\cap L(u_j)=\emptyset$, then $Vdd^cu_1\wedge\ldots \wedge dd^cu_q$ has locally finite mass in $\Omega$. \endstat \begproof{} Same proof as for 3.11, when $\delta>0$ is taken so small that $\Omega_\delta\supset L(u_j)$ for all $1\le j\le q$.\qed \endproof Finally, we show that Monge-Amp\`ere operators can also be defined in the case of plurisubharmonic functions with non compact pole sets, provided that the mutual intersections of the pole sets are of sufficiently small Hausdorff dimension with respect to the dimension $p$ of~$T$. \begstat{(4.5) Theorem} Let $u_1,\ldots,u_q$ be plurisubharmonic functions on~$X$. The currents $u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T$ and $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ are well defined and have locally finite mass in~$X$ as soon as $q\le p$ and $$\cH_{2p-2m+1}\big(L(u_{j_1})\cap\ldots\cap L(u_{j_m})\cap\Supp\,T\big)=0$$ for all choices of indices $j_1<\ldots0$ small. If $0\notin A_j$ for some~$j$, we choose instead $W_j$ to be a small neighborhood of~$0$ such that $\ol W_j\subset (\ol B'\times(1-\delta)B'')\ssm A_j\,$; property (4.8) is then automatically satisfied. Let $\chi_j\ge 0$ be a function with compact support in $W_j$, equal to $1$ near $K\cap A_j$ if $A_j\ni 0$ (resp. equal to $1$ near $0$ if $A_j\not\ni 0$) and let $\chi(z')\ge 0$ be a function equal to $1$ on $1/2\,B'$ with compact support in~$B'$. Then $$\int_{B'\times B''}dd^c(\chi_1u_1)\wedge\ldots\wedge dd^c(\chi_qu_q) \wedge T\wedge\chi(z')\,(dd^c|z'|^2)^s=0$$ because the integrand is $dd^c$ exact and has compact support in $B'\times B''$ thanks to~(4.8). If we expand all factors $dd^c(\chi_ju_j)$, we find a term $$\chi_1\ldots\chi_q\chi(z')dd^cu_1\wedge\ldots\wedge dd^c u_q\wedge T\ge 0$$ which coincides with $dd^cu_1\wedge\ldots\wedge dd^c u_q\wedge T$ on a small neighborhood of $0$ where $\chi_j=\chi=1$. The other terms involve $$d\chi_j\wedge d^cu_j+du_j\wedge d^c\chi_j+u_jdd^c\chi_j$$ for at least one index $j$. However $d\chi_j$ and $dd^c\chi_j$ vanish on some neighborhood $V'_j$ of~$K\cap A_j$ and therefore $u_j$ is bounded on $\smash{\ol B'\times\ol B''}\ssm V'_j$. We then apply the induction hypothesis $({\rm b}_{q-1})$ to the current $$\Theta=dd^cu_1\wedge\ldots\wedge\wh{dd^cu_j}\wedge \ldots\wedge dd^cu_q\wedge T$$ and the usual Chern-Levine-Nirenberg inequality to the product of $\Theta$ with the mixed term $d\chi_j\wedge d^cu_j+du_j\wedge d^c\chi_j$. Remark~3.5 can be applied because $\chi_j$ is smooth and is therefore a difference $\chi^{(1)}_j-\chi^{(2)}_j$ of locally bounded plurisubharmonic functions in $\bC^n$. Let $K'$ be a compact neighborhood of $\smash{ \ol B'\times\ol B''}$ with $K'\subset K^\circ$, and let $V_j$ be a neighborhood of $K\cap A_j$ with $\ol V_j\subset V'_j$. Then with $L':=(\smash{\ol B'\times\ol B''})\ssm V'_j\subset(K'\ssm V_j)^\circ$ we obtain $$\eqalign{ ||(d\chi_j{\wedge}d^cu_j+du_j{\wedge}d^c&\chi_j)\wedge \Theta||_{\ol B'\times \ol B''} =||(d\chi_j{\wedge}d^cu_j+du_j{\wedge}d^c\chi_j)\wedge\Theta||_{L'}\cr &\le C_1||u_j||_{L^\infty(K'\ssm V_j)}||\Theta||_{K'\ssm V_j},\cr ||\Theta||_{K'\ssm V_j}\le||\Theta||_{K'} &\le C_2||u_1||_{L^\infty(K\ssm V_1)}\ldots\wh{||u_j||}\ldots ||u_q||_{L^\infty(K\ssm V_q)}||T||_K.\cr}$$ Now, we may slightly move the unitary basis in $\bC^n$ and get coordinate systems $z^m=(z^m_1,\ldots,z^m_n)$ with the same properties as above, such that the forms $$(dd^c|z^{m\prime}|^2)^s={s!\over\pi^s} \ii\,dz^m_1\wedge d\ol z^m_1\wedge\ldots\wedge\ii\,dz^m_s \wedge d\ol z^m_s,~~~~1\le m\le N$$ define a basis of $\bigwedge^{s,s}(\bC^n)^\star$. It follows that all measures $$dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T\wedge \ii\,dz^m_1\wedge d\ol z^m_1\wedge\ldots\wedge\ii\,dz^m_s\wedge d\ol z^m_s$$ satisfy estimate $({\rm b}_q)$ on a small neighborhood $L$ of~$0$. \endproof \begproof{of Step 2.} We argue in a similar way with the integrals $$\eqalign{ \int_{B'\times B''}&\chi_1u_1dd^c(\chi_2u_2)\wedge\ldots dd^c(\chi_qu_q) \wedge T\wedge\chi(z')(dd^c|z'|^2)^s\wedge dd^c|z_{s+1}|^2\cr &=\int_{B'\times B''}|z_{s+1}|^2dd^c(\chi_1u_1)\wedge \ldots dd^c(\chi_qu_q)\wedge T\wedge\chi(z')(dd^c|z'|^2)^s.\cr}$$ We already know by $({\rm b}_q)$ and Remark~3.5 that all terms in the right hand integral admit the desired bound. For $q=1$, this shows that $({\rm b}_1)\Longrightarrow({\rm a}_1)$. Except for $\chi_1\ldots\chi_q\chi(z')\,u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T$, all terms in the left hand integral involve derivatives of~$\chi_j$. By construction, the support of these derivatives is disjoint from~$A_j$, thus we only have to obtain a bound for $$\int_Lu_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\alpha$$ when $L=\ol B(x_0,r)$ is disjoint from $A_j$ for some $j\ge 2$, say $L\cap A_2=\emptyset$, and $\alpha$ is a constant positive form of type~$(p-q,p-q)$. Then $\ol B(x_0,r+\varepsilon)\subset K^\circ\ssm\ol V_2$ for some $\varepsilon>0$ and some neighborhood $V_2$ of~$K\cap A_2$. By the max construction used e.g. in Prop.~4.1, we can replace $u_2$ by a plurisubharmonic function $\wt u_2$ equal to $u_2$ in $L$ and to $A(|z-x_0|^2-r^2)-M$ in $\ol B(x_0,r+\varepsilon)\ssm B(x_0,r+ \varepsilon/2)$, with $M=||u_2||_{L^\infty(K\ssm V_2)}$ and $A=M/\varepsilon r$. Let $\chi\ge 0$ be a smooth function equal to $1$ on $B(x_0,r+\varepsilon/2)$ with support in $B(x_0,r)$. Then $$\eqalign{ \int_{B(x_0,r+\varepsilon)}u_1&dd^c(\chi\wt u_2)\wedge dd^cu_3\wedge \ldots\wedge dd^cu_q\wedge T\wedge\alpha\cr &=\int_{B(x_0,r+\varepsilon)}\chi\wt u_2dd^cu_1\wedge dd^cu_3\wedge\ldots \wedge dd^cu_q\wedge T\wedge\alpha\cr &\le O(1)~||u_1||_{L^\infty(K\ssm V_1)}\ldots||u_q||_{L^\infty(K\ssm V_q)} ||T||_K\cr}$$ where the last estimate is obtained by the induction hypothesis $({\rm b}_{q-1})$ applied to $dd^cu_1\wedge dd^cu_3\wedge\ldots\wedge dd^cu_q\wedge T$. By construction $$dd^c(\chi\wt u_2)=\chi\,dd^c\wt u_2+(\hbox{\rm smooth terms involving $d\chi$})$$ coincides with $dd^cu_2$ in $L$, and $({\rm a}_{q-1})$ implies the required estimate for the other terms in the left hand integral.\qed \endproof \begstat{(4.9) Proposition} With the assumptions of Th.~$4.5$, the analogue of the monotone convergence Theorem {\rm 3.7~(a,b)} holds. \endstat \begproof{} By the arguments already used in the proof of Th.~3.7 (e.g. by Lemma~3.9), it is enough to show that $$\eqalign{ \int_{B'\times B''}&\chi_1\ldots\chi_q\,u_1\wedge dd^cu_2\wedge \ldots\wedge dd^cu_q\wedge T\wedge\alpha\cr &\le\liminf_{k\to+\infty}\int_{B'\times B''}\chi_1\ldots\chi_q\, u^k_1dd^cu^k_2\wedge\ldots\wedge dd^cu^k_q\wedge T\wedge\alpha\cr}$$ where $\alpha=\chi(z')(dd^c|z'|^2)^s$ is closed. Here the functions $\chi_j$, $\chi$ are chosen as in the proof of Step~1 in~4.7, especially their product has compact support in $B'\times B''$ and $\chi_j=\chi=1$ in a neighborhood of the given point~$x_0$. We argue by induction on $q$ and also on the number $m$ of functions $(u_j)_{j\ge 1}$ which are unbounded near~$x_0$. If $u_j$ is bounded near $x_0$, we take $W_j'' \compact W'_j\compact W_j$ to be small balls of center $x_0$ on which $u_j$ is bounded and we modify the sequence $u_j^k$ on the corona $W_j\ssm W''_j$ so as to make it constant and equal to a smooth function $A|z-x_0|^2+B$ on the smaller corona~$W_j\ssm W'_j$. In that case, we take $\chi_j$ equal to $1$ near~$\smash{\ol W'_j}$ and $\Supp\,\chi_j\subset W_j$. For every $\ell=1,\ldots,q$, we are going to check that $$\eqalign{ \liminf_{k\to+\infty}\int_{B'\times B''} \chi_1u^k_1&dd^c(\chi_2u^k_2)\wedge\ldots\cr dd^c(\chi_{\ell-1}u^k_{\ell-1})\wedge{}&dd^c(\chi_\ell u_\ell)\wedge dd^c(\chi_{\ell+1}u_{\ell+1})\ldots dd^c(\chi_qu_q)\wedge T\wedge\alpha\cr {}\le\liminf_{k\to+\infty}\int_{B'\times B''} \chi_1u^k_1&dd^c(\chi_2u^k_2)\wedge\ldots\cr dd^c(\chi_{\ell-1}u^k_{\ell-1})\wedge{}&dd^c(\chi_\ell u^k_\ell)\wedge dd^c(\chi_{\ell+1}u_{\ell+1})\ldots dd^c(\chi_qu_q)\wedge T\wedge\alpha.\cr}$$ In order to do this, we integrate by parts $\chi_1u^k_1dd^c(\chi_\ell u_\ell)$ into $\chi_\ell u_\ell dd^c(\chi_1u^k_1)$ for $\ell\ge 2$, and we use the inequality $u_\ell\le u^k_\ell$. Of course, the derivatives $d\chi_j$, $d^c\chi_j$, $dd^c\chi_j$ produce terms which are no longer positive and we have to take care of these. However, $\Supp\,d\chi_j$ is disjoint from the unbounded locus of $u_j$ when $u_j$ is unbounded, and contained in $W_j\ssm\smash{\ol W'_j}$ when $u_j$ is bounded. The number $m$ of unbounded functions is therefore replaced by $m-1$ in the first case, whereas in the second case $u^k_j=u_j$ is constant and smooth on $\Supp\,d\chi_j$, so $q$ can be replaced by $q-1$. By induction on $q+m$ (and thanks to the polarization technique~3.5), the limit of the terms involving derivatives of $\chi_j$ is equal on both sides to the corresponding terms obtained by suppressing all indices~$k$. Hence these terms do not give any contribution in the inequalities.\qed \endproof We finally quote the following simple consequences of Th.~4.5 when $T$ is arbitrary and $q=1$, resp. when $T=1$ has bidegree $(0,0)$ and $q$ is arbitrary. \begstat{(4.10) Corollary} Let $T$ be a closed positive current of bidimension $(p,p)$ and let $u$ be a plurisubharmonic function on $X$ such that $L(u)\cap\Supp\,T$ is contained in an analytic set of dimension at most $p-1$. Then $uT$ and $dd^cu\wedge T$ are well defined and have locally finite mass in~$X$.\qed \endstat \begstat{(4.11) Corollary} Let $u_1,\ldots,u_q$ be plurisubharmonic functions on $X$ such that $L(u_j)$ is contained in an analytic set $A_j\subset X$ for every~$j$. Then $dd^cu_1\wedge\ldots\wedge dd^cu_q$ is well defined as soon as $A_{j_1}\cap\ldots\cap A_{j_m}$ has codimension at least $m$ for all choices of indices $j_1<\ldots0$ such that $$[Z_1]\wedge\ldots\wedge[Z_q]=\sum m_k[C_k].$$ The integer $m_k$ is called the multiplicity of intersection of $Z_1,\ldots,Z_q$ along the component~$C_k$. \endstat \begproof{} The wedge product has bidegree $(q,q)$ and support in $C=\bigcup C_k$ where $\codim C=q$, so it must be a sum as above with $m_k\in\bR_+$. We check by induction on $q$ that $m_k$ is a positive integer. If we denote by $A$ some irreducible component of $|Z_1|\cap\ldots\cap |Z_{q-1}|$, we need only check that $[A]\wedge[Z_q]$ is an integral analytic cycle of codimension $q$ with positive coefficients on each component $C_k$ of the intersection. However $[A]\wedge[Z_q]=dd^c(\log|f_q|~[A])$. First suppose that no component of $A\cap f_q^{-1}(0)$ is contained in the singular part $A_\sing$. Then the Lelong-Poincar\'e equation applied on $A_\reg$ shows that $dd^c(\log|f_q|~[A])=\sum m_k[C_k]$ on $X\ssm A_\sing$, where $m_k$ is the vanishing order of $f_q$ along $C_k$ in $A_\reg$. Since $C\cap A_\sing$ has codimension $q+1$ at least, the equality must hold on $X$. In general, we replace $f_q$ by $f_q-\varepsilon$ so that the divisor of $f_q-\varepsilon$ has no component contained in $A_\sing$. Then $dd^c(\log|f_q-\varepsilon|~[A])$ is an integral codimension $q$ cycle with positive multiplicities on each component of $A\cap f_q^{-1}(\varepsilon)$ and we conclude by letting $\varepsilon$ tend to zero.\qed \endproof \titleb{5.}{Generalized Lelong Numbers} The concepts we are going to study mostly concern the behaviour of currents or plurisubharmonic functions in a neighborhood of a point at which we have for instance a logarithmic pole. Since the interesting applications are local, we assume from now on (unless otherwise stated) that $X$ is a Stein manifold, i.e. that $X$ has a strictly plurisubharmonic exhaustion function. Let $\varphi:X\longrightarrow[-\infty,+\infty[$ be a continuous plurisubharmonic function (in general $\varphi$ may have $-\infty$ poles, our continuity assumption means that $e^\varphi$ is continuous). The sets $$\leqalignno{ S(r)&=\{x\in X\,;\,\varphi(x)=r\},&(5.1)\cr B(r)&=\{x\in X\,;\,\varphi(x)0$ the set $E_c=\{x\in X\,;\,\nu(T,x)\ge c\}$ is a closed set of locally finite $\cH_{2p}$ Hausdorff measure in $X$. \endstat \begproof{} By (5.7), we infer $\nu(T,a)=\lim_{r\to 0}\sigma_T\big(\ol B(a,r)\big)p!/\pi^pr^{2p}$. The function $a\mapsto\sigma_T\big(\ol B(a,r)\big)$ is clearly upper semicontinuous. Hence the decreasing limit $\nu(T,a)$ as $r$ decreases to $0$ is also upper semicontinuous in~$a$. This implies that $E_c$ is closed. Now, let $K$ be a compact subset in $X$ and let $\{a_j\}_{1\le j\le N}$, $N=N(\varepsilon)$, be a maximal collection of points in $E_c\cap K$ such that $|a_j-a_k|\ge 2\varepsilon$ for $j\ne k$. The balls $B(a_j,2\varepsilon)$ cover $E_c\cap K$, whereas the balls $B(a_j,\varepsilon)$ are disjoint. If $K_{c,\varepsilon}$ is the set of points which are at distance $\le\varepsilon$ of $E_c\cap K$, we get $$\sigma_T(K_{c,\varepsilon})\ge\sum\sigma_T\big(B(a_j,\varepsilon)\big) \ge N(\varepsilon)\,c\pi^p\varepsilon^{2p}/p!,$$ since $\nu(T,a_j)\ge c$. By the definition of Hausdorff measure, we infer $$\eqalignno{ \cH_{2p}(E_c\cap K)&\le\liminf_{\varepsilon\to 0} \sum\big(\hbox{\rm diam}\,B(a_j,2\varepsilon)\big)^{2p}\cr &\le\liminf_{\varepsilon\to 0}N(\varepsilon)(4\varepsilon)^{2p} \le{p!4^{2p}\over c\pi^p}\sigma_T(E_c\cap K).&\square\cr}$$ \endproof Finally, we conclude this section by proving two simple semi-continuity results for Lelong numbers. \begstat{(5.12) Proposition} Let $T_k$ be a sequence of closed positive currents of bidimension $(p,p)$ converging weakly to a limit~$T$. Suppose that there is a closed set $A$ such that $\Supp\,T_k\subset A$ for all~$k$ and such that $\varphi$ is semi-exhaustive on $A$ with $A\cap B(R)\compact X$. Then for all $rr$, hence the total mass $\mu_r(S(r))=\mu_r(B(s))$ is equal to the difference between the masses of $(dd^c\varphi)^n$ and $\bOne_{X\setminus B(r)}(dd^c\varphi)^n$ over $B(s)$, i.e. $$\mu_r\big(S(r)\big)=\int_{B(r)}(dd^c\varphi)^n.\leqno(6.2)$$ \begstat{(6.3) Example} \rm When $(dd^c\varphi)^n=0$ on $X\setminus\varphi^{-1}(-\infty)$, formula (6.1) can be simplified into $\mu_r=(dd^c\varphi_{\gge r})^n$. This is so for $\varphi(z)=\log|z|$. In this case, the invariance of $\varphi$ under unitary transformations implies that $\mu_r$ is also invariant. As the total mass of $\mu_r$ is equal to $1$ by 5.10 and (6.2), we see that $\mu_r$ is the invariant measure of mass $1$ on the euclidean sphere of radius~$e^r$. \endstat \begstat{(6.4) Proposition} Assume that $\varphi$ is smooth near $S(r)$ and that $d\varphi\ne 0$ on $S(r)$, i.e. $r$ is a non critical value. Then $S(r)=\partial B(r)$ is a smooth oriented real hypersurface and the measure $\mu_r$ is given by the $(2n-1)$-volume form $(dd^c\varphi)^{n-1}\wedge d^c\varphi_{\restriction S(r)}$. \endstat \begproof{} Write $\max\{t,r\}=\lim_{k\to+\infty}\chi_k(t)$ where $\chi$ is a decreasing sequence of smooth convex functions with $\chi_k(t)=r$ for $t\le r-1/k$, $\chi_k(t)=t$ for $t\ge r+1/k$. Theorem 3.6 shows that $(dd^c\chi_k\circ\varphi)^n$ converges weakly to $(dd^c\varphi_{\gge r})^n$. Let $h$ be a smooth function $h$ with compact support near $S(r)$. Let us apply Stokes' theorem with $S(r)$ considered as the boundary of $X\setminus B(r)\,$: $$\leqalignno{ \int_X h(dd^c\varphi_{\gge r})^n&=\lim_{k\to+\infty} \int_X h(dd^c\chi_k\circ\varphi)^n\cr &=\lim_{k\to+\infty}\int_X -dh\wedge(dd^c\chi_k\circ\varphi)^{n-1} \wedge d^c(\chi_k\circ\varphi)\cr &=\lim_{k\to+\infty}\int_X -\chi'_k(\varphi)^n\,dh\wedge(dd^c\varphi)^{n-1} \wedge d^c\varphi\cr &=\int_{X\setminus B(r)}-dh\wedge(dd^c\varphi)^{n-1}\wedge d^c\varphi\cr &=\int_{S(r)}h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi+ \int_{X\setminus B(r)}h\,(dd^c\varphi)^{n-1}\wedge dd^c\varphi.\cr}$$ Near $S(r)$ we thus have an equality of measures $$(dd^c\varphi_{\gge r})^n=(dd^c\varphi)^{n-1}\wedge d^c \varphi_{\restriction S(r)}+\bOne_{X\setminus B(r)}(dd^c\varphi)^n. \eqno{\square}$$ \endproof \begstat{(6.5) Jensen-Lelong formula} Let $V$ be any plurisubharmonic function on~$X$. Then $V$ is $\mu_r$-integrable for every $r\in{}]-\infty,R[$ and $$\mu_r(V)-\int_{B(r)}V(dd^c\varphi)^n= \int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt.$$ \endstat \begproof{} Proposition 3.11 shows that $V$ is integrable with respect to the measure $(dd^c\varphi_{\gge r})^n$, hence $V$ is $\mu_r$-integrable. By definition $$\nu(dd^cV,\varphi,t)=\int_{\varphi(z)0$. Then $B(r)$ is the polydisk of radii $(e^{r/\lambda_1},\ldots,e^{r/\lambda_n})$. If some coordinate $z_j$ is non zero, say $z_1$, we can write $\varphi(z)$ as $\lambda_1\log|z_1|$ plus some function depending only on the $(n-1)$ variables $z_j/z_1^{\lambda_1/\lambda_j}$. Hence $(dd^c\varphi)^n=0$ on $\bC^n\setminus\{0\}$. It will be shown later that $$(dd^c\varphi)^n=\lambda_1\ldots\lambda_n\,\delta_0.\leqno(6.11\,{\rm a})$$ We now determine the measures $\mu_r$. At any point $z$ where not all terms $|z_j|^{\lambda_j}$ are equal, the smallest one can be omitted without changing $\varphi$ in a neighborhood of $z$. Thus $\varphi$ depends only on $(n-1)$-variables and $(dd^c\varphi_{\ge r})^n=0$, $\mu_r=0$ near $z$. It follows that $\mu_r$ is supported by the distinguished boundary $|z_j|=e^{r/\lambda_j}$ of the polydisk $B(r)$. As $\varphi$ is invariant by all rotations $z_j\longmapsto e^{\ii\theta_j}z_j$, the measure $\mu_r$ is also invariant and we see that $\mu_r$ is a constant multiple of $d\theta_1\ldots d\theta_n$. By formula (6.2) and (6.11$\,$a) we get $$\mu_r=\lambda_1\ldots\lambda_n\,(2\pi)^{-n}d\theta_1\ldots d\theta_n. \leqno(6.11\,{\rm b})$$ In particular, the Lelong number $\nu(dd^cV,\varphi)$ is given by $$\nu(dd^cV,\varphi)=\lim_{r\to-\infty}{\lambda_1\ldots\lambda_n\over r} \int_{\theta_j\in[0,2\pi]}\kern-6pt V(e^{r/\lambda_1+\ii\theta_1} ,\ldots,e^{r/\lambda_n+\ii\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}.$$ These numbers have been introduced and studied by (Kiselman 1986). We call them {\it directional Lelong numbers} with coefficients $(\lambda_1,\ldots,\lambda_n)$. For an arbitrary current $T$, we define $$\nu(T,x,\lambda)=\nu\big(T,\log\max|z_j-x_j|^{\lambda_j}\big). \leqno(6.11\,{\rm c})$$ The above formula for $\nu(dd^cV,\varphi)$ combined with the analogue of Harnack's inequality (6.9$\,$d) for polydisks gives $$\leqalignno{ \nu(dd^cV,x,\lambda)&=\lim_{r\to 0}{\lambda_1\ldots\lambda_n\over\log r} \int V(r^{1/\lambda_1}e^{\ii\theta_1},\ldots,r^{1/\lambda_n} e^{\ii\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}\cr &=\lim_{r\to 0}{\lambda_1\ldots\lambda_n\over\log r} \sup_{\theta_1,\ldots,\theta_n}V(r^{1/\lambda_1}e^{\ii\theta_1},\ldots,r^{1/\lambda_n} e^{\ii\theta_n}).&(6.11\,{\rm d})\cr}$$ \endstat \titleb{7.}{Comparison Theorems for Lelong Numbers} Let $T$ be a closed positive current of bidimension $(p,p)$ on a Stein manifold $X$ equipped with a semi-exhaustive plurisubharmonic weight~$\varphi$. We first show that the Lelong numbers $\nu(T,\varphi)$ only depend on the asymptotic behaviour of $\varphi$ near the polar set $S(-\infty)$. In a precise way: \begstat{(7.1) First comparison theorem} Let $\varphi,\psi:X\longrightarrow[-\infty,+\infty[$ be continuous plurisubharmonic functions. We assume that $\varphi,\psi$ are semi-exhaustive on $\Supp\,T$ and that $$\ell:=\limsup{{\psi(x)}\over{\varphi(x)}}<+\infty~~~~{\it as}~~ x\in\Supp\,T~~{\it and}~~\varphi(x)\to -\infty.$$ Then $\nu(T,\psi)\le \ell^p\nu(T,\varphi)$, and the equality holds if $\ell=\lim\psi/\varphi$. \endstat \begproof{} Definition 6.4 shows immediately that $\nu(T,\lambda\varphi)=\lambda^p\nu(T,\varphi)$ for every scalar $\lambda>0$. It is thus sufficient to verify the inequality $\nu(T,\psi)\le\nu(T,\varphi)$ under the hypothesis $\limsup\psi/\varphi<1$. For all $c>0$, consider the plurisubharmonic function $$u_c=\max(\psi-c,\varphi).$$ Let $R_\varphi$ and $R_\psi$ be such that $B_\varphi(R_\varphi)\cap\Supp\,T$ and $B_\psi(R_\psi)\cap\Supp\,T$ be relatively compact in $X$. Let $r0$ large enough, we have $u_c=\varphi$ on $\varphi^{-1}([a,r])$ and Stokes' formula gives $$\nu(T,\varphi,r)=\nu(T,u_c,r)\ge\nu(T,u_c).$$ The hypothesis $\limsup\psi/\varphi<1$ implies on the other hand that there exists $t_0<0$ such that $u_c=\psi-c$ on $\{u_c0$ we have $$\big(dd^c\log\max_{1\le j\le n}|z_j|^{\lambda_j}\big)^n= \Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\lambda_j}\Big)^n= \lambda_1\ldots\lambda_n\,\delta_0.$$ \endstat \begproof{} In fact, our measures vanish on $\bC^n\ssm\{0\}$ by the arguments explained in example~6.11. Hence they are equal to $c\,\delta_0$ for some constant $c\ge 0$ which is simply the Lelong number of the bidimension $(n,n)$-current $T=[X]=1$ with the corresponding weight. The comparison theorem shows that the first equality holds and that $$\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\lambda_j}\Big)^n= \ell^{-n}\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\ell\lambda_j}\Big)^n$$ for all $\ell>0$. By taking $\ell$ large and approximating $\ell\lambda_j$ with $2[\ell\lambda_j/2]$, we may assume that $\lambda_j=2s_j$ is an even integer. Then formula (5.6) gives $$\eqalign{ &\int_{\sum|z_j|^{2s_j}0$, we thus get $$\eqalign{ \int_{Y_r^\star}\pi_0^\star T\wedge\omega_Y^{N+p} &=\sum_{0\le k\le p}\lambda_k\int_{B(0,r)\ssm\{0\}}T\wedge\beta^k\wedge (dd^c\log|z|)^{p-k}\cr &=\sum_{0\le k\le p}\lambda_k2^{-(p-k)}r^{-2(p-k)} \int_{B(0,r)\ssm\{0\}}T\wedge\beta^p<+\infty.\cr}$$ The Skoda-El Mir theorem 2.3 shows that the trivial extension $\wt\pi_0^\star T$ of $\pi_0^\star T$ is a closed positive current on~$Y_R$. Of course, the zero section $\pi^{-1}(0)$ might also carry some extra mass of the desired current~$\pi^\star T$. Since $\pi^{-1}(0)$ has codimension~$q$, this extra mass cannot exist when $q>n-p=\codim\pi^\star T$ and we simply set $\pi^\star T=\wt\pi_0^\star T$. On the other hand, if $q=n-p$, we set $$\pi^\star T:=\wt\pi_0^\star T+\nu(T,0)\,[\pi^{-1}(0)].\leqno(7.12)$$ We can now apply parallel slicing with respect to $\sigma:Y_R\to G(q,n)$, which is a submersion: for almost all $S\in G(q,n)$, there is a well defined slice $T_{\restriction S}= (\pi^\star T)_{\restriction\sigma^{-1}(S)}$. These slices coincide with the usual restrictions of $T$ to $S$ if $T$ is smooth. \begstat{(7.13) Theorem {\rm (Siu~1974)}} For almost all $S\in G(q,n)$ with $q\ge n-p$, the slice $T_{\restriction S}$ satisfies~ $\nu(T_{\restriction S},0)=\nu(T,0)$. \endstat \begproof{} If $q=n-p$, the slice $T_{\restriction S}$ consists of some positive measure with support in $S\ssm\{0\}$ plus a Dirac measure $\nu(T,0)\,\delta_0$ coming from the slice of $\nu(T,0)\,[\pi^{-1}(0)]$. The equality $\nu(T_{\restriction S},0)= \nu(T,0)$ thus follows directly from~(7.12). In the general case $q>n-p$, it is clearly sufficient to prove the following two properties: \smallskip \noindent a)~~ $\nu(T,0,r)=\displaystyle\int_{S\in G(q,n)} \nu(T_{\restriction S},0,r)\,dv(S)$~~ for all $r\in{}]0,R[\,$; \smallskip \noindent b)~~ $\nu(T_{\restriction S},0)\ge\nu(T,0)$~~ for almost all $S$. \smallskip \noindent In fact, a) implies that $\nu(T,0)$ is the average of all Lelong numbers $\nu(T_{\restriction S},0)$ and the conjunction with b) implies that these numbers must be equal to $\nu(T,0)$ for almost all $S$. In order to prove a) and b), we can suppose without loss of generality that $T$ is smooth on $B(0,R)\ssm\{0\}$. Otherwise, we perform a small convolution with respect to the action of ${\rm Gl}_n(\bC)$ on $\bC^n$: $$T_\varepsilon=\int_{g\in{\rm Gl}_n(\bC)}\rho_\varepsilon(g)\, g^\star T\,dv(g)$$ where $(\rho_\varepsilon)$ is a regularizing family with support in an $\varepsilon$-neighborhood of the unit element of ${\rm Gl}_n(\bC)$. Then $T_\varepsilon$ is smooth in $B(0,(1-\varepsilon)R)\ssm\{0\}$ and converges weakly to $T$. Moreover, we have $\nu(T_\varepsilon,0)=\nu(T,0)$ by (7.2) and $\nu(T_{\restriction S},0)\ge\limsup_{\varepsilon\to 0}\nu(T_{\varepsilon, \restriction S},0)$ by (5.12), thus a), b) are preserved in the limit. If $T$ is smooth on $B(0,R)\ssm\{0\}$, the slice $T_{\restriction S}$ is defined for all $S$ and is simply the restriction of $T$ to $S\ssm\{0\}$ (carrying no mass at the origin). \smallskip \noindent a) Here we may even assume that $T$ is smooth at $0$ by performing an ordinary convolution. As $T_{\restriction S}$ has bidegree $(n-p,n-p)$, we have $$\nu(T_{\restriction S},0,r)=\int_{S\cap B(0,r)}T\wedge\alpha_S^{q-(n-p)} =\int_{B(0,r)}T\wedge [S]\wedge\alpha_S^{p+q-n}$$ where $\alpha_S=dd^c\log|w|$ and $w=(w_1,\ldots,w_q)$ are orthonormal coordinates on~$S$. We simply have to check that $$\int_{S\in G(q,n)}[S]\wedge\alpha_S^{p+q-n}\,dv(S)=(dd^c\log|z|)^p.$$ However, both sides are unitary and homothety invariant $(p,p)$-forms with Lelong number $1$ at the origin, so they must coincide by Lemma~7.11. \smallskip \noindent b) We prove the inequality when $S=\bC^q\times\{0\}$. By the comparison theorem~7.1, for every $r>0$ and $\varepsilon>0$ we have $$\leqalignno{ &\int_{B(0,r)}T\wedge\gamma_\varepsilon^p\ge\nu(T,0)~~~~\hbox{\rm where} &(7.14)\cr &\gamma_\varepsilon={1\over 2}dd^c\log(\varepsilon|z_1|^2+\ldots+ \varepsilon|z_q|^2+|z_{q+1}|^2+\ldots+|z_n|^2).\cr}$$ We claim that the current $\gamma_\varepsilon^p$ converges weakly to $$[S]\wedge\alpha_S^{p+q-n}= [S]\wedge\Big({1\over2}dd^c\log(|z_1|^2+\ldots+|z_q|^2)\Big)^{p+q-n}$$ as $\varepsilon$ tends to $0$. In fact, the Lelong number of $\gamma_\varepsilon^p$ at $0$ is $1$, hence by homogeneity $$\int_{B(0,r)}\gamma_\varepsilon^p\wedge(dd^c|z|^2)^{n-p}=(2r^2)^{n-p}$$ for all $\varepsilon,r>0$. Therefore the family $(\gamma_\varepsilon^p)$ is relatively compact in the weak topology. Since $\gamma_0=\lim\gamma_\varepsilon$ is smooth on $\bC^n\ssm S$ and depends only on $n-q$ variables ($n-q\le p$), we have $\lim\gamma_\varepsilon^p=\gamma_0^p=0$ on $\bC^n\ssm S$. This shows that every weak limit of $(\gamma_\varepsilon^p)$ has support in $S$. Each of these is the direct image by inclusion of a unitary and homothety invariant $(p+q-n,p+q-n)$-form on $S$ with Lelong number equal to $1$ at $0$. Therefore we must have $$\lim_{\varepsilon\to 0}\gamma_\varepsilon^p=(i_S)_\star (\alpha_S^{p+q-n})=[S]\wedge\alpha_S^{p+q-n},$$ and our claim is proved (of course, this can also be checked by direct elementary calculations). By taking the limsup in (7.14) we obtain $$\nu(T_{\restriction S},0,r+0)=\int_{\ol B(0,r)}T\wedge[S]\wedge \alpha_S^{p+q-n}\ge\nu(T,0)$$ (the singularity of $T$ at $0$ does not create any difficulty because we can modify $T$ by a $dd^c$-exact form near $0$ to make it smooth everywhere). Property b) follows when $r$ tends to $0$.\qed \endproof \titleb{8.}{Siu's Semicontinuity Theorem} Let $X$, $Y$ be complex manifolds of dimension $n$, $m$ such that $X$ is Stein. Let $\varphi:X\times Y\longrightarrow[-\infty,+\infty[$ be a continuous plurisubharmonic function. We assume that $\varphi$ is {\it semi-exhaustive} with respect to $\Supp\,T$ , i.e. that for every compact subset $L\subset Y$ there exists $R=R(L)<0$ such that $$\{(x,y)\in\Supp\,T\times L\,; \,\varphi(x,y)\le R\}\compact X\times Y.\leqno(8.1)$$ Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$. For every point $y\in Y$, the function $\varphi_y(x):=\varphi(x,y)$ is semi-exhaustive on $\Supp\,T\,$; one can therefore associate with $y$ a generalized Lelong number $\nu(T,\varphi_y)$. Proposition~5.13 implies that the map $y\mapsto\nu(T,\varphi_y)$ is upper semi-continuous, hence the upperlevel sets $$E_c=E_c(T,\varphi)=\{y\in Y ;\nu(T,\varphi_y)\ge c\}\ ,\ c>0\leqno(8.2)$$ are closed. Under mild additional hypotheses, we are going to show that the sets $E_c$ are in fact analytic subsets of $Y$ (Demailly 1987a). \begstat{(8.3) Definition} We say that a function $f(x,y)$ is locally H\"older continuous with respect to $y$ on $X\times Y$ if every point of $X\times Y$ has a neighborhood $\Omega$ on which $$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$ for all $(x,y_1)\in\Omega$, $(x,y_2)\in\Omega$, with some constants $M>0$, $\gamma\in{}]0,1]$, and suitable coordinates on $Y$. \endstat \begstat{(8.4) Theorem {\rm(Demailly 1987a)}} Let $T$ be a closed positive current on $X$ and let $$\varphi:X\times Y\longrightarrow[-\infty,+\infty[$$ be a continuous plurisubharmonic function. Assume that $\varphi$ is semi-exhaustive on $\Supp\,T$ and that $e^{\varphi(x,y)}$ is locally H\"older continuous with respect to $y$ on~$X\times Y$. Then the upperlevel sets $$E_c(T,\varphi)=\{y\in Y;\nu(T,\varphi_y)\ge c\}$$ are analytic subsets of $Y$. \endstat This theorem can be rephrased by saying that $y\longmapsto\nu(T,\varphi_y)$ is upper semi-continuous with respect to the analytic Zariski topology. As a special case, we get the following important result of (Siu~1974): \begstat{(8.5) Corollary} If $T$ is a closed positive current of bidimension $(p,p)$ on a complex manifold~$X$, the upperlevel sets $E_c(T)=\{x\in X\,;\,\nu(T,x)\ge c\}$ of the usual Lelong numbers are analytic subsets of dimension $\le p$. \endstat \begproof{} The result is local, so we may assume that $X\subset\bC^n$ is an open subset. Theorem~8.4 with $Y=X$ and $\varphi(x,y)=\log|x-y|$ shows that $E_c(T)$ is analytic. Moreover, Prop.~5.11 implies $\dim E_c(T)\le p$.\qed \endproof \noindent{\bf(8.6) Generalization.} Theorem 8.4 can be applied more generally to weight functions of the type $$\varphi(x,y)=\max_j~\log\Big(\sum_k|F_{j,k}(x,y)|^{\lambda_{j,k}}\Big)$$ where $F_{j,k}$ are holomorphic functions on $X\times Y$ and where $\gamma_{j,k}$ are positive real constants; in this case $e^\varphi$ is H\"older continuous of exponent $\gamma=\min\{\lambda_{j,k},1\}$ and $\varphi$ is semi-exhaustive with respect to the whole space $X$ as soon as the projection ${\rm pr}_2:\bigcap F_{j,k}^{-1}(0)\longrightarrow Y$ is proper and finite. For example, when $\varphi(x,y)=\log\max|x_j-y_j|^\lambda_j$ on an open subset $X$ of $\bC^n$ , we see that the upperlevel sets for Kiselman's numbers $\nu(T,x,\lambda)$ are analytic in $X$ (a result first proved in (Kiselman~1986). More generally, set $\psi_\lambda(z)=\log\max|z_j|^{\lambda_j}$ and $\varphi(x,y,g)=\psi_\lambda\big(g(x-y)\big)$ where $x,y\in\bC^n$ and $g\in{\rm Gl}(\bC^n)$. Then $\nu(T,\varphi_{y,g})$ is the Kiselman number of $T$ at $y$ when the coordinates have been rotated by $g$. It is clear that $\varphi$ is plurisubharmonic in $(x,y,g)$ and semi-exhaustive with respect to $x$, and that $e^\varphi$ is locally H\"older continuous with respect to $(y,g)$. Thus the upperlevel sets $$E_c=\{(y,g)\in X\times{\rm Gl}(\bC^n)\,;\,\nu(T,\varphi_{y,g})\ge c\}$$ are analytic in $X\times{\rm Gl}(\bC^n)$. However this result is not meaningful on a manifold, because it is not invariant under coordinate changes. One can obtain an invariant version as follows. Let $X$ be a manifold and let $J^k\cO_X$ be the bundle of $k$-jets of holomorphic functions on~$X$. We consider the bundle $S_k$ over $X$ whose fiber $S_{k,y}$ is the set of $n$-tuples of $k$-jets $u=(u_1,\ldots,u_n)\in(J^k\cO_{X,y})^n$ such that $u_j(y)=0$ and $du_1\wedge\ldots\wedge du_n(y)\ne 0$. Let $(z_j)$ be local coordinates on an open set $\Omega\subset X$. Modulo $O(|z-y|^{k+1})$, we can write $$u_j(z)=\sum_{1\le|\alpha|\le k}a_{j,\alpha}(z-y)^\alpha$$ with ${\rm det}(a_{j,(0,\ldots,1_k,\ldots,0)})\ne 0$. The numbers $((y_j),(a_{j,\alpha}))$ define a coordinate system on the total space of $S_{k\,\restriction\Omega}$. For $(x,(y,u))\in X\times S_k$, we introduce the function $$\varphi(x,y,u)=\log\max|u_j(x)|^{\lambda_j}= \log\max\Big|\sum_{1\le|\alpha|\le k}a_{j,\alpha}(x-y)^\alpha \Big|^{\lambda_j}$$ which has all properties required by Th.~8.4 on a neighborhood of the diagonal $x=y$, i.e. a neighborhood of $X\times_X S_k$ in $X\times S_k$. For $k$ large, we claim that Kiselman's directional Lelong numbers $$\nu(T,y,u,\lambda):=\nu(T,\varphi_{y,u})$$ with respect to the coordinate system $(u_j)$ at $y$ do not depend on the selection of the jet representives and are therefore canonically defined on~$S_k$. In fact, a change of $u_j$ by $O(|z-y|^{k+1})$ adds $O(|z-y|^{(k+1)\lambda_j})$ to $e^\varphi$, and we have $e^\varphi\ge O(|z-y|^{\max\lambda_j})$. Hence by (7.1) it is enough to take $k+1\ge\max\lambda_j/\min\lambda_j$. Theorem~8.4 then shows that the upperlevel sets $E_c(T,\varphi)$ are analytic in~$S_k$.\qed \medskip \titled{Proof of the Semicontinuity Theorem~8.4} As the result is local on $Y$, we may assume without loss of generality that $Y$ is a ball in $\bC^m$. After addition of a constant to $\varphi$, we may also assume that there exists a compact subset $K\subset X$ such that $$\{(x,y)\in X\times Y ;\varphi(x,y)\le 0\}\subset K\times Y.$$ By Th.~7.1, the Lelong numbers depend only on the asymptotic behaviour of $\varphi$ near the (compact) polar set $\varphi^{-1}(-\infty)\cap({\rm Supp T}\times Y)$. We can add a smooth strictly plurisubharmonic function on $X\times Y$ to make $\varphi$ strictly plurisuharmonic. Then Richberg's approximation theorem for continuous plurisubharmonic functions shows that there exists a smooth plurisubharmonic function $\wt\varphi$ such that $\varphi\le\wt\varphi\le\varphi+1$. We may therefore assume that $\varphi$ is smooth on $(X\times Y)\setminus\varphi^{-1}(-\infty)$. \medskip \noindent $\bullet$ {\bf First step}: {\it construction of a local plurisubharmonic potential.} Our goal is to generalize the usual construction of plurisubharmonic potentials associated with a closed positive current (Lelong 1967, Skoda 1972a). We replace here the usual kernel $|z-\zeta|^{-2p}$ arising from the hermitian metric of $\bC^n$ by a kernel depending on the weight $\varphi$. Let $\chi\in C^\infty(\bR,\bR)$ be an increasing function such that $\chi(t)=t$ for $t\le-1$ and $\chi(t)=0$ for $t\ge 0$. We consider the half-plane $H=\{z\in\bC\,;\,{\rm Re}z<-1\}$ and associate with $T$ the potential function $V$ on $Y\times H$ defined by $$V(y,z)=-\int^0_{{\rm Re}z}\nu(T,\varphi_y,t)\chi'(t)\,dt.\leqno(8.7)$$ For every $t>\Re z$, Stokes' formula gives $$\nu(T,\varphi_y,t)= \int_{\varphi(x,y)0$. If $Y$ is connected and $U_0\not\equiv-\infty$, we already conclude that the density set $\bigcup_{c>0}E_c$ is pluripolar in~$Y$. \medskip \noindent $\bullet$ {\bf Second step}: {\it application of Kiselman's minimum principle.} Let $a\ge 0$ be arbitrary. The function $$Y\times H\ni(y,z)\longmapsto\rho(y)+V(y,z)-a{\rm Re}z$$ is plurisubharmonic and independent of ${\rm Im}\,z$. By Kiselman's theorem~1.7.8, the Legendre transform $$U_a(y)=\inf_{r<-1}\big\{\rho(y)+V(y,r)-ar\big\}$$ is locally plurisubharmonic or $\equiv-\infty$ on $Y$. \begstat{(8.9) Lemma} Let $y_0\in Y$ be a given point. \medskip \item{\rm a)} If $a>\nu(T,\varphi_{y_0})$, then $U_a$ is bounded below on a neighborhood of $y_0$. \medskip \item{\rm b)} If $a<\nu(T,\varphi_{y_0})$, then $U_a(y_0)=-\infty$. \endstat \begproof{} By definition of $V$ (cf. $(8.7)$) we have $$V(y,r)\le-\nu(T,\varphi_y,r)\int_r^0\chi'(t)dt=r\nu(T,\varphi_y,r) \le r\nu(T,\varphi_y).\leqno(8.10)$$ Then clearly $U_a(y_0)=-\infty$ if $a<\nu(T,\varphi_{y_0})$. On the other hand, if $\nu(T,\varphi_{y_0})-\infty$ if $y\in Y\setminus E_c$ (apply Lemma~8.9). For any integer $k\ge 1$, let $w_k\in C^\infty(Y)$ be a plurisubharmonic regularization of $U_{c-1/k}$ such that $w_k\ge U_{c-1/k}$ on $Y$ and $w_k\le -2^k$ on\break $E_c\cap Y_k$ where $Y_k=\{y\in Y\,;\,d(y,\partial Y)\ge 1/k\}$. Then Lemma~8.9~a) shows that the family $(w_k)$ is uniformly bounded below on every compact subset of $Y\setminus E_c$. We can also choose $w_k$ uniformly bounded above on every compact subset of $Y$ because $U_{c-1/k}\le U_c$. The function $$w=\sum_{k=1}^{+\infty}~2^{-k}w_k$$ satifies our requirements.\qed \endproof \noindent $\bullet$ {\bf Third step}: {\it estimation of the singularities of the potentials} $U_a$. \begstat{(8.12) Lemma} Let $y_0\in Y$ be a given point, $L$ a compact neighborhood of $y_0$, $K\subset X$ a compact subset and $r_0$ a real number $<-1$ such that $$\{(x,y)\in X\times L;\varphi(x,y)\le r_0\}\subset K\times L.$$ Assume that $e^{\varphi(x,y)}$ is locally H\"older continuous in $y$ and that $$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$ for all $(x,y_1,y_2)\in K\times L\times L$. Then, for all $\varepsilon\in{}]0,1[$, there exists a real number $\eta(\varepsilon)>0$ such that all $y\in Y$ with $|y-y_0|<\eta(\varepsilon)$ satisfy $$U_a(y)\le\rho(y)+\big((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\big) \Big(\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\Big).$$ \endstat \begproof{} First, we try to estimate $\nu(T,\varphi_y,r)$ when $y\in L$ is near $y_0$. Set $$\left\{ \eqalign{\psi(x)&=(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2\cr \psi(x)&=\max\bigl(\varphi_y(x),(1-\varepsilon)\varphi_{y_0}(x)+ \varepsilon r-\varepsilon/2\bigr)\cr \psi(x)&=\varphi_y(x)\cr} \eqalign{&{~\rm if~}\cr &{~\rm if~}\cr &{~\rm if~}\cr} \eqalign{&\varphi_{y_0}(x)\le r-1\cr r-1\le&\varphi_{y_0}(x)\le r\cr r\le&\varphi_{y_0}(x)\le r_0\cr}\right.$$ and verify that this definition is coherent when $|y-y_0|$ is small enough. By hypothesis $$|e^{\varphi_y(x)}-e^{\varphi_{y_0}(x)}|\le M|y-y_0|^\gamma.$$ This inequality implies $$\eqalign{\varphi_y(x)&\le\varphi_{y_0}(x)+\log\bigl(1+M|y-y_0|^\gamma e^{-\varphi_{y_0}(x)}\bigr)\cr \varphi_y(x)&\ge\varphi_{y_0}(x)+\log\bigl(1-M|y-y_0|^\gamma e^{-\varphi_{y_0}(x)}\bigr).\cr}$$ In particular, for $\varphi_{y_0}(x)=r$, we have $(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2=r-\varepsilon/2$, thus $$\varphi_y(x)\ge r+\log(1-M|y-y_0|^\gamma e^{-r}).$$ Similarly, for $\varphi_{y_0}(x)=r-1$, we have $(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2= r-1+\varepsilon/2$, thus $$\varphi_y(x)\le r-1+\log(1+M|y-y_0|^\gamma e^{1-r}).$$ The definition of $\psi$ is thus coherent as soon as $M|y-y_0|^\gamma e^{1-r}\le\varepsilon/2$ , i.e. $$\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\le r.$$ In this case $\psi$ coincides with $\varphi_y$ on a neighborhood of $\{\psi=r\}$ , and with $$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2$$ on a neighborhood of the polar set $\psi^{-1}(-\infty)$. By Stokes' formula applied to $\nu(T,\psi,r)$, we infer $$\nu(T,\varphi_y,r)=\nu(T,\psi,r)\ge\nu(T,\psi)= (1-\varepsilon)^p\nu(T,\varphi_{y_0}).$$ From (8.10) we get $V(y,r)\le r\nu(T,\varphi_y,r)$, hence $$\leqalignno{ U_a(y)&\le\rho(y)+V(y,r)-ar\le \rho(y)+r\big(\nu(T,\varphi_y,r)-a\big),\cr U_a(y)&\le\rho(y)+r\bigl((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\bigr). &(8.13)\cr}$$ Suppose $\gamma\log|y-y_0|+\log(2eM/\varepsilon)\le r_0$ , i.e. $|y-y_0|\le(\varepsilon/2eM)^{1/\gamma}e^{r_0/\gamma}\,$; one can then choose $r=\gamma\log|y-y_0|+\log(2eM/\varepsilon)$, and by $(8.13)$ this yields the inequality asserted in Th.~8.12.\qed \endproof \medskip \noindent $\bullet$ {\bf Fourth step}: {\it application of the H\"ormander-Bombieri-Skoda theorem.} The end of the proof relies on the following crucial result, which is a consequence of the H\"ormander-Bombieri-Skoda theorem (Bombieri~1970, Skoda~1972a, Skoda~1976); it will be proved in Chapter~8, see Cor.~8.?.?. \begstat{(8.14) Proposition} Let $u$ be a plurisubharmonic function on a complex mani\-fold~$Y$. The set of points in a neighborhood of which $e^{-u}$ is not integrable is an analytic subset of $Y$.\qed \endstat \begproof{of Theorem~8.4 (end).} The main idea in what follows is due to (Kiselman 1979). For $a,b>0$, we let $Z_{a,b}$ be the set of points in a neighborhood of which $\exp(-U_a/b)$ is not integrable. Then $Z_{a,b}$ is analytic, and as the family $(U_a)$ is increasing in $a$, we have $Z_{a',b'}\supset Z_{a'',b''}$ if $a'\le a''$, $b'\le b''$. Let $y_0\in Y$ be a given point. If $y_0\notin E_c$, then $\nu (T,\varphi_{y_0})0$. On the other hand, if $y_0\in E_c$ and if $a0$ that $$U_a(y)\le(1-\varepsilon)(c-a)\gamma\log|y-y_0|+C(\varepsilon)$$ on a neighborhood of $y_0$. Hence $\exp(-U_a/b)$ is non integrable at $y_0$ as soon as $b<(c-a)\gamma/2m$, where $m=\dim Y$. We obtain therefore $$E_c=\bigcap_{{\scriptstyle am_A}A\cap E_c(T).$$ However, for $c>m_A$, the intersection $A\cap E_c(T)$ is a proper analytic subset of~$A$.\qed \endproof \begstat{(8.16) Proposition} Let $T$ be a closed positive current of bidimension $(p,p)$ and let $A$ be an irreducible $p$-dimensional analytic subset of $X$. Then \hbox{$\bOne_AT=m_A[A]$,} in particular $T-m_A[A]$ is positive. \endstat \begproof{} As the question is local and as a closed positive current of bidimension $(p,p)$ cannot carry any mass on a $(p-1)$-dimensional analytic subset, it is enough to work in a neighborhood of a regular point $x_0\in A$. Hence, by choosing suitable coordinates, we can suppose that $X$ is an open set in $\bC^n$ and that $A$ is the intersection of $X$ with a $p$-dimensional linear subspace. Then, for every point $a\in A$, the inequality $\nu(T,a)\ge m_A$ implies $$\sigma_T\big(B(a,r)\big)\ge m_A\,\pi^pr^{2p}/p!= m_A\sigma_{[A]}\big(B(a,r)\big)$$ for all $r$ such that $B(a,r)\subset X$. Now, set $\Theta=T-m_A[A]$ and $\beta=dd^c|z|^2$. Our inequality says that $\int\bOne_{B(a,r)}\Theta\wedge\beta^p\ge 0$. If we integrate this with respect to some positive continuous function $f$ with compact support in $A$, we get $\int_X g_r\Theta\wedge\beta^p\ge 0$ where $$g_r(z)=\int_A\bOne_{B(a,r)}(z)\,f(a)\,d\lambda(a) =\int_{a\in A\cap B(z,r)}f(a)\,d\lambda(a).$$ Here $g_r$ is continuous on $\bC^n$, and as $r$ tends to $0$ the function $g_r(z)/(\pi^pr^{2p}/p!)$ converges to $f$ on $A$ and to $0$ on $X\ssm A$, with a global uniform bound. Hence we obtain $\int\bOne_Af\,\Theta\wedge\beta^p\ge 0$. Since this inequality is true for all continuous functions $f\ge 0$ with compact support in $A$, we conclude that the measure $\bOne_A\Theta\wedge\beta^p$ is positive. By a linear change of coordinates, we see that $$\bOne_A\Theta\wedge\Big(dd^c\sum_{1\le j\le n} \lambda_j|u_j|^2\Big)^n\ge 0$$ for every basis $(u_1,\ldots,u_n)$ of linear forms and for all coefficients~\hbox{$\lambda_j>0$.} Take $\lambda_1=\ldots=\lambda_p=1$ and let the other $\lambda_j$ tend to~$0$. Then we get \hbox{$\bOne_A\Theta\wedge\ii du_1\wedge d\ol u_1\wedge\ldots\wedge du_p\wedge d\ol u_p\ge 0$.} This implies $\bOne_A\Theta\ge 0$, or equivalently $\bOne_AT\ge m_A[A]$. By Cor.~2.4 we know that $\bOne_AT$ is a closed positive current, thus $\bOne_AT=\lambda[A]$ with $\lambda\ge 0$. We have just seen that $\lambda\ge m_A$. On the other hand, $T\ge\bOne_AT=\lambda[A]$ clearly implies $m_A\ge\lambda$.\qed \endproof \begstat{(8.16) Siu's decomposition formula} If $T$ is a closed positive current of bidimension~$(p,p)$, there is a unique decomposition of $T$ as a (possibly finite) weakly convergent series $$T=\sum_{j\ge 1}\lambda_j[A_j]+R,~~~~\lambda_j>0,$$ where $[A_j]$ is the current of integration over an irreducible $p$-dimensional ana\-lytic set $A_j\subset X$ and where $R$ is a closed positive current with the property that $\dim E_c(R)0$. \endstat \begproof{of uniqueness.} If $T$ has such a decomposition, the $p$-dimensional components of $E_c(T)$ are $(A_j)_{\lambda_j\ge c}$, for $\nu(T,x)=\sum\lambda_j\nu([A_j],x)+\nu(R,x)$ is non zero only on $\bigcup A_j\cup\bigcup E_c(R)$, and is equal to $\lambda_j$ generically on $A_j$ $\big($more precisely, $\nu(T,x)=\lambda_j$ at every regular point of $A_j$ which does not belong to any intersection $A_j\cup A_k$, $k\ne j$ or to $\bigcup E_c(R)\big)$. In particular $A_j$ and $\lambda_j$ are unique. \endproof \begproof{of existence.} Let $(A_j)_{j\ge 1}$ be the countable collection of $p$-dimensional components occurring in one of the sets $E_c(T)$, $c\in\bQ_+^\star$, and let $\lambda_j>0$ be the generic Lelong number of $T$ along $A_j$. Then Prop.~8.16 shows by induction on $N$ that $R_N=T-\sum_{1\le j\le N}\lambda_j[A_j]$ is positive. As $R_N$ is a decreasing sequence, there must be a limit $R=\lim_{N\to+\infty}R_N$ in the weak topology. Thus we have the asserted decomposition. By construction, $R$ has zero generic Lelong number along $A_j$, so $\dim E_c(R)0$.\qed \endproof It is very important to note that some components of lower dimension can actually occur in $E_c(R)$, but they cannot be subtracted because $R$ has bidimension $(p,p)$. A typical case is the case of a bidimension \hbox{$(n-1,n-1)$} current $T=dd^cu$ with $u=\log(|F_j|^{\gamma_1}+\ldots|F_N|^{\gamma_N})$ and $F_j\in\cO(X)$. In general $\bigcup E_c(T)=\bigcap F_j^{-1}(0)$ has dimension~$0$. Let $(Z_k)_{k\ge1}$ be the irreducible components of $Z$ of codimension $p$ exactly. Then there exist multiplicities $\lambda_k>0$ such that $$(dd^cu)^p=\sum_{k\ge 1}\lambda_k[Z_k]+R,$$ where $R$ is a closed positive current such that $\bOne_ZR=0$ and $\codim E_c(R)>p$ for every $c>0$. Moreover the multiplicities $\lambda_k$ are integers if $\gamma_1,\ldots,\gamma_N$ are integers, and $\lambda_k=\gamma_1\ldots\gamma_p$ if $\gamma_1\le\ldots\le\gamma_N$ and some partial Jacobian determinant of $(F_1,\ldots,F_p)$ of order $p$ does not vanish identically along~$Z_k$. \endstat \begproof{} Observe that $(dd^cu)^p$ is well defined thanks to Cor.~4.11. The comparison theorem~7.8 applied with $\varphi(z)=\log|z-x|$, $v_1=\ldots=v_p=u$, $u_1=\ldots=u_p=\varphi$ and $T=1$ shows that the Lelong number of $(dd^cu)^p$ is equal to $0$ at every point of~$X\ssm Z$. Hence $E_c((dd^cu)^p)$ is contained in $Z$ and its $(n-p)$-dimensional components are members of the family $(Z_k)$. The asserted decomposition follows from Siu's formula~8.16. We must have $\bOne_{Z_k}R=0$ for all irreducible components of~$Z$: when $\codim Z_k>p$ this is automatically true, and when $\codim Z_k=p$ this follows from (8.16) and the fact that $\codim E_c(R)>p$. If $\det(\partial F_j/\partial z_k)_{1\le j,k\le p}\ne 0$ at some point $x_0\in Z_k$, then $(Z,x_0)=(Z_k,x_0)$ is a smooth germ defined by the equations $F_1=\ldots=F_p=0$. If we denote $v=\log\sum_{j\le p}|F_j|^{\gamma_j}$ with $\gamma_1\le\ldots\le\gamma_N$, then $u\sim v$ near $Z_k$ and Th.~7.8 implies $\nu((dd^cu)^p,x)=\nu((dd^cv)^p,x)$ for all $x\in Z_k$ near $x_0$. On the other hand, if $G:=(F_1,\ldots,F_p):X\to\bC^p$, Cor.~7.4 gives $$(dd^cv)^p=G^\star\Big(dd^c\log\sum_{1\le j\le p}|z_j|^{\gamma_j}\Big)^p= \gamma_1\ldots\gamma_p\,G^\star\delta_0=\gamma_1\ldots\gamma_p\,[Z_k]$$ near~$x_0$. This implies that the generic Lelong number of $(dd^cu)^p$ along $Z_k$ is $\lambda_k=\gamma_1\ldots\gamma_p$. The integrality of $\lambda_k$ when $\gamma_1,\ldots,\gamma_N$ are integers will be proved in the next section.\qed \endproof \titleb{9.}{Transformation of Lelong Numbers by Direct Images} Let $F:X\to Y$ be a holomorphic map between complex manifolds of respective dimensions $\dim X=n$, $\dim Y=m$, and let $T$ be a closed positive current of bidimension $(p,p)$ on~$X$. If $F_{\restriction\Supp\,T}$ is proper, the direct image $F_\star T$ is defined by $$\langle F_\star T,\alpha\rangle=\langle T,F^\star\alpha\rangle \leqno(9.1)$$ for every test form $\alpha$ of bidegree $(p,p)$ on $Y$. This makes sense because $\Supp\,T\cap F^{-1}(\Supp\,\alpha)$ is compact. It is easily seen that $F_\star T$ is a closed positive current of bidimension $(p,p)$ on $Y$. \begstat{(9.2) Example} \rm Let $T=[A]$ where $A$ is a $p$-dimensional irreducible analytic set in $X$ such that $F_{\restriction A}$ is proper. We know by Remmert's theorem~2.7.8 that $F(A)$ is an analytic set in~$Y$. Two cases may occur. Either $F_{\restriction A}$ is generically finite and $F$ induces an \'etale covering $A\ssm F^{-1}(Z)\longrightarrow F(A)\ssm Z$ for some nowhere dense analytic subset $Z\subset F(A)$, or $F_{\restriction A}$ has generic fibers of positive dimension and $\dim F(A)<\dim A$. In the first case, let $s<+\infty$ be the covering degree. Then for every test form $\alpha$ of bidegree $(p,p)$ on $Y$ we get $$\langle F_\star[A],\alpha\rangle=\int_A F^\star\alpha=\int_{A\ssm F^{-1}(Z)} F^\star\alpha=s\int_{F(A)\ssm Z}\alpha=s\,\langle[F(A)],\alpha\rangle$$ because $Z$ and $F^{-1}(Z)$ are negligible sets. Hence $F_\star[A]= s[F(A)]$. On the other hand, if $\dim F(A)<\dim A=p$, the restriction of $\alpha$ to $F(A)_\reg$ is zero, and therefore so is this the restriction of $F^\star\alpha$ to $A_\reg$. Hence $F_\star[A]=0$.\qed \endstat Now, let $\psi$ be a continuous plurisubharmonic function on $Y$ which is semi-exhaustive on $F(\Supp\,T)$ (this set certainly contains $\Supp F_\star T$). Since $F_{\restriction\Supp\,T}$ is proper, it follows that $\psi\circ F$ is semi-exhaustive on $\Supp\,T$, for $$\Supp\,T\cap\{\psi\circ F0\,;\,\exists C>0,\,|H(z)|\ge C|z-x|^\alpha~ \hbox{\rm near}~x\big\},&({\rm b})\cr \ol\mu_p(F,x)&=\inf_G{\sigma(G\circ F,x)^p\over\mu_p(G,0)},&({\rm c})\cr}$$ where $G$ runs over all germs of maps $(Y,y)\longrightarrow(\bC^n,0)$ such that $G\circ F$ is finite. \endstat \begproof{} If $F^{-1}(y)=\{x_1,\ldots,x_N\}$, there is a neighborhood $W$ of $y$ and disjoint neighborhoods $V_j$ of $x_j$ such that $F^{-1}(W)=\bigcup V_j$. Then $F_\star T=\sum(F_{\restriction V_j})_\star T$ on $W$, so it is enough to consider the case when $F^{-1}(y)$ consists of a single point~$x$. Therefore, we assume that $F:V\to W$ is proper and finite, where $V$, $W$ are neighborhoods of $0$ in $\bC^n$, $\bC^m$ and $F^{-1}(0)=\{0\}$. Let \hbox{$G:(\bC^m,0)\longrightarrow(\bC^n,0)$} be a germ of map such that $G\circ F$ is finite. Hilbert's Nullstellensatz shows that there exists $\alpha>0$ and $C>0$ such that $|G\circ F(z)|\ge C|z|^\alpha$ near~$0$. Then the comparison theorem~7.1 implies $$\nu(G_\star F_\star T,0)=\nu(T,\log|G\circ F|)\le\alpha^p\nu(T,\log|z|)= \alpha^p\nu(T,0).$$ On the other hand, Th.~9.9 applied to $\Theta=F_\star T$ on $W$ gives $$\nu(G_\star F_\star T,0)\ge\mu_p(G,0)\,\nu(F_\star T,0).$$ Therefore $$\nu(F_\star T,0)\le{\alpha^p\over\mu_p(G,0)}\nu(T,0).$$ The infimum of all possible values of $\alpha$ is by definition $\sigma(G\circ F,0)$, thus by taking the infimum over $G$ we obtain $$\nu(F_\star T,0)\le\ol\mu_p(F,0)\,\nu(T,0).\eqno{\square}$$ \endproof \begstat{(9.13) Example} \rm Let $F(z_1,\ldots,z_n)=(z_1^{s_1},\ldots,z_n^{s_n})$, $s_1\le\ldots\le s_n$ as in~9.8. Then we have $$\mu_p(F,0)=s_1\ldots s_p,~~~~~~\ol\mu_p(F,0)=s_{n-p+1}\ldots s_n.$$ To see this, let $s$ be the lowest common multiple of $s_1,\ldots,s_n$ and let $G(z_1,\ldots,z_n)=(z_1^{s/s_1},\ldots,z_n^{s/s_n})$. Clearly $\mu_p(G,0)=(s/s_{n-p+1})\ldots(s/s_n)$ and $\sigma(G\circ F,0)=s$, so we get by definition $\ol\mu_p(F,0)\le s_{n-p+1}\ldots s_n$. Finally, if $T=[A]$ is the current of integration over the $p$-dimensional subspace $A=\{z_1=\ldots=z_{n-p}=0\}$, then $F_\star[A]= s_{n-p+1}\ldots s_n\,[A]$ because $F_{\restriction A}$ has covering degree $s_{n-p+1}\ldots s_n$. Theorem~9.12 shows that we must have $s_{n-p+1}\ldots s_n\le\ol\mu_p(F,0)$, QED. If $\lambda_1\le\ldots\le\lambda_n$ are positive real numbers and $s_j$ is taken to be the integer part of $k\lambda_j$ as $k$ tends to $+\infty$, Theorems~9.9 and 9.12 imply in the limit the following: \endstat \begstat{(9.14) Corollary} For $0<\lambda_1\le\ldots\le\lambda_n$, Kiselman's directional Lelong numbers satisfy the inequalities $$\lambda_1\ldots\lambda_p\,\nu(T,x)\le\nu(T,x,\lambda)\le \lambda_{n-p+1}\ldots\lambda_n\,\nu(T,x).\eqno{\square}$$ \endstat \begstat{(9.15) Remark} \rm It would be interesting to have a direct geometric interpretation of~$\ol\mu_p(F,x)$. In fact, we do not even know whether $\ol\mu_p(F,x)$ is always an integer. \endstat \titleb{10.}{A Schwarz Lemma. Application to Number Theory} In this section, we show how Jensen's formula and Lelong numbers can be used to prove a fairly general Schwarz lemma relating growth and zeros of entire functions in~$\bC^n$. In order to simplify notations, we denote by $|F|_r$ the supremum of the modulus of a function $F$ on the ball of center $0$ and radius~$r$. Then, following (Demailly~1982a), we present some applications with a more arithmetical flavour. \begstat{(10.1) Schwarz lemma} Let $P_1,\ldots,P_N\in\bC[z_1,\ldots,z_n]$ be polynomials of degree~$\delta$, such that their homogeneous parts of degree $\delta$ do not vanish simultaneously except at~$0$. Then there is a constant \hbox{$C\ge 2$} such that for all entire functions $F\in\cO(\bC^n)$ and all $R\ge r\ge 1$ we have $$\log|F|_r\le\log|F|_R-\delta^{1-n}\nu([Z_F],\log|P|)\,\log{R\over Cr}$$ where $Z_F$ is the zero divisor of $F$ and $P=(P_1,\ldots,P_N):\bC^n\longrightarrow \bC^N$. Moreover $$\nu([Z_F],\log|P|)\ge\sum_{w\in P^{-1}(0)}\ord(F,w)\,\mu_{n-1}(P,w)$$ where $\ord(F,w)$ denotes the vanishing order of $F$ at $w$ and $\mu_{n-1}(P,w)$ is the $(n-1)$-multiplicity of $P$ at $w$, as defined in $(9.5)$ and $(9.7)$. \endstat \begproof{} Our assumptions imply that $P$ is a proper and finite map. The last inequality is then just a formal consequence of formula (9.4) and Th.~9.9 applied to~$T=[Z_F]$. Let $Q_j$ be the homogeneous part of degree $\delta$ in~$P_j$. For~$z_0\in B(0,r)$, we introduce the weight functions $$\varphi(z)=\log|P(z)|,~~~~~~\psi(z)=\log|Q(z-z_0)|.$$ Since $Q^{-1}(0)=\{0\}$ by hypothesis, the homogeneity of $Q$ shows that there are constants $C_1,C_2>0$ such that $$C_1|z|^\delta\le|Q(z)|\le C_2|z|^\delta~~~~\hbox{\rm on}~~\bC^n. \leqno(10.2)$$ The homogeneity also implies $(dd^c\psi)^n=\delta^n\,\delta_{z_0}$. We apply the Lelong Jensen formula~6.5 to the measures $\mu_{\psi,s}$ associated with $\psi$ and to~$V=\log|F|$. This gives $$\mu_{\psi,s}(\log|F|)-\delta^n\log|F(z_0)|=\int_{-\infty}^sdt\int_{\{\psi\psi$ near $\{\psi=t-2\}$ and $(1-\varepsilon)\varphi+\varepsilon t-\varepsilon<\psi$ near $\{\psi=t\}$. Then Stokes' theorem implies $$\leqalignno{ \int_{\{\psi-C_5re^{(2-t)/\delta}+\varepsilon$ on $\{\psi=t-2\}$. Hence it is sufficient to take $\varepsilon=C_5re^{(2-t)/\delta}$. This number has to be $<1$, so we take $t\ge s_0\ge 2+\delta\log(C_5r)$. Moreover, (10.5) actually holds only if $P^{-1}(0)\subset\{\psi0$ a number $\omega_t(S)$ equal to the minimal degree of polynomials $P\in\bC[z_1,\ldots,z_n]$ which vanish at order $\ge t$ at every point of~$S$. The obvious subadditivity property $$\omega_{t_1+t_2}(S)\le\omega_{t_1}(S)+\omega_{t_2}(S)$$ easily shows that $$\Omega(S):=\inf_{t>0}{\omega_t(S)\over t}= \lim_{t\to+\infty}{\omega_t(S)\over t}.$$ We call $\omega_1(S)$ the {\it degree} of $S$ (minimal degree of algebraic hypersurfaces containing~$S$) and $\Omega(S)$ the {\it singular degree} of~$S$. If we apply Cor.~10.6 to a polynomial $F$ vanishing at order $t$ on $S$ and fix $r=1$, we get $$\log|F|_R\ge t{\delta+n(n-1)/2\over n!}\log{R\over C}+\log|F|_1$$ with $\delta=\omega_1(S)$, in particular $${\rm deg}\,F\ge t{\omega_1(S)+n(n-1)/2\over n!}.$$ The minimum of ${\rm deg}\,F$ over all such $F$ is by definition~$\omega_t(S)$. If we divide by $t$ and take the infimum over~$t$, we get the interesting inequality $${\omega_t(S)\over t}\ge\Omega(S)\ge{\omega_1(S)+n(n-1)/2\over n!}. \leqno(10.7)$$ \begstat{(10.8) Remark} \rm The constant ${\omega_1(S)+n(n-1)/2\over n!}$ in (10.6) and (10.7) is optimal for $n=1,2$ but not for~$n\ge3$. It can be shown by means of H\"ormander's $L^2$ estimates (Waldschmidt 1978) that for every $\varepsilon>0$ the Schwarz lemma (10.6) holds with coefficient $\Omega(S)-\varepsilon\,$: $$\log|F|_r\le\log|F|_R-\ord(F,S)(\Omega(S)-\varepsilon)\log{R\over C_\varepsilon r},$$ and that $\Omega(S)\ge(\omega_u(S)+1)/(u+n-1)$ for every $u\ge 1\,$; this last inequality is due to (Esnault-Viehweg 1983), who used deep tools of algebraic geometry; (Azhari 1990) reproved it recently by means of H\"ormander's $L^2$ estimates. Rather simple examples (Demailly 1982a) lead to the conjecture $$\Omega(S)\ge{\omega_u(S)+n-1\over u+n-1}~~~~\hbox{for every}~~u\ge 1.$$ The special case $u=1$ of the conjecture was first stated by (Chudnovsky 1979). \endstat Finally, let us mention that Cor.~10.6 contains Bombieri's theorem on algebraic values of meromorphic maps satisfying algebraic differential equations (Bombieri 1970). Recall that an entire function $F\in\cO(\bC^n)$ is said to be of order $\le\rho$ if for every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that $|F(z)|\le C_\varepsilon\exp( |z|^{\rho+\varepsilon})$. A meromorphic function is said to be of order $\le\rho$ if it can be written $G/H$ where $G$, $H$ are entire functions of order $\le\rho$. \begstat{(10.9) Theorem {\rm(Bombieri 1970)}} Let $F_1,\ldots,F_N$ be meromorphic functions on $\bC^n$, such that $F_1,\ldots,F_d$, $n0$, there exist constants $C_1,C_2>0$, $r\ge 1$ and an infinite sequence $G_t$ of entire functions, $t\in T\subset \bN$ $($depending on $m$ and on the choice of the points $w_j)$, such that \smallskip \item{\rm a)} $G_t$ vanishes at order $\ge t$ at all points $w_1,\ldots,w_m\,;$ \smallskip \item{\rm b)} $|G_t|_r\ge (C_1t)^{-t\,[K:\bQ]}\,;$ \smallskip \item{\rm c)} $|G_t|_{R(t)}\le C_2^t$~~ where $R(t)=(t^{d-n}/\log t)^{1/(\rho_1+\ldots+\rho_d+\varepsilon)}$. \smallskip \endstat An application of Cor.~10.6 to $F=G_t$ and $R=R(t)$ gives the desired bound for the degree $\delta$ as $t$ tends to~$+\infty$ and $\varepsilon$ tends to~$0$. If $\delta_0$ is the largest integer which satisfies the inequality of Th.~10.9, we get a contradiction if we take $\delta=\delta_0+1$. This shows that $S$ must be contained in an algebraic hypersurface of degree $\delta\le\delta_0$.\qed \titlea{Chapter IV}{\newline Sheaf Cohomology and Spectral Sequences} \begpet One of the main topics of this book is the computation of various cohomology groups arising in algebraic geometry. The theory of sheaves provides a general framework in which many cohomology theories can be treated in a unified way. The cohomology theory of sheaves will be constructed here by means of Godement's simplicial flabby resolution. However, we have emphasized the analogy with Alexander-Spanier cochains in order to give a simple definition of the cup product. In this way, all the basic properties of cohomology groups (long exact sequences, Mayer Vietoris exact sequence, Leray's theorem, relations with Cech cohomology, De Rham-Weil isomorphism theorem) can be derived in a very elementary way from the definitions. Spectral sequences and hypercohomology groups are then introduced, with two principal examples in view: the Leray spectral sequence and the Hodge-Fr\"olicher spectral sequence. The basic results concerning cohomology groups with constant or locally constant coefficients (invariance by homotopy, Poincar\'e duality, Leray-Hirsch theorem) are also included, in order to present a self-contained approach of algebraic topology. \endpet \titleb{1.}{Basic Results of Homological Algebra} Let us first recall briefly some standard notations and results of homological algebra that will be used systematically in the sequel. Let $R$ be a commutative ring with unit. A {\it differential module} $(K,d)$ is a $R$-module $K$ together with an endomorphism $d:K\to K$, called the {\it differential}, such that $d\circ d=0$. The modules of {\it cycles} and of {\it boundaries} of $K$ are defined respectively by $$Z(K)=\ker d,~~~~B(K)=\Im d.\leqno(1.1)$$ Our hypothesis $d\circ d=0$ implies $B(K)\subset Z(K)$. The {\it homology group} of $K$ is by definition the quotient module $$H(K)=Z(K)/B(K).\leqno(1.2)$$ A {\it morphism of differential modules} $\varphi:K\longrightarrow L$ is a $R$-homomorphism $\varphi:K\longrightarrow L$ such that $d\circ\varphi= \varphi\circ d\,;$ here we denote by the same symbol $d$ the differentials of $K$ and $L$. It is then clear that $\varphi\big(Z(K)\big)\subset Z(L)$ and $\varphi\big(B(K)\big)\subset B(L)$. Therefore, we get an induced morphism on homology groups, denoted $$H(\varphi):H(K)\longrightarrow H(L).\leqno(1.3)$$ It is easily seen that $H$ is a functor, i.e.\ $H(\psi\circ\varphi)= H(\psi)\circ H(\varphi)$. We say that two morphisms $\varphi,\psi:K\longrightarrow L$ are {\it homotopic} if there exists a $R$-linear map $h:K\longrightarrow L$ such that $$d\circ h+h\circ d=\psi-\varphi.\leqno(1.4)$$ Then $h$ is said to be a {\it homotopy} between $\varphi$ and $\psi$. For every cocycle $z\in Z(K)$, we infer $\psi(z)-\varphi(z)=dh(z)$, hence the maps $H(\varphi)$ and $H(\psi)$ coincide. The module $K$ itself is said to be homotopic to $0$ if $\Id_{K}$ is homotopic to $0\,$; then $H(K)=0$. \begstat{(1.5) Snake lemma} Let $$0\longrightarrow K\buildo\varphi\over\longrightarrow L\buildo\psi \over\longrightarrow M\longrightarrow 0$$ be a short exact sequence of morphisms of differential modules. Then there exists a homomorphism $\partial:H(M)\longrightarrow H(K)$, called the connecting homomorphism, and a homology exact sequence $$\eqalign{ &H(K)\buildo H(\varphi)\over {\relbar\joinrel\relbar\joinrel\longrightarrow}H(L) \buildo H(\psi)\over {\relbar\joinrel\relbar\joinrel\longrightarrow}H(M)\cr &~~~~\raise2pt\hbox{$\nwarrow$}\kern-4pt \buildo{\displaystyle\partial}\over{\hbox to 105pt{\hrulefill}} \kern-4pt\raise2pt\hbox{$\swarrow$}\cr}$$ Moreover, to any commutative diagram of short exact sequences $$\cmalign{ 0\longrightarrow&K \longrightarrow&L \longrightarrow&M \longrightarrow 0\cr &\big\downarrow&\big\downarrow&\big\downarrow\cr 0\longrightarrow&\wt K \longrightarrow&\wt L \longrightarrow&\wt M \longrightarrow 0\cr}$$ is associated a commutative diagram of homology exact sequences $$\cmalign{ &H(K)\longrightarrow&H(L)\longrightarrow&H(M)\buildo \partial\over\longrightarrow&H(K)\longrightarrow\cdots\cr &~~~\big\downarrow&~~~\big\downarrow&~~~\big\downarrow&~~~\big\downarrow\cr &H(\wt K)\longrightarrow&H(\wt L)\longrightarrow&H(\wt M)\buildo \partial\over\longrightarrow&H(\wt K)\longrightarrow\cdots.\cr}$$ \endstat \begproof{} We first define the connecting homomorphism $\partial$~: let $m\in Z(M)$ represent a given cohomology class $\{m\}$ in $H(M)$. Then $$\partial\{m\}=\{k\}\in H(K)$$ is the class of any element $k\in\varphi^{-1}d\psi^{-1}(m)$, as obtained through the following construction: $$\cmalign{ &&l\in L&\buildo{\displaystyle\psi}\over{\mapstex 24 }&~~m\in M\cr &&~~\buildo{\begarrow}\over{\Big\downarrow}~d~~&&~~~ \buildo{\begarrow}\over{\Big\downarrow}~d\cr \hfill k\in K~~&\buildo{\displaystyle\varphi} \over{\mapstex 24 }~~&dl\in L& \buildo{\displaystyle\psi}\over{\mapstex 24 }&~~0\in M.\cr}$$ The element $l$ is chosen to be a preimage of $m$ by the surjective map $\psi$~; as $\psi(dl)=d(m)=0$, there exists a unique element $k\in K$ such that $\varphi(k)=dl$. The element $k$ is actually a cocycle in $Z(K)$ because $\varphi$ is injective and $$\varphi(dk)=d\varphi(k)= d(dl)=0~~\Longrightarrow~~dk=0.$$ The map $\partial$ will be well defined if we show that the cohomology class $\{k\}$ depends only on $\{m\}$ and not on the choices made for the representatives $m$ and~$l$. Consider another representative $m'=m+dm_1$. Let $l_1\in L$ such that $\psi(l_1)=m_1$. Then $l$ has to be replaced by an element $l'\in L$ such that $$\psi(l')=m+dm_1=\psi(l+dl_1).$$ It follows that $l'=l+dl_1+\varphi(k_1)$ for some $k_1\in K$, hence $$dl'=dl+d\varphi(k_1)=\varphi(k)+\varphi(dk_1)=\varphi(k'),$$ therefore $k'=k+dk_1$ and $k'$ has the same cohomology class as $k$. Now, let us show that $\ker\partial=\Im H(\psi)$. If $\{m\}$ is in the image of $H(\psi)$, we can take $m=\psi(l)$ with $dl=0$, thus $\partial\{m\}=0$. Conversely, if $\partial\{m\}=\{k\}=0$, we have $k=dk_1$ for some $k_1\in K$, hence $dl=\varphi(k)=d\varphi(k_1)$, $z:=l-\varphi(k_1)\in Z(L)$ and $m=\psi(l)=\psi(z)$ is in $\Im H(\psi)$. We leave the verification of the other equalities $\Im H(\varphi)=\ker H(\psi)$, $\Im\partial=\ker H(\varphi)$ and of the commutation statement to the reader.\qed \endproof In most applications, the differential modules come with a natural $\bZ$-grading. A homological complex is a graded differential module $K_\bu=\bigoplus_{q\in\bZ}K_q$ together with a differential $d$ of degree $-1$, i.e.\ $d=\bigoplus d_q$ with $d_q:K_q\longrightarrow K_{q-1}$ and $d_{q-1}\circ d_q=0$. Similarly, a cohomological complex is a graded differential module $K^\bu=\bigoplus_{q\in\bZ}K^q$ with differentials $d^q:K^q\longrightarrow K^{q+1}$ such that $d^{q+1}\circ d^q=0$ (superscripts are always used instead of subscripts in that case). The corresponding (co)cycle, (co)boundary and (co)homology modules inherit a natural $\bZ$-grading. In the case of cohomology, say, these modules will be denoted $$Z^\bu(K^\bu)=\bigoplus Z^q(K^\bu),~~ B^\bu(K^\bu)=\bigoplus B^q(K^\bu),~~H^\bu(K^\bu)=\bigoplus H^q(K^\bu).$$ Unless otherwise stated, morphisms of complexes are assumed to be of degree $0$, i.e.\ of the form $\varphi^\bu=\bigoplus\varphi^q$ with $\varphi^q:K^q\longrightarrow L^q$. Any short exact sequence $$0\longrightarrow K^\bu\buildo\varphi^\bu\over\longrightarrow L^\bu\buildo\psi^\bu \over\longrightarrow M^\bu\longrightarrow 0$$ gives rise to a corresponding {\it long exact sequence} of cohomology groups $$H^q(K^\bu)\buildo H^q(\varphi^\bu)\over {\relbar\joinrel\relbar\joinrel\longrightarrow}H^q(L^\bu)\buildo H^q(\psi^\bu)\over {\relbar\joinrel\relbar\joinrel\longrightarrow}H^q(M^\bu) \buildo\partial^q\over\longrightarrow H^{q+1}(K^\bu)\buildo H^{q+1}(\varphi^\bu)\over {\relbar\joinrel\relbar\joinrel\longrightarrow\cdots}\leqno(1.6)$$ and there is a similar homology long exact sequence with a connecting homomorphism $\partial_q$ of degree $-1$. When dealing with commutative diagrams of such sequences, the following simple lemma is often useful; the proof consists in a straightforward diagram chasing. \begstat{(1.7) Five lemma} Consider a commutative diagram of $R$-modules $$\cmalign{ &A_1\hfill\longrightarrow&A_2\hfill\longrightarrow&A_3\hfill\longrightarrow&A_4\hfill\longrightarrow&A_5\cr &\big\downarrow\varphi_1&\big\downarrow\varphi_2&\big\downarrow\varphi_3 &\big\downarrow\varphi_4&\big\downarrow\varphi_5\cr &B_1\hfill\longrightarrow&B_2\hfill\longrightarrow&B_3\hfill\longrightarrow&B_4\hfill\longrightarrow&B_5\cr}$$ where the rows are exact sequences. If $\varphi_2$ and $\varphi_4$ are injective and $\varphi_1$ surjective, then $\varphi_3$ is injective. If $\varphi_2$ and $\varphi_4$ is surjective and $\varphi_5$ injective, then $\varphi_3$ is surjective. In particular, $\varphi_3$ is an isomorphism as soon as $\varphi_1,\varphi_2,\varphi_4,\varphi_5$ are isomorphisms. \endstat \titleb{2.}{The Simplicial Flabby Resolution of a Sheaf} Let $X$ be a topological space and let $\cA$ be a sheaf of abelian groups on~$X$ (see \S~II-2 for the definition). All the sheaves appearing in the sequel are assumed implicitly to be sheaves of {\it abelian groups}, unless otherwise stated. The first useful notion is that of resolution. \begstat{(2.1) Definition} A $($cohomological$)$ resolution of $\cA$ is a differential complex of sheaves $(\cL^\bu,d)$ with $\cL^q=0$, $d^q=0$ for $q<0$, such that there is an exact sequence $$0\longrightarrow\cA\buildo j\over\longrightarrow\cL^0\buildo d^0\over\longrightarrow\cL^1\longrightarrow\cdots \longrightarrow\cL^q\buildo d^q\over\longrightarrow\cL^{q+1}\longrightarrow\cdots~.$$ If $\varphi:\cA\longrightarrow\cB$ is a morphism of sheaves and $(\cM^\bu,d)$ a resolution of~$\cB$, a morphism of resolutions $\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$ is a commutative diagram $$\cmalign{ 0\longrightarrow&\cA\buildo j\over\longrightarrow&\cL^0\hfill\buildo d^0\over\longrightarrow&\cL^1\hfill\longrightarrow\cdots \longrightarrow&\cL^q\hfill\buildo d^q\over\longrightarrow&\cL^{q+1}\hfill\longrightarrow&\cdots\cr &\kern-1.7pt\big\downarrow\varphi&\big\downarrow\varphi^0 &\big\downarrow\varphi^1&\big\downarrow\varphi^q &\big\downarrow\varphi^{q+1}&\cr 0\longrightarrow&\cB\hfill\buildo j\over\longrightarrow&\cM^0\buildo d^0\over\longrightarrow&\cM^1\longrightarrow\cdots \longrightarrow&\cM^q\buildo d^q\over\longrightarrow&\cM^{q+1}\hfill\longrightarrow&\cdots~.\cr}$$ \endstat \begstat{(2.2) Example} \rm Let $X$ be a differentiable manifold and $\cE^q$ the sheaf of germs of $C^\infty$ differential forms of degree $q$ with real values. The exterior derivative $d$ defines a resolution $(\cE^\bu,d)$ of the sheaf $\bR$ of locally constant functions with real values. In fact Poincar\'e's lemma asserts that $d$ is locally exact in degree $q\ge 1$, and it is clear that the sections of $\ker d^0$ on connected open sets are constants.\qed \endstat In the sequel, we will be interested by special resolutions in which the sheaves $\cL^q$ have no local ``rigidity''. For that purpose, we introduce flabby sheaves, which have become a standard tool in sheaf theory since the publication of Godement's book (Godement 1957). \begstat{(2.3) Definition} A sheaf $\cF$ is called flabby if for every open subset $U$ of $X$, the restriction map $\cF(X)\longrightarrow\cF(U)$ is onto, i.e.\ if every section of $\cF$ on $U$ can be extended to $X$. \endstat Let $\pi:\cA\longrightarrow X$ be a sheaf on $X$. We denote by $\cA^{[0]}$ the sheaf of germs of sections $X\longrightarrow\cA$ which are {\it not necessarily continuous}. In other words, $\cA^{[0]}(U)$ is the set of all maps $f:U\longrightarrow\cA$ such that $f(x)\in\cA_x$ for all $x\in U$, or equivalently $\cA^{[0]}(U)=\prod_{x\in U}\cA_x$. It is clear that $\cA^{[0]}$ is flabby and there is a canonical injection $$j:\cA\longrightarrow\cA^{[0]}$$ defined as follows: to any $s\in\cA_x$ we associate the germ $\wt s\in\cA^{[0]}_x$ equal to the continuous section $y\longmapsto\wt s(y)$ near $x$ such that $\wt s(x)=s$. In the sequel we merely denote $\wt s:y\longmapsto s(y)$ for simplicity. The sheaf $\cA^{[0]}$ is called the {\it canonical flabby sheaf} associated to $\cA$. We define inductively $$\cA^{[q]}=(\cA^{[q-1]})^{[0]}.$$ The stalk $\cA^{[q]}_x$ can be considered as the set of equivalence classes of maps $f:X^{q+1}\longrightarrow\cA$ such that $f(x_0,\ldots,x_q)\in\cA_{x_q}$, with two such maps identified if they coincide on a set of the form $$x_0\in V,~~~x_1\in V(x_0),~~\ldots~,~~~x_q\in V(x_0,\ldots,x_{q-1}), \leqno(2.4)$$ where $V$ is an open neighborhood of $x$ and $V(x_0,\ldots,x_j)$ an open neighborhood of $x_j$, depending on $x_0,\ldots,x_j$. This is easily seen by induction on~$q$, if we identify a map $f:X^{q+1}\to\cA$ to the map $X\to\cA^{[q-1]}$, $x_0\mapsto f_{x_0}$ such that $f_{x_0}(x_1,\ldots,x_q)=f(x_0,x_1,\ldots,x_q)$. Similarly, $\cA^{[q]}(U)$ is the set of equivalence classes of functions $X^{q+1}\ni(x_0,\ldots,x_q)\longmapsto f(x_0,\ldots,x_q)\in\cA_{x_q}$, with two such functions identified if they coincide on a set of the form $$x_0\in U,~~~x_1\in V(x_0),~~\ldots~,~~~x_q\in V(x_0,\ldots,x_{q-1}). \leqno(2.4')$$ Here, we may of course suppose $V(x_0,\ldots,x_{q-1})\subset\ldots\subset V(x_0,x_1)\subset V(x_0)\subset U$. We define a differential $d^q:\cA^{[q]}\longrightarrow\cA^{[q+1]}$ by $$\leqalignno{ &(d^qf)(x_0,\ldots,x_{q+1})=&(2.5)\cr &\sum_{0\le j\le q}(-1)^jf(x_0,\ldots,\wh{x_j},\ldots,x_{q+1}) +(-1)^{q+1}f(x_0,\ldots,x_q)(x_{q+1}).\cr}$$ The meaning of the last term is to be understood as follows: the element $s=f(x_0,\ldots,x_q)$ is a germ in $\cA_{x_q}$, therefore $s$ defines a continuous section $x_{q+1}\mapsto s(x_{q+1})$ of $\cA$ in a neighborhood $V(x_0,\ldots,x_q)$ of $x_q$. In low degrees, we have the formulas $$\leqalignno{ (js)(x_0)&=s(x_0),~~~s\in\cA_x,\cr (d^0f)(x_0,x_1)&=f(x_1)-f(x_0)(x_1),~~~f\in\cA^{[0]}_x,&(2.6)\cr (d^1f)(x_0,x_1,x_2)&=f(x_1,x_2)-f(x_0,x_2)+f(x_0,x_1)(x_2), ~~~f\in\cA^{[1]}_x.\cr}$$ \begstat{(2.7) Theorem {\rm (Godement 1957)}} The complex $(\cA^{[\bu]},d)$ is a resolution of the sheaf $\cA$, called the simplicial flabby resolution of $\cA$. \endstat \begproof{} For $s\in\cA_x$, the associated continuous germ obviously satisfies\break $s(x_0)(x_1)=s(x_1)$ for $x_0\in V$, $x_1\in V(x_0)$ small enough. The reader will easily infer from this that $d^0\circ j=0$ and $d^{q+1}\circ d^q=0$. In order to verify that $(\cA^{[\bu]},d)$ is a resolution of $\cA$, we show that the complex $$\cdots\longrightarrow 0\longrightarrow\cA_x\buildo j\over\longrightarrow\cA^{[0]}_x\buildo d^0\over\longrightarrow \cdots\longrightarrow\cA^{[q]}_x\buildo d^q\over\longrightarrow\cA^{[q+1]}_x\longrightarrow\cdots$$ is homotopic to zero for every point $x\in X$. Set $\cA^{[-1]}=\cA$, $d^{-1}=j$ and $$\cmalign{ &h^0~:~~\cA^{[0]}_x\longrightarrow\cA_x,~~~~&h^0(f)=f(x)\in\cA_x,\cr &h^q~:~~\cA^{[q]}_x\longrightarrow\cA^{[q-1]}_x,~~~~&h^q(f)(x_0,\ldots,x_{q-1})= f(x,x_0,\ldots,x_{q-1}).\cr}$$ A straightforward computation shows that $(h^{q+1}\circ d^q+d^{q-1}\circ h^q)(f)=f$ for all $q\in\bZ$ and $f\in \cA^{[q]}_x$.\qed \endproof If $\varphi:\cA\longrightarrow\cB$ is a sheaf morphism, it is clear that $\varphi$ induces a morphism of resolutions $$\varphi^{[\bu]}:\cA^{[\bu]}\longrightarrow\cB^{[\bu]}.\leqno(2.8)$$ For every short exact sequence $\cA\to\cB\to\cC$ of sheaves, we get a corresponding short exact sequence of sheaf complexes $$\cA^{[\bu]}\longrightarrow\cB^{[\bu]}\longrightarrow\cC^{[\bu]}.\leqno(2.9)$$ \titleb{3.}{Cohomology Groups with Values in a Sheaf} \titlec{3.A.}{Definition and Functorial Properties} If $\pi:\cA\to X$ is a sheaf of abelian groups, the {\it cohomology groups} of $\cA$ on $X$ are (in a vague sense) algebraic invariants which describe the rigidity properties of the global sections of $\cA$. \begstat{(3.1) Definition} For every $q\in\bZ$, the $q$-th cohomology group of $X$ with values in $\cA$ is $$\eqalign{ H^q(X&,\cA)=H^q\big(\cA^{[\bu]}(X)\big)=\cr &=\ker\big(d^q:\cA^{[q]}(X)\to\cA^{[q+1]}(X)\big)/ \Im(d^{q-1}:\cA^{[q-1]}(X)\to\cA^{[q]}(X)\big)\cr}$$ with the convention $\cA^{[q]}=0$, $d^q=0$, $H^q(X,\cA)=0$ when $q<0$. \endstat For any subset $S\subset X$, we denote by $\cA_{\restriction S}$ the {\it restriction} of $\cA$ to $S$, i.e.\ the sheaf $\cA_{\restriction S}=\pi^{-1}(S)$ equipped with the projection $\pi_{\restriction S}$ onto $S$. Then we write $H^q(S, \cA_{\restriction S})=H^q(S,\cA)$ for simplicity. When $U$ is open, we see that $(\cA^{[q]})_{\restriction U}$ coincides with $(\cA_{\restriction U})^{[q]}$, thus we have $H^q(U,\cA)= H^q\big(\cA^{[\bu]}(U)\big)$. It is easy to show that every exact sequence of sheaves $0\to\cA\to\cL^0\to\cL^1$ induces an exact sequence $$0\longrightarrow\cA(X)\longrightarrow\cL^0(X)\longrightarrow\cL^1(X).\leqno(3.2)$$ If we apply this to $\cL^q=\cA^{[q]}$, $q=0,1$, we conclude that $$H^0(X,\cA)=\cA(X).\leqno(3.3)$$ Let $\varphi:\cA\longrightarrow\cB$ be a sheaf morphism; (2.8) shows that there is an induced morphism $$H^q(\varphi):H^{q}(X,\cA)\longrightarrow H^q(X,\cB)\leqno(3.4)$$ on cohomology groups. Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves. Then we have an exact sequence of {\it groups} $$0\longrightarrow\cA^{[0]}(X)\longrightarrow\cB^{[0]}(X)\longrightarrow\cC^{[0]}(X)\longrightarrow0$$ because $\cA^{[0]}(X)=\prod_{x\in X}\cA_x$. Similarly, (2.9) yields for every $q$ an exact sequence of {\it groups} $$0\longrightarrow\cA^{[q]}(X)\longrightarrow\cB^{[q]}(X)\longrightarrow\cC^{[q]}(X)\longrightarrow0.$$ If we take (3.3) into account, the snake lemma implies: \begstat{(3.5) Theorem} To any exact sequence of sheaves $0\to\cA\to\cB\to\cC\to 0$ is associated a long exact sequence of cohomology groups $$\cmalign{ ~~~0&\longrightarrow~~\cA(X)&\longrightarrow~~\cB(X)&\longrightarrow~~\cC(X)&\longrightarrow~~H^1(X,\cA)&\longrightarrow\cdots\cr \cdots&\longrightarrow H^q(X,\cA)&\longrightarrow H^q(X,\cB)&\longrightarrow H^q(X,\cC)&\longrightarrow H^{q+1}(X,\cA) &\longrightarrow\cdots.\cr}$$ \endstat \begstat{(3.6) Corollary} Let $\cB\to\cC$ be a surjective sheaf morphism and let $\cA$ be its kernel. If $H^1(X,\cA)=0$, then $\cB(X)\longrightarrow\cC(X)$ is surjective.\qed \endstat \titlec{3.B.}{Exact Sequence Associated to a Closed Subset} Let $S$ be a closed subset of $X$ and $U=X\ssm S$. For any sheaf $\cA$ on $X$, the presheaf $$\Omega\longmapsto\cA(S\cap\Omega),~~~\Omega\subset X~~\hbox{\rm open}$$ with the obvious restriction maps satisfies axioms (II-$2.4')$ and (II-$2.4'')$, so it defines a sheaf on $X$ which we denote by $\cA^S$. This sheaf should not be confused with the restriction sheaf $\cA_{\restriction S}$, which is a sheaf on $S$. We easily find $$(\cA^S)_x=\cA_x~~~\hbox{\rm if}~~x\in S,~~~(\cA^S)_x=0~~~\hbox{\rm if}~~x\in U. \leqno(3.7)$$ Observe that these relations would completely fail if $S$ were not closed. The restriction morphism $f\mapsto f_{\restriction S}$ induces a surjective sheaf morphism $\cA\to\cA^S$. We let $\cA_U$ be its kernel, so that we have the relations $$(\cA_U)_x=0~~~\hbox{\rm if}~~x\in S,~~~(\cA_U)_x=\cA_x~~~\hbox{\rm if}~~x\in U. \leqno(3.8)$$ From the definition, we obtain in particular $$\cA^S(X)=\cA(S),~~~\cA_U(X)=\{\hbox{\rm sections of}~\cA(X)~ \hbox{\rm vanishing on}~S\}.\leqno(3.9)$$ Theorem 3.5 applied to the exact sequence $0\to\cA_U\to\cA\to\cA^S\to 0$ on $X$ gives a long exact sequence $$\cmalign{ 0&\longrightarrow~~\cA_U(X)&\longrightarrow~~\cA(X)&\longrightarrow~~\cA(S)&\longrightarrow~~H^1(X,\cA_U)&\cdots\cr &\longrightarrow H^q(X,\cA_U)&\longrightarrow H^q(X,\cA)&\longrightarrow H^q(X,\cA^S)&\longrightarrow H^{q+1}(X,\cA_U) &\cdots\cr}\leqno(3.9)$$ \titlec{3.C.}{Mayer-Vietoris Exact Sequence} Let $U_1$, $U_2$ be open subsets of $X$ and $U=U_1\cup U_2$, $V=U_1\cap U_2$. For any sheaf $\cA$ on $X$ and any $q$ we have an exact sequence $$0\longrightarrow\cA^{[q]}(U)\longrightarrow\cA^{[q]}(U_1)\oplus\cA^{[q]}(U_2)\longrightarrow \cA^{[q]}(V)\longrightarrow 0$$ where the injection is given by $f\longmapsto(f_{\restriction U_1}, f_{\restriction U_2})$ and the surjection by $(g_1,g_2)\longmapsto g_{2\restriction V}-g_{1\restriction V}$~; the surjectivity of this map follows immediately from the fact that $\cA^{[q]}$ is flabby. An application of the snake lemma yields: \begstat{(3.11) Theorem} For any sheaf $\cA$ on $X$ and any open sets $U_1,U_2\subset X$, set $U=U_1\cup U_2$, $V=U_1\cap U_2$. Then there is an exact sequence $$H^q(U,\cA)\longrightarrow H^q(U_1,\cA)\oplus H^q(U_2,\cA)\longrightarrow H^q(V,\cA) \longrightarrow H^{q+1}(U,\cA)\cdots\eqno\square$$ \endstat \titleb{4.}{Acyclic Sheaves} Given a sheaf $\cA$ on $X$, it is usually very important to decide whether the cohomology groups $H^q(U,\cA)$ vanish for $q\ge 1$, and if this is the case, for which type of open sets~$U$. Note that one cannot expect to have $H^0(U,\cA)=0$ in general, since a sheaf always has local sections. \begstat{(4.1) Definition} A sheaf $\cA$ is said to be acyclic on an open subset $U$ if $H^q(U,\cA)=0$ for $q\ge 1$. \endstat \titlec{4.A.}{Case of Flabby Sheaves} We are going to show that flabby sheaves are acyclic. First we need the following simple result. \begstat{(4.2) Proposition} Let $\cA$ be a sheaf with the following property: for every section $f$ of $\cA$ on an open subset $U\subset X$ and every point $x\in X$, there exists a neighborhood $\Omega$ of $x$ and a section $h\in\cA(\Omega)$ such that $h=f$ on $U\cap\Omega$. Then $\cA$ is flabby. \endstat A consequence of this proposition is that flabbiness is a local property: a sheaf $\cA$ is flabby on $X$ if and only if it is flabby on a neighborhood of every point of $X$. \begproof{} Let $f\in\cA(U)$ be given. Consider the set of pairs $(v,V)$ where $v$ in $\cB(V)$ is an extension of $f$ on an open subset $V\supset U$. This set is inductively ordered, so there exists a maximal extension $(v,V)$ by Zorn's lemma. The assumption shows that $V$ must be equal to $X$.\qed \endproof \begstat{(4.3) Proposition} Let $0\longrightarrow\cA\buildo j\over\longrightarrow\cB\buildo p \over\longrightarrow\cC\longrightarrow 0$ be an exact sequence of sheaves. If $\cA$ is flabby, the sequence of groups $$0\longrightarrow\cA(U)\buildo j\over\longrightarrow\cB(U)\buildo p\over\longrightarrow\cC(U)\longrightarrow 0$$ is exact for every open set $U$. If $\cA$ and $\cB$ are flabby, then $\cC$ is flabby. \endstat \begproof{} Let $g\in\cC(U)$ be given. Consider the set $E$ of pairs $(v,V)$ where $V$ is an open subset of $U$ and $v\in\cB(V)$ is such that $p(v)=g$ on $V$. It is clear that $E$ is inductively ordered, so $E$ has a maximal element $(v,V)$, and we will prove that $V=U$. Otherwise, let $x\in U\ssm V$ and let $h$ be a section of $\cB$ in a neighborhood of $x$ such that $p(h_x)=g_x$. Then $p(h)=g$ on a neighborhood $\Omega$ of $x$, thus $p(v-h)=0$ on $V\cap\Omega$ and $v-h=j(u)$ with $u\in\cA(V\cap\Omega)$. If $\cA$ is flabby, $u$ has an extension $\wt u\in\cA(X)$ and we can define a section $w\in\cB(V\cup\Omega)$ such that $p(w)=g$ by $$w=v~~\hbox{\rm on}~~V,~~~~w=h+j(\wt u)~~\hbox{\rm on}~~\Omega,$$ contradicting the maximality of $(v,V)$. Therefore $V=U$, $v\in\cB(U)$ and $p(v)=g$ on $U$. The first statement is proved. If $\cB$ is also flabby, $v$ has an extension $\wt v\in\cB(X)$ and $\wt g=p(\wt v)\in\cC(X)$ is an extension of $g$. Hence $\cC$ is flabby.\qed \endproof \begstat{(4.4) Theorem} A flabby sheaf $\cA$ is acyclic on all open sets $U\subset X$. \endstat \begproof{} Let $\cZ^q=\ker\big(d^q:\cA^{[q]}\to\cA^{[q+1]}\big)$. Then $\cZ^0=\cA$ and we have an exact sequence of sheaves $$0\longrightarrow\cZ^q\longrightarrow\cA^{[q]}\buildo d^q\over\longrightarrow\cZ^{q+1}\longrightarrow 0$$ because $\Im d^q=\ker d^{q+1}=\cZ^{q+1}$. Proposition 4.3 implies by induction on $q$ that all sheaves $\cZ^q$ are flabby, and yields exact sequences $$0\longrightarrow\cZ^q(U)\longrightarrow\cA^{[q]}(U)\buildo d^q\over\longrightarrow\cZ^{q+1}(U)\longrightarrow 0.$$ For $q\ge 1$, we find therefore $$\eqalign{ \ker\big(d^q:\cA^{[q]}(U)\to\cA^{[q+1]}(U)\big)&=\cZ^q(U)\cr &=\Im\big(d^{q-1}:\cA^{[q-1]}(U)\to\cA^{[q]}(U)\big),\cr}$$ that is, $H^q(U,\cA)=H^q\big(\cA^{[\bu]}(U)\big)=0$.\qed \endproof \titlec{4.B.}{Soft Sheaves over Paracompact Spaces} We now discuss another general situation which produces acyclic sheaves. Recall that a topological space $X$ is said to be {\it paracompact} if $X$ is Hausdorff and if every open covering of $X$ has a locally finite refinement. For instance, it is well known that every metric space is paracompact. A paracompact space $X$ is always {\it normal}\/; in particular, for any locally finite open covering $(U_\alpha)$ of $X$ there exists an open covering $(V_\alpha)$ such that $\ol V_\alpha\subset U_\alpha$. We will also need another closely related concept. \begstat{(4.5) Definition} We say that a subspace $S$ is strongly paracompact in $X$ if $S$ is Hausdorff and if the following property is satisfied: for every covering $(U_\alpha)$ of $S$ by open sets in $X$, there exists another such covering $(V_\beta)$ and a neighborhood $W$ of $S$ such that each set $W\cap\ol V_\beta$ is contained in some $U_\alpha$, and such that every point of $S$ has a neighborhood intersecting only finitely many sets $V_\beta$. \endstat It is clear that a strongly paracompact subspace $S$ is itself paracompact. Conversely, the following result is easy to check: \begstat{(4.6) Lemma} A subspace $S$ is strongly paracompact in $X$ as soon as one of the following situations occurs: \medskip \item{\rm a)} $X$ is paracompact and $S$ is closed; \smallskip \item{\rm b)} $S$ has a fundamental family of paracompact neighborhoods in $X\,$; \smallskip \item{\rm c)} $S$ is paracompact and has a neighborhood homeomorphic to some product $S\times T$, in which $S$ is embedded as a slice $S\times\{t_0\}$.\qed \endstat \begstat{(4.7) Theorem} Let $\cA$ be a sheaf on $X$ and $S$ a strongly paracompact subspace of $X$. Then every section $f$ of $\cA$ on $S$ can be extended to a section of $\cA$ on some open neighborhood $\Omega$ of $\cA$. \endstat \begproof{} Let $f\in\cA(S)$. For every point $z\in S$ there exists an open neighborhood $U_z$ and a section $\smash{\wt f}_z\in\cA(U_z)$ such that $\smash{\wt f}_z(z)=f(z)$. After shrinking $U_z$, we may assume that $\smash{\wt f}_z$ and $f$ coincide on $S\cap U_z$. Let $(V_\alpha)$ be an open covering of $S$ that is locally finite near $S$ and $W$ a neighborhood of $S$ such that $W\cap\ol V_\alpha\subset U_{z(\alpha)}$ (Def.\ 4.5). We let $$\Omega=\big\{x\in W\cap\bigcup V_\alpha\,;\, \smash{\wt f}_{z(\alpha)}(x)=\smash{\wt f}_{z(\beta)}(x),~\forall \alpha,\beta~\hbox{\rm with~}x\in\ol V_\alpha\cap\ol V_\beta\big\}.$$ Then $(\Omega\cap V_\alpha)$ is an open covering of $\Omega$ and all pairs of sections $\smash{\wt f}_{z(\alpha)}$ coincide in pairwise intersections. Thus there exists a section $F$ of $\cA$ on $\Omega$ which is equal to $\smash{\wt f}_{z(\alpha)}$ on $\Omega\cap V_\alpha$. It remains only to show that $\Omega$ is a neighborhood of $S$. Let $z_0\in S$. There exists a neighborhood $U'$ of $z_0$ which meets only finitely many sets $V_{\alpha_1},\ldots,V_{\alpha_p}$. After shrinking $U'$, we may keep only those $V_{\alpha_l}$ such that $z_0\in\ol V_{\alpha_l}$. The sections $\smash{\wt f}_{z(\alpha_l)}$ coincide at $z_0$, so they coincide on some neighborhood $U''$ of this point. Hence $W\cap U''\subset\Omega$, so $\Omega$ is a neighborhood of $S$.\qed \endproof \begstat{(4.8) Corollary} If $X$ is paracompact, every section $f\in\cA(S)$ defined on a closed set $S$ extends to a neighborhood $\Omega$ of $S$.\qed \endstat \begstat{(4.9) Definition} A sheaf $\cA$ on $X$ is said to be soft if every section $f$ of $\cA$ on a closed set $S$ can be extended to $X$, i.e.\ if the restriction map $\cA(X)\longrightarrow\cA(S)$ is onto for every closed set $S$. \endstat \begstat{(4.10) Example} \rm On a paracompact space, every flabby sheaf $\cA$ is soft: this is a consequence of Cor.\ 4.8. \endstat \begstat{(4.11) Example} \rm On a paracompact space, the Tietze-Urysohn extension theorem shows that the sheaf $\cC_X$ of germs of continuous functions on $X$ is a soft sheaf of rings. However, observe that $\cC_X$ is not flabby as soon as $X$ is not discrete. \endstat \begstat{(4.12) Example} \rm If $X$ is a paracompact differentiable manifold, the sheaf $\cE_X$ of germs of $C^\infty$ functions on $X$ is a soft sheaf of rings.\qed \endstat Until the end of this section, we assume that $X$ is a {\it paracompact topo\-lo\-gical space}. We first show that softness is a local property. \begstat{(4.13) Proposition} A sheaf $\cA$ is soft on $X$ if and only if it is soft in a neighborhood of every point $x\in X$. \endstat \begproof{} If $\cA$ is soft on $X$, it is soft on any closed neighborhood of a given point. Conversely, let $(U_\alpha)_{\alpha\in I}$ be a locally finite open covering of $X$ which refine some covering by neighborhoods on which $\cA$ is soft. Let $(V_\alpha)$ be a finer covering such that $\ol V_\alpha\subset U_\alpha$, and $f\in\cA(S)$ be a section of $\cA$ on a closed subset $S$ of $X$. We consider the set $E$ of pairs $(g,J)$, where $J\subset I$ and where $g$ is a section over $F_J:=S\cup\bigcup_{\alpha\in J}\ol V_\alpha$, such that $g=f$ on $S$. As the family $(\ol V_\alpha)$ is locally finite, a section of $\cA$ over $F_J$ is continuous as soon it is continuous on $S$ and on each $\ol V_\alpha$. Then $(f,\emptyset)\in E$ and $E$ is inductively ordered by the relation $$(g',J')\longrightarrow(g'',J'')~~~\hbox{\rm if}~~J'\subset J''~~\hbox{\rm and}~~ g'=g''~~\hbox{\rm on}~~F_{J'}$$ No element $(g,J)$, $J\ne I$, can be maximal: the assumption shows that $\smash{g_{\restriction F_J\cap\ol V_\alpha}}$ has an extension to $\ol V_\alpha$, thus such a $g$ has an extension to $F_{J\cup\{\alpha\}}$ for any $\alpha\notin J$. Hence $E$ has a maximal element $(g,I)$ defined on $F_I=X$.\qed \endproof \begstat{(4.14) Proposition} Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves. If $\cA$ is soft, the map $\cB(S)\to\cC(S)$ is onto for any closed subset $S$ of $X$. If $\cA$ and $\cB$ are soft, then $\cC$ is soft. \endstat By the above inductive method, this result can be proved in a way similar to its analogue for flabby sheaves. We therefore obtain: \begstat{(4.15) Theorem} On a paracompact space, a soft sheaf is acyclic on all closed subsets.\qed \endstat \begstat{(4.16) Definition} The {\it support} of a section $f\in\cA(X)$ is defined by $$\Supp\,f=\big\{x\in X\,;\,f(x)\ne 0\big\}.$$ \endstat $\Supp\,f$ is always a closed set: as $\cA\to X$ is a local homeomorphism, the equality $f(x)=0$ implies $f=0$ in a neighborhood of $x$. \begstat{(4.17) Theorem} Let $(U_\alpha)_{\alpha\in I}$ be an open covering of $X$. If $\cA$ is soft and $f\in\cA(X)$, there exists a partition of $f$ subordinate to $(U_\alpha)$, i.e.\ a family of sections $f_\alpha\in\cA(X)$ such that $(\Supp\,f_\alpha)$ is locally finite, $\Supp\,f_\alpha\subset U_\alpha$ and $\sum f_\alpha=f$ on $X$. \endstat \begproof{} Assume first that $(U_\alpha)$ is locally finite. There exists an open covering $(V_\alpha)$ such that $\ol V_\alpha\subset U_\alpha$. Let $(f_\alpha)_{\alpha\in J}$, $J\subset I$, be a maximal family of sections $f_\alpha\in\cA(X)$ such that $\Supp\,f_\alpha\subset U_\alpha$ and $\sum_{\alpha\in J}f_\alpha=f$ on $S=\bigcup_{\alpha\in J}\ol V_\alpha$. If $J\ne I$ and $\beta\in I\ssm J$, there exists a section $f_\beta\in\cA(X)$ such that $$f_\beta=0~~~\hbox{\rm on}~~X\ssm U_\beta~~~\hbox{\rm and}~~~ f_\beta=f-\sum_{\alpha\in J}f_\alpha~~~\hbox{\rm on}~~S\cup\ol V_\beta$$ because $(X\ssm U_\beta)\cup S\cup\ol V_\beta$ is closed and $f-\sum f_\alpha=0$ on $(X\ssm U_\alpha)\cap S$. This is a contradiction unless $J=I$. In general, let $(V_j)$ be a locally finite refinement of $(U_\alpha)$, such that $V_j\subset U_{\rho(j)}$, and let $(f'_j)$ be a partition of $f$ subordinate to $(V_j)$. Then $f_\alpha=\sum_{j\in\rho^{-1}(\alpha)}f'_j$ is the required partition of $f$.\qed \endproof Finally, we discuss a special situation which occurs very often in practice. Let $\cR$ be a sheaf of commutative rings on $X$~; the rings $\cR_x$ are supposed to have a unit element. Assume that $\cA$ is a sheaf of modules over $\cR$. It is clear that $\cA^{[0]}$ is a $\cR^{[0]}$-module, and thus also a $\cR$-module. Therefore all sheaves $\cA^{[q]}$ are $\cR$-modules and the cohomology groups $H^q(U,\cA)$ have a natural structure of $\cR(U)$-module. \begstat{(4.18) Lemma} If $\cR$ is soft, every sheaf $\cA$ of $\cR$-modules is soft. \endstat \begproof{} Every section $f\in\cA(S)$ defined on a closed set $S$ has an extension to some open neighborhood $\Omega$. Let $\psi\in\cR(X)$ be such that $\psi=1$ on $S$ and $\psi=0$ on $X\ssm\Omega$. Then $\psi f$, defined as $0$ on $X\ssm\Omega$, is an extension of $f$ to~$X$.\qed \endproof \begstat{(4.19) Corollary} Let $\cA$ be a sheaf of $\cE_X$-modules on a paracompact differentiable manifold $X$. Then $H^q(X,\cA)=0$ for all $q\ge 1$. \endstat \titleb{5.}{\v Cech Cohomology} \titlec{5.A.}{Definitions} In many important circumstances, cohomology groups with values in a sheaf $\cA$ can be computed by means of the complex of \v Cech cochains, which is directly related to the spaces of sections of $\cA$ on sufficiently fine coverings of $X$. This more concrete approach was historically the first one used to define sheaf cohomology (Leray 1950, Cartan 1950); however \v Cech cohomology does not always coincide with the ``good" cohomology on non paracompact spaces. Let $\cU=(U_\alpha)_{\alpha\in I}$ be an open covering of $X$. For the sake of simplicity, we denote $$U_{\alpha_0\alpha_1\ldots\alpha_q}=U_{\alpha_0}\cap U_{\alpha_1}\cap \ldots\cap U_{\alpha_q}.$$ The group $C^q(\cU,\cA)$ of {\it \v Cech $q$-cochains} is the set of families $$c=(c_{\alpha_0\alpha_1\ldots\alpha_q})\in\prod_{(\alpha_0,\ldots,\alpha_q) \in I^{q+1}}\cA(U_{\alpha_0\alpha_1\ldots\alpha_q}).$$ The group structure on $C^q(\cU,\cA)$ is the obvious one deduced from the addition law on sections of $\cA$. The {\it \v Cech differential} $\delta^q:C^q(\cU,\cA)\longrightarrow C^{q+1}(\cU,\cA)$ is defined by the formula $$(\delta^qc)_{\alpha_0\ldots\alpha_{q+1}}= \sum_{0\le j\le q+1}(-1)^j\,c_{\alpha_0\ldots\wh{\alpha_j}\ldots\alpha_{q+1}\; \restriction U_{\alpha_0\ldots\alpha_{q+1}}},\leqno(5.1)$$ and we set $C^q(\cU,\cA)=0$, $\delta^q=0$ for $q<0$. In degrees $0$ and $1$, we get for example $$\leqalignno{ &q=0,~~~c=(c_\alpha),~~~\hfill(\delta^0c)_{\alpha\beta} =c_\beta-c_{\alpha\;\restriction U_{\alpha\beta}},&(5.2)\cr &q=1,~~~c=(c_{\alpha\beta}),~~~(\delta^1c)_{\alpha\beta\gamma} =c_{\beta\gamma}-c_{\alpha\gamma}+c_{\alpha\beta\; \restriction U_{\alpha\beta\gamma}}.&(5.2')\cr}$$ Easy verifications left to the reader show that $\delta^{q+1}\circ\delta^q=0$. We get therefore a cochain complex $\big(C^\bu(\cU,\cA),\delta\big)$, called the {\it complex of \v Cech cochains} relative to the covering $\cU$. \begstat{(5.3) Definition} The \v Cech cohomology group of $\cA$ relative to $\cU$ is $$\check H^q(\cU,\cA)=H^q\big(C^\bu(\cU,\cA)\big).$$ \endstat Formula (5.2) shows that the set of \v Cech $0$-cocycles is the set of families $(c_\alpha)\in\prod\cA(U_\alpha)$ such that $c_\beta=c_\alpha$ on $U_\alpha\cap U_\beta$. Such a family defines in a unique way a global section $f\in\cA(X)$ with $f_{\restriction U_\alpha}=c_\alpha$. Hence $$\check H^0(\cU,\cA)=\cA(X).\leqno(5.4)$$ Now, let $\cV=(V_\beta)_{\beta\in J}$ be another open covering of $X$ that is finer than $\cU$~; this means that there exists a map $\rho:J\to I$ such that $V_\beta\subset U_{\rho(\beta)}$ for every $\beta\in J$. Then we can define a morphism $\rho^\bu:C^\bu(\cU,\cA)\longrightarrow C^\bu(\cV,\cA)$ by $$(\rho^q c)_{\beta_0\ldots\beta_q}=c_{\rho(\beta_0)\ldots\rho(\beta_q)\; \restriction V_{\beta_0\ldots\beta_q}}~;\leqno(5.5)$$ the commutation property $\delta\rho^\bu=\rho^\bu\delta$ is immediate. If $\rho':J\to I$ is another refinement map such that $V_\beta\subset U_{\rho'(\beta)}$ for all $\beta$, the morphisms $\rho^\bu$, $\rho^{\prime\bu}$ are homotopic. To see this, we define a map $h^q:C^q(\cU,\cA)\longrightarrow C^{q-1}(\cV,\cA)$ by $$(h^q c)_{\beta_0\ldots\beta_{q-1}}=\sum_{0\le j\le q-1}(-1)^j c_{\rho(\beta_0)\ldots\rho(\beta_j)\rho'(\beta_j)\ldots\rho'(\beta_{q-1})\; \restriction V_{\beta_0\ldots\beta_{q-1}}}.$$ The homotopy identity $\delta^{q-1}\circ h^q+h^{q+1}\circ\delta^q= \rho^{\prime q}-\rho^q$ is easy to verify. Hence $\rho^\bu$ and $\rho^{\prime\bu}$ induce a map depending only on $\cU$, $\cV$~: $$H^q(\rho^\bu)=H^q(\rho^{\prime\bu})~:~~\check H^q(\cU,\cA)\longrightarrow \check H^q(\cV,\cA).\leqno(5.6)$$ Now, we want to define a {\it direct limit} $\check H^q(X,\cA)$ of the groups $\check H^q(\cU,\cA)$ by means of the refinement mappings $(5.6)$. In order to avoid set theoretic difficulties, the coverings used in this definition will be considered as subsets of the power set $\cP(X)$, so that the collection of all coverings becomes actually a set. \begstat{(5.7) Definition} The \v Cech cohomology group $\check H^q(X,\cA)$ is the direct limit $$\check H^q(X,\cA)=\lim_{\displaystyle\,\longrightarrow\atop\scriptstyle\cU} ~~\check H^q(\cU,\cA)$$ when $\cU$ runs over the collection of all open coverings of $X$. Explicitly, this means that the elements of $\check H^q(X,\cA)$ are the equivalence classes in the disjoint union of the groups $\check H^q(\cU,\cA)$, with an element in $\check H^q(\cU,\cA)$ and another in $\check H^q(\cV,\cA)$ identified if their images in $\check H^q(\cW,\cA)$ coincide for some refinement $\cW$ of the coverings $\cU$ and $\cV$. \endstat If $\varphi:\cA\to\cB$ is a sheaf morphism, we have an obvious induced morphism $\varphi^\bu:C^\bu(\cU,\cA)\longrightarrow C^\bu(\cU,\cB)$, and therefore we find a morphism $$H^q(\varphi^\bu):\check H^q(\cU,\cA)\longrightarrow\check H^q(\cU,\cB).$$ Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves. We have an exact sequence of groups $$0\longrightarrow C^q(\cU,\cA)\longrightarrow C^q(\cU,\cB)\longrightarrow C^q(\cU,\cC),\leqno(5.8)$$ but in general the last map is not surjective, because every section in $\cC(U_{\alpha_0,\ldots,\alpha_q})$ need not have a lifting in $\cB(U_{\alpha_0,\ldots,\alpha_q})$. The image of $C^\bu(\cU,\cB)$ in $C^\bu(\cU,\cC)$ will be denoted $C^\bu_\cB(\cU,\cC)$ and called the complex of {\it liftable cochains} of $\cC$ in $\cB$. By construction, the sequence $$0\longrightarrow C^q(\cU,\cA)\longrightarrow C^q(\cU,\cB)\longrightarrow C^q_\cB(\cU,\cC)\longrightarrow 0\leqno(5.9)$$ is exact, thus we get a corresponding long exact sequence of cohomology $$\check H^q(\cU,\cA)\longrightarrow\check H^q(\cU,\cB)\longrightarrow\check H^q_\cB(\cU,\cC) \longrightarrow\check H^{q+1}(\cU,\cA)\longrightarrow\cdots.\leqno(5.10)$$ If $\cA$ is flabby, Prop.\ 4.3 shows that we have $C^q_\cB(\cU,\cC)=C^q(\cU,\cC)$, hence $\check H^q_\cB(\cU,\cC)= \check H^q(\cU,\cC)$. \begstat{(5.11) Proposition} Let $\cA$ be a sheaf on $X$. Assume that either \medskip \item{\rm a)} $\cA$ is flabby, or~$:$ \medskip \item{\rm b)} $X$ is paracompact and $\cA$ is a sheaf of modules over a soft sheaf of rings $\cR$ on $X$. \medskip \noindent Then $\check H^q(\cU,\cA)=0$ for every $q\ge 1$ and every open covering $\cU=(U_\alpha)_{\alpha\in I}$ of~$X$. \endstat \begproof{} b) Let $(\psi_\alpha)_{\alpha\in I}$ be a partition of unity in $\cR$ subordinate to $\cU$ (Prop.\ 4.17). We define a map $h^q:C^q(\cU,\cA)\longrightarrow C^{q-1}(\cU,\cA)$ by $$(h^qc)_{\alpha_0\ldots\alpha_{q-1}}=\sum_{\nu\in I}\psi_\nu\, c_{\nu\alpha_0\ldots\alpha_{q-1}}\leqno(5.12)$$ where $\psi_\nu\,c_{\nu\alpha_0\ldots\alpha_{q-1}}$ is extended by $0$ on $U_{\alpha_0\ldots\alpha_{q-1}}\cap\complement U_\nu$. It is clear that $$(\delta^{q-1}h^qc)_{\alpha_0\ldots\alpha_q}=\sum_{\nu\in I} \psi_\nu\big(c_{\alpha_0\ldots\alpha_q}-(\delta^qc)_{\nu\alpha_0\ldots\alpha_q} \big),$$ i.e.\ $\delta^{q-1}h^q+h^{q+1}\delta^q=\Id$. Hence $\delta^q c=0$ implies $\delta^{q-1}h^qc=c$ if $q\ge1$. \medskip \noindent{a)} First we show that the result is true for the sheaf $\cA^{[0]}$. One can find a family of sets $L_\nu\subset U_\nu$ such that $(L_\nu)$ is a partition of $X$. If $\psi_\nu$ is the characteristic func\-tion of $L_\nu$, Formula (5.12) makes sense for any cochain $c\in C^q(\cU,\cA^{[0]})$ because $\cA^{[0]}$ is a module over the ring $\bZ^{[0]}$ of germs of arbitrary functions $X\to\bZ$. Hence $\check H^q(\cU,\cA^{[0]})=0$ for $q\ge 1$. We shall prove this property for all flabby sheaves by induction on $q$. Consider the exact sequence $$0\longrightarrow\cA\longrightarrow\cA^{[0]}\longrightarrow\cC\longrightarrow 0$$ where $\cC=\cA^{[0]}/\cA$. By the remark after (5.10), we have exact sequences $$\eqalign{ &\cA^{[0]}(X)\longrightarrow\cC(X)\longrightarrow\check H^1(\cU,\cA)\longrightarrow\check H^1(\cU,\cA^{[0]})=0,\cr &\check H^q(\cU,\cC)\longrightarrow\check H^{q+1}(\cU,\cA)\longrightarrow\check H^{q+1}(\cU,\cA^{[0]}) =0.\cr}$$ Then $\cA^{[0]}(X)\longrightarrow\cC(X)$ is surjective by Prop.\ 4.3, thus $\check H^1(\cU,\cA)=0$. By 4.3 again, $\cC$ is flabby; the induction hypothesis $\check H^q(\cU,\cC)=0$ implies that $\check H^{q+1}(\cU,\cA)=0$.\qed \endproof \titlec{5.B.}{Leray's Theorem for Acyclic Coverings} We first show the existence of a natural morphism from \v Cech cohomology to ordinary cohomology. Let $\cU=(U_\alpha)_{\alpha\in I}$ be a covering of $X$. Select a map $\lambda:X\to I$ such that $x\in U_{\lambda(x)}$ for every $x\in X$. To every cochain $c\in C^q(\cU,\cA)$ we associate the section $\lambda^qc=f\in\cA^{[q]}(X)$ such that $$f(x_0,\ldots,x_q)=c_{\lambda(x_0)\ldots\lambda(x_q)}(x_q)\in\cA_{x_q}~; \leqno(5.13)$$ note that the right hand side is well defined as soon as $$x_0\in X,~~~x_1\in U_{\lambda(x_0)},~~\ldots~,~~~x_q\in U_{\lambda(x_0)\ldots\lambda(x_{q-1})}.$$ A comparison of (2.5) and (5.13) immediately shows that the section of $\cA^{[q+1]}(X)$ associated to $\delta^qc$ is $$\sum_{0\le j\le q+1}(-1)^j\,c_{\lambda(x_0)\ldots\wh{\lambda(x_j)}\ldots \lambda(x_{q+1})}(x_{q+1})=(d^qf)(x_0,\ldots,x_{q+1}).$$ In this way we get a morphism of complexes $\lambda^\bu:C^\bu(\cU,\cA)\longrightarrow\cA^{[\bu]}(X)$. There is a corresponding morphism $$H^q(\lambda^\bu):\check H^q(\cU,\cA)\longrightarrow H^q(X,\cA).\leqno(5.14)$$ If $\cV=(V_\beta)_{\beta\in J}$ is a refinement of $\cU$ such that $V_\beta\subset U_{\rho(\beta)}$ and $x\in V_{\mu(x)}$ for all $x,\beta$, we get a commutative diagram $$\eqalign{ \check H^q(&\cU,\cA)\buildo H^q(\rho^\bu)\over {\relbar\joinrel\relbar\joinrel\relbar\joinrel\longrightarrow}\check H^q(\cV,\cA)\cr {\scriptstyle H^q(\lambda^\bu)} &\searrow\qquad\qquad\qquad\swarrow{\scriptstyle H^q(\mu^\bu)}\cr &\qquad H^q(X,\cA)\cr}$$ with $\lambda=\rho\circ\mu$. In particular, (5.6) shows that the map $H^q(\lambda^\bu)$ in (5.14) does not depend on the choice of $\lambda$~: if $\lambda'$ is another choice, then $H^q(\lambda^\bu)$ and $H^q(\lambda^{\prime\bu})$ can be both factorized through the group $\check H^q(\cV,\cA)$ with $V_x=U_{\lambda(x)}\cap U_{\lambda'(x)}$ and $\mu=\Id_X$. By the universal property of direct limits, we get an induced morphism $$\check H^q(X,\cA)\longrightarrow H^q(X,\cA).\leqno(5.15)$$ Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves. There is a commutative diagram $$\cmalign{ 0&\longrightarrow C^\bu(\cU,\cA)&\longrightarrow C^\bu(\cU,\cB)&\longrightarrow C^\bu_\cB(\cU,\cC)&\longrightarrow 0\cr &\qquad\quad~\big\downarrow&\qquad\quad~\big\downarrow &\qquad\quad~\big\downarrow&\cr 0&\longrightarrow\cA^{[\bu]}(X)&\longrightarrow\cB^{[\bu]}(X)&\longrightarrow\cC^{[\bu]}(X)&\longrightarrow 0\cr}$$ where the vertical arrows are given by the morphisms $\lambda^\bu$. We obtain therefore a commutative diagram $$\cmalign{ &\kern-10pt\check H^q(\cU,\cA)&\longrightarrow\check H^q(\cU,\cB)&\longrightarrow\check H^q_\cB(\cU,\cC)&\longrightarrow\check H^{q+1}(\cU,\cA)&\longrightarrow\check H^{q+1}(\cU,\cB)\cr &\quad\big\downarrow&\qquad\quad~~\big\downarrow&\qquad\quad~~ \big\downarrow&\qquad\quad~~~\big\downarrow&\qquad\quad~~~\big\downarrow\cr &\kern-10pt H^q(X,\cA)&\longrightarrow H^q(X,\cB)&\longrightarrow H^q(X,\cC)&\longrightarrow H^{q+1}(X,\cA)&\longrightarrow H^{q+1}(X,\cB).\cr}\leqno(5.16)$$ \begstat{(5.17) Theorem {\rm(Leray)}} Assume that $$H^s(U_{\alpha_0\ldots\alpha_t},\cA)=0$$ for all indices $\alpha_0,\ldots,\alpha_t$ and $s\ge 1$. Then {\rm(5.14)} gives an isomorphism $\check H^q(\cU,\cA)\simeq H^q(X,\cA)$. \endstat We say that the covering $\cU$ is {\it acyclic} (with respect to $\cA$) if the hypo\-thesis of Th.~5.17 is satisfied. Leray's theorem asserts that the cohomology groups of $\cA$ on $X$ can be computed by means of an arbitrary acyclic covering (if such a covering exists), without using the direct limit procedure. \begproof{} By induction on $q$, the result being obvious for $q=0$. Consider the exact sequence $0\to\cA\to\cB\to\cC\to 0$ with $\cB=\cA^{[0]}$ and $\cC=\cA^{[0]}/\cA$. As $\cB$ is acyclic, the hypothesis on $\cA$ and the long exact sequence of cohomology imply $H^s(U_{\alpha_0\ldots\alpha_t},\cC)=0$ for $s\ge 1$, $t\ge 0$. Moreover $C^\bu_\cB(\cU,\cC)=C^\bu(\cU,\cC)$ thanks to Cor.\ 3.6. The induction hypothesis in degree $q$ and diagram (5.16) give $$\cmalign{ &\check H^q(\cU,\cB)&\longrightarrow\check H^q(\cU,\cC)&\longrightarrow\check H^{q+1}(\cU,\cA)&\longrightarrow 0\cr &\quad~~\big\downarrow\simeq&\qquad\quad~~\big\downarrow\simeq &\qquad\quad~~~\big\downarrow&\cr &H^q(X,\cB)&\longrightarrow H^q(X,\cC)&\longrightarrow H^{q+1}(X,\cA)&\longrightarrow 0,\cr}$$ hence $\check H^{q+1}(\cU,\cA)\longrightarrow H^{q+1}(X,\cA)$ is also an isomorphism.\qed \endproof \begstat{(5.18) Remark} \rm The morphism $H^1(\lambda^\bu):\check H^1(\cU,\cA)\longrightarrow H^1(X,\cA)$ is always injective. Indeed, we have a commutative diagram $$\cmalign{ &\check H^0(\cU,\cB)&\longrightarrow\check H^0_\cB(\cU,\cC)&\longrightarrow\check H^1(\cU,\cA) &\longrightarrow 0\cr &\quad~~\big\downarrow =&\qquad\quad~~\hookdown &\qquad\quad~~\big\downarrow&\cr &H^0(X,\cB)&\longrightarrow H^0(X,\cC)&\longrightarrow H^1(X,\cA)&\longrightarrow 0,\cr}$$ where $\check H^0_\cB(\cU,\cC)$ is the subspace of $\cC(X)=H^0(X,\cC)$ consisting of sections which can be lifted in $\cB$ over each $U_\alpha$. As a consequence, the refinement mappings $$H^1(\rho^\bu):\check H^1(\cU,\cA)\longrightarrow\check H^1(\cV,\cA)$$ are also injective.\qed \endstat \titlec{5.C.}{\v Cech Cohomology on Paracompact Spaces} We will prove here that \v Cech cohomology theory coincides with the ordinary one on paracompact spaces. \begstat{(5.19) Proposition} Assume that $X$ is paracompact. If $$0\longrightarrow\cA\longrightarrow\cB\longrightarrow\cC\longrightarrow 0$$ is an exact sequence of sheaves, there is an exact sequence $$\check H^q(X,\cA)\longrightarrow \check H^q(X,\cB)\longrightarrow \check H^q(X,\cC)\longrightarrow \check H^{q+1}(X,\cA)\longrightarrow\cdots$$ which is the direct limit of the exact sequences {\rm(5.10)} over all coverings~$\cU$. \endstat \begproof{} We have to show that the natural map $$\lim_{\displaystyle\longrightarrow}~~\check H^q_\cB(\cU,\cC)\longrightarrow \lim_{\displaystyle\longrightarrow}~~\check H^q(\cU,\cC)$$ is an isomorphism. This follows easily from the following lemma, which says essentially that every cochain in $\cC$ becomes liftable in $\cB$ after a refinement of the covering. \endproof \begstat{(5.20) Lifting lemma} Let $\cU=(U_\alpha)_{\alpha\in I}$ be an open covering of $X$ and $c\in C^q(\cU,\cC)$. If $X$ is paracompact, there exists a finer covering $\cV=(V_\beta)_{\beta\in J}$ and a refinement map $\rho:J\to I$ such that $\rho^q c\in C^q_\cB(\cV,\cC)$. \endstat \begproof{} Since $\cU$ admits a locally finite refinement, we may assume that $\cU$ itself is locally finite. There exists an open covering $\cW=(W_\alpha)_{\alpha\in I}$ of $X$ such that $\ol W_\alpha\subset U_\alpha$. For every point $x\in X$, we can select an open neighborhood $V_x$ of $x$ with the following properties: \medskip \noindent{a)} if $x\in W_\alpha$, then $V_x\subset W_\alpha$~; \medskip \noindent{b)} if $x\in U_\alpha$ or if $V_x\cap W_\alpha\ne\emptyset$, then $V_x\subset U_\alpha$~; \medskip \noindent{c)}$\,$ if $x\in U_{\alpha_0\ldots\alpha_q}$, then $c_{\alpha_0\ldots\alpha_q}\in C^q(U_{\alpha_0\ldots\alpha_q},\cC)$ admits a lifting in $\cB(V_x)$. \medskip \noindent Indeed, a) (resp. c)) can be achieved because $x$ belongs to only finitely many sets $W_\alpha$ (resp. $U_\alpha$), and so only finitely many sections of $\cC$ have to be lifted in $\cB$. b) can be achieved because $x$ has a neighborhood $V'_x$ that meets only finitely many sets $U_\alpha$~; then we take $$V_x\subset V'_x\cap\bigcap_{U_\alpha\ni x}U_\alpha\cap \bigcap_{U_\alpha\not\ni x}(V'_x\ssm\ol W_\alpha).$$ Choose $\rho:X\to I$ such that $x\in W_{\rho(x)}$ for every $x$. Then a) implies $V_x\subset W_{\rho(x)}$, so $\cV=(V_x)_{x\in X}$ is finer than $\cU$, and $\rho$ defines a refinement map. If $V_{x_0\ldots x_q}\ne\emptyset$, we have $$V_{x_0}\cap W_{\rho(x_j)}\supset V_{x_0}\cap V_{x_j}\ne\emptyset~~~ \hbox{\rm for}~~0\le j\le q,$$ thus $V_{x_0}\subset U_{\rho(x_0)\ldots\rho(x_q)}$ by b). Now, c) implies that the section $c_{\rho(x_0)\ldots\rho(x_q)}$ admits a lifting in $\cB(V_{x_0})$, and in particular in $\cB(V_{x_0\ldots x_q})$. Therefore $\rho^qc$ is liftable in $\cB$.\qed \endproof \begstat{(5.21) Theorem} If $X$ is a paracompact space, the canonical morphism \hbox{$\check H^q(X,\cA)\simeq H^q(X,\cA)$} is an isomorphism. \endstat \begproof{} Argue by induction on $q$ as in Leray's theorem, with the \v Cech coho\-mology exact sequence over $\cU$ replaced by its direct limit in~(5.16).\qed \endproof In the next chapters, we will be concerned only by paracompact spaces, and most often in fact by manifolds that are either compact or countable at infinity. In these cases, we will not distinguish $H^q(X,\cA)$ and $\check H^q(X,\cA)$. \titlec{5.D.}{Alternate \v Cech Cochains} For explicit calculations, it is sometimes useful to consider a slightly modified \v Cech complex which has the advantage of producing much smaller cochain groups. If $\cA$ is a sheaf and $\cU=(U_\alpha)_{\alpha\in I}$ an open covering of $X$, we let $AC^q(\cU,\cA)\subset C^q(\cU,\cA)$ be the subgroup of {\it alternate \v Cech cochains}, consisting of \v Cech cochains $c=(c_{\alpha_0\ldots\alpha_q})$ such that $$\left\{\eqalign{ c_{\alpha_0\ldots\alpha_q}&=0\phantom{(\sigma)\,c_{\alpha_0\ldots\alpha_q},} ~~~\hbox{\rm if}~~\alpha_i=\alpha_j,~~i\ne j,\cr c_{\alpha_{\sigma(0)}\ldots\alpha_{\sigma(q)}}&=\varepsilon(\sigma) \,c_{\alpha_0\ldots\alpha_q}\cr}\right.\leqno(5.22)$$ for any permutation $\sigma$ of $\{1,\ldots,q\}$ of signature $\varepsilon(\sigma)$. Then the \v Cech differential (5.1) of an alternate cochain is still alternate, so $AC^\bu(\cU,\cA)$ is a subcomplex of $C^\bu(\cU,\cA)$. We are going to show that the inclusion induces an isomorphism in cohomology: $$H^q\big(AC^\bu(\cU,\cA)\big)\simeq H^q\big(C^\bu(\cU,\cA)\big)= \check H^q(\cU,\cA).\leqno(5.23)$$ Select a total ordering on the index set $I$. For each such ordering, we can define a projection $\pi^q:C^q(\cU,\cA)\longrightarrow AC^q(\cU,\cA)\subset C^q(\cU,\cA)$ by $$c\longmapsto \hbox{\rm alternate}~\wt c~~\hbox{\rm such that}~ \wt c_{\alpha_0\ldots\alpha_q}=c_{\alpha_0\ldots\alpha_q}~~\hbox{\rm whenever}~~ \alpha_0<\ldots<\alpha_q.$$ As $\pi^\bu$ is a morphism of complexes, it is enough to verify that $\pi^\bu$ is homotopic to the identity on $C^\bu(\cU,\cA)$. For a given multi-index $\alpha=(\alpha_0,\ldots,\alpha_q)$, which may contain repeated indices, there is a unique permutation $\big(m(0),\ldots,m(q)\big)$ of $(0,\ldots,q)$ such that $$\alpha_{m(0)}\le\ldots\le\alpha_{m(q)}~~~~\hbox{\rm and}~~m(l)n$. \endstat \begproof{} In fact, we have $AC^q(\cU,\cA)=0$ for $q>n$.\qed \endproof \titleb{6.}{The De Rham-Weil Isomorphism Theorem} In \S~3 we defined cohomology groups by means of the simplicial flabby resolution. We show here that any resolution by acyclic sheaves could have been used instead. Let $(\cL^{\bu},d)$ be a resolution of a sheaf $\cA$. We assume in addition that all $\cL^q$ are acyclic on $X$, i.e.\ $H^s(X,\cL^q)=0$ for all $q\ge 0$ and $s\ge 1$. Set $\cZ^q=\ker d^q$. Then $\cZ^0=\cA$ and for every $q\ge 1$ we get a short exact sequence $$0\longrightarrow\cZ^{q-1}\longrightarrow\cL^{q-1}\buildo d^{q-1}\over\longrightarrow\cZ^q\longrightarrow 0.$$ Theorem 3.5 yields an exact sequence $$(6.1)~H^s(X,\cL^{q-1})\!{\buildo d^{q-1}\over\longrightarrow}H^s(X,\cZ^q)\!{\buildo \partial^{s,q}\over\longrightarrow}H^{s+1}(X,\cZ^{q-1}){\to}H^{s+1}(X,\cL^{q-1}){=}0.$$ If $s\ge 1$, the first group is also zero and we get an isomorphism $$\partial^{s,q}:H^s(X,\cZ^q)\buildo\simeq\over\longrightarrow H^{s+1}(X,\cZ^{q-1}).$$ For $s=0$ we have $H^0(X,\cL^{q-1})=\cL^{q-1}(X)$ and $H^0(X,\cZ^q)=\cZ^q(X)$ is the $q$-cocycle group of $\cL^\bu(X)$, so the connecting map $\partial^{0,q}$ gives an isomorphism $$H^q\big(\cL^\bu(X)\big)=\cZ^q(X)/d^{q-1}\cL^{q-1}(x) \buildo\wt\partial^{0,q}\over\longrightarrow H^1(X,\cZ^{q-1}).$$ The composite map $\partial^{q-1,1}\circ\cdots\circ\partial^{1,q-1}\circ \wt\partial^{0,q}$ therefore defines an isomorphism $$\leqalignno{\quad\qquad&H^q\big(\cL^\bu(X)\big)\!\buildo\wt\partial^{0,q} \over\longrightarrow\! H^1(X,\cZ^{q-1})\!\buildo\partial^{1,q-1}\over\longrightarrow\!\cdots\! \buildo\partial^{q-1,1}\over\longrightarrow\! H^q(X,\cZ^0){=}H^q(X,\cA).&(6.2)\cr}$$ This isomorphism behaves functorially with respect to morphisms of reso\-lutions. Our assertion means that for every sheaf morphism $\varphi:\cA\to\cB$ and every morphism of resolutions $\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$, there is a commutative diagram $$\cmalign{ &H^s\big(&\cL^\bu(X)\big)\hfill\longrightarrow H^s(&X,\cA)\cr &&\big\downarrow H^s(\varphi^\bu)\hfill&\big\downarrow H^s(\varphi)\hfill\cr &H^s\big(&\cM^\bu(X)\big)\longrightarrow H^s(&X,\cB).\cr}\leqno(6.3)$$ If $\cW^q=\ker\big(d^q:\cM^q\to\cM^{q+1}\big)$, the functoriality comes from the fact that we have commutative diagrams $$\cmalign{ 0\to&\cZ^{q-1}\hfill\to&\cL^{q-1}\hfill\to&\cZ^q\to 0~,~~~~ H^s(&X,\cZ^q)\hfill\buildo\partial^{s,q}\over\longrightarrow H^{s+1}(&X,\cZ^{q-1})\cr &\big\downarrow\varphi^{q-1}\hfill&~\big\downarrow\varphi^{q-1}\hfill &\big\downarrow\varphi^q\hfill&\big\downarrow H^s(\varphi^q)\hfill &\big\downarrow H^{s+1}(\varphi^{q-1})\hfill\cr 0\to&\cW^{q-1}\to&\cM^{q-1}\to&\cW^q\to 0~,~~~~ H^s(&X,\cW^q)\buildo\partial^{s,q}\over\longrightarrow H^{s+1}(&X,\cW^{q-1}).\cr}$$ \begstat{(6.4) De Rham-Weil isomorphism theorem} If $(\cL^\bu,d)$ is a resolution of $\cA$ by sheaves $\cL^q$ which are acyclic on $X$, there is a functorial isomorphism $$H^q\big(\cL^\bu(X)\big)\longrightarrow H^q(X,\cA).\eqno\square$$ \endstat \begstat{(6.5) Example: De Rham cohomology} \rm Let $X$ be a $n$-dimensional paracompact differential manifold. Consider the resolution $$0\to\bR\to\cE^0\buildo d\over\to\cE^1\to\cdots\to\cE^q\buildo d\over\to \cE^{q+1}\to\cdots\to\cE^n\to 0$$ given by the exterior derivative $d$ acting on germs of $C^\infty$ differential $q$-forms (c.f. Example 2.2). The {\it De Rham cohomology groups} of $X$ are precisely $$H^q_{\DR}(X,\bR)=H^q\big(\cE^\bu(X)\big).\leqno(6.6)$$ All sheaves $\cE^q$ are $\cE_X$-modules, so $\cE^q$ is acyclic by Cor.\ 4.19. Therefore, we get an isomorphism $$H^q_{\DR}(X,\bR)\buildo\simeq\over\longrightarrow H^q(X,\bR)\leqno(6.7)$$ from the De Rham cohomology onto the cohomology with values in the constant sheaf $\bR$. Instead of using $C^\infty$ differential forms, one can consider the resolution of $\bR$ given by the exterior derivative $d$ acting on currents: $$0\to\bR\to\cD'_n\buildo d\over\to\cD'_{n-1}\to\cdots\to\cD'_{n-q} \buildo d\over\to\cD'_{n-q-1}\to\cdots\to\cD'_0\to 0.$$ The sheaves $\cD'_q$ are also $\cE_X$-modules, hence acyclic. Thanks to (6.3), the inclusion $\cE^q\subset\cD_{n-q}'$ induces an isomorphism $$H^q\big(\cE^\bu(X)\big)\simeq H^q\big(\cD'_{n-\bu}(X)\big),\leqno(6.8)$$ both groups being isomorphic to $H^q(X,\bR)$. The isomorphism between cohomology of differential forms and singular cohomology (another topological invariant) was first established by (De Rham 1931). The above proof follows essentially the method given by (Weil 1952), in a more abstract setting. As we will see, the isomorphism $(6.7)$ can be put under a very explicit form in terms of \v Cech cohomology. We need a simple lemma. \endstat \begstat{(6.9) Lemma} Let $X$ be a paracompact differentiable manifold. There are arbitrarily fine open coverings $\cU=(U_\alpha)$ such that all intersections $U_{\alpha_0\ldots\alpha_q}$ are diffeomorphic to convex sets. \endstat \begproof{} Select locally finite coverings $\Omega'_j\subset\!\subset\Omega_j$ of $X$ by open sets diffeomorphic to concentric euclidean balls in $\bR^n$. Let us denote by $\tau_{jk}$ the transition diffeomorphism from the coordinates in $\Omega_k$ to those in $\Omega_j$. For any point $a\in\Omega'_j$, the function $x\mapsto|x-a|^2$ computed in terms of the coordinates of $\Omega_j$ becomes $|\tau_{jk}(x)-\tau_{jk}(a)|^2$ on any patch $\Omega_k\ni a$. It is clear that these functions are strictly convex at $a$, thus there is a euclidean ball $B(a,\varepsilon)\subset\Omega'_j$ such that all functions are strictly convex on $B(a,\varepsilon)\cap\Omega'_k \subset\Omega_k$ (only a finite number of indices $k$ is involved). Now, choose $\cU$ to be a (locally finite) covering of $X$ by such balls $U_\alpha=B(a_\alpha,\varepsilon_\alpha)$ with $U_\alpha\subset\Omega'_{\rho(\alpha)}$. Then the intersection $U_{\alpha_0\ldots\alpha_q}$ is defined in $\Omega_k$, $k=\rho(\alpha_0)$, by the equations $$|\tau_{jk}(x)-\tau_{jk}(a_{\alpha_m})|^2<\varepsilon_{\alpha_m}^2$$ where $j=\rho(\alpha_m)$, $0\le m\le q$. Hence the intersection is convex in the open coordinate chart $\Omega_{\rho(\alpha_0)}$.\qed \endproof Let $\Omega$ be an open subset of $\bR^n$ which is starshaped with respect to the origin. Then the De Rham complex $\bR\longrightarrow\cE^\bu(\Omega)$ is acyclic: indeed, Poincar\'e's lemma yields a homotopy operator $k^q:\cE^q(\Omega)\longrightarrow\cE^{q-1}(\Omega)$ such that $$\eqalign{ &k^qf_x(\xi_1,\ldots,\xi_{q-1})=\int_0^1t^{q-1}\,f_{tx}(x,\xi_1,\ldots,\xi_{q-1})\,dt, ~~~x\in\Omega,~~\xi_j\in\bR^n,\cr &k^0f=f(0)\in\bR~~~\hbox{\rm for}~~f\in\cE^0(\Omega).\cr}$$ Hence $H^q_{\DR}(\Omega,\bR)=0$ for $q\ge 1$. Now, consider the resolution $\cE^\bu$ of the constant sheaf $\bR$ on $X$, and apply the proof of the De Rham-Weil isomorphism theorem to \v Cech cohomology groups over a covering $\cU$ chosen as in Lemma 6.9. Since the intersections $U_{\alpha_0\ldots\alpha_s}$ are convex, all \v Cech cochains in $C^s(\cU,\cZ^q)$ are liftable in $\cE^{q-1}$ by means of $k^q$. Hence for all $s=1,\ldots,q$ we have isomorphisms $\partial^{s,q-s}:\check H^s(\cU,\cZ^{q-s})\longrightarrow\check H^{s+1}(\cU,\cZ^{q-s-1})$ for $s\ge 1$ and we get a resulting isomorphism $$\partial^{q-1,1}\circ\cdots\circ\partial^{1,q-1}\circ \wt\partial^{0,q}:H^q_{\DR}(X,\bR)\buildo\simeq\over\longrightarrow\check H^q(\cU,\bR)$$ We are going to compute the connecting homomorphisms $\partial^{s,q-s}$ and their inverses explicitly. Let $c$ in $C^s(\cU,\cZ^{q-s})$ such that $\delta^sc=0$. As $c_{\alpha_0\ldots\alpha_s}$ is $d$-closed, we can write $c=d(k^{q-s}c)$ where the cochain $k^{q-s}c\in C^s(\cU,\cE^{q-s-1})$ is defined as the family of sections $k^{q-s}c_{\alpha_0\ldots\alpha_s}\in\cE^{q-s-1}(U_{\alpha_0\ldots\alpha_s})$. Then $d(\delta^sk^{q-s}c)=\delta^s(dk^{q-s}c)=\delta^sc=0$ and $$\partial^{s,q-s}\{c\}=\{\delta^sk^{q-s}c\}\in\check H^{s+1}(\cU,\cZ^{q-s-1}).$$ The isomorphism $H^q_{\DR}(X,\bR)\buildo\simeq\over\longrightarrow\check H^q(\cU,\bR)$ is thus defined as follows: to the cohomology class $\{f\}$ of a closed $q$-form $f\in\cE^q(X)$, we associate the cocycle $(c^0_\alpha)= (f_{\restriction U_\alpha})\in C^0(\cU,\cZ^q)$, then the cocycle $$c^1_{\alpha\beta}=k^qc^0_\beta-k^qc^0_\alpha\in C^1(\cU,\cZ^{q-1}),$$ and by induction cocycles $(c^s_{\alpha_0\ldots\alpha_s})\in C^s(\cU,\cZ^{q-s})$ given by $$c^{s+1}_{\alpha_0\ldots\alpha_{s+1}}=\sum_{0\le j\le s+1}(-1)^j\, k^{q-s}c^s_{\alpha_0\ldots\wh{\alpha_j}\ldots\alpha_{s+1}}~~~~ \hbox{\rm on}~~U_{\alpha_0\ldots\alpha_{s+1}}.\leqno(6.10)$$ The image of $\{f\}$ in $\check H^q(\cU,\bR)$ is the class of the $q$-cocycle $(c^q_{\alpha_0\ldots\alpha_q})$ in $C^q(\cU,\bR)$. Conversely, let $(\psi_\alpha)$ be a $C^\infty$ partition of unity subordinate to $\cU$. Any \v Cech cocycle $c\in C^{s+1}(\cU,\cZ^{q-s-1})$ can be written $c=\delta^s\gamma$ with $\gamma\in C^s(\cU,\cE^{q-s-1})$ given by $$\gamma_{\alpha_0\ldots\alpha_s}=\sum_{\nu\in I}\psi_\nu\, c_{\nu\alpha_0\ldots\alpha_s},$$ (c.f. Prop.\ 5.11~b)), thus $\{c'\}=(\partial^{s,q-s})^{-1}\{c\}$ can be represented by the cochain $c'=d\gamma\in C^s(\cU,\cZ^{q-s})$ such that $$c'_{\alpha_0\ldots\alpha_s}=\sum_{\nu\in I}d\psi_\nu\wedge c_{\nu\alpha_0\ldots\alpha_s}=(-1)^{q-s-1}\sum_{\nu\in I} c_{\nu\alpha_0\ldots\alpha_s}\wedge d\psi_\nu.$$ For a reason that will become apparent later, we shall in fact modify the sign of our isomorphism $\partial^{s,q-s}$ by the factor $(-1)^{q-s-1}$. Starting from a class $\{c\}\in\check H^q(\cU,\bR)$, we obtain inductively $\{b\}\in\check H^0(\cU,\cZ^q)$ such that $$b_{\alpha_0}=\sum_{\nu_0,\ldots,\nu_{q-1}}c_{\nu_0\ldots\nu_{q-1}\alpha_0}\, d\psi_{\nu_0}\wedge\ldots\wedge d\psi_{\nu_{q-1}}~~~~\hbox{\rm on}~~U_{\alpha_0}, \leqno(6.11)$$ corresponding to $\{f\}\in H^q_{\DR}(X,\bR)$ given by the explicit formula $$f=\sum_{\nu_q}\psi_{\nu_q}b_{\nu_q}= \sum_{\nu_0,\ldots,\nu_q}c_{\nu_0\ldots\nu_q}\,\psi_{\nu_q} d\psi_{\nu_0}\wedge\ldots\wedge d\psi_{\nu_{q-1}}.\leqno(6.12)$$ The choice of sign corresponds to (6.2) multiplied by $(-1)^{q(q-1)/2}$. \begstat{(6.13) Example: Dolbeault cohomology groups} \rm Let $X$ be a $\bC$-analytic manifold of dimension $n$, and let $\cE^{p,q}$ be the sheaf of germs of $C^\infty$ differential forms of type $(p,q)$ with complex values. For every $p=0,1,\ldots,n$, the Dolbeault-Grothendieck Lemma I-2.9 shows that $(\cE^{p,\bu},d'')$ is a resolution of the sheaf $\Omega^p_X$ of germs of holomorphic forms of degree $p$ on $X$. The {\it Dolbeault cohomology groups} of $X$ already considered in chapter~1 can be defined by $$H^{p,q}(X,\bC)=H^q\big(\cE^{p,\bu}(X)\big).\leqno(6.14)$$ The sheaves $\cE^{p,q}$ are acyclic, so we get the {\it Dolbeault isomorphism theorem}, originally proved in (Dolbeault 1953), which relates $d''$-cohomology and sheaf cohomology: $$H^{p,q}(X,\bC)\buildo\simeq\over\longrightarrow H^q(X,\Omega^p_X).\leqno(6.15)$$ The case $p=0$ is especially interesting: $$H^{0,q}(X,\bC)\simeq H^q(X,\cO_X).\leqno(6.16)$$ As in the case of De Rham cohomology, there is an inclusion $\cE^{p,q}\subset\cD'_{n-p,n-q}$ and the complex of currents $(\cD'_{n-p,n-\bu},d'')$ defines also a resolution of $\Omega^p_X$. Hence there is an isomorphism: $$H^{p,q}(X,\bC)=H^q\big(\cE^{p,\bu}(X)\big)\simeq H^q\big(\cD'_{n-p,n-\bu}(X)\big).\leqno(6.17)$$ \endstat \titleb{7.}{Cohomology with Supports} As its name indicates, cohomology with supports deals with sections of sheaves having supports in prescribed closed sets. We first introduce what is an admissible family of supports. \begstat{(7.1) Definition} A family of supports on a topological space $X$ is a collection $\Phi$ of closed subsets of $X$ with the following two properties: \medskip \item{\rm a)} If $F\,,\,F'\in\Phi$, then $F\cup F'\in\Phi~;$ \medskip \item{\rm b)} If $F\in\Phi$ and $F'\subset F$ is closed, then $F'\in\Phi.$ \endstat \begstat{(7.2) Example} \rm Let $S$ be an arbitrary subset of $X$. Then the family of all closed subsets of $X$ contained in $S$ is a family of supports. \endstat \begstat{(7.3) Example} \rm The collection of all compact (non necessarily Hausdorff) subsets of $X$ is a family of supports, which will be denoted simply $c$ in the sequel.\qed \endstat \begstat{(7.4) Definition} For any sheaf $\cA$ and any family of supports $\Phi$ on $X$, $\cA_\Phi(X)$ will denote the set of all sections $f\in\cA(X)$ such that $\Supp\,f\in\Phi$. \endstat It is clear that $\cA_\Phi(X)$ is a subgroup of $\cA(X)$. We can now introduce cohomology groups with arbitrary supports. \begstat{(7.5) Definition} The cohomology groups of $\cA$ with supports in $\Phi$ are $$H^q_\Phi(X,\cA)=H^q\big(\cA^{[\bu]}_\Phi(X)\big).$$ The cohomology groups with compact supports will be denoted $H^q_c(X,\cA)$ and the cohomology groups with supports in a subset $S$ will be denoted $H^q_S(X,\cA)$. \endstat In particular $H^0_\Phi(X,\cA)=\cA_\Phi(X)$. If $0\to\cA\to\cB\to\cC\to 0$ is an exact sequence, there are corresponding exact sequences $$\cmalign{ &\hfil 0~\longrightarrow&~~\cA^{[q]}_\Phi(X)&\longrightarrow~~\cB^{[q]}_\Phi(X) &\longrightarrow~~\cC^{[q]}_\Phi(X)&\longrightarrow\cdots\cr &&~~H_\Phi^q(X,\cA)&\longrightarrow H_\Phi^q(X,\cB)&\longrightarrow H_\Phi^q(X,\cC)&\longrightarrow H_\Phi^{q+1}(X,\cA)\longrightarrow\cdots.\cr}\leqno(7.6)$$ When $\cA$ is flabby, there is an exact sequence $$0\longrightarrow\cA_\Phi(X)\longrightarrow\cB_\Phi(X)\longrightarrow\cC_\Phi(X)\longrightarrow 0\leqno(7.7)$$ and every $g\in\cC_\Phi(X)$ can be lifted to $v\in\cB_\Phi(X)$ without enlarging the support: apply the proof of Prop.\ 4.3 to a maximal lifting which extends $w=0$ on $W=\complement(\Supp\,g)$. It follows that a flabby sheaf $\cA$ is $\Phi$-acyclic, i.e.\ $H^q_\Phi(X,\cA)=0$ for all $q\ge 1$. Similarly, assume that $X$ is paracompact and that $\cA$ is soft, and suppose that $\Phi$ has the following additional property: every set $F\in\Phi$ has a neighborhood $G\in\Phi$. An adaptation of the proofs of Prop.\ 4.3 and 4.13 shows that (7.7) is again exact. Therefore every soft sheaf is also $\Phi$-acyclic in that case. As a consequence of (7.6), any resolution $\cL^\bu$ of $\cA$ by $\Phi$-acyclic sheaves provides a canonical De Rham-Weil isomorphism $$H^q\big(\cL^\bu_\Phi(X)\big)\longrightarrow H^q_\Phi(X,\cA).\leqno(7.8)$$ \begstat{(7.9) Example: De Rham cohomology with compact support} \rm In the special case of the De Rham resolution $\bR\longrightarrow\cE^\bu$ on a paracompact manifold, we get an isomorphism $$H^q_{\DR,c}(X,\bR):=H^q\big((\cD^\bu(X)\big)\buildo\simeq\over\longrightarrow H^q_c(X,\bR),\leqno(7.10)$$ where $\cD^q(X)$ is the space of smooth differential $q$-forms with compact support in $X$. These groups are called the {\it De Rham cohomology groups} of $X$ with compact support. When $X$ is oriented, $\dim X=n$, we can also consider the complex of compactly supported currents: $$0\longrightarrow\cE'_n(X)\buildo d\over\longrightarrow\cE'_{n-1}(X)\longrightarrow\cdots\longrightarrow\cE'_{n-q}(X) \buildo d\over\longrightarrow\cE'_{n-q-1}(X)\longrightarrow\cdots.$$ Note that $\cD^\bu(X)$ and $\cE'_{n-\bu}(X)$ are respectively the subgroups of compactly supported sections in $\cE^\bu$ and $\cD'_{n-\bu}$, both of which are acyclic resolutions~of~$\bR$. Therefore the inclusion $\cD^\bu(X)\subset\cE'_{n-\bu}(X)$ induces an isomorphism $$H^q\big(\cD^\bu(X)\big)\simeq H^q\big(\cE'_{n-\bu}(X)\big),\leqno(7.11)$$ both groups being isomorphic to $H^q_c(X,\bR)$.\qed \endstat Now, we concentrate our attention on cohomology groups with compact support. We assume until the end of this section that $X$ is a {\it locally compact} space. \begstat{(7.12) Proposition} There is an isomorphism $$H^q_c(X,\cA)=\lim_{\displaystyle\,\longrightarrow\atop\scriptstyle U\subset\!\subset X}~~H^q(\ol U,\cA_U)$$ where $\cA_U$ is the sheaf of sections of $\cA$ vanishing on $X\ssm U$ (c.f. \S 3). \endstat \begproof{} By definition $$H^q_c(X,\cA)=H^q\big(\cA^{[\bu]}_c(X)\big) =\lim_{\displaystyle\,\longrightarrow\atop\scriptstyle U\subset\!\subset X}~~ H^q\big((\cA^{[\bu]})_U(\ol U)\big)$$ since sections of $(\cA^{[\bu]})_U(\ol U)$ can be extended by $0$ on $X\ssm U$. However, $(\cA^{[\bu]})_U$ is a resolution of $\cA_U$ and $\smash{(\cA^{[q]})_U}$ is a $\smash{\bZ^{[q]}}$-module, so it is acyclic on $\smash{\ol U}$. The De Rham-Weil isomorphism theorem implies $$H^q\big((\cA^{[\bu]})_U(\ol U)\big)=H^q(\ol U,\cA_U)$$ and the proposition follows. The reader should take care of the fact that $(\cA^{[q]})_U$ does not coincide in general with $(\cA_U)^{[q]}$.\qed \endproof The cohomology groups with compact support can also be defined by means of \v Cech cohomology. \begstat{(7.13) Definition} Assume that $X$ is a separable locally compact space. If $\cU=(U_\alpha)$ is a locally finite covering of $X$ by relatively compact open subsets, we let $C^q_c(\cU,\cA)$ be the subgroups of cochains such that only finitely many coefficients $c_{\alpha_0\ldots\alpha_q}$ are non zero. The \v Cech cohomology groups with compact support are defined by $$\eqalign{ &\check H^q_c(\cU,\cA)=H^q\big(C^\bu_c(\cU,\cA)\big)\cr &\check H^q_c(X,\cA)=\lim_{{\displaystyle\longrightarrow}\atop{\scriptstyle\cU}} H^q\big(C^\bu_c(\cU,\cA)\big)\cr}$$ \endstat For such coverings $\cU$, Formula (5.13) yields canonical morphisms $$H^q(\lambda^\bu)~:~~\check H^q_c(\cU,\cA)\longrightarrow H^q_c(X,\cA).\leqno(7.14)$$ Now, the lifting Lemma 5.20 is valid for cochains with compact supports, and the same proof as the one given in \S 5 gives an isomorphism $$\check H^q_c(X,\cA)\simeq H^q_c(X,\cA).\leqno(7.15)$$ \titleb{8.}{Cup Product} Let $\cR$ be a sheaf of commutative rings and $\cA$, $\cB$ sheaves of $\cR$-modules on a space $X$. We denote by $\cA\otimes_\cR\cB$ the sheaf on $X$ defined by $$(\cA\otimes_\cR\cB)_x=\cA_x\otimes_{\cR_x}\cB_x,\leqno(8.1)$$ with the weakest topology such that the range of any section given by $\cA(U)\otimes_{\cR(U)}\cB(U)$ is open in $\cA\otimes_\cR\cB$ for any open set $U\subset X$. Given $f\in\cA^{[p]}_x$ and $g\in\cB^{[q]}_x$, the {\it cup product} $f\smallsmile g\in(\cA\otimes_\cR\cB)^{[p+q]}_x$ is defined by $$f\smallsmile g(x_0,\ldots,x_{p+q})=f(x_0,\ldots,x_p)(x_{p+q})\otimes g(x_p,\ldots,x_{p+q}). \leqno(8.2)$$ A simple computation shows that $$d^{p+q}(f\smallsmile g)=(d^pf)\smallsmile g+(-1)^p\,f\smallsmile(d^qg). \leqno(8.3)$$ In particular, $f\smallsmile g$ is a cocycle if $f,g$ are cocycles, and we have $$(f+d^{p-1}f')\smallsmile(g+d^{q-1}g')=f\smallsmile g+d^{p+q-1} \big(f'\smallsmile g+(-1)^pf\smallsmile g'+f'\smallsmile dg'\big).$$ Consequently, there is a well defined $\cR(X)$-bilinear morphism $$H^p(X,\cA)\times H^q(X,\cB)\longrightarrow H^{p+q}(X,\cA\otimes_\cR\cB)\leqno(8.4)$$ which maps a pair $(\{f\},\{g\})$ to $\{f\smallsmile g\}$. Let $0\to\cB\to\cB'\to\cB''\to 0$ be an exact sequence of sheaves. Assume that the sequence obtained after taking the tensor product by $\cA$ is also exact: $$0\longrightarrow\cA\otimes_\cR\cB\longrightarrow\cA\otimes_\cR\cB'\longrightarrow\cA\otimes_\cR\cB''\longrightarrow 0.$$ Then we obtain connecting homomorphisms $$\eqalign{ &\partial^q~:~~H^q(X,\cB'')\longrightarrow H^{q+1}(X,\cB),\cr &\partial^q~:~~H^q(X,\cA\otimes_\cR\cB'')\longrightarrow H^{q+1}(X,\cA\otimes_\cR\cB). \cr}$$ For every $\alpha\in H^p(X,\cA)$, $\beta''\in H^q(X,\cB'')$ we have $$\leqalignno{ \partial^{p+q}(\alpha\smallsmile\beta'')&=(-1)^p\,\alpha\smallsmile(\partial^q\beta''), &(8.5)\cr \partial^{p+q}(\beta''\smallsmile\alpha)&=(\partial^q\beta'')\smallsmile\alpha, &(8.5')\cr}$$ where the second line corresponds to the tensor product of the exact sequence by $\cA$ on the right side. These formulas are deduced from (8.3) applied to a repre\-sentant $f\in\cA^{[p]}(X)$ of $\alpha$ and to a lifting $g'\in \cB^{\prime[q]}(X)$ of a representative $g''$ of $\beta''$ (note that $d^pf=0$). \begstat{(8.6) Associativity and anticommutativity} Let $i:\cA\otimes_\cR\cB\longrightarrow\cB\otimes_\cR\cA$ be the canonical isomorphism $s\otimes t\mapsto t\otimes s$. For all $\alpha\in H^p(X,\cA)$, $\beta\in H^q(X,\cB)$ we have $$\beta\smallsmile\alpha=(-1)^{pq}\,i(\alpha\smallsmile\beta).$$ If $\cC$ is another sheaf of $\cR$-modules and $\gamma\in H^r(X,\cC)$, then $$(\alpha\smallsmile\beta)\smallsmile\gamma= \alpha\smallsmile(\beta\smallsmile\gamma).$$ \endstat \begproof{} The associativity property is easily seen to hold already for all cochains $$(f\smallsmile g)\smallsmile h=f\smallsmile(g\smallsmile h),~~~f\in\cA^{[p]}_x,~~g\in\cB^{[q]}_x,~~ h\in\cC^{[r]}_x.$$ The commutation property is obvious for $p=q=0$, and can be proved in general by induction on $p+q$. Assume for example $q\ge 1$. Consider the exact sequence $$0\longrightarrow\cB\longrightarrow\cB'\longrightarrow\cB''\longrightarrow 0$$ where $\cB'=\cB^{[0]}$ and $\cB''=\cB^{[0]}/\cB$. This exact sequence splits on each stalk (but not globally, nor even locally): a left inverse $\smash{\cB^{[0]}_x}\to\cB_x$ of the inclusion\break is given by $g\mapsto g(x)$. Hence the sequence remains exact after taking the tensor product with $\cA$. Now, as $\cB'$ is acyclic, the connecting homomorphism $H^{q-1}(X,\cB'')\longrightarrow H^q(X,\cB)$ is onto, so there is $\beta''\in H^{q-1}(X,\cB'')$ such that $\beta=\partial^{q-1}\beta''$. Using (8.$5'$), (8.5) and the induction hypothesis, we find $$\eqalignno{ \beta\smallsmile\alpha&=\partial^{p+q-1}(\beta''\smallsmile\alpha)=\partial^{p+q-1}\big( (-1)^{p(q-1)}\,i(\alpha\smallsmile\beta'')\big)\cr &=(-1)^{p(q-1)}\,i\partial^{p+q-1}(\alpha\smallsmile\beta'')= (-1)^{p(q-1)}(-1)^p\,i(\alpha\smallsmile\beta).&\square\cr}$$ \endproof Theorem 8.6 shows in particular that $H^\bu(X,\cR)$ is a graded associative and supercommutative algebra, i.e.\ $\beta\smallsmile\alpha=(-1)^{pq}\,\alpha\smallsmile\beta$ for all classes $\alpha\in H^p(X,\cR)$, $\beta\in H^q(X,\cR)$. If $\cA$ is a $\cR$-module, then $H^\bu(X,\cA)$ is a graded $H^\bu(X,\cR)$-module. \begstat{(8.7) Remark} \rm The cup product can also be defined for \v Cech cochains. Given $c\in C^p(\cU,\cA)$ and $c'\in C^q(\cU,\cB)$, the cochain $c\smallsmile c'\in C^{p+q}(\cU,\cA\otimes_\cR\cB)$ is defined by $$(c\smallsmile c')_{\alpha_0\ldots\alpha_{p+q}}=c_{\alpha_0\ldots\alpha_p}\otimes c'_{\alpha_p\ldots\alpha_{p+q}}~~~\hbox{\rm on}~~U_{\alpha_0\ldots\alpha_{p+q}}.$$ Straightforward calculations show that $$\delta^{p+q}(c\smallsmile c')=(\delta^pc)\smallsmile c'+(-1)^p\,c\smallsmile(\delta^qc')$$ and that there is a commutative diagram $$\cmalign{ \check H^p(\cU,\cA)&\times\check H^q(\cU,\cB)&\longrightarrow \check H^{p+q}(\cU&,\cA\otimes_\cR\cB)\cr &\big\downarrow&&\big\downarrow\cr H^p(X,\cA)&\times H^q(X,\cB)&\longrightarrow H^{p+q}(X&,\cA\otimes_\cR\cB),\cr}$$ where the vertical arrows are the canonical morphisms $H^s(\lambda^\bu)$ of (5.14). Note that the commutativity already holds in fact on cochains. \endstat \begstat{(8.8) Remark} \rm Let $\Phi$ and $\Psi$ be families of supports on $X$. Then $\Phi\cap\Psi$ is again a family of supports, and Formula (8.2) defines a bilinear map $$H_\Phi^p(X,\cA)\times H_\Psi^q(X,\cB)\longrightarrow H_{\Phi\cap\Psi}^{p+q} (X,\cA\otimes_\cR\cB)\leqno(8.9)$$ on cohomology groups with supports. This follows immediately from the fact that $\Supp(f\smallsmile g)\subset\Supp\,f\cap\Supp\,g$. \endstat \begstat{(8.10) Remark} \rm Assume that $X$ is a differentiable manifold. Then the cohomology complex $H^\bu_{\DR}(X,\bR)$ has a natural structure of supercommutative algebra given by the wedge product of differential forms. We shall prove the following compatibility statement: \medskip \noindent{\it Let $H^q(X,\bR)\longrightarrow H^q_{\DR}(X,\bR)$ be the De Rham-Weil isomorphism given by Formula {\rm (6.12)}. Then the cup product $c'\smallsmile c''$ is mapped on the wedge product $f'\wedge f''$ of the corresponding De Rham cohomology classes.} \medskip \noindent By remark 8.7, we may suppose that $c',c''$ are \v Cech cohomology classes of respective degrees $p,q$. Formulas (6.11) and (6.12) give $$\eqalign{ f'_{\restriction U_{\nu_p}}&=\sum_{\nu_0,\ldots,\nu_{p-1}} c'_{\nu_0\ldots\nu_{p-1}\nu_p}\,d\psi_{\nu_0}\wedge\ldots\wedge d\psi_{\nu_{p-1}},\cr f''&=\sum_{\nu_p,\ldots,\nu_{p+q}} c''_{\nu_p\ldots\nu_{p+q}}\,\psi_{\nu_{p+q}}d\psi_{\nu_p}\wedge\ldots\wedge d\psi_{\nu_{p+q-1}}.\cr}$$ We get therefore $$f'\wedge f''=\sum_{\nu_0,\ldots,\nu_{p+q}} c'_{\nu_0\ldots\nu_p}\,c''_{\nu_p\ldots\nu_{p+q}}\, \psi_{\nu_{p+q}}d\psi_{\nu_0}\wedge\ldots\wedge\psi_{\nu_{p+q-1}},$$ which is precisely the image of $c\smallsmile c'$ in the De Rham cohomology.\qed \endstat \titleb{9.}{Inverse Images and Cartesian Products} \titlec{9.A.}{Inverse Image of a Sheaf} Let $F:X\to Y$ be a continuous map between topological spaces $X,Y$, and let $\pi:\cA\to Y$ be a sheaf of abelian groups. Recall that {\it inverse image} $F^{-1}\cA$ is defined as the sheaf-space $$F^{-1}\cA=\cA\times_Y X=\big\{(s,x)\,;\,\pi(s)=F(x)\big\}$$ with projection $\pi'=\pr_2:F^{-1}\cA\to X$. The stalks of $F^{-1}\cA$ are given by $$(F^{-1}\cA)_x=\cA_{F(x)},\leqno(9.1)$$ and the sections $\sigma\in F^{-1}\cA(U)$ can be considered as continuous mappings \hbox{$\sigma:U\to\cA$} such that $\pi\circ\sigma=F$. In particular, any section $s\in\cA(U)$ has a {\it pull-back} $$F^\star s=s\circ F\in F^{-1}\cA\big(F^{-1}(U)\big).\leqno(9.2)$$ For any $v\in\cA^{[q]}_y$, we define $F^\star v\in(F^{-1}\cA)^{[q]}_x$ by $$F^\star v(x_0,\ldots,x_q)=v\big(F(x_0),\ldots,F(x_q)\big)\in(F^{-1}\cA)_{x_q}= \cA_{F(x_q)}\leqno(9.3)$$ for $x_0\in V(x)$, $x_1\in V(x_0),\ldots,x_q\in V(x_0,\ldots,x_{q-1})$. We get in this way a morphism of complexes $F^\star:\cA^{[\bu]}(Y)\longrightarrow (F^{-1}\cA)^{[\bu]}(X)$. On cohomology groups, we thus have an induced morphism $$F^\star~:~~H^q(Y,\cA)\longrightarrow H^q(X,F^{-1}\cA\cA).\leqno(9.4)$$ Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves on $X$. Thanks to property (9.1), there is an exact sequence $$0\longrightarrow F^{-1}\cA\longrightarrow F^{-1}\cB\longrightarrow F^{-1}\cC\longrightarrow 0.$$ It is clear on the definitions that the morphism $F^\star$ in (9.4) commutes with the associated cohomology exact sequences. Also, $F^\star$ preserves the cup product, i.e.\ $F^\star(\alpha\smallsmile\beta)= F^\star\alpha\smallsmile F^\star\beta$ whenever $\alpha,\beta$ are cohomology classes with values in sheaves $\cA$, $\cB$ on $X$. Furthermore, if $G:Y\to Z$ is a continuous map, we have $$(G\circ F)^\star=F^\star\circ G^\star.\leqno(9.5)$$ \begstat{(9.6) Remark} \rm Similar definitions can be given for \v Cech cohomology. If $\cU=(U_\alpha)_{\alpha\in I}$ is an open covering of $Y$, then $F^{-1}\cU=\big(F^{-1}(U_\alpha)\big)_{\alpha\in I}$ is an open covering of $X$. For $c\in C^q(\cU,\cA)$, we set $$(F^\star c)_{\alpha_0\ldots\alpha_q}=c_{\alpha_0\ldots\alpha_q}\circ F \in C^q(F^{-1}\cU,F^{-1}\cA).$$ This definition is obviously compatible with the morphism from \v Cech cohomology to ordinary cohomology. \endstat \begstat{(9.7) Remark} \rm Let $\Phi$ be a family of supports on $Y$. We define $F^{-1}\Psi$ to be the family of closed sets $K\subset X$ such that $F(K)$ is contained in some set $L\in\Psi$. Then (9.4) can be generalized in the form $$F^\star~:~~H_\Psi^q(Y,\cA)\longrightarrow H_{F^{-1}\Psi}^q(X,F^{-1}\cA). \leqno(9.8)$$ \endstat \begstat{(9.9) Remark} \rm Assume that $X$ and $Y$ are paracompact differentiable manifolds and that $F:X\to Y$ is a $C^\infty$ map. If $(\psi_\alpha)_{\alpha\in I}$ is a partition of unity subordinate to $\cU$, then $(\psi_\alpha\circ F)_{\alpha\in I}$ is a partition of unity on $X$ subordinate to $F^{-1}\cU$. Let $c\in C^q(\cU,\bR)$. The differential form associated to $F^\star c$ in the De Rham cohomology is $$\eqalign{ g&=\sum_{\nu_0,\ldots,\nu_q}c_{\nu_0\ldots\nu_q}(\psi_{\nu_q}\circ F) d(\psi_{\nu_0}\circ F)\wedge\ldots\wedge d(\psi_{\nu_{q-1}}\circ F)\cr &=F^\star\Big(\sum_{\nu_0,\ldots,\nu_q}c_{\nu_0\ldots\nu_q}\,\psi_{\nu_q} d\psi_{\nu_0}\wedge\ldots\wedge d\psi_{\nu_{q-1}}\Big).\cr}$$ Hence we have a commutative diagram $$\cmalign{ &H^q_{\DR}(Y,\bR)&\buildo\simeq\over\longrightarrow&\check H^q(Y,\bR) &\buildo\simeq\over\longrightarrow&H^q(Y,\bR)\cr &\quad~~\big\downarrow F^\star&&\quad~~\big\downarrow F^\star &&\quad~~\big\downarrow F^\star\cr &H^q_{\DR}(X,\bR)&\buildo\simeq\over\longrightarrow&\check H^q(X,\bR) &\buildo\simeq\over\longrightarrow&H^q(X,\bR).\cr}$$ \endstat \titlec{9.B.}{Cohomology Groups of a Subspace} Let $\cA$ be a sheaf on a topological space $X$, let $S$ be a subspace of $X$ and $i_S:S\lhra X$ the inclusion. Then $i_S^{-1}\cA$ is the restriction of $\cA$ to $S$, so that $H^q(S,\cA)=H^q(S,i_S^{-1}\cA)$ by definition. For any two subspaces $S'\subset S$, the inclusion of $S'$ in $S$ induces a restriction morphism $$H^q(S,\cA)\longrightarrow H^q(S',\cA).$$ \begstat{(9.10) Theorem} Let $\cA$ be a sheaf on $X$ and $S$ a strongly paracompact subspace in $X$. When $\Omega$ ranges over open neighborhoods of $S$, we have $$H^q(S,\cA)= \lim_{\displaystyle\longrightarrow\atop\scriptstyle\Omega\supset S}~~H^q(\Omega,\cA).$$ \endstat \begproof{} When $q=0$, the property is equivalent to Prop.\ 4.7. The general case follows by induction on $q$ if we use the long cohomology exact sequences associated to the short exact sequence $$0\longrightarrow\cA\longrightarrow\cA^{[0]}\longrightarrow\cA^{[0]}/\cA\longrightarrow 0$$ on $S$ and on its neighborhoods $\Omega$ (note that the restriction of a flabby sheaf to $S$ is soft by Prop.\ 4.7 and the fact that every closed subspace of a strongly paracompact subspace is strongly paracompact).\qed \endproof \titlec{9.C.}{Cartesian Product} We use here the formalism of inverse images to deduce the cartesian product from the cup product. Let $R$ be a fixed commutative ring and $\cA\to X$, $\cB\to Y$ sheaves of $R$-modules. We define the {\it external tensor product} by $$\cA\stimes_R\cB=\pr_1^{-1}\cA\otimes_R\pr_2^{-1}\cB \leqno(9.11)$$ where $\pr_1$, $\pr_2$ are the projections of $X\times Y$ onto $X$, $Y$ respectively. The sheaf $\cA\stimes_R\;\cB$ is thus the sheaf on $X\times Y$ whose stalks are $$(\cA\stimes_R\cB)_{(x,y)}=\cA_x\otimes_R\cB_y.\leqno(9.12)$$ For all cohomology classes $\alpha\in H^p(X,\cA)$, $\beta\in H^q(Y,\cB)$ the {\it cartesian product} $\alpha\times\beta\in H^{p+q}(X\times Y, \cA\stimes_R\cB)$ is defined by $$\alpha\times\beta=(\pr_1^\star\alpha)\smallsmile(\pr_2^\star \beta).\leqno(9.13)$$ More generally, let $\Phi$ and $\Psi$ be families of supports in $X$ and $Y$ respectively. If $\Phi\times\Psi$ denotes the family of all closed subsets of $X\times Y$ contained in products $K\times L$ of elements $K\in\Phi$, $L\in\Psi$, the cartesian product defines a $R$-bilinear map $$H_\Phi^p(X,\cA)\times H_\Psi^q(Y,\cB)\longrightarrow H_{\Phi\times\Psi}^{p+q} (X\times Y,\cA\stimes_R\cB).\leqno(9.14)$$ If $\cA'\to X$, $\cB'\to Y$ are sheaves of abelian groups and if $\alpha'$, $\beta'$ are cohomology classes of degree $p'$, $q'$ with values in $\cA'$, $\cB'$, one gets easily $$(\alpha\times\beta)\smallsmile(\alpha'\times\beta')=(-1)^{qp'} (\alpha\smallsmile\alpha')\times(\beta\smallsmile\beta').\leqno(9.15)$$ Furthermore, if $F:X'\to X$ and $G:Y'\to Y$ are continuous maps, then $$(F\times G)^\star(\alpha\times\beta)=(F^\star\alpha)\times(G^\star\beta). \leqno(9.16)$$ \titleb{10.}{Spectral Sequence of a Filtered Complex} \titlec{10.A.}{Construction of the Spectral Sequence} The theory of spectral sequences consists essentially in computing the homo\-logy groups of a differential module $(K,d)$ by ``successive approximations", once a filtration $F_p(K)$ is given in $K$ and the cohomology groups of the graded modules $G_p(K)$ are known. Let us first recall some standard definitions and notations concerning filtrations. \begstat{(10.1) Definition} Let $R$ be a commutative ring. A filtration of a $R$-module $M$ is a sequence of submodules $M_p\subset M$, $p\in\bZ$, also denoted $M_p=F_p(M)$, such that $M_{p+1}\subset M_p$ for all $p\in\bZ$, $\bigcup M_p=M$ and $\bigcap M_p=\{0\}$. The associated graded module is $$G(M)=\bigoplus_{p\in\bZ}G_p(M),~~~~G_p(M)=M_p/M_{p+1}.$$ \endstat Let $(K,d)$ be a differential module equipped with a filtration $(K_p)$ by differential submodules (i.e.\ $dK_p\subset K_p$ for every $p$). For any number \hbox{$r\in\bN\cup\{\infty\}$}, we define $Z^p_r,\,B^p_r\subset G_p(K)=K_p/K_{p+1}$ by $$\leqalignno{ \qquad\qquad Z^p_r&=K_p\cap d^{-1}K_{p+r}~~\hbox{\rm mod}~K_{p+1},~~~ Z^p_\infty=K_p\cap d^{-1}\{0\}~~\hbox{\rm mod}~K_{p+1},&(10.2)\cr \qquad\qquad B^p_r&=K_p\cap dK_{p-r+1}~~\hbox{\rm mod}~K_{p+1},~~~ B^p_\infty=K_p\cap dK~~~\hbox{\rm mod}~K_{p+1}.&(10.2')\cr}$$ \begstat{(10.3) Lemma} For every $p$ and $r$, there are inclusions $$\ldots\subset B^p_r\subset B^p_{r+1}\subset\ldots\subset B^p_\infty \subset Z^p_\infty\subset\ldots\subset Z^p_{r+1}\subset Z^p_r\subset\ldots$$ and the differential $d$ induces an isomorphism $$\wt d~:~~Z^p_r/Z^p_{r+1}\longrightarrow B^{p+r}_{r+1}/B^{p+r}_r.$$ \endstat \begproof{} It is clear that $(Z^p_r)$ decreases with $r$, that $(B^p_r)$ increases with $r$, and that $B^p_\infty\subset Z^p_\infty$. By definition $$\eqalign{ Z^p_r&=(K_p\cap d^{-1}K_{p+r})/(K_{p+1}\cap d^{-1}K_{p+r}),\cr B^p_r&=(K_p\cap dK_{p-r+1})/(K_{p+1}\cap dK_{p-r+1}).\cr}$$ The differential $d$ induces a morphism $$Z^p_r\longrightarrow (dK_p\cap K_{p+r})/(dK_{p+1}\cap K_{p+r})$$ whose kernel is $(K_p\cap d^{-1}\{0\})/(K_{p+1}\cap d^{-1}\{0\})=Z^p_\infty$, whence isomorphisms $$\eqalign{ \wh d~:~~&Z^p_r/Z^p_\infty\longrightarrow(K_{p+r}\cap dK_p)/(K_{p+r}\cap dK_{p+1}),\cr \wt d~:~~&Z^p_r/Z^p_{r+1}\longrightarrow(K_{p+r}\cap dK_p)/ (K_{p+r}\cap dK_{p+1}+K_{p+r+1}\cap dK_p).\cr}$$ The right hand side of the last arrow can be identified to $B^{p+r}_{r+1}/B^{p+r}_r$, for $$\eqalignno{ B^{p+r}_r&=(K_{p+r}\cap dK_{p+1})/(K_{p+r+1}\cap dK_{p+1}),\cr B^{p+r}_{r+1}&=(K_{p+r}\cap dK_p)/(K_{p+r+1}\cap dK_p).&\square\cr}$$ \endproof Now, for each $r\in\bN$, we define a complex $E^\bu_r=\bigoplus_{p\in\bZ} E^p_r$ with a differential $d_r:E^p_r\longrightarrow E^{p+r}_r$ of degree $r$ as follows: we set $E^p_r=Z^p_r/B^p_r$ and take $$d_r~:~~Z^p_r/B^p_r\lraww Z^p_r/Z^p_{r+1}\buildo{\displaystyle\wt d} \over\longrightarrow B^{p+r}_{r+1}/B^{p+r}_r\lhra Z^{p+r}_r/B^{p+r}_r\leqno(10.4)$$ where the first arrow is the obvious projection and the third arrow the obvious inclusion. Since $d_r$ is induced by $d$, we actually have $d_r\circ d_r=0$~; this can also be seen directly by the fact that $B^{p+r}_{r+1}\subset Z^{p+r}_{r+1}$. \begstat{(10.5) Theorem and definition} There is a canonical isomorphism $E^\bu_{r+1}\simeq H^\bu(E^\bu_r)$. The sequence of differential complexes $(E^\bu_r,d^\bu_r)$ is called the spectral sequence of the filtered differential module $(K,d)$. \endstat \begproof{} Since $\wt d$ is an isomorphism in (10.4), we have $$\ker\,d_r=Z^p_{r+1}/B^p_r,~~~~\Im d_r=B^{p+r}_{r+1}/B^{p+r}_r.$$ Hence the image of $d_r:E^{p-r}_r\longrightarrow E^p_r$ is $B^p_{r+1}/B^p_r$ and $$H^p(E^\bu_r)=(Z^p_{r+1}/B^p_r)/(B^p_{r+1}/B^p_r) \simeq Z^p_{r+1}/B^p_{r+1}=E^p_{r+1}.\eqno\square$$ \endproof \begstat{(10.6) Theorem} Consider the filtration of the homology module $H(K)$ defined by $$F_p\big(H(K)\big)=\Im\big(H(K_p)\longrightarrow H(K)\big).$$ Then there is a canonical isomorphism $$E^p_\infty=G_p\big(H(K)\big).$$ \endstat \begproof{} Clearly $F_p\big(H(K)\big)=(K_p\cap d^{-1}\{0\})/(K_p\cap dK)$, whereas $$\eqalign{ Z^p_\infty&=(K_p\cap d^{-1}\{0\})/(K_{p+1}\cap d^{-1}\{0\}),~~ B^p_\infty=(K_p\cap dK)/(K_{p+1}\cap dK),\cr E^p_\infty&=Z^p_\infty/B^p_\infty= (K_p\cap d^{-1}\{0\})/(K_{p+1}\cap d^{-1}\{0\}+K_p\cap dK).\cr}$$ It follows that $E^p_\infty\simeq F_p\big(H(K)\big)/ F_{p+1}\big(H(K)\big)$.\qed \endproof In most applications, the differential module $K$ has a natural grading compatible with the filtration. Let us consider for example the case of a cohomology complex $K^\bu=\bigoplus_{l\in\bZ}K^l$. The filtration $K^\bu_p=F_p(K^\bu)$ is said to be {\it compatible} with the differential complex structure if each $K^\bu_p$ is a subcomplex of $K^\bu$, i.e.\ $$K^\bu_p=\bigoplus_{l\in\bZ}K^l_p$$ where $(K^l_p)$ is a filtration of $K^l$. Then we define $Z^{p,q}_r$, $B^{p,q}_r$, $E^{p,q}_r$ to be the sets of elements of $Z^p_r$, $B^p_r$, $E^p_r$ of total degree $p+q$. Therefore \medskip\noindent $\cmalign{ (10.7)\hfill&Z^{p,q}_r&=K^{p+q}_p\cap d^{-1}K^{p+q+1}_{p+r} ~~~\hbox{\rm mod}~~K^{p+q}_{p+1}~,~~~~&Z^p_r=\bigoplus Z^{p,q}_r,\cr (10.7')\hfill&B^{p,q}_r&=K^{p+q}_p\cap dK^{p+q-1}_{p-r+1} ~~~\hbox{\rm mod}~~K^{p+q}_{p+1}~,~~~~&B^p_r=\bigoplus B^{p,q}_r,\cr (10.7'')\hfill&E^{p,q}_r&=Z^{p,q}_r/B^{p,q}_r~,&E^p_r=\bigoplus E^{p,q}_r,\cr}$ \medskip\noindent and the differential $d_r$ has bidegree $(r,-r+1)$, i.e.\ $$d_r~:~~E^{p,q}_r\longrightarrow E^{p+r\,,\,q-r+1}_r.\leqno(10.8)$$ For an element of pure bidegree $(p,q)$, $p$ is called the {\it filtering degree}, $q$ the {\it complementary degree} and $p+q$ the {\it total degree}. \begstat{(10.9) Definition} A filtration $(K^\bu_p)$ of a complex $K^\bu$ is said to be regular if for each degree $l$ there are indices $\nu(l)\le N(l)$ such that $K^l_p=K^l$ for $p<\nu(l)$ and $K^l_p=0$ for $p>N(l)$. \endstat If the filtration is regular, then (10.7) and $(10.7')$ show that $$\eqalign{ Z^{p,q}_r=Z^{p,q}_{r+1}=\ldots=Z^{p,q}_\infty~~~~\hbox{\rm for}~~ &r>N(p+q+1)-p,\cr B^{p,q}_r=B^{p,q}_{r+1}=\ldots=B^{p,q}_\infty~~~~\hbox{\rm for}~~ &r>p+1-\nu(p+q-1),\cr}$$ therefore $E^{p,q}_r=E^{p,q}_\infty$ for $r\ge r_0(p,q)$. We say that the spectral sequence {\it converges} to its limit term, and we write symbolically $$E^{p,q}_r\Longrightarrow H^{p+q}(K^\bu)\leqno(10.10)$$ to express the following facts: there is a spectral sequence whose terms of the $r$-th generation are $E^{p,q}_r$, the sequence converges to a limit term $E^{p,q}_\infty$, and $E^{p,l-p}_\infty$ is the term $G_p\big(H^l(K^\bu)\big)$ in the graded module associated to some filtration of $H^l(K^\bu)$. \begstat{(10.11) Definition} The spectral sequence is said to collapse in $E^\bu_r$ if all terms $Z^{p,q}_k$, $B^{p,q}_k$, $E^{p,q}_k$ are constant for $k\ge r$, or equivalently if $d_k=0$ in all bidegrees for $k\ge r$. \endstat \begstat{(10.12) Special case} \rm Assume that there exists an integer $r\ge 2$ and an index $q_0$ such that $E^{p,q}_r=0$ for $q\ne q_0$. Then this property remains true for larger values of $r$, and we must have $d_r=0$. It follows that the spectral sequence collapses in $E^\bu_r$ and that $$H^l(K^\bu)=E^{l-q_0,q_0}_r.$$ Similarly, if $E^{p,q}_r=0$ for $p\ne p_0$ and some $r\ge 1$ then $$H^l(K^\bu)=E^{p_0,l-p_0}_r.\eqno\square$$ \endstat \titlec{10.B.}{Computation of the First Terms} Consider an arbitrary spectral sequence. For $r=0$, we have $Z^p_0=K_p/K_{p+1}$, $B^p_0=\{0\}$, thus $$E^p_0=K_p/K_{p+1}=G_p(K).\leqno(10.13)$$ The differential $d_0$ is induced by $d$ on the quotients, and $$E^p_1=H\big(G_p(K)\big).\leqno(10.14)$$ Now, there is a short exact sequence of differential modules $$0\longrightarrow G_{p+1}(K)\longrightarrow K_p/K_{p+2}\longrightarrow G_p(K)\longrightarrow 0.$$ We get therefore a connecting homomorphism $$E^p_1=H\big(G_p(K)\big)\buildo\partial\over\longrightarrow H\big(G_{p+1}(K)\big)=E^{p+1}_1.\leqno(10.15)$$ We claim that $\partial$ coincides with the differential $d_1$~: indeed, both are induced by $d$. When $K^\bu$ is a filtered cohomology complex, $d_1$ is the connecting homomorphism $$E^{p,q}_1=H^{p+q}\big(G_p(K^\bu)\big)\buildo\partial\over\longrightarrow H^{p+q+1}\big(G_{p+1}(K^\bu)\big)=E^{p+1,q}_1.\leqno(10.16)$$ \titleb{11.}{Spectral Sequence of a Double Complex} A double complex is a bigraded module $K^{\bu,\bu}=\bigoplus K^{p,q}$ together with a differential $d=d'+d''$ such that $$d':K^{p,q}\longrightarrow K^{p+1,q},~~~d'':K^{p,q+1}\longrightarrow K^{p,q+1},\leqno(11.1)$$ and $d\circ d=0$. In particular, $d'$ and $d''$ satisfy the relations $$d^{\prime2}=d^{\prime\prime2}=0,~~~d'd''+d''d'=0.\leqno(11.2)$$ The {\it simple complex associated} to $K^{\bu,\bu}$ is defined by $$K^l=\bigoplus_{p+q=l}K^{p,q}$$ together with the differential $d$. We will suppose here that both graduations of $K^{\bu,\bu}$ are positive, i.e.\ $K^{p,q}=0$ for $p<0$ or $q<0$. The {\it first filtration} of $K^\bu$ is defined by $$K^l_p=\bigoplus_{i+j=l,~i\ge p}K^{i,j}=\bigoplus_{p\le i\le l}K^{i,l-i}. \leqno(11.3)$$ The graded module associated to this filtration is of course $G_p(K^l)=K^{p,l-p}$, and the differential induced by $d$ on the quotient coincides with $d''$ because $d'$ takes $K^l_p$ to $K^{l+1}_{p+1}$. Thus we have a spectral sequence beginning by $$E^{p,q}_0=K^{p,q},~~~d_0=d'',~~~E^{p,q}_1=H^q_{d''}(K^{p,\bu}). \leqno(11.4)$$ By (10.16), $d_1$ is the connecting homomorphism associated to the short exact sequence $$0\longrightarrow K^{p+1,\bu}\longrightarrow K^{p,\bu}\oplus K^{p+1,\bu}\longrightarrow K^{p,\bu}\longrightarrow 0$$ where the differential is given by ($d$ mod $K^{p+2,\bu}$) for the central term and by $d''$ for the two others. The definition of the connecting homomorphism in the proof of Th. 1.5 shows that $$d_1=\partial:~~H^q_{d''}(K^{p,\bu})\longrightarrow H^q_{d''}(K^{p+1,\bu})$$ is induced by $d'$. Consequently, we find $$E^{p,q}_2=H^p_{d'}(E_1^{\bu,q})= H^p_{d'}\big(H^q_{d''}(K^{\bu,\bu})\big).\leqno(11.5)$$ For such a spectral sequence, several interesting additional features can be pointed out. For all $r$ and $l$, there is an injective homomorphism $$E^{0,l}_{r+1}\lhra E^{0,l}_r$$ whose image can be identified with the set of $d_r$-cocycles in $E^{0,l}_r$~; the coboundary group is zero because $E^{p,q}_r=0$ for $q<0$. Similarly, $E^{l,0}_r$ is equal to its cocycle submodule, and there is a surjective homomorphism $$E^{l,0}_r\lraww E^{l,0}_{r+1}\simeq E^{l,0}_r/d_rE^{l-r,r-1}_r.$$ Furthermore, the filtration on $H^l(K^\bu)$ begins at $p=0$ and stops at $p=l$, i.e.\ $$F_0\big(H^l(K^\bu)\big)=H^l(K^\bu),~~~F_p\big(H^l(K^\bu)\big)=0~~~ \hbox{\rm for}~~p>l.\leqno(11.6)$$ Therefore, there are canonical maps $$\eqalign{ &H^l(K^\bu)\lraww G_0\big(H^l(K^\bu)\big)=E^{0,l}_\infty\lhra E^{0,l}_r,\cr &E^{l,0}_r\lraww E^{l,0}_\infty=G_l\big(H^l(K^\bu)\big)\lhra H^l(K^\bu).} \leqno(11.7)$$ These maps are called the {\it edge homomorphisms} of the spectral sequence. \begstat{(11.8) Theorem} There is an exact sequence $$0\longrightarrow E^{1,0}_2\longrightarrow H^1(K^\bu)\longrightarrow E^{0,1}_2\buildo d_2\over\longrightarrow E^{2,0}_2 \longrightarrow H^2(K^\bu)$$ where the non indicated arrows are edge homomorphisms. \endstat \begproof{} By 11.6, the graded module associated to $H^1(K^\bu)$ has only two components, and we have an exact sequence $$0\longrightarrow E^{1,0}_\infty\longrightarrow H^1(K^\bu)\longrightarrow E^{0,1}_\infty\longrightarrow 0.$$ However $E^{1,0}_\infty=E^{1,0}_2$ because all differentials $d_r$ starting from $E^{1,0}_r$ or abuting to $E^{1,0}_r$ must be zero for $r\ge 2$. Similarly, $E^{0,1}_\infty=E^{0,1}_3$ and $E^{2,0}_\infty=E^{2,0}_3$, thus there is an exact sequence $$0\longrightarrow E^{0,1}_\infty\longrightarrow E^{0,1}_2\buildo d_2\over\longrightarrow E^{2,0}_2\longrightarrow E^{2,0}_\infty\longrightarrow 0.$$ A combination of the two above exact sequences yields $$0\longrightarrow E^{1,0}_2\longrightarrow H^1(K^\bu)\longrightarrow E^{0,1}_2\buildo d_2\over\longrightarrow E^{2,0}_2\longrightarrow E^{2,0}_\infty\longrightarrow 0.$$ Taking into account the injection $E^{2,0}_\infty\lhra H^2(K^\bu)$ in (11.7), we get the required exact sequence.\qed \endproof \begstat{(11.9) Example} \rm Let $X$ be a complex manifold of dimension $n$. Consider the double complex $K^{p,q}=C^\infty_{p,q}(X,\bC)$ together with the exterior derivative $d=d'+d''$. Then there is a spectral sequence which starts from the Dolbeault cohomology groups $$E^{p,q}_1=H^{p,q}(X,\bC)$$ and which converges to the graded module associated to a filtration of the De Rham cohomology groups: $$E^{p,q}_r\Longrightarrow H^{p+q}_{\DR}(X,\bC).$$ This spectral sequence is called the {\it Hodge-Fr\"olicher spectral sequence} (Fr\"o\-licher 1955). We will study it in much more detail in chapter 6 when $X$ is compact.\qed \endstat Frequently, the spectral sequence under consideration can be obtained from two distinct double complexes and one needs to compare the final cohomology groups. The following lemma can often be applied. \begstat{(11.10) Lemma} Let $K^{p,q}\longrightarrow L^{p,q}$ be a morphism of double complexes $($i.e.\ a double sequence of maps commuting with $d'$ and $d'')$. Then there are induced morphisms $${}_KE^{\bu,\bu}_r\longrightarrow{}_LE^{\bu,\bu}_r,~~~~r\ge 0$$ of the associated spectral sequences. If one of these morphisms is an isomorphism for some $r$, then $H^l(K^\bu)\longrightarrow H^l(L^\bu)$ is an isomorphism. \endstat \begproof{} If the $r$-terms are isomorphic, they have the same cohomology groups, thus ${}_KE^{\bu,\bu}_{r+1}\simeq{}_LE^{\bu,\bu}_{r+1}$ and ${}_KE^{\bu,\bu}_\infty\simeq{}_LE^{\bu,\bu}_\infty$ in the limit. The lemma follows from the fact that if a morphism of graded modules $\varphi:M\longrightarrow M'$ induces an isomorphism $G_\bu(M)\longrightarrow G_\bu(M')$, then $\varphi$ is an isomorphism.\qed \endproof \titleb{12.}{Hypercohomology Groups} Let $(\cL^\bu,\delta)$ be a complex of sheaves $$0\longrightarrow\cL^0\buildo\delta^0\over\longrightarrow\cL^1\longrightarrow\cdots\longrightarrow\cL^q\buildo\delta^q \over\longrightarrow\cdots$$ on a topological space $X$. We denote by $\cH^q=\cH^q(\cL^\bu)$ the $q$-th sheaf of cohomology of this complex; thus $\cH^q$ is a sheaf of abelian groups over $X$. Our goal is to define ``generalized cohomology groups'' attached to $\cL^\bu$ on~$X$, in such a way that these groups only depend on the cohomology sheaves~$\cH^q$. For this, we associate to $\cL^\bu$ the double complex of groups $$K_\cL^{p,q}=(\cL^q)^{[p]}(X)\leqno(12.1)$$ with differential $d'=d^p$ given by (2.5), and with $d''=(-1)^p (\delta^q)^{[p]}$. As~$(\delta^q)^{[\bu]}:(\cL^q)^{[\bu]}\longrightarrow (\cL^{q+1})^{[\bu]}$ is a morphism of complexes, we get the expected relation $d'd''+d''d'=0$. \begstat{(12.2) Definition} The groups $H^q(K_\cL^\bu)$ are called the hypercohomology groups of $\cL^\bu$ and are denoted $\bH^q(X,\cL^\bu)$. \endstat Clearly $\bH^0(X,\cL^\bu)=\cH^0(X)$ where $\cH^0=\ker\,\delta^0$ is the first cohomology sheaf of $\cL^\bu$. If $\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$ is a morphism of sheaf complexes, there is an associated morphism of double complexes $\varphi^{\bu,\bu}:K^{\bu,\bu}_\cL\longrightarrow K^{\bu,\bu}_\cM$, hence a natural morphism $$\bH^q(\varphi^\bu)~:~~\bH^q(X,\cL^\bu)\longrightarrow\bH^q(X,\cM^\bu). \leqno(12.3)$$ We first list a few immediate properties of hypercohomology groups, whose proofs are left to the reader. \begstat{(12.4) Proposition} The following properties hold: \medskip \item{\rm a)} If $\cL^q=0$ for $q\ne 0$, then $\bH^q(X,\cL^\bu)=H^q(X,\cL^0)$. \medskip \item{\rm b)} If $\cL^\bu[s]$ denotes the complex $\cL^\bu$ shifted of $s$ indices to the right, i.e.\ $\cL^\bu[s]^q=\cL^{q-s}$, then $\bH^q(X,\cL^\bu[s])=\bH^{q-s}(X,\cL^\bu)$. \medskip \item{\rm c)} If $0\longrightarrow\cL^\bu\longrightarrow\cM^\bu\longrightarrow\cN^\bu\longrightarrow0$ is an exact sequence of sheaf complexes, there is a long exact sequence $$\cdots\bH^q(X,\cL^\bu)\longrightarrow\bH^q(X,\cM^\bu)\longrightarrow\bH^q(X,\cN^\bu) \buildo\partial\over\longrightarrow\bH^{q+1}(X,\cL^\bu)\cdots.\eqno\square$$ \smallskip \endstat If $\cL^\bu$ is a sheaf complex, the spectral sequence associated to the first filtration of $K_\cL^\bu$ is given by $$E^{p,q}_1=H^q_{d''}(K_\cL^{p,\bu})=H^q\big((\cL^\bu)^{[p]}(X)\big).$$ However by (2.9) the functor $\cA\longmapsto\cA^{[p]}(X)$ preserves exact sequences. Therefore, we get $$\leqalignno{ E^{p,q}_1&=\big(\cH^q(\cL^\bu)\big)^{[p]}(X),&(12.5)\cr E^{p,q}_2&=H^p\big(X,\cH^q(\cL^\bu)\big),&(12.5')\cr}$$ since $E_2^{p,q}=H^p_{d'}(E_1^{\bu,q})$. If $\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$ is a morphism, an application of Lemma 11.10 to the $E_2$-term of the associated first spectral sequences of $K_\cL^{\bu,\bu}$ and $K_\cM^{\bu,\bu}$ yields: \begstat{(12.6) Corollary} If $\varphi^\bu:\cL^\bu\longrightarrow\cM^\bu$ is a quasi-isomorphism $\big($this means that $\varphi^\bu$ induces an isomorphism $\cH^\bu(\cL^\bu)\longrightarrow\cH^\bu(\cM^\bu)~\big)$, then $$\bH^l(\varphi^\bu)~:~~\bH^l(X,\cL^\bu)\longrightarrow\bH^l(X,\cM^\bu)$$ is an isomorphism. \endstat Now, we may reverse the roles of the indices $p,q$ and of the differentials $d',d''$. The {\it second filtration} $F_p(K^l_\cL)=\bigoplus_{j\ge p}K_\cL^{l-j,j}$ is associated to a spectral sequence such that $\smash{\wt E}^{p,q}_1=H^q_{d'}(K_\cL^{\bu,p})= \smash{H^q_{d'}\big((\cL^p)^{[\bu]}(X)\big)}$, hence $$\leqalignno{ \wt E^{p,q}_1&=H^q(X,\cL^p),&(12.7)\cr \wt E^{p,q}_2&=H^p_{\delta}\big(H^q(X,\cL^\bu)\big).&(12.7')\cr}$$ These two spectral sequences converge to limit terms which are the graded modules associated to filtrations of $\bH^\bu(X,\cL^\bu)$~; these filtrations are in gene\-ral different. Let us mention two interesting special cases. \medskip \noindent{$\bullet$} Assume first that the complex $\cL^\bu$ is a resolution of a sheaf $\cA$, so that $\cH^0=\cA$ and $\cH^q=0$ for $q\ge 1$. Then we find $$E^{p,0}_2=H^p(X,\cA),~~~~E^{p,q}_2=0~~\hbox{\rm for}~~q\ge 1.$$ It follows that the first spectral sequence collapses in $E^\bu_2$, and 10.12 implies $$\bH^l(X,\cL^\bu)\simeq H^l(X,\cA).\leqno(12.8)$$ \noindent{$\bullet$} Now, assume that the sheaves $\cL^q$ are acyclic. The second spectral sequence gives $$\leqalignno{ \wt E^{p,0}_2=H^p&\big(\cL^\bu(X)\big),~~~~ \wt E^{p,q}_2=0~~\hbox{\rm for}~~q\ge 1,\cr &\bH^l(X,\cL^\bu)\simeq H^l\big(\cL^\bu(X)\big).&(12.9)\cr}$$ If both conditions hold, i.e.\ if $\cL^\bu$ is a resolution of a sheaf $\cA$ by acyclic sheaves, then (12.8) and (12.9) can be combined to obtain a new proof of the De Rham-Weil isomorphism $H^l(X,\cA)\simeq H^l\big(\cL^\bu(X)\big)$. \titleb{13.}{Direct Images and the Leray Spectral Sequence} \titlec{13.A.}{Direct Images of a Sheaf} Let $X,Y$ be topological spaces, $F:X\to Y$ a continuous map and $\cA$ a sheaf of abelian groups on $X$. Recall that the {\it direct image} $F_\star\cA$ is the presheaf on $Y$ defined for any open set $U\subset Y$ by $$(F_\star\cA)(U)=\cA\big(F^{-1}(U)\big).\leqno(13.1)$$ Axioms (II-$2.4'$ and (II-$2.4'')$ are clearly satisfied, thus $F_\star\cA$ is in fact a sheaf. The following result is obvious: $$\cA~~\hbox{\rm is flabby}~~~\Longrightarrow~~~F_\star\cA~~ \hbox{\rm is flabby.}\leqno(13.2)$$ Every sheaf morphism $\varphi:\cA\to\cB$ induces a corresponding morphism $$F_\star\varphi~:~~F_\star\cA\longrightarrow F_\star\cB,$$ so $F_\star$ is a functor on the category of sheaves on $X$ to the category of sheaves on $Y$. This functor is exact on the left: indeed, to every exact sequence\break $0\to\cA\to\cB\to\cC$ is associated an exact sequence $$0\longrightarrow F_\star\cA\longrightarrow F_\star\cB\longrightarrow F_\star\cC,$$ but $F_\star\cB\to F_\star\cC$ need not be onto if $\cB\to\cC$ is. All this follows immediately from the considerations of \S 3. In particular, the simplicial flabby resolution $(\cA^{[\bu]},d)$ yields a complex of sheaves $$0\longrightarrow F_\star\cA^{[0]}\longrightarrow F_\star\cA^{[1]}\longrightarrow\cdots\longrightarrow F_\star\cA^{[q]}\buildo F_\star d^q\over\longrightarrow F_\star\cA^{[q+1]}\longrightarrow\cdots. \leqno(13.3)$$ \begstat{(13.4) Definition} The $q$-th direct image of $\cA$ by $F$ is the $q$-th cohomology sheaf of the sheaf complex $(13.3)$. It is denoted $$R^qF_\star\cA=\cH^q(F_\star\cA^{[\bu]}).$$ \endstat As $F_\star$ is exact on the left, the sequence $0\to F_\star\cA\to F_\star\cA^{[0]}\to F_\star\cA^{[1]}$ is exact, thus $$R^0F_\star\cA=F_\star\cA.\leqno(13.5)$$ We now compute the stalks of $R^qF_\star\cA$. As the kernel or cokernel of a sheaf morphism is obtained stalk by stalk, we have $$(R^qF_\star\cA)_y=H^q\big((F_\star\cA^{[\bu]})_y\big)= \lim_{\displaystyle\longrightarrow\atop\scriptstyle U\ni y}~~ H^q\big(F_\star\cA^{[\bu]}(U)\big).$$ The very definition of $F_\star$ and of sheaf cohomology groups implies $$H^q\big(F_\star\cA^{[\bu]}(U)\big)=H^q\big(\cA^{[\bu]}(F^{-1}(U))\big)= H^q\big(F^{-1}(U),\cA\big),$$ hence we find $$(R^qF_\star\cA)_y=\lim_{\displaystyle\longrightarrow\atop\scriptstyle U\ni y}~~ H^q\big(F^{-1}(U),\cA\big),\leqno(13.6)$$ i.e.\, $R^qF_\star\cA$ is the sheaf associated to the presheaf $U\mapsto H^q\big(F^{-1}(U),\cA\big)$. We~must stress here that the stronger relation $R^qF_\star\cA(U)=H^q\big(F^{-1}(U),\cA\big)$ need not be true in general. If the fiber $F^{-1}(y)$ is strongly paracompact in $X$ and if the family of open sets $F^{-1}(U)$ is a fundamental family of neighborhoods of $F^{-1}(y)$ (this situation occurs for example if $X$ and $Y$ are locally compact spaces and $F$ a proper map, or if $F=\pr_1:X=Y\times S\longrightarrow Y$ where $S$ is compact), Th. 9.10 implies the more natural relation $$(R^qF_\star\cA)_y=H^q\big(F^{-1}(y),\cA\big).\leqno(13.6')$$ Let $0\to\cA\to\cB\to\cC\to 0$ be an exact sequence of sheaves on $X$. Apply the long exact sequence of cohomology on every open set $F^{-1}(U)$ and take the direct limit over~$U$. We get an exact sequence of sheaves: $$\cmalign{ &~~~0&\longrightarrow~~F_\star\cA&\longrightarrow~~F_\star\cB&\longrightarrow~~F_\star\cC&\longrightarrow~~ R^1F_\star\cA&\longrightarrow\cdots\cr &\cdots&\longrightarrow R^qF_\star\cA&\longrightarrow R^q F_\star\cB&\longrightarrow R^q F_\star\cC&\longrightarrow R^{q+1}F_\star\cA&\longrightarrow\cdots.\cr}\leqno(13.7)$$ \titlec{13.B.}{Leray Spectral Sequence} For any continuous map~$F:X\to Y$, the Leray spectral sequence relates the cohomology groups of a sheaf $\cA$ on $X$ and those of its direct images $R^qF_\star\cA$ on~$Y$. Consider the two spectral sequences $E^\bu_r$, $\wt E^\bu_r$ associated with the complex of sheaves $\cL^\bu=F_\star\cA^{[\bu]}$ on~$Y$, as in \S~12. By definition we have $\cH^q(\cL^\bu)=R^qF_\star\cA$. By $(12.5')$ the second term of the first spectral sequence is $$E^{p,q}_2=H^p(Y,R^qF_\star\cA),$$ and this spectral sequence converges to the graded module associated to a filtration of $\bH^l(Y,F_\star\cA^{[\bu]})$. On the other hand, (13.2) implies that $F_\star\cA^{[q]}$ is flabby. Hence, the second special case (12.9) can be applied: $$\bH^l(Y,F_\star\cA^{[\bu]})\simeq H^l\big(F_\star\cA^{[\bu]}(Y)\big) =H^l\big(\cA^{[\bu]}(X)\big)=H^l(X,\cA).$$ We may conclude this discussion by the following \begstat{(13.8) Theorem} For any continuous map $F:X\to Y$ and any sheaf $\cA$ of abelian groups on $X$, there exists a spectral sequence whose $E^\bu_2$ term is $$E_2^{p,q}=H^p(Y,R^qF_\star\cA),$$ which converges to a limit term $E^{p,l-p}_\infty$ equal to the graded module associated with a filtration of $H^l(X,\cA)$. The edge homomorphism $$H^l(Y,F_\star\cA)\lraww E^{l,0}_\infty\lhra H^l(X,\cA)$$ coincides with the composite morphism $$F^{\#}:~~H^l(Y,F_\star\cA)\buildo F^\star\over\longrightarrow H^l(X,F^{-1}F_\star\cA) \buildo H^l(\mu_F)\over{\relbar\joinrel\relbar\joinrel\longrightarrow} H^l(X,\cA)$$ where $\mu_F:F^{-1}F_\star\cA\longrightarrow\cA$ is the canonical sheaf morphism. \endstat \begproof{} Only the last statement remains to be proved. The morphism $\mu_F$ is defined as follows: every element $s\in (F^{-1}F_\star\cA)_x= (F_\star\cA)_{F(x)}$ is the germ of a section $\wt s\in F_\star\cA(V)=\cA\big(F^{-1}(V)\big)$ on a neighborhood $V$ of $F(x)$. Then $F^{-1}(V)$ is a neighborhood of $x$ and we let $\mu_F s$ be the germ of $\wt s$ at $x$. Now, we observe that to any commutative diagram of topological spaces and continuous maps is associated a commutative diagram involving the direct image sheaves and their cohomology groups: $$\cmalign{ \hfill X&\buildo F\over\longrightarrow&Y\kern80pt\hfill H^l(X,\cA)&\buildo~~F^\#\over\longleftarrow &H^l(Y,F_\star\cA)\cr {\scriptstyle G}\big\downarrow&&\big\downarrow{\scriptstyle H}\hfill {\scriptstyle G^\#}\big\uparrow\qquad&&\qquad\big\uparrow {\scriptstyle H^\#}\cr \hfill X'&\buildo F'\over\longrightarrow&Y'\hfill H^l(X',G_\star\cA)&\buildo~~F^{\prime\#}\over\longleftarrow &H^l(Y',F'_\star G_\star\cA).\cr}$$ There is a similar commutative diagram in which $F^\#$ and $F^{\prime\#}$ are replaced by the edge homomorphisms of the spectral sequences of $F$ and $F'$~: indeed there is a natural morphism $H^{-1}F'_\star\cB\longrightarrow F_\star G^{-1}\cB$ for any sheaf $\cB$ on $X'$, so we get a morphism of sheaf complexes $$H^{-1}F'_\star(G_\star\cA)^{[\bu]}\longrightarrow F_\star G^{-1}(G_\star\cA)^{[\bu]}\longrightarrow F_\star(G^{-1}G_\star\cA)^{[\bu]}\longrightarrow F_\star\cA^{[\bu]},$$ hence also a morphism of the spectral sequences associated to both ends. The special case $X'=Y'=Y$, $G=F$, $F'=H=\Id_Y$ then shows that our statement is true for $F$ if it is true for $F'$. Hence we may assume that $F$ is the identity map; in this case, we need only show that the edge homomorphism of the spectral sequence of $F_\star\cA^{[\bu]}=\cA^{[\bu]}$ is the identity map. This is an immediate consequence of the fact that we have a quasi-isomorphism $$(\cdots\to 0\to\cA\to 0\to\cdots)\longrightarrow\cA^{[\bu]}.\eqno\square$$ \endproof \begstat{(13.9) Corollary} If $R^qF_\star\cA=0$ for $q\ge 1$, there is an isomorphism $H^l(Y,F_\star\cA)\simeq H^l(X,\cA)$ induced by $F^{\#}$. \endstat \begproof{} We are in the special case 10.12 with $E^{p,q}_2=0$ for $q\ne 0$, so $$H^l(Y,F_\star\cA)=E^{l,0}_2\simeq H^l(X,\cA).\eqno\square$$ \endproof \begstat{(13.10) Corollary} Let $F:X\longrightarrow Y$ be a proper finite-to-one map. For any sheaf $\cA$ on $X$, we have $R^qF_\star\cA=0$ for $q\ge 1$ and there is an isomorphism $H^l(Y,F_\star\cA)\simeq H^l(X,\cA)$. \endstat \begproof{} By definition of higher direct images, we have $$(R^qF_\star\cA)_y=\lim_{\displaystyle\,\longrightarrow\atop\scriptstyle{U\ni y}} ~~H^q\big(\cA^{[\bu]}\big(F^{-1}(U)\big)\big).$$ If $F^{-1}(y)=\{x_1,\ldots,x_m\}$, the assumptions imply that $\big(F^{-1}(U)\big)$ is a fundamental system of neighborhoods of $\{x_1,\ldots,x_m\}$. Therefore $$(R^qF_\star\cA)_y=\bigoplus_{1\le j\le m}H^q\big(\cA^{[\bu]}_{x_j}\big) =\cases{\bigoplus\cA_{x_j}&for~~$q=0$,\cr 0&for~~$q\ge 1$,\cr}$$ and we conclude by Cor.\ 13.9.\qed \endproof Corollary 13.10 can be applied in particular to the inclusion $j:S\to X$ of a {\it closed} subspace $S$. Then $j_\star\cA$ coincides with the sheaf $\cA^S$ defined in \S 3 and we get $H^q(S,\cA)=H^q(X,\cA^S)$. It is very important to observe that the property $R^qj_\star\cA=0$ for $q\ge 1$ need not be true if $S$ is not closed. \titlec{13.C.}{Topological Dimension} As a first application of the Leray spectral sequence, we are going to derive some properties related to the concept of {\it topological dimension}. \begstat{(13.11) Definition} A non empty space $X$ is said to be of topological dimension $\le n$ if $H^q(X,\cA)=0$ for any $q>n$ and any sheaf $\cA$ on $X$. We let $\topdim X$ be the smallest such integer $n$ if it exists, and $+\infty$ otherwise. \endstat \begstat{(13.12) Criterion} For a paracompact space $X$, the following conditions are equivalent: \medskip \item{\rm a)} $\topdim X\le n~;$ \medskip \item{\rm b)} the sheaf $\cZ^n=\ker(\cA^{[n]}\longrightarrow\cA^{[n+1]})$ is soft for every sheaf $\cA~;$ \medskip \item{\rm c)} every sheaf $\cA$ admits a resolution $0\to\cL^0\to\cdots \to\cL^n\to 0$ of length $n$ by soft sheaves.\smallskip \endstat \begproof{} b) $\Longrightarrow$ c). Take $\cL^q=\cA^{[q]}$ for $qn$. \medskip \noindent{a)} $\Longrightarrow$ b). Let $S$ be a closed set and $U=X\ssm S$. As in Prop.\ 7.12, $(\cA^{[\bu]})_U$ is an acyclic resolution of $\cA_U$. Clearly $\ker\big((\cA^{[n]})_U\to(\cA^{[n+1]})_U\big)=\cZ^n_U$, so the isomorphisms (6.2) obtained in the proof of the De Rham-Weil theorem imply $$H^1(X,\cZ^n_U)\simeq H^{n+1}(X,\cA_U)=0.$$ By (3.10), the restriction map $\cZ^n(X)\longrightarrow\cZ^n(S)$ is onto, so $\cZ^n$ is soft.\qed \endproof \begstat{(13.13) Theorem} The following properties hold: \medskip \item{\rm a)} If $X$ is paracompact and if every point of $X$ has a neighborhood of topological dimension $\le n$, then $\topdim X\le n$. \medskip \item{\rm b)} If $S\subset X$, then $\topdim S\le \topdim X$ provided that $S$ is closed or $X$ metrizable. \medskip \item{\rm c)}If $X,Y$ are metrizable spaces, one of them locally compact, then $$\topdim(X\times Y)\le\topdim X+\topdim Y.$$ \item{\rm d)} If $X$ is metrizable and locally homeomorphic to a subspace of $\bR^n$, then $\topdim X\le n$.\smallskip \endstat \begproof{} a) Apply criterion 13.12~b) and the fact that softness is a local property (Prop.\ 4.12). \medskip \noindent{b)} When $S$ is closed in $X$, the property follows from Cor.\ 13.10. When $X$ is metrizable, any subset $S$ is strongly paracompact. Let $j:S\longrightarrow X$ be the injection and $\cA$ a sheaf on $S$. As $\cA=(j_\star\cA)_{\restriction S}$, we have $$H^q(S,\cA)=H^q(S,j_\star\cA)= \lim_{\displaystyle\longrightarrow\atop\scriptstyle\Omega\supset S}H^q(\Omega, j_\star\cA)$$ by Th. 9.10. We may therefore assume that $S$ is open in $X$. Then every point of $S$ has a neighborhood which is closed in $X$, so we conclude by a) and the first case of~b). \medskip \noindent{c)} Thanks to a) and b), we may assume for example that $X$ is compact. Let $\cA$ be a sheaf on $X\times Y$ and $\pi:X\times Y\longrightarrow Y$ the second projection. Set $n_X=\topdim X$, $n_Y=\topdim Y$. In virtue of $(13.6')$, we have $R^q\pi_\star\cA=0$ for $q>n_X$. In the Leray spectral sequence, we obtain therefore $$E^{p,q}_2=H^p(Y,R^q\pi_\star\cA)=0~~~\hbox{\rm for}~~p>n_Y~~\hbox{\rm or}~~ q>n_X,$$ thus $E^{p,l-p}_\infty=0$ when $l>n_X+n_Y$ and we infer $H^l(X\times Y,\cA)=0$. \medskip \noindent{d)} The unit interval $[0,1]\subset\bR$ is of topological dimension $\le 1$, because $[0,1]$ admits arbitrarily fine coverings $$\cU_k=\big(~[0,1]~\cap~](\alpha-1)/k,(\alpha+1)/k[~\big)_{0\le \alpha\le k} \leqno(13.14)$$ for which only consecutive open sets $U_\alpha$, $U_{\alpha+1}$ intersect; we may therefore apply Prop.\ 5.24. Hence $\bR^n\simeq~]0,1[^n\subset[0,1]^n$ is of topological dimension $\le n$ by b) and c). Property d) follows.\qed \endproof \titleb{14.}{Alexander-Spanier Cohomology} \titlec{14.A.}{Invariance by Homotopy} Alexander-Spanier's theory can be viewed as the special case of sheaf cohomology theory with {\it constant coefficients}, i.e.\ with values in constant sheaves. \begstat{(14.1) Definition} Let $X$ be a topological space, $R$ a commutative ring and $M$ a $R$-module. The constant sheaf $X\times M$ is denoted $M$ for simplicity. The Alexander-Spanier $q$-th cohomology group with values in $M$ is the sheaf cohomology group $H^q(X,M)$. \endstat In particular $H^0(X,M)$ is the set of locally constant functions $X\to M$, so $H^0(X,M)\simeq M^E$, where $E$ is the set of connected components of $X$. We will not repeat here the properties of Alexander-Spanier cohomology groups that are formal consequences of those of general sheaf theory, but we focus our attention instead on new features, such as invariance by homotopy. \begstat{(14.2) Lemma} Let $I$ denote the interval $[0,1]$ of real numbers. Then $H^0(I,M)=M$ and $H^q(I,M)=0$ for $q\ne 0$. \endstat \begproof{} Let us employ alternate \v Cech cochains for the coverings $\cU_n$ defined in (13.14). As $I$ is paracompact, we have $$H^q(I,M)=\lim_{\displaystyle\longrightarrow}~~\check H^q(\cU_n,M).$$ However, the alternate \v Cech complex has only two non zero components and one non zero differential: $$\eqalign{ &AC^0(\cU_n,M)=\big\{(c_\alpha)_{0\le\alpha\le n}\big\}=M^{n+1},\cr &AC^1(\cU_n,M)=\big\{(c_{\alpha\,\alpha+1})_{0\le\alpha\le n-1}\big\}=M^n,\cr &d^0:(c_\alpha)\longmapsto(c'_{\alpha\,\alpha+1})=(c_{\alpha+1}-c_\alpha). \cr}$$ We see that $d^0$ is surjective, and that $\ker d^0=\big\{(m,m,\ldots,m)\big\}=M$.\qed \endproof For any continuous map $f:X\longrightarrow Y$, the inverse image of the constant sheaf $M$ on $Y$ is $f^{-1}M=M$. We get therefore a morphism $$f^\star: H^q(Y,M)\longrightarrow H^q(X,M),\leqno(14.3)$$ which, as already mentioned in \S 9, is compatible with cup product. \begstat{(14.4) Proposition} For any space $X$, the projection $\pi:X\times I\longrightarrow X$ and the injections $i_t:X\longrightarrow X\times I$, $x\longmapsto (x,t)$ induce inverse isomorphisms $$H^q(X,M)~{\raise-4pt\hbox{ $\scriptstyle\pi^\star\atop\displaystyle\relbar\joinrel\longrightarrow$} \atop\raise4pt\hbox{ $\displaystyle\longleftarrow\joinrel\relbar\atop\scriptstyle i_t^\star$}} ~H^q(X\times I,M).$$ In particular, $i_t^\star$ does not depend on $t$. \endstat \begproof{} As $\pi\circ i_t=\Id$, we have $i_t^\star\circ\pi^\star= \Id$, so it is sufficient to check that $\pi^\star$ is an isomorphism. However $(R^q\pi_\star M)_x=H^q(I,M)$ in virtue of $(13.6')$, so we get $$R^0\pi_\star M=M,~~~~R^q\pi_\star M=0~~~\hbox{\rm for}~~q\ne 0$$ and conclude by Cor.\ 13.9.\qed \endproof \begstat{(14.4) Theorem} If $f,g:X\longrightarrow Y$ are homotopic maps, then $$f^\star=g^\star:H^q(Y,M)\longrightarrow H^q(X,M).$$ \endstat \begproof{} Let $H:X\times I\longrightarrow Y$ be a homotopy between $f$ and $g$, with $f=H\circ i_0$ and $g=H\circ i_1$. Proposition 14.3 implies $$f^\star=i_0^\star\circ H^\star=i_1^\star\circ H^\star=g^\star.\eqno\square$$ \endproof We denote $f\sim g$ the homotopy equivalence relation. Two spaces $X,Y$ are said to be homotopically equivalent ($X\sim Y$) if there exist continuous maps $u:X\longrightarrow Y$, $v:Y\longrightarrow X$ such that $v\circ u\sim\Id_X$ and $u\circ v\sim\Id_Y$. Then $H^q(X,M)\simeq H^q(Y,M)$ and $u^\star, v^\star$ are inverse isomorphisms. \begstat{(14.5) Example} \rm A subspace $S\subset X$ is said to be a $($strong$)$ deformation retract of $X$ if there exists a {\it retraction} of $X$ onto $S$, i.e.\ a map $r:X\longrightarrow S$ such that $r\circ j=\Id_S$ ($j=$ inclusion of $S$ in $X$), which is a {\it deformation} of $\Id_X$, i.e.\ there exists a homotopy $H:X\times I\longrightarrow X$ relative to $S$ between $\Id_X$ and $j\circ r$~: $$H(x,0)=x,~~H(x,1)=r(x)~~\hbox{\rm on}~~X,~~~~H(x,t)=x~~\hbox{\rm on}~~S\times I.$$ Then $X$ and $S$ are homotopically equivalent. In particular $X$ is said to be {\it contractible} if $X$ has a deformation retraction onto a point $x_0$. In this case $$H^q(X,M)=H^q(\{x_0\},M)=\cases{M&for $q=0$\cr 0&for $q\ne 0$.}$$ \endstat \begstat{(14.6) Corollary} If $X$ is a compact differentiable manifold, the cohomology groups $H^q(X,R)$ are finitely generated over $R$. \endstat \begproof{} Lemma 6.9 shows that $X$ has a finite covering $\cU$ such that the intersections $U_{\alpha_0\ldots\alpha_q}$ are contractible. Hence $\cU$ is acyclic, $H^q(X,R)=H^q\big(C^\bu(\cU,R)\big)$ and each \v Cech cochain space is a finitely generated (free) module.\qed \endproof \begstat{(14.7) Example: Cohomology Groups of Spheres} \rm Set $$S^n=\big\{x\in\bR^{n+1}~;~x_0^2+x_1^2+\ldots+x_n^2=1\big\},~~~n\ge 1.$$ We will prove by induction on $n$ that $$H^q(S^n,M)=\cases{M&for $q=0$ or $q=n$\cr 0&otherwise.\cr}\leqno(14.8)$$ As $S^n$ is connected, we have $H^0(S^n,M)=M$. In order to compute the higher cohomology groups, we apply the Mayer-Vietoris exact sequence 3.11 to the covering $(U_1,U_2)$ with $$U_1=S^n\ssm\{(-1,0,\ldots,0)\},~~~~U_2=S^n\ssm\{(1,0,\ldots,0)\}.$$ Then $U_1,U_2\approx\bR^n$ are contractible, and $U_1\cap U_2$ can be retracted by deformation on the equator $S^n\cap\{x_0=0\}\approx S^{n-1}$. Omitting $M$ in the notations of cohomology groups, we get exact sequences $$\leqalignno{&~~~~~~ H^0(U_1)\oplus H^0(U_2)\longrightarrow H^0(U_1\cap U_2)\longrightarrow H^1(S^n)\longrightarrow 0,&(14.9')\cr &~~~~~~0\longrightarrow H^{q-1}(U_1\cap U_2)\longrightarrow H^q(S^n)\longrightarrow 0,~~~q\ge 2. &(14.9'')\cr}$$ For $n=1$, $U_1\cap U_2$ consists of two open arcs, so we have $$H^0(U_1)\oplus H^0(U_2)= H^0(U_1\cap U_2)=M\times M,$$ and the first arrow in $(14.9')$ is $(m_1,m_2)\longmapsto(m_2-m_1,m_2-m_1)$. We infer easily that $H^1(S^1)=M$ and that $$H^q(S^1)=H^{q-1}(U_1\cap U_2)=0~~~\hbox{\rm for}~~q\ge 2.$$ For $n\ge 2$, $U_1\cap U_2$ is connected, so the first arrow in $(14.9')$ is onto and $H^1(S^n)=0$. The second sequence $(14.9'')$ yields $H^q(S^n)\simeq H^{q-1}(S^{n-1})$. An induction concludes the proof.\qed \endstat \titlec{14.B.}{Relative Cohomology Groups and Excision Theorem} Let $X$ be a topological space and $S$ a subspace. We denote by $M^{[q]}(X,S)$ the subgroup of sections $u\in M^{[q]}(X)$ such that $u(x_0,\ldots,x_q)=0$ when $$(x_0,\ldots,x_q)\in S^q,~~~x_1\in V(x_0),~\ldots,~x_q\in V(x_0,\ldots,x_{q-1}). $$ Then $M^{[\bu]}(X,S)$ is a subcomplex of $M^{[\bu]}(X)$ and we define the {\it relative cohomology group} of the pair $(X,S)$ with values in $M$ as $$H^q(X,S\,;\,M)=H^q\big(M^{[\bu]}(X,S)\big).\leqno(14.10)$$ By definition of $M^{[q]}(X,S)$, there is an exact sequence $$0\longrightarrow M^{[q]}(X,S)\longrightarrow M^{[q]}(X)\longrightarrow(M_{\restriction S})^{[q]}(S)\longrightarrow 0. \leqno(14.11)$$ The reader should take care of the fact that $(M_{\restriction S})^{[q]}(S)$ does not coincide with the module of sections $M^{[q]}(S)$ of the sheaf $M^{[q]}$ on $X$, except if $S$ is open. The snake lemma shows that there is an ``exact sequence of the pair": $$(14.12)~~H^q(X,S\,;\,M)\to H^q(X,M)\to H^q(S,M)\to H^{q+1}(X,S\,;\,M) \cdots.$$ We have in particular $H^0(X,S\,;\,M)=M^E$, where $E$ is the set of connected components of $X$ which do not meet $S$. More generally, for a triple $(X,S,T)$ with $X\supset S\supset T$, there is an ``exact sequence of the triple": $$\leqalignno{ 0&\longrightarrow M^{[q]}(X,S)\longrightarrow M^{[q]}(X,T)\longrightarrow M^{[q]}(S,T)\longrightarrow 0,&(14.12')\cr ~~~~H^q(X,S\,;\,&M)\longrightarrow H^q(X,T\,;\,M)\longrightarrow H^q(S,T\,;\,M)\longrightarrow H^{q+1}(X,S\,;\,M).\cr}$$ The definition of the cup product in (8.2) shows that $\alpha\smallsmile \beta$ vanishes on $S\cup S'$ if $\alpha,\beta$ vanish on $S$, $S'$ respectively. Therefore, we obtain a bilinear map $$H^p(X,S\,;\,M)\times H^q(X,S'\,;\,M')\longrightarrow H^{p+q}(X,S\cup S'\,;\,M\otimes M').\leqno(14.13)$$ If $f:(X,S)\longrightarrow(Y,T)$ is a morphism of pairs, i.e.\ a continuous map $X\to Y$ such that $f(S)\subset T$, there is an induced pull-back morphism $$f^\star:~~H^q(Y,T\,;\,M)\longrightarrow H^q(X,S\,;\,M)\leqno(14.14)$$ which is compatible with the cup product. Two morphisms of pairs $f,g$ are said to be homotopic when there is a pair homotopy $H:(X\times I,S\times I)\longrightarrow(Y,T)$. An application of the exact sequence of the pair shows that $$\pi^\star:H^q(X,S\,;\,M)\longrightarrow H^q(X\times I,S\times I\,;\,M)$$ is an isomorphism. It follows as above that $f^\star=g^\star$ as soon as $f,g$ are homotopic. \begstat{(14.15) Excision theorem} For subspaces $\ol T\subset S^\circ$ of $X$, the restriction morphism $H^q(X,S\,;\,M)\longrightarrow H^q(X\ssm T,S\ssm T\,;\,M)$ is an isomorphism. \endstat \begproof{} Under our assumption, it is not difficult to check that the surjective restriction map $M^{[q]}(X,S)\longrightarrow M^{[q]}(X\ssm T, S\ssm T)$ is also injective, because the kernel consists of sections $u\in M^{[q]}(X)$ such that $u(x_0,\ldots,x_q)=0$ on $(X\ssm T)^{q+1}\cup S^{q+1}$, and this set is a neighborhood of the diagonal of $X^{q+1}$.\qed \endproof \begstat{(14.16) Proposition} If $S$ is open or strongly paracompact in $X$, the relative cohomology groups can be written in terms of cohomology groups with supports in $X\ssm S\,$: $$H^q(X,S\,;\,M)\simeq H^q_{X\ssm S}(X,M).$$ In particular, if $X\ssm S$ is relatively compact in $X$, we have $$H^q(X,S\,;\,M)\simeq H^q_c(X\ssm S,M).$$ \endstat \begproof{} We have an exact sequence $$0\longrightarrow M^{[\bu]}_{X\ssm S}(X)\longrightarrow M^{[\bu]}(X)\longrightarrow M^{[\bu]}(S)\longrightarrow 0\leqno(14.17)$$ where $M^{[\bu]}_{X\ssm S}(X)$ denotes sections with support in $X\ssm S$. If $S$ is open, then $M^{[\bu]}(S)=(M_{\restriction S})^{[\bu]}(S)$, hence $M^{[\bu]}_{X\ssm S}(X)=M^{[\bu]}(X,S)$ and the result follows. If $S$ is strongly paracompact, Prop.\ 4.7 and Th. 9.10 show that $$H^q\big(M^{[\bu]}(S)\big)=H^q\big( \lim_{\displaystyle\longrightarrow\atop\scriptstyle\Omega\supset S}M^{[\bu]}(\Omega)\big) =\lim_{\displaystyle\longrightarrow\atop\scriptstyle\Omega\supset S}H^q(\Omega,M) =H^q(S,M_{\restriction S}).$$ If we consider the diagram $$\cmalign{ 0&\longrightarrow&M^{[\bu]}_{X\ssm S}(X)&\longrightarrow&M^{[\bu]}(X)&\longrightarrow &M^{[\bu]}(S)&\longrightarrow 0\cr &&~~~~~~\big\downarrow&&~~~\big\downarrow \Id&&~~~\big\downarrow \restriction S&\cr 0&\longrightarrow&M^{[\bu]}(X,S)&\longrightarrow&M^{[\bu]}(X)&\longrightarrow &(M_{\restriction S})^{[\bu]}(S)&\longrightarrow 0\cr}$$ we see that the last two vertical arrows induce isomorphisms in cohomology. Therefore, the first one also does.\qed \endproof \begstat{(14.18) Corollary} Let $X,Y$ be locally compact spaces and \hbox{$f,g:X\to Y$} proper maps. We say that $f,g$ are properly homotopic if they are homotopic through a proper homotopy $H:X\times I\longrightarrow Y$. Then $$f^\star=g^\star~:~~H^q_c(Y,M)\longrightarrow H^q_c(X,M).$$ \endstat \begproof{} Let $\wh X=X\cup\{\infty\}$, $\wh Y=Y\cup\{\infty\}$ be the Alexandrov compactifications of $X$, $Y$. Then $f,g,H$ can be extended as continuous maps $$\wh f,\wh g~:~~\wh X\longrightarrow\wh Y,~~~~\wh H~:~~\wh X\times I\longrightarrow\wh Y$$ with $\wh f(\infty)=\wh g(\infty)=H(\infty,t)=\infty$, so that $\wh f,\wh g$ are homotopic as maps $(\wh X,\infty)\longrightarrow (\wh Y,\infty)$. Proposition 14.16 implies $H^q_c(X,M)=H^q(\wt X,\infty\,;\,M)$ and the result follows.\qed \endproof \titleb{15.}{K\"unneth Formula} \titlec{15.A.}{Flat Modules and Tor Functors} The goal of this section is to investigate homological properties related to tensor products. We work in the category of modules over a commutative ring $R$ with unit. All tensor products appearing here are tensor products over $R$. The starting point is the observation that tensor product with a given module is a right exact functor: if $0\to A\to B\to C\to 0$ is an exact sequence and $M$ a $R$-module, then $$A\otimes M\longrightarrow B\otimes M\longrightarrow C\otimes M\longrightarrow 0$$ is exact, but the map $A\otimes M\longrightarrow B\otimes M$ need not be injective. A counterexample is given by the sequence $$0\longrightarrow\bZ\buildo 2\times\over\longrightarrow\bZ\longrightarrow\bZ/2\bZ\longrightarrow0~~~~\hbox{\rm over}~~R=\bZ$$ tensorized by $M=\bZ/2\bZ$. However, the injectivity holds if $M$ is a free\break $R$-module. More generally, one says that $M$ is a {\it flat $R$-module} if the tensor product by $M$ preserves injectivity, or equivalently, if $\otimes M$ is a left exact functor. A {\it flat resolution} $C_\bu$ of a $R$-module $A$ is a homology exact sequence $$\cdots\longrightarrow C_q\longrightarrow C_{q-1}\longrightarrow\cdots\longrightarrow C_1\longrightarrow C_0\longrightarrow A\longrightarrow 0$$ where $C_q$ are flat $R$-modules and $C_q=0$ for $q<0$. Such a resolution always exists because every module $A$ is a quotient of a free module $C_0$. Inductively, we take $C_{q+1}$ to be a free module such that $\ker(C_q\to C_{q-1})$ is a quotient of $C_{q+1}$. In terms of homology groups, we have $H_0(C_\bu)=A$ and $H_q(C_\bu)=0$ for $q\ne 0$. Given $R$-modules $A,B$ and free resolutions $d':C_\bu\longrightarrow A$, $d'':D_\bu\longrightarrow B$, we consider the double homology complex $$K_{p,q}=C_p\otimes D_q,~~~~d=d'\otimes\Id+(-1)^p\Id\otimes d''$$ and the associated first and second spectral sequences. Since $C_p$ is free, we have $$E^1_{p,q}=H_q(C_p\otimes D_\bu)=\cases{C_p\otimes B&for~~$q=0$,\cr 0&for~~$q\ne0$.\cr}$$ Similarly, the second spectral sequence also collapses and we have $$H_l(K_\bu)=H_l(C_\bu\otimes B)=H_l(A\otimes D_\bu).$$ This implies in particular that the homology groups $H_l(K_\bu)$ do not depend on the choice of the resolutions $C_\bu$ or $D_\bu$. \begstat{(15.1) Definition} The $q$-th torsion module of $A$ and $B$ is $$\Tor_q(A,B)=H_q(K_\bu)=H_q(C_\bu\otimes B)=H_q(A\otimes D_\bu).$$ \endstat Since the definition of $K_\bu$ is symmetric with respect to $A$ and $B$, we have $\Tor_q(A,B)\simeq\Tor_q(B,A)$. By the right-exactness of $\otimes B$, we find in particular $\Tor_0(A,B)=A\otimes B$. Moreover, if $B$ is flat, $\otimes B$ is also left exact, thus $\Tor_q(A,B)=0$ for all $q\ge 1$ and all modules $A$. If $0\to A\to A'\to A''\to 0$ is an exact sequence, there is a corresponding exact sequence of homology complexes $$0\longrightarrow A\otimes D_\bu\longrightarrow A'\otimes D_\bu\longrightarrow A''\otimes D_\bu\longrightarrow 0,$$ thus a long exact sequence $$\cmalign{ &\longrightarrow\Tor_q(A,B)&\longrightarrow\Tor_q(A',B)&\longrightarrow\Tor_q(A'',B)&\longrightarrow \Tor_{q-1}(A,B)\cr \hfill\cdots&\longrightarrow~~~A\otimes B&\longrightarrow~~~A'\otimes B&\longrightarrow~~~A''\otimes B &\longrightarrow~~~0.\cr}\leqno(15.2)$$ It follows that $B$ is flat if and only if $\Tor_1(A,B)=0$ for every $R$-module $A$. Suppose now that $R$ is a {\it principal ring}. Then every module $A$ has a free resolution $0\to C_1\to C_0\to A\to 0$ because the kernel of any surjective map $C_0\to A$ is free (every submodule of a free module is free). It follows that one always has $\Tor_q(A,B)=0$ for $q\ge 2$. In this case, we denote $\Tor_1(A,B)=A\star B$ and call it the {\it torsion product} of $A$ and $B$. The above exact sequence (15.2) reduces to $$0\to A\star B\to A'\star B\to A''\star B\to A\otimes B\to A'\otimes B\to A''\otimes B\to 0.\leqno(15.3)$$ In order to compute $A\star B$, we may restrict ourselves to finitely generated modules, because every module is a direct limit of such modules and the $\star$ product commutes with direct limits. Over a principal ring $R$, every finitely generated module is a direct sum of a free module and of cyclic modules $R/aR$. It is thus sufficient to compute $R/aR\star R/bR$. The obvious free resolution $\smash{R\buildo a\times\over\longrightarrow R}$ of $R/aR$ shows that $R/aR\star R/bR$ is the kernel of the map $\smash{R/bR\buildo a\times\over\longrightarrow R/bR}$. Hence $$R/aR\star R/bR\simeq R/(a\wedge b)R\leqno(15.4)$$ where $a\wedge b$ denotes the greatest common divisor of $a$ and $b$. It follows that a module $B$ is flat if and only if it is torsion free. If $R$ is a field, every $R$-module is free, thus $A\star B=0$ for all $A$ and $B$. \titlec{15.B.}{K\"unneth and Universal Coefficient Formulas} The algebraic K\"unneth formula describes the cohomology groups of the tensor product of two differential complexes. \begstat{(15.5) Algebraic K\"unneth formula} Let $(K^\bu,d')$, $(L^\bu,d'')$ be complexes of $R$-modules and $(K\otimes L)^\bu$ the simple complex associated to the double complex $(K\otimes L)^{p,q}=K^p\otimes L^q$. If $K^\bu$ or $L^\bu$ is torsion free, there is a split exact sequence $$0\to\bigoplus_{p+q=l}H^p(K^\bu)\otimes H^q(L^\bu)\buildo\mu\over\to H^l\big((K\otimes L)^\bu\big)\to\bigoplus_{p+q=l+1}H^p(K^\bu)\star \buildu{\displaystyle\to 0\phantom{\Big|}}\under{H^q(L^\bu)}$$ where the map $\mu$ is defined by $\mu(\{k^p\}\times\{l^q\})=\{k^p\otimes l^q\}$ for all cocycles $\{k^p\}\in H^p(K^\bu)$, $\{l^q\}\in H^q(L^\bu)$. \endstat \begstat{(15.6) Corollary} If $R$ is a field, or if one of the graded modules $H^\bu(K^\bu)$, $H^\bu(L^\bu)$ is torsion free, then $$H^l\big((K\otimes L)^\bu\big)\simeq\bigoplus_{p+q=l}H^p(K^\bu)\otimes H^q(L^\bu).$$ \endstat \begproof{} Assume for example that $K^\bu$ is torsion free. There is a short exact sequence of complexes $$0\longrightarrow Z^\bu\longrightarrow K^\bu\buildo d'\over\longrightarrow B^{\bu+1}\longrightarrow 0$$ where $Z^\bu,B^\bu\subset K^\bu$ are respectively the graded modules of cocycles and coboundaries in $K^\bu$, considered as subcomplexes with zero differential. As $B^{\bu+1}$ is torsion free, the tensor product of the above sequence with $L^\bu$ is still exact. The corresponding long exact sequence for the associated simple complexes yields: $$\leqalignno{ H^l\big((B\otimes L)^\bu\big)&\longrightarrow H^l\big((Z\otimes L)^\bu\big)\longrightarrow H^l\big((K\otimes L)^\bu\big)\buildo d'\over\longrightarrow H^{l+1} \big((B\otimes L)^\bu\big)\cr &\longrightarrow H^{l+1}\big((Z\otimes L)^\bu\big)\cdots.&(15.7)\cr}$$ The first and last arrows are connecting homomorphisms; in this situation, they are easily seen to be induced by the inclusion $B^\bu\subset Z^\bu$. Since the differential of $Z^\bu$ is zero, the simple complex $(Z\otimes L)^\bu$ is isomorphic to the direct sum $\bigoplus_p Z^p\otimes L^{\bu-p}$, where $Z^p$ is torsion free. Similar properties hold for $(B\otimes L)^\bu$, hence $$H^l\big((Z\otimes L)^\bu\big)=\bigoplus_{p+q=l}Z^p\otimes H^q(L^\bu),~~~~ H^l\big((B\otimes L)^\bu\big)=\bigoplus_{p+q=l}B^p\otimes H^q(L^\bu).$$ The exact sequence $$0\longrightarrow B^p\longrightarrow Z^p\longrightarrow H^p(K^\bu)\longrightarrow 0$$ tensorized by $H^q(L^\bu)$ yields an exact sequence of the type (15.3): $$\eqalign{ 0\to H^p(K^\bu)\star H^q(L^\bu)\to B^p\otimes&H^q(L^\bu)\to Z^p\otimes H^q(L^\bu)\cr &\to H^p(K^\bu)\otimes H^q(L^\bu)\to 0.\cr}$$ By the above equalities, we get $$\eqalign{ 0\longrightarrow\bigoplus_{p+q=l}H^p(K^\bu)\star H^q&(L^\bu)\longrightarrow H^l\big((B\otimes L)^\bu\big)\longrightarrow H^l\big((Z\otimes L)^\bu\big)\cr &\longrightarrow\bigoplus_{p+q=l}H^p(K^\bu)\otimes H^q(L^\bu)\longrightarrow 0.\cr}$$ In our initial long exact sequence (15.7), the cokernel of the first arrow is thus $\bigoplus_{p+q=l}H^p(K^\bu)\otimes H^q(L^\bu)$ and the kernel of the last arrow is the torsion sum $\bigoplus_{p+q=l+1}H^p(K^\bu)\star H^q(L^\bu)$. This gives the exact sequence of the lemma. We leave the computation of the map $\mu$ as an exercise for the reader. The splitting assertion can be obtained by observing that there always exists a torsion free complex $\wt K^\bu$ that splits (i.e.\ $\wt Z^\bu\subset\wt K^\bu$ splits), and a morphism $\wt K^\bu\longrightarrow K^\bu$ inducing an isomorphism in cohomology; then the projection $\wt K^\bu\longrightarrow\wt Z^\bu$ yields a projection $$\eqalign{ H^l\big((\wt K\otimes L)^\bu\big)\longrightarrow H^l\big((\wt Z\otimes L)^\bu\big) \simeq\bigoplus_{p+q=l}\wt Z^p&\otimes H^q(L^\bu)\cr &\longrightarrow\bigoplus_{p+q=l}H^p(\wt K^\bu)\otimes H^q(L^\bu).\cr}$$ To construct $\wt K^\bu$, let $\wt Z^\bu\longrightarrow Z^\bu$ be a surjective map with $\wt Z^\bu$ free, $\wt B^\bu$ the inverse image of $B^\bu$ in $\wt\cZ^\bu$ and $\wt K^\bu=\wt Z^\bu\oplus\wt B^{\bu+1}$, where the differential $\wt K^\bu\longrightarrow \wt K^{\bu+1}$ is given by $\wt Z^\bu\longrightarrow 0$ and $\wt B^{\bu+1}\subset\wt Z^{\bu+1}\oplus 0$~; as $\wt B^\bu$ is free, the map $\wt B^{\bu+1}\longrightarrow B^{\bu+1}$ can be lifted to a map $\wt B^{\bu+1}\longrightarrow K^\bu$, and this lifting combined with the composite $\wt Z^\bu\to Z^\bu\subset K^\bu$ yields the required complex morphism $\wt K^\bu=\wt Z^\bu\oplus\wt B^{\bu+1}\longrightarrow K^\bu$.\qed \endproof \begstat{(15.8) Universal coefficient formula} Let $K^\bu$ be a complex of $R$-modules and $M$ a $R$-module such that either $K^\bu$ or $M$ is torsion free. Then there is a split exact sequence $$0\longrightarrow H^p(K^\bu)\otimes M\longrightarrow H^p(K^\bu\otimes M)\longrightarrow H^{p+1}(K^\bu)\star M \longrightarrow 0.$$ \endstat Indeed, this is a special case of Formula~15.5 when the complex $L^\bu$ is reduced to one term $L^0=M$. In general, it is interesting to observe that the spectral sequence of $K^\bu\otimes L^\bu$ collapses in $E_2$ if $K^\bu$ is torsion free: $H^k\big((K\otimes L)^\bu\big)$ is in fact the direct sum of the terms $E^{p,q}_2=H^p\big(K^\bu\otimes H^q(L^\bu)\big)$ thanks to (15.8). \titlec{15.C. K\"unneth Formula for Sheaf Cohomology} Here we apply the general algebraic machinery to compute cohomology groups over a product space $X\times Y$. The main argument is a combination of the Leray spectral sequence with the universal coefficient formula for sheaf cohomology. \begstat{(15.9) Theorem} Let $\cA$ be a sheaf of $R$-modules over a topological space $X$ and $M$ a $R$-module. Assume that either $\cA$ or $M$ is torsion free and that either $X$ is compact or $M$ is finitely generated. Then there is a split exact sequence $$0\longrightarrow H^p(X,\cA)\otimes M\longrightarrow H^p(X,\cA\otimes M)\longrightarrow H^{p+1}(X,\cA)\star M \longrightarrow 0.$$ \endstat \begproof{} If $M$ is finitely generated, we get $(\cA\otimes M)^{[\bu]}(X)=\cA^{[\bu]}(X)\otimes M$ easily, so the above exact sequence is a consequence of Formula~15.8. If $X$ is compact, we may consider \v Cech cochains $C^q(\cU,\cA\otimes M)$ over finite coverings. There is an obvious morphism $$C^q(\cU,\cA)\otimes M\longrightarrow C^q(\cU,\cA\otimes M)$$ but this morphism need not be surjective nor injective. However, since $$(\cA\otimes M)_x=\cA_x\otimes M= \lim_{\displaystyle\longrightarrow\atop\scriptstyle V\ni x}~~\cA(V)\otimes M,$$ the following properties are easy to verify: \medskip \item{a)} If $c\in C^q(\cU,\cA\otimes M)$, there is a refinement $\cV$ of $\cU$ and $\rho:\cV\longrightarrow\cU$ such that $\rho^\star c\in C^q(\cV,\cA\otimes M)$ is in the image of $C^q(\cV,\cA)\otimes M$. \medskip \item{b)} If a tensor $t\in C^q(\cU,\cA)\otimes M$ is mapped to $0$ in $C^q(\cU,\cA\otimes M)$, there is a refinement $\cV$ of $\cU$ such that $\rho^\star t\in C^q(\cV,\cA)\otimes M$ equals $0$. \medskip \noindent From a) and b) it follows that $$\check H^q(X,\cA\otimes M)= \lim_{\displaystyle\longrightarrow\atop\scriptstyle\cU}~~ H^q\big(C^\bu(\cU,\cA\otimes M)\big) =\lim_{\displaystyle\longrightarrow\atop\scriptstyle\cU}~~ H^q\big(C^\bu(\cU,\cA)\otimes M\big)$$ and the desired exact sequence is the direct limit of the exact sequences of Formula~15.8 obtained for $K^\bu=C^\bu(\cU,\cA)$.\qed \endproof \begstat{(15.10) Theorem {\rm(K\"unneth)}} Let $\cA$ and $\cB$ be sheaves of $R$-modules over topological spaces $X$ and $Y$. Assume that $\cA$ is torsion free, that $Y$ is compact and that either $X$ is compact or the cohomology groups $H^q(Y,\cB)$ are finitely generated $R$-modules. There is a split exact sequence $$\eqalign{ 0\longrightarrow\bigoplus_{p+q=l}H^p(X,\cA)&\otimes H^q(Y,\cB)\buildo\mu\over\longrightarrow H^l(X\times Y,\cA\stimes\cB)\cr &\longrightarrow\bigoplus_{p+q=l+1}H^p(X,\cA)\star H^q(Y,\cB)\longrightarrow 0\cr}$$ where $\mu$ is the map given by the cartesian product $\bigoplus\alpha_p\otimes\beta_q\longmapsto\sum \alpha_p\times\beta_q.$ \endstat \begproof{} We compute $H^l(X,\cA\stimes\cB)$ by means of the Leray spectral sequence of the projection $\pi:X\times Y\longrightarrow X$. This means that we are considering the differential sheaf $\cL^q=\pi_\star(\cA\stimes\cB)^{[q]}$ and the double complex $$K^{p,q}=(\cL^q)^{[p]}(X).$$ By $(12.5')$ we have ${}_K E_2^{p,q}=H^p\big(X,\cH^q(\cL^\bu)\big)$. As $Y$ is compact, the cohomology sheaves $\cH^q(\cL^\bu)=R^q\pi_\star(\cA\stimes\cB)$ are given by $$R^q\pi_\star(\cA\stimes\cB)_x\!=\!H^q(\{x\}\times Y,\cA\stimes \cB_{\restriction\{x\}\times Y})\!=\!H^q(Y,\cA_x\otimes\cB)\!=\! \cA_x\otimes H^q(Y,\cB)$$ thanks to the compact case of Th. 15.9 where $M=\cA_x$ is torsion free. We obtain therefore $$\eqalign{ &R^q\pi_\star(\cA\stimes\cB)=\cA\otimes H^q(Y,\cB),\cr &{}_KE^{p,q}_2=H^p\big(X,\cA\otimes H^q(Y,\cB)\big).\cr}$$ Theorem 15.9 shows that the $E_2$-term is actually given by the desired exact sequence, but it is not a priori clear that the spectral sequence collapses in $E_2$. In order to check this, we consider the double complex $$C^{p,q}=\cA^{[p]}(X)\otimes\cB^{[q]}(Y)$$ and construct a natural morphism $C^{\bu,\bu}\longrightarrow K^{\bu,\bu}$. We may consider $K^{p,q}=\big(\pi_\star(\cA\stimes\cB)^{[q]}\big)^{[p]}(X)$ as the set of equivalence classes of functions $$h\big(\xi_0,\ldots,\xi_p)\in\pi_\star(\cA\stimes\cB)^{[q]}_{\xi_p} =\lim_{\displaystyle\longrightarrow}~~(\cA\stimes\cB)^{[q]}\big(\pi^{-1} \big(V(\xi_p)\big)\big)$$ or more precisely $$\eqalign{ &h\big(\xi_0,\ldots,\xi_p\,;\,(x_0,y_0),\ldots,(x_q,y_q)\big)\in\cA_{x_q}\otimes\cB_{y_q} ~~~~\hbox{\rm with}\cr &\xi_0\in X,~~~\xi_j\in V(\xi_0,\ldots,\xi_{j-1}),~~~1\le j\le p,\cr &(x_0,y_0)\in V(\xi_0,\ldots,\xi_p)\times Y,\cr &(x_j,y_j)\in V\big(\xi_0,\ldots,\xi_p\,;\,(x_0,y_0),\ldots,(x_{j-1},y_{j-1})\big), ~~~1\le j\le q.\cr}$$ Then $f\otimes g\in C^{p,q}$ is mapped to $h\in K^{p,q}$ by the formula $$h\big(\xi_0,\ldots,\xi_p\,;\,(x_0,y_0),\ldots,(x_q,y_q)\big)= f(\xi_0,\ldots,\xi_p)(x_q)\otimes g(y_0,\ldots,y_q).$$ As $\cA^{[p]}(X)$ is torsion free, we find $${}_CE^{p,q}_1=\cA^{[p]}(X)\otimes H^q(Y,\cB).$$ Since either $X$ is compact or $H^q(Y,\cB)$ finitely generated, Th. 15.9 yields $${}_CE^{p,q}_2=H^p\big(X,\cA\otimes H^q(Y,\cB)\big)\simeq{}_KE^{p,q}_2$$ hence $H^l(K^\bu)\simeq H^l(C^\bu)$ and the algebraic K\"unneth formula 15.5 concludes the proof.\qed \endproof \begstat{(15.11) Remark} \rm The exact sequences of Th. 15.9 and of K\"unneth's theorem also hold for cohomology groups with compact support, provided that $X$ and $Y$ are locally compact and $\cA$ (or $\cB$) is torsion free. This is an immediate consequence of Prop.\ 7.12 on direct limits of cohomology groups over compact subsets. \endstat \begstat{(15.12) Corollary} When $\cA$ and $\cB$ are torsion free constant sheaves, e.g. $\cA=\cB=\bZ$ or $\bR$, the K\"unneth formula holds as soon as $X$ or $Y$ has the same homotopy type as a finite cell complex. \endstat \begproof{} If $Y$ satisfies the assumption, we may suppose in fact that $Y$ is a finite cell complex by the homotopy invariance. Then $Y$ is compact and $H^\bu(Y,\cB)$ is finitely generated, so Th.~15.10 can be applied.\qed \endproof \titleb{16.}{Poincar\'e duality} \titlec{16.A.}{Injective Modules and Ext Functors} The study of duality requires some algebraic preliminaries on the ~Hom~ functor and its derived functors $\Ext^q$. Let $R$ be a commutative ring with unit, $M$ a $R$-module and $$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$$ an exact sequence of $R$-modules. Then we have exact sequences $$\eqalign{ 0\longrightarrow&\Hom_R(M,A)\longrightarrow\Hom_R(M,B)\longrightarrow\Hom_R(M,C),\cr &\Hom_R(A,M)\longleftarrow\Hom_R(B,M)\longleftarrow\Hom_R(C,M) \longleftarrow 0,\cr}$$ i.e.\ $\Hom(M,\bu)$ is a covariant left exact functor and $\Hom(\bu,M)$ a contravariant right exact functor. The module $M$ is said to be {\it projective} if $\Hom(M,\bu)$ is also right exact, and {\it injective} if $\Hom(\bu,M)$ is also left exact. Every free\break $R$-module is projective. Conversely, if $M$ is projective, any surjective morphism $F\longrightarrow M$ from a free module $F$ onto $M$ must split $\big(\Id_M$ has a preimage in $\Hom(M,F)\big)$; if $R$ is a principal ring, ``projective" is therefore equivalent to ``free". \begstat{(16.1) Proposition} Over a principal ring $R$, a module $M$ is injective if and only if it is divisible, i.e.\ if for every $x\in M$ and $\lambda\in R\ssm\{0\}$, there exists $y\in M$ such that $\lambda y=x$. \endstat \begproof{} If $M$ is injective, the exact sequence $0\longrightarrow R\buildo\lambda\times\over\longrightarrow R\longrightarrow R/\lambda R\longrightarrow 0$ shows that $$M=\Hom(R,M)\buildo\lambda\times\over\longrightarrow\Hom(R,M)=M$$ must be surjective, thus $M$ is divisible. Conversely, assume that $R$ is divisible. Let $f:A\longrightarrow M$ be a morphism and $B\supset A$. Zorn's lemma implies that there is a maximal extension $\smash{\wt f:\wt A}\longrightarrow M$ of $f$ where $A\subset\smash{\wt A}\subset B$. If $\smash{\wt A}\ne B$, select $x\in B\ssm\smash{\wt A}$ and consider the ideal $I$ of elements $\lambda\in R$ such that $\lambda x\in\smash{\wt A}$. As $R$ is principal we have $I=\lambda_0R$ for some $\lambda_0$. If $\lambda_0\ne 0$, select $y\in M$ such that $\lambda_0y=\smash{\wt f} (\lambda_0x)$ and if $\lambda_0=0$ take $y$ arbitrary. Then $\wt f$ can be extended to $\smash{\wt A}+Rx$ by letting $\smash{\wt f}(x)=y$. This is a contradiction, so we must have $\smash{\wt A}=B$.\qed \endproof \begstat{(16.2) Proposition} Every module $M$ can be embedded in an injective \hbox{module $\smash{\wt M}$.} \endstat \begproof{} Assume first $R=\bZ$. Then set $$M'=\Hom_\bZ(M,\bQ/\bZ),~~~~M''=\Hom_\bZ(M',\bQ/\bZ)\subset\bQ/\bZ^{M'}.$$ Since $\bQ/\bZ$ is divisible, $\bQ/\bZ$ and $\bQ/\bZ^{M'}$ are injective. It is therefore sufficient to show that the canonical morphism $M\longrightarrow M''$ is injective. In fact, for any $x\in M\ssm\{0\}$, the subgroup $\bZ x$ is cyclic (finite or infinite), so there is a non trivial morphism $\bZ x\longrightarrow\bQ/\bZ$, and we can extend this morphism into a morphism $u:M\longrightarrow\bQ/\bZ$. Then $u\in M'$ and $u(x)\ne 0$, so $M\longrightarrow M''$ is injective. Now, for an arbitrary ring $R$, we set $\wt M=\Hom_\bZ\big(R, \bQ/\bZ^{M'}\big)$. There are $R$-linear embeddings $$M=\Hom_R(R,M)\lhra\Hom_\bZ(R,M)\lhra\Hom_\bZ\big(R,\bQ/\bZ^{M'}\big) =\wt M$$ and since $\Hom_R(\bu,\wt M)\simeq\Hom_\bZ\big(\bu,\bQ/\bZ^{M'}\big)$, it is clear that $\smash{\wt M}$ is injective over the ring $R$.\qed \endproof As a consequence, any module has a (cohomological) resolution by injective modules. Let $A,B$ be given $R$-modules, let $d':B\to D^\bu$ be an injective resolution of $B$ and let $d'':C_\bu\to A$ be a free (or projective) resolution of~$A$. We consider the cohomology double complex $$K^{p,q}=\Hom(C_q,D^p),~~~~d=d'+(-1)^p(d'')^\dagger$$ ($\dagger$ means transposition) and the associated first and second spectral sequences. Since $\Hom(\bu,D^p)$ and $\Hom(C_q,\bu)$ are exact, we get $$\eqalign{ E^{p,0}_1&=\Hom(A,D^p),~~~~\wt E^{p,0}_1=\Hom(C_p,B),\cr E^{p,q}_1&=\wt E^{p,q}_1=0~~~~\hbox{\rm for}~~q\ne 0.\cr}$$ Therefore, both spectral sequences collapse in $E_1$ and we get $$H^l(K^\bu)=H^l\big(\Hom(A,D^\bu)\big)=H^l\big(\Hom(C_\bu,B)\big)~;$$ in particular, the cohomology groups $H^l(K^\bu)$ do not depend on the choice of the resolutions $C_\bu$ or $D^\bu$. \begstat{(16.3) Definition} The $q$-th extension module of $A$, $B$ is $$\Ext^q_R(A,B)=H^q(K^\bu)=H^q\big(\Hom(A,D^\bu)\big) =H^q\big(\Hom(C_\bu,B)\big).$$ \endstat By the left exactness of $\Hom(A,\bu)$, we get in particular $\Ext^0(A,B)=\Hom(A,B)$. If $A$ is projective or $B$ injective, then clearly $\Ext^q(A,B)=0$ for all $q\ge 1$. Any exact sequence $0\to A\to A'\to A''\to 0$ is converted into an exact sequence by $\Hom(\bu,D^\bu)$, thus we get a long exact sequence $$\eqalign{ 0&\longrightarrow\Hom(A'',B)\longrightarrow\Hom(A',B)\longrightarrow\Hom(A,B)\longrightarrow\Ext^1(A'',B)\cdots\cr &\longrightarrow\Ext^q(A'',B)\longrightarrow\Ext^q(A',B)\longrightarrow\Ext^q(A,B) \longrightarrow\Ext^{q+1}(A'',B)\cdots\cr}$$ Similarly, any exact sequence $0\to B\to B'\to B''\to 0$ yields $$\eqalign{ 0&\longrightarrow\Hom(A,B)\longrightarrow\Hom(A,B')\longrightarrow\Hom(A,B'')\longrightarrow\Ext^1(A,B)\cdots\cr &\longrightarrow\Ext^q(A,B)\longrightarrow\Ext^q(A,B')\longrightarrow\Ext^q(A,B'') \longrightarrow\Ext^{q+1}(A,B)\cdots\cr}$$ Suppose now that $R$ is a principal ring. Then the resolutions $C_\bu$ or $D^\bu$ can be taken of length $1$ (any quotient of a divisible module is divisible), thus $\Ext^q(A,B)$ is always $0$ for $q\ge 2$. In this case, we simply denote $\Ext^1(A,B)= \Ext(A,B)$. When $A$ is finitely generated, the computation of $\Ext(A,B)$ can be reduced to the cyclic case, since $\Ext(A,B)=0$ when $A$ is free. For $A=R/aR$, the obvious free resolution $R\smash{\buildo a\times\over\longrightarrow}R$ gives $$\Ext_R(R/aR,B)=B/aB.\leqno(16.4)$$ \begstat{(16.5) Lemma} Let $K_\bu$ be a homology complex and let $M\to M^\bu$ be an injective resolution of a $R$-module $M$. Let $L^\bu$ be the simple complex associated to $L^{p,q}=\Hom_R(K_q,M^p)$. There is a split exact sequence $$0\longrightarrow\Ext\big(H_{q-1}(K_\bu),M\big)\longrightarrow H^q(L^\bu)\longrightarrow \Hom\big(H_q(K_\bu),M\big)\longrightarrow 0.$$ \endstat \begproof{} As the functor $\Hom_R(\bu,M^p)$ is exact, we get $$\eqalign{ {}_LE^{p,q}_1&=\Hom\big(H_q(K_\bu),M^p\big),\cr {}_LE^{p,q}_2&=\cases{ \Hom\big(H_q(K_\bu),M\big)&for~~$p=0$,\cr \Ext\big(H_q(K_\bu),M\big)&for~~$p=1$,\cr 0&for~~$p\ge 2$.\cr}\cr}$$ The spectral sequence collapses in $E_2$, therefore we get $$\eqalign{ G_0\big(H^q(L^\bu)\big)&=\Hom\big(H_q(K_\bu),M\big),\cr G_1\big(H^q(L^\bu)\big)&=\Ext\big(H_{q-1}(K_\bu),M\big)\cr}$$ and the expected exact sequence follows. By the same arguments as at the end of the proof of Formula~15.5, we may assume that $K_\bu$ is split, so that there is a projection $K_q\longrightarrow Z_q$. Then the composite morphism $$\eqalign{ \Hom\big(H_q(K_\bu),M\big)=\Hom(Z_q/B_q,M)\longrightarrow \Hom&(K_q/B_q,M)\cr &\subset Z^q(L^\bu)\longrightarrow H^q(L^\bu)\cr}$$ defines a splitting of the exact sequence.\qed \endproof \titlec{16.B.}{Poincar\'e Duality for Sheaves} Let $\cA$ be a sheaf of abelian groups on a locally compact topological space $X$ of finite topological dimension $n=\topdim X$. By 13.12~c), $\cA$ admits a soft resolution $\cL^\bu$ of length $n$. If $M\to M^0\to M^1\to 0$ is an injective resolution of a $R$-module $M$, we introduce the double complex of presheaves $\cF^{p,q}_{\cA,M}$ defined by $$\cF_{\cA,M}^{p,q}(U)=\Hom_R\big(\cL^{n-q}_c(U),M^p\big),\leqno(16.6)$$ where the restriction map $\cF_{\cA,M}^{p,q}(U)\longrightarrow\cF_{\cA,M}^{p,q}(V)$ is the adjoint of the inclusion $\cL^{n-q}_c(V)\longrightarrow\cL^{n-q}_c(U)$ when $V\subset U$. As $\cL^{n-q}$ is soft, any $f\in\cL^{n-q}_c(U)$ can be written as $f=\sum f_\alpha$ with $(f_\alpha)$ subordinate to any open covering $(U_\alpha)$ of $U$~; it follows easily that $\cF_{\cA,M}^{p,q}$ satisfy axioms (II-2.4) of sheaves. The injectivity of $M^p$ implies that $\cF_{\cA,M}^{p,q}$ is a flabby sheaf. By Lemma~16.5, we get a split exact sequence $$\leqalignno{ 0\longrightarrow\Ext\big(H^{n-q+1}_c(X,\cA),M\big)\longrightarrow H^q&\big(\cF_{\cA,M}^\bu(X)\big)\cr &\longrightarrow\Hom\big(H^{n-q}_c(X,\cA),M\big)\longrightarrow 0.&(16.7)\cr}$$ This can be seen as an abstract Poincar\'e duality formula, relating the cohomology groups of a differential sheaf $\cF^\bu_{\cA,M}$ ``dual" of $\cA$ to the dual of the cohomology with compact support of $\cA$. In concrete applications, it still remains to compute $H^q\big(\cF^\bu_{\cA,M}(X)\big)$. This can be done easily when $X$ is a manifold and $\cA$ is a constant or locally constant sheaf. \titlec{16.C.}{Poincar\'e Duality on Topological Manifolds} Here, $X$ denotes a paracompact topological manifold of dimension $n$. \begstat{(16.8) Definition} Let $L$ be a $R$-module. A locally constant sheaf of stalk $L$ on $X$ is a sheaf $\cA$ such that every point has a neighborhood $\Omega$ on which $\cA_{\restriction\Omega}$ is $R$-isomorphic to the constant sheaf $L$. \endstat Thus, a locally constant sheaf $\cA$ can be seen as a discrete fiber bundle over $X$ whose fibers are $R$-modules and whose transition automorphisms are $R$-linear. If $X$ is locally contractible, a locally constant sheaf of stalk $L$ is given, up to isomorphism, by a representation $\rho:\pi_1(X)\longrightarrow\Aut_R(L)$ of the fundamental group of $X$, up to conjugation; denoting by $\smash{\wt X}$ the universal covering of $X$, the sheaf $\cA$ associated to $\rho$ can be viewed as the quotient of $\smash{\wt X}\times L$ by the diagonal action of $\pi_1(X)$. We leave the reader check himself the details of these assertions: in fact similar arguments will be explained in full details in \S V-6 when properties of flat vector bundles are discussed. Let $\cA$ be a locally constant sheaf of stalk $L$, let $\cL^\bu$ be a soft resolution of $\cA$ and $\cF^{p,q}_{\cA,M}$ the associated flabby sheaves. For an arbitrary open set \hbox{$U\subset X$}, Formula (16.7) gives a (non canonical) isomorphism $$H^q\big(\cF_{\cA,M}^\bu(U)\big)\simeq\Hom\big(H^{n-q}_c(U,\cA),M\big)\oplus \Ext\big(H^{n-q+1}_c(U,\cA),M\big)$$ and in the special case $q=0$ a canonical isomorphism $$H^0\big(\cF_{\cA,M}^\bu(U)\big)=\Hom\big(H^n_c(U,\cA),M\big).\leqno(16.9)$$ For an open subset $\Omega$ homeomorphic to $\bR^n$, we have $\cA_{\restriction\Omega}\simeq L$. Proposition 14.16 and the exact sequence of the pair yield $$H^q_c(\Omega,L)\simeq H^q(S^n,\{\infty\}\,;\,L)= \cases{L&for~~$q=n$,\cr 0&for~~$q\ne n$.\cr}$$ If $\Omega\simeq\bR^n$, we find $$H^0\big(\cF_{\cA,M}^\bu(\Omega)\big)\simeq\Hom(L,M),~~~~ H^1\big(\cF_{\cA,M}^\bu(\Omega)\big)\simeq\Ext(L,M)$$ and $H^q\big(\cF_{\cA,M}^\bu(\Omega)\big)=0$ for $q\ne 0,1$. Consider open sets $V\subset\Omega$ where $V$ is a deformation retract of $\Omega$. Then the restriction map $H^q\big(\cF_{\cA,M}^\bu(\Omega)\big)\longrightarrow H^q\big(\cF_{\cA,M}^\bu(V)\big)$ is an isomorphism. Taking the direct limit over all such neighborhoods $V$ of a given point $x\in\Omega$, we see that $\cH^0(\cF_{\cA,M}^\bu)$ and $\cH^1(\cF_{\cA,M}^\bu)$ are locally constant sheaves of stalks $\Hom(L,M)$ and $\Ext(L,M)$, and that $\cH^q(\cF_{\cA,M}^\bu)=0$ for $q\ne 0,1$. When $\Ext(L,M)=0$, the complex $\cF^\bu_{\cA,M}$ is thus a flabby resolution of $\cH^0=\cH^0(\cF^\bu_{\cA,M})$ and we get isomorphisms $$\leqalignno{ &H^q\big(\cF^\bu_{\cA,M}(X)\big)=H^q(X,\cH^0),&(16.10)\cr &\cH^0(U)=H^0(\cF^\bu_{\cA,M}(U)\big)=\Hom\big(H^n_c(U,\cA),M\big). &(16.11)\cr}$$ \begstat{(16.12) Definition} The locally constant sheaf $\tau_X= \cH^0(\cF^\bu_{\bZ,\bZ})$ of stalk $\bZ$ defined by $$\tau_X(U)=\Hom_\bZ\big(H^n_c(U,\bZ),\bZ\big)$$ is called the orientation sheaf (or dualizing sheaf) of $X$. \endstat This sheaf is given by a homomorphism $\pi_1(X)\longrightarrow\{1,-1\}$~; it is not difficult to check that $\tau_X$ coincides with the trivial sheaf $\bZ$ if and only if $X$ is orientable (cf.\ exercice 18.?). In general, $H^n_c(U,\cA)=H^n_c(U,\bZ)\otimes_\bZ\cA(U)$ for any small open set $U$ on which $\cA$ is trivial, thus $$\cH^0(\cF^\bu_{\cA,M})=\tau_X\otimes_\bZ\Hom(\cA,M).$$ A combination of (16.7) and $(16.10)$ then gives: \begstat{(16.13) Poincar\'e duality theorem} Let $X$ be a topological manifold, let $\cA$ be a locally constant sheaf over $X$ of stalk $L$ and let $M$ be a $R$-module such that $\Ext(L,M)=0$. There is a split exact sequence $$\eqalign{ 0\longrightarrow\Ext\big(H^{n-q+1}_c(X,\cA),M\big)\longrightarrow H^q&\big(X,\tau_X\otimes\Hom(\cA,M)\big)\cr &\longrightarrow\Hom\big(H^{n-q}_c(X,\cA),M\big)\longrightarrow 0.\cr}$$ In particular, if either $X$ is orientable or $R$ has characteristic 2, then $$\eqalignno{ 0\longrightarrow\Ext\big(H^{n-q+1}_c(X,R),R\big)\longrightarrow H^q(X,R)\longrightarrow \Hom\big(H^{n-q}_c&(X,R),R\big)\cr &\longrightarrow 0.&\square\cr}$$ \endstat \begstat{(16.14) Corollary} Let $X$ be a connected topological manifold, $n=\dim X$. Then for any $R$-module $L$ \medskip \item{\rm a)} $H^n_c(X,\tau_X\otimes L)\simeq L~;$ \medskip \item{\rm b)} $H^n_c(X,L)\simeq L/2L$~~if $X$ is not orientable. \endstat \begproof{} First assume that $L$ is free. For $q=0$ and $\cA=\tau_X\otimes L$, the Poincar\'e duality formula gives an isomorphism $$\Hom\big(H^n_c(X,\tau_X\otimes L),M\big)\simeq\Hom(L,M)$$ and the isomorphism is functorial with respect to morphisms $M\longrightarrow M'$. Taking $M=L$ or $M=H^n_c(X,\tau_X\otimes L)$, we easily obtain the existence of inverse morphisms $H^n_c(X,\tau_X\otimes L)\longrightarrow L$ and $L\longrightarrow H^n_c(X,\tau_X\otimes L)$, hence equality~a). Similarly, for $\cA=L$ we get $$\Hom\big(H^n_c(X,L),M\big)\simeq H^0\big(X,\tau_X\otimes\Hom(L,M)\big).$$ If $X$ is non orientable, then $\tau_X$ is non trivial and the global sections of the sheaf $\tau_X\otimes\Hom(L,M)$ consist of $2$-torsion elements of $\Hom(L,M)$, that is $$\Hom\big(H^n_c(X,L),M\big)\simeq\Hom(L/2L,M).$$ Formula b) follows. If $L$ is not free, the result can be extended by using a free resolution $0\to L_1\to L_0\to L\to 0$ and the associated long exact sequence.\qed \endproof \begstat{(16.15) Remark} \rm If $X$ is a connected non compact $n$-dimensional manifold, it can be proved (exercise 18.?) that $H^n(X,\cA)=0$ for every locally constant sheaf $\cA$ on $X$.\qed \endstat Assume from now on that $X$ is oriented. Replacing $M$ by $L\otimes M$ and using the obvious morphism $M\longrightarrow\Hom(L,L\otimes M)$, the Poincar\'e duality theorem yields a morphism $$H^q(X,M)\longrightarrow\Hom\big(H^{n-q}_c(X,L),L\otimes M\big),\leqno(16.16)$$ in other words, a bilinear pairing $$H^{n-q}_c(X,L)\times H^q(X,M)\longrightarrow L\otimes M.\leqno(16.16')$$ \begstat{(16.17) Proposition} Up to the sign, the above pairing is given by the cup product, modulo the identification $H^n_c(X,L\otimes M)\simeq L\otimes M$. \endstat \begproof{} By functoriality in $L$, we may assume $L=R$. Then we make the following special choices of resolutions: $$\eqalign{ \cL^q&=R^{[q]}~~~\hbox{\rm for}~~q0$ is a smooth density. If $E$ is a euclidean or hermitian vector bundle, we have a Hilbert space $L^2(M,E)$ of global sections $u$ of~$E$ with measurable coefficients, satisfying the $L^2$ estimate $$\|u\|^2=\int_M|u(x)|^2\,dV(x)<+\infty.\leqno(1.4)$$ We denote by $$\Ll u,v\Gg=\int_M\langle u(x),v(x)\rangle\,dV(x),\qquad u,v\in L^2(M,E)\leqno(1.4')$$ the corresponding $L^2$ inner product. \begstat{(1.5) Definition} If $P:C^\infty(M,E)\to C^\infty(M,F)$ is a differential operator and both $E$, $F$ are euclidean or hermitian, there exists a unique differential operator $$P^\star:C^\infty(M,F)\to C^\infty(M,E),$$ called the {\it formal adjoint} of $P$, such that for all sections $u\in C^\infty(M,E)$ and $v\in C^\infty(M,F)$ there is an identity $$\Ll Pu,v\Gg=\Ll u,P^\star v\Gg,\qquad\hbox{\rm whenever $\Supp u\cap\Supp v\compact M$}.$$ \endstat \begproof{} The uniqueness is easy, using the density of the set of elements $u\in C^\infty(M,E)$ with compact support in $L^2(M,E)$. Since uniqueness is clear, it is enough, by a partition of unity argument, to show the existence of $P^\star$ locally. Now, let $Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x)$ be the expansion of $P$ with respect to trivializations of $E$, $F$ given by orthonormal frames over some coordinate open set $\Omega\subset M$. When \hbox{$\Supp u\cap\Supp v\compact\Omega$} an integration by parts yields $$\eqalign{ \Ll Pu,v\Gg&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu} a_{\alpha\lambda\mu}D^\alpha u_\mu(x)\ol v_\lambda(x)\,\gamma(x)\, dx_1,\ldots,dx_m\cr &=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}(-1)^{|\alpha|} u_\mu(x)\ol{D^\alpha(\gamma(x)\,\ol a_{\alpha\lambda\mu}v_\lambda(x)} \,dx_1,\ldots,dx_m\cr &=\int_\Omega\langle u,\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1} D^\alpha\big(\gamma(x)\,{}^t\ol a_\alpha v(x)\big)\rangle\,dV(x).\cr}$$ Hence we see that $P^\star$ exists and is uniquely defined by $$P^\star v(x)=\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}D^\alpha \big(\gamma(x)\,{}^t\ol a_\alpha v(x)\big).\hfill\square\leqno(1.6)$$ \endproof It follows immediately from (1.6) that the principal symbol of $P^\star$ is $$\sigma_{P^\star}(x,\xi)=(-1)^\delta\sum_{|\alpha|=\delta} {}^t\ol a_\alpha\xi^\alpha=(-1)^\delta\sigma_P(x,\xi)^\star.\leqno(1.7)$$ \begstat{(1.8) Definition} A differential operator $P$ is said to be elliptic if $$\sigma_P(x,\xi)\in\Hom(E_x,F_x)$$ is injective for every $x\in M$ and $\xi\in T^\star_{M,x}\ssm\{0\}$. \endstat \titleb{\S 2.}{Formalism of PseudoDifferential Operators} We assume throughout this section that $(M,g)$ is a compact Riemannian manifold. For any positive integer $k$ and any hermitian bundle $F\to M$, we denote by $W^k(M,F)$ the Sobolev space of sections $s:M\to F$ whose derivatives up to order $k$ are in $L^2$. Let $\|~~\|_k$ be the norm of the Hilbert space $W^k(M,F)$. Let $P$ be an elliptic differential operator of order $d$ acting on $C^\infty(M,F)$. We need the following basic facts of elliptic $PDE$ theory, see e.g.\ (H\"ormander 1963). \begstat{(2.1) Sobolev lemma} For $k>l+{m\over 2}$, $W^k(M,F)\subset C^l(M,F)$. \endstat \begstat{(2.2) Rellich lemma} For every integer $k$, the inclusion $$W^{k+1}(M,F)\lhra W^k(M,F)$$ is a compact linear operator. \endstat \begstat{(2.3) G\aa rding's inequality} Let $\wt P$ be the extension of $P$ to sections with distribution coefficients. For any $u\in W^0 (M,F)$ such that $\wt P u\in W^k(M,F)$, then $u\in W^{k+d}(M,F)$ and $$\|u\|_{k+d}\le C_k(\|\wt Pu\|_k+\|u\|_0),$$ where $C_k$ is a positive constant depending only on $k$. \endstat \begstat{(2.4) Corollary} The operator $P:C^\infty(M,F)\to C^\infty (M,F)$ has the following properties: \smallskip \item{\rm i)} $\ker P$ is finite dimensional. \smallskip \item{\rm ii)} $P\big(C^\infty(M,F)\big)$ is closed and of finite codimension; furthermore, if $P^\star$ is the formal adjoint of $P$, there is a decomposition $$C^\infty(M,F)=P\big(C^\infty(M,F)\big)\oplus\ker P^\star$$ as an orthogonal direct sum in $W^0(M,F)=L^2(M,F)$.\smallskip \endstat \begproof{} (i) G\aa rding's inequality shows that $\|u\|_{k+d}\le C_k\|u\|_0$ for any $u$ in $\ker P$. Thanks to the Sobolev lemma, this implies that $\ker P$ is closed in $W^0(M,F)$. Moreover, the unit closed $\|~~\|_0$-ball of $\ker P$ is contained in the $\|~~\|_d$-ball of radius $C_0$, thus compact by the Rellich lemma. Riesz' theorem implies that $\dim\ker P<+\infty$. \medskip\noindent(ii) We first show that the extension $$\wt P:W^{k+d}(M,F)\to W^k(M,F)$$ has a closed range for any $k$. For every $\varepsilon>0$, there exists a finite number of elements $v_1,\ldots,v_N\in W^{k+d}(M,F)$, $N=N(\varepsilon)$, such that $$\|u\|_0\le\varepsilon\|u\|_{k+d}+\sum^N_{j=1}|\Ll u,v_j\Gg_0|~; \leqno(2.5)$$ indeed the set $$K_{(v_j)}=\Big\{ u\in W^{k+d}(M,F)~ ;~\varepsilon\|u\|_{k+d}+ \sum^N_{j=1} |\Ll u,v_j\Gg_0|\le 1\Big\}$$ is relatively compact in $W^0(M,F)$ and $\bigcap_{(v_j)}\ol K_{(v_j)}=\{ 0\}$. It follows that there exist elements $(v_j)$ such that $\ol K_{(v_j)}$ is contained in the unit ball of $W^0(M,F)$,~$QED$. Substitute $||u||_0$ by the upper bound (2.5) in G\aa rding's inequality; we get $$(1-C_k\varepsilon)\|u\|_{k+d}\le C_k\Big(\|\wt P u\|_k+\sum^N_{j=1} |\Ll u,v_j\Gg_0|\Big).$$ Define $G=\big\{u\in W^{k+d}(M,F)~;~u\perp v_j,~1\le j\le n\}$ and choose $\varepsilon=1/2C_k$. We obtain $$\|u\|_{k+d}\le 2C_k\|\wt P u\|_k,~~\forall u\in G.$$ This implies that $\wt P(G)$ is closed. Therefore $$\wt P\big(W^{k+d}(M,F)\big)=\wt P(G)+{\rm Vect} \big(\wt P(v_1),\ldots,\wt P(v_N)\big)$$ is closed in $W^k(M,F)$. Take in particular $k=0$. Since $C^\infty(M,F)$ is dense in $W^d(M,F)$, we see that in $W^0(M,F)$ $$\left(\wt P\big(W^d(M,F)\big)\right)^\perp= \Big(P\big(C^\infty(M,F)\big)\Big)^\perp=\ker\wt{P^\star}.$$ We have proved that $$W^0(M,F)=\wt P\big(W^d(M,F)\big)\oplus\ker\wt{P^\star}. \leqno(2.6)$$ Since $P^\star$ is also elliptic, it follows that $\ker\wt{P^\star}$ is finite dimensional and that $\ker\wt{P^\star}=\ker P^\star$ is contained in $C^\infty(M,F)$. Thanks to G\aa rding's inequality, the decomposition formula (2.6) yields $$\leqalignno{ W^k(M,F)&=\wt P\big(W^{k+d}(M,F)\big)\oplus\ker P^\star,&(2.7)\cr C^\infty(M,F)&=P\big(C^\infty(M,F)\big)\oplus\ker P^\star.&(2.8)\cr}$$ \endproof \titleb{\S 3.}{Hodge Theory of Compact Riemannian Manifolds} \titlec{\S 3.1.}{Euclidean Structure of the Exterior Algebra} Let $(M,g)$ be an oriented Riemannian $C^\infty$-manifold, $\dim_{\bR} M=m$, and $E\to M$ a hermitian vector bundle of rank $r$ over $M$. We denote respectively by $(\xi_1,\ldots,\xi_m)$ and $(e_1,\ldots,e_r)$ orthonormal frames of $T_M$ and $E$ over an open subset $\Omega\subset M$, and by $(\xi^\star_1,\ldots,\xi^\star_m)$,~$(e^\star_1,\ldots,e^\star_r)$ the corresponding dual frames of $T^\star_M,~E^\star$. Let $dV$ stand for the Riemannian volume form on $M$. The exterior algebra $\Lambda T^\star_M$ has a natural inner product $\langle\bu,\bu\rangle$ such that $$\langle u_1\wedge\ldots\wedge u_p,v_1\wedge\ldots\wedge v_p\rangle= \det(\langle u_j,v_k\rangle)_{1\le j,k\le p},~~~~u_j,v_k\in T^\star_M \leqno(3.1)$$ for all $p$, with $\Lambda T^\star_M=\bigoplus\Lambda^p T^\star_M$ as an orthogonal sum. Then the covectors $\xi^\star_I=\xi^\star_{i_1}\wedge \cdots\wedge\xi^\star_{i_p},~ i_10$. This simple remark already implies that compact K\"ahler manifolds must satisfy some restrictive topological conditions: \begstat{(4.3) Consequence} \smallskip \item{\rm a)} If $(X,\omega)$ is compact K\"ahler and if $\{\omega\}$ denotes the cohomology class of $\omega$ in $H^2(X,\bR)$, then $\{\omega\}^n\ne 0$. \smallskip \item{\rm b)} If $X$ is compact K\"ahler, then $H^{2k}(X,\bR)\ne 0$ for $0\le k\le n$. In fact, $\{\omega\}^k$ is a non zero class in $H^{2k}(X,\bR)$. \endstat \begstat{(4.4) Example} \rm The complex projective space $\bP^n$ is K\"ahler. A natural K\"ahler metric $\omega$ on $\bP^n$, called the {\it Fubini-Study metric}, is defined by $$p^\star\omega={\ii\over 2\pi} d'd''\log\big(|\zeta_0|^2+|\zeta_1|^2 +\cdots+|\zeta_n|^2\big)$$ where $\zeta_0,\zeta_1,\ldots,\zeta_n$ are coordinates of $\bC^{n+1}$ and where \hbox{$p:\bC^{n+1}\setminus\{0\}\to\bP^n$} is the projection. Let $z=(\zeta_1/\zeta_0,\ldots,\zeta_n/\zeta_0)$ be non homogeneous coordinates on $\bC^n\subset\bP^n$. Then (V-15.8) and (V-15.12) show that $$\omega={\ii\over 2\pi}d'd''\log(1+|z|^2)={\ii\over 2\pi}c\big(\cO(1)\big), ~~~~\int_{\bP^n}\omega^n=1.$$ Furthermore $\{\omega\}\in H^2(\bP^n,\bZ)$ is a generator of the cohomology algebra $H^\bu(\bP^n,\bZ)$ in virtue of Th.~V-15.10. \endstat \begstat{(4.5) Example} \rm A {\it complex torus} is a quotient $X=\bC^n/\Gamma$ by a lattice $\Gamma$ of rank~$2n$. Then $X$ is a compact complex manifold. Any positive definite hermitian form $\omega=\ii\sum h_{jk}dz_j\wedge d\ol z_k$ with constant coefficients defines a K\"ahler metric on~$X$. \endstat \begstat{(4.6) Example} \rm Every (complex) submanifold $Y$ of a K\"ahler manifold $(X,\omega)$ is K\"ahler with metric $\omega_{\restriction Y}$. Especially, all submanifolds of $\bP^n$ are K\"ahler. \endstat \begstat{(4.7) Example} \rm Consider the complex surface $$X=(\bC^2\setminus\{ 0\})/\Gamma$$ where $\Gamma=\{\lambda^n~;~n\in\bZ\}$, $\lambda<1$, acts as a group of homotheties. Since $\bC^2\setminus\{ 0\}$ is diffeomorphic to $\bR^\star_+\times S^3$, we have $X\simeq S^1\times S^3$. Therefore $H^2(X,\bR)=0$ by K\"unneth's formula IV-15.10, and property 4.3 b) shows that $X$ is not K\"ahler. More generally, one can obtaintake $\Gamma$ to be an infinite cyclic group generated by a holomorphic contraction of $\bC^2$, of the form $$\pmatrix{z_1\cr z_2\cr}\longmapsto\pmatrix{\lambda_1z_1\cr\lambda_2z_2\cr}, \qquad\hbox{\rm resp.}\quad\pmatrix{z_1\cr z_2\cr}\longmapsto \pmatrix{\lambda z_1\cr\lambda z_2+z_1^p\cr},$$ where $\lambda,\lambda_1,\lambda_2$ are complex numbers such that $0<|\lambda_1|\le|\lambda_2|<1$, \hbox{$0<|\lambda|<1$}, and $p$ a positive integer. These non K\"ahler surfaces are called {\it Hopf surfaces}. \endstat The following Theorem shows that a hermitian metric $\omega$ on $X$ is K\"ahler if and only if the metric $\omega$ is tangent at order $2$ to a hermitian metric with constant coefficients at every point of~$X$. \begstat{(4.8) Theorem} Let $\omega$ be a $C^\infty$ positive definite $(1,1)$-form on $X$. In order that $\omega$ be K\"ahler, it is necessary and sufficient that to every point $x_0\in X$ corresponds a holomorphic coordinate system $(z_1,\ldots,z_n)$ centered at $x_0$ such that $$\omega=\ii\sum_{1\le l,m\le n}\omega_{lm}\,dz_l\wedge d\ol z_m,~~~~ \omega_{lm}=\delta_{lm}+O(|z|^2).\leqno(4.9)$$ If $\omega$ is K\"ahler, the coordinates $(z_j)_{1\le j\le n}$ can be chosen such that $$\omega_{lm}=\langle{\partial\over\partial z_l},{\partial\over\partial z_m} \rangle=\delta_{lm}-\sum_{1\le j,k\le n}c_{jklm}\,z_j\ol z_k+O(|z|^3) ,\leqno(4.10)$$ where $(c_{jklm})$ are the coefficients of the Chern curvature tensor $$\Theta(T_X)_{x_0}=\sum_{j,k,l,m} c_{jklm}\,dz_j\wedge d\ol z_k\otimes \Big({\partial\over\partial z_l}\Big)^\star \otimes{\partial\over\partial z_m}\leqno(4.11)$$ associated to $(T_X,\omega)$ at $x_0$. Such a system $(z_j)$ will be called a geodesic coordinate system at $x_0$. \endstat \begproof{} It is clear that (4.9) implies $d_{x_0}\omega=0$, so the condition is sufficient. Assume now that $\omega$ is K\"ahler. Then one can choose local coordinates $(x_1,\ldots,x_n)$ such that $(dx_1,\ldots,dx_n)$ is an $\omega$-orthonormal basis of $T_{x_0}^\star X$. Therefore $$\leqalignno{ \omega&=\ii\sum_{1\le l,m\le n}\wt\omega_{lm}\,dx_l\wedge d\ol x_m, ~~~~{\rm where}\cr \qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|)=\delta_{lm}+\sum_{1\le j\le n} (a_{jlm}x_j+a'_{jlm}\ol x_j)+O(|x|^2).&(4.12)\cr}$$ Since $\omega$ is real, we have $a'_{jlm}=\ol a_{jml}\,$; on the other hand the K\"ahler condition $\partial\omega_{lm}/\partial x_j= \partial\omega_{jm}/\partial x_l$ at $x_0$ implies $a_{jlm}=a_{ljm}$. Set now $$z_m=x_m+{1\over 2}\sum_{j,l}a_{jlm}x_jx_l,~~~~1\le m\le n.$$ Then $(z_m)$ is a coordinate system at $x_0$, and $$\eqalign{ dz_m&=dx_m+\sum_{j,l}a_{jlm}x_jdx_l,\cr \ii\sum_mdz_m\wedge d\ol z_m&=\ii\sum_m dx_m\wedge d\ol x_m +\ii\sum_{j,l,m}a_{jlm}x_j\,dx_l\wedge d\ol x_m\cr &\phantom{=\ii\sum_m dx_m~\wedge d\ol x_m} +\ii\sum_{j,l,m}\ol a_{jlm}\ol x_j\,dx_m\wedge d\ol x_l+O(|x|^2)\cr &=\ii\sum_{l,m}\wt\omega_{lm}\,dx_l\wedge\ol dx_m+O(|x|^2)= \omega+O(|z|^2).\cr}$$ Condition (4.9) is proved. Suppose the coordinates $(x_m)$ chosen from the beginning so that (4.9) holds with respect to $(x_m)$. Then the Taylor expansion (4.12) can be refined into $$\leqalignno{ \qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|^2)&(4.13)\cr &=\delta_{lm}+\sum_{j,k} \big(a_{jklm}x_j\ol x_k+a'_{jklm}x_jx_k+a''_{jklm}\ol x_j\ol x_k\big)+ O(|x|^3).}$$ These new coefficients satisfy the relations $$a'_{jklm}=a'_{kjlm},~~~~a''_{jklm}=\ol a'_{jkml},~~~~\ol a_{jklm}=a_{kjml}.$$ The K\"ahler condition $\partial\omega_{lm}/\partial x_j=\partial \omega_{jm}/\partial x_l$ at $x=0$ gives the equality $a'_{jklm}=a'_{lkjm}\,;$ in particular $a'_{jklm}$ is invariant under all permutations of~$j,k,l$. If we set $$z_m=x_m+{1\over 3}\sum_{j,k,l}a'_{jklm}\,x_jx_kx_l,~~~~1\le m\le n,$$ then by (4.13) we find $$\leqalignno{ \qquad dz_m&=dx_m+\sum_{j,k,l}a'_{jklm}\,x_jx_k\,dx_l,~~~~1\le m\le n,\cr \omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ol z_m+\ii\sum_{j,k,l,m}a_{jklm}\,x_j \ol x_k\,dx_l\wedge d\ol x_m+O(|x|^3),\cr \omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ol z_m+\ii\sum_{j,k,l,m}a_{jklm}\,z_j \ol z_k\,dz_l\wedge d\ol z_m+O(|z|^3).&(4.14)\cr}$$ It is now easy to compute the Chern curvature tensor $\Theta(T_X)_{x_0}$ in terms of the coefficients $a_{jklm}$. Indeed $$\eqalign{ \langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle&= \delta_{lm}+\sum_{j,k}a_{jklm}\,z_j\ol z_k+O(|z|^3),\cr d'\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle&= \Big\{D'{\partial\over\partial z_l},{\partial\over\partial z_m}\Big\}= \sum_{j,k}a_{jklm}\,\ol z_k\,dz_j+O(|z|^2),\cr \Theta(T_X)\cdot{\partial\over\partial z_l}&=D''D'\Big({\partial\over\partial z_l}\Big) =-\sum_{j,k,m}a_{jklm}\,dz_j\wedge d\ol z_k\otimes{\partial\over\partial z_m} +O(|z|),\cr}$$ therefore $c_{jklm}=-a_{jklm}$ and the expansion (4.10) follows from (4.14).\qed \endproof \begstat{(4.15) Remark} \rm As a by-product of our computations, we find that on a K\"ahler manifold the coefficients of $\Theta(T_X)$ satisfy the symmetry relations $$\ol c_{jklm}=c_{kjml},~~~~c_{jklm}=c_{lkjm}=c_{jmlk}=c_{lmjk}.$$ \endstat \titleb{\S 5.}{Basic Results of K\"ahler Geometry} \titlec{\S 5.1.}{Operators of Hermitian Geometry} Let $(X,\omega)$ be a hermitian manifold and let $z_j=x_j+\ii y_j$, $1\le j\le n$, be analytic coordinates at a point $x\in X$ such that $\omega(x)=\ii\sum dz_j\wedge d\ol z_j$ is diagonalized at this point. The associated hermitian form is the $h(x)=2\sum dz_j\otimes d\ol z_j$ and its real part is the euclidean metric $2\sum (dx_j)^2+(dy_j)^2$. It follows from this that $|dx_j|=|dy_j|=1/\sqrt{2}$, $|dz_j|=|d\ol z_j|=1$, and that $(\partial/\partial z_1,\ldots,\partial/\partial z_n)$ is an orthonormal basis of $(T^\star_x X,\omega)$. Formula (3.1) with $u_j,v_k$ in the orthogonal sum $(\bC\otimes T_X)^\star=T^\star_X\oplus\ol{T^\star_X}$ defines a natural inner product on the exterior algebra $\Lambda^\bu(\bC\otimes T_X)^\star$. The norm of a form $$u=\sum_{I,J}~~u_{I,J} dz_I\wedge d\ol z_J\in\Lambda (\bC\otimes T_X)^\star$$ at the given point $x$ is then equal to $$|u(x)|^2=\sum_{I,J} ~|u_{I,J}(x)|^2.\leqno(5.1)$$ The Hodge $\star $ operator (3.2) can be extended to $\bC$-valued forms by the formula $$u\wedge{}\star\ol v=\langle u,v\rangle\,dV.\leqno(5.2)$$ It follows that $\star $ is a $\bC$-linear isometry $$\star~:~~\Lambda^{p,q} T^\star_X\longrightarrow\Lambda^{n-q,n-p} T^\star_X.$$ The usual operators of hermitian geometry are the operators $d,~\delta= -\star d\,\star ,~\Delta=d\delta+\delta d$ already defined, and their complex counterparts $$\left\{\eqalign{ d&=d'+d'',\cr \delta&=d^{\prime\star}+d^{\prime\prime\star},~~~~d^{\prime\star}=(d')^\star=-\star d''\star,~~~~ d^{\prime\prime\star}=(d'')^\star=-\star d'\star,\cr \Delta'&=d'd^{\prime\star}+d^{\prime\star}d',~~~~ \Delta''=d''d^{\prime\prime\star}+d^{\prime\prime\star}d''.\cr}\right.\leqno(5.3)$$ Another important operator is the operator $L$ of type (1,1) defined by $$Lu=\omega\wedge u\leqno(5.4)$$ and its adjoint $\Lambda=\star ^{-1}L\,\star ~:$ $$\langle u,\Lambda v\rangle=\langle Lu,v\rangle.\leqno(5.5)$$ \titlec{\S 5.2.}{Commutation Identities} If $A,B$ are endomorphisms of the algebra $C^\infty_{\bu,\bu}(X,\bC)$, their graded commutator (or graded Lie bracket) is defined by $$[A,B]=AB-(-1)^{ab} BA\leqno(5.6)$$ where $a,b$ are the degrees of $A$ and $B$ respectively. If $C$ is another endomorphism of degree $c$, the following {\it Jacobi identity} is easy to check: $$(-1)^{ca}\big[A,[B,C]\big]+(-1)^{ab}\big[B,[C,A]\big] +(-1)^{bc}\big[C,[A,B]\big]=0.\leqno(5.7)$$ For any $\alpha\in\Lambda^{p,q}T^\star_X$, we still denote by $\alpha$ the endomorphism of type $(p,q)$ on $\Lambda^{\bu,\bu} T^\star_X$ defined by $u\mapsto\alpha\wedge u$. Let $\gamma\in\Lambda^{1,1} T^\star_X$ be a real (1,1)-form. There exists an $\omega$-orthogonal basis $(\zeta_1,\zeta_2,\ldots,\zeta_n)$ in $T_X$ which diagonalizes both forms $\omega$ and $\gamma\,$: $$\omega=\ii\sum_{1\le j\le n}\zeta^\star_j\wedge\ol\zeta^\star_j,~~~~ \gamma=\ii\sum_{1\le j\le n}\gamma_j\,\zeta^\star_j\wedge\ol\zeta^\star_j,~~ \gamma_j\in\bR.$$ \begstat{(5.8) Proposition} For every form $u=\sum u_{J,K}\,\zeta^\star_J \wedge\ol\zeta^\star_K$, one has $$[\gamma,\Lambda]u=\sum_{J,K}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K} \gamma_j-\sum_{1\le j\le n}\gamma_j\Big) u_{J,K}\,\zeta^\star_J\wedge\ol\zeta^\star_K.$$ \endstat \begproof{} If $u$ is of type $(p,q)$, a brute-force computation yields $$\eqalignno{ \Lambda u&=\ii(-1)^p\sum_{J,K,l} u_{J,K}\,(\zeta_l\ort\zeta^\star_J) \wedge(\ol\zeta_l\ort\ol\zeta^\star_K),~~~~1\le l\le n,\cr \gamma\wedge u&=\ii(-1)^p\sum_{J,K,m}\gamma_m u_{J,K}\,\zeta^\star_m\wedge \zeta^\star_J\wedge\ol\zeta^\star_m\wedge\ol\zeta^\star_K,~~~~1\le m\le n,\cr [\gamma,\Lambda] u&=\sum_{J,K,l,m}\gamma_m\,u_{J,K}\Big( \big(\zeta^\star_l\wedge(\zeta_m\ort\zeta^\star_J)\big)\wedge \big(\ol\zeta^\star_l\wedge(\ol\zeta_m\ort\ol\zeta^\star_K)\big)\cr &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -\big(\zeta_m\ort(\zeta^\star_l\wedge\zeta^\star_J)\big)\wedge \big(\ol\zeta_m\ort(\ol\zeta^\star_l\wedge\ol\zeta^\star_K)\big)\Big)\cr &=\sum_{J,K,m}\gamma_m\,u_{J,K}\Big( \zeta^\star_m\wedge(\zeta_m\ort\zeta^\star_J)\wedge\ol\zeta^\star_K\cr &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +\zeta^\star_J\wedge\ol\zeta^\star_m\wedge(\ol\zeta_m\ort\ol\zeta^\star_K)- \zeta^\star_J\wedge\ol\zeta^\star_K\Big)\cr &=\sum_{J,K}\bigg(\sum_{m\in J}\gamma_m+\sum_{m\in K}\gamma_m-\sum_{1\le m\le n}\gamma_m\bigg) u_{J,K}\,\zeta^\star_J\wedge\ol\zeta^\star_K. &\square\cr}$$ \endproof \begstat{(5.9) Corollary} For every $u\in\Lambda^{p,q}T^\star_X$, we have $$[L,\Lambda]u=(p+q-n) u.$$ \endstat \begproof{} Indeed, if $\gamma=\omega$, we have $\gamma_1=\cdots=\gamma_n=1$. \qed \endproof This result can be generalized as follows: for every $u\in\Lambda^k(\bC\otimes T_X)^\star$, we have $$[L^r,\Lambda]u=r(k-n+r-1)\,L^{r-1}u.\leqno(5.10)$$ In fact, it is clear that $$\eqalign{ [L^r,\Lambda]u&=\sum_{0\le m\le r-1}\kern-8pt L^{r-1-m}[L,\Lambda]L^m u\cr &=\sum_{0\le m\le r-1}\kern-8pt (2m+k-n)L^{r-1-m}L^m u =\big(r(r-1)+r(k-n)\big) L^{r-1} u.\cr}$$ \titlec{\S 5.3.}{Primitive Elements and Hard Lefschetz Theorem} In this subsection, we prove a fundamental decomposition theorem for the representation of the unitary group $U(T_X)\simeq U(n)$ acting on the spaces $\Lambda^{p,q}T^\star_X$ of $(p,q)$-forms. It turns out that the representation is never irreducible if $02n$ and therefore we have $L^{n+1}u=0$. This gives $$(n+1)\cdots\big(n+1-(s-1)\big)\cdot (-k)(-k+1)\cdots(-k+s-1)L^{n+1-s}u=0.$$ The integral coefficient is $\ne 0$ if $s\le k$, hence: \begstat{(5.13) Corollary} If $u\in\Prim^kT^\star_X$, then $L^su=0$ for $s\ge (n+1-k)_+$. \endstat \begstat{(5.14) Corollary} $\Prim^kT^\star_X=0$ for $n+1\le k\le 2n$. \endstat \begproof{} Apply Corollary 5.13 with $s=0$.\qed \endproof \begstat{(5.15) Primitive decomposition formula} For every $u\in\Lambda^k(\bC\otimes T_X)^\star$, there is a unique decomposition $$u=\sum_{r\ge (k-n)_+} L^ru_r,~~~~u_r\in\Prim^{k-2r}T^\star_X.$$ Furthermore~ $u_r=\Phi_{k,r} (L,\Lambda)u$~ where $\,\Phi_{k,r}\,$ is a non commutative polynomial in $L,\Lambda$ with rational coefficients. As a consequence, there are direct sum decompositions of $U(n)$-representations $$\eqalign{ &\Lambda^k(\bC\otimes T_X)^\star=\bigoplus_{r\ge (k-n)_+} L^r\Prim^{k-2r}T^\star_X,\cr &\Lambda^{p,q}T_X^\star=\bigoplus_{r\ge(p+q-n)_+} L^r\Prim^{p-r,q-r}T^\star_X.\cr}$$ \endstat \begproof{of the uniqueness of the decomposition} Assume that $u=0$ and that $u_r\ne 0$ for some $r$. Let $s$ be the largest integer such that $u_s\ne 0$. Then $$\Lambda^s u=0=\sum_{(k-n)_+\le r\le s}\Lambda^s L^r u_r= \sum_{(k-n)_+\le r\le s}\Lambda^{s-r}\Lambda^r L^r u_r.$$ But formula (5.12) shows that $\Lambda^r L^r u_r=c_{k,r} u_r$ for some non zero integral coefficient $c_{k,r}=r!(n-k+r+1)\cdots (n-k+2r)$. Since $u_r$ is primitive we get $\Lambda^sL^ru_r=0$ when $rn$, $\Lambda^{k-n}u=0$. Then $v={}\star u\in\Lambda^{2n-k} (\bC\otimes T_X)^\star$ and $2n-k0$, and which have a zero of order $\ge|m_j|$ at $a_j$ if $m_j<0$. Clearly $$E(\Delta)=E\otimes\cO(\Delta),~~~~\cO(\Delta+\Delta')=\cO(\Delta)\otimes \cO(\Delta').\leqno(10.3)$$ For any point $a\in X$ and any integer $m>0$, there is an exact sequence $$0\longrightarrow E\longrightarrow E(m[a])\longrightarrow\cS\longrightarrow 0$$ where $\cS=E(m[a])/E$ is a sheaf with only one non zero stalk $\cS_a$ isomorphic to $\bC^m$. Indeed, if $z$ is a holomorphic coordinate near $a$, the stalk $\cS_a$ corresponds to the polar parts $\sum_{-m\le k<0} c_kz^k$ in the power series expansions of germs of meromorphic sections at point $a$. We get an exact sequence $$H^0\big(X,E(m[a])\big)\longrightarrow\bC^m\longrightarrow H^1(X,E).$$ When $m$ is chosen larger than $\dim H^1(X,E)$, we see that $E(m[a])$ has a non zero section and conclude: \begstat{(10.4) Theorem} Let $a$ be a given point on a curve. Then every line bundle $E$ has non zero meromorphic sections $f$ with a pole at $a$ and no other poles. \endstat If $\Delta$ is the divisor of a meromorphic section $f$ of $E$, we have $E\simeq\cO(\Delta)$, so the map $${\rm Div}(X)\longrightarrow H^1(X,\cO^\star),~~~~\Delta\longmapsto\cO(\Delta)$$ is onto (cf.\ (V-13.8)). On the other hand, ${\rm Div}$ is clearly a soft sheaf, thus $H^1(X,{\rm Div})=0$. The long cohomology sequence associated to the exact sequence $1\to\cO^\star\to\cM^\star\to{\rm Div}\to 0$ implies: \begstat{(10.5) Corollary} On any complex curve, one has $H^1(X,\cM^\star)=0$ and there is an exact sequence $$0\longrightarrow\bC^\star\longrightarrow\cM^\star(X)\longrightarrow{\rm Div}(X)\longrightarrow H^1(X,\cO^\star)\longrightarrow 0.$$ \endstat The first Chern class $c_1(E)\in H^2(X,\bZ)$ can be interpreted as an integer. This integer is called the {\it degree} of~$E$. If $E\simeq\cO(\Delta)$ with $\Delta=\sum m_ja_j$, formula V-13.6 shows that the image of $c_1(E)$ in $H^2(X,\bR)$ is the De Rham cohomology class of the associated current $[\Delta]=\sum m_j\delta_{a_j}$, hence $$c_1(E)=\int_X[\Delta]=\sum m_j.\leqno(10.6)$$ If $\sum m_ja_j$ is the divisor of a meromorphic function, we have $\sum m_j=0$ because the associated bundle $E=\cO(\sum m_ja_j)$ is trivial. \begstat{(10.7) Theorem} Let $E$ be a line bundle on a complex curve~$X$. Then \smallskip \item{\rm a)} $H^0(X,E)=0$~~if~~$c_1(E)<0$ or if~~$c_1(E)=0$ and $E$ is non trivial$\,;$ \smallskip \item{\rm b)} For every positive $(1,1)$-form $\omega$ on $X$ with $\int_X\omega=1$, $E$ has a hermitian metric such that ${\ii\over 2\pi} \Theta(E)=c_1(E)\,\omega$. In particular, $E$ has a metric of positive $($resp. negative$)$ curvature if and only if $c_1(E)>0$ $($resp. if and only if $c_1(E)<0)$.\smallskip \endstat \begproof{} a) If $E$ has a non zero holomorphic section $f$, then its degree is $c_1(E)=\int_X{\rm div}\,f\ge 0$. In fact, we even have $c_1(E)>0$ unless $f$ does not vanish, in which case $E$ is trivial. b) Select an arbitrary hermitian metric $h$ on $E$. Then $c_1(E)\,\omega-{\ii\over 2\pi}\Theta_h(E)$ is a real $(1,1)$-form cohomologous to zero (the integral over $X$ is zero), so Lemma~8.6~c) implies $$c_1(E)\,\omega-{\ii\over 2\pi}\Theta_h(E)=\ii d'd''\varphi$$ for some real function $\varphi\in C^\infty(X,\bR)$. If we replace the initial metric of $E$ by $h'=h\,e^{-\varphi}$, we get a metric of constant curvature $c_1(E)\,\omega$.\qed \endproof \begstat{(10.8) Riemann-Roch formula} Let $E$ be a holomorphic line bundle and let $h^q(E)=\dim H^q(X,E)$. Then $$h^0(E)-h^1(E)=c_1(E)-g+1.$$ Moreover $h^1(E)=h^0(K\otimes E^\star)$, where $K=\Omega^1_X$ is the canonical line bundle of~$X$. \endstat \begproof{} We claim that for every line bundle $F$ and every divisor $\Delta$ we have the equality $$h^0\big(F(\Delta)\big)-h^1\big(F(\Delta)\big)=h^0(F)-h^1(F)+\int_X[\Delta]. \leqno(10.9)$$ If we write $E=\cO(\Delta)$ and apply the above equality with $F=\cO$, the Riemann-Roch formula results from (10.6), (10.9) and from the equalities $$h^0(\cO)=\dim H^0(X,\cO)=1,~~~~h^1(\cO)=\dim H^1(X,\cO)=g.$$ However, (10.9) need only be proved when $\Delta\ge 0\,$: otherwise we are reduced to this case by writing $\Delta=\Delta_1-\Delta_2$ with $\Delta_1,\Delta_2\ge 0$ and by applying the result to the pairs $(F,\Delta_1)$ and $\big(F(\Delta),\Delta_2\big)$. If $\Delta=\sum m_ja_j \ge 0$, there is an exact sequence $$0\longrightarrow F\longrightarrow F(\Delta)\longrightarrow\cS\longrightarrow 0$$ where $\cS_{a_j}\simeq\bC^{m_j}$ and the other stalks are zero. Let $m=\sum m_j=\int_X[\Delta]$. The sheaf $\cS$ is acyclic, because its support $\{a_j\}$ is of dimension $0$. Hence there is an exact sequence $$0\longrightarrow H^0(F)\longrightarrow H^0\big(F(\Delta)\big)\longrightarrow\bC^m\longrightarrow H^1(F)\longrightarrow H^1\big(F(\Delta)\big)\longrightarrow 0$$ and (10.9) follows. The equality $h^1(E)=h^0(K\otimes E^\star)$ is a consequence of the Serre duality theorem $$\big(H^{0,1}(X,E)\big)^\star\simeq H^{1,0}(X,E^\star),~~~~\hbox{\rm i.e.}~~ \big(H^1(X,E)\big)^\star\simeq H^0(X,K\otimes E^\star).\eqno{\square}$$ \endproof \begstat{(10.10) Corollary (Hurwitz' formula)} $c_1(K)=2g-2$. \endstat \begproof{} Apply Riemann-Roch to $E=K$ and observe that $$\left.\eqalign{ h^0(K)&=\dim H^0(X,\Omega^1_X)=g\cr h^1(K)&=\dim H^1(X,\Omega^1_X)=h^{1,1}=b_2=1\cr}\right.\leqno(10.11)$$ \endproof \begstat{(10.12) Corollary} For every $a\in X$ and every $m\in\bZ$ $$h^0\big(K(-m[a])\big)=h^1\big(\cO(m[a])\big)=h^0\big(\cO(m[a])\big)-m+g-1.$$ \endstat \titlec{\S 10.2.}{Jacobian of a Curve} By the Neron-Severi sequence (9.6), there is an exact sequence $$0\longrightarrow\Jac(X)\longrightarrow H^1(X,\cO^\star)\buildo c_1\over\longrightarrow\bZ\longrightarrow 0, \leqno(10.13)$$ where the Jacobian $\Jac(X)$ is a $g$-dimensional torus. Choose a base point $a\in X$. For every point $x\in X$, the line bundle $\cO([x]-[a])$ has zero first Chern class, so we have a well-defined map $$\Phi_a:X\longrightarrow\Jac(X),~~~~x\longmapsto\cO([x]-[a]).\leqno(10.14)$$ Observe that the Jacobian $\Jac(X)$ of a curve coincides by definition with the Albanese variety $\Alb(X)$. \begstat{(10.15) Lemma} The above map $\Phi_a$ coincides with the Albanese map \hbox{$\alpha:X\longrightarrow\Alb(X)$} defined in $(9.12)$. \endstat \begproof{} By holomorphic continuation, it is enough to prove that \hbox{$\Phi_a(x)=\alpha(x)$} when $x$ is near $a$. Let $z$ be a complex coordinate and let $D'\subset\!\subset D$ be open disks centered at~$a$. Relatively to the covering $$U_1=D,~~~~U_2=X\setminus\ol{D'},$$ the line bundle $\cO([x]-[a])$ is defined by the \v Cech cocycle $c\in C^1({\cal U},\cO^\star)$ such that $$c_{12}(z)={z-x\over z-a}~~~{\rm on}~~U_{12}=D\setminus\ol{D'}.$$ On the other hand, we compute $\alpha(x)$ by Formula~(9.14). The path integral current $I_{[a,x]}\in\cD'_1(X)$ is equal to $0$ on~$U_2$. Lemma~I-2.10 implies \hbox{$d''(dz/2\pi\ii z)=\delta_0\,d\ol z \wedge dz/2i=\delta_0$} according to the usual identification of distributions and currents of degree $0$, thus $$I^{0,1}_{[a,x]}=d''\Big({dz\over2\pi iz}\star I^{0,1}_{[a,x]}\Big)~~~ {\rm on}~~U_1.$$ Therefore $\{I^{0,1}_{[a,x]}\}\in H^{0,1}(X,\bC)$ is equal to the \v Cech cohomology class $\c['\}$ in $H^1(X,\cO)$ represented by the cocycle $$c'_{12}(z)={dw\over2\pi iw}\star I^{0,1}_{[a,x]}={1\over 2\pi i}\int_a^x {dw\over w-z}={1\over 2\pi\ii}\log{z-x\over z-a}~~~{\rm on}~~U_{12}$$ and we have $c=\exp(2\pi\ii c')$ in $H^1(X,\cO^\star)$.\qed \endproof The nature of $\Phi_a$ depends on the value of the genus $g$. A careful examination of $\Phi_a$ will enable us to determine all curves of genus $0$ and $1$. \begstat{(10.16) Theorem} The following properties are equivalent: \smallskip \item{\rm a)} $g=0\,;$ \smallskip \item{\rm b)} $X$ has a meromorphic function $f$ having only one simple pole $p\,;$ \smallskip \item{\rm c)} $X$ is biholomorphic to $\bP^1$. \endstat \begproof{} c) $\Longrightarrow$ a) is clear. \medskip \noindent a) $\Longrightarrow$ b). Since $g=0$, we have $\Jac(X)=0$. If $p,p'\in X$ are distinct points, the bundle $\cO([p']-[p])$ has zero first Chern class, therefore it is trivial and there exists a meromorphic function $f$ with ${\rm div}\,f=[p']-[p]$. In particular $p$ is the only pole of $f$, and this pole is simple. \medskip \noindent b) $\Longrightarrow$ c). We may consider $f$ as a map $X\longrightarrow\bP^1= \bC\cup\{\infty\}$. For every value $w\in\bC$, the function $f-w$ must have exactly one simple zero $x\in X$ because $\int_X{\rm div}(f-w)=0$ and $p$ is a simple pole. Therefore $f:X\to\bP^1$ is bijective and $X$ is biholomorphic to $\bP^1$.\qed \endproof \begstat{(10.17) Theorem} The map $\Phi_a$ is always injective for $g\ge 1$. \smallskip \item{\rm a)} If $g=1$, $\Phi_a$ is a biholomorphism. In particular every curve of genus $1$ is biholomorphic to a complex torus $\bC/\Gamma$. \smallskip \item{\rm b)} If $g\ge 2$, $\Phi_a$ is an embedding. \endstat \begproof{} If $\Phi_a$ is not injective, there exist points $x_1\ne x_2$ such that $\cO([x_1]-[x_2])$ is trivial; then there is a meromorphic function $f$ such that ${\rm div}\,f=[x_1]-[x_2]$ and Th.~10.16 implies that $g=0$. When $g=1$, $\Phi_a$ is an injective map $X\longrightarrow\Jac(X)\simeq\bC/\Gamma$, thus $\Phi_a$ is open. It follows that $\Phi_a(X)$ is a compact open subset of $\bC/\Gamma$, so $\Phi_a(X)=\bC/\Gamma$ and $\Phi_a$ is a biholomorphism of $X$ onto $\bC/\Gamma$. In order to prove that $\Phi_a$ is an embedding when $g\ge 2$, it is sufficient to show that the holomorphic $1$-forms $u_1,\ldots,u_g$ do not all vanish at a given point $x\in X$. In fact, $X$ has no non constant meromorphic function having only a simple pole at $x$, thus $h^0\big(\cO([x])\big)=1$ and Cor.~10.12 implies $$h^0\big(K(-[x])\big)=g-10\leqno(10.20)$$ unless $s_1=1$, $s_2=2$, $\ldots$, $s_g=g$. Now, we have $$h^0\big(K(-m[a])\big)={\rm card}\{j\,;\,s_j>m\} =g-{\rm card}\{j\,;\,s_j\le m\}$$ and Cor.~10.12 gives $$h^0\big(\cO(m[a])\big)=m+1-{\rm card}\{j\,;\,s_j\le m\}.\leqno(10.21)$$ If $a$ is not a zero of $W(u_1,\ldots,u_g)$, we find $$\cases{ h^0\big(\cO(m[a])\big)=1&for~~$m\le g$,\cr h^0\big(\cO(m[a])\big)=m+1-g&for~~$m>g$.\cr}\leqno(10.22)$$ The zeroes of $W(u_1,\ldots,u_g)$ are called the {\it Weierstrass points} of $X$, and the associated {\it Weierstrass sequence} is the sequence $w_m=h^0\big(\cO(m[a])\big)$, $m\in\bN$. We have $w_{m-1}\le w_m\le w_{m-1}+1$ and $s_1<\ldots0$, thus $u\equiv 0$. Hodge theory implies $H^{p,q}(X,E)=0$.\qed \endproof \titleb{3.}{Kodaira-Akizuki-Nakano Vanishing Theorem} The main goal of vanishing theorems is to find natural geometric or algebraic conditions on a bundle $E$ that will ensure that some cohomology groups with values in $E$ vanish. In the next three sections, we prove various vanishing theorems for cohomology groups of a hermitian {\it line bundle} $E$ over a {\it compact} complex manifold $X$. \begstat{(3.1) Definition} A hermitian holomorphic line bundle $E$ on $X$ is said to be positive $($resp. negative$)$ if the hermitian matrix $\big(c_{jk}(z)\big)$ of its Chern curvature form $$\ii\Theta(E)=\ii\sum_{1\le j,k\le n}c_{jk}(z)\,dz_j\wedge d\ol z_k$$ is positive $($resp. negative$)$ definite at every point $z\in X$. \endstat Assume that $X$ has a K\"ahler metric $\omega$. Let $$\gamma_1(x)\le\ldots\le\gamma_n(x)$$ be the eigenvalues of $\ii\Theta(E)_x$ with respect to $\omega_x$ at each point $x\in X$, and let $$\ii\Theta(E)_x=\ii\sum_{1\le j\le n}\gamma_j(x)\,\zeta_j\wedge\ol\zeta_j, ~~~~\zeta_j\in T^\star_x X$$ be a diagonalization of $\ii\Theta(E)_x$. By Prop.~VI-8.3 we have $$\leqalignno{ \langle [\ii\Theta(E),\Lambda]u,u\rangle&=\sum_{J,K} \Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}\gamma_j -\sum_{1\le j\le n}\gamma_j\Big)|u_{J,K}|^2\cr &\ge(\gamma_1+\ldots+\gamma_q-\gamma_{p+1}-\ldots-\gamma_n)|u|^2 &(3.2)\cr}$$ for any form $u=\sum_{J,K}u_{J,K}\,\zeta_J\wedge\ol\zeta_K \in\Lambda^{p,q}T^\star X$. \begstat{(3.3) Akizuki-Nakano vanishing theorem (1954)} Let $E$ be a holomorphic line bundle on $X$. \smallskip \item{\rm a)} If $E$ is positive, then~ $H^{p,q}(X,E)=0~~$ for~ $p+q\ge n+1.$ \smallskip \item{\rm b)} If $E$ is negative, then~ $H^{p,q}(X,E)=0~~$ for~ $p+q\le n-1.$ \endstat \begproof{} In case a), choose $\omega=\ii\Theta(E)$ as a K\"ahler metric on $X$. Then we have $\gamma_j(x)=1$ for all $j$ and $x$, so that $$\Ll [\ii\Theta(E),\Lambda]u,u\Gg\ge (p+q-n)||u||^2$$ for any $u\in\Lambda^{p,q}T^\star X\otimes E$. Assertion a) follows now from Corollary~2.3. Property~b) is proved similarly, by taking $\omega=-\ii\Theta(E)$. One can also derive b) from a) by Serre duality (Theorem~VI-11.3).\qed \endproof When $p=0$ or $p=n$, Th.~3.3 can be generalized to the case where $\ii\Theta(E)$ degenerates at some points. We use here the standard notations $$\Omega^p_X=\Lambda^pT^\star X,~~~~K_X=\Lambda^nT^\star X,~~~~ n=\dim_\bC X~;\leqno(3.4)$$ $K_X$ is called the {\it canonical line bundle} of $X$. \begstat{(3.5) Theorem (Grauert-Riemenschneider 1970)} Let $(X,\omega)$ be a compact and connected K\"ahler manifold and $E$ a line bundle on $X$. \smallskip \item{\rm a)} If $\ii\Theta(E)\ge 0$ on $X$ and $\ii\Theta(E)>0$ in at least one point $x_0\in X$, then \smallskip \centerline{$H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge 1.$} \smallskip \item{\rm b)} If $\ii\Theta(E)\le 0$ on $X$ and $\ii\Theta(E)<0$ in at least one point $x_0\in X$, then \smallskip \centerline{$H^q(X,E)=0~~~~\hbox{\rm for}~~q\le n-1.$} \endstat It will be proved in Volume II, by means of holomorphic Morse inequalities, that the K\"ahler assumption is in fact unnecessary. This improvement is a deep result first proved by (Siu 1984) with a different ad hoc method. \begproof{} For $p=n$, formula (3.2) gives $$\Ll [\ii\Theta(E),\Lambda]u,u\Gg\ge (\gamma_1+\ldots+\gamma_q)|u|^2\leqno(3.6)$$ and a) follows from Cor.~2.5. Now b) is a consequence of a) by Serre duality.\qed \endproof \titleb{4.}{Girbau's Vanishing Theorem} Let $E$ be a line bundle over a compact connected K\"ahler manifold $(X,\omega)$. Girbau's theorem deals with the (possibly everywhere) degenerate semi-positive case. We first state the corresponding generalization of Th.~4.5. \begstat{(4.1) Theorem} If $\ii\Theta(E)$ is semi-positive and has at least $n-s+1$ positive eigenvalues at a point $x_0\in X$ for some integer $s\in\{1,\ldots,n\}$, then $$H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge s.$$ \endstat \begproof{} Apply 2.5 and inequality (3.6), and observe that $\gamma_q(x_0)>0$ for all $q\ge s$.\qed \endproof \begstat{(4.2) Theorem (Girbau 1976)} If $\ii\Theta(E)$ is semi-positive and has at least $n-s+1$ positive eigenvalues at every point $x\in X$, then $$H^{p,q}(X,E)=0~~~~\hbox{\rm for}~~p+q\ge n+s.$$ \endstat \begproof{} Let us consider on $X$ the new K\"ahler metric $$\omega_\varepsilon=\varepsilon\omega +\ii\Theta(E),~~~~\varepsilon>0,$$ and let $\ii\Theta(E)=\ii\sum\gamma_j\,\zeta_j\wedge\ol\zeta_j$ be a diagonalization of $\ii\Theta(E)$ with respect to $\omega$ and with $\gamma_1\le\ldots\le \gamma_n$. Then $$\omega_\varepsilon=\ii\sum~(\varepsilon+\gamma_j)\,\zeta_j\wedge\ol\zeta_j.$$ The eigenvalues of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ are given therefore by $$\gamma_{j,\varepsilon}=\gamma_j/(\varepsilon+\gamma_j)\in[0,1[, ~~~~1\le j\le n.\leqno(4.3)$$ On the other hand, the hypothesis is equivalent to $\gamma_s>0$ on $X$. For $j\ge s$ we have $\gamma_j\ge\gamma_s$, thus $$\gamma_{j,\varepsilon}={1\over 1+\varepsilon/\gamma_j}\ge{1\over 1+ \varepsilon/\gamma_s}\ge1-\varepsilon/\gamma_s,~~~~s\le j\le n.~\leqno(4.4)$$ Let us denote the operators and inner products associated to $\omega_\varepsilon$ with $\varepsilon$ as an index. Then inequality (3.2) combined with (4.4) implies $$\eqalign{ \langle[\ii\Theta(E),\Lambda_\varepsilon]u,u\rangle_\varepsilon &\ge\Big(\big(q-s+1)\big)(1-\varepsilon/\gamma_s)-(n-p)\Big)|u|^2\cr &=\big(p+q-n-s+1-(q-s+1)\varepsilon/\gamma_s\big)|u|^2.\cr}$$ Theorem 4.2 follows now from Cor.~2.3 if we choose $$\varepsilon<{p+q-n-s+1\over q-s+1}~\min_{x\in X}\gamma_s(x).\eqno{\square}$$ \endproof \begstat{(4.5) Remark} \rm The following example due to (Ramanujam 1972, 1974) shows that Girbau's result is no longer true for $p0$ (that is, $s=1$) is due to (Kodaira 1953) and (Serre 1956). \begstat{(5.1) Theorem} Let $F$ be a holomorphic vector bundle over a compact complex manifold $X$, $s$ a positive integer and $E$ a hermitian line bundle such that $\ii\Theta(E)$ has at least $n-s+1$ positive eigenvalues at every point $x\in X$. Then there exists an integer $k_0\ge 0$ such that $$H^q(X,E^k\otimes F)=0~~~~\hbox{\rm for}~~q\ge s~~\hbox{\rm and}~~k\ge k_0.$$ \endstat \begproof{} The main idea is to construct a hermitian metric $\omega_\varepsilon$ on $X$ in such a way that all negative eigenvalues of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ will be of small absolute value. Let $\omega$ denote a fixed hermitian metric on $X$ and let $\gamma_1\le\ldots\le\gamma_n$ be the corresponding eigenvalues of $\ii\Theta(E)$. \begstat{(5.2) Lemma} Let $\psi\in C^\infty(\bR,\bR)$. If $A$ is a hermitian $n\times n$ matrix with eigenvalues $\lambda_1\le\ldots\le\lambda_n$ and corresponding eigenvectors $v_1,\ldots, v_n$, we define $\psi[A]$ as the hermitian matrix with eigenvalues $\psi(\lambda_j)$ and eigenvectors $v_j$, $1\le j\le n$. Then the map $A\longmapsto\psi[A]$ is $C^\infty$ on $\Herm(\bC^n)$. \endstat \begproof{} Although the result is very well known, we give here a short proof. Without loss of generality, we may assume that $\psi$ is compactly supported. Then we have $$\psi[A]={1\over 2\pi}\int_{-\infty}^{+\infty}\wh\psi(t)e^{itA}dt$$ where $\wh\psi$ is the rapidly decreasing Fourier transform of $\psi$. The equality\break $\int_0^t(t-u)^pu^q\,du=p!\,q!/(p+q+1)!~$ and obvious power series developments yield $$D_A(e^{itA})\cdot B=i\int_0^t e^{\ii(t-u)A}\,B\,e^{iuA}du.$$ Since $e^{iuA}$ is unitary, we get $\|D_A(e^{itA})\|\le|t|$. A differentiation under the integral sign and Leibniz' formula imply by induction on $k$ the bound \hbox{$\|D_A^k(e^{itA})\|\le|t|^k$.} Hence $A\longmapsto\psi[A]$ is smooth.\qed \endproof Let us consider now the positive numbers $$t_0=\inf_X \gamma_s>0,~~~~M=\sup_X\max_j|\gamma_j|>0.$$ We select a function $\psi_\varepsilon\in C^\infty(\bR,\bR)$ such that $$\psi_\varepsilon(t)=t~~\hbox{\rm for}~t\ge t_0, ~~\psi_\varepsilon(t)\ge t~~\hbox{\rm for}~0\le t\le t_0, ~~\psi_\varepsilon(t)=M/\varepsilon~~\hbox{\rm for}~t\le 0.$$ By Lemma~5.2, $\omega_\varepsilon:=\psi_\varepsilon[\ii\Theta(E)]$ is a smooth hermitian metric on $X$. Let us write $$\ii\Theta(E)=\ii\sum_{1\le j\le n}\gamma_j\,\zeta_j\wedge\ol\zeta_j,~~~~ \omega_\varepsilon=\ii\sum_{1\le j\le n}\psi_\varepsilon(\gamma_j)\, \zeta_j\wedge\ol\zeta_j$$ in an orthonormal basis $(\zeta_1,\ldots,\zeta_n)$ of $T^\star X$ for $\omega$. The eigenvalues of $\ii\Theta(E)$ with respect to $\omega_\varepsilon$ are given by $\gamma_{j,\varepsilon}=\gamma_j/\psi_\varepsilon(\gamma_j)$ and the construction of $\psi_\varepsilon$ shows that $-\varepsilon\le \gamma_{j,\varepsilon}\le1$, $1\le j\le n$, and $\gamma_{j,\varepsilon}=1$ for $s\le j\le n$. Now, we have $$H^q(X,E^k\otimes F)\simeq H^{n,q}(X,E^k\otimes G)$$ where $G=F\otimes K^\star_X$. Let $e$, $(g_\lambda)_{1\le\lambda\le r}$ and $(\zeta_j)_{1\le j\le n}$ denote orthonormal frames of $E$, $G$ and $(T^\star X,\omega_\varepsilon)$ respectively. For $$u=\sum_{|J|=q,\lambda}u_{J,\lambda}\,\zeta_1\wedge\ldots\wedge\zeta_n \wedge\ol\zeta_J\otimes e^k\otimes g_\lambda \in\Lambda^{n,q}T^\star X\otimes E^k\otimes G,$$ inequality (3.2) yields $$\langle[\ii\Theta(E),\Lambda_\varepsilon]u,u\rangle_\varepsilon=\sum_{J,\lambda} \Big(\sum_{j\in J}\gamma_{j,\varepsilon}\Big)|u_{J,\lambda}|^2\ge \big(q-s+1-(s-1)\varepsilon\big)|u|^2.$$ Choosing $\varepsilon=1/s$ and $q\ge s$, the right hand side becomes $\ge (1/s)|u|^2$. Since $\Theta(E^k\otimes G)=k\Theta(E)\otimes\Id_G+\Theta(G)$, there exists an integer $k_0$ such that $$\big[\ii\Theta(E^k\otimes G),\Lambda_\varepsilon\big]+T_{\omega_\varepsilon}~~~~ \hbox{\rm acting~on~~}\Lambda^{n,q}T^\star X\otimes E^k\otimes G$$ is positive definite for $q\ge s$ and $k\ge k_0$. The proof is complete.\qed \endproof \titleb{6.}{Positivity Concepts for Vector Bundles} Let $E$ be a hermitian holomorphic vector bundle of rank $r$ over $X$, where $\dim_{\bC}X = n$. Denote by $(e_1,\ldots,e_r)$ an orthonormal frame of $E$ over a coordinate patch $\Omega \subset X$ with complex coordinates $(z_1,\ldots,z_n)$, and $$\ii\Theta(E) = \ii\sum_{1\le j,k\le n,~1\le \lambda,\mu\le r} c_{jk\lambda\mu}\,dz_j \wedge d\ol z_k \otimes e^\star_\lambda \otimes e_\mu, ~~~~\ol c_{jk\lambda\mu} = c_{kj\mu\lambda} \leqno (6.1)$$ the Chern curvature tensor. To $\ii\Theta(E)$ corresponds a natural hermitian form $\theta_E$ on $TX\otimes E$ defined by $$\theta_E = \sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}(dz_j \otimes e^\star_\lambda) \otimes (\ol{dz_k \otimes e^\star_\mu}),$$ and such that $$\theta_E(u,u) = \sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}(x)\,u_{j\lambda} \ol u_{k\mu},~~~~u\in T_xX\otimes E_x.(6.2)$$ \begstat{(6.3) Definition (Nakano 1955)} $E$ is said to be Nakano positive $($resp. Nakano semi-negative$)$ if $\theta_E$ is positive definite $($resp. semi-negative$)$ as a hermitian form on $TX\otimes E$, i.e.\ if for every $u\in TX\otimes E,~u\ne 0,$ we have $$\theta_E(u,u) > 0\quad(\hbox{\it resp.}\le 0).$$ We write $>_\Nak$~~$($resp.~$\le_\Nak)$ for Nakano positivity $($resp. semi-negativity$)$. \endstat \begstat{(6.4) Definition (Griffiths 1969)} $E$ is said to be Griffiths positive $($resp. Griffiths semi-negative$)$ if for all $\xi \in T_xX$, $\xi\ne 0$ and $s\in E_x$, $s\ne 0$ we have $$\theta_E(\xi\otimes s,\xi\otimes s) > 0\quad (\hbox{\it resp.}\le 0).$$ We write $>_\Grif$ $($resp. $\le_\Grif)$ for Griffiths positivity $($resp. semi-negativity$)$. \endstat It is clear that Nakano positivity implies Griffiths positivity and that both concepts coincide if $r=1$ (in the case of a line bundle, $E$ is merely said to be positive). One can generalize further by introducing additional concepts of positivity which interpolate between Griffiths positivity and Nakano positivity. \begstat{(6.5) Definition} Let $T$ and $E$ be complex vector spaces of dimensions $n,r$ respectively, and let $\Theta$ be a hermitian form on $T\otimes E$. \smallskip \item{\rm a)} A tensor $u\in T\otimes E$ is said to be of rank $m$ if $m$ is the smallest $\ge 0$ integer such that $u$ can be written $$u = \sum_{j=1}^m \xi_j \otimes s_j,~~~~\xi_j\in T,~s_j\in E.$$ \smallskip \item{\rm b)} $\Theta$ is said to be $m$-positive $($resp. $m$-semi-negative$)$ if $\Theta(u,u)>0$ $($resp. $\Theta(u,u)\le 0)$ for every tensor $u\in T\otimes E$ of rank $\le m$, $u\ne 0$. In this case, we write $$\Theta >_m 0~~~~(\hbox{\it resp.}~~\Theta \le_m 0).$$\smallskip \endstat We say that the bundle $E$ is $m$-positive if $\theta_E >_m 0$. Griffiths positivity corresponds to $m=1$ and Nakano positivity to $m\ge \min(n,r)$. \begstat{(6.6) Proposition} A bundle $E$ is Griffiths positive if and only if $E^\star$ is Griffiths negative. \endstat \begproof{} By (V-4.$3'$) we get $\ii\Theta(E^\star) = -\ii\Theta(E)^\dagger$, hence $$\theta_{E^\star}(\xi_1\otimes\ol s_2,\xi_2\otimes\ol s_1)= -\theta_E(\xi_1\otimes s_1,\xi_2\otimes s_2),~~~~\forall\xi_1,\xi_2\in TX,~ \forall s_1,s_2\in E,$$ where $\ol s_j=\langle\bu,s_j\rangle\in E^\star$. Proposition 6.6 follows immediately.\qed \endproof It should be observed that the corresponding duality property for Nakano positive bundles is {\it not true}. In fact, using (6.1) we get $$\ii\Theta(E^\star) = -\ii\sum_{j,k,\lambda,\mu} c_{jk\mu\lambda} dz_j\wedge d\ol z_k \otimes e^{\star\star}_\lambda \otimes e^\star_\mu,$$ $$\theta_{E^\star}(v,v)=-\sum_{j,k,\mu,\lambda}c_{jk\mu\lambda}v_{j\lambda} \ol v_{k\mu},\leqno(6.7)$$ for any $v=\sum v_{j\lambda}\,(\partial/\partial z_j)\otimes e^\star_\lambda \in TX\otimes E^\star$. The following example shows that Nakano positivity or negativity of $\theta_E$ and $\theta_{E^\star}$ are unrelated. \begstat{(6.8) Example} \rm Let $H$ be the rank $n$ bundle over $\bP^n$ defined in \S~V-15. For any $u=\sum u_{j\lambda}(\partial/\partial z_j)\otimes \wt e_\lambda\in TX\otimes H$, $v=\sum v_{j\lambda}(\partial/\partial z_j) \otimes\wt e_\lambda^\star\in TX\otimes H^\star$, $1\le j,\lambda\le n$, formula (V-15.9) implies $$\left\{ \eqalign{ \theta_H(u,u)&=\sum u_{j\lambda}\ol u_{\lambda j}\cr \theta_{H^\star}(v,v)&=\sum v_{jj}\ol v_{\lambda\lambda}= \big|\sum v_{jj}\big|^2.\cr}\right.\leqno(6.9)$$ It is then clear that $H\ge_\Grif 0$ and $H^\star\le_\Nak 0$ , but $H$ is neither $\ge_\Nak0$ nor $\le_\Nak0$. \endstat \begstat{(6.10) Proposition} Let $0\to S \to E \to Q \to 0$ be an exact sequence of hermitian vector bundles. Then \medskip \item{\rm a)} $E\ge_\Grif 0~~\Longrightarrow~~Q\ge_\Grif 0,$ \smallskip \item{\rm b)} $E\le_\Grif 0~~\Longrightarrow~~S\le_\Grif 0,$ \smallskip \item{\rm c)} $E\le_\Nak0~~\Longrightarrow~~S\le_\Nak 0,$ \smallskip \noindent and analogous implications hold true for strict positivity. \endstat \begproof{} If $\beta$ is written $\sum dz_j \otimes \beta_j$,~$\beta_j \in \hom(S,Q)$, then formulas (V-14.6) and (V-14.7) yield $$\eqalign{ \ii\Theta(S) &= \ii\Theta(E)_{\restriction S} - \sum dz_j \wedge d\ol z_k \otimes \beta^\star_k \beta_j,\cr \ii\Theta(Q) &= \ii\Theta(E)_{\restriction Q} + \sum dz_j \wedge d\ol z_k \otimes \beta_j\beta^\star_k.\cr}$$ Since $\beta\cdot(\xi\otimes s)= \sum \xi_j\beta_j\cdot s$ and $\beta^\star \cdot(\xi\otimes s)= \sum \ol \xi_k\beta^\star_k\cdot s$ we get $$\theta_S(\xi\otimes s,\xi'\otimes s') = \theta_E(\xi\otimes s,\xi'\otimes s') - \sum_{j,k} \xi_j \ol \xi'_k \langle \beta_j\cdot s,\beta_k\cdot s'\rangle,$$ $$\theta_S(u,u) = \theta_E(u,u) - |\beta\cdot u|^2,$$ $$\theta_Q(\xi\otimes s,\xi'\otimes s') = \theta_E(\xi\otimes s,\xi'\otimes s') + \sum_{j,k} \xi_j \ol \xi'_k \langle \beta^\star_k\cdot s,\beta^\star_j\cdot s'\rangle,$$ $$\theta_Q(\xi\otimes s,\xi\otimes s) = \theta_E(\xi\otimes s,\xi\otimes s) +|\beta^\star\cdot(\xi\otimes s)|^2.\eqno{\square}$$ \endproof Since $H$ is a quotient bundle of the trivial bundle $\soul V$, Example~6.8 shows that $E\ge_\Nak 0$ {\it does not imply} $Q\ge_\Nak 0$. \titleb{7.}{Nakano Vanishing Theorem} Let $(X,\omega)$ be a compact K\"ahler manifold, $\dim_\bC X=n$, and $E\longrightarrow X$ a hermitian vector bundle of rank $r$. We are going to compute explicitly the hermitian operator $[\ii\Theta(E),\Lambda]$ acting on $\Lambda^{p,q}T^\star X \otimes E$. Let $x_0\in X$ and $(z_1,\ldots,z_n)$ be local coordinates such that $(\partial/\partial z_1,\ldots,\partial/\partial z_n)$ is an orthonormal basis of $(TX,\omega)$ at $x_0$. One can write $$\eqalign{ \omega_{x_0}&=\ii\sum_{1\le j\le n}dz_j\wedge d\ol z_j,\cr \ii\Theta(E)_{x_0}&=\ii\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}\,dz_j\wedge d\ol z_k \otimes e_\lambda^\star\otimes e_\mu\cr}$$ where $(e_1,\ldots,e_r)$ is an orthonormal basis of $E_{x_0}$. Let $$u=\sum_{|J|=p,\,|K|=q,\,\lambda}u_{J,K,\lambda}\,dz_J\wedge d\ol z_K \otimes e_\lambda\in\big(\Lambda^{p,q}T^\star X\otimes E\big)_{x_0}.$$ A simple computation as in the proof of Prop.~VI-8.3 gives $$\eqalign{ \Lambda u&=\ii(-1)^p\sum_{J,K,\lambda,s}u_{J,K,\lambda}\, \Big({\partial\over\partial z_s}\ort dz_J\Big)\wedge \Big({\partial\over\partial\ol z_s}\ort d\ol z_K\Big)\otimes e_\lambda,\cr \ii\Theta(E)\wedge u&=\ii(-1)^p\sum_{j,k,\lambda,\mu,J,K}c_{jk\lambda\mu}\, u_{J,K,\lambda}\,dz_j\wedge dz_J\wedge d\ol z_k\wedge d\ol z_K\otimes e_\mu,\cr [\ii\Theta(E),\Lambda]u&=\sum_{j,k,\lambda,\mu,J,K} c_{jk\lambda\mu}\,u_{J,K,\lambda}\,dz_j\wedge\Big({\partial\over\partial z_k} \ort dz_J\Big)\wedge d\ol z_K\otimes e_\mu\cr &+\sum_{j,k,\lambda,\mu,J,K}c_{jk\lambda\mu}\,u_{J,K,\lambda}\,dz_J\wedge d\ol z_k\wedge\Big({\partial\over\partial\ol z_j}\ort d\ol z_K\Big)\otimes e_\mu\cr &-\sum_{j,\lambda,\mu,J,K}c_{jj\lambda\mu}\,u_{J,K,\lambda}\,dz_J\wedge d\ol z_K \otimes e_\mu.\cr}$$ We extend the definition of $u_{J,K,\lambda}$ to non increasing multi-indices $J=(j_s)$, $K=(k_s)$ by deciding that $u_{J,K,\lambda}=0$ if $J$ or $K$ contains identical components repeated and that $u_{J,K,\lambda}$ is alternate in the indices $(j_s)$, $(k_s)$. Then the above equality can be written $$\eqalign{ \langle[\ii\Theta(E),\Lambda]u,u\rangle&= \sum c_{jk\lambda\mu}\,u_{J,jS,\lambda}{\ol u_{J,kS,\mu}}\cr &+\sum c_{jk\lambda\mu}\,u_{kR,K,\lambda}{\ol u_{jR,K,\mu}}\cr &-\sum c_{jj\lambda\mu}\,u_{J,K,\lambda}{\ol u_{J,K,\mu}},\cr}$$ extended over all indices $j,k,\lambda,\mu,J,K,R,S$ with $|R|=p-1$, $|S|=q-1$. This hermitian form appears rather difficult to handle for general $(p,q)$ because of sign compensation. Two interesting cases are $p=n$ and $q=n$. \medskip \noindent$\bu$ For $u=\sum u_{K,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge d\ol z_K\otimes e_\lambda$ of type $(n,q)$, we get $$\langle[\ii\Theta(E),\Lambda]u,u\rangle=\sum_{|S|=q-1}\sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}\,u_{jS,\lambda}\ol u_{kS,\mu},\leqno(7.1)$$ because of the equality of the second and third summations in the general formula. Since $u_{jS,\lambda}=0$ for $j\in S$, the rank of the tensor $(u_{jS,\lambda})_{j,\lambda}\in\bC^n\otimes\bC^r$ is in fact $\le\min\{n-q+1,r\}$. We obtain therefore: \begstat{(7.2) Lemma} Assume that $E>_m0$ in the sense of Def.~$6.5$. Then the hermitian operator $[\ii\Theta(E),\Lambda]$ is positive definite on $\Lambda^{n,q}T^\star X\otimes E$ for $q\ge 1$ and $m\ge\min\{n-q+1,r\}.$ \endstat \begstat{(7.3) Theorem} Let $X$ be a compact connected K\"ahler manifold of dimension $n$ and $E$ a hermitian vector bundle of rank $r$. If $\theta_E\ge_m 0$ on $X$ and $\theta_E>_m0$ in at least one point, then $$H^{n,q}(X,E)=H^q(X,K_X\otimes E)=0~~~~\hbox{\rm for}~~q\ge 1~~\hbox{\rm and}~~ m\ge\min\{n-q+1,r\}.$$ \endstat \vskip-6pt\noindent$\bu$ Similarly, for $u=\sum u_{J,\lambda}\,dz_J\wedge d\ol z_1\wedge\ldots\wedge d\ol z_n\otimes e_\lambda$ of type $(p,n)$, we get $$\langle[\ii\Theta(E),\Lambda]u,u\rangle=\sum_{|R|=p-1}\sum_{j,k,\lambda,\mu} c_{jk\lambda\mu}\,u_{kR,\lambda}\ol u_{jR,\mu},$$ because of the equality of the first and third summations in the general formula. The indices $j,k$ are twisted, thus $[\ii\Theta(E),\Lambda]$ defines a positive hermitian form under the assumption $\ii\Theta(E)^\dagger>_m 0$, i.e.\ $\ii\Theta(E^\star)<_m 0$, with \hbox{$m\ge\min\{n-p+1,r\}$.} Serre duality $\big(H^{p,0}(X,E)\big)^\star=H^{n-p,n}(X,E^\star)$~gives: \begstat{(7.4) Theorem} Let $X$ and $E$ be as above. If $\theta_E\le_m 0$ on $X$ and $\theta_E<_m 0$ in at least one point, then $$H^{p,0}(X,E)=H^0(X,\Omega^p_X\otimes E)=0~~~~\hbox{\rm for}~~p_\Grif 0~~\Longrightarrow~~E\otimes \det E >_\Nak 0.$$ \endstat To prove this result, we first use (V-4.$2'$) and (V-4.6). If $\End(E\otimes\det E)$ is identified to $\hom(E,E)$, one can write $$\Theta(E\otimes \det E) = \Theta(E)+\Tr_E(\Theta(E)) \otimes \Id_E,$$ $$\theta_{E\otimes \det E} = \theta_E+\Tr_E\theta_E \otimes h,$$ where $h$ denotes the hermitian metric on $E$ and where $\Tr_E\theta_E$ is the hermitian form on $TX$ defined by $$\Tr_E \theta_E(\xi,\xi) = \sum_{1\le \lambda\le r} \theta_E(\xi\otimes e_\lambda,\xi\otimes e_\lambda),~\xi\in TX,$$ for any orthonormal frame $(e_1,\ldots,e_r)$ of $E$. Theorem 8.1 is now a consequence of the following simple property of hermitian forms on a tensor product of complex vector spaces. \begstat{(8.2) Proposition} Let $T,E$ be complex vector spaces of respective dimensions $n,r,\,$ and $h$ a hermitian metric on $E$. Then for every hermitian form $\Theta$ on $T\otimes E$ $$\Theta >_\Grif 0~~\Longrightarrow~~ \Theta + \Tr_E\Theta \otimes h >_\Nak 0.$$ \endstat We first need a lemma analogous to Fourier inversion formula for discrete Fourier transforms. \begstat{(8.3) Lemma} Let $q$ be an integer $\ge 3$, and $x_\lambda,~y_\mu,~1\le \lambda,\mu \le r$, be complex numbers. Let $\sigma$ describe the set $U^r_q$ of $r$-tuples of $q$-th roots of unity and put $$x'_\sigma = \sum_{1\le \lambda\le r} x_\lambda \ol \sigma_\lambda,~~~~ y'_\sigma = \sum_{1\le \mu\le r} y_\mu \ol \sigma_\mu,~~~~ \sigma\in U^r_q.$$ Then for every pair $(\alpha,\beta),~1\le \alpha,\beta\le r$, the following identity holds: $$q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ol y'_\sigma \sigma_\alpha \ol \sigma_\beta=\cases{ x_\alpha \ol y_\beta&if~~$\alpha\ne \beta,$\cr \displaystyle\sum_{1\le \mu\le r} x_\mu \ol y_\mu&if~~$\alpha=\beta.$\cr}$$ \endstat \begproof{} The coefficient of $x_\lambda\ol y_\mu$ in the summation $q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ol y'_\sigma \sigma_\alpha \ol \sigma_\beta$ is given by $$q^{-r} \sum_{\sigma\in U^r_q}\sigma_\alpha \ol \sigma_\beta\ol \sigma_\lambda \sigma_\mu.$$ This coefficient equals 1 when the pairs $\{ \alpha,\mu\}$ and $\{\beta,\lambda\}$ are equal (in which case $\sigma_\alpha \ol \sigma_\beta \ol \sigma_\lambda \sigma_\mu = 1$ for any one of the $q^r$ elements of $U^r_q$). Hence, it is sufficient to prove that $$\sum_{\sigma\in U^r_q} \sigma_\alpha \ol \sigma_\beta \ol \sigma_\lambda \sigma_\mu = 0$$ when the pairs $\{ \alpha,\mu\}$ and $\{ \beta,\lambda\}$ are distinct. If $\{ \alpha,\mu\} \ne \{\beta,\lambda\}$, then one of the elements of one of the pairs does not belong to the other pair. As the four indices $\alpha,\beta,\lambda,\mu$ play the same role, we may suppose for example that $\alpha\notin \{\beta,\lambda\}$. Let us apply to $\sigma$ the substitution $\sigma\mapsto \tau$, where $\tau$ is defined by $$\tau_\alpha = e^{2\pi\ii/q}\sigma_\alpha,~\tau_\nu = \sigma_\nu \quad \hbox{\rm for}\quad \nu\ne \alpha.$$ We get $$\sum_\sigma\sigma_\alpha\ol\sigma_\beta\ol\sigma_\lambda\sigma_\mu=\sum_\tau =\cases{ \displaystyle e^{2\pi\ii/q}\sum_\sigma&if~~$\alpha\ne\mu,$\cr \displaystyle e^{4\pi\ii/q}\sum_\sigma&if~~$\alpha=\mu,$\cr}$$ Since $q\ge 3$ by hypothesis, it follows that $$\sum_\sigma\sigma_\alpha\ol\sigma_\beta\ol\sigma_\lambda\sigma_\mu=0.$$ \endproof \begproof{of Proposition 8.2.} Let $(t_j)_{1\le j\le n}$ be a basis of $T$, $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis of $E$ and $\xi=\sum_j\xi_jt_j\in T$, $u=\sum_{j,\lambda}u_{j\lambda}\,t_j\otimes e_\lambda \in T\otimes E$. The coefficients $c_{jk\lambda\mu}$ of $\Theta$ with respect to the basis $t_j\otimes e_\lambda$ satisfy the symmetry relation $\ol c_{jk\lambda\mu}=c_{kj\mu\lambda}$, and we have the formulas $$\eqalign{\Theta(u,u) &=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}u_{j\lambda}\ol u_{k\mu},\cr \Tr_E\Theta(\xi,\xi)&=\sum_{j,k,\lambda}c_{jk\lambda\lambda}\xi_j\ol\xi_k,\cr (\Theta+\Tr_E\Theta\otimes h)(u,u)&=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu} u_{j\lambda} \ol u_{k\mu}+ c_{jk\lambda\lambda}u_{j\mu}\ol u_{k\mu}.\cr}$$ For every $\sigma\in U^r_q$ (cf.\ Lemma~8.3), put $$\eqalign{ u'_{j\sigma}&=\sum_{1\le\lambda\le r}u_{j\lambda}\ol\sigma_\lambda\in\bC,\cr \wh u_\sigma&=\sum_j u'_{j\sigma}t_j\in T\quad,\quad \wh e_\sigma=\sum_\lambda\sigma_\lambda e_\lambda\in E.\cr}$$ Lemma 8.3 implies $$\eqalign{ q^{-r} \sum_{\sigma\in U^r_q}\Theta(\wh u_\sigma\otimes\wh e_\sigma, \wh u_\sigma\otimes\wh e_\sigma)&= q^{-r} \sum_{\sigma\in U^r_q} c_{jk\lambda\mu} u'_{j\sigma} \ol u'_{k\sigma}\sigma_\lambda \ol \sigma_\mu\cr &= \sum_{j,k,\lambda\ne \mu} c_{jk\lambda\mu} u_{j\lambda} \ol u_{k\mu} + \sum_{j,k,\lambda,\mu} c_{jk\lambda\lambda} u_{j\mu} \ol u_{k\mu}.\cr}$$ The Griffiths positivity assumption shows that the left hand side is $\ge 0$, hence $$(\Theta + \Tr_E \Theta \otimes h)(u,u) \ge \sum_{j,k,\lambda} c_{jk\lambda\lambda} u_{j\lambda} \ol u_{k\lambda} \ge 0$$ with strict positivity if $\Theta >_\Grif 0$ and $u\ne 0$.\qed \endproof \begstat{(8.4) Example} \rm Take $E=H$ over $\bP^n=P(V)$. The exact sequence $$0 \longrightarrow \cO(-1) \longrightarrow \soul V \longrightarrow H \longrightarrow 0$$ implies $\det \soul V=\det H \otimes \cO(-1)$. Since $\det \soul V$ is a trivial bundle, we get (non canonical) isomorphisms $$\eqalign{ \det H &\simeq \cO(1),\cr T\bP^n &= H\otimes \cO(1) \simeq H\otimes \det H.\cr}$$ We already know that $H\ge_\Grif 0$, hence $T\bP^n \ge_\Nak 0$. A direct computation based on (6.9) shows that $$\eqalign{ \theta_{T\bP^n}(u,u) &= (\theta_H + \Tr_H \theta_H \otimes h)(u,u)\cr &= \sum_{1\le j,k\le n} u_{jk} \ol u_{kj} + u_{jk} \ol u_{jk} = {1\over 2} \sum_{1\le j,k\le n} |u_{jk} + u_{kj}|^2.\cr}$$ In addition, we have $T\bP^n>_\Grif 0$. However, the Serre duality theorem gives $$\eqalign{ H^q(\bP^n,K_{\bP^n}\otimes T\bP^n)^\star &\simeq H^{n-q}(\bP^n,T^\star\bP^n)\cr &=H^{1,n-q}(\bP^n,\bC)=\cases{ \bC&if~~$q=n-1$,\cr 0 &if~~$q\ne n-1$.\cr}\cr}$$ For $n\ge 2$, Th.~7.3 implies that $T\bP^n$ has no hermitian metric such that $\theta_{T\bP^n}\ge_20$ on $\bP^n$ and $\theta_{T\bP^n}>_20$ in one point. This shows that the notion of $2$-positivity is actually stronger than $1$-positivity (i.e.\ Griffiths positivity). \endstat \begstat{(8.5) Remark} \rm Since $\Tr_H\theta_H = \theta_{\cO(1)}$ is positive and $\theta_{T\bP^n}$ is not $>_\Nak 0$ when $n\ge 2$, we see that Prop.~8.2 is best possible in the sense that there cannot exist any constant $c<1$ such that $$\Theta>_\Grif0~~~\Longrightarrow~~~\Theta + c \Tr_E \Theta\otimes h \ge_\Nak 0.$$ \endstat \titleb{9.}{Applications to Griffiths Positive Bundles} We first need a preliminary result. \begstat{(9.1) Proposition} Let $T$ be a complex vector space and $(E,h)$ a hermitian vector space of respective dimensions $n,r$ with $r\ge 2$. Then for any hermitian form $\Theta$ on $T\otimes E$ and any integer $m\ge 1$ $$\Theta>_\Grif0~~~\Longrightarrow~~~m\Tr_E\Theta\otimes h-\Theta>_m0.$$ \endstat \begproof{} Let us distinguish two cases. \medskip \noindent a)~ $m=1$.~ Let $u\in T \otimes E$ be a tensor of rank 1. Then $u$ can be written $u = \xi_1 \otimes e_1$ with $\xi_1\in T,~\xi_1 \ne 0$, and $e_1 \in E,~|e_1| = 1$. Complete $e_1$ into an orthonormal basis $(e_1,\ldots,e_r)$ of $E$. One gets immediately $$\eqalign{ (\Tr_E \Theta \otimes h)(u,u) &= \Tr_E \Theta (\xi_1,\xi_1) = \sum_{1\le \lambda\le r} \Theta(\xi_1\otimes e_\lambda , \xi_1 \otimes e_\lambda)\cr &> \Theta (\xi_1\otimes e_1,\xi_1\otimes e_1) = \Theta (u,u).\cr}$$ \medskip \noindent b)~ $m\ge 2$.~ Every tensor $u\in T\otimes E$ of $\hbox{\rm rank~} \le m$ can be written $$u=\sum_{1\le \lambda\le q} \xi_\lambda \otimes e_\lambda\quad, \quad \xi_\lambda \in T,$$ with $q = \min(m,r)$ and $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis of $E$. Let $F$ be the vector subspace of $E$ generated by $(e_1,\ldots,e_q)$ and $\Theta_F$ the restriction of $\Theta$ to $T\otimes F$. The first part shows that $$\Theta':= \Tr_F \Theta_F \otimes h - \Theta_F >_\Grif 0.$$ Proposition 9.2 applied to $\Theta'$ on $T\otimes F$ yields $$\Theta'+\Tr_F \Theta'\otimes h=q\Tr_F\Theta_F\otimes h-\Theta_F>_q 0.$$ Since $u\in T\otimes F$ is of $\hbox{\rm rank~} \le q \le m$, we get (for $u\ne 0$) $$\eqalignno{ \Theta(u,u) = \Theta_F(u,u) &< q(\Tr_F \Theta_F \otimes h)(u,u)\cr &=q\sum_{1\le j,\lambda\le q}\Theta(\xi_j\otimes e_\lambda, \xi_j\otimes e_\lambda)\le m\Tr_E\Theta\otimes h(u,u).&\square\cr}$$ \endproof Proposition 9.1 is of course also true in the semi-positive case. From these facts, we deduce \begstat{(9.2) Theorem} Let $E$ be a Griffiths $($semi-$)$positive bundle of rank $r\ge 2$. Then for any integer $m\ge 1$ $$E^\star \otimes (\det E)^m >_m 0~~~~(\hbox{\it resp.}~~\ge_m 0).$$ \endstat \begproof{} Apply Prop.~8.1 to $\Theta = -\theta_{E^\star} >_\Grif 0$ and observe that $$\theta_{\det E} = - \theta_{\det E^\star} = \Tr_{E^\star} \Theta.$$ \endproof \begstat{(9.3) Theorem} Let $0\to S \to E \to Q \to 0$ be an exact sequence of hermitian vector bundles. Then for any $m\ge 1$ $$E>_m0~~~\Longrightarrow~~~S\otimes (\det Q)^m >_m 0.$$ \endstat \begproof{} Formulas (V-14.6) and (V-14.7) imply $$\ii\Theta(S) >_m \ii\beta^\star \wedge \beta\quad,\quad \ii\Theta(Q) >_m \ii\beta \wedge \beta^\star,$$ $$\ii\Theta(\det Q) = \Tr_Q(\ii\Theta(Q)) > \Tr_Q(\ii\beta \wedge \beta^\star).$$ If we write $\beta = \sum dz_j \otimes \beta_j$ as in the proof of Prop.~6.10, then $$\eqalign{ \Tr_Q(\ii\beta \wedge \beta^\star) &= \sum idz_j \wedge d\ol z_k \Tr_Q(\beta_j\beta^\star_k)\cr &=\sum idz_j \wedge d\ol z_k \Tr_S(\beta^\star_k\beta_j) = \Tr_S(-\ii\beta^\star \wedge \beta).\cr}$$ Furthermore, it has been already proved that $-\ii\beta^\star \wedge \beta \ge_\Nak 0$. By Prop.~8.1 applied to the corresponding hermitian form $\Theta$ on $TX \otimes S$, we get $$m \Tr_S(-\ii\beta^\star \wedge \beta) \otimes \Id_S + \ii\beta^\star\wedge \beta \ge_m 0,$$ and Th.~9.3 follows. \endproof \begstat{(9.4) Corollary} Let $X$ be a compact $n$-dimensional complex manifold, $E$ a vector bundle of rank $r\ge 2$ and $m\ge 1$ an integer. Then \smallskip \item{\rm a)} $E>_\Grif0 \Longrightarrow H^{n,q}(X,E\otimes \det\,E)=0$~~~for~~$q\ge 1\,;$ \smallskip \item{\rm b)} $E>_\Grif0\Longrightarrow H^{n,q}\big (X,E^\star\otimes(\det\,E)^m\big)=0$~~~for~~$q\ge 1$\hfill\break and~~$m\ge\min\{n-q+1,r\}\,;$ \smallskip \item{\rm c)} Let~ $0\to S\to E\to Q\to 0$ be an exact sequence of vector bundles and $m=\min\{n-q+1,\rk\,S\}$, $q\ge 1$. If $E>_m0$ and if $L$ is a line bundle such that $L\otimes(\det Q)^{-m}\ge 0$, then $$H^{n,q}(X,S\otimes L)=0.$$ \endstat \begproof{} Immediate consequence of Theorems 7.3, 8.1, 9.2 and 9.3.\qed \endproof Note that under our hypotheses $\omega=\ii\,\Tr_E\,\Theta(E)= \ii\Theta(\Lambda^rE)$ is a K\"ahler metric on $X$. Corollary 2.5 shows that it is enough in a), b), c) to assume semi-positivity and strict positivity in one point (this is true a priori only if $X$ is supposed in addition to be K\"ahler, but this hypothesis can be removed by means of Siu's result mentioned after~(4.5). a) is in fact a special case of a result of (Griffiths 1969), which we will prove in full generality in volume II (see the chapter on vanishing theorems for ample vector bundles); property b) will be also considerably strengthened there. Property c) is due to (Skoda 1978) for $q=0$ and to (Demailly 1982c) in general. Let us take the tensor product of the exact sequence in c) with $(\det\,Q)^l$. The corresponding long cohomology exact sequence implies that the natural morphism $$H^{n,q}\big(X,E\otimes(\det\,Q)^l\big)\longrightarrow H^{n,q}\big(X,Q\otimes(\det\,Q)^l\big)$$ is surjective for $q\ge 0$ and $l,m\ge\min\{n-q,\rk\,S\}$, bijective for $q\ge 1$ and $l,m\ge\min\{n-q+1,\rk\,S\}$. \titleb{10.}{Cohomology Groups of $\cO(k)$ over $\bP^n$} As an illustration of the above results, we compute now the cohomology groups of all line bundles $\cO(k)\to\bP^n$. This precise evaluation will be needed in the proof of a general vanishing theorem for vector bundles, due to Le Potier (see volume II). As in \S V-15, we consider a complex vector space $V$ of dimension $n+1$ and the exact sequence $$0\longrightarrow\cO(-1)\longrightarrow\soul V\longrightarrow H\longrightarrow 0\leqno(10.1)$$ of vector bundles over $\bP^n=P(V)$. We thus have \hbox{ $\det\soul V=\det H\otimes\cO(-1)$}, and as $TP(V)=H\otimes\cO(1)$ by Th.~V-15.7, we find $$K_{P(V)}=\det T^\star P(V)=\det H^\star\otimes\cO(-n)=\det\soul V^\star \otimes\cO(-n-1)\leqno(10.2)$$ where~ $\det\soul V$ is a trivial line bundle. Before going further, we need some notations. For every integer $k\in\bN$, we consider the homological complex $C^{\bu,k} (V^\star)$ with differential $\gamma$ such that $$\left\{\eqalign{ C^{p,k}(V^\star) =&~\Lambda^pV^\star\otimes S^{k-p}V^\star,~~~~0\le p\le k,\cr =&~0~~~~~\hbox{\rm otherwise},\cr \gamma~:~~\Lambda^pV^\star&\otimes S^{k-p}V^\star\longrightarrow\Lambda^{p-1}V^\star \otimes S^{k-p+1}V^\star,\cr}\right.\leqno(10.3)$$ where $\gamma$ is the linear map obtained by contraction with the Euler vector field $\Id_V\in V\otimes V^\star$, through the obvious maps $V\otimes\Lambda^pV^\star\longrightarrow\Lambda^{p-1}V^\star$ and $V^\star\otimes S^{k-p}V^\star\longrightarrow S^{k-p+1}V^\star$. If $(z_0,\ldots,z_n)$ are coordinates on $V$, the module $C^{p,k}(V^\star)$ can be identified with the space of $p$-forms $$\alpha(z)=\sum_{|I|=p}\alpha_I(z)\,dz_I$$ where the $\alpha_I$ 's are homogeneous polynomials of degree $k-p$. The differential $\gamma$ is given by contraction with the Euler vector field $\xi=\sum_{0\le j\le n}z_j\,\partial/\partial z_j$. Let us denote by $Z^{p,k}(V^\star)$ the space of $p$-cycles of $C^{\bu,k}(V^\star)$, i.e.\ the space of forms $\alpha\in C^{p,k}(V^\star)$ such that $\xi\ort\alpha=0$. The exterior derivative $d$ also acts on $C^{\bu,k}(V^\star)\,;$ we have $$d~:~~C^{p,k}(V^\star)\longrightarrow C^{p+1,k}(V^\star),$$ and a trivial computation shows that $d\gamma+\gamma d=k\cdot\Id_ {C^{\bu,k}(V^\star)}.$ \begstat{(10.4) Theorem} For $k\ne 0$, $C^{\bu,k}(V^\star)$ is exact and there exist canonical isomorphisms $$C^{\bu,k}(V^\star)=\Lambda^pV^\star\otimes S^{k-p}V^\star\simeq Z^{p,k}(V^\star)\oplus Z^{p-1,k}(V^\star).$$ \endstat \begproof{} The identity $d\gamma+\gamma d=k\cdot\Id$ implies the exactness. The isomorphism is given by ${1\over k}\gamma d\oplus\gamma$ and its inverse by $\cP_1+{1\over k}d\circ\cP_2$.\qed \endproof Let us consider now the canonical mappings $$\pi~:~~V\setminus\{0\}\longrightarrow P(V),~~~~\mu'~:~~V\setminus\{0\}\longrightarrow\cO(-1)$$ defined in \S V-15. As $T_{[z]}P(V)\simeq V/\bC\xi(z)$ for all $z\in V\setminus\{0\}$, every form $\alpha\in Z^{p,k}(V^\star)$ defines a holomorphic section of $\pi^\star\big(\Lambda^pT^\star P(V)\big)$, $\alpha(z)$ being homogeneous of degree $k$ with respect to $z$. Hence $\alpha(z)\otimes\mu'(z)^{-k}$ is a holomorphic section of $\pi^\star\big(\Lambda^pT^\star P(V)\otimes\cO(k)\big)$, and since its homogeneity degree is $0$, it induces a holomorphic section of $\Lambda^pT^\star P(V)\otimes\cO(k)$. We thus have an injective morphism $$Z^{p,k}(V^\star)\longrightarrow H^{p,0}\big(P(V),\cO(k)\big).\leqno(10.5)$$ \begstat{(10.6) Theorem} The groups $H^{p,0}\big(P(V),\cO(k)\big)$ are given by \smallskip \item{\rm a)} $H^{p,0}\big(P(V),\cO(k)\big)\simeq Z^{p,k}(V^\star)$ ~~for~~$k\ge p\ge 0,$ \smallskip \item{\rm b)} $H^{p,0}\big(P(V),\cO(k)\big)=0$ ~~for~~$k\le p$~~and~~$(k,p)\ne(0,0)$. \endstat \begproof{} Let $s$ be a holomorphic section of $\Lambda^pT^\star P(V) \otimes\cO(k)$. Set $$\alpha(z)=(d\pi_z)^\star\big(s([z])\otimes\mu'(z)^k\big),~~~~z\in V\setminus\{0\}.$$ Then $\alpha$ is a holomorphic $p$-form on $V\setminus\{0\}$ such that $\xi\ort\alpha=0$, and the coefficients of $\alpha$ are homogeneous of degree $k-p$ on $V\setminus\{0\}$ (recall that $d\pi_{\lambda z}=\lambda^{-1} d\pi_z$). It follows that $\alpha=0$ if $kp\,;$ \smallskip \item{${\rm \ol c)}$} $H^{p,n}\big(P(V),\cO(k)\big)\simeq Z^{n-p,-k}(V)$ ~~for~~$k<-n+p\,;$ \smallskip \item{${\rm \ol d)}$} $H^{p,p}\big(P(V),\bC\big)=\bC,~~~~0\le p\le n.$ \endstat \begproof{} $\bullet$ ${\rm \ol d)}$ is already known, and so is a) when $k=0$ (Th.~VI-13.3). \medskip \noindent$\bullet$ b) and ${\rm \ol b)}$ follow from Th.~10.6, and c), ${\rm \ol c)}$ are equivalent to b), ${\rm \ol b)}$ via Serre duality: $$H^{p,q}\big(P(V),\cO(k)\big)^\star=H^{n-p,n-q}\big(P(V),\cO(-k)\big),$$ thanks to the canonical isomorphism $\big(Z^{p,k}(V)\big)^\star=Z^{p,k} (V^\star)$. \medskip \noindent$\bullet$ Let us prove now property a) when $k\ne 0$ and property d). By Serre duality, we may assume $k>0$. Then $$\Lambda^pT^\star P(V)\simeq K_{P(V)}\otimes\Lambda^{n-p}TP(V).$$ It is very easy to verify that $E\ge_\Nak0$ implies $\Lambda^sE \ge_\Nak0$ for every integer $s$. Since $TP(V)\ge_\Nak0$, we get therefore $$F=\Lambda^{n-p}TP(V)\otimes\cO(k)>_\Nak0~~~~\hbox{\rm for}~~k>0,$$ and the Nakano vanishing theorem implies $$\eqalignno{ H^{p,q}\big(P(V),\cO(k)\big)&=H^q\big(P(V),\Lambda^pT^\star P(V)\otimes \cO(k)\big)\cr &=H^q\big(P(V),K_{P(V)}\otimes F\big)=0,~~~~q\ge 1.&\square\cr}$$ \endproof \titleb{11.}{Ample Vector Bundles} \titlec{11.A.}{Globally Generated Vector Bundles} All definitions concerning ampleness are purely algebraic and do not involve differential geometry. We shall see however that ampleness is intimately connected with the differential geometric notion of positivity. For a general discussion of properties of ample vector bundles in arbitrary characteristic, we refer to (Hartshorne 1966). \begstat{(11.1) Definition} Let $E\to X$ be a holomorphic vector bundle over an arbitrary complex manifold~$X$. \smallskip \item{\rm a)} $E$ is said to be globally generated if for every $x\in X$ the evaluation map $H^0(X,E) \to E_x$ is onto. \smallskip \item{\rm b)} $E$ is said to be semi-ample if there exists an integer $k_0$ such that $S^kE$ is globally generated for $k\ge k_0$. \smallskip \endstat Any quotient of a trivial vector bundle is globally generated, for example the tautological quotient vector bundle $Q$ over the Grassmannian $G_r(V)$ is globally generated. Conversely, every globally generated vector bundle $E$ of rank $r$ is isomorphic to the quotient of a trivial vector bundle of rank $\le n+r$, as shown by the following result. \begstat{(11.2) Proposition} If a vector bundle $E$ of rank $r$ is globally generated, then there exists a finite dimensional subspace $V\subset H^0(X,E)$, $\dim V\le n+r$, such that $V$ generates all fibers $E_x$, $x\in X$. \endstat \begproof{} Put an arbitrary hermitian metric on $E$ and consider the Fr\'echet space ${\cal F}=\big(H^0(X,E)\big)^{n+r}$ of $(n+r)$-tuples of holomorphic sections of $E$, endowed with the topology of uniform convergence on compact subsets of~$X$. For every compact set $K\subset X$, we set $$A(K)= \{(s_1,\ldots,s_{n+r})\in{\cal F}~{\rm which~do~not~generate~}E~{\rm on}~K\}.$$ It is enough to prove that $A(K)$ is of first category in ${\cal F}\,:$ indeed, Baire's theorem will imply that $A(X)= \bigcup A(K_\nu)$ is also of first category, if $(K_\nu)$ is an exhaustive sequence of compact subsets of $X$. It is clear that $A(K)$ is closed, because $A(K)$ is characterized by the closed condition $$\min_K\sum_{i_1<\cdotsd ~~{\rm if}~~r\ge 2.$$ \endstat \begstat{(11.9) Proposition} If $E$ is very ample of rank $r$, there exists a subspace $V$ of $H^0(X,E)$, $\dim V\le\max\big(nr+n+r,2(n+r)\big)$, such that all the evaluation maps $V\to E_x\oplus E_y$, $x\ne y$, and $V\to (J^1E)_x$, $x\in X$, are surjective. \endstat \begproof{} The arguments are exactly the same as in the proof of Prop.~11.4, if we consider instead the bundles $J^1E\longrightarrow X$ and $E\times E\longrightarrow X\times X\setminus\Delta_X$ of respective ranks $r(n+1)$ and $2r$, and sections $s'_1,\ldots,s'_N\in H^0(X,E)$ generating these bundles.\qed \endproof \begstat{(11.10) Proposition} Let $E\to X$ be a holomorphic vector bundle. \smallskip \item{\rm a)} If $V\subset H^0(X,E)$ generates $J^1E\longrightarrow X$ and $E\times E\longrightarrow X\times X\setminus\Delta_X$, then $\psi_V$ is an embedding. \smallskip \item{\rm b)} Conversely, if rank$\,E=1$ and if there exists $V\subset H^0(X,E)$ generating $E$ such that $\psi_V$ is an embedding, then $E$ is very ample.\smallskip \endstat \begproof{} b) is immediate, because $E=\psi_V^\star(\cO(1))$ and $\cO(1)$ is very ample. Note that the result is false for $r\ge 2$ as shown by the example $E=Q$ over $X=G_r(V)$. \medskip \noindent a) Under the assumption of a), it is clear since $S_x=\{s\in V~;~ s(x)=0\}$ that $S_x=S_y$ implies $x=y$, hence $\psi_V$ is injective. Therefore, it is enough to prove that the map $x\mapsto S_x$ has an injective differential. Let $x\in X$ and $W\subset V$ such that $S_x\oplus W=V$. Choose a coordinate system in a neighborhood of $x$ in $X$ and a small tangent vector $h\in T_xX$. The element $S_{x+h}\in G_r(V)$ is the graph of a small linear map $u=O(|h|):S_x\to W$. Thus we have $$S_{x+h}=\{s'=s+t\in V~;~s\in S_x,~t=u(s)\in W,~s'(x+h)=0\}.$$ Since $s(x)=0$ and $|t|=O(|h|)$, we find $$s'(x+h)=s'(x)+d_xs'\cdot h+O(|s'|\cdot|h|^2)=t(x)+d_xs\cdot h+ O(|s|\cdot|h|^2),$$ thus $s'(x+h)=0$ if and only if $t(x)=-d_xs\cdot h+O(|s|\cdot|h|^2)$. Thanks to the fiber isomorphism $\Psi_V:E_x\longrightarrow V/S_x\simeq W$,~ $t(x)\longmapsto t$ mod $S_x$, we get: $$u(s)=t=\Psi_V(t(x))=-\Psi_V\big(d_xs\cdot h+O(|s|\cdot|h|^2)\big).$$ Recall that $T_yG_r(V)=\hom(S_y,Q_y)=\hom(y,V/y)$ (see V-16.5) and use these identifications at $y=S_x$. It follows that $$(d_x\psi_V)\cdot h=u=\big(S_x\longrightarrow V/S_x,~s\longmapsto -\Psi_V (d_xs\cdot h)\big),\leqno(11.11)$$ Now hypothesis a) implies that $S_x\ni s\longmapsto d_xs\in\hom(T_xX,E_x)$ is onto, hence $d_x\psi_V$ is injective.\qed \endproof \begstat{(11.12) Corollary} If $E$ is an ample line bundle, then $E>0$. \endstat \begproof{} If $E$ is very ample, diagram $(11.4')$ shows that $E$ is the pull-back of $\cO(1)$ by the embedding $\psi_V$, hence $i\Theta(E)= \psi_V^\star\big(\ii\Theta(\cO(1))\big)>0$ with the induced metric. The ample case follows by extracting roots.\qed \endproof \begstat{(11.13) Corollary} If $E$ is a very ample vector bundle, then $E$ carries a hermitian metric such that $E^\star<_\Nak 0$, in particular $E>_\Grif 0$. \endstat \begproof{} Choose $V$ as in Prop.~11.9 and select an arbitrary hermitian metric on $V$. Then diagram 11.4 yields $E=\psi_V^\star Q$, hence $\theta_E=\Psi_V^\star\theta_Q$. By formula (V-16.9) we have for every $\xi\in TG_r(V)=\hom(S,Q)$ and $t\in Q\,:$ $$\theta_Q(\xi\otimes t,\xi\otimes t)=\sum_{j,k,l} \xi_{jk}\ol\xi_{lk}t_l\ol t_j =\sum_k\Big|\sum_j\ol t_j\xi_{jk}\Big|^2=|\langle\bu,t\rangle\circ\xi|^2.$$ Let $h\in T_xX$, $t\in E_x$. Thanks to formula (11.11), we get $$\eqalign{ \theta_E(h\otimes t&,h\otimes t)=\theta_Q\big((d_x\psi_V\cdot h)\otimes \Psi_V(t),(d_x\psi_V\cdot h)\otimes\Psi_V(t)\big)\cr &=\big|\langle\bu,\Psi_V(t)\rangle\circ(d_x\psi_V\cdot h)\big|^2 =\big|S_x\ni s\longmapsto\langle\Psi_V(d_xs\cdot h),\Psi_V(t)\rangle\big|^2\cr &=\big|S_x\ni s\longmapsto\langle d_xs\cdot h,t\rangle\big|^2\ge 0.\cr}$$ As $S_x\ni s\mapsto d_xs\in T^\star X\otimes E$ is surjective, it follows that \hbox{$\theta_E(h\otimes t,h\otimes t)\ne 0$} when $h\ne 0$, $t\ne 0$. Now, $d_xs$ defines a linear form on $TX\otimes E^\star$ and the above formula for the curvature of $E$ clearly yields $$\theta_{E^\star}(u,u)=-|S_x\ni s\longmapsto d_xs\cdot u|^2<0~~~~{\rm if}~ u\ne 0.\eqno{\square}$$ \endproof \begstat{(11.14) Problem (Griffiths 1969)} If $E$ is an ample vector bundle over a compact manifold~$X$, then is $E>_\Grif 0$~? \endstat Griffiths' problem has been solved in the affirmative when $X$ is a curve (Umemura~1973), see also (Campana-Flenner~1990), but the general case is still unclear and seems very deep. The next sections will be concerned with the important result of Kodaira asserting the equivalence between positivity and ampleness for line bundles. \titleb{12.}{Blowing-up along a Submanifold} Here we generalize the blowing-up process already considered in Remark 4.5 to arbitrary manifolds. Let $X$ be a complex $n$-dimensional manifold and $Y$ a closed submanifold with $\codim_XY=s$. \begstat{(12.1) Notations} The normal bundle of $Y$ in $X$ is the vector bundle over $Y$ defined as the quotient $NY=(TX)_{\restriction Y}/TY$. The fibers of $NY$ are thus given by $N_yY=T_yX/T_yY$ at every point $y\in Y$. We also consider the projectivized normal bundle $P(NY)\longrightarrow Y$ whose fibers are the projective spaces $P(N_yY)$ associated to the fibers of $NY$. \endstat The {\it blow-up of $X$ with center $Y$} (to be constructed later) is a complex $n$-dimensional manifold $\wt X$ together with a holomorphic map $\sigma:\wt X\longrightarrow X$ such that: \smallskip \item{i)} $E:=\sigma^{-1}(Y)$ is a smooth {\it hypersurface} in $\wt X$, and the restriction \hbox{$\sigma:E\to Y$} is a holomorphic fiber bundle isomorphic to the projec\-ti\-vized normal bundle $P(NY)\to Y$. \smallskip \item{ii)} $\sigma:\wt X\setminus E\longrightarrow X\setminus Y$ is a biholomorphism. \smallskip \noindent In order to construct $\wt X$ and $\sigma$, we first define the set-theoretic underlying objects as the disjoint sums $$\cmalign{ \wt X&=(X\setminus Y)\amalg E,~~~~ &{\rm where}~~E:=P(NY),\cr \sigma&=\Id_{X\setminus Y}\amalg{}\,\,\pi,~~~~ &{\rm where}~~\pi:E\longrightarrow Y.\cr}$$ \input epsfiles/fig_7_1.tex \vskip16mm \centerline{{\bf VII-1} Blow-up of one point in $X$.} \vskip6mm \noindent This means intuitively that we have replaced each point $y\in Y$ by the projective space of all directions normal to $Y$. When $Y$ is reduced to a single point, the geometric picture is given by Fig.~1 below. In general, the picture is obtained by slicing $X$ transversally to $Y$ near each point and by blowing-up each slice at the intersection point with~$Y$. It remains to construct the manifold structure on $\wt X$ and in particular to describe what are the holomorphic functions near a point of~$E$. Let $f,g$ be holomorphic functions on an open set $U\subset X$ such that \hbox{$f=g=0$} on $Y\cap U$. Then $df$ and $dg$ vanish on $TY_{\restriction Y\cap U}$, hence $df$ and $dg$ induce linear forms on $NY_{\restriction Y\cap U}$. The holomorphic function $h(z)=f(z)/g(z)$ on the open set $$U_g:=\big\{z\in U~;~g(z)\ne 0\big\}\subset U\setminus Y$$ can be extended in a natural way to a function $\wt h$ on the set $$\wt U_g=U_g\cup\big\{(z,[\xi])\in P(NY)_{\restriction Y\cap U}~;~dg_z(\xi) \ne 0\big\}\subset\wt X$$ by letting $$\wt h(z,[\xi])={df_z(\xi)\over dg_z(\xi)},~~~~(z,[\xi])\in P(NY)_{\restriction Y\cap U}.$$ Using this observation, we now define the manifold structure on $\wt X$ by giving explicitly an atlas. Every coordinate chart of $X\setminus Y$ is taken to be also a coordinate chart of $\wt X$. Furthermore, for every point $y_0\in Y$, there exists a neighborhood $U$ of $y_0$ in $X$ and a coordinate chart $\tau(z)=(z_1,\ldots,z_n):U\to\bC^n$ centered at $y_0$ such that $\tau(U)=B'\times B''$ for some balls \hbox{$B'\subset\bC^s$}, $B''\subset\bC^{n-s}$, and such that \hbox{$Y\cap U=\tau^{-1}(\{0\}\times B'')=\{z_1{=}\ldots{=}z_s{=}0\}$.} It~follows that $(z_{s+1},\ldots,z_n)$ are local coordinates on $Y\cap U$ and that the vector fields $(\partial/\partial z_1,\ldots,\partial/\partial z_s)$ yield a holomorphic frame of $NY_{\restriction Y\cap U}$. Let us denote by $(\xi_1,\ldots,\xi_s)$ the corresponding coordinates along the fibers of $NY$. Then $(\xi_1,\ldots,\xi_s,z_{s+1},\ldots,z_n)$ are coordinates on the total space $NY$. For every $j=1,\ldots,s$, we set $$\wt U_j=\wt U_{z_j}=\big\{z\in U\setminus Y~;~z_j\ne 0\big\}\cup \big\{(z,[\xi])\in P(NY)_{\restriction Y\cap U}~;~\xi_j\ne 0\big\}.$$ Then $(\wt U_j)_{1\le j\le s}$ is a covering of $\wt U=\sigma^{-1}(U)$ and for each $j$ we define a coordinate chart $\wt\tau_j=(w_1,\ldots,w_n): \wt U_j\longrightarrow\bC^n$ by $$w_k:=\Big({z_k\over z_j}\Big)^\sim~~~~{\rm for}~~1\le k\le s,k\ne j\,;~~~~ w_k:=z_k~~~{\rm for}~~k>s~~{\rm or}~~k=j.$$ For $z\in U\setminus Y$, resp. $(z,[\xi])\in P(NY)_{\restriction Y\cap U}$, we get $$\eqalign{ \wt\tau_j(z)&=(w_1,\ldots,w_n)=\Big({z_1\over z_j},\ldots,{z_{j-1}\over z_j},z_j, {z_{j+1}\over z_j},\ldots,{z_s\over z_j},z_{s+1},\ldots,z_n\Big),\cr \wt\tau_j(z,[\xi])&=(w_1,\ldots,w_n)=\Big({\xi_1\over \xi_j},\ldots,{\xi_{j-1}\over \xi_j},\,0\,,{\xi_{j+1}\over\xi_j} ,\ldots,{\xi_s\over \xi_j},\xi_{s+1},\ldots,\xi_n\Big).\cr}$$ With respect to the coordinates $(w_k)$ on $\wt U_j$ and $(z_k)$ on $U$, the map $\sigma$ is given by $$\leqalignno{ \wt U_j&\buildo\sigma\over\longrightarrow U&(12.2)\cr w&\buildo\sigma_j\over\longmapsto (w_1w_j,\ldots,w_{j-1}w_j\,;\,w_j\,;\,w_{j+1}w_j,\ldots,w_sw_j\,;\,w_{s+1},\ldots,w_n)\cr}$$ where $\sigma_j=\tau\circ\sigma\circ\wt\tau_j^{-1}$, thus $\sigma$ is holomorphic. The range of the coordinate chart $\wt\tau_j$ is $\wt\tau_j(\wt U_j)=\sigma_j^{-1}\big(\tau(U)\big)$, so it is actually open in $\bC^n$. Furthermore $E\cap\wt U_j$ is defined by the single equation $w_j=0$, thus $E$ is a smooth hypersurface in $\wt X$. It remains only to verify that the coordinate changes $w\longmapsto w'$ associated to any coordinate change $z\longmapsto z'$ on $X$ are holomorphic. For that purpose, it is sufficient to verify that $(f/g)^\sim$ is holomorphic in $(w_1,\ldots,w_n)$ on $\wt U_j\cap\wt U_g$. As $g$ vanishes on $Y\cap U$, we can write $g(z)=\sum_{1\le k\le s} z_kA_k(z)$ for some holomorphic functions $A_k$ on $U$. Therefore $${g(z)\over z_j}=A_j(\sigma_j(w))+\sum_{k\ne j}w_kA_k(\sigma_j(w))$$ has an extension $(g/z_j)^\sim$ to $\wt U_j$ which is a holomorphic function of the variables $(w_1,\ldots,w_n)$. Since $(g/z_j)^\sim(z,[\xi])= dg_z(\xi)/\xi_j$ on $E\cap\wt U_j$, it is clear that $$\wt U_j\cap\wt U_g=\big\{w\in\wt U_j~;~(g/z_j)^\sim(w)\ne 0\big\}.$$ Hence $\wt U_j\cap\wt U_g$ is open in $\wt U_g$ and $(f/g)^\sim= (f/z_j)^\sim/(g/z_j)^\sim$ is holomorphic on $\wt U_j\cap\wt U_g$. \begstat{(12.3) Definition} The map $\sigma:\wt X\to X$ is called the blow-up of $X$ with center $Y$ and $E=\sigma^{-1}(Y)\simeq P(NY)$ is called the exceptional divisor of $\wt X$. \endstat According to (V-13.5), we denote by $\cO(E)$ the line bundle on $\wt X$ associated to the divisor $E$ and by $h\in H^0(\wt X,\cO(E))$ the canonical section such that ${\rm div}(h)=[E]$. On the other hand, we denote by $\cO_{P(NY)}(-1)\subset\pi^\star(NY)$ the tautological line subbundle over $E=P(NY)$ such that the fiber above the point $(z,[\xi])$ is $\bC\xi\subset N_zY$. \begstat{(12.4) Proposition} $\cO(E)$ enjoys the following properties: \smallskip \item{\rm a)} $\cO(E)_{\restriction E}$ is isomorphic to $\cO_{P(NY)}(-1)$. \smallskip \item{\rm b)} Assume that $X$ is compact. For every positive line bundle $L$ over $X$, the line bundle $\cO(-E)\otimes\sigma^\star(L^k)$ over $\wt X$ is positive for $k>0$ large enough.\smallskip \endstat \begproof{} a) The canonical section $h\in H^0(\wt X,\cO(E))$ vanishes at order $1$ along~$E$, hence the kernel of its differential $$dh:(T\wt X)_{\restriction E}\longrightarrow\cO(E)_{\restriction E}$$ is $TE$. We get therefore an isomorphism $NE\simeq \cO(E)_{\restriction E}$. Now, the map \hbox{$\sigma:\wt X\to X$} satisfies $\sigma(E)\subset Y$, so its differential $d\sigma:T\wt X\longrightarrow\sigma^\star(TX)$ is such that $d\sigma(TE)\subset\sigma^\star(TY)$. Therefore $d\sigma$ induces a morphism $$NE\longrightarrow\sigma^\star(NY)=\pi^\star(NY)\leqno(12.5)$$ of vector bundles over $E$. The vector field $\partial/ \partial w_j$ yields a non vanishing section of $NE$ on $\wt U_j$, and $(12.2)$ implies $$d\sigma_j\Big({\partial\over\partial w_j}\Big)= {\partial\over\partial z_j}+\sum_{1\le k\le s,k\ne j}w_k {\partial\over\partial z_k}~~~~/\!/~~~ \sum_{1\le k\le s}\xi_k{\partial\over\partial z_k}$$ at every point $(z,[\xi])\in E$. This shows that (12.5) is an isomorphism of $NE$ onto $\cO_{P(NY)}(-1)\subset\pi^\star(NY)$, hence $$\cO(E)_{\restriction E}\simeq NE\simeq\cO_{P(NY)}(-1).\leqno(12.6)$$ {\rm b)} Select an arbitrary hermitian metric on $TX$ and consider the induced metrics on $NY$ and on $\cO_{P(NY)}(1)\longrightarrow E=P(NY)$. The restriction of $\cO_{P(NY)}(1)$ to each fiber $P(N_zY)$ is the standard line bundle $\cO(1)$ over $\bP^{s-1}\,;$ thus by (V-15.10) this restriction has a positive definite curvature form. Extend now the metric of $\cO_{P(NY)}(1)$ on $E$ to a metric of $\cO(-E)$ on $X$ in an arbitrary way. If $F=\cO(-E)\otimes\sigma^\star(L^k)$, then $\Theta(F)=\Theta(\cO(-E))+ k\,\sigma^\star \Theta(L)$, thus for every $t\in T\wt X$ we have $$\theta_F(t,t)=\theta_{\cO(-E)}(t,t)+ k\,\theta_L\big(d\sigma(t),d\sigma(t)\big).$$ By the compactness of the unitary tangent bundle to $\wt X$ and the positivity of $\theta_L$, it is sufficient to verify that $\theta_{\cO(-E)}(t,t)>0$ for every unit vector $t\in T_z\wt X$ such that $d\sigma(t)=0$. However, from the computations of a), this can only happen when $z\in E$ and $t\in TE$, and in that case $d\sigma(t)=d\pi(t)=0$, so $t$ is tangent to the fiber $P(N_zY)$. Therefore $$\theta_{\cO(-E)}(t,t)=\theta_{\cO_{P(NY)}(1)}(t,t)>0.\eqno\square$$ \endproof \begstat{(12.7) Proposition} The canonical line bundle of $\wt X$ is given by $$K_{\wt X}=\cO\big((s-1)E\big)\otimes\sigma^\star K_X,~~~~\hbox{\it where}~~ s=\codim_XY.$$ \endstat \begproof{} $K_X$ is generated on $U$ by the holomorphic $n$-form $dz_1\wedge\ldots\wedge dz_n$. Using (12.2), we see that $\sigma^\star K_X$ is generated on $\wt U_j$ by $$\sigma^\star(dz_1\wedge\ldots\wedge dz_n)=w_j^{s-1}\,dw_1\wedge\ldots \wedge dw_n.$$ Since the divisor of the section $h\in H^0(\wt X,\cO(E))$ is the hypersurface $E$ defined by the equation $w_j=0$ in $\wt U_j$, we have a well defined line bundle isomorphism $$\sigma^\star K_X\longrightarrow \cO\big((1-s)E\big)\otimes K_{\wt X},~~~~~ \alpha\longmapsto h^{1-s}\sigma^\star(\alpha).\eqno\square$$ \endproof \titleb{13.}{Equivalence of Positivity and Ampleness for Line Bundles} We have seen in section 11 that every ample line bundle carries a hermitian metric of positive curvature. The converse will be a consequence of the following result. \begstat{(13.1) Theorem} Let $L\longrightarrow X$ be a positive line bundle and $L^k$ the \hbox{$k$-th} tensor power of $L$. For every $N$-tuple $(x_1,\ldots,x_N)$ of distinct points of $X$, there exists a constant $C>0$ such that the evaluation maps $$H^0(X,L^k)\longrightarrow(J^mL^k)_{x_1}\oplus\cdots\oplus(J^mL^k)_{x_N}$$ are surjective for all integers $m\ge 0$, $k\ge C(m+1)$. \endstat \begstat{(13.2) Lemma} Let $\sigma:\wt X\longrightarrow X$ be the blow-up of $X$ with center the finite set $Y=\{x_1,\ldots,x_N\}$, and let $\cO(E)$ be the line bundle associated to the exceptional divisor $E$. Then $$H^1(\wt X,\cO(-mE)\otimes\sigma^\star L^k)=0$$ for $m\ge 1$, $k\ge Cm$ and $C\ge 0$ large enough. \endstat \begproof{} By Prop.~12.7 we get $K_{\wt X}=\cO\big((n-1)E\big)\otimes \sigma^\star K_X$ and $$H^1\big(\wt X,\cO(-mE)\otimes\sigma^\star L^k\big) =H^{n,1}\big(\wt X,K_{\wt X}^{-1}\otimes\cO(-mE)\otimes\sigma^\star L^k\big) =H^{n,1}\big(\wt X,F\big)$$ where $F=\cO\big(-(m+n-1)E\big)\otimes\sigma^\star(K_X^{-1}\otimes L^k)$, so the conclusion will follow from the Kodaira-Nakano vanishing theorem if we can show that $F>0$ when $k$ is large enough. Fix an arbitrary hermitian metric on $K_X$. Then $$\Theta(F)=(m+n-1)\Theta(\cO(-E))+ \sigma^\star\big(k\Theta(L)-\Theta(K_X)\big).$$ There is $k_0\ge 0$ such that $\ii\big(k_0\Theta(L)-\Theta(K_X)\big)>0$ on~$X$, and Prop.~12.4 implies the existence of $C_0>0$ such that $\ii\big(\Theta(\cO(-E))+C_0\sigma^\star \Theta(L)\big)>0$ on~$\wt X$. Thus $\ii\Theta(F)>0$ for $m\ge 2-n$ and $k\ge k_0+C_0(m+n-1)$.\qed \endproof \begproof{of Theorem 13.1.} Let $v_j\in H^0(\Omega_j,L^k)$ be a holomorphic section of $L^k$ in a neighborhood $\Omega_j$ of $x_j$ having a prescribed $m$-jet at $x_j$. Set $$v(x)=\sum_j\psi_j(x)v_j(x)$$ where $\psi_j=1$ in a neighborhood of $x_j$ and $\psi_j$ has compact support in $\Omega_j$. Then $d''v=\sum d''\psi_j\cdot v_j$ vanishes in a neighborhood of $x_1,\ldots,x_N$. Let $h$ be the canonical section of $\cO(E)^{-1}$ such that ${\rm div}(h)=[E]$. The $(0,1)$-form $\sigma^\star d''v$ vanishes in a neighborhood of $E=h^{-1}(0)$, hence $$w=h^{-(m+1)}\sigma^\star d''v\in C^\infty_{0,1}\big(\wt X,\cO(-(m+1)E)\otimes \sigma^\star L^k\big).$$ and $w$ is a $d''$-closed form. By Lemma~13.2 there exists a smooth section $u\in C^\infty_{0,0}\big(\wt X,\cO(-(m+1)E)\otimes\sigma^\star L^k\big)$ such that $d''u=w=h^{-(m+1)}\sigma^\star d''v$. This implies $$\sigma^\star v-h^{m+1}u\in H^0(\wt X,\sigma^\star L^k),$$ and since $\sigma^\star L$ is trivial near $E$, there exists a section $g\in H^0(X,L^k)$ such that $\sigma^\star g=\sigma^\star v-h^{m+1}u$. As $h$ vanishes at order 1 along $E$, the $m$-jet of $g$ at $x_j$ must be equal to that of $v$ (or $v_j$).\qed \endproof \begstat{(13.3) Corollary} For any holomorphic line bundle $L\longrightarrow X$, the following conditions are equivalent: \smallskip \item{\rm a)} $L$ is ample; \smallskip \item{\rm b)} $L>0$, i.e.\ $L$ possesses a hermitian metric such that $\ii\Theta(L)>0$. \endstat \begproof{} a) $\Longrightarrow$ b) is given by Cor.~11.12, whereas b) $\Longrightarrow$ a) is a consequence of Th.~13.1 for $m=1$.\qed \endproof \titleb{14.}{Kodaira's Projectivity Criterion} The following fundamental projectivity criterion is due to (Kodaira 1954). \begstat{(14.1) Theorem} Let $X$ be a compact complex manifold, $\dim_\bC X=n$. The following conditions are equivalent. \smallskip \item{\rm a)} $X$ is projective algebraic, i.e.\ $X$ can be embedded as an algebraic submanifold of the complex projective space $\bP^N$ for $N$ large. \smallskip \item{\rm b)} $X$ carries a positive line bundle $L$. \smallskip \item{\rm c)} $X$ carries a Hodge metric, i.e.\ a K\"ahler metric $\omega$ with rational cohomology class $\{\omega\}\in H^2(X,\bQ)$.\smallskip \endstat \begproof{} a) $\Longrightarrow$ b). Take $L=\cO(1)_{\restriction X}$. \medskip \noindent b) $\Longrightarrow$ c). Take $\omega={\ii\over 2\pi}\Theta(L)\,;$ then $\{\omega\}$ is the image of $c_1(L)\in H^2(X,\bZ)$. \medskip \noindent c) $\Longrightarrow$ b). We can multiply $\{\omega\}$ by a common denominator of its coefficients and suppose that $\{\omega\}$ is in the image of $H^2(X,\bZ)$. Then Th.~V-13.9~b) shows that there exists a hermitian line bundle $L$ such that ${\ii\over 2\pi}\Theta(L)=\omega >0$. \medskip \noindent b) $\Longrightarrow$ a). Corollary 13.3 shows that $F=L^k$ is very ample for some integer $k>0$. Then Prop.~11.9 enables us to find a subspace $V$ of $H^0(X,F)$, $\dim V\le 2n+2$, such that $\psi_V:X\longrightarrow G_1(V)=P(V^\star)$ is an embedding. Thus $X$ can be embedded in $\bP^{2n+1}$ and Chow's theorem~II-7.10 shows that the image is an algebraic set in~$\bP^{2n+1}$.\qed \endproof \begstat{(14.2) Remark} \rm The above proof shows in particular that every $n$-dimen\-sional projective manifold $X$ can be embedded in $\bP^{2n+1}$. This can be shown directly by using generic projections $\bP^N\to\bP^{2n+1}$ and Whitney type arguments as in~11.2. \endstat \begstat{(14.3) Corollary} Every compact Riemann surface $X$ is isomorphic to an algebraic curve in $\bP^3$. \endstat \begproof{} Any positive smooth form $\omega$ of type $(1,1)$ is K\"ahler, and $\omega$ is in fact a Hodge metric if we normalize its volume so that $\int_X\omega=1$.\qed \endproof \noindent This example can be somewhat generalized as follows. \begstat{(14.4) Corollary} Every K\"ahler manifold $(X,\omega)$ such that $H^2(X,\cO)=0$ is projective. \endstat \begproof{} By hypothesis $H^{0,2}(X,\bC)=0=H^{2,0}(X,\bC)$, hence $$H^2(X,\bC)=H^{1,1}(X,\bC)$$ admits a basis $\{\alpha_1\},\ldots,\{\alpha_N\}\in H^2(X,\bQ)$ where $\alpha_1,\ldots,\alpha_N$ are harmonic real $(1,1)$-forms. Since $\{\omega\}$ is real, we have $\{\omega\}=\lambda_1\{\alpha_1\}+\ldots+\lambda_N\{\alpha_N\}$, $\lambda_j\in\bR$, thus $$\omega=\lambda_1\alpha_1+\ldots+\lambda_N\alpha_N$$ because $\omega$ itself is harmonic. If $\mu_1,\ldots,\mu_N$ are rational numbers sufficiently close to $\lambda_1,\ldots,\lambda_N$, then $\wt\omega:=\mu_1\alpha_1+\cdots\mu_N\alpha_N$ is close to $\omega$, so $\wt\omega$ is a positive definite $d$-closed $(1,1)$-form, and $\{\wt\omega\}\in H^2(X,\bQ)$.\qed \endproof We obtain now as a consequence the celebrated Riemann criterion characterizing {\it abelian varieties} (${}={}$projective algebraic complex tori). \begstat{(14.5) Corollary} A complex torus $X=\bC^n/\Gamma$ $(\Gamma$ a lattice of $\bC^n)$ is an abelian variety if and only if there exists a positive definite hermitian form $h$ on $\bC^n$ such that $$\Im\big(h(\gamma_1,\gamma_2)\big)\in\bZ~~~~{\rm for~all}~~\gamma_1,\gamma_2 \in\Gamma.$$ \endstat \begproof{(Sufficiency of the condition).} Set $\omega=-\Im h$. Then $\omega$ defines a constant K\"ahler metric on $\bC^n$, hence also on $X=\bC^n/\Gamma$. Let $(a_1,\ldots,a_{2n})$ be an integral basis of the lattice $\Gamma$. We denote by $T_j$, $T_{jk}$ the real $1$- and $2$-tori $$T_{j}=(\bR/\bZ)a_j,~~~1\le j\le n,~~~~T_{jk}=T_j\oplus T_k,~~~ 1\le j0$. In that case, for every $v\in\cH_2$ such that $Sv=0$, there exists $u\in\cH_1$ such that $Tu=v$ and $$\|u\|_1^2\le{1\over C}\|v\|_2^2.$$ In particular $$\ol{\Im T}=\Im T=\Ker S,~~~~ \ol{\Im S^\star}=\Im S^\star=\Ker T^\star.$$ \endstat \begproof{} Since $S$ is closed, the kernel $\Ker S$ is closed in $\cH_2$. The relation $(\Ker S)^\perp=\ol{\Im S^\star}$ implies $$\cH_2=\Ker S\oplus\ol{\Im S^\star}\leqno(1.4)$$ and similarly $\cH_2=\Ker T^\star\oplus\ol{\Im T}$. However, the assumption $S\circ T=0$ shows that $\ol{\Im T}\subset\Ker S$, therefore $$\Ker S=(\Ker S\cap\Ker T^\star)\oplus\ol{\Im T}.\leqno(1.5)$$ The first two equalities in Th.~1.2 are then equivalent to the conjunction of (1.4) and (1.5). Now, under assumption (1.3), we are going to show that the equation $Tu=v$ is always solvable if $Sv=0$. Let $x\in\Dom T^\star$. One can write $$x=x'+x''~~~~\hbox{\rm where}~~x'\in\Ker S~~\hbox{\rm and}~~x''\in(\Ker S)^\perp \subset(\Im T)^\perp=\Ker T^\star.$$ Since $x,x''\in\Dom T^\star$, we have also $x'\in\Dom T^\star$. We get $$\langle v,x\rangle_2=\langle v,x'\rangle_2+\langle v,x''\rangle_2= \langle v,x'\rangle_2$$ because $v\in\Ker S$ and $x''\in(\Ker S)^\perp$. As $Sx'=0$ and $T^\star x''=0$, the Cauchy-Schwarz inequality combined with (1.3) implies $$|\langle v,x\rangle_2|^2\le\|v\|^2_2~\|x'\|^2_2\le {1\over C}\|v\|^2_2~\|T^\star x'\|^2_1={1\over C}\|v\|^2_2~\|T^\star x\|^2_1.$$ This shows that the linear form $T^\star_X\ni x\longmapsto\langle x,v\rangle_2$ is continuous on \hbox{$\Im T^\star\subset\cH_1$} with norm $\le C^{-1/2}\|v\|_2$. By the Hahn-Banach theorem, this form can be extended to a continuous linear form on $\cH_1$ of norm $\le C^{-1/2}\|v\|_2$, i.e.\ we can find $u\in\cH_1$ such that $\|u\|_1\le C^{-1/2}\|v\|_2$ and $$\langle x,v\rangle_2=\langle T^\star x,u\rangle_1,~~~~ \forall x\in\Dom T^\star.$$ This means that $u\in\Dom(T^\star)^\star=\Dom T$ and $v=Tu$. We have thus shown that $\Im T=\Ker S$, in particular $\Im T$ is closed. The dual \hbox{equality} \hbox{$\Im S^\star=\Ker T^\star$} follows by considering the dual pair $(S^\star,T^\star)$.\qed \endproof \titleb{2.}{Complete Riemannian Manifolds} Let $(M,g)$ be a riemannian manifold of dimension $m$, with metric $$g(x)=\sum g_{jk}(x)\,dx_j\otimes dx_k,~~~~1\le j,k\le m.$$ The length of a path $\gamma~:~[a,b]\longrightarrow M$ is by definition $$\ell(\gamma)=\int_a^b|\gamma'(t)|_gdt=\int_a^b\Big( \sum_{j,k}g_{jk}\big(\gamma(t)\big)\,\gamma'_j(t)\gamma'_k(t)\Big)^{1/2}dt.$$ The geodesic distance of two points $x,y\in M$ is $$\delta(x,y)=\inf_\gamma\ell(\gamma)~~~~\hbox{\rm over paths $\gamma$ with} ~~\gamma(a)=x,~~\gamma(b)=y,$$ if $x,y$ are in the same connected component of $M$, $\delta(x,y)=+\infty$ otherwise. It~is easy to check that $\delta$ satisfies the usual axioms of distances: for the separation axiom, use the fact that if $y$ is outside some closed coordinate ball $\ol B$ of radius $r$ centered at $x$ and if $g\ge c|dx|^2$ on $\ol B$, then $\delta(x,y)\ge c^{1/2}r$. In~addition, $\delta$ satisfies the axiom: $$\hbox{for every $x,y\in M$,}~~~~\inf_{z\in M} \max\{\delta(x,z),\delta(y,z)\}={1\over 2}\delta(x,y).\qquad\leqno(2.1)$$ In fact for every $\varepsilon>0$ there is a path $\gamma$ such that $\gamma(a)=x$, $\gamma(b)=y$, \hbox{$\ell(\gamma)<\delta(x,y)+\varepsilon$} and we can take $z$ to be at mid-distance between $x$ and $y$ along~$\gamma$. A metric space $E$ with a distance $\delta$ satisfying the additional axiom~(2.1) will be called a {\it geodesic} metric space. It is then easy to see by dichotomy that any two points $x,y\in E$ can be joined by a chain of points \hbox{$x=x_0$}, \hbox{$x_1,\ldots,x_N=y$} such that $\delta(x_j,x_{j+1})<\varepsilon$ and \hbox{$\sum\delta(x_j,x_{j+1})<\delta(x,y)+\varepsilon$}. \begstat{(2.2) Lemma {\rm(Hopf-Rinow)}} Let $(E,\delta)$ be a geodesic metric space. Then the following properties are equivalent: \smallskip \item{\rm a)} $E$ is locally compact and complete$\,;$ \smallskip \item{\rm b)} all closed geodesic balls $\ol B(x_0,r)$ are compact. \endstat \begproof{} Since any Cauchy sequence is bounded, it is immediate that b) implies~a). We now check that a)~$\Longrightarrow$~b). Fix $x_0$ and define $R$ to be the supremum of all $r>0$ such that $\ol B(x_0,r)$ is compact. Since $E$ is locally compact, we have $R>0$. Suppose that $R<+\infty$. Then $\ol B(x_0,r)$ is compact for every~$r0$ be the smallest eigenvalue of this operator at $x\in X$. Then $\lambda$ is continuous on $X$ and we have $$\int_X\langle A_{E,\omega}^{-1}g,g\rangle\,dV\le\int_X\lambda(x)^{-1} |g(x)|^2\,dV.$$ The above situation occurs for example if $\omega$ is complete K\"ahler, $E>_m0$ and $p=n$, $q\ge 1$, $m\ge\min\{n-q+1,r\}$ (apply Lemma~VII-7.2). \endstat \titleb{5.}{Estimates on Weakly Pseudoconvex Manifolds} We first introduce a large class of complex manifolds on which the $L^2$ estimates will be easily tractable. \begstat{(5.1) Definition} A complex manifold $X$ is said to be weakly pseudo\-convex if there exists an exhaustion function $\psi\in C^\infty(X,\bR)$ such that $\ii d'd''\psi\ge 0$ on $X$, i.e.\ $\psi$ is plurisubharmonic. \endstat For domains $\Omega\subset\bC^n$, the above weak pseudoconvexity notion is equivalent to pseudoconvexity (cf.\ Th.~I-4.14). Note that every compact manifold is also weakly pseudoconvex (take $\psi\equiv 0$). Other examples that will appear later are Stein manifolds, or the total space of a Griffiths semi-negative vector bundle over a compact manifold (cf.\ Prop.~IX-?.?). \begstat{(5.2) Theorem} Every weakly pseudoconvex K\"ahler manifold $(X,\omega)$ carries a complete K\"ahler metric $\wh\omega$. \endstat \begproof{} Let $\psi\in C^\infty(X,\bR)$ be an exhaustive plurisubharmonic function on $X$. After addition of a constant to $\psi$, we can assume $\psi\ge 0$. Then $\wh\omega=\omega+\ii d'd''(\psi^2)$ is K\"ahler and $$\wh\omega=\omega+2i\psi d'd''\psi+2\ii d'\psi\wedge d''\psi\ge \omega+2\ii d'\psi\wedge d''\psi.$$ Since $d\psi=d'\psi+d''\psi$, we get $|d\psi|_{\wh\omega}=\sqrt{2} |d'\psi|_{\wh\omega}\le 1$ and Lemma~2.4 shows that $\wh\omega$ is complete.\qed \endproof Observe that we could have set more generally $\wh\omega=\omega+\ii d'd'' (\chi\circ\psi)$ where $\chi$ is a convex increasing function. Then $$\leqalignno{ \wh\omega&=\omega+i(\chi'\circ\psi)d'd''\psi+i(\chi''\circ\psi)d'\psi\wedge d''\psi\cr &\ge\omega+\ii d'(\rho\circ\psi)\wedge d''(\rho\circ\psi)&(5.3)\cr}$$ where $\rho(t)=\int_0^t\sqrt{\chi''(u)}\,du$. We thus have $|d'(\rho\circ\psi)|_{\wh\omega}\le 1$ and $\wh\omega$ will be complete as soon as $\lim_{t\to+\infty}\rho(t)=+\infty$, i.e.\ $$\int_0^{+\infty}\sqrt{\chi''(u)}\,du=+\infty.\leqno(5.4)$$ One can take for example $\chi(t)=t-\log(t)$ for $t\ge 1$. It follows from the above considerations that almost all vanishing theorems for positive vector bundles over compact manifolds are also valid on weakly pseudoconvex manifolds. Let us mention here the analogues of some results proved in Chapter~7. \begstat{(5.5) Theorem} For any $m$-positive vector bundle of rank $r$ over a weakly pseudoconvex manifold $X$, we have $H^{n,q}(X,E)=0$ for all $q\ge 1$ and \hbox{$m\ge\min\{n-q+1,r\}$}. \endstat \begproof{} The curvature form $\ii\Theta(\det E)$ is a K\"ahler metric on $X$, hence $X$ possesses a complete K\"ahler metric $\omega$. Let $\psi\in C^\infty(X,\bR)$ be an exhaustive plurisubharmonic function. For any convex increasing function $\chi\in C^\infty(\bR,\bR)$, we denote by $E_\chi$ the holomorphic vector bundle $E$ together with the modified metric $|u|_\chi^2=|u|^2\,\exp\big(-\chi\circ\psi(x)\big)$, $u\in E_x$. We get $$\ii\Theta(E_\chi)=\ii\Theta(E)+\ii d'd''(\chi\circ\psi)\otimes\,\Id_E \ge_m \ii\Theta(E),$$ thus $A_{E_\chi,\omega}\ge A_{E,\omega}>0$ in bidegree $(n,q)$. Let $g$ be a given form of bidegree $(n,q)$ with $L^2_\loc$ coefficients, such that $D''g=0$. The integrals $$\int_X\langle A_{E_\chi,\omega}^{-1}g,g\rangle_\chi\,dV\le \int_X\langle A_{E,\omega}^{-1}g,g\rangle\,e^{-\chi\circ\psi}\,dV,~~~~ \int_X |g|^2\,e^{-\chi\circ\psi}\,dV$$ become convergent if $\chi$ grows fast enough. We can thus apply Th.~4.5 to $(X,E_\chi,\omega)$ and find a $(n,q-1)$ form $f$ such that $D''f=g$. If $g$ is smooth, Remark~4.6 shows that $f$ can also be chosen smooth.\qed \endproof \begstat{(5.6) Theorem} If $E$ is a positive line bundle over a weakly pseudoconvex manifold $X$, then $H^{p,q}(X,E)=0$ for $p+q\ge n+1$. \endstat \begproof{} The proof is similar to that of Th.~5.5, except that we use here the K\"ahler metric $$\omega_\chi=\ii\Theta(E_\chi)=\omega+\ii d'd''(\chi\circ\psi),~~~~ \omega=\ii\Theta(E),$$ which depends on $\chi$. By (5.4) $\omega_\chi$ is complete as soon as $\chi$ is a convex increasing function that grows fast enough. Apply now Th.~4.5 to $(X,E_\chi,\omega_\chi)$ and observe that $A_{E_\chi,\omega_\chi}=[\ii\Theta(E_\chi),\Lambda_\chi]= (p+q-n)\,\Id$~ in bidegree $(p,q)$ in virtue of Cor.~VI-8.4 It remains to show that for every form $g\in C^\infty_{p,q}(X,E)$ there exists a choice of $\chi$ such that $g\in L^2_{p,q}(X,E_\chi,\omega_\chi)$. By (5.3) the norm of a scalar form with respect to $\omega_\chi$ is less than its norm with respect to $\omega$, hence $|g|_\chi^2\le|g|^2\,\exp(-\chi\circ\psi)$. On the other hand $$dV_\chi\le C\big(1+\chi'\circ\psi+\chi''\circ\psi\big)^n\,dV$$ where $C$ is a positive continuous function on $X$. The following lemma implies that we can always choose $\chi$ in order that the integral of $|g|_\chi^2\,dV_\chi$ converges on $X$. \endproof \begstat{(5.7) Lemma} For any positive function $\lambda\in C^\infty\big( [0,+\infty[,\bR\big)$, there exists a smooth convex function $\chi\in C^\infty\big([0,+\infty[,\bR\big)$ such that $\chi,\chi',\chi''\ge\lambda$ and \hbox{$(1+\chi'+\chi'')^ne^{-\chi}\le 1/\lambda$}. \endstat \begproof{} We shall construct $\chi$ such that $\chi''\ge \chi'\ge\chi\ge\lambda$ and \hbox{$\chi''/\chi^2\le C$} for some constant $C$. Then $\chi$ satisties the conclusion of the lemma after addition of a constant. Without loss of generality, we may assume that $\lambda$ is increasing and $\lambda\ge 1$. We define $\chi$ as a power series $$\chi(t)=\sum_{k=0}^{+\infty}\,a_0a_1\ldots a_k\,t^k,$$ where $a_k>0$ is a decreasing sequence converging to $0$ very slowly. Then $\chi$ is real analytic on $\bR$ and the inequalities $\chi''\ge\chi'\ge\chi$ are realized if we choose $a_k\ge 1/k$, $k\ge1$. Select a strictly increasing sequence of integers $(N_p)_{p\ge 1}$ so large that ${1\over p}\lambda(p+1)^{1/N_p}\in[1/p,1/(p-1)[$. We set $$\eqalign{ a_0&=\ldots=a_{N_1-1}=e\,\lambda(2),\cr a_k&={1\over p}\,\lambda(p+1)^{1/N_p}\,e^{1/\sqrt{k}},~~~~N_p\le k0$ be small. Select $\rho$ such that $\exp(-\rho\circ\psi(x))\le\varepsilon\gamma_s^{0,\omega}(x)$ at every point. Then for $j\ge s$ we get $$\gamma_j^{\chi,\alpha}\ge{\gamma_j^{0,\omega}\over \gamma_j^{0,\omega}+\varepsilon\gamma_j^{0,\omega}}={1\over 1+\varepsilon} \ge 1-\varepsilon,$$ and Th.~VI-8.3 implies $$\eqalign{ \langle\big[\ii\Theta(E_\chi),\Lambda_\alpha\big]u,u\rangle_\alpha&\ge \big(\gamma_1^{\chi,\alpha}+\cdots+\gamma_p^{\chi,\alpha}- \gamma_{q+1}^{\chi,\alpha}-\ldots-\gamma_n^{\chi,\alpha}\big)|u|^2\cr &\ge\big((p-s+1)(1-\varepsilon)-(n-q)\big)|u|^2\cr &\ge\big(1-(p-s+1)\varepsilon\big)|u|^2.\cr}$$ It remains however to control the torsion term $T_\alpha$. As $\omega$ is K\"ahler, trivial computations yield $$\eqalign{ d'\alpha&=-\rho'\circ\psi~\exp(-\rho\circ\psi)~d'\psi\wedge\omega,\cr d'd''\alpha&=\exp(-\rho\circ\psi)~\big[\big((\rho'\circ\psi)^2- \rho''\circ\psi\big)d'\psi\wedge d''\psi-\rho'\circ\psi~d'd''\psi\big] \wedge\omega.\cr}$$ Since $$\alpha\ge i(\chi'\circ\psi~d'd''\psi+\chi''\circ\psi~d'\psi\wedge d''\psi) +\exp(-\rho\circ\psi)\omega,$$ we get the upper bounds $$\eqalign{ |d'\alpha|_\alpha&\le\rho'\circ\psi~|d'\psi|_\alpha~|\exp(-\rho\circ\psi) \omega|_\alpha\le\rho'\circ\psi~(\chi''\circ\psi)^{-{1\over 2}}\cr |d'd''\alpha|_\alpha&\le{(\rho'\circ\psi)^2+\rho''\circ\psi\over\chi''\circ\psi} +{\rho'\circ\psi\over\chi'\circ\psi}.\cr}$$ It is then clear that we can choose $\chi$ growing sufficiently fast in order that $|T_\alpha|_\alpha\le\varepsilon$. If $\varepsilon$ is chosen sufficiently small, we get $A_{E_\chi,\alpha}\ge{1\over 2}\,\Id$, and the conclusion is obtained in the same way as for Th.~5.6.\qed \endproof \titleb{6.}{H\"ormander's Estimates for non Complete K\"ahler Metrics} Our aim here is to derive also estimates for a non complete K\"ahler metric, for example the standard metric of $\bC^n$ on a bounded domain $\Omega\subset\!\subset\bC^n$. A result of this type can be obtained in the situation described at the end of Remark~4.8. The underlying idea is due to (H\"ormander 1966), although we do not apply his so called ``three weights" technique, but use instead an approximation of the given metric $\omega$ by complete K\"ahler metrics. \begstat{(6.1) Theorem} Let $(X,\wh\omega)$ be a complete K\"ahler manifold, $\omega$ another K\"ahler metric, possibly non complete, and $E\longrightarrow X$ a $m$-semi-positive vector bundle. Let $g\in L^2_{n,q}(X,E)$ be such that $D''g=0$ and $$\int_X\langle A_q^{-1}g,g\rangle\,dV<+\infty$$ with respect to $\omega$, where $A_q$ stands for the operator $\ii\Theta(E)\wedge\Lambda$ in bidegree $(n,q)$ and $q\ge 1$, $m\ge\min\{n-q+1,r\}$. Then there exists $f\in L^2_{n,q-1}(X,E)$ such that $D''f=g$ and $$\|f\|^2\le\int_X\langle A_q^{-1}g,g\rangle\,dV.$$ \endstat \begproof{} For every $\varepsilon>0$, the K\"ahler metric $$\omega_\varepsilon=\omega+\varepsilon\wh\omega$$ is complete. The idea of the proof is to apply the $L^2$ estimates to $\omega_\varepsilon$ and to let $\varepsilon$ tend to zero. Let us put an index $\varepsilon$ to all objects depending on $\omega_\varepsilon$. It follows from Lemma~6.3 below that $$|u|_\varepsilon^2\,dV_\varepsilon\le|u|^2\,dV,~~~~ \langle A_{q,\varepsilon}^{-1}u,u\rangle_\varepsilon\,dV_\varepsilon\le \langle A_q^{-1}u,u\rangle\,dV\leqno(6.2)$$ for every $u\in\Lambda^{n,q}T^\star_X\otimes E$. If these estimates are taken for granted, Th.~4.5 applied to $\omega_\varepsilon$ yields a section $f_\varepsilon\in L^2_{n,q-1}(X,E)$ such that $D''f_\varepsilon=g$ and $$\int_X|f_\varepsilon|_\varepsilon^2\,dV_\varepsilon\le \int_X\langle A_{q,\varepsilon}^{-1}g,g\rangle_\varepsilon\,dV_\varepsilon\le \int_X\langle A_q^{-1}g,g\rangle\,dV.$$ This implies that the family $(f_\varepsilon)$ is bounded in $L^2$ norm on every compact subset of $X$. We can thus find a weakly convergent subsequence $(f_{\varepsilon_\nu})$ in $L^2_\loc$. The weak limit $f$ is the solution we are looking for.\qed \endproof \begstat{(6.3) Lemma} Let $\omega$, $\gamma$ be hermitian metrics on $X$ such that $\gamma\ge\omega$. For every $u\in\Lambda^{n,q} T^\star_X\otimes E$, $q\ge 1$, we have $$|u|_\gamma^2\,dV_\gamma\le|u|^2\,dV,~~~~ \langle A_{q,\gamma}^{-1}u,u\rangle_\gamma\,dV_\gamma\le \langle A_q^{-1}u,u\rangle\,dV$$ where an index $\gamma$ means that the corresponding term is computed in terms of $\gamma$ instead of $\omega$. \endstat \begproof{} Let $x_0\in X$ be a given point and $(z_1,\ldots,z_n)$ coordinates such that $$\omega=\ii\sum_{1\le j\le n}dz_j\wedge d\ol z_j,~~~~ \gamma=\ii\sum_{1\le j\le n}\gamma_j\,dz_j\wedge d\ol z_j~~~ \hbox{\rm at}~~x_0,$$ where $\gamma_1\le\ldots\le\gamma_n$ are the eigenvalues of $\gamma$ with respect to $\omega$ (thus $\gamma_j\ge 1$). We have $|dz_j|_\gamma^2=\gamma_j^{-1}$ and $|dz_K|_\gamma^2= \gamma_K^{-1}$ for any multi-index $K$, with the notation $\gamma_K=\prod_{j\in K}\gamma_j$. For every $\,u=\sum u_{K,\lambda}dz_1\wedge\ldots\wedge dz_n\wedge d\ol z_K \otimes e_\lambda$, $|K|=q$, $1\le\lambda\le r$, the computations of \S~VII-7 yield $$\leqalignno{ |u|_\gamma^2&=\sum_{K,\lambda}~(\gamma_1\ldots\gamma_n)^{-1} \gamma_K^{-1}\,|u_{K,\lambda}|^2,~~~~ dV_\gamma=\gamma_1\ldots\gamma_n\,dV,\cr |u|_\gamma^2\,dV_\gamma&=\sum_{K,\lambda}\gamma_K^{-1}\, |u_{K,\lambda}|^2\,dV\le|u|^2\,dV,\cr \Lambda_\gamma u&=\sum_{|I|=q-1}\sum_{j,\lambda}\ii(-1)^{n+j-1}\gamma_j^{-1}\, u_{jI,\lambda}\,(\wh{dz_j})\wedge d\ol z_I \otimes e_\lambda,\cr}$$ where $(\wh{dz_j})$ means $dz_1\wedge\ldots\wh{dz_j}\ldots \wedge dz_n$, $$\leqalignno{ A_{q,\gamma}u&=\sum_{|I|=q-1}\sum_{j,k,\lambda,\mu}\gamma_j^{-1}\, c_{jk\lambda\mu}\,u_{jI,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge d\ol z_{kI}\otimes e_\mu,\cr \llap{\hbox{$\langle A_{q,\gamma}u,u\rangle_\gamma$}}&= (\gamma_1\ldots\gamma_n)^{-1}\sum_{|I|=q-1}\gamma_I^{-1} \sum_{j,k,\lambda,\mu}\gamma_j^{-1}\gamma_k^{-1}c_{jk\lambda\mu}\, u_{jI,\lambda}\ol u_{kI,\mu}\cr &\ge(\gamma_1\ldots\gamma_n)^{-1}\sum_{|I|=q-1}\gamma_I^{-2} \sum_{j,k,\lambda,\mu}\gamma_j^{-1}\gamma_k^{-1}c_{jk\lambda\mu}\, u_{jI,\lambda}\ol u_{kI,\mu}\cr &=\gamma_1\ldots\gamma_n\,\langle A_qS_\gamma u,S_\gamma u\rangle\cr}$$ where $S_\gamma$ is the operator defined by $$S_\gamma u=\sum_K~(\gamma_1\ldots\gamma_n\gamma_K)^{-1}\, u_{K,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge d\ol z_K\otimes e_\lambda.$$ We get therefore $$\eqalign{ |\langle u,v\rangle_\gamma|^2=|\langle u,S_\gamma v\rangle|^2&\le \langle A_q^{-1}u,u\rangle\langle A_q S_\gamma v,S_\gamma v\rangle\cr &\le(\gamma_1\ldots\gamma_n)^{-1}\langle A_q^{-1}u,u\rangle \langle A_{q,\gamma}v,v\rangle_\gamma,\cr}$$ and the choice $v=A_{q,\gamma}^{-1}u$ implies $$\langle A_{q,\gamma}^{-1}u,u\rangle_\gamma\le (\gamma_1\ldots\gamma_n)^{-1}\,\langle A_q^{-1}u,u\rangle\,;$$ this relation is equivalent to the last one in the lemma.\qed \endproof An important special case is that of a semi-positive line bundle $E$. If we let $0\le\lambda_1(x)\le\ldots\le\lambda_n(x)$ be the eigenvalues of $\ii\Theta(E)_x$ with respect to $\omega_x$ for all $x\in X$, formula VI-8.3 implies $$\leqalignno{ \langle A_qu,u\rangle&\ge(\lambda_1+\cdots+\lambda_q)|u|^2,\cr \int_X\langle A_q^{-1}g,g\rangle\,dV&\le\int_X~{1\over\lambda_1+\cdots+ \lambda_q}\,|g|^2\,dV.&(6.4)\cr}$$ A typical situation where these estimates can be applied is the case when $E$ is the trivial line bundle $X\times\bC$ with metric given by a weight $e^{-\varphi}$. One can assume for example that $\varphi$ is plurisubharmonic and that $\ii d'd''\varphi$ has at least $n-q+1$ positive eigenvalues at every point, i.e.\ $\lambda_q>0$ on $X$. This situation leads to very important $L^2$ estimates, which are precisely those given by (H\"ormander~1965, 1966). We state here a slightly more general result. \begstat{(6.5) Theorem} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler manifold, $E$ a hermitian line bundle on $X$, $\varphi\in C^\infty(X,\bR)$ a weight function such that the eigenvalues $\lambda_1\le\ldots\le\lambda_n$ of $\ii\Theta(E)+\ii d'd''\varphi$ are $\ge0$. Then for every form $g$ of type $(n,q)$, $q\ge 1$, with $L^2_\loc$ (resp. $C^\infty$) coefficients such that $D''g=0$ and $$\int_X~{1\over\lambda_1+\cdots+\lambda_q}\,|g|^2\,e^{-\varphi}\,dV<+\infty,$$ we can find a $L^2_\loc$ (resp. $C^\infty$) form $f$ of type $(n,q-1)$ such that $D''f=g$ and $$\int_X~|f|^2\,e^{-\varphi}\,dV\le\int_X~{1\over\lambda_1+\cdots+ \lambda_q}\,|g|^2\,e^{-\varphi}\,dV.$$ \endstat \begproof{} Apply the general estimates to the bundle $E_\varphi$ deduced from $E$ by multiplication of the metric by $e^{-\varphi}\,;$ we have $\ii\Theta(E_\varphi)=\ii\Theta(E)+\ii d'd''\varphi$. It is not necessary here to assume in addition that $g\in L^2_{n,q} (X,E_\varphi)$. In fact, $g$ is in $L^2_\loc$ and we can exhaust X by the relatively compact weakly pseudoconvex domains $$X_c=\big\{x\in X~;~\psi(x)0$, there is a section $F\in H^0(X,K_X\otimes L)$ such that \hbox{$F_{\restriction Y}=f_{\restriction Y}$} and $$\int_X{|F|^2\over(1+|\sigma|^2)^{p+\varepsilon}}\,dV\le \Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,dV.$$ \endstat The proof will involve a weight with logarithmic singularities along $Y$. We must therefore apply the existence theorem over $X\ssm Y$. This requires to know whether $X\ssm Y$ has a complete K\"ahler metric. \begstat{(7.2) Lemma} Let $(X,\omega)$ be a K\"ahler manifold, and $Y=\sigma^{-1}(0)$ an analytic subset defined by a section of a hermitian vector bundle $E\to X$. If $X$ is weakly pseudoconvex and exhausted by $X_c=\{x\in X~;~\psi(x)0$ and let $\eta$ tend to $0$, we see that we can assume $f$ defined in a neighborhood of $\ol U$. Let $h$ be the continuous section of $L$ such that $h=(1-|\sigma|^{p+1})f$ on $U=\{|\sigma|<1\}$ and $h=0$ on $X\ssm U$. We have $h_{\restriction Y}=f_{\restriction Y}$ and $$d''h=-{p+1\over 2}\,|\sigma|^{p-1}\,\{\sigma,D'\sigma\}\,f~~~~{\rm on}~~U, ~~~~d''h=0~~~~{\rm on}~~X\ssm U.$$ We consider $g=d''h$ as a $(n,1)$-form with values in the hermitian line bundle $L_\varphi=L$, endowed with the weight $e^{-\varphi}$ given by $$\varphi=p\log|\sigma|^2+\varepsilon\log(1+|\sigma|^2).$$ Notice that $\varphi$ is singular along $Y$. The Cauchy-Schwarz inequality implies $\ii\{D'\sigma,\sigma\} \wedge\{\sigma,D'\sigma\}\le\ii\{D'\sigma,D'\sigma\}$ as in Lemma~7.2, and we find $$\eqalign{ \ii d'd''\log(1+|\sigma|^2)&={(1+|\sigma|^2)\ii\{D'\sigma,D'\sigma\}- \ii\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over(1+|\sigma|^2)^2}\cr &\quad{}-{\{\ii\Theta(E)\sigma,\sigma\}\over1+|\sigma|^2} \ge{\ii\{D'\sigma,D'\sigma\}\over (1+|\sigma|^2)^2}-{\{\ii\Theta(E)\sigma,\sigma\}\over1+|\sigma|^2}.\cr}$$ The inequality $\ii d'd''\log|\sigma|^2\ge-\{\ii\Theta(E)\sigma,\sigma\}/|\sigma|^2$ obtained in Lemma~7.2 and the above one imply $$\eqalign{\ii\Theta(L_\varphi) &=\ii\Theta(L)+p\,\ii d'd''\log|\sigma|^2+\varepsilon\,\ii d'd''\log(1+|\sigma|^2)\cr &\ge \ii\Theta(L)-\Big({p\over|\sigma|^2}+{\varepsilon\over 1+|\sigma|^2}\Big)\{\ii\Theta(E) \sigma,\sigma\}+\varepsilon{\ii\{D'\sigma,D'\sigma\}\over (1+|\sigma|^2 )^2}\cr &\ge\varepsilon{\ii\,\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over|\sigma|^2 \,(1+|\sigma|^2 )^2},\cr}$$ thanks to the hypothesis on the curvature of $L$ and the Cauchy-Schwarz inequality. Set $\xi=(p+1)/2\,|\sigma|^{p-1}\{D'\sigma,\sigma\} =\sum\xi_j\,dz_j$ in an $\omega$-orthonormal basis $\partial/\partial z_j$, and let $\wh\xi=\sum\xi_j\partial/\partial\ol z_j$ be the dual $(0,1)$-vector field. For every $(n,1)$-form $v$ with values in $L_\varphi$, we find $$\eqalign{ \big|\langle d''h,v\rangle\big|&=\big|\langle\ol\xi\wedge f,v\rangle\big|= \big|\langle f,\wh\xi\ort v\rangle\big|\le|f|\,|\wh\xi\ort v|,\cr \wh\xi\ort v&=\sum-\ii\xi_j\,dz_j\wedge\Lambda v=-\ii\xi\wedge\Lambda v,\cr |\langle d''h,v\rangle|^2&\le|f|^2\,|\wh\xi\ort v|^2= |f|^2\langle-\ii\xi\wedge\Lambda v,\wh\xi\ort v\rangle\cr &=|f|^2\langle-\ii\ol\xi\wedge\xi\wedge\Lambda v,v\rangle= |f|^2\langle[\ii\xi\wedge\ol\xi,\Lambda]v,v\rangle\cr &\le{(p+1)^2\over 4\varepsilon}\,|\sigma|^{2p}\,(1+|\sigma|^2)^2\,|f|^2\, \langle[\ii\Theta(L_\varphi),\Lambda]v,v\rangle.}$$ Thus, in the notations of Th.~6.1, the form $g=d''h$ satisfies $$\langle A_1^{-1}g,g\rangle\le {(p+1)^2\over4\varepsilon}\,|\sigma|^{2p}(1+|\sigma|^2)^2\,|f|^2\le {(p+1)^2\over\varepsilon}\,|f|^2\,e^\varphi,$$ where the last equality results from the fact that $(1+|\sigma|^2)^2\le 4$ on the support of $g$. Lemma 7.2 shows that the existence theorem 6.1 can be applied on each set $X_c\ssm Y$. Letting $c$ tend to infinity, we infer the existence of a $(n,0)$-form $u$ with values in $L$ such that $d''u=g$ on $X\ssm Y$ and $$\eqalign{ \int_{X\ssm Y}|u|^2\,e^{-\varphi}\,dV&\le \int_{X\ssm Y}\langle A_1^{-1}g,g\rangle e^{-\varphi},~~~~{\rm thus}\cr \int_{X\ssm Y}{|u|^2\over|\sigma|^{2p}(1+|\sigma|^2)^\varepsilon}\,dV &\le{(p+1)^2\over\varepsilon}\int_U|f|^2\,dV.\cr}$$ This estimate implies in particular that $u$ is locally $L^2$ near $Y$. As $g$ is conti\-nuous over $X$, Lemma~7.3 below shows that the equality \hbox{$d''u=g=d''h$} extends to $X$, thus $F=h-u$ is holomorphic everywhere. Hence \hbox{$u=h-F$} is continuous on $X$. As $|\sigma(x)|\le C\,d(x,Y)$ in a neighborhood of every point of $Y$, we see that $|\sigma|^{-2p}$ is non integrable at every point $x_0\in Y_{\rm reg}$, because $\codim Y\le p$. It follows that $u=0$ on $Y$, so $$F_{\restriction Y}=h_{\restriction Y}=f_{\restriction Y}.$$ The final $L^2$-estimate of Th.~7.1 follows from the inequality $$|F|^2=|h-u|^2\le(1+|\sigma|^{-2p})\,|u|^2+(1+|\sigma|^{2p})\,|f|^2$$ which implies $${|F|^2\over(1+|\sigma|^2)^p}\le{|u|^2\over|\sigma|^{2p}}+|f|^2. \eqno{\square}$$ \endproof \begstat{(7.3) Lemma} Let $\Omega$ be an open subset of $\bC^n$ and $Y$ an analytic subset of $\Omega$. Assume that $v$ is a $(p,q-1)$-form with $L^2_\loc$ coefficients and $w$ a $(p,q)$-form with $L^1_\loc$ coefficients such that $d''v=w$ on $\Omega\ssm Y$ $($in the sense of distribution theory$)$. Then $d''v=w$ on $\Omega$. \endstat \begproof{} An induction on the dimension of $Y$ shows that it is sufficient to prove the result in a neighborhood of a regular point $a\in Y$. By using a local analytic isomorphism, the proof is reduced to the case where $Y$ is contained in the hyperplane $z_1=0$, with $a=0$. Let $\lambda\in C^\infty(\bR,\bR)$ be a function such that $\lambda(t)=0$ for $t\le{1\over 2}$ and $\lambda(t)=1$ for $t\ge 1$. We must show that $$\int_\Omega w\wedge\alpha=(-1)^{p+q}\int_\Omega v\wedge d''\alpha \leqno(7.4)$$ for all $\alpha\in\cD_{n-p,n-q}(\Omega)$. Set $\lambda_\varepsilon(z)= \lambda(|z_1|/\varepsilon)$ and replace $\alpha$ in the integral by $\lambda_\varepsilon\alpha$. Then $\lambda_\varepsilon\alpha\in \cD_{n-p,n-q}(\Omega\ssm Y)$ and the hypotheses imply $$\int_\Omega w\wedge\lambda_\varepsilon\alpha=(-1)^{p+q} \int_\Omega v\wedge d''(\lambda_\varepsilon\alpha)=(-1)^{p+q} \int_\Omega v\wedge (d''\lambda_\varepsilon\wedge\alpha+\lambda_\varepsilon d''\alpha).$$ As $w$ and $v$ have $L^1_\loc$ coefficients on $\Omega$, the integrals of $w\wedge\lambda_\varepsilon\alpha$ and $v\wedge\lambda_\varepsilon d''\alpha$ converge respectively to the integrals of $w\wedge\alpha$ and $v\wedge d''\alpha$ as $\varepsilon$ tends to $0$. The remaining term can be estimated by means of the Cauchy-Schwarz inequality: $$\Big|\int_\Omega v\wedge d''\lambda_\varepsilon\wedge\alpha\Big|^2\le \int_{|z_1|\le\varepsilon}|v\wedge\alpha|^2\,dV.~ \int_{\Supp\alpha}|d''\lambda_\varepsilon|^2\,dV\,;$$ as $v\in L^2_\loc(\Omega)$, the integral $\int_{|z_1|\le\varepsilon}|v\wedge\alpha|^2\,dV$ converges to $0$ with $\varepsilon$, whereas $$\int_{\Supp\alpha}|d''\lambda_\varepsilon|^2\,dV\le {C\over\varepsilon^2}\,{\rm Vol}\big(\Supp\,\alpha \cap\{|z_1|\le\varepsilon\}\big)\le C'.$$ Equality (7.4) follows when $\varepsilon$ tends to $0$.\qed \endproof \begstat{(7.5) Corollary} Let $\Omega\subset\bC^n$ be a weakly pseudoconvex domain and let $\varphi$, $\psi$ be plurisubharmonic functions on $\Omega$, where $\psi$ is supposed to be finite and continuous. Let $\sigma=(\sigma_1,\ldots, \sigma_r)$ be a family of holomorphic functions on $\Omega$, let \hbox{$Y=\sigma^{-1}(0)$}, $p={}$ maximal codimension of $Y$ and set \smallskip \item{\rm a)} $U=\{z\in\Omega\,;\,|\sigma(z)|^20$ and every holomorphic function $f$ on $U$, there exists a holomorphic function $F$ on $\Omega$ such that $F_{\restriction Y}=f_{\restriction Y}$ and $$\leqalignno{ \int_\Omega {|F|^2\,e^{-\varphi+p\psi}\over (1+|\sigma|^2e^{\psi})^{p+\varepsilon}}\,dV&\le \Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,e^{-\varphi+p\psi}\,dV, ~~~~{\it resp.}&{\rm a)}\cr \int_\Omega {|F|^2\,e^{-\varphi}\over (e^\psi+|\sigma|^2)^{p+\varepsilon}}\,dV&\le \Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,e^{-\varphi-(p+ \varepsilon)\psi}\,dV.&{\rm b)}\cr}$$ \endstat \begproof{} After taking convolutions with smooth kernels on pseudoconvex subdomains $\Omega_c\subset\!\subset\Omega$, we may assume $\varphi$, $\psi$ smooth. In either case a) or~b), apply Th.~7.1 to \medskip \noindent a)~~$E=\Omega\times\bC^r$ with the weight $e^\psi$, $L=\Omega\times\bC$ with the weight $e^{-\varphi+p\psi}$, and $U=\{|\sigma|^2e^\psi<1\}$. Then $$\ii\Theta(E)=-\ii d'd''\psi\otimes\Id_E\le 0,~~~~\ii\Theta(L)=\ii d'd''\varphi- p\,\ii d'd''\psi\ge p\,\ii\Theta(E).$$ \smallskip \noindent b)~~$E=\Omega\times\bC^r$ with the weight $e^{-\psi}$, $L=\Omega\times\bC$ with the weight $e^{-\varphi-(p+\varepsilon)\psi}$, and $U=\{|\sigma|^2e^{-\psi}<1\}$. Then $$\ii\Theta(E)=\ii d'd''\psi\otimes\Id_E\ge 0,~~~~\ii\Theta(L)=\ii d'd''\varphi+ (p+\varepsilon)\,\ii d'd''\psi\ge(p+\varepsilon)\,\ii\Theta(E).$$ The condition on $\Theta(L)$ is satisfied in both cases and $K_{\Omega}$ is trivial.\qed \endproof \begstat{(7.6) H\"ormander-Bombieri-Skoda theorem} Let $\Omega\subset\bC^n$ be a weakly pseudoconvex domain and $\varphi$ a plurisubharmonic function on $\Omega$. For every $\varepsilon>0$ and every point $z_0\in\Omega$ such that $e^{-\varphi}$ is integrable in a neighborhood of $z_0$, there exists a holomorphic function $F$ on $\Omega$ such that $F(z_0)=1$ and $$\int_\Omega{|F(z)|^2\,e^{-\varphi(z)}\over(1+|z|^2)^{n+\varepsilon}}\,dV< +\infty.$$ \endstat (Bombieri~1970) originally stated the theorem with the exponent $3n$ instead of $n+\varepsilon\,;$ the improved exponent $n+\varepsilon$ is due to (Skoda~1975). The example $\Omega=\bC^n$, $\varphi(z)=0$ shows that one cannot replace $\varepsilon$ by $0$. \begproof{} Apply Cor.~7.5 b) to $f\equiv 1$, $\sigma(z)=z-z_0$, $p=n$ and $\psi\equiv\log r^2$ where $U=B(z_0,r)$ is a ball such that $\int_U e^{-\varphi}\,dV<+\infty$.\qed \endproof \begstat{(7.7) Corollary} Let $\varphi$ be a plurisubharmonic function on a complex ma\-ni\-fold $X$. Let $A$ be the set of points $z\in X$ such that $e^{-\varphi}$ is not locally integrable in a neighborhood of $z$. Then $A$ is an analytic subset of $X$. \endstat \begproof{} Let $\Omega\subset X$ be an open coordinate patch isomorphic to a ball of~$\bC^n$, with coordinates $(z_1,\ldots,z_n)$. Define $E\subset H^0(\Omega,\cO)$ to be the Hilbert space of holomorphic functions $f$ on $\Omega$ such that $$\int_\Omega |f(z)|^2e^{-\varphi(z)}\,dV(z)<+\infty.$$ Then $A\cap\Omega=\bigcap_{f\in E}f^{-1}(0)$. In fact, every $f$ in $E$ must obviously vanish on~$A\,;$ conversely, if $z_0\notin A$, Th.~7.6 shows that there exists $f\in E$ such that $f(z_0)\ne 0$. By Th.~II-5.5, we conclude that $A$ is analytic.\qed \endproof \titleb{8.}{Applications to Hypersurface Singularities} We first give some basic definitions and results concerning multiplicities of divisors on a complex manifold. \begstat{(8.1) Proposition} Let $X$ be a complex manifold and $\Delta=\sum\lambda_j[Z_j]$ a divisor on $X$ with real coefficients $\lambda_j\ge 0$. Let $x\in X$ and $f_j=0$, $1\le j\le N$, irreducible equations of $Z_j$ on a neighborhood $U$ of $x$. \smallskip \item{\rm a)}~~The multiplicity of $\Delta$ at $x$ is defined by $$\mu(\Delta,x)=\sum\lambda_j\,{\rm ord}_x f_j.$$ \smallskip \item{\rm b)} $\Delta$ is said to have normal crossings at a point $x\in\Supp\,\Delta$ if all hypersurfaces $Z_j$ containing $x$ are smooth at $x$ and intersect transversally, i.e.\ if the linear forms $df_j$ defining the corresponding tangent spaces $T_xZ_j$ are linearly independent at $x$. The set ${\rm nnc}(\Delta)$ of non normal crossing points is an analytic subset of $X$. \smallskip \item{\rm c)} The non-integrability locus ${\rm nil}(\Delta)$ is defined as the set of points $x\in X$ such that $\prod|f_j|^{-2\lambda_j}$ is non integrable near $x$. Then ${\rm nil}(\Delta)$ is an analytic subset of $X$ and there are inclusions $$\{x\in X\,;\,\mu(\Delta,x)\ge n\}\subset{\rm nil}(\Delta)\subset \{x\in X\,;\,\mu(\Delta,x)\ge 1\}.$$ \item{}Moreover ${\rm nil}(\Delta)\subset{\rm nnc}(\Delta)$ if all coefficients of $\Delta$ \hbox{satisfy $\lambda_j<1$}. \endstat \begproof{} b) The set ${\rm nnc}(\Delta)\cap U$ is the union of the analytic sets $$f_{j_1}=\ldots=f_{j_p}=0,~~~~df_{j_1}\wedge\ldots\wedge df_{j_p}=0,$$ for each subset $\{j_1,\ldots,j_p\}$ of the index set $\{1,\ldots,N\}$. Thus ${\rm nnc}(\Delta)$ is analytic. \medskip\noindent c) The analyticity of ${\rm nil}(\Delta)$ follows from Cor.~7.7 applied to the plurisubharmonic function $\varphi=\sum 2\lambda_j\, \log|f_j|$. Assume first that $\lambda_j<1$ and that $\Delta$ has normal crossings at $x$. Let $f_{j_1}(x)=\ldots=f_{j_s}(x)=0$ and \hbox{$f_j(x)\ne 0$} for $j\ne j_l$. Then, we can choose local coordinates $(w_1,\ldots,w_n)$ on $U$ such that $w_1=f_{j_1}(z)$, $\ldots$, $w_s=f_{j_s}(z)$, and we have $$\int_{U}{d\lambda(z)\over\prod|f_j(z)|^{2\lambda_j}} \le\int_{U}{C\,d\lambda(w)\over |w_1|^{2\lambda_1}\ldots|w_s|^{2\lambda_s}}<+\infty.$$ It follows that ${\rm nil}(\Delta)\subset{\rm nnc}(\Delta)$. Let us prove now the statement relating ${\rm nil}(\Delta)$ with multiplicity sets. Near any point $x$, we have $|f_j(z)|\le C_j|z-x|^{m_j}$ with $m_j={\rm ord}_x f_j$, thus $$\prod|f_j|^{-2\lambda_j}\ge C\,|z-x|^{-2\mu(\Delta,x)}.$$ It follows that $x\in{\rm nil}(\Delta)$ as soon as $\mu(\Delta,x)\ge n$. On the other hand, we are going to prove that $\mu(\Delta,x)<1$ implies $x\notin{\rm nil}(\Delta)$, i.e.\ $\prod|f_j|^{-2\lambda_j}$ integrable near $x$. We may assume $\lambda_j$ rational; otherwise replace each $\lambda_j$ by a slightly larger rational number in such a way that $\mu(\Delta,x)<1$ is still true. Set $f=\prod f_j^{k\lambda_j}$ where $k$ is a common denominator. The result is then a consequence of the following lemma.\qed \endproof \begstat{(8.2) Lemma} If $f\in\cO_{X,x}$ is not identically $0$, there exists a neighbor\-hood $U$ of $x$ such that $\int_U|f|^{-2\lambda}\,dV$ converges for all $\lambda<1/m$, $m={\rm ord}_x f$. \endstat \begproof{} One can assume that $f$ is a Weierstrass polynomial $$f(z)=z_n^m+a_1(z')z_n^{m-1}+\cdots+a_m(z'),~~~~a_j(z')\in\cO_{n-1},~~~~ a_j(0)=0,$$ with respect to some coordinates $(z_1,\ldots,z_n)$ centered at $x$. Let $v_j(z')$, \hbox{$1\le j\le m$,} denote the roots $z_n$ of $f(z)=0$. On a small neighborhood $U$ of $x$ we have $|v_j(z')|\le 1$. The inequality between arithmetic and geometric mean implies $$\eqalign{ \int_{\{|z_n|\le 1\}}|f(z)|^{-2\lambda}\,dx_ndy_n&=\int_{\{|z_n|\le 1\}} \prod_{1\le j\le m}|z_n-v_j(z')|^{-2\lambda}\,dx_ndy_n\cr &\le{1\over m}\int_{\{|z_n|\le 1\}}\sum_{1\le j\le m}|z_n-v_j(z') |^{-2m\lambda}\,dx_ndy_n\cr &\le\int_{\{|z_n|\le 2\}}{dx_ndy_n\over|z_n|^{2m\lambda}},\cr}$$ so the Lemma follows from the Fubini theorem.\qed \endproof Another interesting application concerns the study of multiplicities of singular points for algebraic hypersurfaces in $\bP^n$. Following (Waldschmidt~1975), we introduce the following definition. \begstat{(8.3) Definition} Let $S$ be a finite subset of $\bP^n$. For any integer $t\ge 1$, we define $\omega_t(S)$ as the minimum of the degrees of non zero homogeneous polynomials $P\in\bC[z_0,\ldots,z_n]$ which vanish at order $t$ at every point of $S$, i.e.\ $D^\alpha P(w)=0$ for every $w\in S$ and every multi-index $\alpha=(\alpha_0,\ldots,\alpha_n)$ of length $|\alpha|a$ this means that $E\cap Y=\emptyset\,)$. \endstat \begproof{} By induction on $n$, it suffices to show that $\dim(E\cap H)\le a-1$ for a generic hyperplane $H\subset\bP^n$. Let $E_j$ be the (finite) family of irreducible components of $E$, and $w_j\in E_j$ an arbitrary point. Then $E\cap H=\bigcup E_j\cap H$ and we have $\dim E_j\cap H<\dim E_j\le a$ as soon as $H$ avoids all points $w_j$.\qed \endproof \begproof{of Theorem 8.6.} By Bertini's lemma, there exists a linear subspace $Y\subset\bP^n$ of codimension $a+1$ such that ${\rm nil}(\Delta)\cap Y=\emptyset$. We consider $P$ as a section of the line bundle $\cO(D)$ over $\bP^n$, where $D=\deg\,P$~ (cf.\ Th.~V-15.5). There are sections $\sigma_1,\ldots,\sigma_{a+1}$ of $\cO(1)$ such that $Y=\sigma^{-1}(0)$. We shall apply Th.~7.1 to $E=\cO(1)$ with its standard hermitian metric, and to $L=\cO(k)$ equipped with the additional weight $\varphi=\alpha\log|P|^2$. We may assume that the open set $U=\{|\sigma|<1\}$ is such that ${\rm nil}(\Delta)\cap\ol U=\emptyset$, otherwise it suffices to multiply $\sigma$ by a large constant. This implies that the polynomial $Q=\prod P_j^{[k_j\alpha]}$ satisfies $$\int_U|Q|^2\,e^{-\varphi}\,dV= \int_U\prod|P_j|^{-2(k_j\alpha-[k_j\alpha])}\,dV<+\infty.$$ Set $\omega=ic\big(\cO(1)\big)$. We have $\ii d'd''\log|P|^2\ge-ic\big(\cO(D) \big)=-D\omega$ by the Lelong-Poincar\'e equation, thus $\ii\Theta(L_\varphi)\ge(k-\alpha D)\omega$. The desired curvature inequality $\ii\Theta(L_\varphi)\ge(a+1+\varepsilon)\ii\Theta(E)$ is satisfied if $k-\alpha D\ge (a+1+\varepsilon)$. We thus take $$k=[\alpha D]+a+2.$$ The section $f\in H^0(U,K_{\bP^n}\otimes L)=H^0\big(U,\cO(k-n-1)\big)$ is taken to be a multiple of $Q$ by some polynomial. This is possible provided that $$k-n-1\ge\deg\,Q~~~\Longleftrightarrow~~~ \alpha D+a+2-n-1\ge\sum[k_j\alpha]\,\deg\,P_j,$$ or equivalently, as $D=\sum k_j\,\deg\,P_j$, $$\sum(k_j\alpha-[k_j\alpha])\,\deg\,P_j\ge n-a-1.\leqno(8.10)$$ Then we get $f\in H^0(U,K_{\bP^n}\otimes L)$ such that $\int_U |f|^2\,e^{-\varphi}\,dV<+\infty$. Theorem 7.1 implies the existence of $F\in H^0(\bP^n,K_{\bP^n}\otimes L)$, i.e.\ of a polynomial $F$ of degree $k-n-1$, such that $$\int_{\bP^n}|F|^2e^{-\varphi}\,dV=\int_{\bP^n}{|F|^2\over|P|^{2\alpha}} \,dV<+\infty\,;$$ observe that $|\sigma|$ is bounded, for we are on a compact manifold. Near any $w\in S$, we have $|P(z)|\le C|z-w|^{t_2}$, thus \hbox{$|P(z)|^{2\alpha}\le C|z-w|^{2(t_1+n-1)}$}. This implies that the above integral can converge only if $F$ vanishes at order $\ge t_1$ at each point $w\in S$. Therefore $$\omega_{t_1}(S)\le\deg\,F=k-n-1= [\alpha D]+a+1-n\le\alpha\omega_{t_2}(S)+a+1-n,$$ which is the desired inequality. However, the above proof only works under the additional assumption (8.10). Assume on the contrary that $$\beta=\sum(k_j\alpha-[k_j\alpha])\,\deg\,P_j0$. Then for every $D''$-closed form $f$ of type $(n,k)$ with values in $Q\otimes L$ such that $\|f\|<+\infty$, there exists a $D''$-closed form $h$ of type $(n,k)$ with values in $E\otimes L$ such that $f=g\cdot h$ and $$\|h\|^2\le(1+m/\varepsilon)\,\|f\|^2.$$ \endstat The idea of the proof is essentially due to (Skoda~1978), who actually proved the special case $k=0$. The general case appeared in (Demailly~1982c). \begproof{} Let $j:S\to E$ be the inclusion morphism, $g^\star:Q\to E$ and $j^\star:E\to S$ the adjoints of $g,j$, and $$D_E = \pmatrix{D_S&-\beta^\star\cr \beta & D_Q\cr},~~ \beta\in C^\infty_{1,0}\big(X,\hom(S,Q)\big),~~ \beta^\star\in C^\infty_{0,1}\big(X,\hom(Q,S)\big),$$ the matrix of $D_E$ with respect to the orthogonal splitting $E\simeq S\oplus Q$ (cf.\ \S V-14). Then $g^\star f$ is a lifting of $f$ in $E\otimes L$. We shall try to find $h$ under the form $$h=g^\star f+ju,~~~~u\in L^2_{n,k}(X,S\otimes L).$$ As the images of $S$ and $Q$ in $E$ are orthogonal, we have $|h|^2=|f|^2+|u|^2$ at every point of $X$. On the other hand $D''_{Q\otimes L}f=0$ by hypothesis and $D''g^\star=-j\circ\beta^\star$ by V-14.3~d), hence $$D''_{E\otimes L}h=-j(\beta^\star\wedge f)+j\,D''_{S\otimes L}= j(D''_{S\otimes L}-\beta^\star\wedge f).$$ We are thus led to solve the equation $$D''_{S\otimes L}u=\beta^\star\wedge f,\leqno(9.4)$$ and for that, we apply Th.~4.5 to the $(n,k+1)$-form $\beta^\star\wedge f$. One observes now that the curvature of $S\otimes L$ can be expressed in terms of $\beta$. This remark will be used to prove: \endproof \begstat{(9.5) Lemma} $\langle A_k^{-1}(\beta^\star\wedge f), (\beta^\star\wedge f)\rangle\le(m/\varepsilon)\,|f|^2$. \endstat If the Lemma is taken for granted, Th.~4.5 yields a solution $u$ of~(9.4) in $L^2_{n,q}(X,S\otimes L)$ such that $\|u\|^2\le(m/\varepsilon)\,\|f\|^2$. As \hbox{$\|h\|^2=\|f\|^2+\|u\|^2$}, the proof of Th.~9.3 is complete.\qed \begproof{of Lemma~9.5.} Exactly as in the proof of Th.~VII-10.3, formulas (V-14.6) and (V-14.7) yield $$\ii\Theta(S)\ge_m \ii\beta^\star\wedge\beta,~~~~ \ii\Theta(\det Q)\ge\Tr_Q(\ii\beta\wedge \beta^\star)=\Tr_S(-\ii\beta^\star\wedge\beta).$$ Since $C^\infty_{1,1}(X,\Herm\,S)\ni\Theta:=-\ii\beta^\star\wedge\beta \ge_\Grif0$, Prop.~VII-10.1 implies $$m\,\Tr_S(-\ii\beta^\star\wedge\beta)\otimes\Id_S+\ii\beta^\star\wedge \beta\ge_m 0.$$ From the hypothesis on the curvature of $L$ we get $$\eqalign{ \ii\Theta(S\otimes L)&\ge_m \ii\Theta(S)\otimes\Id_L+(m+\varepsilon)\, \ii\Theta(\det Q)\otimes\Id_{S\otimes L}\cr &\ge_m\big(\ii\beta^\star\wedge\beta+(m+\varepsilon)\,\Tr_S(-\ii\beta^\star \wedge\beta)\otimes\Id_S\big)\otimes\Id_L\cr &\ge_m(\varepsilon/m)\,(-\ii\beta^\star\wedge\beta)\otimes\Id_S\otimes \Id_L.\cr}$$ For any $v\in\Lambda^{n,k+1}T^\star_X\otimes S\otimes L$, Lemma~VII-7.2 implies $$\langle A_{k,S\otimes L}v,v\rangle\ge(\varepsilon/m)\, \langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle,\leqno(9.6)$$ because $\rk(S\otimes L)=s$ and $m=\min\{n-k,s\}$. Let $(dz_1,\ldots,dz_n)$ be an orthonormal basis of $T^\star_X$ at a given point $x_0\in X$ and set $$\beta=\sum_{1\le j\le n}dz_j\otimes\beta_j,~~~~\beta_j\in\hom(S,Q).$$ The adjoint of the operator~ $\beta^\star\wedge\bu=\sum d\ol z_j\wedge \beta_j^\star\,\bu$ ~is the contraction $\beta\ort \bu$ defined by $$\beta\ort v=\sum{\partial\over\partial\ol z_j}\ort (\beta_j v)= \sum -\ii dz_j\wedge\Lambda(\beta_j v)=-\ii\beta\wedge\Lambda v.$$ We get consequently $\langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle =|\beta\ort v|^2$ and (9.6) implies $$|\langle\beta^\star\wedge f,v\rangle|^2=|\langle f,\beta\ort v\rangle|^2\le |f|^2\,|\beta\ort v|^2\le(m/\varepsilon) \langle A_{k,S\otimes L}v,v\rangle\,|f|^2.\eqno{\square}$$ \endproof If $X$ has a plurisubharmonic exhaustion function $\psi$, we can select a convex increasing function $\chi\in C^\infty(\bR,\bR)$ and multiply the metric of $L$ by the weight $\exp(-\chi\circ\psi)$ in order to make the $L^2$ norm of $f$ converge. Theorem 9.3 implies therefore: \begstat{(9.7) Corollary} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler manifold, let $g:E\to Q$ be a surjective bundle morphism with $r=\rk\,E$, $q=\rk\,Q$, let \hbox{$m=\min\{n-k,r-q\}$} and let $L\to X$ be a hermitian line bundle. Suppose that $E\ge_m 0$ and $$\ii\Theta(L)-(m+\varepsilon)\,\ii\Theta(\det Q)\ge 0$$ for some $\varepsilon>0$. Then $g$ induces a surjective map $$H^k(X,K_X\otimes E\otimes L)\longrightarrow H^k(X,K_X\otimes Q\otimes L).$$ \endstat The most remarkable feature of this result is that it does not require any strict positivity assumption on the curvature (for instance $E$ can be a flat bundle). A careful examination of the proof shows that it amounts to verify that the image of the coboundary morphism $$-\beta^\star\wedge\bu~:~H^k(X,K_X\otimes Q\otimes L)\longrightarrow H^{k+1}(X,K_X\otimes S\otimes L)$$ vanishes; however the cohomology group $H^{k+1}(X,K_X\otimes S\otimes L)$ itself does not vanish in general as it would do under a strict positivity assumption (cf.\ Th.~VII-9.4). We want now to get also estimates when $Q$ is endowed with a metric given a priori, that can be distinct from the quotient metric of $E$ by $g$. Then the map $g^\star(gg^\star)^{-1}~:~Q\longrightarrow E$ is the lifting of $Q$ orthogonal to $S=\Ker g$. The quotient metric $|\bu|'$ on $Q$ is therefore defined in terms of the original metric $|\bu|$ by $$|v|^{\prime 2}=|g^\star(gg^\star)^{-1}v|^2=\langle(gg^\star)^{-1}v,v\rangle =\det(gg^\star)^{-1}\,\langle\wt{gg^\star}v,v\rangle$$ where $\wt{gg^\star}\in\End(Q)$ denotes the endomorphism of $Q$ whose matrix is the transposed of the comatrix of $gg^\star$. For every $w\in\det Q$, we find $$|w|^{\prime2}=\det(gg^\star)^{-1}\,|w|^2.$$ If $Q'$ denotes the bundle $Q$ with the quotient metric, we get $$\ii\Theta(\det Q')=\ii\Theta(\det Q)+\ii d'd''\log\det(gg^\star).$$ In order that the hypotheses of Th.~9.3 be satisfied, we are led to define a new metric $|\bu|'$ on $L$ by $|u|^{\prime 2}=|u|^2\,\big(\det(gg^\star) \big)^{-m-\varepsilon}$. Then $$\ii\Theta(L')=\ii\Theta(L)+(m+\varepsilon)\,\ii d'd''\log\det(gg^\star)\ge (m+\varepsilon)\,\ii\Theta(\det Q').$$ Theorem 9.3 applied to $(E,Q',L')$ can now be reformulated: \begstat{(9.8) Theorem} Let $X$ be a complete K\"ahler manifold equipped with a K\"ahler metric $\omega$ on~$X$, let $E\to Q$ be a surjective morphism of hermitian vector bundles and let $L\to X$ be a hermitian line bundle. Set $r=\rk\,E$, $q=\rk\,Q$ and $m=\min\{n-k,r-q\}$ and suppose $E\ge_m 0$, $$\ii\Theta(L)-(m+\varepsilon)\ii\Theta(\det Q)\ge 0$$ for some $\varepsilon>0$. Then for every $D''$-closed form $f$ of type $(n,k)$ with values in $Q\otimes L$ such that $$I=\int_X\langle\wt{gg^\star}f,f\rangle\,(\det gg^\star)^{-m-1-\varepsilon}\, dV<+\infty,$$ there exists a $D''$-closed form $h$ of type $(n,k)$ with values in $E\otimes L$ such that $f=g\cdot h$ and $$\int_X|h|^2\,(\det gg^\star)^{-m-\varepsilon}\,dV \le(1+m/\varepsilon)\,I.\eqno\square$$ \endstat Our next goal is to extend Th.~9.8 in the case when $g~:~E\longrightarrow Q$ is only {\it generically} surjective; this means that the analytic set $$Y=\{x\in X~;~g_x~:~E_x\longrightarrow Q_x~{\rm is~not~surjective~}\}$$ defined by the equation $\Lambda^q g=0$ is nowhere dense in $X$. Here $\Lambda^q g$ is a section of the bundle $\hom(\Lambda^qE,\det Q)$. \begstat{(9.9) Theorem} The existence statement and the estimates of Th.~$9.8$ remain true for a generically surjective morphism $g:E\to Q$ provided that $X$ is weakly pseudoconvex. \endstat \begproof{} Apply Th.~9.8 to each relatively compact domain $X_c\ssm Y$ (these domains are complete K\"ahler by Lemma~7.2). From a sequence of solutions on $X_c\ssm Y$ we can extract a subsequence converging weakly on $X\ssm Y$ as c tends to $+\infty$. One gets a form $h$ satisfying the estimates, such that $D''h=0$ on $X\ssm Y$ and $f=g\cdot h$. In order to see that $D''h=0$ on $X$, it suffices to apply Lemma~7.3 and to observe that $h$ has $L^2_\loc$ coefficients on $X$ by our estimates.\qed \endproof A very special but interesting case is obtained for the trivial bundles $E=\Omega\times\bC^r$, $Q=\Omega\times\bC$ over a pseudoconvex open set \hbox{$\Omega\subset\bC^n$}. Then the morphism $g$ is given by a $r$-tuple $(g_1,\ldots,g_r)$ of holomorphic functions on $\Omega$. Let us take $k=0$ and $L=\Omega\times\bC$ with the metric given by a weight $e^{-\varphi}$. If we observe that $\wt{gg^\star}=\Id$ when $\rk\,Q=1$, Th.~9.8 applied on \hbox{$X=\Omega\ssm g^{-1}(0)$} and Lemmas~7.2, 7.3 give: \begstat{(9.10) Theorem {\rm(Skoda 1978)}} Let $\Omega$ be a complete K\"ahler open subset of $\bC^n$ and $\varphi$ a plurisubharmonic function on $\Omega$. Set $m=\min\{n,r-1\}$. Then for every holomorphic function $f$ on $\Omega$ such that $$I=\int_{\Omega\ssm Z}|f|^2\,|g|^{-2(m+1+\varepsilon)}e^{-\varphi}\,dV <+\infty,$$ where $Z=g^{-1}(0)$, there exist holomorphic functions $(h_1,\ldots,h_r)$ on $\Omega$ such that $f=\sum g_jh_j$ and $$\int_{\Omega\ssm Y}|h|^2\,|g|^{-2(m+\varepsilon)}e^{-\varphi}\,dV \le(1+m/\varepsilon)I.\eqno{\square}$$ \endstat This last theorem can be used in order to obtain a quick solution of the Levi problem mentioned in \S I-4. It can be used also to prove a result of (Diederich-Pflug~1981), relating the pseudoconvexity property and the existence of complete K\"ahler metrics for domains of $\bC^n$. \begstat{(9.11) Theorem} Let $\Omega\subset\bC^n$ be an open subset. Then: \smallskip \item{\rm a)} $\Omega$ is a domain of holomorphy if and only if $\Omega$ is pseudoconvex~$;$ \smallskip \item{\rm b)} If $\omcirc=\Omega$ and if $\Omega$ has a complete K\"ahler metric $\wh\omega$, then $\Omega$ is pseudoconvex.\smallskip \endstat Note that statement b) can be false if the assumption $\omcirc=\Omega$ is omitted: in fact $\bC^n\ssm\{0\}$ is complete K\"ahler by Lemma~7.2, but it is not pseudoconvex if $n\ge 2$. \begproof{} b) By Th.~I-4.12, it is enough to verify that $\Omega$ is a domain of holomorphy, i.e.\ that for every connected open subset $U$ such that $U\cap\partial\Omega\ne\emptyset$ and every connected component W of $U\cap\Omega$ there exists a holomorphic function $h$ on $\Omega$ such that $h_{\restriction W}$ cannot be continued to~$U$. Since $\omcirc=\Omega$, the set $U\ssm\ol\Omega$ is not empty. We select $a\in U\ssm\ol\Omega$. Then the integral $$\int_\Omega|z-a|^{-2(n+\varepsilon)}\,dV(z)$$ converges. By Th.~9.10 applied to $f(z)=1$, $g_j(z)=z_j-a_j$ and $\varphi=0$, there exist holomorphic functions $h_j$ on $\Omega$ such that $\sum(z_j-a_j)\,h_j(z)=1$. This shows that at least one of the functions $h_j$ cannot be analytically continued at $a\in U$. \medskip \noindent a) Assume that $\Omega$ is pseudoconvex. Given any open connected set $U$ such that $U\cap\partial\Omega\ne\emptyset$, choose $a\in U\cap\partial\Omega$. By Th.~I-4.14~c) the function $$\varphi(z)=(n+\varepsilon)(\log(1+|z|^2)-2\log d(z,\complement\Omega)\big)$$ is plurisubharmonic on $\Omega$. Then the integral $$\int_\Omega|z-a|^{-2(n+\varepsilon)}\,e^{-\varphi(z)}\,dV(z) \le\int_\Omega(1+|z|^2)^{-n-\varepsilon}\,dV(z)$$ converges, and we conclude as for b).\qed \endproof \titleb{10.}{Application of Skoda's $L^2$ Estimates to Local Algebra} We apply here Th.~9.10 to the study of ideals in the ring $\cO_n=\bC\{z_1,\ldots,z_n\}$ of germs of holomorphic functions on $(\bC^n,0)$. Let $\cI=(g_1,\ldots,g_r)\ne(0)$ be an ideal of $\cO_n$. \begstat{(10.1) Definition} Let $k\in\bR_+$. We associate to $\cI$ the following ideals: \smallskip \item{\rm a)} the ideal $\ol\cI^{(k)}$ of germs $u\in\cO_n$ such that $|u|\le C|g|^k$ for some constant $C\ge 0$, where $|g|^2=|g_1|^2+\cdots+|g_r|^2$. \smallskip \item{\rm b)} the ideal $\wh\cI^{(k)}$ of germs $u\in\cO_n$ such that $$\int_\Omega|u|^2\,|g|^{-2(k+\varepsilon)}\,dV<+\infty$$ on a small ball $\Omega$ centered at $0$, if $\varepsilon>0$ is small enough.\smallskip \endstat \begstat{(10.2) Proposition} For all $k,l\in\bR_+$ we have \smallskip \item{\rm a)} $\ol\cI^{(k)}\subset\wh\cI^{(k)}\,;$ \smallskip \item{\rm b)} $\cI^k\subset\ol\cI^{(k)}$ if $k\in\bN\,;$ \smallskip \item{\rm c)} $\ol\cI^{(k)}.\ol\cI^{(l)}\subset\ol\cI^{(k+l)}\,;$ \smallskip \item{\rm d)} $\ol\cI^{(k)}.\wh\cI^{(l)}\subset\wh\cI^{(k+l)}.$ \endstat All properties are immediate from the definitions except a) which is a consequence of Lemma~8.2. Before stating the main result, we need a simple lemma. \begstat{(10.3) Lemma} If $\cI=(g_1,\ldots,g_r)$ and $r>n$, we can find elements $\wt g_1,\ldots,\wt g_n\in\cI$ such that $C^{-1}|g|\le|\wt g|\le C|g|$ on a neighborhood of $0$. Each $\wt g_j$ can be taken to be a linear combination $$\wt g_j=a_j.\,g=\sum_{1\le k\le r}a_{jk}g_k,~~~~a_j\in\bC^r\ssm\{0\}$$ where the coefficients $([a_1],\ldots,[a_n])$ are chosen in the complement of a proper analytic subset of $(\bP^{r-1})^n$. \endstat It follows from the Lemma that the ideal $\cJ=(\wt g_1,\ldots,\wt g_n)\subset\cI$ satisfies $\smash{\ol\cJ}^{(k)}=\smash{\ol\cI}^{(k)}$ and $\smash{\wh\cJ}^{(k)}=\smash{\wh\cI}^{(k)}$ for all $k$. \begproof{} Assume that $g\in\cO(\Omega)^r$. Consider the analytic subsets in $\Omega\times(\bP^{r-1})^n$ defined by $$\eqalign{ A&=\big\{(z,[w_1],\ldots,[w_n])\,;\,w_j.\,g(z)=0\big\},\cr A^\star&=\bigcup{\rm irreducible~components~of}~A~{\rm not~contained~in~} g^{-1}(0)\times(\bP^{r-1})^n.\cr}$$ For $z\notin g^{-1}(0)$ the fiber $A_z=\{([w_1],\ldots,[w_n])\,;\,w_j.\,g(z)=0\} =A^\star_z$ is a product of $n$ hyperplanes in $\bP^{r-1}$, hence $A\cap(\Omega\ssm g^{-1}(0))\times(\bP^{r-1})^n$ is a fiber bundle with base $\Omega\ssm g^{-1}(0)$ and fiber $(\bP^{r-2})^n$. As $A^\star$ is the closure of this set in $\Omega\times(\bP^{r-1})^n$, we have $$\dim A^\star=n+n(r-2)=n(r-1)=\dim(\bP^{r-1})^n.$$ It follows that the zero fiber $$A^\star_0=A^\star\cap\big(\{0\}\times(\bP^{r-1})^n\big)$$ is a proper subset of $\{0\}\times(\bP^{r-1})^n$. Choose $(a_1,\ldots,a_n)\in(\bC^r\ssm\{0\})^n$ such that \hbox{ $(0,[a_1],\ldots,[a_n])$} is not in~$A^\star_0$. By an easy compactness argument the set \hbox{$A^\star\cap\big(\ol B(0,\varepsilon)\times(\bP^{r-1})^n\big)$} is disjoint from the neighborhood \hbox{$B(0,\varepsilon)\times\prod[B(a_j,\varepsilon)]$} of \hbox{$(0,[a_1],\ldots,[a_n])$} for $\varepsilon$ small enough. For $z\in B(0,\varepsilon)$ we have \hbox{$|a_j.\,g(z)|\ge\varepsilon|g(z)|$} for some~$j$, otherwise the inequa\-lity $|a_j.\,g(z)|<\varepsilon|g(z)|$ would imply the existence of $h_j\in\bC^r$ with $|h_j|<\varepsilon$ and $a_j.\,g(z)=h_j.\,g(z)$. Since $g(z)\ne 0$, we would have $$(z,[a_1-h_1],\ldots,[a_n-h_n])\in A^\star\cap\big(B(0,\varepsilon)\times (\bP^{r-1})^n\big),$$ a contradiction. We obtain therefore $$\varepsilon|g(z)|\le\max|a_j.\,g(z)|\le(\max|a_j|)\,|g(z)|~~~~{\rm on}~~ B(0,\varepsilon).\eqno{\square}$$ \endproof \begstat{(10.4) Theorem {\rm(Brian\c con-Skoda~1974)}} Set $p=\min\{n-1,r-1\}$. Then \smallskip \item{\rm a)} $\wh\cI^{(k+1)}=\cI\,\wh\cI^{(k)}=\ol\cI\,\wh\cI^{(k)}$~~ for $k\ge p$. \smallskip \item{\rm b)} $\ol\cI^{(k+p)}\subset\wh\cI^{(k+p)}\subset\cI^k$~~ for all $k\in\bN$. \endstat \begproof{} a) The inclusions $\cI\,\wh\cI^{(k)}\subset\ol\cI\,\wh\cI^{(k)} \subset\wh\cI^{(k+1)}$ are obvious thanks to Prop.~10.2, so we only have to prove that $\wh\cI^{(k+1)}\subset\cI\,\wh\cI^{(k)}$. Assume first that $r\le n$. Let $f\in\wh\cI^{(k+1)}$ be such that $$\int_\Omega|f|^2\,|g|^{-2(k+1+\varepsilon)}\,dV<+\infty.$$ For $k\ge p-1$, we can apply Th.~9.10 with $m=r-1$ and with the weight $\varphi=(k-m)\log|g|^2$. Hence $f$ can be written $f=\sum g_jh_j$ with $$\int_\Omega|h|^2\,|g|^{-2(k+\varepsilon)}\,dV<+\infty,$$ thus $h_j\in\wh\cI^{(k)}$ and $f\in\cI\,\wh\cI^{(k)}$. When $r>n$, Lemma~10.3 shows that there is an ideal $\cJ\subset\cI$ with $n$ generators such that $\smash{\wh\cJ}^{(k)}=\smash{\wh\cI}^{(k)}$. We find $$\wh\cI^{(k+1)}=\wh\cJ^{(k+1)}\subset\cJ\,\wh\cJ^{(k)}\subset \cI\,\wh\cI^{(k)}~~~~{\rm for}~~k\ge n-1.$$ \medskip \noindent b) Property a) implies inductively $\wh\cI^{(k+p)}=\cI^k\,\wh\cI^{(p)}$ for all $k\in\bN$. This gives in particular $\wh\cI^{(k+p)}\subset\cI^k$.\qed \endproof \begstat{(10.5) Corollary} \smallskip \item{\rm a)} The ideal $\ol\cI$ is the integral closure of $\cI$, i.e.\ by definition the set of germs $u\in\cO_n$ which satisfy an equation $$u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^s,~~~~1\le s\le d.$$ \smallskip \item{\rm b)} Similarly, $\ol\cI^{(k)}$ is the set of germs $u\in\cO_n$ which satisfy an equation $$u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^{]ks[},~~~~1\le s\le d,$$ where $]t[$ denotes the smallest integer $\ge t$.\smallskip \endstat As the ideal $\ol\cI^{(k)}$ is finitely generated, property b) shows that there always exists a rational number $l\ge k$ such that $\smash{\ol\cI}^{(l)}=\smash{\ol\cI}^{(k)}$. \begproof{} a) If $u\in\cO_n$ satisfies a polynomial equation with coefficients $a_s\in\cI^s$, then clearly $|a_s|\le C_s\,|g|^s$ and Lemma~II-4.10 implies $|u|\le C\,|g|$. Conversely, assume that $u\in\ol\cI$. The ring $\cO_n$ is Noetherian, so the ideal $\smash{\wh\cI}^{(p)}$ has a finite number of generators $v_1,\ldots,v_N$. For every $j$ we have $uv_j\in\ol\cI\,\smash{\wh\cI}^{(p)}= \cI\,\smash{\wh\cI}^{(p)}$, hence there exist elements $b_{jk}\in\cI$ such that $$uv_j=\sum_{1\le k\le N}b_{jk}v_k.$$ The matrix $(u\delta_{jk}-b_{jk})$ has the non zero vector $(v_j)$ in its kernel, thus $u$ satisfies the equation $\det(u\delta_{jk}-b_{jk})=0$, which is of the required type. \medskip \noindent b) Observe that $v_1,\ldots,v_N$ satisfy simultaneously some integrability condition $\int_\Omega|v_j|^{-2(p+\varepsilon)}<+\infty$, thus $\smash{\wh\cI}^{(p)}=\smash{\wh\cI}^{(p+\eta)}$ for $\eta\in[0,\varepsilon[$. Let $u\in\smash{\ol\cI}^{(k)}$. For every integer $m\in\bN$ we have $$u^mv_j\in\ol\cI^{(km)}\,\wh\cI^{(p+\eta)}\subset \wh\cI^{(km+\eta+p)}.$$ If $k\notin\bQ$, we can find $m$ such that $d(km+\varepsilon/2,\bZ) <\varepsilon/2$, thus $km+\eta\in\bN$ for some $\eta\in{}]0,\varepsilon[$. If $k\in\bQ$, we take $m$ such that $km\in\bN$ and $\eta=0$. Then $$u^mv_j\in\wh\cI^{(N+p)}=\cI^N\,\wh\cI^{(p)}~~~~{\rm with}~~ N=km+\eta\in\bN,$$ and the reasoning made in a) gives $\det(u^m\delta_{jk}-b_{jk})=0$ for some \hbox{$b_{jk}\in\cI^N$}. This is an equation of the type described in b), where the coefficients $a_s$ vanish when $s$ is not a multiple of $m$ and $a_{ms}\in\cI^{Ns}\subset\cI^{]kms[}$.\qed \endproof Let us mention that Brian\c con and Skoda's result 10.4~b) is optimal for $k=1$. Take for example $\cI=(g_1,\ldots,g_r)$ with $g_j(z)=z_j^r$, $1\le j\le r$, and $f(z)=z_1\ldots z_r$. Then $|f|\le C|g|$ and 10.4~b) yields $f^r\in\cI\,;$ however, it is easy to verify that $f^{r-1}\notin\cI$. The theorem also gives an answer to the following conjecture made by J.~Mather. \begstat{(10.6) Corollary} Let $f\in\cO_n$ and $\cI_f=(z_1\partial f/ \partial z_1,\ldots,z_n\partial f/\partial z_n)$. Then $f\in\ol\cI_f$, and for every integer $k\ge 0$, $f^{k+n-1}\in\cI^k_f$. \endstat The Corollary is also optimal for $k=1\,:$ for example, one can verify that the function $f(z)=(z_1\ldots z_n)^3+z_1^{3n-1}+\ldots+z_n^{3n-1}$ is such that $f^{n-1}\notin\cI_f$. \begproof{} Set $g_j(z)=z_j\,\partial f/\partial z_j$, $1\le j\le n$. By 10.4~b), it suffices to show that $|f|\le C|g|$. For every germ of analytic curve $\bC\ni t\longmapsto\gamma(t)$, $\gamma\not\equiv 0$, the vanishing order of $f\circ\gamma(t)$ at $t=0$ is the same as that of $$t\,{d(f\circ\gamma)\over dt}=\sum_{1\le j\le n}t\,\gamma_j'(t)\, {\partial f\over\partial z_j}\big(\gamma(t)\big).$$ We thus obtain $$|f\circ\gamma(t)|\le C_1\,|t|~\Big|{d(f\circ\gamma)\over dt}\Big| \le C_2\,\sum_{1\le j\le n}|t\,\gamma_j'(t)|~\Big| {\partial f\over\partial z_j}\big(\gamma(t)\big)\Big| \le C_3\,|g\circ\gamma(t)|$$ and conclude by the following elementary lemma.\qed \endproof \begstat{(10.7) Lemma} Let $f,g_1,\ldots,g_r\in\cO_n$ be germs of holomorphic functions vanishing at $0$. Then we have $|f|\le C|g|$ for some constant $C$ if and only if for every germ of analytic curve $\gamma$ through $0$ there exists a constant $C_\gamma$ such that \hbox{$|f\circ\gamma|\le C_\gamma|g\circ\gamma|$}. \endstat \begproof{} If the inequality $|f|\le C|g|$ does not hold on any neighborhood of $0$, the germ of analytic set $(A,0)\subset(\bC^{n+r},0)$ defined by $$g_j(z)=f(z)z_{n+j},~~~~1\le j\le r,$$ contains a sequence of points $\big(z_\nu,g_j(z_\nu)/f(z_\nu)\big)$ converging to $0$ as $\nu$ tends to $+\infty$, with $f(z_\nu)\ne 0$. Hence $(A,0)$ contains an irreducible component on which $f\not\equiv 0$ and there is a germ of curve \hbox{$\wt\gamma=(\gamma,\gamma_{n+j}): (\bC,0)\to(\bC^{n+r},0)$} contained in $(A,0)$ such that $f\circ\gamma \not\equiv 0$. We get \hbox{$g_j\circ\gamma=(f\circ\gamma)\gamma_{n+j}$}, hence \hbox{$|g\circ\gamma(t)|\le C|t|\,|f\circ\gamma(t)|$} and the inequality \hbox{$|f\circ\gamma|\le C_\gamma|g\circ\gamma|$} does not hold.\qed \endproof \titleb{11.}{Integrability of Almost Complex Structures} Let $M$ be a $C^\infty$ manifold of real dimension $m=2n$. An {\it almost complex structure} on $M$ is by definition an endomorphism $J\in\End(TM)$ of class $C^\infty$ such that $J^2=-\Id$. Then $TM$ becomes a complex vector bundle for which the scalar multiplication by $i$ is given by $J$. The pair $(M,J)$ is said to be an {\it almost complex manifold}. For such a manifold, the complexified tangent space $T_\bC M=\bC\otimes_\bR TM$ splits into conjugate complex subspaces $$T_\bC M=T^{1,0}M\oplus T^{0,1}M,~~~~\dim_\bC T^{1,0}M=\dim_\bC T^{0,1}M=n, \leqno(11.1)$$ where $T^{1,0}M$, $T^{0,1}M\subset T_\bC M$ are the eigenspaces of $\Id\otimes J$ corresponding to the eigenvalues $i$ and $-i$. The complexified exterior algebra $\bC\otimes_\bR\Lambda^\bu T^\star M= \Lambda^\bu T_\bC^\star M$ has a corresponding splitting $$\Lambda^k T^\star_\bC M=\bigoplus_{p+q=k}\Lambda^{p,q} T_\bC^\star M \leqno(11.2)$$ where we denote by definition $$\Lambda^{p,q}T^\star_\bC M=\Lambda^p(T^{1,0}M)^\star\otimes_\bC \Lambda^q(T^{0,1}M)^\star.\leqno(11.3)$$ As for complex manifolds, we let $C^s_{p,q}(M,E)$ be the space of differential forms of class $C^s$ and bidegree $(p,q)$ on $M$ with values in a complex vector bundle $E$. There is a natural antisymmetric bilinear map $$\theta~:~~C^\infty(M,T^{1,0}M)\times C^\infty(M,T^{1,0}M)\longrightarrow C^\infty(M,T^{0,1}M)$$ which associates to a pair $(\xi,\eta)$ of $(1,0)$-vector fields the $(0,1)$-component of the Lie bracket $[\xi,\eta]$. Since $$[\xi,f\eta]=f[\xi,\eta]+(\xi.f)\,\eta,~~~~\forall f\in C^\infty(M,\bC)$$ we see that $\theta(\xi,f\eta)=f\,\theta(\xi,\eta)$. It follows that $\theta$ is in fact a $(2,0)$-form on $M$ with values in $T^{0,1}M$. If $M$ is a {\it complex analytic manifold} and $J$ its natural almost complex structure, we have in fact $\theta=0$, because $[\partial/\partial z_j,\partial/\partial z_k]=0$, $1\le j,k\le n$, for any holomorphic local coordinate system $(z_1,\ldots,z_n)$. \begstat{(11.4) Definition} The form $\theta\in C^\infty_{2,0}(M,T^{0,1}M)$ is called the torsion form of $J$. The almost complex structure $J$ is said to be integrable if $\theta=0$. \endstat \begstat{(11.5) Example} \rm If $M$ is of real dimension $m=2$, every almost complex structure is integrable, because $n=1$ and alternate $(2,0)$-forms must be zero. Assume that $M$ is a smooth oriented surface. To any Riemannian metric $g$ we can associate the endomorphism $J\in\End(TM)$ equal to the rotation of $+\pi/2$. A change of orientation changes $J$ into the conjugate structure $-J$. Conversely, if $J$ is given, $TM$ is a complex line bundle, so $M$ is oriented, and a Riemannian metric $g$ is associated to $J$ if and only if $g$ is $J$-hermitian. As a consequence, there is a one-to-one correspondence between conformal classes of Riemannian metrics on $M$ and almost complex structures corresponding to a given orientation.\qed \endstat If $(M,J)$ is an almost complex manifold and $u\in C^\infty_{p,q}(M,\bC)$, we let $d'u,~d''u$ be the components of type $(p+1,q)$ and $(p,q+1)$ in the exterior derivative $du$. Let $(\xi_1,\ldots,\xi_n)$ be a frame of $T^{1,0}M_{\restriction\Omega}$. The torsion form $\theta$ can be written $$\theta=\sum_{1\le j\le n}\alpha_j\otimes\ol\xi_j,~~~~ \alpha_j\in C^\infty_{2,0}(\Omega,\bC).$$ Then $\theta$ yields conjugate operators $\theta',\theta''$ on $\Lambda^\bu T_\bC^\star M$ such that $$\theta'u=\sum_{1\le j\le n}\alpha_j\wedge(\ol\xi_j\ort\,u),~~~~ \theta''u=\sum_{1\le j\le n}\ol\alpha_j\wedge(\xi_j\ort\,u).\leqno(11.6)$$ If $u$ is of bidegree $(p,q)$, then $\theta'u$ and $\theta''u$ are of bidegree $(p+2,q-1)$ and $(p-1,q+2)$. It is clear that $\theta'$, $\theta''$ are derivations, i.e.\ $$\theta'(u\wedge v)=(\theta'u)\wedge v+(-1)^{\deg\,u}u\wedge(\theta'v)$$ for all smooth forms $u,v$, and similarly for $\theta''$. \begstat{(11.7) Proposition} We have~ $d=d'+d''-\theta'-\theta''$. \endstat \begproof{} Since all operators occuring in the formula are derivations, it is sufficient to check the formula for forms of degree 0 or~1. If $u$ is of degree 0, the result is obvious because $\theta'u=\theta''u=0$ and $du$ can only have components of types $(1,0)$ or $(0,1)$. If $u$ is a $1$-form and $\xi,\eta$ are complex vector fields, we have $$du(\xi,\eta)=\xi.u(\eta)-\eta.du(\xi)-u([\xi,\eta]).$$ When $u$ is of type $(0,1)$ and $\xi,\eta$ of type $(1,0)$, we find $$(du)^{2,0}(\xi,\eta)=-u\big(\theta(\xi,\eta)\big)~$$ thus $(du)^{2,0}=-\theta'u$, and of course $(du)^{1,1}=d'u$, $(du)^{0,2}=d''u$, $\theta''u=0$ by definition. The case of a $(1,0)$-form $u$ follows by conjugation.\qed \endproof Proposition 11.7 shows that $J$ is integrable if and only if $d=d'+d''$. In this case, we infer immediately $$d^{\prime 2}=0,~~~~d'd''+d''d'=0,~~~~d^{\prime\prime 2}=0.$$ For an integrable almost complex structure, we thus have the same formalism as for a complex analytic structure, and indeed we shall prove: \begstat{(11.8) Newlander-Nirenberg theorem (1957)} Every integrable almost complex structure $J$ on $M$ is defined by a unique analytic structure. \endstat The proof we shall give follows rather closely that of (H\"ormander~1966), which was itself based on previous ideas of (Kohn~1963, 1964). A function $f\in C^1(\Omega,\bC)$, $\Omega\subset M$, is said to be $J$-holomorphic if $d''f=0$. Let $f_1,\ldots,f_p\in C^1(\Omega,\bC)$ and let $h$ be a function of class $C^1$ on an open subset of $\bC^p$ containing the range of $f=(f_1,\ldots,f_p)$. An easy computation gives $$d''(h\circ f)=\sum_{1\le j\le p}\Big({\partial h\over\partial z_j} \circ f\Big)d''f_j+\Big({\partial h\over\partial\ol z_j}\circ f\Big) \ol{d'f_j},\leqno(11.9)$$ in particular $h\circ f$ is $J$-holomorphic as soon as $f_1,\ldots,f_p$ are $J$-holomorphic and $h$ holomorphic in the usual sense. Constructing a complex analytic structure on $M$ amounts to show the existence of $J$-holomorphic complex coordinates $(z_1,\ldots,z_n)$ on a neighborhood $\Omega$ of every point $a\in M$. Formula (11.9) then shows that all coordinate changes $h:(z_k)\mapsto(w_k)$ are holomorphic in the usual sense, so that $M$ is furnished with a complex analytic atlas. The uniqueness of the analytic structure associated to $J$ is clear, since the holomorphic functions are characterized by the condition $d''f=0$. In order to show the existence, we need a lemma. \begstat{(11.10) Lemma} For every point $a\in M$ and every integer $s\ge 1$, there exist $C^\infty$ complex coordinates $(z_1,\ldots,z_n)$ centered at $a$ such that $$d''z_j={\rm O}(|z|^s),~~~~1\le j\le n.$$ \endstat \begproof{} By induction on $s$. Let $(\xi^\star_1,\ldots,\xi^\star_n)$ be a basis of $\Lambda^{1,0}T^\star_\bC M$. One can find complex functions $z_j$ such that $dz_j(a)=\xi^\star_j$, i.e.\ $$d'z_j(a)=\xi^\star_j,~~~~d''z_j(a)=0.$$ Then $(z_1,\ldots,z_n)$ satisfy the conclusions of the Lemma~for $s=1$. If $(z_1,\ldots,z_n)$ are already constructed for the integer $s$, we have a Taylor expansion $$d''z_j=\sum_{1\le k\le n}P_{jk}(z,\ol z)\,\ol{d'z_k}+{\rm O}(|z|^{s+1})$$ where $P_{jk}(z,w)$ is a homogeneous polynomial in $(z,w)\in\bC^n\times \bC^n$ of total degree $s$. As $J$ is integrable, we have $$\eqalign{ 0=d^{\prime\prime 2}z_j&=\sum_{1\le k,l\le n}{\partial P_{jk}\over \partial z_l}\,d''z_l\wedge\ol{d'z_k}+{\partial P_{jk}\over \partial\ol z_l}\,\ol{d'z_l}\wedge\ol{d'z_k}+{\rm O}(|z|^s)\cr &=\sum_{1\le k1$ and that $x_1=0$ if $p=1$.\qed \endproof \begstat{(1.2) Artin-Rees lemma} Let $F$ be a finitely generated $R$-module and let $E$ be a submodule. There exists an integer $s$ such that $$E\cap\gm^k F=\gm^{k-s}(E\cap\gm^sF)~~~~\hbox{\rm for}~~k\ge s.$$ \endstat \begproof{} Let $R_t$ be the polynomial ring $R[\gm t]=R+\gm t+\cdots+\gm^k t^k+\cdots$ where $t$ is an indeterminate. If $g_1,\ldots, g_p$ is a set of generators of the ideal $\gm$ over $R$, we see that the ring $R_t$ is generated by $g_1t,\ldots,g_pt$ over $R$, hence $R_t$ is also noetherian. Now, we consider the $R_t$-modules $$E_t=\bigoplus E\,t^k,~~~~F_t=\bigoplus~(\gm^k F)\,t^k.$$ Then $F_t$ is generated over $R_t$ by the generators of $F$ over $R$, hence the submodule $E_t\cap F_t$ is finitely generated. Let $s$ be the highest exponent of $t$ in a set of generators $P_1(t),\ldots,P_N(t)$ of $E_t\cap F_t$. If we identify the components of $t^k$ in the extreme terms of the equality $$\bigoplus~\big(E\cap\gm^kF\big)\,t^k=E_t\cap F_t= \sum_j\Big(\bigoplus_k\gm^k\,t^k\Big)P_j(t),$$ we get $$E\cap\gm^kF\subset\sum_{l\le s}\gm^{k-l}(E\cap\gm^lF)\subset \gm^{k-s}(E\cap\gm^sF).$$ The opposite inclusion is clear.\qed \endproof \begstat{(1.3) Krull lemma} Let $F$ be a finitely generated $R$-module and let $E$ be a submodule. Then \smallskip \item{\rm a)} $\bigcap_{k\ge 0}\gm^k F=\{0\}$. \smallskip \item{\rm b)} $\bigcap_{k\ge 0}(E+\gm^k F)=E$. \endstat \begproof{} a) Put $G=\bigcap_{k\ge 0}\gm^k F\subset F$. By the Artin-Rees lemma, there exists $s\in\bN$ such that $G\cap\gm^k F= \gm^{k-s}(G\cap\gm^sF)$. Taking $k=s+1$, we find $G\subset\gm G$, hence $\gm G=G$ and $G=\{0\}$ by the Nakayama lemma. \medskip \noindent b) By applying a) to the quotient module $F/E$ we get $\bigcap\gm^k(F/E)=\{0\}$. Property b) follows.\qed \endproof Now assume that $R=\cO_n=\bC\{z_1,\ldots,z_n\}$ and $\gm=(z_1,\ldots,z_n)$. Then $\cO_n/\gm^k$ is a finite dimensional vector space generated by the monomials $z^\alpha$, $|\alpha|0$ on $]c,+\infty[$. Then $\chi\circ\psi=0$ on $X_c$, so that $\chi\circ\psi$ is weakly $q$-convex everywhere in virtue of (2.3). In the weakly $q$-convex case, we may therefore always assume $K=\emptyset$. The following properties are almost immediate consequences of the definition: \begstat{(2.9) Theorem} \smallskip \item{\rm a)} A scheme $X$ is strongly $($weakly$)$ $q$-convex if and only if the reduced space $X_\red$ is strongly $($weakly$)$ $q$-convex. \smallskip \item{\rm b)} If $X$ is strongly $($weakly$)$ $q$-convex, every closed analytic subset $Y$ of $X_\red$ is strongly $($weakly$)$ $q$-convex. \smallskip \item{\rm c)} If $X$ is strongly $($weakly$)$ $q$-convex, every sublevel set $X_c$ containing the exceptional compact set $K$ is strongly $($weakly$)$ $q$-convex. \smallskip \item{\rm d)} If $U_j$ is a weakly $q_j$-convex open subset of $X$, $1\le j\le s$, the intersection $U=U_1\cap\ldots\cap U_s$ is weakly $q$-convex with $q-1=\sum(q_j-1)$~$;$ $U$ is strongly $q$-convex $($resp. $q$-complete$)$ as soon as one of the sets $U_j$ is strongly $q_j$-convex $($resp. $q_j$-complete$)$.\smallskip \endstat \begproof{} a) is clear, since Def.~2.5 does not involve the structure sheaf $\cO_X$. In cases b) and c), let $\psi$ be an exhaustion of the required type on $X$. Then $\psi_{\restriction Y}$ and $1/(c-\psi)$ are exhaustions on $Y$ and $X_c$ respectively (this is so only if $Y$ is closed). Moreover, these functions are strongly (weakly) $q$-convex on $Y\ssm(K\cap Y)$ and $X_c\ssm K$, thanks to Prop.~2.1 and 2.2. For property d), note that a sum $\psi=\psi_1+\cdots+\psi_s$ of exhaustion functions on the sets $U_j$ is an exhaustion on $U$, choose the $\psi_j$'s weakly $q_j$-convex everywhere, and apply Prop.~2.2.\qed \endproof \begstat{(2.10) Corollary} Any finite intersection $U=U_1\cap\ldots\cap U_s$ of weakly $1$-convex open subsets is weakly $1$-convex. The set $U$ is strongly $1$-convex $($resp. $1$-complete$)$ as soon as one of the sets $U_j$ is strongly $1$-convex $($resp. $1$-complete$)$. \endstat \titlec{2.B.}{Neighborhoods of q-complete subspaces} We prove now a rather useful result asserting the existence of $q$-complete neighborhoods for $q$-complete subvarieties. The case $q=1$ goes back to (Siu~1976), who used a much more complicated method. The first step is an approximation-extension theorem for strongly $q$-convex functions. \begstat{(2.11) Proposition} Let $Y$ be an analytic set in a complex space $X$ and $\psi$ a strongly $q$-convex $C^\infty$ function on $Y$. For every continuous function $\delta>0$ on $Y$, there exists a strongly $q$-convex $C^\infty$ function $\varphi$ on a neighborhood $V$ of $Y$ such that $\psi\le\varphi_{\restriction Y}<\psi+\delta$. \endstat \begproof{} Let $Z_k$ be a stratication of $Y$ as given by Prop.~II.5.6, i.e.\ $Z_k$ is an increasing sequence of analytic subsets of $Y$ such that $Y=\bigcup Z_k$ and $Z_k\ssm Z_{k-1}$ is a smooth $k$-dimensional manifold (possibly empty for some $k$'s). We shall prove by induction on $k$ the following statement: {\it There exists a $C^\infty$ function $\varphi_k$ on $X$ which is strongly $q$-convex along $Y$ and on a closed neighborhood $\ol V_k$ of $Z_k$ in $X$, such that $\psi\le\varphi_{k\restriction Y}<\psi+\delta$.} We first observe that any smooth extension $\varphi_{-1}$ of $\psi$ to $X$ satisfies the requirements with $Z_{-1}=V_{-1}=\emptyset$. Assume that $V_{k-1}$ and $\varphi_{k-1}$ have been constructed. Then $Z_k\ssm V_{k-1}\subset Z_k\ssm Z_{k-1}$ is contained in $Z_{k,\reg}$. The closed set $Z_k\ssm V_{k-1}$ has a locally finite covering $(A_\lambda)$ in $X$ by open coordinate patches $A_\lambda\subset\Omega_\lambda\subset\bC^{N_\lambda}$ in which $Z_k$ is given by equations $z'_\lambda=(z_{\lambda,k+1},\ldots, z_{\lambda,N_\lambda})=0$. Let $\theta_\lambda$ be $C^\infty$ functions with compact support in $A_\lambda$ such that $0\le\theta_\lambda \le 1$ and $\sum\theta_\lambda=1$ on $Z_k\ssm V_{k-1}$. We set $$\varphi_k(x)=\varphi_{k-1}(x)+\sum~\theta_\lambda(x)\,\varepsilon_\lambda^3 \,\log(1+\varepsilon_\lambda^{-4}|z'_\lambda|^2)~~~~\hbox{\rm on}~~X.$$ For $\varepsilon_\lambda>0$ small enough, we will have $\psi\le\varphi_{k-1\restriction Y}\le\varphi_{k\restriction Y}< \psi+\delta$. Now, we check that $\varphi_k$ is still strongly $q$-convex along $Y$ and near any $x_0\in\ol V_{k-1}$, and that $\varphi_k$ becomes strongly $q$-convex near any $x_0\in Z_k\ssm V_{k-1}$. We may assume that $x_0\in\Supp \theta_\mu$ for some $\mu$, otherwise $\varphi_k$ coincides with $\varphi_{k-1}$ in a neighborhood of $x_0$. Select $\mu$ and a small neighborhood $W\compact\Omega_\mu$ of $x_0$ such that \medskip \item{a)} if $x_0\in Z_k\ssm V_{k-1}$, then $\theta_\mu(x_0)>0$ and $A_\mu\cap W\compact\{\theta_\mu>0\}\,$; \smallskip \item{b)} if $x_0\in A_\lambda$ for some $\lambda$ (there is only a finite set $I$ of such $\lambda$'s), then\break $A_\mu\cap W\compact A_\lambda$ and $z_{\lambda\restriction A_\mu\cap W}$ has a holomorphic extension $\wt z_\lambda$ to $\ol W\,$; \smallskip \item{c)} if $x_0\in\ol V_{k-1}$, then $\varphi_{k-1 \restriction A_\mu\cap W}$ has a strongly $q$-convex extension $\wt\varphi_{k-1}$ to $\ol W\,;$ \smallskip \item{d)} if $x_0\in Y\ssm\ol V_{k-1}$, then $\varphi_{k-1 \restriction Y\cap W}$ has a strongly $q$-convex extension $\wt\varphi_{k-1}$ to $\ol W\,.$ \medskip \noindent{}Otherwise take an arbitrary smooth extension $\wt\varphi_{k-1}$ of $\varphi_{k-1\restriction A_\mu\cap W}$ to $\ol W$ and let $\smash{\wt\theta_\lambda}$ be an extension of $\theta_{\lambda \restriction A_\mu\cap W}$ to $\ol W$. Then $$\wt\varphi_k=\wt\varphi_{k-1}+\sum\wt\theta_\lambda\,\varepsilon_\lambda^3\, \log(1+\varepsilon_\lambda^{-4}|{\wt z}^{\,\prime}_\lambda|^2)$$ is an extension of $\varphi_{k\restriction A_\mu\cap W}$ to $\ol W$, resp. of $\varphi_{k\restriction Y\cap W}$ to $\ol W$ in case d). As the function $\log(1+\varepsilon_\lambda^{-4} |{\wt z}^{\,\prime}_\lambda|^2)$ is plurisubharmonic and as its first derivative $\langle{\wt z}^{\,\prime}_\lambda, d{\wt z}^{\,\prime}_\lambda\rangle\,(\varepsilon_\lambda^4+ |{\wt z}^{\,\prime}_\lambda|^2)^{-1}$ is bounded by $O(\varepsilon_\lambda^{-2})$, we see that $$id'd''\wt\varphi_k\ge id'd''\wt\varphi_{k-1}- O({\scriptstyle\sum}\varepsilon_\lambda).$$ Therefore, for $\varepsilon_\lambda$ small enough, $\wt\varphi_k$ remains $q$-convex on $\ol W$ in cases c) and d). Since all functions ${\wt z}^{\,\prime}_\lambda$ vanish along $Z_k\cap W$, we have $$id'd''\wt\varphi_k\ge id'd''\wt\varphi_{k-1} +\sum_{\lambda\in I}\theta_\lambda\,\varepsilon_\lambda^{-1}\,id'd'' |{\wt z}^{\,\prime}_\lambda|^2\ge id'd''\wt\varphi_{k-1} +\theta_\mu\,\varepsilon_\mu^{-1}\,id'd''|z'_\mu|^2$$ at every point of $Z_k\cap W$. Moreover $id'd''\wt\varphi_{k-1}$ has at most $(q-1)$-negative eigenvalues on $TZ_k$ since $Z_k\subset Y$, whereas $id'd''|z'_\mu|^2$ is positive definite in the normal directions to $Z_k$ in $\Omega_\mu$. In case a), we thus find that $\wt\varphi_k$ is strongly $q$-convex on $\smash{\ol W}$ for $\varepsilon_\mu$ small enough; we also observe that only finitely many conditions are required on each $\varepsilon_\lambda$ if we choose a locally finite covering of $\bigcup\Supp\theta_\lambda$ by neighborhoods $W$ as above. Therefore, for $\varepsilon_\lambda$ small enough, $\varphi_k$ is strongly $q$-convex on a neighborhood $\smash{\ol V'_k}$ of $Z_k\ssm V_{k-1}$. The function $\varphi_k$ and the set $V_k=V'_k\cup V_{k-1}$ satisfy the requirements at order $k$. It is clear that we can choose the sequence $\varphi_k$ stationary on every compact subset of $X\,$; the limit $\varphi$ and the open set $V=\bigcup V_k$ fulfill the proposition.\qed \endproof The second step is the existence of almost plurisubharmonic functions having poles along a prescribed analytic set. By an almost plurisubharmonic function on a manifold, we mean a function that is locally equal to the sum of a plurisubharmonic function and of a smooth function, or equivalently, a function whose complex Hessian has bounded negative part. On a complex space, we require that our function can be locally extended as an almost plurisubharmonic function in the ambient space of an embedding. \begstat{(2.12) Lemma} Let $Y$ be an analytic subvariety in a complex space $X$. There is an almost plurisubharmonic function $v$ on $X$ such that $v=-\infty$ on $Y$ with logarithmic poles and $v\in C^\infty(X\ssm Y)$. \endstat \begproof{} Since $\cI_Y\subset\cO_X$ is a coherent subsheaf, there is a locally finite covering of $X$ by patches $A_\lambda$ isomorphic to analytic sets in balls $B(0,r_\lambda)\subset\bC^{N_\lambda}$, such that $\cI_Y$ admits a system of generators $g_\lambda=(g_{\lambda,j})$ on a neighborhood of each set $\ol A_\lambda$. We set $$\eqalign{ &v_\lambda(z)=\log|g_\lambda(z)|^2-{1\over r^2_\lambda -|z-z_\lambda|^2}~~~~\hbox{\rm on}~~A_\lambda,\cr &v(z)=M_{(1,\ldots,1)}\big(\ldots\,,v_\lambda(z),\,\ldots\big)~~~~\hbox{\rm for}~~\lambda~~\hbox{\rm such that}~~A_\lambda\ni z,\cr}$$ where $M_\eta$ is the regularized max function defined in I-3.37. As the generators $(g_{\lambda,j})$ can be expressed in terms of one another on a neighborhood of $\ol A_\lambda\cap \ol A_\mu$, we see that the quotient $|g_\lambda|/|g_\mu|$ remains bounded on this set. Therefore none of the values $v_\lambda(z)$ for $A_\lambda\ni z$ and $z$ near $\partial A_\lambda$ contributes to the value of $v$, since $1/(r_\lambda^2-|z-z_\lambda|^2)$ tends to $+\infty$ on $\partial A_\lambda$. It follows that $v$ is smooth on $X\ssm Y\,$; as each $v_\lambda$ is almost plurisubharmonic on $A_\lambda$, we also see that $v$ is almost plurisubharmonic on $X$.\qed \endproof \begstat{(2.13) Theorem} Let $X$ be a complex space and $Y$ a strongly $q$-complete analytic subset. Then $Y$ has a fundamental family of strongly $q$-complete neighborhoods $V$ in $X$. \endstat \begproof{} By Prop.~2.11 applied to a strongly $q$-convex exhaustion of $Y$ and $\delta=1$, there exists a strongly $q$-convex function $\varphi$ on a neighborhood $W_0$ of $Y$ such that $\varphi_{\restriction Y}$ is an exhaustion. Let $W_1$ be a neighborhood of $Y$ such that $\ol W_1\subset W_0$ and such that $\varphi_{\restriction\ol W_1}$ is an exhaustion. We are going to show that every neighborhood $W\subset W_1$ of $Y$ contains a strongly $q$-complete neighborhood $V$. If $v$ is the function given by Lemma~2.12, we set $$\wt v=v+\chi\circ\varphi~~~~\hbox{\rm on}~~\ol W$$ where $\chi:\bR\to\bR$ is a smooth convex increasing function. If $\chi$ grows fast enough, we get $\wt v>0$ on $\partial W$ and the $(q-1)$-codimensional subspace on which $id'd''\varphi$ is positive definite (in some ambient space) is also positive definite for $id'd''\wt v$ provided that $\chi'$ be large enough to compensate the bounded negative part of $id'd''v$. Then $\wt v$ is strongly $q$-convex. Let $\theta$ be a smooth convex increasing function on $]-\infty,0[$ such that $\theta(t)=0$ for $t<-3$ and $\theta(t)=-1/t$ on $]-1,0[$. The open set $V=\{z\in W\,;\,\wt v(z)<0\}$ is a neighborhood of $Y$ and $\smash{\wt\psi}=\varphi+\theta\circ\wt v$ is a strongly $q$-convex exhaustion of~$V$.\qed \endproof \titlec{2.C.}{Runge Open Subsets} In order to extend the classical Runge theorem into an approximation result for sheaf cohomology groups, we need the concept of a $q$-Runge open subset. \begstat{(2.14) Definition} An open subset $U$ of a complex space $X$ is said to be $q$-Runge $($resp. $q$-Runge complete$)$ in $X$ if for every compact subset $L\subset U$ there exists a smooth exhaustion function $\psi$ on $X$ and a sublevel set $X_b$ of $\psi$ such that $L\subset X_b\compact U$ and $\psi$ is strongly $q$-convex on $X\ssm\ol X_b$ $($resp. on the whole space $X)$. \endstat \begstat{(2.15) Example} \rm If $X$ is strongly $q$-complete and if $\psi$ is a strongly $q$-convex exhaustion function of $X$, then every sublevel set $X_c$ of $\psi$ is $q$-Runge complete in $X\,$: every compact set $L\subset X_c$ satisfies $L\subset X_b\compact X_c$ for some $b0$ and Lemma~2.11 gives a strongly $q$-convex function $\varphi$ on a neighborhood $W_0$ of $Y$ so that $\psi\le\varphi_{\restriction Y}<\psi+\delta$. The neighborhood $V$ and the function $\smash{\wt\psi}= \varphi+\theta\circ\wt v$ constructed in the proof of Th.~2.13 are the desired ones: we have $\psi\le\smash{\wt\psi_{\restriction Y}} =\varphi_{\restriction Y}<\psi+\delta$, thus $$L\subset Y\cap V_{b-\delta}\subset\{\psi0$ a given number. There exists a hermitian metric $\omega$ on $M$ such that the eigenvalues $\gamma_1\le\ldots\le\gamma_n$ of the Hessian form $id'd''\psi$ with respect to $\omega$ satisfy $\gamma_1\ge-\varepsilon$ and $\gamma_q=\ldots=\gamma_n=1$. \endstat \begproof{} Let $\omega_0$ be a fixed hermitian metric, $A_0\in C^\infty(\End TM)$ the hermitian endomorphism associated to the hermitian form $id'd''\psi$ with respect to $\omega_0$, and $\gamma^0_1\le\ldots\le\gamma^0_n$ the eigenvalues of $A_0$ (or $id'd''\psi$). We can choose a function $\eta\in C^\infty(M,\bR)$ such that $0<\eta(x)\le\gamma^0_q(x)$ at each point $x\in M$. Select a positive function $\theta\in C^\infty(\bR,\bR)$ such that $$\theta(t)\ge |t|/\varepsilon~~\hbox{\rm for}~~t\le 0,~~~ \theta(t)\ge t~~\hbox{\rm for}~~t\ge 0,~~~\theta(t)=t~~\hbox{\rm for}~~t\ge 1.$$ We let $\omega$ be the hermitian metric defined by the hermitian endomorphism $$A(x)=\eta(x)\,\theta[(\eta(x))^{-1}A_0(x)]$$ where $\theta[\eta^{-1}A_0]\in C^\infty(\End TM)$ is defined as in Lemma~VII-6.2. By construction, the eigenvalues of $A(x)$ are $\alpha_j(x)=\eta(x)\theta\big(\gamma^0_j(x)/\eta(x)\big)>0$ and we have $$\cmalign{ &\alpha_j(x)\ge|\gamma^0_j(x)|/\varepsilon~~~~ &\hbox{\rm for}~~\gamma^0_j(x)\le 0,\cr &\alpha_j(x)\ge\gamma^0_j(x)~~~~&\hbox{\rm for}~~\gamma^0_j(x)\ge 0,\cr &\alpha_j(x)=\gamma^0_j(x)~~~~&\hbox{\rm for}~~j\ge q~~~ \big(\hbox{\rm then}~\gamma^0_j(x)\ge\eta(x)\big).\cr}$$ The eigenvalues of $id'd''\psi$ with respect to $\omega$ are $\gamma_j(x)=\gamma^0_j(x)/\alpha_j(x)$ and they have the required properties.\qed \endproof On a hermitian manifold $(M,\omega)$, we consider the Laplace operator $\Delta_\omega$ defined by $$\Delta_\omega v=\hbox{\rm Trace}_\omega(id'd''v)=\sum_{1\le j,k\le n} \omega^{jk}(z){\partial^2v\over\partial z_j\partial\ol z_k}\leqno(3.2)$$ where $(\omega^{jk})$ is the conjugate of the inverse matrix of $(\omega_{jk})$. Note that $\Delta_\omega$ may differ from the usual Laplace-Beltrami operator if $\omega$ is not K\"ahler. We say that $v$ is strongly $\omega$-subharmonic if $\Delta_\omega v>0$. This property implies clearly that $v$ is strongly $n$-convex; however, as $$\eqalign{ \Delta_\omega\chi(v_1,\ldots,v_s)=\sum_j&{\partial\chi\over\partial t_j}(v_1,\ldots, v_s)\,\Delta_\omega v_j\cr &{}+\sum_{j,k}{\partial^2\chi\over\partial t_j\partial t_k}(v_1,\ldots,v_s)\,\langle d'v_j,d'v_k\rangle_\omega,\cr}$$ subharmonicity has the advantage of being preserved by all convex increasing transformations. Conversely, if $\psi$ is strongly $n$-convex and $\omega$ chosen as in Lemma~3.1 with $\varepsilon$ small enough, we get $\Delta_\omega\psi\ge 1-(n-1)\varepsilon>0$, thus $\psi$ is strongly subharmonic for a suitable metric $\omega$. \begstat{(3.3) Lemma} Let $U,W\subset M$ be open sets such that for every connected component $U_s$ of $U$ there is a connected component $W_{t(s)}$ of $W$ such that $W_{t(s)}\cap U_s\ne\emptyset$ and $W_{t(s)}\ssm\smash{\ol U_s}\ne\emptyset$. Then there exists a function $v\in C^\infty(M,\bR)$, $v\ge 0$, with support contained in $\smash{\ol U\cup\ol W}$, such that $v$ is strongly $\omega$-subharmonic and $>0$ on $U$. \endstat \begproof{} We first prove that the result is true when $U,W$ are small cylinders with the same radius and axis. Let $a_0\in M$ be a given point and $z_1,\ldots,z_n$ holomorphic coordinates centered at $a_0$. We set $\Re z_j=x_{2j-1}$, $\Im z_j=x_{2j}$, $x'=(x_2,\ldots,x_{2n})$ and $\omega=\sum\wt\omega_{jk}(x)dx_j\otimes dx_k$. Let $U$ be the cylinder $|x_1|0$ such that $$\sum\wt\omega^{jk}(x)\xi_j\xi_k\ge c|\xi|^2~~~\hbox{\rm and}~~~ \sum|\wt\omega^{jk}(x)|\le C~~~\hbox{\rm on}~~\ol U.$$ Let $\chi\in C^\infty(\bR,\bR)$ be a nonnegative function equal to $0$ on $]-\infty,-r]\cup[r+\varepsilon,+\infty[$ and strictly convex on $]-r,r]$. We take explicitly $\chi(x_1)=(x_1+r) \exp(-1/(x_1+r)^2\big)$ on $]-r,r]$ and $$v(x)=\chi(x_1)\exp\big(1/(|x'|^2-r^2)\big)~~~\hbox{\rm on}~~U\cup W,~~~ v=0~~~\hbox{\rm on}~~M\ssm(U\cup W).$$ We have $v\in C^\infty(M,\bR)$, $v>0$ on $U$, and a simple computation gives $$\eqalign{ &{\Delta_\omega v(x)\over v(x)}= \wt\omega^{11}(x)\big(4(x_1+r)^{-5}-2(x_1+r)^{-3}\big)\cr &~~+\sum_{j>1}\wt \omega^{1j}(x)\big(1+2(x_1+r)^{-2}\big)(-2x_j)(r^2-|x'|^2)^{-2}\cr &~~+\sum_{j,k>1}\wt\omega^{jk}(x)\Big( x_jx_k\big(4-8(r^2-|x'|^2) \big)-2(r^2-|x'|^2)^2\delta_{jk}\Big)(r^2-|x'|^2)^{-4}.\cr}$$ For $r$ small, we get $$\eqalign{ {\Delta_\omega v(x)\over v(x)}\ge 2c(x_1+r)^{-5} &-C_1(x_1+r)^{-2}|x'|(r^2-|x'|^2)^{-2}\cr &+(2c|x'|^2-C_2r^4)(r^2-|x'|^2)^{-4}\cr}$$ with constants $C_1,C_2$ independent of $r$. The negative term is bounded by $C_3(x_1+r)^{-4}+c|x'|^2( r^2-|x'|^2)^{-4}$, hence $$\Delta_\omega v/v(x)\ge c(x_1+r)^{-5}+(c|x'|^2-C_2r^4)(r^2-|x'|^2)^{-4}.$$ The last term is negative only when $|x'|0$ on $U_j$. By induction, we can find constants $C_j>0$ such that $v_0+C_1v_1+\cdots+C_j v_j$ is strongly $\omega$-subharmonic on $U_0\cup\ldots\cup U_j$ and $\omega$-subharmonic on $M\ssm\ol W_j$. Then $$w_{z_0}=v_0+C_1v_1+\ldots+C_Nv_N\ge 0$$ is $\omega$-subharmonic on $U$ and strongly $\omega$-subharmonic $>0$ on a neighborhood $\Omega_0$ of the given point $z_0$. Select a denumerable covering of $U$ by such neighborhoods $\Omega_p$ and set $v(z)=\sum\varepsilon_pw_{z_p}(z)$ where $\varepsilon_p$ is a sequence converging sufficiently fast to $0$ so that $v\in C^\infty(M,\bR)$. Then $v$ has the required properties. In the general case, we find for each pair $(U_s,W_{t(s)})$ a function $v_s$ with support in $\smash{\ol U_s\cup\ol W_{t(s)}}$, strongly $\omega$-subharmonic and $>0$ on $U_s$. Any convergent series $v=\sum\varepsilon_sv_s$ yields a function with the desired properties.\qed \endproof \begstat{(3.4) Lemma} Let $X$ be a connected, locally connected and locally compact topological space. If $U$ is a relatively compact open subset of $X$, we let $\wt U$ be the union of $U$ with all compact connected components of $X\ssm U$. Then $\wt U$ is open and relatively compact in $X$, and $X\ssm\smash{\wt U}$ has only finitely many connected components, all non compact. \endstat \begproof{} A rather easy exercise of general topology. Intuitively, $\wt U$ is obtained by ``filling the holes" of $U$ in $X$.\qed \endproof \begstat{(3.5) Theorem {\rm(Greene-Wu~1975)}} Every $n$-dimensional connected non compact complex manifold $M$ has a strongly subharmonic exhaustion function with respect to any hermitian metric $\omega$. In particular, $M$ is strongly $n$-complete. \endstat \begproof{} Let $\varphi\in C^\infty(M,\bR)$ be an arbitrary exhaustion function. There exists a sequence of connected smoothly bounded open sets $\Omega'_\nu\compact M$ such that $\smash{\ol\Omega'_\nu}\subset\Omega'_{\nu+1}$ and $M=\bigcup\Omega'_\nu$. Let $\Omega_\nu=\smash{\wt\Omega'_\nu}$ be the relatively compact open set given by Lemma~3.4. Then $\smash{\ol\Omega_\nu}\subset\Omega_{\nu+1}$, $M=\bigcup\Omega_\nu$ and $M\ssm\Omega_\nu$ has no compact connected component. We set $$U_1=\Omega_2,~~~~U_\nu=\Omega_{\nu+1}\ssm\ol{\Omega_{\nu-2}} ~~~\hbox{\rm for}~~\nu\ge 2.$$ Then $\partial U_\nu=\partial\Omega_{\nu+1}\cup\partial\Omega_{\nu-2}\,$; any connected component $U_{\nu,s}$ of $U_\nu$ has its boundary $\partial U_{\nu,s}\not\subset\partial\Omega_{\nu-2}$, otherwise $\ol U_{\nu,s}$ would be open and closed in $M\ssm\Omega_{\nu-2}$, hence $\ol U_{\nu,s}$ would be a compact component of $M\ssm\Omega_{\nu-2}$. Therefore $\partial U_{\nu,s}$ intersects $\partial\Omega_{\nu+1}\subset U_{\nu+1}$. If $\partial U_{\nu+1,t(s)}$ is a connected component of $U_{\nu+1}$ containing a point of $\partial U_{\nu,s}$, then $U_{\nu+1,t(s)}\cap U_{\nu,s}\ne \emptyset$ and $U_{\nu+1,t(s)}\ssm\ol U_{\nu,s}\ne\emptyset$. Lemma 7 implies that there is a nonnegative function $v_\nu\in C^\infty(M,\bR)$ with support in $U_\nu\cup U_{\nu+1}$, which is strongly $\omega$-subharmonic on $U_\nu$. An induction yields constants $C_\nu$ such that $$\psi_\nu=\varphi+C_1v_1+\cdots+C_\nu v_\nu$$ is strongly $\omega$-subharmonic on $\ol{\Omega_\nu}\subset U_0\cup \ldots\cup U_\nu$, thus $\psi=\varphi+\sum C_\nu v_\nu$ is a strongly $\omega$-subharmonic exhaustion function on $M$.\qed \endproof By an induction on the dimension, the above result can be generalized to an arbi\-trary complex space (or analytic scheme), as was first shown by T.~Ohsawa. \begstat{(3.6) Theorem {\rm(Ohsawa~1984)}} Let $X$ be a complex space of maximal dimension $n$. \smallskip \item{\rm a)} $X$ is always strongly $(n+1)$-complete. \smallskip \item{\rm b)} If $X$ has no compact irreducible component of dimension $n$, then $X$ is strongly $n$-complete. \smallskip \item{\rm c)} If $X$ has only finitely many irreducible components of dimension $n$, then $X$ is strongly $n$-convex.\smallskip \endstat \begproof{} We prove a) and b) by induction on $n=\dim X$. For $n=0$, property b) is void and a) is obvious (any function can then be considered as strongly $1$-convex). Assume that a) has been proved in dimension $\le n-1$. Let $X'$ be the union of $X_\sing$ and of the irreducible components of $X$ of dimension at most $n-1$, and $M=X\ssm X'$ the $n$-dimensional part of $X_\reg$. As $\dim X'\le n-1$, the induction hypothesis shows that $X'$ is strongly \hbox{$n$-complete}. By Th.~2.13, there exists a strongly $n$-convex exhaustion function $\varphi'$ on a neighborhood $V'$ of $X'$. Take a closed neighborhood $\ol V\subset V'$ and an arbitrary exhaustion $\varphi$ on $X$ that extends $\varphi'_{\smash{\restriction\ol V}}$. Since every function on a $n$-dimensional manifold is strongly $(n+1)$-convex, we conclude that $X$ is at worst $(n+1)$-complete, as stated in a). \medskip In case b), the hypothesis means that the connected components $M_j$ of $M=X\ssm X'$ have non compact closure $\ol M_j$ in $X$. On the other hand, Lemma~3.1 shows that there exists a hermitian metric $\omega$ on $M$ such that $\varphi_{\restriction M\cap V}$ is strongly $\omega$-subharmonic. Consider the open sets $U_{j,\nu}\subset M_j$ provided by Lemma~3.7 below. By the arguments already used in Th.~3.5, we can find a strongly $\omega$-subharmonic exhaustion $\psi=\varphi+\sum_{j,\nu}C_{j,\nu}v_{j,\nu}$ on $X$, with $v_{j,\nu}$ strongly $\omega$-subharmonic on $U_{j,\nu}$, $\Supp v_{j,\nu}\subset U_{j,\nu}\cup U_{j,\nu+1}$ and $C_{j,\nu}$ large. Then $\psi$ is strongly $n$-convex on $X$. \endproof \begstat{(3.7) Lemma} For each $j$, there exists a sequence of open sets $U_{j,\nu}\compact M_j$, $\nu\in\bN$, such that \smallskip \item{\rm a)} $M_j\ssm V'\subset\bigcup_\nu U_{j,\nu}$~ and $(U_{j,\nu})$ is locally finite in $\ol M_j\,;$ \smallskip \item{\rm b)} for every connected component $U_{j,\nu,s}$ of $U_{j,\nu}$ there is a connected component $U_{j,\nu+1,t(s)}$ of $U_{j,\nu+1}$ such that $U_{j,\nu+1,t(s)}\cap U_{j,\nu,s}\ne\emptyset$ and $U_{j,\nu+1,t(s)}\ssm\smash{\ol U_{j,\nu,s}}\ne\emptyset$.\smallskip \endstat \begproof{} By Lemma~3.4 applied to the space $\ol M_j$, there exists a sequence of relatively compact connected open sets $\Omega_{j,\nu}$ in $\ol M_j$ such that $\ol M_j\ssm\Omega_{j,\nu}$ has no compact connected component, $\ol\Omega_{j,\nu}\subset\Omega_{j,\nu+1}$ and $\ol M_j=\bigcup\Omega_{j,\nu}$. We define a compact set $K_{j,\nu}\subset M_j$ and an open set $W_{j,\nu}\subset\ol M_j$ containing $K_{j,\nu}$ by $$K_{j,\nu}=(\ol\Omega_{j,\nu}\ssm\Omega_{j,\nu-1})\ssm V', ~~~~W_{j,\nu}=\Omega_{j,\nu+1}\ssm\ol\Omega_{j,\nu-2}.$$ By induction on $\nu$, we construct an open set $U_{j,\nu}\compact W_{j,\nu}\ssm X'\subset M_j$ and a finite set $F_{j,\nu}\subset \partial U_{j,\nu}\ssm\ol\Omega_{j,\nu}$. We let $F_{j,-1}=\emptyset$. If these sets are already constructed for $\nu-1$, the compact set $K_{j,\nu}\cup F_{j,\nu-1}$ is contained in the open set $W_{j,\nu}$, thus contained in a finite union of connected components $W_{j,\nu,s}$. We can write $K_{j,\nu}\cup F_{j,\nu-1}= \bigcup L_{j,\nu,s}$ where $L_{j,\nu,s}$ is contained in $W_{j,\nu,s}\ssm X'\subset M_j$. The open set $W_{j,\nu,s}\ssm X'$ is connected and non contained in $\ol\Omega_{j,\nu}\cup L_{j,\nu,s}$, otherwise its closure $\ol W_{j,\nu,s}$ would have no boundary point $\in\partial\Omega_{j,\nu+1}$, thus would be open and compact in $\ol M_j\ssm\Omega_{j,\nu-2}$, contradiction. We select a point $a_s\in (W_{j,\nu,s}\ssm X')\ssm( \ol\Omega_{j,\nu}\cup L_{j,\nu,s})$ and a smoothly bounded connected open set $U_{j,\nu,s}\compact W_{j,\nu,s}\ssm X'$ containing $L_{j,\nu,s}$ with $a_s\in\partial U_{j,\nu,s}$. Finally, we set $U_{j,\nu}=\bigcup_s U_{j,\nu,s}$ and let $F_{j,\nu}$ be the set of all points $a_s$. By construction, we have $U_{j,\nu}\supset K_{j,\nu}\cup F_{j,\nu-1}$, thus $\bigcup U_{j,\nu}\supset\bigcup K_{j,\nu}=M_j\ssm V'$, and $\partial U_{j,\nu,s}\ni a_s$ with $a_s\in F_{j,\nu}\subset U_{j,\nu+1}$. Property b) follows.\qed \endproof \begproof{of Theorem 3.6~c) (end).} Let $Y\subset X$ be the union of $X_\sing$ with all irreducible components of $X$ that are non compact or of dimension $b+1\}.$$ For every point $x\in P$ with $x\in T_\alpha$, there exists a connected open set $V_x\compact M\ssm L$ containing $x$ such that $\partial V_x\ni y_\alpha$ ($M\ssm L$ is a neighborhood of $M\ssm U$ and we can consider a tubular neighborhood of a path from $x$ to $y_\alpha$ in $M\ssm L$). The compact set $P$ can be covered by a finite number of open sets $V_{x_j}$. Then Lemma~3.3 yields functions $v_j$ with support in $\smash{\ol V_{x_j}\cup\ol W}$ which are strongly $\omega$-subharmonic on $V_{x_j}$. Let $\chi$ be a convex increasing function such that $\chi(t)=0$ on $]-\infty,b]$ and $\chi'(t)>0$ on $]b,+\infty[$. Consider the function $$\psi=\varphi+\sum C_jv_j+\chi\circ\varphi.$$ First, choose $C_j$ large enough so that $\psi\ge b$ on $P$. Then choose $\chi$ increasing fast enough so that $\psi$ is strongly $\omega$-subharmonic on $\ol W$. Then $\psi$ is a strongly $n$-convex exhaustion function on $M$, and as $\psi\ge\varphi$ on $M$ and $\psi=\varphi$ on $L$, we see that $$L\subset\{x\in M\,;\,\psi(x)0~~~~\hbox{\rm if}~~ \varepsilon\le 1/q.$$ We choose $\chi_0$ increasing fast enough so that all the eigenvalues of the above curvature tensor are $\ge 1$ when $\chi=\chi_0$. Then for every $g\in C^\infty_{n,k}(M,E)$ with $D''g=0$ the equation $D''f=g$ can be solved with an estimate $$\int_M|f|^2e^{-\chi\circ\psi}dV\le\int_M|g|^2e^{-\chi\circ\psi}dV, $$ where $\chi=\chi_0+\chi_1$ and where $\chi_1$ is a convex increasing function chosen so that the integral of $g$ converges. This gives a). In order to prove b), let $h\in C^\infty_{n,q-1}(U,E)$ be such that $D''h=0$ and let $L$ be an arbitrary compact subset of $U$. Thanks to Def.~2.14, we can choose $\psi$ such that there is a sublevel set $M_b$ with $L\subset M_b\compact U$. Select $b_0c$, and verify assertions 4.10 b), c) for all restriction maps $$H^k(X_d,\cS)\longrightarrow H^k(X_c,\cS),~~~~k\ge q-1.$$ The basic idea, already contained in (Andreotti-Grauert~1962), is to deform $X_c$ into $X_d$ through a sequence of strongly $q$-convex open sets $(G_j)$ such that $G_{j+1}$ is obtained from $G_j$ by making a small bump. \begstat{(4.12) Lemma} There exist a sequence of strongly $q$-convex open sets\break $G_0\subset\ldots\subset G_s$ and a sequence of strongly $q$-complete open sets $U_0,\ldots,U_{s-1}$ in $X$ such that \smallskip \item{\rm a)} $G_0=X_c$,~~~$G_s=X_d$,~~~$G_{j+1}=G_j\cup U_j$~~for~ $0\le j\le s-1\,;$ \smallskip \item{\rm b)} $G_j=\{x\in X\,;\,\psi_j(x)0$ so small that $$\psi_j=\psi-\varepsilon\sum_{0\le k0.$$ Then $\psi_{j,\eta}$ is an exhaustion of $U_j$. As $\varphi_j$ is bounded below, we have $$L\subset\{\psi_{j,\eta}c$, the restriction map $$H^k(X_d,\cS)\longrightarrow H^k(X_c,\cS)$$ is an isomorphism when $k\ge q$ and has a dense range when $k=q-1$. \endstat \begproof{} Thanks to Lemma~4.12, we are led to consider the restriction maps $$H^k(G_{j+1},\cS)\longrightarrow H^k(G_j,\cS).\leqno(4.14)$$ Let us apply the Mayer-Vietoris exact sequence IV-3.11 to $G_{j+1}=G_j\cup U_j$. For $k\ge q$ we have $H^k(U_j,\cS)=H^k(G_j\cap U_j,\cS)=0$ by Lemma~4.4. Hence we get an exact sequence $$\cmalign{ H^{q-1}(G_{j+1},\cS)&\longrightarrow H^{q-1}(G_j,\cS)\oplus H^{q-1}(U_j,\cS)&\longrightarrow H^{q-1}(G_j\cap U_j,\cS)\longrightarrow\cr H^k(G_{j+1},\cS)&\longrightarrow~~~~~~~~~~~~~H^k(G_j,\cS)&\longrightarrow~~~0~~~\longrightarrow\cdots,~~~~ k\ge q.\cr}$$ In this sequence, all the arrows are induced by restriction maps, so they define continuous linear operators. We already infer that the map (4.14) is bijective for $k>q$ and surjective for $k=q$. There exist a $\cS$-acyclic covering $\cV=(V_\alpha)$ of $X_d$ and a finite family $\cV'=(V'_{\alpha_1},\ldots,V'_{\alpha_p})$ of open sets such that $V'_{\alpha_j}\compact V_{\alpha_j}$ and $\bigcup V'_{\alpha_j} \supset\smash{\ol X_c}$. Let $\cW$ be a locally finite $\cS$-acyclic covering of $X_c$ which refines $\cV'\cap X_c=(V'_{\alpha_j}\cap X_c)$. The refinement map $$C^\bu(\cV,\cS)\longrightarrow C^\bu(\cV'\cap X_c,\cS)\longrightarrow C^\bu(\cW,\cS)$$ is compact because the first arrow is, and it induces a surjective map $$H^k(X_d,\cS)\longrightarrow H^k(X_c,\cS)~~~~\hbox{\rm for}~~k\ge q.$$ By Schwartz' theorem 1.9, we conclude that $H^k(X_c,\cS)$ is Hausdorff and finite dimensional for $k\ge q$. This is equally true for $H^q(G_j,\cS)$ because $G_j$ is also a global sublevel set $\{x\in X\,;\,\psi_j(x)b$ such that $X_c\compact U$. For every $d\in\bR$, we denote by $\cW_d\subset\cW$ the collection of sets $W_\alpha\in\cW$ such that $W_\alpha\subset X_d$. Then $\cW_d$ is a $\cS$-acyclic covering of $X_d$. We consider the sequence of \v Cech complexes $$E^\bu_\nu=C^\bu(\cW_{c+\nu},\cS),~~~~\nu\in\bN$$ together with the surjective projection maps $E^\bu_{\nu+1}\longrightarrow E^\bu_\nu$, and their inverse limit $E^\bu=C^\bu(\cW,\cS)$. Then we have $H^k(E^\bu)=H^k(X,\cS)$ and $H^k(E^\bu_\nu)=H^k(X_{c+\nu},\cS)$. Propositions 1.11 (a,b,c) and 4.13 imply that $H^k(X,\cS)\longrightarrow H^k(X_c,\cS)$ is bijective for $k\ge q$ and has a dense range for $k=q-1$. It already follows that $H^k(X,\cS)$ is Hausdorff for $k\ge q$. Now, take an increasing sequence of open sets $X_{c_\nu}$ equal to sublevel sets of a sequence of exhaustions $\psi_\nu$, such that $U=\bigcup X_{c_\nu}$. Then all groups $H^k(X_{c_\nu},\cS)$ are in bijection with $H^k(X,\cS)$ for $k\ge q$, and the image of $H^{q-1}(X_{c_{\nu+1}},\cS)$ in $H^{q-1}(X_{c_\nu},\cS)$ is dense because it contains the image of $H^{q-1}(X,\cS)$. Proposition 1.11 (a,b,c) again shows that $H^k(U,\cS)\longrightarrow H^k(X_{c_0},\cS)$ is bijective for $k\ge q$, and d) shows that $H^{q-1}(X,\cS)\longrightarrow H^{q-1}(U,\cS)$ has a dense range. The theorem follows.\qed \endproof A combination of Andreotti-Grauert's theorem with Th.~3.6 yields the following important consequence. \begstat{(4.15) Corollary} Let $\cS$ be a coherent sheaf over an analytic scheme $(X,\cO_X)$ with $\dim X\le n$. \smallskip \item{\rm a)} We have~~$H^k(X,\cS)=0$~~for all $k\ge n+1\,;$ \smallskip \item{\rm b)} If $X$ has no compact irreducible component of dimension $n$, then we have $H^n(X,\cS)=0$. \smallskip \item{\rm c)} If $X$ has only finitely many $n$-dimensional compact irreducible components, then $H^n(X,\cS)$ is finite dimensional. \qed\smallskip \endstat The special case of 4.15 b) when $X$ is smooth and $\cS$ locally free has been first proved by (Malgrange~1955), and the general case is due to (Siu~1969). Another consequence is the following approximation theorem for coherent sheaves over manifolds, which results from Prop.~3.8. \begstat{(4.16) Proposition} Let $\cS$ be a coherent sheaf over a non compact connected complex manifold $M$ with $\dim M=n$. Let $U\subset M$ be an open subset such that the complement $M\ssm U$ has no compact connected component. Then the restriction map $H^{n-1}(M,\cS)\longrightarrow H^{n-1}(U,\cS)$ has a dense range.\qed \endstat \titleb{5.}{Grauert's Direct Image Theorem} The goal of this section is to prove the following fundamental result on direct images of coherent analytic sheaves, due to (Grauert~1960). \begstat{(5.1) Direct image theorem} Let $X$, $Y$ be complex analytic schemes and let $F:X\to Y$ be a proper analytic morphism. If $\cS$ is a coherent $\cO_X$-module, the direct images $R^qF_\star\cS$ are coherent $\cO_Y$-modules. \endstat We give below a beautiful proof due to (Kiehl-Verdier~1971), which is much simpler than Grauert's original proof; this proof rests on rather deep properties of nuclear modules over nuclear Fr\'echet algebras. We first introduce the basic concept of topological tensor product. Our presentation owes much to the seminar lectures by (Douady-Verdier~1973). \titlec{5.A.}{Topological Tensor Products and Nuclear Spaces} The algebra of holomorphic functions on a product space $X\times Y$ is a completion $\cO(X)\hotimes\cO(Y)$ of the algebraic tensor product $\cO(X)\otimes\cO(Y)$. We are going to describe the construction and the basic properties of the required topological tensor products $\hotimes$. Let $E$, $F$ be (real or complex) vector spaces equipped with semi-norms $p$ and $q$, respectively. Then $E\otimes F$ can be equipped with any one of the two natural semi-norms $p\otimes_\pi q$, $p\otimes_\varepsilon q$ defined by $$\eqalign{ p\otimes_\pi q(t)&=\inf\Big\{\sum_{1\le j\le N}p(x_j)\,q(y_j)\,;~ t=\!\!\sum_{1\le j\le N}x_j\otimes y_j\,,~x_j\in E\,,~y_j\in F~\Big\},\cr p\otimes_\varepsilon q(t)&=\sup_{||\xi||_p\le 1,\,||\eta||_q\le 1} \big|\xi\otimes\eta(t)\big|,~~~~\xi\in E',~~\eta\in F'\,;\cr}$$ the inequalities in the last line mean that $\xi$, $\eta$ satisfy $|\xi(x)| \le p(x)$ and $|\eta(y)|\le q(y)$ for all $x\in E$, $y\in F$. Then clearly $p\otimes_\varepsilon q\le p\otimes_\pi q$, for $$p\otimes_\varepsilon q\Big(\sum x_j\otimes y_j\Big)\le \sum p\otimes_\varepsilon q(x_j\otimes y_j)\le\sum p(x_j)\,q(y_j).$$ Given $x\in E$, $y\in F$, the Hahn-Banach theorem implies that there exist $\xi$,~$\eta$ such that $||\xi||_p=||\eta||_q=1$ with $\xi(x)=p(x)$ and $\eta(y)=q(y)$, hence $p\otimes_\varepsilon q(x\otimes y)\ge p(x)\,q(y)$. On the other hand $p\otimes_\pi q(x\otimes y)\le p(x)\,q(y)$, thus $$p\otimes_\varepsilon q(x\otimes y)=p\otimes_\pi q(x\otimes y)=p(x)\,q(y).$$ \begstat{(5.2) Definition} Let $E$, $F$ be locally convex topological vector spaces. The topological tensor product $E\hotimes_\pi F$ $($resp. $E\hotimes_\varepsilon F\,)$ is the Hausdorff completion of $E\otimes F$, equipped with the family of semi-norms $p\otimes_\pi q$ $($resp. $p\otimes_\varepsilon q)$ associated to fundamental families of semi-norms on $E$ and $F$. \endstat Since we may also write $$p\otimes_\pi q(t)=\inf\Big\{\sum|\lambda_j|\,;~ t=\sum\lambda_j\,x_j\otimes y_j\,,~p(x_j)\le 1\,,~q(y_j)\le 1~\Big\}$$ where the $\lambda_j$'s are scalars, we see that the closed unit ball $B(p\hotimes_\pi q)$ in $E\hotimes_\pi F$ is the closed convex hull of $B(p)\otimes B(q)$. From this, we easily infer that the topological dual space $(E\hotimes_\pi F)'$ is isomorphic to the space of continuous bilinear forms on $E\times F$. Another simple consequence of this interpretation of $B(p\hotimes_\pi q)$ is example a) below. \begstat{\bf(5.3) Examples} \smallskip\rm \noindent{a)} For all discrete spaces $I$, $J$, there is an isometry $$\ell^1(I)\hotimes_\pi\ell^1(J)\simeq\ell^1(I\times J).$$ \noindent{b)} For Banach spaces $(E,p)$, $(F,q)$, the closed unit ball in $E\hotimes_\varepsilon F$ is dual to the unit ball $B(p'\hotimes_\pi q')$ of $E'\hotimes_\pi F'$ through the natural pairing extending the algebraic pairing of $E\otimes F$ and $E'\otimes F'$. If $c_0(I)$ denotes the space of bounded sequences on $I$ converging to zero at infinity, we have $c_0(I)'=\ell^1(I)$, hence by duality $c_0(I)\hotimes_\varepsilon c_0(J)$ is isometric to $c_0(I\times J)$. \medskip \noindent{c)} If $X$, $Y$ are compact topological spaces and if $C(X)$, $C(Y)$ are their algebras of continuous functions with the sup norm, then $$C(X)\hotimes_\varepsilon C(Y)\simeq C(X\times Y).$$ Indeed, $C(X)'$ is the space of finite Borel measures equipped with the mass norm. Thus for $f\in C(X)\otimes C(Y)$, the $\otimes_\varepsilon$-seminorm is given by $$||f||_\varepsilon=\sup_{||\mu||\le 1,\,||\nu||\le 1}\mu\otimes\nu(f)= \sup_{X\times Y}|f|\,;$$ the last equality is obtained by taking Dirac measures $\delta_x$, $\delta_y$ for $\mu$, $\nu$ (the inequality $\le$ is obvious). Now $C(X)\otimes C(Y)$ is dense in $C(X\times Y)$ by the Stone-Weierstrass theorem, hence its completion is $C(X\times Y)$, as desired.\qed \endstat Let $f:E_1\to E_2$ and $g:F_1\to F_2$ be continuous morphisms. For all semi-norms $p_2$, $q_2$ on $E_2$, $F_2$, there exist semi-norms $p_1$, $q_1$ on $E_1$, $F_1$ and constants $||f||=||f||_{p_1,p_2}$, $||g||=||g||_{q_1,q_2}$ such that $p_2\circ f\le ||f||\,p_1$ and $q_2\circ g\le ||g||\,q_1$. Then we find $$(p_2\otimes_\pi q_2)\circ(f\otimes g)\le ||f||\,||g||\,p_1\otimes_\pi q_1$$ and a similar formula with $p_j\otimes_\varepsilon q_j$. It follows that there are well defined continuous maps \medskip\noindent $\cmalign{ (5.4')\hfill\qquad &f\hotimes_\pi g&{}:E_1\hotimes_\pi F_1&{}\longrightarrow E_2\hotimes_\pi F_2,\cr (5.4'')\hfill\qquad &f\hotimes_\varepsilon g&{}:E_1\hotimes_\varepsilon F_1&{}\longrightarrow E_2\hotimes_\varepsilon F_2.\cr}$ \medskip\noindent Another simple fact is that $\hotimes_\pi$ preserves open morphisms: \begstat{(5.5) Proposition} If $f:E_1\to E_2$ and $g:F_1\to F_2$ are epimorphisms, then $f\hotimes_\pi g:E_1\hotimes_\pi F_1\longrightarrow E_2\hotimes_\pi F_2$ is an epimorphism. \endstat \begproof{} Recall that when $E$ is locally convex complete and $F$ Hausdorff, a morphism $u:E\to F$ is open if and only if $\ol{u(V)}$ is a neighborhood of $0$ for every neighborhood of $0$ (this can be checked essentially by the same proof as 1.8~b)). Here, for any semi-norms $p$, $q$ on $E_1$, $F_1$ the closure of $f\hotimes_\pi g\big(B(p\hotimes_\pi q)\big)$ contains the closed convex hull of $f\big(B(p)\big)\otimes g\big(B(q)\big)$ in which $f\big(B(p)\big)$ and $g\big(B(q)\big)$ are neighborhoods of~$0$, so it is a neighborhood of $0$ in $E\hotimes_\pi F$.\qed \endproof If $E_1\subset E_2$ is a closed subspace, every continuous semi-norm $p_1$ on $E_1$ is the restriction of a continuous semi-norm on $E_2$, and every linear form $\xi_1\in E_1'$ such that $||\xi_1||_{p_1}\le 1$ can be extended to a linear form $\xi_2\in E_2$ such that $||\xi_2||_{p_2}=||\xi_1||_{p_1}$ (Hahn-Banach theorem); similar properties hold for a closed subspace $F_1\subset F_2$. We infer that $$(p_2\otimes_\varepsilon q_2)_{\restriction E_1\otimes F_1}= p_1\otimes_\varepsilon q_1\,,$$ thus $E_1\hotimes_\varepsilon F_1$ is a closed subspace of $E_2\hotimes_\varepsilon F_2$. In other words: \begstat{(5.6) Proposition} If $f:E_1\to E_2$ and $g:F_1\to F_2$ are monomorphisms, then $f\hotimes_\varepsilon g:E_1\hotimes_\varepsilon F_1\longrightarrow E_2\hotimes_\varepsilon F_2$ is a monomorphism.\qed \endstat Unfortunately, 5.5 fails for $\hotimes_\varepsilon$ and 5.6 fails for $\hotimes_\pi$, even with Fr\'echet or Banach spaces. It follows that neither $\hotimes_\pi$ nor $\hotimes_\varepsilon$ are exact functors in the category of Fr\'echet spaces. In order to circumvent this difficulty, it is necessary to work in a suitable subcategory. \begstat{(5.7) Definition} A morphism $f:E\to F$ of complete locally convex spaces is said to be nuclear if $f$ can be written as $$f(x)=\sum\lambda_j\,\xi_j(x)\,y_j$$ where $(\lambda_j)$ is a sequence of scalars with $\sum|\lambda_j|<+\infty$, $\xi_j\in E'$ an equi\-con\-tinuous sequence of linear forms and $y_j\in F$ a bounded sequence. \endstat When $E$ and $F$ are Banach spaces, the space of nuclear morphisms is isomorphic to $E'\hotimes_\pi F$ and the nuclear norm $||f||_\nu$ is defined to be the norm in this space, namely $$||f||_\nu=\inf\Big\{\sum|\lambda_j|\,;~f=\sum\lambda_j\,\xi_j\otimes y_j,~ ||\xi_j||\le 1,~||y_j||\le 1~\Big\}.\leqno(5.8)$$ For general spaces $E$, $F$, the equicontinuity of $(\xi_j)$ means that there is a semi-norm $p$ on $E$ and a constant $C$ such that $|\xi_j(x)|\le C\,p(x)$ for all $j$. Then the definition shows that $f:E\to F$ is nuclear if and only if $f$ can be factorized as $E\to E_1\to F_1\to F$ where $E_1\to F_1$ is a nuclear morphism of Banach spaces: indeed we need only take $E_1$ be equal to the Hausdorff completion $\wh E_p$ of $(E,p)$ and let $F_1$ be the subspace of $F$ generated by the closed balanced convex hull of~$\{y_j\}$ ($=$ unit ball in $F_1)\,$; moreover, if $u:S\to E$ and $v:F\to T$ are continuous, the nuclearity of $f$ implies the nuclearity of $v\circ f\circ u\,$; its nuclear decomposition is then $v\circ f\circ u= \sum\lambda_j\,(\xi_j\circ u)\otimes v(y_j)$. \begstat{(5.9) Remark} \rm Every nuclear morphism is compact: indeed, we may assume in Def.~5.7 that $(y_j)$ converges to $0$ and $\sum|\lambda_j|\le 1$, otherwise we replace $y_j$ by $\varepsilon_j y_j$ with $\varepsilon_j$ converging to zero such that $\sum|\lambda_j/\varepsilon_j|\le 1\,$; then, if $U\subset F$ is a neighborhood of $0$ such that $|\xi_j(U)|\le 1$ for all $j$, the image $f(U)$ is contained in the closed convex hull of the compact set $\{y_j\}\cup\{0\}$, which is compact. \endstat \begstat{(5.10) Proposition} If $E,F,G$ are Banach spaces and if $f:E\to F$ is nuclear, there is a continuous morphism $$f\hotimes\Id_G:E\hotimes_\varepsilon G\longrightarrow F\hotimes_\pi G$$ extending $f\otimes\Id_G$, such that $||f\hotimes\Id_G||\le||f||_\nu$. \endstat \begproof{} If $f=\sum\lambda_j\,\xi_j\otimes y_j$ as in (5.8), then for any $t\in E\otimes G$ we have $$(f\otimes\Id_G)(t)=\sum\lambda_j\big(\xi_j\otimes\Id_G(t)\big) \otimes y_j$$ where $(\xi_j\otimes\Id_G)(t)\in G$ has norm $$||(\xi_j\otimes\Id_G)(t)||= \sup_{\eta\in G',\,||\eta||\le 1}\big|\eta\big(\xi_j\otimes\Id_G(t)\big)\big| =\sup_\eta\big|\xi_j\otimes\eta(t)\big|\le ||t||_\varepsilon.$$ Therefore $||f\otimes\Id_G(t)||_\pi\le\sum|\lambda_j|\,||t||_\varepsilon$, and the infimum over all decompositions of $f$ yields $$||f\otimes\Id_G(t)||_\pi\le||f||_\nu||t||_\varepsilon.$$ Proposition 5.10 follows.\qed \endproof If $E$ is a Fr\'echet space and $(p_j)$ an increasing sequence of semi-norms on $E$ defining the topology of $E$, we have $$E=\lim_{{\displaystyle\longleftarrow}}\wh E_{p_j},$$ where $\wh E_{p_j}$ is the Hausdorff completion of $(E,p_j)$ and $\wh E_{p_{j+1}}\to\wh E_{p_j}$ the canonical morphism. Here $\wh E_{p_j}$ is a Banach space for the induced norm $\wh p_j$. \begstat{(5.11) Definition} A Fr\'echet space $E$ is said to be nuclear if the topology of $E$ can be defined by an increasing sequence of semi-norms $p_j$ such that each canonical morphism $$\wh E_{p_{j+1}}\longrightarrow\wh E_{p_j}$$ of Banach spaces is nuclear. \endstat If $E,F$ are arbitrary locally convex spaces, we always have a continuous morphism $E\hotimes_\pi F\to E\hotimes_\varepsilon F$, because $p\otimes_\varepsilon q\le p\otimes_\pi q$. If $E$, say, is nuclear, this morphism yields in fact an isomorphism $E\hotimes_\varepsilon F \simeq E\hotimes_\pi F\,$: indeed, by Prop.~5.10, we have $p_j\hotimes_\pi q\le C_j\,p_{j+1}\hotimes_\varepsilon q$ where $C_j$ is the nuclear norm of $\wh E_{p_{j+1}}\to\wh E_{p_j}$. Hence, when $E$ or $F$ is nuclear, we will identify $E\hotimes_\pi F$ and $E\hotimes_\varepsilon F$ and omit $\varepsilon$ or $\pi$ in the notation $E\hotimes F$. \begstat{(5.12) Example} \rm Let $D=\prod D(0,R_j)$ be a polydisk in $\bC^n$. For any $t\in{}]0,1[$, we equip $\cO(D)$ with the semi-norm $$p_t(f)=\sup_{tD}|f|.$$ The completion of $\big(\cO(D),p_t\big)$ is the Banach space $E_t$ of holomorphic functions on $tD$ which are continuous up to the boundary. We claim that for $t'1$,\cr \ker(E\hotimes_A B_q\to E\hotimes_A L_q)&for $k=1$.\cr}$$ Hence we obtain inductively $$\eqalign{ \toor_q^A(E,F)&=\toor_q^A(E,B_{-1})\simeq\ldots\simeq\toor_1^A(E,B_{q-2})\cr &\simeq\ker(E\hotimes_A B_{q-1}\to E\hotimes_A L_{q-1})\cr}$$ and a commutative diagram $$\eqalign{ &E\hotimes_A L_{q+1}\longrightarrow E\hotimes_A L_q\longrightarrow E\hotimes_A B_{q-1}\longrightarrow 0\cr &\qquad\searrow\qquad\qquad\nearrow\cr &~~~~\qquad E\hotimes_A B_q\cr}$$ in which the horizontal line is exact (thanks to the surjectivity of the left oblique arrow and the exactness of the sequence with $E\hotimes_A B_q$ as first term). Therefore $\ker(E\hotimes_A B_{q-1}\to E\hotimes_A L_{q-1})$ can be interpreted as the kernel of $E\hotimes_A L_q\to E\hotimes_A L_{q-1}$ modulo the image of $E\hotimes_A L_{q+1}\to E\hotimes_A L_q$,\break and this is is precisely the definition of $H_q(E\hotimes_A L_\bu)$.\qed \endproof Now, we are ready to introduce the crucial concept of transversality. \begstat{(5.32) Definition} We say that two nuclear $A$-modules $E$, $F$ are transverse if $E\hotimes_A F$ is Hausdorff and if $\toor_q^A(E,F)=0$ for $q\ge 1$. \endstat For example, a nuclearly free $A$-module $E=A\hotimes V$ is transverse to any nuclear $A$-module~$F$. Before proving further general properties, we give a fundamental example. \begstat{(5.33) Proposition} Let $X$, $Y$ be Stein spaces and let $U'\subset U\compact X$, $V\compact Y$ be Stein open subsets. If $\cF$ is a coherent sheaf over $X\times Y$, then $\cO(U')$ and $\cF(U\times V)$ are transverse over $\cO(U)$. Moreover $$\cO(U')\hotimes_{\cO(U)}\cF(U\times V)=\cF(U'\times V).$$ \endstat \begproof{} Let $\cL_\bu\to\cF$ be a free resolution of $\cF$ over $U\times V\,$; such a resolution exists by Cartan's theorem~A. Then $\cL_\bu(U\times V)$ is a resolution of \hbox{$\cF(U\times V)$} which is nuclearly free over $\cO(U)$, for $\cO(U\times V)=\cO(U)\hotimes\cO(V)\,$; in particular, we get $$\eqalign{ \cO(U')\hotimes_{\cO(U)}\cO(U\times V)&=\cO(U')\hotimes\cO(V)=\cO(U'\times V),\cr \cO(U')\hotimes_{\cO(U)}\cL_\bu(U\times V)&=\cL_\bu(U'\times V).\cr}$$ But $\cL_\bu(U'\times V)$ is a resolution of $\cF(U'\times V)$, so $$\toor_q^{\cO(U)}\big(\cO(U'),\cF(U\times V)\big)=\cases{ \cF(U'\times V)&for $q=0$,\cr 0&for $q\ge 1$.\cr}\eqno{\square}$$ \endproof \begstat{(5.34) Properties} \smallskip \item{\rm a)} If $0\to E_1\to E_2\to E_3\to 0$ is an exact sequence of nuclear $A$-modules and if $E_2$, $E_3$ are transverse to $F$, then $E_1$ is transverse to $F$. \smallskip \item{\rm b)} Let $A\to A_1\to A_2$ be homomorphisms of nuclear algebras and let $E$ be a nuclear $A$-module. if $A_1$ and $A_2$ are transverse to $E$ over $A$, then $A_2$ is tranverse to $A_1\hotimes_A E$ over $A_1$. \smallskip \item{\rm c)} Let $E^\bu$ be a complex of nuclear $A$-modules, bounded on the right side, and let $M$ be a nuclear $A$-module which is transverse to all $E^n$. If $E^\bu$ is acyclic in degrees $\ge k$, then $M\hotimes_A E^\bu$ is also acyclic in degrees $\ge k$. \smallskip \item{\rm d)} Let $E^\bu$, $F^\bu$ be complexes of nuclear $A$-modules, bounded on the right side. Let $f^\bu:E^\bu\to F^\bu$ be a $A$-linear morphism and let $M$ be a nuclear $A$-module which is transverse to all $E^q$ and $F^q$. If $f^\bu$ induces an isomorphism $H^q(f^\bu):H^q(E^\bu)\to H^q(F^\bu)$ in degrees $q\ge k$ and an epimorphism in degree $q=k-1$, then $$\Id_M\hotimes_A f^\bu:M\hotimes_A E^\bu\to M\hotimes_A F^\bu$$ has the same property.\smallskip \endstat \begproof{} a) is an immediate consequence of the $\toor$ exact sequence. \medskip \noindent{}To prove b), we need only check that if $A_1$ is transverse to $E$ over $A$, then $$\toor_q^{A_1}(A_2,A_1\hotimes_A E)=\toor_q^A(A_2,E),~~~~\forall n\ge 0.$$ Indeed, if $L_\bu=A\hotimes V_\bu$ is a nuclearly free resolution of $E$ over $A$, then $A_1\hotimes_A L_\bu=A_1\hotimes V_\bu$ is a nuclearly free resolution of $A_1\hotimes_A E$ over $A_1$, since $H_q(A_1\hotimes_A L_\bu)= \toor_q^A(A_1,E)=0$ for $q\ge 1$. Hence $$\eqalign{ \toor_q^{A_1}(A_2,A_1\hotimes_A{}\!E)&=H_q\big(A_2\hotimes_{A_1}{}\!(A_1 \hotimes_A L_\bu)\big)=H_q\big(A_2\hotimes_{A_1}{}\!(A_1\hotimes V_\bu)\big)\cr &=H_q(A_2\hotimes V_\bu)=H_q(A_2\hotimes_A L_\bu)=\toor_q^A(A_2,E).\cr}$$ c) The short exact sequences $0\to Z^q(E^\bu)\lhra E^q\buildo d^q\over\longrightarrow Z^{q+1}(E^\bu)\to 0$ show by backward induction on $q$ that $M$ is transverse to $Z^q(E^\bu)$ for \hbox{$q\ge k-1$}. Hence for $q\ge k-1$ we obtain an exact sequence $$0\longrightarrow M\hotimes_A Z^q(E^\bu)\lhra M\hotimes_A E^q\buildo d^q\over\longrightarrow M\hotimes_A Z^{q+1}(E^\bu)\longrightarrow 0,$$ which gives in particular $Z^q(M\hotimes_A E^\bu)=B^q(M\hotimes_A E^\bu)= M\hotimes_A Z^q(E^\bu)$ for $q\ge k$, as desired. \medskip \noindent{d)} is obtained by applying c) to the {\it mapping cylinder} $C(f^\bu)$, as defined in the following lemma (the proof is straightforward and left to the reader).\qed \endproof \begstat{(5.35) Lemma} If $f^\bu:E^\bu\to F^\bu$ is a morphism of complexes, the mapping cylinder $C^\bu=C(f^\bu)$ is the complex defined by $C^q=E^q\oplus F^{q-1}$ with differential $$\pmatrix{d^q_E&0\cr -f^q&d^{q-1}_F\cr}:E^q\oplus F^{q-1}\longrightarrow E^{q+1}\oplus F^q.$$ Then there is a short exact sequence $0\to F^{\bu-1}\to C^\bu\to E^\bu\to 0$ and the associated connecting homomorphism $\partial^q:H^q(E^\bu)\to H^q(F^\bu)$ is equal to $H^q(f^\bu)\,$; in particular, $C^\bu$ is acyclic in degree $q$ if and only if $H^q(f^\bu)$ is injective and $H^{q-1}(f^\bu)$ is surjective.\qed \endstat \titlec{5.D.}{$A$-Subnuclear Morphisms and Perturbations} We now introduce a notion of nuclearity relatively to an algebra $A$. This notion is needed for example to describe the properties of the $\cO(S)$-linear restriction map $\cO(S\times U)\to\cO(S\times U')$ when $U'\compact U$. \begstat{(5.36) Definition} Let $E$ and $F$ be Fr\'echet $A$-modules over a Fr\'echet algebra $A$ and let $f:E\to F$ be a $A$-linear map. We say that \smallskip \item{\rm a)} $f$ is $A$-nuclear if there exist a scalar sequence $(\lambda_j)$ with $\sum|\lambda_j|<+\infty$, an equicontinuous family of $A$-linears maps $\xi_j:E\to A$ and a bounded sequence $y_j$ in $F$ such that for all $x\in E$ $$f(x)=\sum\lambda_j\,\xi_j(x)y_j.$$ \item{\rm b)} $f$ is $A$-subnuclear if there exists a Fr\'echet $A$-module $M$ and an epimorphism $p:M\to E$ such that $f\circ p$ is $A$-nuclear; if $E$ is nuclear, we also require $M$ to be nuclear. \smallskip \endstat If $f:E\to F$ is $A$-nuclear and if $u:S\to E$ and $v:F\to T$ are continuous $A$-linear maps then $v\circ f\circ u$ is $A$-nuclear; the same is true for $A$-subnuclear maps. If $V$ and $W$ are nuclear spaces and if $u:V\to W$ is $\bC$-nuclear, then $\Id_A\hotimes u:A\hotimes V\to A\hotimes W$ is $A$-nuclear. From this we infer: \begstat{(5.37) Proposition} Let $S$, $Z$ be Stein spaces and let $U'\compact U\compact Z$ be Stein open subsets. Then the restriction $\rho:\cO(S\times U)\to\cO(S\times U')$ is \hbox{$\cO(S)$-nuclear}. If $\cF$ is a coherent sheaf over $Y\times Z$ with $Y$ Stein and $S\compact Y$, then the restriction map $\rho:\cF(S\times U)\to\cF(S\times U')$ is $\cO(S)$-subnuclear. \endstat \begproof{} As $\cO(S\times U)=\cO(S)\hotimes\cO(U)$ and $\cO(U)\to\cO(U')$ is $\bC$-nuclear, only the second statement needs a proof. By Cartan's theorem A, there exists a free resolution $\cL_\bu\to \cF$ over $S\times U$. Then there is a commutative diagram $$\cmalign{ \hfill\cL_0(S&\times U)&\longrightarrow\cF(S&\times U)\cr \hfill\rho&\big\downarrow&&\big\downarrow\rho\cr \hfill\cL_0(S&\times U')&\longrightarrow\cF(S&\times U')\cr}$$ in which the top horizontal arrow is an $\cO(S)$-epimorphism and the left vertical arrow is an $\cO(S)$-nuclear map; its composition with the bottom horizontal arrow is thus also $\cO(S)$-nuclear.\qed \endproof Let $f:E\to F$ be a $A$-linear morphism of Fr\'echet $A$-modules. Suppose that $f(E)\subset F_1$ where $F_1$ is a {\it closed} $A$-submodule of $F$ and let $f_1:E\to F_1$ be the map induced by $f$. If $f$ is $A$-nuclear, it is not true in general that $f_1$ is $A$-nuclear or $A$-subnuclear, even if $A$, $E$, $F$ are nuclear. However: \begstat{(5.38) Proposition} With the above notations, suppose $A$, $E$, $F$ nuclear. Let $B$ be a nuclear Fr\'echet algebra and let $\rho:A\to B$ be a $\bC$-nuclear \hbox{homomorphism}. Suppose that $B$ is transverse to $E$, $F$ and $F/F_1$ over $A$. If~$f:E\to F$ is $A$-subnuclear, then $\Id_B\hotimes_A f_1:B\hotimes_A E\to B \hotimes_A F_1$ is $B$-subnuclear. \endstat \begproof{} We first show that $\rho\hotimes_A f_1:E=A\hotimes_A E\to B\hotimes_A F_1$ is $\bC$-nuclear. Since a quotient of a $\bC$-nuclear map is $\bC$-nuclear by Cor.~5.16~b), we may suppose for this that $f$ is $A$-nuclear. Write $$\eqalign{ f(x)&=\sum\lambda_j\,\xi_j(x)y_j,~~~~\xi_j:E\to A,~~\sum|\lambda_j|<+\infty, ~~y_j\in F,\cr \rho(t)&=\sum\mu_k\,\eta_k(t)b_k,~~~~\eta_k:A\to\bC,~~\sum|\mu_k|<+\infty, ~~b_k\in B\cr}$$ as in the definition of ($A$-)nuclearity. Then $\rho\hotimes_A f:E\longrightarrow B\hotimes_A F$ is \hbox{$\bC$-nuclear}: for any $x\in E$, we have $\rho(\xi_j(x))=\xi_j(x)\rho(1)$ in the $A$-module structure of $B$, hence $$\eqalign{ \rho\hotimes_A f(x)=\rho\otimes f(1\otimes x) &=\sum\lambda_j\,\rho(\xi_j(x))\hotimes_A y_j\cr &=\sum\lambda_j\mu_k\,(\eta_k\circ\xi_j)(x)\,b_k\hotimes_A y_j.\cr}$$ By our transversality assumptions, $B\hotimes_A F_1$ is a closed subspace of $B\hotimes_A F$. As $\Im(\rho\hotimes_A f)\subset B\hotimes_A F_1$, the induced map $\rho\hotimes_A f_1:E\to B\hotimes_A F_1$ is $\bC$-nuclear by Cor.~5.16~a). Finally, there is a commutative diagram $$\cmalign{ B&\hotimes E&\buildo\Id_B\hotimes(\rho\hotimes_A f_1)\over{\larex 60 } B\hotimes(&B\hotimes_A F_1)\cr &~\big\downarrow&&~\big\downarrow\cr B&\hotimes_A E&\buildo\Id_B\hotimes_A f_1\over{\larex 60 } \hfill B&\hotimes_A F_1\cr}$$ in which the vertical arrows are $B$-linear epimorphisms. The top horizontal arrow is $B$-nuclear by the $\bC$-nuclearity of $\rho\hotimes_A f_1$, hence $\Id_B\hotimes_A f_1$ is \hbox{$B$-subnuclear}.\qed \endproof Example~5.12 suggests the following definition (which is somewhat less general than some other in current use, but sufficient for our purposes). \begstat{(5.39) Definition} We say that a Fr\'echet algebra $A$ is fully nuclear if the topology of $A$ is defined by an increasing family $(p_t)_{t\in{}]0,1[}$ of multiplicative semi-norms $\big($that is, $p_t(xy)\le p_t(x)\,p_t(y)\,\big)$, such that the Banach algebra homomorphism $\wh A_{p_{t'}}\to\wh A_{p_t}$ is nuclear for all $tN}\lambda_j\,\xi_j(x)y_j.$$ For $N$ large enough, we have $||v''||<1$, hence $\Id_S-v''$ is an automorphism and $\Coker(\Id_S-v'-v'')$ is generated by the classes of $y_1,\ldots,y_N$.\qed \endproof \begproof{of Theorem 5.42.} a) We may suppose that $E$ is nuclearly free and that $u$ is $A$-nuclear, otherwise we replace $f$, $u$ by their composition with \hbox{$A\hotimes M\longrightarrow M\buildo p\over\longrightarrow E$,} where $M$ is nuclear and $p:M\to E$ is an epimorphism such that $u\circ p$ is $A$-nuclear. \medskip \noindent{b)} As in (5.9), there is a $A$-nuclear decomposition $u(x)=\sum\lambda_j\,\xi_j(x)y_j$ where $(y_j)$ converges to $0$ in~$F$. Since $f$ is an epimorphism, we can find a sequence $(x_j)$ converging to $0$ in $E$ such that $f(x_j)=y_j$. Hence we have $u=f\circ v$ where $v(x)=\sum\lambda_j\,\xi_j(x)x_j$ is a $A$-nuclear endomorphism of $E$, and the cokernel of $f-u$ is the image by $f$ of the cokernel of $\Id_E-v$. \medskip \noindent{c)} By a), b) we may suppose that $F=E=A\hotimes M$, $f=\Id_E$ and that $u$ is $A$-nuclear. Let $B$ be the Banach algebra $B=\wh A_{p_t}$. Then \hbox{$B\hotimes_A E=B\hotimes M$} is a Fr\'echet $B$-module and $\Id_B\hotimes_A u$ is $B$-nuclear. By Lemma~5.42, \hbox{$\Id_B\hotimes_A\Id_E-\Id_B\hotimes_A u$} has a finitely generated cokernel over $B$. Now, there is an obvious morphism $B\to A_t$, hence by taking the tensor product with $A_t\hotimes_B\bu$ we get $$A_t\hotimes_B(B\hotimes_A E)=A_t\hotimes_B(B\hotimes M)= A_t\hotimes M=A_t\hotimes_A E=E_t$$ and we see that $$\Id_{E_t}-u_t=\Id_{A_t}\hotimes_A\Id_E-\Id_{A_t}\hotimes_A u$$ has a finitely generated cokernel over $A_t$.\qed \endproof \titlec{5.E.}{Proof of the Direct Image Theorem} We first prove a functional analytic version of the result, which appears as a relative version of Schwartz' theorem~1.9. \begstat{(5.44) Theorem} Let $A$ be a fully nuclear algebra, $E^\bu$ and $F^\bu$ complexes of fully $A$-transverse nuclear $A$-modules. Let $f^\bu:E^\bu\to F^\bu$ be a morphism of complexes such that each $f^q$ is $A$-subnuclear. Suppose that $E^\bu$ and $F^\bu$ are bounded on the right and that $H^q(f^\bu)$ is an isomorphism for each $q$. Then for every $t<1$, there is a complex $L^\bu$ of finitely generated free $A_t$-modules and a complex morphism $h^\bu:L^\bu\to E_t^\bu$ which induces an isomorphism on cohomology. \endstat \begproof{} a) We first show the following statement: \smallskip \item{}{\it Suppose that $E^\bu_t$ and $F^\bu_t$ are acyclic in degrees $>q$. Then for every $t'N$. Fix a sequence $t<\ldotsq$.} \smallskip Suppose that $L^k$ has been constructed for $k\ge q$. Consider the mapping cylinder $N^\bu_q=C(f^\bu_{t_q}\circ h^\bu_{\gge q,\,t_q})$ and the complex morphism $$M^\bu_q\longrightarrow N^\bu_q,~~~~ L^k_{\gge q,\,t_q}\oplus E^{k-1}_{t_q}\longrightarrow L^k_{\gge q,\,t_q}\oplus F^{k-1}_{t_q}$$ given by $\Id\oplus f^{k-1}_{t_q}$. This morphism is $A_{t_q}$-subnuclear in each degree and induces an isomorphism in cohomology (compare the cohomology of the short exact sequences associated to each mapping cylinder, with the obvious morphism between them). Moreover, $M^\bu_q$ and $N^\bu_q$ are acyclic in degrees \hbox{$k>q$}. By step a), the cohomology space $H^q(M^\bu_{q,\,t_{q-1}})$ is a finitely generated \hbox{$A_{t_{q-1}}$-module}. Therefore, we can find a finitely generated free $A_{t_{q-1}}$-module $L^{q-1}$ and a morphism $$d^{q-1}\oplus h^{q-1}:L^{q-1}\to M^q_{q,\,t_{q-1}}=L^q_{t_{q-1}}\oplus E^{q-1}_{t_{q-1}}$$ such that the image is contained in $Z^q(M^\bu_{q,\,t_{q-1}})$ and generates the cohomology space $H^q(M^\bu_{q,\,t_{q-1}})$. As $M^{q-1}_{q,\,t_{q-1}}=E^{q-2}_{t_{q-1}}$, this means that $M^\bu_{q-1}$ is also acyclic in degree $q$. Thus $L^{q-1}$, together with the maps $(d^{q-1},h^{q-1})$ satisfies the induction hypotheses for $q-1$, and $L^\bu_t$ together with the induced map $h^\bu_t:L^\bu_t\to E^\bu_t$ is the required morphism of complexes.\qed \endproof \begproof{of theorem 5.1.} Let $X$, $Y$ be complex analytic schemes, let \hbox{$F:X\to Y$} be a proper analytic morphism and let $\cS$ be a coherent sheaf over $X$. Fix a point $y_0\in Y$, a neighborhood of $y_0$ which is isomorphic to a closed analytic subscheme of a Stein open set $W\subset\bC^n$ and a polydisk $D^0=D(y_0,R_0)\compact W$. The compact set $K=F^{-1}(\ol D^0\cap Y)$ can be covered by finitely many open subsets $U_\alpha^0\compact X$ which possess embeddings as closed analytic subschemes of Stein open sets $\Omega_\alpha^0\subset\bC^{N_\alpha}$. Let $\Omega'_\alpha\compact\Omega_\alpha\compact\Omega_\alpha^0$ be Stein open subsets such that $U_\alpha=U^0_\alpha\cap\Omega_\alpha$ and $U'_\alpha=U^0_\alpha\cap\Omega'_\alpha$ still cover $K$. Let $i_\alpha:U^0_\alpha\to\Omega^0_\alpha$ and $j:Y\cap D^0\to D^0$ be the embeddings and $\cS_\alpha=\big(i_\alpha\times(j\circ F)\big)_\star\cS$ the image sheaf of $\cS$ on $\Omega^0_\alpha\times D^0$. Let $D\compact D^0$ be a concentric polydisk. Then $\cS\big(U_\alpha\cap F^{-1}(D)\big)= \cS_\alpha(\Omega_\alpha\times D)$ is a fully transverse $\cO(D)$-module by Ex.~5.41, and so is $\cS\big(U'_\alpha\cap F^{-1}(D)\big)= \cS_\alpha(\Omega'_\alpha\times D)$. Moreover, the restriction map $$\cS\big(U_\alpha\cap F^{-1}(D)\big)\longrightarrow\cS\big(U'_\alpha\cap F^{-1}(D)\big)$$ is $\cO(D)$-subnuclear by Prop.~5.37 applied to $\cF=\cS_\alpha$. For every Stein open set $V\subset D$, Prop.~5.33 shows that $$\cO(V)\hotimes_{\cO(D)}\cS\big(U_\alpha\cap F^{-1}(D)\big) =\cS\big(U_\alpha\cap F^{-1}(V)\big).$$ Denote by $\cU\cap F^{-1}(D)$ the collection $\big(U_\alpha\cap F^{-1}(D) \big)$ and use a similar notation with $\cU'=(U'_\alpha)$. As $\cU\cap F^{-1}(D)$, $\cU'\cap F^{-1}(D)$ are Stein coverings of $F^{-1}(D)$, the Leray theorem applied to the alternate \v Cech complex of $\cS$ over $\cU\cap F^{-1}(D)$ and $\cU'\cap F^{-1}(D)$ gives an isomorphism $$H^\bu\big(AC^\bu(\cU\cap F^{-1}(D),\cS)\big)= H^\bu\big(AC^\bu(\cU'\cap F^{-1}(D),\cS)\big)=H^\bu\big(F^{-1}(D),\cS\big).$$ By the above discussion, $AC^\bu(\cU\cap F^{-1}(D),\cS)$ and $AC^\bu(\cU'\cap F^{-1}(D),\cS)$ are finite complexes of fully transverse nuclear $\cO(D)$-modules, the restriction map $$AC^\bu(\cU\cap F^{-1}(D),\cS)\longrightarrow AC^\bu(\cU'\cap F^{-1}(D),\cS)$$ is $\cO(D)$-subnuclear and induces an isomorphism on cohomology groups. Set $D=D(y_0,R)$ and $D_t=D(y_0,tR)$. Theorem~5.44 shows that for every $t<1$ there is a complex of finitely generated free $\cO$-modules $\cL^\bu$ and a \hbox{$\cO(D_t)$-linear} morphism of complexes $$\cL^\bu(D_t)\to AC^\bu(\cU\cap F^{-1}(D_t),\cS)$$ which induces an isomorphism on cohomology. Let $V\subset D_t$ be an arbitrary Stein open set. By Prop.~5.34~d) applied with $M=\cO(V)$, we conclude that $\cL^\bu(V)\to AC^\bu(\cU\cap F^{-1}(V),\cS)$ induces an isomorphism on cohomology. If we take the direct limit as $V$ runs over all Stein neighborhoods of a point $y\in Y\cap D_t$, we see that $\cH^q(\cL^\bu)\simeq R^qF_\star\cS$ over $Y\cap D_t$, hence $R^qF_\star\cS$ is $\cO_Y$-coherent near $y_0$.\qed \endproof \end