% Complex Analytic and Differential Geometry, Chapter I % J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France \input analgeom.mac \def\aff{{\rm aff}} \def\BM{{\rm BM}} \def\Diag{{\rm Diag}} \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 $\bbbc^n$. 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 $\bbbr^n$. 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 $m\in\bbbn$ and $k\in\bbbn\cup\{\infty,\omega\}$. 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 $\bbbr^m$. An {\it atlas} of class $C^k$ is a collection of homeomorphisms $\tau_\alpha:U_\alpha\lra V_\alpha$, $\alpha\in I$, called {\it differentiable charts}, such that $(U_\alpha)_{\alpha\in I}$ is an open covering of $M$ and $V_\alpha$ an open subset of $\bbbr^m$, and such that for all $\alpha,\beta\in I$ the {\it transition map} $$\tau_{\alpha\beta}=\tau_\alpha\circ\tau_\beta^{-1}: \tau_\beta(U_\alpha\cap U_\beta)\lra\tau_\alpha(U_\alpha\cap U_\beta) \leqno(1.1)$$ 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 $\tau_\alpha(x)=(x^\alpha_1\ld x^\alpha_m)$ 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 $s\in\bbbn\cup\{\infty,\omega\}$, $0\le s\le k$, we denote by $C^s(\Omega,\bbbr)$ 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, $C^s(\Omega,\bbbr)$ 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 $$C^1(\Omega,\bbbr)\ni f\longmapsto \xi\cdot f=\sum_{1\le j\le m}\xi_j\,{\partial f\over\partial x_j}(a)$$ in any local coordinate system $(x_1\ld x_m)$ 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 $(dx_1\ld dx_m)$ is the dual basis of $(\partial/\partial x_1\ld \partial/\partial x_m)$ 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 $I=(i_1\ld i_p)$ 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 $\Omega\subset\bbbr^m$. We first show that $T$ is cohomologous to a smooth form. In fact, let $\psi\in\ci(\bbbr^m)$ be a cut-off function such that $\Supp\psi\subset\ovl\Omega$, $0<\psi\le 1$ 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 $\bbbr^m$ which leaves $\complement\Omega$ invariant pointwise, so $F_v(\Omega)=\Omega$. This diffeomorphism is homotopic to the identity through the homotopy $H_v(t,x)=F_{tv}(x):[0,1]\times\Omega \lra\Omega$ 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 $\Omega\subset\bbbr^m$ 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 $\Omega\subset\bbbc$ be an open set and let $z=x+\ii y$ be the complex variable, where $x,y\in\bbbr$. If $f$ is a function of class $C^1$ on $\Omega$, we have $$df={\partial f\over\partial x}\,dx+{\partial f\over\partial y}\,dy ={\partial f\over\partial z}\,dz+{\partial f\over\partial\ovl z}\,d\ovl z$$ with the usual notations $${\partial\over\partial z}={1\over 2}\Big({\partial\over\partial x} -\ii{\partial\over\partial y}\Big),~~~~ {\partial\over\partial\ovl z}={1\over 2}\Big({\partial\over\partial x} +\ii{\partial\over\partial y}\Big).\leqno(3.1)$$ The function $f$ is holomorphic on $\Omega$ if $df$ is $\bbbc$-linear, that is, $\partial f/\partial\ovl z=0$. \begstat{(3.2) Cauchy formula} Let $K\subset\bbbc$ be a compact set with piecewise $C^1$ boun\-dary $\partial K$. Then for every $f\in C^1(K,\bbbc)$ $$f(w)={1\over 2\pi\ii}\int_{\partial K}{f(z)\over z-w}\,dz- \int_K{1\over\pi(z-w)}\,{\partial f\over\partial\ovl z}\,d\lambda(z),~~~~ w\in K^\circ$$ where $d\lambda(z)={\ii\over 2}dz\wedge d\ovl z=dx\wedge dy$ is the Lebesgue measure on $\bbbc$. \endstat \begproof{} Assume for simplicity $w=0$. As the function $z\mapsto 1/z$ is locally integrable at $z=0$, we get $$\eqalign{ \int_K{1\over\pi z}\,{\partial f\over\partial\ovl z}\,d\lambda(z) &=\lim_{\varepsilon\to 0}\int_{K\ssm D(0,\varepsilon)} {1\over\pi z}\,{\partial f\over\partial\ovl z}\,{\ii\over 2}dz\wedge d\ovl z\cr &=\lim_{\varepsilon\to 0}\int_{K\ssm D(0,\varepsilon)} d\Big[{1\over 2\pi\ii}\,f(z)\,{dz\over z}\Big]\cr &={1\over 2\pi\ii}\int_{\partial K}f(z)\,{dz\over z}- \lim_{\varepsilon\to 0}{1\over 2\pi\ii}\int_{\partial D(0,\varepsilon)} f(z)\,{dz\over z}\cr}$$ 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 $\partial/\partial\ovl z$ on $\bbbc$, i.e.\ $\partial E/\partial\ovl z=\delta_0$ $($Dirac measure at $0)$. As a consequence, if $v$ is a distribution with compact support in $\bbbc$, then the convolution $u=(1/\pi z)\star v$ is a solution of the equation $\partial u/\partial\ovl z=v$. \endstat \begproof{} Apply (3.2) with $w=0$, $f\in\cD(\bbbc)$ and $K\supset{\rm Supp}\,f$, so that $f=0$ on the boundary $\partial K$ and $f(0)=\langle 1/\pi z,-\partial f/\partial\ovl z\rangle$.\qed \endproof \begstat{(3.5) Remark} \rm It should be observed that this formula cannot be used to solve the equation $\partial u/\partial\ovl z=v$ 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 $$\langle v,z^n\rangle=-\langle u,\partial z^n/\partial\ovl z\rangle=0$$ 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 $\Omega\subset\bbbc^n$ be an open set. A function $f:\Omega\to\bbbc$ 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 $z_1\ld z_{j-1}$, $z_{j+1}\ld z_n$ 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 $(z_{0,1}\ld z_{0,n})$ and (multi)radius $R=(R_1,\ld R_n)$ is defined as the product of the disks of center $z_{0,j}$ and radius $R_j>0$ in each factor~$\bbbc$~: $$D(z_0,R)=D(z_{0,1},R_1)\times\ldots\times D(z_{0,n},R_n)\subset\bbbc^n. \leqno(3.6)$$ 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 $\partial D(z_0,R)= \bigcup_j\{z\in\ovl D(z_0,R)\,;\,|z_j-z_{0,j}|=R_j\}$ when $n\ge 2$. By induction on $n$, we easily get the \begstat{(3.8) Cauchy formula on polydisks} If $\ovl D(z_0,R)$ is a closed polydisk contained in $\Omega$ and $f\in\cO(\Omega)$, then for all $w\in D(z_0,R)$ we have $$f(w)={1\over(2\pi\ii)^n}\int_{\Gamma(z_0,R)} {f(z_1\ld z_n)\over(z_1-w_1)\ldots(z_n-w_n)}\,dz_1\ldots dz_n.\eqno\qed$$ \endstat The expansion $(z_j-w_j)^{-1}=\sum (w_j-z_{0,j})^{\alpha_j} (z_j-z_{0,j})^{-\alpha_j-1}$, $\alpha_j\in\bbbn$, $1\le j\le n$, shows that $f$ can be expanded as a convergent power series $f(w)=\sum_{\alpha\in\bbbn^n}a_\alpha(w-z_0)^\alpha$ 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 $$a_\alpha= {1\over(2\pi\ii)^n}\int_{\Gamma(z_0,R)}{f(z_1\ld z_n)\,dz_1\ldots dz_n \over(z_1-z_{0,1})^{\alpha_1+1}\ldots(z_n-z_{0,n})^{\alpha_n+1}} ={f^{(\alpha)}(z_0)\over\alpha!}.\leqno(3.9)$$ As a consequence, $f$ is holomorphic over $\Omega$ if and only if $f$ is $\bbbc$-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 $\alpha\in\bbbn^n$, then $f=0$ on~$\Omega$.\qed \endstat Another consequence of (3.9) is the {\it Cauchy inequality} $$|f^{(\alpha)}(z_0)|\le{\alpha!\over R^\alpha}\sup_{\Gamma(z_0,R)}|f|,~~~~ \ovl D(z_0,R)\subset\Omega,\leqno(3.11)$$ From this, it follows that every bounded holomorphic function on $\bbbc^n$ is constant (Liouville's theorem), and more generally, every holomorphic function $F$ on $\bbbc^n$ 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 $C^0(\Omega,\bbbc)$. Then $\cO(\Omega)$ is closed in $C^0(\Omega,\bbbc)$. 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 $\dim_\bbbc X=n$ is a differentiable manifold equipped with a holomorphic atlas $(\tau_\alpha)$ with values in $\bbbc^n$~; 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 $$d\tau_\alpha(x):T_{X,x}\lra\bbbc^n,\qquad U_\alpha\ni x\,;$$ the induced complex structure on $T_{X,x}$ is indeed independent of $\alpha$ since the differentials $d\tau_{\alpha\beta}$ are $\bbbc$-linear isomorphisms. We denote by $T^\bbbr_X$ the underlying real tangent space and by $J\in{\rm End}(T^\bbbr_X)$ the {\it almost complex structure}, i.e.\ the operator of multiplication by $\ii=\sqrt{-1}$. If $(z_1\ld z_n)$ are complex analytic coordinates on an open subset $\Omega\subset X$ and $z_k=x_k+\ii y_k$, then $(x_1,y_1\ld x_n,y_n)$ define real coordinates on $\Omega$, and $T^\bbbr_{X\restriction\Omega}$ 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 $\bbbc\otimes T_X=\bbbc\otimes_\bbbr T^\bbbr_X= T^\bbbr_X\oplus\ii T^\bbbr_X$ 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 $$~{\partial\over\partial z_k}={1\over 2}\Big({\partial\over\partial x_k}- \ii{\partial\over\partial y_k}\Big),~~~~ {\partial\over\partial\ovl z_k}={1\over 2}\Big({\partial\over\partial x_k}+ \ii{\partial\over\partial y_k}\Big),~~~~1\le k\le n\leqno(3.13)$$ 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 $\ovl{T_X}$ (with conjugate complex structure $-J$), via the $\bbbc$-linear embeddings $$\cmalign{T_X&\lra T^{1,0}_X\subset\bbbc\otimes T_X, \hfill\ovl{T_X}&\lra T^{0,1}_X\subset\bbbc\otimes T_X\cr \hfill\xi&\longmapsto{1\over2}(\xi-\ii J\xi),~~~~~ \xi&\longmapsto{1\over2}(\xi+\ii J\xi).\cr}$$ We thus have a canonical decomposition $\bbbc\otimes T_X=T^{1,0}_X\oplus T^{0,1}_X\simeq T_X\oplus\ovl{T_X}$, and by duality a decomposition $$\Hom_\bbbr(T^\bbbr_X;\bbbc)\simeq\Hom_\bbbc(\bbbc\otimes T_X;\bbbc)\simeq T^\star_X\oplus\ovl{T^\star_X}$$ where $T^\star_X$ is the space of $\bbbc$-linear forms and $\ovl{T^\star_X}$ the space of conjugate $\bbbc$-linear forms. With these notations, $(dx_k,dy_k)$ is a basis of $\Hom_\bbbr(T_\bbbr X,\bbbc)$, $(dz_j)$ a basis of $T^\star_X$, $(d\ovl z_j)$ a basis of $\ovl{T^\star_X}$, and the differential of a function $f\in C^1(\Omega,\bbbc)$ can be written $$df=\sum_{k=1}^n {\partial f\over\partial x_k}\,dx_k+{\partial f\over\partial y_k}\,dy_k =\sum_{k=1}^n {\partial f\over\partial z_k}\,dz_k+{\partial f\over\partial\ovl z_k}\, d\ovl z_k.\leqno(3.14)$$ The function $f$ is holomorphic on $\Omega$ if and only if $df$ is $\bbbc$-linear, i.e.\ if and only if $f$ satisfies the {\it Cauchy-Riemann equations} $\partial f/\partial\ovl z_k=0$ 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 $\bbbc\otimes_\bbbr\Lambda^\bu_\bbbr(T_X^\bbbr)^\star =\Lambda^\bu_\bbbc(\bbbc\otimes T_X)^\star$ is given by $$\Lambda^k(\bbbc\otimes T_X)^\star=\Lambda^k\big(T_X\oplus\ovl{T_X}\big)^\star =\bigoplus_{p+q=k}\Lambda^{p,q}T^\star_X,~~~~0\le k\le 2n$$ where the exterior products are taken over $\bbbc$, and where the components $\Lambda^{p,q}T^\star_X$ are defined by $$\Lambda^{p,q}T^\star_X=\Lambda^p T^\star_X\otimes \Lambda^q\ovl{T^\star_X}.\leqno(3.15)$$ 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 $$u(z)=\sum_{|I|=p,|J|=q}u_{I,J}(z)\,dz_I\wedge d\ovl z_J,~~~~ u_{I,J}\in C^s(\Omega,\bbbc).$$ 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 $\Lambda^k T^\star_\bbbc X$ in its $(p,q)$ components. Formula (3.14) shows that the exterior derivative $d$ splits into $d=d'+d''$, where $$\leqalignno{ d'&:\ci(X,\Lambda^{p,q}T^\star_X)\lra\ci(X,\Lambda^{p+1,q}T^\star_X),\cr d''&:\ci(X,\Lambda^{p,q}T^\star_X)\lra\ci(X,\Lambda^{p,q+1}T^\star_X),\cr d'u&=\sum_{I,J}\sum_{1\le k\le n}{\partial u_{I,J}\over\partial z_k}\,dz_k \wedge dz_I\wedge d\ovl z_J,&(3.16')\cr d''u&=\sum_{I,J}\sum_{1\le k\le n.}{\partial u_{I,J}\over\partial\ovl z_k} \,d\ovl z_k\wedge dz_I\wedge d\ovl z_J.&(3.16'')\cr}$$ 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 $p=0,1\ld n$ a complex, called the {\it Dolbeault complex} $$\ci(X,\Lambda^{p,0}T^\star_X)\buildo d''\over\lra\cdots\lra \ci(X,\Lambda^{p,q}T^\star_X)\buildo d''\over\lra \ci(X,\Lambda^{p,q+1}T^\star_X)$$ and corresponding {\it Dolbeault cohomology groups} $$H^{p,q}(X,\bbbc)={\Ker d^{\prime\prime\,p,q}\over \Im d^{\prime\prime\,p,q-1}},\leqno(3.18)$$ with the convention that the image of $d''$ is zero for $q=0$. The coho\-mo\-logy group $H^{p,0}(X,\bbbc)$ consists of $(p,0)$-forms $u=\sum_{|I|=p}u_I(z)\,dz_I$ such that $\partial u_I/\partial\ovl z_k=0$ 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 $F:X_1\lra X_2$ 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{$\bbbc$-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 $$F^\star: H^{p,q}(X_2,\bbbc)\lra H^{p,q}(X_1,\bbbc).$$ The rules of complex differential calculus can be easily extended to currents. We use the following notation. \begstat{(3.20) Definition} There are decompositions $$\cD^k(X,\bbbc)=\bigoplus_{p+q=k}\cD^{p,q}(X,\bbbc),~~~~ \cD'_k(X,\bbbc)=\bigoplus_{p+q=k}\cD'_{p,q}(X,\bbbc).$$ The space $\cD'_{p,q}(X,\bbbc)$ is called the space of currents of bidimension $(p,q)$ and bidegree $(n-p,n-q)$ on~$X$, and is also denoted $\cD^{\prime\,n-p,n-q}(X,\bbbc)$. \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~$\bbbr^m$. We first recall a construction of the Newton kernel. Let $d\lambda=dx_1\ldots dx_m$ be the Lebesgue measure on $\bbbr^m$. We denote by $B(a,r)$ the euclidean open ball of center $a$ and radius $r$ in $\bbbr^m$ 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 $\int_{\bbbr^m}e^{-|x|^2}d\lambda(x)$ 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 $\bbbr^m$ and satisfies $\Delta N=\delta_0$. When $m=2$, this follows from Cor.~3.4 and the fact that $\Delta=4\partial^2/\partial z\partial\ovl z$. 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~ $I_m=\int_{\bbbr^m}(|x|^2+1)^{-1-m/2}\,d\lambda(x)$. The last equality is easily seen by performing the change of variable $y=\varepsilon x$ in the integral $$\int_{\bbbr^m}\varepsilon^2(|x|^2+\varepsilon^2)^{-1-m/2} \,f(x)\,d\lambda(x)=\int_{\bbbr^m}(|y|^2+1)^{-1-m/2} \,f(\varepsilon y)\,d\lambda(y),$$ 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 $\bbbc^n$ with $L^1_\loc$ coefficients defined by $$\leqalignno{ k_\BM(z)&=c_n\sum_{1\le j\le n}(-1)^j {\ovl z_j\,dz_1\wedge\ldots dz_n\wedge d\ovl z_1\wedge\ldots \widehat{d\ovl z_j}\ldots\wedge d\ovl z_n\over|z|^{2n}},&(3.23)\cr c_n&=(-1)^{n(n-1)/2}{(n-1)!\over(2\pi\ii)^n}.\cr}$$ \begstat{(3.24) Lemma} $d''k_\BM=\delta_0$ on $\bbbc^n$. \endstat \begproof{} Since the Lebesgue measure on $\bbbc^n$ is $$d\lambda(z)=\bigwedge_{1\le j\le n}{\ii\over 2}dz_j\wedge d\ovl z_j =\Big({\ii\over 2}\Big)^n(-1)^{\textstyle{n(n-1)\over 2}} dz_1\wedge\ldots dz_n\wedge d\ovl z_1\wedge\ldots d\ovl z_n,$$ we find $$\eqalignno{ d''k_\BM&=-{(n-1)!\over\pi^n}\sum_{1\le j\le n}{\partial\over\partial \ovl z_j}\Big({\ovl z_j\over |z|^{2n}}\Big)d\lambda(z)\cr &=-{1\over n(n-1)\alpha_{2n}}\sum_{1\le j\le n}{\partial^2\over\partial z_j\partial\ovl z_j}\Big({1\over |z|^{2n-2}}\Big)d\lambda(z)\cr &=\Delta N(z)d\lambda(z)=\delta_0.&\square\cr}$$ \endproof We let $K_\BM(z,\zeta)$ be the pull-back of $k_\BM$ by the map $\pi:\bbbc^n\times\bbbc^n\to\bbbc^n$, $(z,\zeta)\longmapsto z-\zeta$. 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 $\Delta\subset\bbbc^n\times\bbbc^n$. \begstat{(3.26) Koppelman formula} Let $\Omega\subset\bbbc^n$ be a bounded open set with piecewise $C^1$ boundary. Then for every $(p,q)$-form $v$ of class $C^1$ on $\ovl\Omega$ 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 $v\in{}^s\cD^{p,q}(\bbbc^n)$ be a form of class $C^s$ with compact support such that $d''v=0$, $q\ge 1$. Then the $(p,q-1)$-form $$u(z)=\int_{\bbbc^n}K_\BM^{p,q-1}(z,\zeta)\wedge v(\zeta)$$ 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 $H^{0,1}_c(\bbbc^n,\bbbc)$ vanishes for $n\ge 2$. \endstat \begproof{} Apply the Koppelman formula on a sufficiently large ball $\ovl\Omega=\ovl B(0,R)$ 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 $\bbbc^n\ssm\ovl B(0,R)$. Now, this complement is a union of complex lines when $n\ge 2$, hence $u=0$ on $\bbbc^n\ssm\ovl B(0,R)$ by Liouville's theorem.\qed \endproof \begstat{(3.28) Hartogs extension theorem} Let $\Omega$ be an open set in $\bbbc^n$, $n\ge 2$, 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 $f_0=(1-\psi)f\in\ci(\Omega)$, 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~$\bbbc^n$, such that $d''v=0$. By Cor.~3.27, there is a smooth function $u$ with compact support in $\bbbc^n$ 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 $\bbbc^n\ssm\Supp\psi$. 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 $\wt f\in\cO(\Omega)\cap C^0(\ovl\Omega)$, 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 $\bbbc^n$ and $v\in{}^s\cE^{p,q}(\Omega,\bbbc)$, $[$resp.\ $v\in{}^s\cD^{\prime\,p,q}(\Omega,\bbbc)]$, 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 ${}^s\cE^{p,q-1}(\omega,\bbbc)$ $[$resp.\ a current $u\in{}^s\cD^{\prime\,p,q-1}(\smash{\omega},\bbbc)]$ such that $d''u=v$ on~$\omega$. \vskip0pt \endstat \begproof{} We assume that $\Omega$ is a ball $B(0,r)\subset\bbbc^n$ 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 $\bbbc^n\times\bbbc^n\times\bbbc^n$ the kernel $$K(z,w,\zeta)=c_n\sum_{j=1}^n{(-1)^j(w_j-\ovl\zeta_j) \over((z-\zeta)\cdot(w-\ovl\zeta))^n} \bigwedge\limits_k(dz_k-d\zeta_k)\wedge \bigwedge\limits_{k\ne j}(dw_k-d\ovl\zeta_k).$$ By construction, $K_\BM(z,\zeta)$ is the result of the substitution $w=\ovl z$ in $K(z,w,\zeta)$, i.e. $K_\BM=h^\star K$ where $h(z,\zeta)=(z,\ovl z,\zeta)$. 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 $g(z)=(z,\ovl z)$ 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 $$U=\big\{(z,w)\in\omega\times\omega\,;\,\forall\zeta\notin\omega,~ \Re(z-\zeta)\cdot(w-\ovl\zeta)>0\big\},$$ which contains the ``conjugate-diagonal'' points $(z,\ovl z)$ as well as the points $(z,0)$ and $(0,w)$ in $\omega\times\omega$. Moreover $U$ clearly has convex slices $(\{z\}\times\bbbc^n)\cap U$ and $(\bbbc^n\times\{w\})\cap U$. 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 $\ovl\zeta_j|\zeta|^{-2n}$. 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 $f\in\cD^{\prime\,p,0}(X,\bbbc)$ satisfies $d''f\in\cE^{p,1}(X,\bbbc)$, then $f\in\cE^{p,0}(X,\bbbc)$. \endstat \begproof{} The result is local, so we may assume that $X=\Omega$ is a neighborhood of $0$ in $\bbbc^n$. The $(p,1)$-form $v=d''f\in \cE^{p,1}(X,\bbbc)$ satisfies $d''v=0$, hence there exists $u\in\cE^{p,0}(\smash{\wt\Omega},\bbbc)$ such that $d''u=d''f$. Then $f-u$ is holomorphic and $f=(f-u)+u\in\cE^{p,0} (\smash{\wt\Omega},\bbbc)$.\qed \endproof \titleb{\S 4.}{Subharmonic Functions} A {\it harmonic} (resp.\ {\it subharmonic}) function on an open subset of $\bbbr^m$ 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 $\bbbr^m\ssm\{0\}$. 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 $\Omega\compact\bbbr^m$ 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 $\Omega\compact\bbbr^m$ is a function $G_\Omega(x,y):\ovl\Omega\times \ovl\Omega\to[-\infty,0]$ with the following properties: \medskip \item{\rm a)} $G_\Omega(x,y)$ is $\ci$ on $\ovl\Omega\times\ovl\Omega \ssm\Diag_\Omega$~~ $(\Diag_\Omega={}$ 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 $$G_r(x,y)=N(x-y)-N\Big({|y|\over r}\Big(x-{r^2\over|y|^2}\,y\Big)\Big),~~~~ x,y\in\ovl B(0,r).\leqno(4.2)$$ 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 $r^2y/|y|^2\notin\ovl B(0,r)$ 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 $\Omega\compact \bbbr^m$, 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 $\Omega=B(0,r)$, $u\in C^2\big(\ovl B(0,r),\bbbr\big)$ 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 $u\in C^0\big(\ovl B(a,r),\bbbr\big)\cap C^2\big(B(a,r),\bbbr\big)$ 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 $v\in C^0\big(S(a,r),\bbbr\big)$ 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 $\ovl B(a,r)$ 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 $\ovl B(a,r)$ 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 $$\eqalign{ \int_{B(0,r)}\Delta u(y)\,G_r(0,y)\,d\lambda(y)&=-{1\over\sigma_{m-1}}\int_0^r {dt\over t^{m-1}}\int_{|y|0$} (resp.\ $\Delta u(a)<0$), Formula (4.10) shows that $\mu_S(u\,;a,r)>u(a)$ (resp.\ $\mu_S(u\,;a,r)-\infty$ almost everywhere on $W$. If $x\in\ovl W$, one can choose $a\in W$ such that $|a-x|-\infty$. Then $B(a,r)$ is a neighborhood of $x$, $\smash{\ovl B}(a,r)\subset\Omega$ and $\mu_B(u\,;a,r)\ge u(a)>-\infty$. Therefore $x\in W$, $W$ is also closed. We must have $W=\Omega$ or $W=\emptyset$~; in the last case $u\equiv-\infty$ by the mean value inequality.\qed \endproof \begstat{(4.18) Theorem} Let $u\in\Sh(\Omega)$ be such that $u\not\equiv-\infty$ on each connected component of $\Omega$. Then \medskip \item{\rm a)} $r\longmapsto\mu_S(u\,;a,r)$, $r\longmapsto\mu_B(u\,;a,r)$ are non decreasing functions in the interval $]0,d(a,\complement\Omega)[\,$, and $\mu_B(u\,;a,r)\le\mu_S(u\,;a,r)$. \smallskip \item{\rm b)} For any family $(\rho_\varepsilon)$ of smoothing kernels, $u\star\rho_\varepsilon\in\Sh(\Omega_\varepsilon)\cap \ci(\Omega_\varepsilon,\bbbr)$, the family $(u\star\rho_\varepsilon)$ is non decreasing in $\varepsilon$ and $\lim_{\varepsilon\to 0}u\star \rho_\varepsilon=u$.\vskip0pt \endstat \begproof{} We first verify statements a) and b) when $u\in C^2(\Omega,\bbbr)$. Then $\Delta u\ge 0$ and $\mu_S(u\,;a,r)$ is non decreasing in virtue of (4.10). By (4.9), we find that $\mu_B(u\,;a,r)$ is also non decreasing and that $\mu_B(u\,;a,r)\le \mu_S(u\,;a,r)$. Furthermore, Formula (4.11) shows that $\varepsilon\longmapsto u\star\rho_\varepsilon(a)$ is non decreasing (provided that $\rho_\varepsilon$ is radially symmetric). In the general case, we first observe that property 4.12~c) is equivalent to the inequality $$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\ci(\Omega_\varepsilon,\bbbr)$ 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 \ci(\Omega_\varepsilon,\bbbr)$. 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')},$$ where $K'=\Supp\psi$, hence the sequence of measures $(\mu_j)$ is uniformly bounded in mass on every compact subset of~$\Omega$. By weak compactness, there is a subsequence $(\mu_{j_\nu})$ which converges weakly to a positive measure $\mu$ on~$\Omega$. We claim that $f\star (\psi\mu_{j_\nu})$ converges to $f\star(\psi\mu)$ in $L^1_\loc(\bbbr^m)$ for every function $f\in L^1_\loc(\bbbr^m)$. In fact, this is clear if $f\in C^\infty(\bbbr^m)$, and in general we use an approximation of $f$ by a smooth function $g$ together with the estimate $$\|(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\bbbr^m$$ to get the conclusion. We apply this when $f=N$ is the Newton kernel. Then $h_j=u_j-N\star(\psi\mu_j)$ is harmonic on $\omega$ and bounded in $L^1(\omega)$. As $h_j=h_j\star\rho_\varepsilon$ for any smoothing kernel $\rho_\varepsilon$, we see that all derivatives $D^\alpha h_j=h_j\star (D^\alpha\rho_\varepsilon)$ are in fact uniformly locally bounded in~$\omega$. Hence, after extracting a new subsequence, we may suppose that $h_{j_\nu}$ converges uniformly to a limit $h$ on~$\omega$. Then $u_{j_\nu}=h_{j_\nu}+N\star(\psi\mu_{j_\nu})$ converges to $u=h+N\star(\psi\mu)$ in $L^1_\loc(\omega)$, as desired.\qed \endproof We conclude this subsection by stating a generalized version of the Green-Riesz formula. \begstat{(4.22) Proposition} Let $u\in\Sh(\Omega)\cap L^1_\loc(\Omega)$ and $\ovl B(0,r)\subset\Omega$. \medskip \item{\rm a)} The Green-Riesz formula still holds true for such an $u$, namely, for every $x\in B(0,r)$ $$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).$$ \item{\rm b)} {\rm (Harnack inequality)}\hfill\break If $u\ge 0$ on $\ovl B(0,r)$, then for all $x\in B(0,r)$ $$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).$$ If $u\le 0$ on $\ovl B(0,r)$, then for all $x\in B(0,r)$ $$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_{\bbbr^m}\mu\star\rho_\varepsilon(y)\,\wt G_x(y)\,d\lambda(y)\cr &=\int_{\bbbr^m}\mu(y)\,\wt G_x\star\rho_\varepsilon(y)\,d\lambda(y).\cr}$$ However $\wt G_x$ is continuous on $\bbbr^m\ssm\{x\}$ and subharmonic in a neighborhood of~$x$, hence $\wt G_x\star\rho_\varepsilon$ converges uniformly to $\wt G_x$ on every compact subset of $\bbbr^m\ssm\{x\}$, and converges pointwise monotonically in a neighborhood of~$x$. The desired equality follows by the monotone convergence theorem. Finally, b) is a consequence of a), for the integral involving $\Delta u$ is nonpositive and $${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}}$$ by (4.6) combined with the obvious inequality $(r-|x|)^m\le |x-y|^m\le (r+|x|)^m$.\qed \endproof \titled{\S 4.C.3. Upper Envelopes and Choquet's Lemma.} Let $\Omega\subset\bbbr^n$ and let $(u_\alpha)_{\alpha\in I}$ be a family of upper semicontinuous functions $\Omega\lra[-\infty,+\infty[$. We assume that $(u_\alpha)$ is locally uniformly bounded above. Then the upper envelope $$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.$$ As every ball $B(z,\varepsilon)$ is a union of balls $B(z_j,\varepsilon_j)$, we easily conclude that $v^\star\ge u^\star$, hence $v^\star=u^\star$. Therefore: \begstat{(4.23) Choquet's lemma} Every family $(u_\alpha)$ has a countable subfamily $(v_j)=(u_{\alpha(j)})$ such that its upper envelope $v$ satisfies $v\le u\le u^\star=v^\star$.\qed \endstat \begstat{(4.24) Proposition} If all $u_\alpha$ are subharmonic, the upper regularization $u^\star$ is subharmonic and equal almost everywhere to $u$. \endstat \begproof{} By Choquet's lemma we may assume that $(u_\alpha)$ is countable. Then $u=\sup u_\alpha$ is a Borel function. As each $u_\alpha$ satisfies the mean value inequality on every ball $\ovl B(z,r)\subset\Omega$, we get $$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)$$ and $u^\star$ is subharmonic. By the upper semicontinuity of $u^\star$ and the above inequality we find $u^\star(z)=\lim_{r\to 0}\mu_B(u\,;\,z,r)$, thus $u^\star=u$ almost everywhere by Lebesgue's lemma.\qed \endproof \titleb{\S 5.}{Plurisubharmonic Functions} \titlec{\S 5.A.}{Definition and Basic Properties} Plurisubharmonic functions have been introduced independently by (Lelong 1942) and (Oka 1942) for the study of holomorphic convexity. They are the complex counterparts of subharmonic functions. \begstat{(5.1) Definition} A function $u:\Omega\lra[-\infty,+\infty[$ defined on an open subset $\Omega\subset\bbbc^n$ is said to be plurisubharmonic if \medskip \item{\rm a)} $u$ is upper semicontinuous~$;$ \smallskip \item{\rm b)} for every complex line $L\subset\bbbc^n$, $u_{\restriction\Omega\cap L}$ is subharmonic on $\Omega\cap L$. \smallskip \noindent The set of plurisubharmonic functions on $\Omega$ is denoted by $\Psh(\Omega)$. \vskip0pt \endstat An equivalent way of stating property b) is: for all $a\in\Omega$, $\xi\in\bbbc^n$, $|\xi|0$. All functions $u'_j(z)=u_j(z)-\gamma_{z_0}(z-z_0)+\varepsilon|z-z_0|^2$ are plurisubharmonic near $z_0$. It follows that $$M_\eta(u'_1\ld u'_p)=u-\gamma_{z_0}(z-z_0)+\varepsilon|z-z_0|^2$$ is also plurisubharmonic near $z_0$. Since $\varepsilon>0$ was arbitrary, e) follows.\qed \endproof \begstat{(5.19) Corollary} Let $u_\alpha\in\ci(\ovl\Omega_\alpha)\cap \Psh(\Omega_\alpha)$ where $\Omega_\alpha\compact X$ is a locally finite open covering of $X$. Assume that $u_\beta(z)<\max\{u_\alpha(z)\}$ at every point $z\in\partial\Omega_\beta$, when $\alpha$ runs over the indices such that $\Omega_\alpha\ni z$. Choose a family $(\eta_\alpha)$ of positive numbers so small that $u_\beta(z)+\eta_\beta\le\max_{\Omega_\alpha\ni z} \{u_\alpha(z)-\eta_\alpha\}$ for all $\beta$ and $z\in\partial\Omega_\beta$. Then the function defined by $$\wt u(z)=M_{(\eta_\alpha)}\big(u_\alpha(z)\big)~~~~ {\rm for~\alpha~such~that}~~\Omega_\alpha\ni z$$ is smooth and plurisubharmonic on $X$.\qed \endstat \begstat{(5.20) Definition} A function $u\in\Psh(X)$ is said to be strictly plurisubharmonic if $u\in L^1_\loc(X)$ and if for every point $x_0\in X$ there exists a neighborhood $\Omega$ of $x_0$ and $c>0$ such that $u(z)-c|z|^2$ is plurisubharmonic on $\Omega$, i.e.\ $\sum~({\partial^2u/\partial z_j\partial\ovl z_k})\xi_j\ovl\xi_k\ge c|\xi|^2$ $($as distributions on $\Omega)$ for all $\xi\in\bbbc^n$. \endstat \begstat{(5.21) Theorem {\rm(Richberg 1968)}} Let $u\in\Psh(X)$ be a continuous function which is strictly plurisubharmonic on an open subset $\Omega\subset X$, with $Hu\ge\gamma$ for some continuous positive hermitian form $\gamma$ on $\Omega$. For any continuous function $\lambda\in C^0(\Omega)$, $\lambda>0$, there exists a plurisubharmonic function $\wt u$ in $C^0(X)\cap\ci(\Omega)$ such that $u\le\wt u\le u+\lambda$ on $\Omega$ and $\wt u=u$ on $X\ssm\Omega$, which is strictly plurisubharmonic on $\Omega$ and satisfies $H\wt u\ge(1-\lambda)\gamma$. In particular, $\wt u$ can be chosen strictly plurisubharmonic on $X$ if $u$ has the same property. \endstat \begproof{} Let $(\Omega_\alpha)$ be a locally finite open covering of $\Omega$ by relatively compact open balls contained in coordinate patches of $X$. Choose concentric balls $\Omega''_\alpha\subset\Omega'_\alpha\subset\Omega_\alpha$ of respective radii $r''_\alpha\eta_\alpha$ on $\ovl\Omega''_\alpha$, we have $u_\alphau+\eta_\alpha$ on $\ovl\Omega''_\alpha$, so that the condition required in Corollary 5.19 is satisfied. We define $$\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$$ and $\wt u_\alpha=u$ on the complement.\qed \endproof \titlec{\S 5.F.}{Polar and Pluripolar Sets.} Polar and pluripolar sets are sets of $-\infty$ poles of subharmonic and plurisubharmonic functions. Although these functions possess a large amount of flexi\-bility, pluripolar sets have some properties which remind their loose relationship with holomorphic functions. \begstat{(5.22) Definition} A set $A\subset\Omega\subset\bbbr^m$ $($resp.\ $A\subset X,$ ${\rm dim}_\bbbc X=n)$ is said to be polar $($resp.\ pluripolar$)$ if for every point $x\in\Omega$ there exist a connected neighborhood $W$ of $x$ and $u\in\Sh(W)$ $($resp.\ $u\in\Psh(W))$, $u\not\equiv-\infty$, such that~ $A\cap W\subset\{x\in W~;~u(x)=-\infty\}$. \endstat Theorem~4.17 implies that a polar or pluripolar set is of zero Lebesgue measure. Now, we prove a simple extension theorem. \begstat{(5.23) Theorem} Let $A\subset\Omega$ be a closed polar set and $v\in\Sh(\Omega\ssm A)$ such that $v$ is bounded above in a neighborhood of every point of $A$. Then $v$ has a unique extension $\wt v\in\Sh(\Omega)$. \endstat \begproof{} The uniqueness is clear because $A$ has zero Lebesgue measure. On the other hand, every point of $A$ has a neighborhood $W$ such that $$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\bbbn}\sum_{a\in I_\nu}|g_{\nu,a}(z)|^{2p(\nu)}$$ converges uniformly to a real analytic function $\psi\in\Psh(X)$ (see Exercise~8.11). By construction $\psi(z)\ge\nu$ for $z\in L_\nu$, hence $\psi$ is an exhaustion.\qed \endproof \begstat{(6.15) Example} \rm The converse to Theorem~6.14 does not hold. In fact let $X=\bbbc^2/\Gamma$ be the quotient of $\bbbc^2$ by the free abelian group of rank $2$ generated by the affine automorphisms $$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\bbbr.$$ Since $\Gamma$ acts properly discontinuously on $\bbbc^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~$\bbbc^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\ovl D(0,|w|)$, $K={}$ unit square in $\bbbc$. 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 $\bbbc$ 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\bbbq$ or $\theta_1\notin\bbbq$ (if $\theta_1,\theta_2\in\bbbq$ 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\bbbq$. \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\bbbc^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 $\bbbc^n$. Hence an open set $\Omega\subset\bbbc^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\ld 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 $\ovl 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\ld 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\ci(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\bbbc^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')}),$$ and if $\varphi$, $\psi$ are plurisubharmonic exhaustions of $S$, $S'$, then $\varphi+\psi\circ F$ is a plurisubharmonic exhaustion of $S\cap F^{-1}(S')$. \medskip\noindent e) $\varphi(z):=\psi(z)+1/(c-u(z))$ is a (strictly) plurisubharmonic exhaustion function on~$\Omega$.\qed \endproof \titleb{\S 7.}{Pseudoconvex Open Sets in $\bbbc^n$} \titlec{\S 7.A.}{Geometric Characterizations of Pseudoconvex Open Sets} We first discuss some characterizations of pseudoconvex open sets in~$\bbbc^n$. We will need the following elementary criterion for plurisubharmonicity. \begstat{(7.1) Criterion} Let $v:\Omega\lra[-\infty,+\infty[$ be an upper semicontinuous function. Then $v$ is plurisubharmonic if and only if for every closed disk $\ovl\Delta=z_0+\ovl D(1)\eta\subset\Omega$ and every polynomial $P\in\bbbc[t]$ such that $v(z_0+t\eta)\le\Re P(t)$ for $|t|=1$, then $v(z_0)\le\Re P(0)$. \endstat \begproof{} The condition is necessary because $t\longmapsto v(z_0+t\eta)-\Re P(t)$ is subharmonic in a neighborhood of $\ovl D(1)$, so it satisfies the maximum principle on $D(1)$ by Th.~4.14. Let us prove now the sufficiency. The upper semicontinuity of $v$ implies $v=\lim_{\nu\to+\infty} v_\nu$ on $\partial\Delta$ where $(v_\nu)$ is a strictly decreasing sequence of continuous functions on $\partial\Delta$. As trigonometric polynomials are dense in $C^0(S^1,\bbbr)$, we may assume $v_\nu(z_0+e^{\ii\theta}\eta)=\Re P_\nu (e^{\ii\theta})$, $P_\nu\in\bbbc[t]$. Then $v(z_0+t\eta)\le\Re P_\nu(t)$ for $|t|=1$, and the hypothesis implies $$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.$$ Taking the limit when $\nu$ tends to $+\infty$ shows that $v$ satisfies the mean value inequality (5.2).\qed \endproof For any $z\in\Omega$ and $\xi\in\bbbc^n$, we denote by $$\delta_\Omega(z,\xi)=\sup\big\{r>0~;~z+D(r)\,\xi\subset\Omega\big\}$$ the distance from $z$ to $\partial\Omega$ in the complex direction $\xi$. \begstat{(7.2) Theorem} Let $\Omega\subset\bbbc^n$ be an open subset. The following properties are equivalent: \medskip \item{\rm a)} $\Omega$ is strongly pseudoconvex $($according to Def.~6.13~{\rm b)}$;$ \smallskip \item{\rm b)} $\Omega$ is weakly pseudoconvex\/$;$ \smallskip \item{\rm c)} $\Omega$ has a plurisubharmonic exhaustion function $\psi$. \smallskip \item{\rm d)} $-\log\delta_\Omega(z,\xi)$ is plurisubharmonic on $\Omega\times\bbbc^n~;$ \smallskip \item{\rm e)} $-\log d(z,\complement\Omega)$ is plurisubharmonic on $\Omega$. \medskip\noindent If one of these properties hold, $\Omega$ is said to be a pseudoconvex open set. \endstat \begproof{} The implications a) $\Longrightarrow$ b) $\Longrightarrow$ c) are obvious. For the implication c) $\Longrightarrow$ d), we use Criterion 7.1. Consider a disk $\ovl\Delta=(z_0,\xi_0)+\ovl D(1)\,(\eta,\alpha)$ in $\Omega\times\bbbc^n$ and a polynomial $P\in\bbbc[t]$ such that $$-\log\delta_\Omega(z_0+t\eta,\xi_0+t\alpha)\le\Re P(t)~~~ \hbox{\rm for}~~|t|=1.$$ We have to verify that the inequality also holds when $|t|<1$. Consider the holomorphic mapping $h:\bbbc^2\lra\bbbc^n$ defined by $$h(t,w)=z_0+t\eta+we^{-P(t)}(\xi_0+t\alpha).$$ By hypothesis $$\eqalign{ &h\big(\ovl D(1)\times\{0\}\big)={\rm pr}_1(\ovl\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}$$ and the desired conclusion is that $h\big(\ovl D(1)\times D(1)\big) \subset\Omega$. Let $J$ be the set of radii $r\ge 0$ such that $h\big(\ovl D(1)\times\ovl D(r)\big)\subset\Omega$. Then $J$ is an open interval $[0,R[$, $R>0$. If $R<1$, we get a contradiction as follows. Let $\psi\in\Psh(\Omega)$ be an exhaustion function and $$K=h\big(\partial D(1)\times\ovl D(R)\big)\compact\Omega,~~~~ c=\sup_K\psi.$$ As $\psi\circ h$ is plurisubharmonic on a neighborhood of $\ovl D(1)\times D(R)$, the maximum principle applied with respect to $t$ implies $$\psi\circ h(t,w)\le c~~~\hbox{\rm on}~~\ovl D(1)\times D(R),$$ hence $h\big(\ovl D(1)\times D(R)\big)\subset\Omega_c\compact\Omega$ and $h\big(\ovl D(1)\times\ovl D(R+\varepsilon)\big)\subset\Omega$ for some $\varepsilon>0$, a contradiction. \medskip \noindent{d)} $\Longrightarrow$ e). The function~ $-\log d(z,\complement \Omega)$ is continuous on $\Omega$ and satisfies the mean value inequality because $$-\log d(z,\complement\Omega)=\sup_{\xi\in\ovl 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\}$$ is a continuous strictly plurisubharmonic exhaustion function. Richberg's theorem 5.21 implies that there exists $\psi\in\ci(\Omega)$ strictly plurisubharmonic such that $u\le\psi\le u+1$. Then $\psi$ is the required exhaustion function.\qed \endproof \begstat{(7.3) Proposition} \smallskip \item{\rm a)} Let $\Omega\subset\bbbc^n$ and $\Omega'\subset\bbbc^p$ be pseudoconvex. Then $\Omega\times\Omega'$ is pseudoconvex. For every holomorphic map $F:\Omega\to\bbbc^p$ the inverse image $F^{-1}(\Omega')$ is pseudoconvex. \smallskip \item{\rm b)} If $(\Omega_\alpha)_{\alpha\in I}$ is a family of pseudoconvex open subsets of $\bbbc^n$, the interior of the intersection $\Omega=\big(\bigcap_{\alpha\in I} \Omega_\alpha\big)^\circ$ is pseudoconvex. \smallskip \item{\rm c)} If $(\Omega_j)_{j\in\bbbn}$ is a non decreasing sequence of pseudoconvex open subsets of $\bbbc^n$, then $\Omega=\bigcup_{j\in\bbbn}\Omega_j$ is pseudoconvex. \vskip0pt \endstat \begproof{} a) Let $\varphi,\psi$ be smooth plurisubharmonic exhaustions of $\Omega,\Omega'$. Then $(z,w)\longmapsto\varphi(z)+\psi(w)$ is an exhaustion of $\Omega\times\Omega'$ and $z\longmapsto\varphi(z)+ \psi(F(z))$ is an exhaustion of $F^{-1}(\Omega')$. \medskip \noindent{b)} We have $-\log d(z,\complement\Omega)=\sup_{\alpha\in I} -\log d(z,\complement\Omega_\alpha)$, so this function is pluri\-sub\-harmonic. \medskip \noindent{c)} The limit $-\log d(z,\complement\Omega)={{\lim}{\downarrow}\,}_{j\to+\infty} -\log d(z,\complement\Omega_j)$ is plurisubharmonic, hence $\Omega$ is pseudoconvex. This result cannot be generalized to strongly pseudoconvex manifolds: J.E.~Fornaess in (Fornaess 1977) has constructed an increasing sequence of $2$-dimensional Stein (even affine algebraic) manifolds $X_\nu$ whose union is not Stein; see Exercise~8.16.\qed \endproof \begstat{(7.4) Examples} \smallskip\rm \noindent{a)} An {\it analytic polyhedron} in $\bbbc^n$ is an open subset of the form $$P=\{z\in\bbbc^n\,;\,|f_j(z)|<1,~1\le j\le N\}$$ where $(f_j)_{1\le j\le N}$ is a family of analytic functions on $\bbbc^n$. By 7.3~a), every analytic polyhedron is pseudoconvex. \medskip \noindent{b)} Let $\omega\subset\bbbc^{n-1}$ be pseudoconvex and let $u:\omega\lra[-\infty,+\infty[$ be an upper semicontinuous function. Then the {\it Hartogs domain} $$\Omega=\big\{(z_1,z')\in\bbbc\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$$ are pseudoconvex. \medskip \noindent{c)} An open set $\Omega\subset\bbbc^n$ is called a {\it tube} of base $\omega$ if $\Omega=\omega+\ii\bbbr^n$ for some open subset $\omega\subset\bbbr^n$. Then of course $-\log d(z,\complement\Omega)=-\log(x,\complement\omega)$ depends only on the real part $x=\Re z$. By Th.~5.13, this function is plurisubharmonic if and only if it is locally convex in $x$. Therefore $\Omega$ if pseudoconvex if and only if every connected component of $\omega$ is convex. \medskip \noindent{d)} An open set $\Omega\subset\bbbc^n$ is called a {\it Reinhardt domain} if $(e^{\ii\theta_1}z_1\ld e^{\ii\theta_n}z_n)$ is in $\Omega$ for every $z=(z_1\ld z_n)\in\Omega$ and $\theta_1\ld\theta_n\in\bbbr^n$. For such a domain, we consider the {\it logarithmic indicatrix} $$\omega^\star=\Omega^\star\cap\bbbr^n~~~\hbox{\rm with}~~~ \Omega^\star=\{\zeta\in\bbbc^n\,;\,(e^{\zeta_1}\ld e^{\zeta_n})\in\Omega\}.$$ It is clear that $\Omega^\star$ is a tube of base $\omega^\star$. Therefore every connected component of $\omega^\star$ must be convex if $\Omega$ is pseudoconvex. The converse is not true: $\Omega=\bbbc^n\ssm\{0\}$ is not pseudoconvex for $n\ge 2$ although $\omega^\star=\bbbr^n$ is convex. However, the Reinhardt open set $$\Omega^\bullet=\big\{(z_1\ld z_n)\in(\bbbc\ssm\{0\})^n\,;\, (\log|z_1|\ld\log|z_n|)\in\omega^\star\big\}\subset\Omega$$ is easily seen to be pseudoconvex if $\omega^\star$ is convex: if $\chi$ is a convex exhaustion of $\omega^\star$, then $\psi(z)=\chi(\log|z_1|\ld\log|z_n|)$ is a plurisubharmonic exhaustion of $\Omega^\bullet$. Similarly, if $\omega^\star$ is convex and such that $x\in\omega^\star \Longrightarrow y\in\omega^\star$ for $y_j\le x_j$, we can take $\chi$ increasing in all variables and tending to $+\infty$ on $\partial\omega^\star$, hence the set $$\wt\Omega=\big\{(z_1\ld z_n)\in\bbbc^n\,;\,|z_j|\le e^{x_j}~~ \hbox{\rm for~some~~}x\in\omega^\star\big\}$$ is a pseudoconvex Reinhardt open set containing $0$.\qed \endstat \titlec{\S 7.B.}{Kiselman's Minimum Principle} We already know that a maximum of plurisubharmonic functions is plurisubharmonic. However, if $v$ is a plurisubharmonic function on $X\times\bbbc^n$, the partial minimum function on $X$ defined by $u(\zeta)=\inf_{z\in\Omega}v(\zeta,z)$ need not be plurisubharmonic. A simple counterexample in $\bbbc\times\bbbc$ is given by $$v(\zeta,z)=|z|^2+2\Re(z\zeta)=|z+\ovl\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\bbbc^3\,;\,\log|t|+v(\zeta,z)<0\}$$ and if $\Omega'\subset\bbbc^2$ is the image of $\Omega$ by the projection map $(t,\zeta,z)\longmapsto(t,\zeta)$, then $\Omega'=\{(t,\zeta)\in\bbbc^2\,;\,\log|t|+u(\zeta)<0\}$ is not pseudoconvex. However, the minimum property holds true when $v(\zeta,z)$ depends only on $\Re z$~: \begstat{(7.5) Theorem {\rm (Kiselman 1978)}} Let $\Omega\subset\bbbc^p\times\bbbc^n$ be a pseudoconvex open set such that each slice $$\Omega_\zeta=\{z\in\bbbc^n\,;\,(\zeta,z)\in\Omega\},~~~~\zeta\in\bbbc^p,$$ is a convex tube $\omega_\zeta+\ii\bbbr^n$, $\omega_\zeta\subset\bbbr^n$. For every plurisubharmonic function $v(\zeta,z)$ on $\Omega$ that does not depend on $\Im z$, the function $$u(\zeta)=\inf_{z\in\Omega_\zeta}~v(\zeta,z)$$ is plurisubharmonic or locally $\equiv-\infty$ on $\Omega'= {\rm pr}_{\bbbc^n}(\Omega)$. \endstat \begproof{} The hypothesis implies that $v(\zeta,z)$ is convex in $x=\Re z$. In addition, we first assume that $v$ is smooth, plurisubharmonic in $(\zeta,z)$, strictly convex in $x$ and $\lim_{x\to\{\infty\}\cup\partial\omega_\zeta}v(\zeta,x)=+\infty$ for every $\zeta\in\Omega'$. Then $x\longmapsto v(\zeta,x)$ has a unique minimum point $x=g(\zeta)$, solution of the equations $\partial v/\partial x_j(x,\zeta)=0$. As the matrix $(\partial^2 v/\partial x_j\partial x_k)$ is positive definite, the implicit function theorem shows that $g$ is smooth. Now, if $\bbbc\ni w\longmapsto\zeta_0+wa$, $a\in\bbbc^n$, $|w|\le 1$ is a complex disk $\Delta$ contained in $\Omega$, there exists a holomorphic function $f$ on the unit disk, smooth up to the boundary, whose real part solves the Dirichlet problem $$\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.$$ The last equality holds because $\Re f=g$ on $\partial\Delta$ and $v(\zeta,z)=v(\zeta,\Re z)$ by hypothesis. As $u(\zeta_0)\le v(\zeta_0,f(0))$ and $u(\zeta)=v(\zeta,g(\zeta))$, we see that $u$ satisfies the mean value inequality, thus $u$ is plurisubharmonic. Now, this result can be extended to arbitrary functions $v$ as follows: let $\psi(\zeta,z)\ge 0$ be a continuous plurisubharmonic function on $\Omega$ which is independent of $\Im z$ and is an exhaustion of $\Omega\cap(\bbbc^p\times\bbbr^n)$, e.g. $$\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)$$ is a decreasing sequence of plurisubharmonic functions on $\Omega_j$ satisfying our previous conditions. As $v=\lim v_j$, we see that $u=\lim u_j$ is plurisubharmonic.\qed \endproof \begstat{(7.6) Corollary} Let $\Omega\subset\bbbc^p\times\bbbc^n$ be a pseudoconvex open set such that all slices $\Omega_\zeta$, $\zeta\in\bbbc^p$, are convex tubes in $\bbbc^n$. Then the projection $\Omega'$ of $\Omega$ on $\bbbc^p$ is pseudoconvex. \endstat \begproof{} Take $v\in\Psh(\Omega)$ equal to the function $\psi$ defined in the proof of Th.~7.5. Then $u$ is a plurisubharmonic exhaustion of $\Omega'$.\qed \endproof \titlec{\S 7.C.}{Levi Form of the Boundary} For an arbitrary domain in $\bbbc^n$, we first show that pseudoconvexity is a local property of the boundary. \begstat{(7.7) Theorem} Let $\Omega\subset\bbbc^n$ be an open subset such that every point $z_0\in\partial\Omega$ has a neighborhood $V$ such that $\Omega\cap V$ is pseudoconvex. Then $\Omega$ is pseudoconvex. \endstat \begproof{} As $d(z,\complement\Omega)$ coincides with $d\big(z,\complement (\Omega\cap V)\big)$ in a neighborhood of $z_0$, we see that there exists a neighborhood $U$ of $\partial\Omega$ such that $-\log d(z,\complement\Omega)$ is plurisubharmonic on $\Omega\cap U$. Choose a convex increasing function $\chi$ such that $$\chi(r)>\sup_{(\Omega\ssm U)\cap\ovl 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\}$$ coincides with $\chi(|z|)$ in a neighborhood of $\Omega\ssm U$. Therefore $\psi\in\Psh(\Omega)$, and $\psi$ is clearly an exhaustion.\qed \endproof Now, we give a geometric characterization of the pseudoconvexity property when $\partial\Omega$ is of class $C^2$. Let $\rho\in C^2(\ovl\Omega)$ be a {\it defining function} of $\Omega$, i.e.\ a function such that $$\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)$$ It is easy to see that ${}^hT_{\partial\Omega,z}$ is the complex hyperplane of vectors $\xi\in\bbbc^n$ such that $$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\ovl z_k}\,\xi_j\ovl\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\ovl\eta_k{\partial\over\partial\ovl 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\ovl z_k}\, \xi_j\ovl\eta_k+\xi_j{\partial\ovl\eta_k\over\partial z_j} {\partial\rho\over\partial\ovl z_k}+\ovl\eta_k{\partial\xi_j\over\partial\ovl 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\ovl\eta_k\over\partial z_j} {\partial\over\partial\ovl z_k}+\ovl\eta_k{\partial\xi_j\over\partial\ovl z_k} {\partial\over\partial\ovl z_j},\cr \Re\big(J[\xi',\eta''].\rho\big)&= \Im\sum\xi_j{\partial\ovl\eta_k\over\partial z_j} {\partial\rho\over\partial\ovl z_k}+\ovl\eta_k {\partial\xi_j\over\partial\ovl z_k}{\partial\rho\over\partial z_j}=-2\Im\sum {\partial^2\rho\over\partial z_j\partial\ovl z_k}\,\xi_j\ovl\eta_k,\cr \langle[\xi,\eta],J\nu\rangle&={4\over|\nabla\rho|}\Im\sum {\partial^2\rho\over\partial z_j\partial\ovl z_k}\,\xi_j\ovl\eta_k= 4\Im L_{\partial\Omega}(\xi,\eta).&\square\cr}$$ \endproof \begstat{(7.12) Theorem} An open subset $\Omega\subset\bbbc^n$ with $C^2$ boundary is pseudoconvex if and only if the Levi form $L_{\partial\Omega}$ is semipositive at every point of $\partial\Omega$. \endstat \begproof{} Set $\delta(z)=d(z,\complement\Omega)$, $z\in\ovl\Omega$. Then $\rho=-\delta$ is $C^2$ near $\partial\Omega$ and satisfies (7.9). If $\Omega$ is pseudoconvex, the plurisubharmonicity of $-\log(-\rho)$ means that for all $z\in\Omega$ near $\partial\Omega$ and all $\xi\in\bbbc^n$ one has $$\sum_{1\le j,k\le n}\Big({1\over|\rho|}\,{\partial^2\rho\over\partial z_j \partial\ovl z_k}+{1\over\rho^2}\,{\partial\rho\over\partial z_j}\, {\partial\rho\over\partial\ovl z_k}\Big)\xi_j\ovl\xi_k\ge 0.$$ Hence $\sum~(\partial^2\rho/\partial z_j\partial\ovl z_k)\xi_j\ovl\xi_k\ge 0$ if $\sum~(\partial\rho/\partial z_j)\xi_j=0$, and an easy argument shows that this is also true at the limit on $\partial\Omega$. Conversely, if $\Omega$ is not pseudoconvex, Th.~7.2 and 7.7 show that $-\log\delta$ is not plurisubharmonic in any neighborhood of $\partial\Omega$. Hence there exists $\xi\in\bbbc^n$ such that $$c=\Big({\partial^2\over\partial t\partial\ovl 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)$$ with $a,b\in\bbbc$. Now, choose $z_0\in\partial\Omega$ such that $\delta(z)=|z-z_0|$ and set $$h(t)=z+t\xi+e^{at+bt^2}(z_0-z),~~~~t\in\bbbc.$$ Then we get $h(0)=z_0$ and $$\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\ovl t}\,\delta(h(t))=\sum {\partial^2\delta\over\partial z_j\partial\ovl z_k}(z_0)\,h'_j(0)\ovl{h'_k(0)}>0 ,$$ hence $h'(0)\in {}^hT_{\partial\Omega,z_0}$ and $L_{\partial\Omega,z_0} (h'(0))<0$.\qed \endproof \begstat{(7.13) Definition} The boundary $\partial\Omega$ is said to be weakly $($resp.\ strongly$)$ pseudoconvex if $L_{\partial\Omega}$ is semipositive $($resp.\ positive definite$)$ on~$\partial\Omega$. The boundary is said to be Levi flat if $L_{\partial\Omega}\equiv 0$. \endstat \begstat{(7.14) Remark} \rm Lemma 7.11 shows that $\partial\Omega$ is Levi flat if and only if the subbundle ${}^hT_{\partial\Omega}\subset T_{\partial\Omega}$ is integrable (i.e.\ stable under the Lie bracket). Assume that $\partial\Omega$ is of class $C^k$, $k\ge 2$. Then ${}^hT_{\partial\Omega}$ is of class $C^{k-1}$. By Frobenius' theorem, the integrability condition implies that ${}^hT_{\partial\Omega}$ is the tangent bundle to a $C^k$ foliation of $\partial\Omega$ whose leaves have real dimension $2n-2$. But the leaves themselves must be complex analytic since ${}^hT_{\partial\Omega}$ is a complex vector space (cf.\ Lemma~7.15 below). Therefore $\partial\Omega$ is Levi flat if and only if it is foliated by complex analytic hypersurfaces. \endstat \begstat{(7.15) Lemma} Let $Y$ be a $C^1$-submanifold of a complex analytic manifold $X$. If the tangent space $T_{Y,x}$ is a complex subspace of $T_{X,x}$ at every point $x\in Y$, then $Y$ is complex analytic. \endstat \begproof{} Let $x_0\in Y$. Select holomorphic coordinates $(z_1\ld z_n)$ on $X$ centered at $x_0$ such that $T_{Y,x_0}$ is spanned by $\partial/\partial z_1\ld\partial/\partial z_p$. Then there exists a neighborhood $U=U'\times U''$ of $x_0$ such that $Y\cap U$ is a graph $$z''=h(z'),~~~~z'=(z_1\ld z_p)\in U',~~z''=(z_{p+1}\ld z_n)$$ with $h\in C^1(U')$ and $dh(0)=0$. The differential of $h$ at $z'$ is the composite of the projection of $\bbbc^p\times\{0\}$ on $T_{Y,(z',h(z'))}$ along $\{0\}\times\bbbc^{n-p}$ and of the second projection $\bbbc^n\to\bbbc^{n-p}$. Hence $dh(z')$ is $\bbbc$-linear at every point and $h$ is holomorphic.\qed \endproof \titleb{\S 8.}{Exercises}\begpet \titled{8.1.} Let $\Omega\subset\bbbc^n$ be an open set such that $$z\in\Omega,~~\lambda\in\bbbc,~~|\lambda|\le 1\Longrightarrow \lambda z\in\Omega.$$ Show that $\Omega$ is a union of polydisks of center $0$ (with arbitrary linear changes of coordinates) and infer that the space of polynomials $\bbbc[z_1\ld z_n]$ is dense in $\cO(\Omega)$ for the topology of uniform convergence on compact subsets and in $\cO(\Omega)\cap C^0(\ovl\Omega)$ for the topology of uniform convergence on~$\ovl\Omega$.\newline {\it Hint}\/: consider the Taylor expansion of a function $f\in\cO(\Omega)$ at the origin, writing it as a series of homogeneous polynomials. To deal with the case of $\ovl\Omega$, first apply a dilation to~$f$. \titled{8.2.}Let $B\subset\bbbc^n$ be the unit euclidean ball, $S=\partial B$ and $f\in\cO(B)\cap C^0(\ovl B)$. Our goal is to check the following Cauchy formula: $$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\ld 0)$ and $f(z)$ is a monomial $z^\alpha$. \smallskip \item{b)} Show that the integral $\int_B z^\alpha\ovl z_1^k\,d\lambda(z)$ vanishes unless $\alpha=(k,0\ld 0)$. Compute the value of the remaining integral by the Fubini theorem, as well as the integrals $\int_S z^\alpha \ovl 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\lra M$ is the inclusion.\newline {\it Hint}\/: if $x_1=\ldots=x_q=0$ are equations of $Y$ in a coordinate system $(x_1\ld 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\ovl z_j$ be a closed current of bidegree $(0,1)$ with compact support in $\bbbc^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 $\bbbc^n\ssm K$.\newline {\it Hint}\/: observe that $S$ is holomorphic on $\bbbc^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\ovl z_J$, $q\ge 1$, be a smooth form of bidegree $(0,q)$ on a polydisk $\Omega=D(0,R)\subset\bbbc^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\ovl z_J$ appearing in $v$ only involve $d\ovl z_1$, $\ldots$, $d\ovl 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\ovl z_k+g$ where $f$, $g$ only involve $d\ovl z_1$, $\ldots$, $d\ovl z_{k-1}$. Then consider $v-d''F$ where $$F=\sum_{|J|=q-1}F_Jd\ovl z_J,~~~~F_J(z)=(\psi(z_k)f_J)\star_k \Big({1\over \pi z_k}\Big),$$ where $\star_k$ denotes the partial convolution with respect to~$z_k$, $\psi(z_k)$ is a cut-off function equal to 1 on $D(0,r_k+\varepsilon)$ and $f=\sum_{|J|=q-1}f_Jd\ovl z_J$. \titled{8.6.} Construct locally bounded non continuous subharmonic functions on $\bbbc$.\newline {\it Hint}\/: consider $e^u$ where $u(z)=\sum_{j\ge 1}2^{-j}\log|z-1/j|$. \titled{8.7.} Let $\omega$ be an open subset of $\bbbr^n$, $n\ge 2$, and $u$ a subharmonic function which is not locally $-\infty$. \smallskip \item{a)} For every open set $\omega\compact\Omega$, show that there is a positive measure $\mu$ with support in $\ovl\omega$ and a harmonic function $h$ on $\omega$ such that $u=N\star\mu+h$ on $\omega$. \smallskip \item{b)} Use this representation to prove the following properties: $u\in L^p_\loc$ for all $p0$ small, show that $\Omega$ contains the Hartogs figure $$\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 $\xi\in\bbbc^n$ 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 $\Omega=\omega+\ii\bbbr^n$ be a connected tube in $\bbbc^n$ of base $\omega$. \smallskip \item{a)} Assume first that $n=2$. Let $T\subset\bbbr^2$ 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 $T+\ii\bbbr^2$.\newline {\it Hint}\/: let $\pi:\bbbc^2\lra\bbbr^2$ 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 $[0,1]\cdot M_\varepsilon+\ii\bbbr^n$. \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 $[a,b]+\ii\bbbr^n$. \titled{8.16.} For each integer $\nu\ge 1$, consider the algebraic variety $$X_\nu=\big\{(z,w,t)\in\bbbc^3\,;\,wt=p_\nu(z)\big\},~~~~ p_\nu(z)=\prod_{1\le k\le\nu}(z-1/k),$$ 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 $\pi_\nu:X_\nu\to \bbbc^2$ be the projection to the first two coordinates. Since $\pi_{\nu+1}\circ j_\nu=\pi_\nu$, there exists a holomorphic map $\pi:X\to\bbbc^2$, $\pi=\lim\pi_\nu$. Show that $$\bbbc^2\ssm\pi(X)=\big\{(z,0)\in\bbbc^2\,;\,z\ne 1/\nu,~\forall\nu\in\bbbn,~ \nu\ge 1\big\},$$ and especially, that $(0,0)\notin\pi(X)$. \item{c)} Consider the compact set $$K=\pi^{-1}\big(\{(z,w)\in\bbbc^2\,;\,|z|\le 1,~|w|=1\}\big).$$ 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 $\chi:\bbbr\to\bbbr$ such that $\delta+\chi\circ\psi$ is strictly plurisubharmonic. \endpet \end