% Complex Analytic and Differential Geometry, Chapter VIII % J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France \input analgeom.mac \def\Dom{\hbox{\rm Dom}\,} \def\Herm{{\rm Herm}} \def\Grif{{\rm Grif}} \def\Ric{{\rm Ricci}} \def\wtP{\smash{\wt P}} \def\omcirc{({\ovl\Omega})^\circ} \def\Gr{{\rm Gr}} \def\Ll{\langle\!\langle} \def\Gg{\rangle\!\rangle} \titlea{Chapter VIII}{\newline $L^2$ Estimates on Pseudoconvex Manifolds} \begpet The main goal of this chapter is to show that the differential geometric technique that has been used in order to prove vanishing theorems also yields very precise $L^2$ estimates for the solutions of equations $d''u=v$ on pseudoconvex manifolds. The central idea, due to (H\"ormander~1965), is to introduce weights of the type $e^{-\varphi}$ where $\varphi$ is a function satisfying suitable convexity conditions. This method leads to generalizations of many standard vanishing theorems to weakly pseudoconvex manifolds. As a special case, we obtain the original H\"ormander estimates for pseudoconvex domains of $\bbbc^n$, and give some applications to algebraic geometry (H\"ormander-Bombieri-Skoda theorem, properties of zero sets of polynomials in $\bbbc^n$). We also derive the Ohsawa-Takegoshi extension theorem for $L^2$ holomorphic functions and Skoda's $L^2$ estimates for surjective bundle morphisms (Skoda 1972a, 1978, Demailly~1982c). Skoda's estimates can be used to obtain a quick solution of the Levi problem, and have important applications to local algebra and Nullstellensatz theorems. Finally, $L^2$ estimates are used to prove the Newlander-Nirenberg theorem on the analyticity of almost complex structures. We apply it to establish Kuranishi's theorem on deformation theory of compact complex manifolds.\endpet \titleb{1.}{Non Bounded Operators on Hilbert Spaces} A few preliminaries of functional analysis will be needed here. Let $\cH_1$, $\cH_2$ be complex Hilbert spaces. We consider a linear operator $T$ defined on a subspace $\Dom T\subset\cH_1$ (called the domain of $T$) into $\cH_2$. The operator $T$ is said to be {\it densely defined} if $\Dom T$ is dense in $\cH_1$, and {\it closed} if its graph $$\Gr\,T=\big\{(x,Tx)~;~x\in\Dom T\big\}$$ is closed in $\cH_1\times\cH_2$. Assume now that $T$ is closed and densely defined. The adjoint $T^\star$ of $T$ (in Von Neumann's sense) is constructed as follows: $\Dom T^\star$ is the set of $y\in\cH_2$ such that the linear form $$\Dom T\ni x\longmapsto\langle Tx,y\rangle_2$$ is bounded in $\cH_1$-norm. Since $\Dom T$ is dense, there exists for every $y$ in $\Dom T^\star$ a unique element $T^\star y\in\cH_1$ such that $\langle Tx,y\rangle_2=\langle x,T^\star y\rangle_1$ for all $x\in \Dom T^\star$. It is immediate to verify that $\Gr\,T^\star= \big(\Gr(-T)\big)^\perp$ in $\cH_1\times\cH_2$. It follows that $T^\star$ is closed and that every pair $(u,v)\in\cH_1\times\cH_2$ can be written $$(u,v)=(x,-Tx)+(T^\star y,y),~~~~x\in\Dom T,~~y\in\Dom T^\star.$$ Take in particular $u=0$. Then $$x+T^\star y=0,~~~~v=y-Tx=y+TT^\star y,~~~~ \langle v,y\rangle_2=\|y\|^2_2+\|T^\star y\|_1^2.$$ If $v\in(\Dom T^\star)^\perp$ we get $\langle v,y\rangle_2=0$, thus $y=0$ and $v=0$. Therefore $T^\star$ is densely defined and our discussion implies: \begstat{(1.1) Theorem {\rm(Von Neumann 19??)}} If $T:\cH_1\lra\cH_2$ is a closed and densely defined operator, then its adjoint $T^\star$ is also closed and densely defined and $(T^\star)^\star=T$. Furthermore, we have the relation $\Ker T^\star=(\Im T)^\perp$ and its dual $(\Ker T)^\perp=\ovl{\Im T^\star}$.\qed \endstat Consider now two closed and densely defined operators $T$, $S\,:$ $$\cH_1\buildo T\over\lra\cH_2\buildo S\over\lra\cH_3$$ such that $S\circ T=0$. By this, we mean that the range $T(\Dom T)$ is contained in $\Ker S\subset \Dom S$, in such a way that there is no problem for defining the composition $S\circ T$. The starting point of all $L^2$ estimates is the following abstract existence theorem. \begstat{(1.2) Theorem} There are orthogonal decompositions $$\eqalign{ \cH_2&=(\Ker S\cap\Ker T^\star)\oplus\ovl{\Im T} \oplus\ovl{\Im S^\star},\cr \Ker S&=(\Ker S\cap\Ker T^\star)\oplus\ovl{\Im T}.\cr}$$ In order that $\Im T=\Ker S$, it suffices that $$\|T^\star x\|^2_1+\|Sx\|^2_3\ge C\|x\|^2_2,~~~~ \forall x\in\Dom S\cap\Dom T^\star\leqno(1.3)$$ for some constant $C>0$. In that case, for every $v\in\cH_2$ such that $Sv=0$, there exists $u\in\cH_1$ such that $Tu=v$ and $$\|u\|_1^2\le{1\over C}\|v\|_2^2.$$ In particular $$\ovl{\Im T}=\Im T=\Ker S,~~~~ \ovl{\Im S^\star}=\Im S^\star=\Ker T^\star.$$ \endstat \begproof{} Since $S$ is closed, the kernel $\Ker S$ is closed in $\cH_2$. The relation $(\Ker S)^\perp=\ovl{\Im S^\star}$ implies $$\cH_2=\Ker S\oplus\ovl{\Im S^\star}\leqno(1.4)$$ and similarly $\cH_2=\Ker T^\star\oplus\ovl{\Im T}$. However, the assumption $S\circ T=0$ shows that $\ovl{\Im T}\subset\Ker S$, therefore $$\Ker S=(\Ker S\cap\Ker T^\star)\oplus\ovl{\Im T}.\leqno(1.5)$$ The first two equalities in Th.~1.2 are then equivalent to the conjunction of (1.4) and (1.5). Now, under assumption (1.3), we are going to show that the equation $Tu=v$ is always solvable if $Sv=0$. Let $x\in\Dom T^\star$. One can write $$x=x'+x''~~~~\hbox{\rm where}~~x'\in\Ker S~~\hbox{\rm and}~~x''\in(\Ker S)^\perp \subset(\Im T)^\perp=\Ker T^\star.$$ Since $x,x''\in\Dom T^\star$, we have also $x'\in\Dom T^\star$. We get $$\langle v,x\rangle_2=\langle v,x'\rangle_2+\langle v,x''\rangle_2= \langle v,x'\rangle_2$$ because $v\in\Ker S$ and $x''\in(\Ker S)^\perp$. As $Sx'=0$ and $T^\star x''=0$, the Cauchy-Schwarz inequality combined with (1.3) implies $$|\langle v,x\rangle_2|^2\le\|v\|^2_2~\|x'\|^2_2\le {1\over C}\|v\|^2_2~\|T^\star x'\|^2_1={1\over C}\|v\|^2_2~\|T^\star x\|^2_1.$$ This shows that the linear form $T^\star_X\ni x\longmapsto\langle x,v\rangle_2$ is continuous on \hbox{$\Im T^\star\subset\cH_1$} with norm $\le C^{-1/2}\|v\|_2$. By the Hahn-Banach theorem, this form can be extended to a continuous linear form on $\cH_1$ of norm $\le C^{-1/2}\|v\|_2$, i.e.\ we can find $u\in\cH_1$ such that $\|u\|_1\le C^{-1/2}\|v\|_2$ and $$\langle x,v\rangle_2=\langle T^\star x,u\rangle_1,~~~~ \forall x\in\Dom T^\star.$$ This means that $u\in\Dom(T^\star)^\star=\Dom T$ and $v=Tu$. We have thus shown that $\Im T=\Ker S$, in particular $\Im T$ is closed. The dual \hbox{equality} \hbox{$\Im S^\star=\Ker T^\star$} follows by considering the dual pair $(S^\star,T^\star)$.\qed \endproof \titleb{2.}{Complete Riemannian Manifolds} Let $(M,g)$ be a riemannian manifold of dimension $m$, with metric $$g(x)=\sum g_{jk}(x)\,dx_j\otimes dx_k,~~~~1\le j,k\le m.$$ The length of a path $\gamma~:~[a,b]\lra M$ is by definition $$\ell(\gamma)=\int_a^b|\gamma'(t)|_gdt=\int_a^b\Big( \sum_{j,k}g_{jk}\big(\gamma(t)\big)\,\gamma'_j(t)\gamma'_k(t)\Big)^{1/2}dt.$$ The geodesic distance of two points $x,y\in M$ is $$\delta(x,y)=\inf_\gamma\ell(\gamma)~~~~\hbox{\rm over paths $\gamma$ with} ~~\gamma(a)=x,~~\gamma(b)=y,$$ if $x,y$ are in the same connected component of $M$, $\delta(x,y)=+\infty$ otherwise. It~is easy to check that $\delta$ satisfies the usual axioms of distances: for the separation axiom, use the fact that if $y$ is outside some closed coordinate ball $\ovl B$ of radius $r$ centered at $x$ and if $g\ge c|dx|^2$ on $\ovl B$, then $\delta(x,y)\ge c^{1/2}r$. In~addition, $\delta$ satisfies the axiom: $$\hbox{for every $x,y\in M$,}~~~~\inf_{z\in M} \max\{\delta(x,z),\delta(y,z)\}={1\over 2}\delta(x,y).\qquad\leqno(2.1)$$ In fact for every $\varepsilon>0$ there is a path $\gamma$ such that $\gamma(a)=x$, $\gamma(b)=y$, \hbox{$\ell(\gamma)<\delta(x,y)+\varepsilon$} and we can take $z$ to be at mid-distance between $x$ and $y$ along~$\gamma$. A metric space $E$ with a distance $\delta$ satisfying the additional axiom~(2.1) will be called a {\it geodesic} metric space. It is then easy to see by dichotomy that any two points $x,y\in E$ can be joined by a chain of points \hbox{$x=x_0$}, \hbox{$x_1\ld x_N=y$} such that $\delta(x_j,x_{j+1})<\varepsilon$ and \hbox{$\sum\delta(x_j,x_{j+1})<\delta(x,y)+\varepsilon$}. \begstat{(2.2) Lemma {\rm(Hopf-Rinow)}} Let $(E,\delta)$ be a geodesic metric space. Then the following properties are equivalent: \smallskip \item{\rm a)} $E$ is locally compact and complete$\,;$ \smallskip \item{\rm b)} all closed geodesic balls $\ovl B(x_0,r)$ are compact. \endstat \begproof{} Since any Cauchy sequence is bounded, it is immediate that b) implies~a). We now check that a)~$\Longrightarrow$~b). Fix $x_0$ and define $R$ to be the supremum of all $r>0$ such that $\ovl B(x_0,r)$ is compact. Since $E$ is locally compact, we have $R>0$. Suppose that $R<+\infty$. Then $\ovl B(x_0,r)$ is compact for every~$r0$ be the smallest eigenvalue of this operator at $x\in X$. Then $\lambda$ is continuous on $X$ and we have $$\int_X\langle A_{E,\omega}^{-1}g,g\rangle\,dV\le\int_X\lambda(x)^{-1} |g(x)|^2\,dV.$$ The above situation occurs for example if $\omega$ is complete K\"ahler, $E>_m0$ and $p=n$, $q\ge 1$, $m\ge\min\{n-q+1,r\}$ (apply Lemma~VII-7.2). \endstat \titleb{5.}{Estimates on Weakly Pseudoconvex Manifolds} We first introduce a large class of complex manifolds on which the $L^2$ estimates will be easily tractable. \begstat{(5.1) Definition} A complex manifold $X$ is said to be weakly pseudo\-convex if there exists an exhaustion function $\psi\in\ci(X,\bbbr)$ such that $\ii d'd''\psi\ge 0$ on $X$, i.e.\ $\psi$ is plurisubharmonic. \endstat For domains $\Omega\subset\bbbc^n$, the above weak pseudoconvexity notion is equivalent to pseudoconvexity (cf.\ Th.~I-4.14). Note that every compact manifold is also weakly pseudoconvex (take $\psi\equiv 0$). Other examples that will appear later are Stein manifolds, or the total space of a Griffiths semi-negative vector bundle over a compact manifold (cf.\ Prop.~IX-?.?). \begstat{(5.2) Theorem} Every weakly pseudoconvex K\"ahler manifold $(X,\omega)$ carries a complete K\"ahler metric $\wh\omega$. \endstat \begproof{} Let $\psi\in\ci(X,\bbbr)$ be an exhaustive plurisubharmonic function on $X$. After addition of a constant to $\psi$, we can assume $\psi\ge 0$. Then $\wh\omega=\omega+\ii d'd''(\psi^2)$ is K\"ahler and $$\wh\omega=\omega+2i\psi d'd''\psi+2\ii d'\psi\wedge d''\psi\ge \omega+2\ii d'\psi\wedge d''\psi.$$ Since $d\psi=d'\psi+d''\psi$, we get $|d\psi|_{\wh\omega}=\sqrt{2} |d'\psi|_{\wh\omega}\le 1$ and Lemma~2.4 shows that $\wh\omega$ is complete.\qed \endproof Observe that we could have set more generally $\wh\omega=\omega+\ii d'd'' (\chi\circ\psi)$ where $\chi$ is a convex increasing function. Then $$\leqalignno{ \wh\omega&=\omega+i(\chi'\circ\psi)d'd''\psi+i(\chi''\circ\psi)d'\psi\wedge d''\psi\cr &\ge\omega+\ii d'(\rho\circ\psi)\wedge d''(\rho\circ\psi)&(5.3)\cr}$$ where $\rho(t)=\int_0^t\sqrt{\chi''(u)}\,du$. We thus have $|d'(\rho\circ\psi)|_{\wh\omega}\le 1$ and $\wh\omega$ will be complete as soon as $\lim_{t\to+\infty}\rho(t)=+\infty$, i.e.\ $$\int_0^{+\infty}\sqrt{\chi''(u)}\,du=+\infty.\leqno(5.4)$$ One can take for example $\chi(t)=t-\log(t)$ for $t\ge 1$. It follows from the above considerations that almost all vanishing theorems for positive vector bundles over compact manifolds are also valid on weakly pseudoconvex manifolds. Let us mention here the analogues of some results proved in Chapter~7. \begstat{(5.5) Theorem} For any $m$-positive vector bundle of rank $r$ over a weakly pseudoconvex manifold $X$, we have $H^{n,q}(X,E)=0$ for all $q\ge 1$ and \hbox{$m\ge\min\{n-q+1,r\}$}. \endstat \begproof{} The curvature form $\ii\Theta(\det E)$ is a K\"ahler metric on $X$, hence $X$ possesses a complete K\"ahler metric $\omega$. Let $\psi\in\ci(X,\bbbr)$ be an exhaustive plurisubharmonic function. For any convex increasing function $\chi\in\ci(\bbbr,\bbbr)$, we denote by $E_\chi$ the holomorphic vector bundle $E$ together with the modified metric $|u|_\chi^2=|u|^2\,\exp\big(-\chi\circ\psi(x)\big)$, $u\in E_x$. We get $$\ii\Theta(E_\chi)=\ii\Theta(E)+\ii d'd''(\chi\circ\psi)\otimes\,\Id_E \ge_m \ii\Theta(E),$$ thus $A_{E_\chi,\omega}\ge A_{E,\omega}>0$ in bidegree $(n,q)$. Let $g$ be a given form of bidegree $(n,q)$ with $L^2_\loc$ coefficients, such that $D''g=0$. The integrals $$\int_X\langle A_{E_\chi,\omega}^{-1}g,g\rangle_\chi\,dV\le \int_X\langle A_{E,\omega}^{-1}g,g\rangle\,e^{-\chi\circ\psi}\,dV,~~~~ \int_X |g|^2\,e^{-\chi\circ\psi}\,dV$$ become convergent if $\chi$ grows fast enough. We can thus apply Th.~4.5 to $(X,E_\chi,\omega)$ and find a $(n,q-1)$ form $f$ such that $D''f=g$. If $g$ is smooth, Remark~4.6 shows that $f$ can also be chosen smooth.\qed \endproof \begstat{(5.6) Theorem} If $E$ is a positive line bundle over a weakly pseudoconvex manifold $X$, then $H^{p,q}(X,E)=0$ for $p+q\ge n+1$. \endstat \begproof{} The proof is similar to that of Th.~5.5, except that we use here the K\"ahler metric $$\omega_\chi=\ii\Theta(E_\chi)=\omega+\ii d'd''(\chi\circ\psi),~~~~ \omega=\ii\Theta(E),$$ which depends on $\chi$. By (5.4) $\omega_\chi$ is complete as soon as $\chi$ is a convex increasing function that grows fast enough. Apply now Th.~4.5 to $(X,E_\chi,\omega_\chi)$ and observe that $A_{E_\chi,\omega_\chi}=[\ii\Theta(E_\chi),\Lambda_\chi]= (p+q-n)\,\Id$~ in bidegree $(p,q)$ in virtue of Cor.~VI-8.4 It remains to show that for every form $g\in\ci_{p,q}(X,E)$ there exists a choice of $\chi$ such that $g\in L^2_{p,q}(X,E_\chi,\omega_\chi)$. By (5.3) the norm of a scalar form with respect to $\omega_\chi$ is less than its norm with respect to $\omega$, hence $|g|_\chi^2\le|g|^2\,\exp(-\chi\circ\psi)$. On the other hand $$dV_\chi\le C\big(1+\chi'\circ\psi+\chi''\circ\psi\big)^n\,dV$$ where $C$ is a positive continuous function on $X$. The following lemma implies that we can always choose $\chi$ in order that the integral of $|g|_\chi^2\,dV_\chi$ converges on $X$. \endproof \begstat{(5.7) Lemma} For any positive function $\lambda\in\ci\big( [0,+\infty[,\bbbr\big)$, there exists a smooth convex function $\chi\in\ci\big([0,+\infty[,\bbbr\big)$ such that $\chi,\chi',\chi''\ge\lambda$ and \hbox{$(1+\chi'+\chi'')^ne^{-\chi}\le 1/\lambda$}. \endstat \begproof{} We shall construct $\chi$ such that $\chi''\ge \chi'\ge\chi\ge\lambda$ and \hbox{$\chi''/\chi^2\le C$} for some constant $C$. Then $\chi$ satisties the conclusion of the lemma after addition of a constant. Without loss of generality, we may assume that $\lambda$ is increasing and $\lambda\ge 1$. We define $\chi$ as a power series $$\chi(t)=\sum_{k=0}^{+\infty}\,a_0a_1\ldots a_k\,t^k,$$ where $a_k>0$ is a decreasing sequence converging to $0$ very slowly. Then $\chi$ is real analytic on $\bbbr$ and the inequalities $\chi''\ge\chi'\ge\chi$ are realized if we choose $a_k\ge 1/k$, $k\ge1$. Select a strictly increasing sequence of integers $(N_p)_{p\ge 1}$ so large that ${1\over p}\lambda(p+1)^{1/N_p}\in[1/p,1/(p-1)[$. We set $$\eqalign{ a_0&=\ldots=a_{N_1-1}=e\,\lambda(2),\cr a_k&={1\over p}\,\lambda(p+1)^{1/N_p}\,e^{1/\sqrt{k}},~~~~N_p\le k0$ be small. Select $\rho$ such that $\exp(-\rho\circ\psi(x))\le\varepsilon\gamma_s^{0,\omega}(x)$ at every point. Then for $j\ge s$ we get $$\gamma_j^{\chi,\alpha}\ge{\gamma_j^{0,\omega}\over \gamma_j^{0,\omega}+\varepsilon\gamma_j^{0,\omega}}={1\over 1+\varepsilon} \ge 1-\varepsilon,$$ and Th.~VI-8.3 implies $$\eqalign{ \langle\big[\ii\Theta(E_\chi),\Lambda_\alpha\big]u,u\rangle_\alpha&\ge \big(\gamma_1^{\chi,\alpha}+\cdots+\gamma_p^{\chi,\alpha}- \gamma_{q+1}^{\chi,\alpha}-\ldots-\gamma_n^{\chi,\alpha}\big)|u|^2\cr &\ge\big((p-s+1)(1-\varepsilon)-(n-q)\big)|u|^2\cr &\ge\big(1-(p-s+1)\varepsilon\big)|u|^2.\cr}$$ It remains however to control the torsion term $T_\alpha$. As $\omega$ is K\"ahler, trivial computations yield $$\eqalign{ d'\alpha&=-\rho'\circ\psi~\exp(-\rho\circ\psi)~d'\psi\wedge\omega,\cr d'd''\alpha&=\exp(-\rho\circ\psi)~\big[\big((\rho'\circ\psi)^2- \rho''\circ\psi\big)d'\psi\wedge d''\psi-\rho'\circ\psi~d'd''\psi\big] \wedge\omega.\cr}$$ Since $$\alpha\ge i(\chi'\circ\psi~d'd''\psi+\chi''\circ\psi~d'\psi\wedge d''\psi) +\exp(-\rho\circ\psi)\omega,$$ we get the upper bounds $$\eqalign{ |d'\alpha|_\alpha&\le\rho'\circ\psi~|d'\psi|_\alpha~|\exp(-\rho\circ\psi) \omega|_\alpha\le\rho'\circ\psi~(\chi''\circ\psi)^{-{1\over 2}}\cr |d'd''\alpha|_\alpha&\le{(\rho'\circ\psi)^2+\rho''\circ\psi\over\chi''\circ\psi} +{\rho'\circ\psi\over\chi'\circ\psi}.\cr}$$ It is then clear that we can choose $\chi$ growing sufficiently fast in order that $|T_\alpha|_\alpha\le\varepsilon$. If $\varepsilon$ is chosen sufficiently small, we get $A_{E_\chi,\alpha}\ge{1\over 2}\,\Id$, and the conclusion is obtained in the same way as for Th.~5.6.\qed \endproof \titleb{6.}{H\"ormander's Estimates for non Complete K\"ahler Metrics} Our aim here is to derive also estimates for a non complete K\"ahler metric, for example the standard metric of $\bbbc^n$ on a bounded domain $\Omega\subset\!\subset\bbbc^n$. A result of this type can be obtained in the situation described at the end of Remark~4.8. The underlying idea is due to (H\"ormander 1966), although we do not apply his so called ``three weights" technique, but use instead an approximation of the given metric $\omega$ by complete K\"ahler metrics. \begstat{(6.1) Theorem} Let $(X,\wh\omega)$ be a complete K\"ahler manifold, $\omega$ another K\"ahler metric, possibly non complete, and $E\lra X$ a $m$-semi-positive vector bundle. Let $g\in L^2_{n,q}(X,E)$ be such that $D''g=0$ and $$\int_X\langle A_q^{-1}g,g\rangle\,dV<+\infty$$ with respect to $\omega$, where $A_q$ stands for the operator $\ii\Theta(E)\wedge\Lambda$ in bidegree $(n,q)$ and $q\ge 1$, $m\ge\min\{n-q+1,r\}$. Then there exists $f\in L^2_{n,q-1}(X,E)$ such that $D''f=g$ and $$\|f\|^2\le\int_X\langle A_q^{-1}g,g\rangle\,dV.$$ \endstat \begproof{} For every $\varepsilon>0$, the K\"ahler metric $$\omega_\varepsilon=\omega+\varepsilon\wh\omega$$ is complete. The idea of the proof is to apply the $L^2$ estimates to $\omega_\varepsilon$ and to let $\varepsilon$ tend to zero. Let us put an index $\varepsilon$ to all objects depending on $\omega_\varepsilon$. It follows from Lemma~6.3 below that $$|u|_\varepsilon^2\,dV_\varepsilon\le|u|^2\,dV,~~~~ \langle A_{q,\varepsilon}^{-1}u,u\rangle_\varepsilon\,dV_\varepsilon\le \langle A_q^{-1}u,u\rangle\,dV\leqno(6.2)$$ for every $u\in\Lambda^{n,q}T^\star_X\otimes E$. If these estimates are taken for granted, Th.~4.5 applied to $\omega_\varepsilon$ yields a section $f_\varepsilon\in L^2_{n,q-1}(X,E)$ such that $D''f_\varepsilon=g$ and $$\int_X|f_\varepsilon|_\varepsilon^2\,dV_\varepsilon\le \int_X\langle A_{q,\varepsilon}^{-1}g,g\rangle_\varepsilon\,dV_\varepsilon\le \int_X\langle A_q^{-1}g,g\rangle\,dV.$$ This implies that the family $(f_\varepsilon)$ is bounded in $L^2$ norm on every compact subset of $X$. We can thus find a weakly convergent subsequence $(f_{\varepsilon_\nu})$ in $L^2_\loc$. The weak limit $f$ is the solution we are looking for.\qed \endproof \begstat{(6.3) Lemma} Let $\omega$, $\gamma$ be hermitian metrics on $X$ such that $\gamma\ge\omega$. For every $u\in\Lambda^{n,q} T^\star_X\otimes E$, $q\ge 1$, we have $$|u|_\gamma^2\,dV_\gamma\le|u|^2\,dV,~~~~ \langle A_{q,\gamma}^{-1}u,u\rangle_\gamma\,dV_\gamma\le \langle A_q^{-1}u,u\rangle\,dV$$ where an index $\gamma$ means that the corresponding term is computed in terms of $\gamma$ instead of $\omega$. \endstat \begproof{} Let $x_0\in X$ be a given point and $(z_1\ld z_n)$ coordinates such that $$\omega=\ii\sum_{1\le j\le n}dz_j\wedge d\ovl z_j,~~~~ \gamma=\ii\sum_{1\le j\le n}\gamma_j\,dz_j\wedge d\ovl z_j~~~ \hbox{\rm at}~~x_0,$$ where $\gamma_1\le\ldots\le\gamma_n$ are the eigenvalues of $\gamma$ with respect to $\omega$ (thus $\gamma_j\ge 1$). We have $|dz_j|_\gamma^2=\gamma_j^{-1}$ and $|dz_K|_\gamma^2= \gamma_K^{-1}$ for any multi-index $K$, with the notation $\gamma_K=\prod_{j\in K}\gamma_j$. For every $\,u=\sum u_{K,\lambda}dz_1\wedge\ldots\wedge dz_n\wedge d\ovl z_K \otimes e_\lambda$, $|K|=q$, $1\le\lambda\le r$, the computations of \S~VII-7 yield $$\leqalignno{ |u|_\gamma^2&=\sum_{K,\lambda}~(\gamma_1\ldots\gamma_n)^{-1} \gamma_K^{-1}\,|u_{K,\lambda}|^2,~~~~ dV_\gamma=\gamma_1\ldots\gamma_n\,dV,\cr |u|_\gamma^2\,dV_\gamma&=\sum_{K,\lambda}\gamma_K^{-1}\, |u_{K,\lambda}|^2\,dV\le|u|^2\,dV,\cr \Lambda_\gamma u&=\sum_{|I|=q-1}\sum_{j,\lambda}\ii(-1)^{n+j-1}\gamma_j^{-1}\, u_{jI,\lambda}\,(\wh{dz_j})\wedge d\ovl z_I \otimes e_\lambda,\cr}$$ where $(\wh{dz_j})$ means $dz_1\wedge\ldots\wh{dz_j}\ldots \wedge dz_n$, $$\leqalignno{ A_{q,\gamma}u&=\sum_{|I|=q-1}\sum_{j,k,\lambda,\mu}\gamma_j^{-1}\, c_{jk\lambda\mu}\,u_{jI,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge d\ovl z_{kI}\otimes e_\mu,\cr \llap{\hbox{$\langle A_{q,\gamma}u,u\rangle_\gamma$}}&= (\gamma_1\ldots\gamma_n)^{-1}\sum_{|I|=q-1}\gamma_I^{-1} \sum_{j,k,\lambda,\mu}\gamma_j^{-1}\gamma_k^{-1}c_{jk\lambda\mu}\, u_{jI,\lambda}\ovl u_{kI,\mu}\cr &\ge(\gamma_1\ldots\gamma_n)^{-1}\sum_{|I|=q-1}\gamma_I^{-2} \sum_{j,k,\lambda,\mu}\gamma_j^{-1}\gamma_k^{-1}c_{jk\lambda\mu}\, u_{jI,\lambda}\ovl u_{kI,\mu}\cr &=\gamma_1\ldots\gamma_n\,\langle A_qS_\gamma u,S_\gamma u\rangle\cr}$$ where $S_\gamma$ is the operator defined by $$S_\gamma u=\sum_K~(\gamma_1\ldots\gamma_n\gamma_K)^{-1}\, u_{K,\lambda}\,dz_1\wedge\ldots\wedge dz_n\wedge d\ovl z_K\otimes e_\lambda.$$ We get therefore $$\eqalign{ |\langle u,v\rangle_\gamma|^2=|\langle u,S_\gamma v\rangle|^2&\le \langle A_q^{-1}u,u\rangle\langle A_q S_\gamma v,S_\gamma v\rangle\cr &\le(\gamma_1\ldots\gamma_n)^{-1}\langle A_q^{-1}u,u\rangle \langle A_{q,\gamma}v,v\rangle_\gamma,\cr}$$ and the choice $v=A_{q,\gamma}^{-1}u$ implies $$\langle A_{q,\gamma}^{-1}u,u\rangle_\gamma\le (\gamma_1\ldots\gamma_n)^{-1}\,\langle A_q^{-1}u,u\rangle\,;$$ this relation is equivalent to the last one in the lemma.\qed \endproof An important special case is that of a semi-positive line bundle $E$. If we let $0\le\lambda_1(x)\le\ldots\le\lambda_n(x)$ be the eigenvalues of $\ii\Theta(E)_x$ with respect to $\omega_x$ for all $x\in X$, formula VI-8.3 implies $$\leqalignno{ \langle A_qu,u\rangle&\ge(\lambda_1+\cdots+\lambda_q)|u|^2,\cr \int_X\langle A_q^{-1}g,g\rangle\,dV&\le\int_X~{1\over\lambda_1+\cdots+ \lambda_q}\,|g|^2\,dV.&(6.4)\cr}$$ A typical situation where these estimates can be applied is the case when $E$ is the trivial line bundle $X\times\bbbc$ with metric given by a weight $e^{-\varphi}$. One can assume for example that $\varphi$ is plurisubharmonic and that $\ii d'd''\varphi$ has at least $n-q+1$ positive eigenvalues at every point, i.e.\ $\lambda_q>0$ on $X$. This situation leads to very important $L^2$ estimates, which are precisely those given by (H\"ormander~1965, 1966). We state here a slightly more general result. \begstat{(6.5) Theorem} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler manifold, $E$ a hermitian line bundle on $X$, $\varphi\in\ci(X,\bbbr)$ a weight function such that the eigenvalues $\lambda_1\le\ldots\le\lambda_n$ of $\ii\Theta(E)+\ii d'd''\varphi$ are $\ge0$. Then for every form $g$ of type $(n,q)$, $q\ge 1$, with $L^2_\loc$ (resp. $\ci$) coefficients such that $D''g=0$ and $$\int_X~{1\over\lambda_1+\cdots+\lambda_q}\,|g|^2\,e^{-\varphi}\,dV<+\infty,$$ we can find a $L^2_\loc$ (resp. $\ci$) form $f$ of type $(n,q-1)$ such that $D''f=g$ and $$\int_X~|f|^2\,e^{-\varphi}\,dV\le\int_X~{1\over\lambda_1+\cdots+ \lambda_q}\,|g|^2\,e^{-\varphi}\,dV.$$ \endstat \begproof{} Apply the general estimates to the bundle $E_\varphi$ deduced from $E$ by multiplication of the metric by $e^{-\varphi}\,;$ we have $\ii\Theta(E_\varphi)=\ii\Theta(E)+\ii d'd''\varphi$. It is not necessary here to assume in addition that $g\in L^2_{n,q} (X,E_\varphi)$. In fact, $g$ is in $L^2_\loc$ and we can exhaust X by the relatively compact weakly pseudoconvex domains $$X_c=\big\{x\in X~;~\psi(x)0$, there is a section $F\in H^0(X,K_X\otimes L)$ such that \hbox{$F_{\restriction Y}=f_{\restriction Y}$} and $$\int_X{|F|^2\over(1+|\sigma|^2)^{p+\varepsilon}}\,dV\le \Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,dV.$$ \endstat The proof will involve a weight with logarithmic singularities along $Y$. We must therefore apply the existence theorem over $X\ssm Y$. This requires to know whether $X\ssm Y$ has a complete K\"ahler metric. \begstat{(7.2) Lemma} Let $(X,\omega)$ be a K\"ahler manifold, and $Y=\sigma^{-1}(0)$ an analytic subset defined by a section of a hermitian vector bundle $E\to X$. If $X$ is weakly pseudoconvex and exhausted by $X_c=\{x\in X~;~\psi(x)0$ and let $\eta$ tend to $0$, we see that we can assume $f$ defined in a neighborhood of $\ovl U$. Let $h$ be the continuous section of $L$ such that $h=(1-|\sigma|^{p+1})f$ on $U=\{|\sigma|<1\}$ and $h=0$ on $X\ssm U$. We have $h_{\restriction Y}=f_{\restriction Y}$ and $$d''h=-{p+1\over 2}\,|\sigma|^{p-1}\,\{\sigma,D'\sigma\}\,f~~~~{\rm on}~~U, ~~~~d''h=0~~~~{\rm on}~~X\ssm U.$$ We consider $g=d''h$ as a $(n,1)$-form with values in the hermitian line bundle $L_\varphi=L$, endowed with the weight $e^{-\varphi}$ given by $$\varphi=p\log|\sigma|^2+\varepsilon\log(1+|\sigma|^2).$$ Notice that $\varphi$ is singular along $Y$. The Cauchy-Schwarz inequality implies $\ii\{D'\sigma,\sigma\} \wedge\{\sigma,D'\sigma\}\le\ii\{D'\sigma,D'\sigma\}$ as in Lemma~7.2, and we find $$\eqalign{ \ii d'd''\log(1+|\sigma|^2)&={(1+|\sigma|^2)\ii\{D'\sigma,D'\sigma\}- \ii\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over(1+|\sigma|^2)^2}\cr &\quad{}-{\{\ii\Theta(E)\sigma,\sigma\}\over1+|\sigma|^2} \ge{\ii\{D'\sigma,D'\sigma\}\over (1+|\sigma|^2)^2}-{\{\ii\Theta(E)\sigma,\sigma\}\over1+|\sigma|^2}.\cr}$$ The inequality $\ii d'd''\log|\sigma|^2\ge-\{\ii\Theta(E)\sigma,\sigma\}/|\sigma|^2$ obtained in Lemma~7.2 and the above one imply $$\eqalign{\ii\Theta(L_\varphi) &=\ii\Theta(L)+p\,\ii d'd''\log|\sigma|^2+\varepsilon\,\ii d'd''\log(1+|\sigma|^2)\cr &\ge \ii\Theta(L)-\Big({p\over|\sigma|^2}+{\varepsilon\over 1+|\sigma|^2}\Big)\{\ii\Theta(E) \sigma,\sigma\}+\varepsilon{\ii\{D'\sigma,D'\sigma\}\over (1+|\sigma|^2 )^2}\cr &\ge\varepsilon{\ii\,\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over|\sigma|^2 \,(1+|\sigma|^2 )^2},\cr}$$ thanks to the hypothesis on the curvature of $L$ and the Cauchy-Schwarz inequality. Set $\xi=(p+1)/2\,|\sigma|^{p-1}\{D'\sigma,\sigma\} =\sum\xi_j\,dz_j$ in an $\omega$-orthonormal basis $\partial/\partial z_j$, and let $\wh\xi=\sum\xi_j\partial/\partial\ovl z_j$ be the dual $(0,1)$-vector field. For every $(n,1)$-form $v$ with values in $L_\varphi$, we find $$\eqalign{ \big|\langle d''h,v\rangle\big|&=\big|\langle\ovl\xi\wedge f,v\rangle\big|= \big|\langle f,\wh\xi\ort v\rangle\big|\le|f|\,|\wh\xi\ort v|,\cr \wh\xi\ort v&=\sum-\ii\xi_j\,dz_j\wedge\Lambda v=-\ii\xi\wedge\Lambda v,\cr |\langle d''h,v\rangle|^2&\le|f|^2\,|\wh\xi\ort v|^2= |f|^2\langle-\ii\xi\wedge\Lambda v,\wh\xi\ort v\rangle\cr &=|f|^2\langle-\ii\ovl\xi\wedge\xi\wedge\Lambda v,v\rangle= |f|^2\langle[\ii\xi\wedge\ovl\xi,\Lambda]v,v\rangle\cr &\le{(p+1)^2\over 4\varepsilon}\,|\sigma|^{2p}\,(1+|\sigma|^2)^2\,|f|^2\, \langle[\ii\Theta(L_\varphi),\Lambda]v,v\rangle.}$$ Thus, in the notations of Th.~6.1, the form $g=d''h$ satisfies $$\langle A_1^{-1}g,g\rangle\le {(p+1)^2\over4\varepsilon}\,|\sigma|^{2p}(1+|\sigma|^2)^2\,|f|^2\le {(p+1)^2\over\varepsilon}\,|f|^2\,e^\varphi,$$ where the last equality results from the fact that $(1+|\sigma|^2)^2\le 4$ on the support of $g$. Lemma 7.2 shows that the existence theorem 6.1 can be applied on each set $X_c\ssm Y$. Letting $c$ tend to infinity, we infer the existence of a $(n,0)$-form $u$ with values in $L$ such that $d''u=g$ on $X\ssm Y$ and $$\eqalign{ \int_{X\ssm Y}|u|^2\,e^{-\varphi}\,dV&\le \int_{X\ssm Y}\langle A_1^{-1}g,g\rangle e^{-\varphi},~~~~{\rm thus}\cr \int_{X\ssm Y}{|u|^2\over|\sigma|^{2p}(1+|\sigma|^2)^\varepsilon}\,dV &\le{(p+1)^2\over\varepsilon}\int_U|f|^2\,dV.\cr}$$ This estimate implies in particular that $u$ is locally $L^2$ near $Y$. As $g$ is conti\-nuous over $X$, Lemma~7.3 below shows that the equality \hbox{$d''u=g=d''h$} extends to $X$, thus $F=h-u$ is holomorphic everywhere. Hence \hbox{$u=h-F$} is continuous on $X$. As $|\sigma(x)|\le C\,d(x,Y)$ in a neighborhood of every point of $Y$, we see that $|\sigma|^{-2p}$ is non integrable at every point $x_0\in Y_{\rm reg}$, because $\codim Y\le p$. It follows that $u=0$ on $Y$, so $$F_{\restriction Y}=h_{\restriction Y}=f_{\restriction Y}.$$ The final $L^2$-estimate of Th.~7.1 follows from the inequality $$|F|^2=|h-u|^2\le(1+|\sigma|^{-2p})\,|u|^2+(1+|\sigma|^{2p})\,|f|^2$$ which implies $${|F|^2\over(1+|\sigma|^2)^p}\le{|u|^2\over|\sigma|^{2p}}+|f|^2. \eqno{\square}$$ \endproof \begstat{(7.3) Lemma} Let $\Omega$ be an open subset of $\bbbc^n$ and $Y$ an analytic subset of $\Omega$. Assume that $v$ is a $(p,q-1)$-form with $L^2_\loc$ coefficients and $w$ a $(p,q)$-form with $L^1_\loc$ coefficients such that $d''v=w$ on $\Omega\ssm Y$ $($in the sense of distribution theory$)$. Then $d''v=w$ on $\Omega$. \endstat \begproof{} An induction on the dimension of $Y$ shows that it is sufficient to prove the result in a neighborhood of a regular point $a\in Y$. By using a local analytic isomorphism, the proof is reduced to the case where $Y$ is contained in the hyperplane $z_1=0$, with $a=0$. Let $\lambda\in\ci(\bbbr,\bbbr)$ be a function such that $\lambda(t)=0$ for $t\le{1\over 2}$ and $\lambda(t)=1$ for $t\ge 1$. We must show that $$\int_\Omega w\wedge\alpha=(-1)^{p+q}\int_\Omega v\wedge d''\alpha \leqno(7.4)$$ for all $\alpha\in\cD_{n-p,n-q}(\Omega)$. Set $\lambda_\varepsilon(z)= \lambda(|z_1|/\varepsilon)$ and replace $\alpha$ in the integral by $\lambda_\varepsilon\alpha$. Then $\lambda_\varepsilon\alpha\in \cD_{n-p,n-q}(\Omega\ssm Y)$ and the hypotheses imply $$\int_\Omega w\wedge\lambda_\varepsilon\alpha=(-1)^{p+q} \int_\Omega v\wedge d''(\lambda_\varepsilon\alpha)=(-1)^{p+q} \int_\Omega v\wedge (d''\lambda_\varepsilon\wedge\alpha+\lambda_\varepsilon d''\alpha).$$ As $w$ and $v$ have $L^1_\loc$ coefficients on $\Omega$, the integrals of $w\wedge\lambda_\varepsilon\alpha$ and $v\wedge\lambda_\varepsilon d''\alpha$ converge respectively to the integrals of $w\wedge\alpha$ and $v\wedge d''\alpha$ as $\varepsilon$ tends to $0$. The remaining term can be estimated by means of the Cauchy-Schwarz inequality: $$\Big|\int_\Omega v\wedge d''\lambda_\varepsilon\wedge\alpha\Big|^2\le \int_{|z_1|\le\varepsilon}|v\wedge\alpha|^2\,dV.~ \int_{\Supp\alpha}|d''\lambda_\varepsilon|^2\,dV\,;$$ as $v\in L^2_\loc(\Omega)$, the integral $\int_{|z_1|\le\varepsilon}|v\wedge\alpha|^2\,dV$ converges to $0$ with $\varepsilon$, whereas $$\int_{\Supp\alpha}|d''\lambda_\varepsilon|^2\,dV\le {C\over\varepsilon^2}\,{\rm Vol}\big(\Supp\,\alpha \cap\{|z_1|\le\varepsilon\}\big)\le C'.$$ Equality (7.4) follows when $\varepsilon$ tends to $0$.\qed \endproof \begstat{(7.5) Corollary} Let $\Omega\subset\bbbc^n$ be a weakly pseudoconvex domain and let $\varphi$, $\psi$ be plurisubharmonic functions on $\Omega$, where $\psi$ is supposed to be finite and continuous. Let $\sigma=(\sigma_1\ld \sigma_r)$ be a family of holomorphic functions on $\Omega$, let \hbox{$Y=\sigma^{-1}(0)$}, $p={}$ maximal codimension of $Y$ and set \smallskip \item{\rm a)} $U=\{z\in\Omega\,;\,|\sigma(z)|^20$ and every holomorphic function $f$ on $U$, there exists a holomorphic function $F$ on $\Omega$ such that $F_{\restriction Y}=f_{\restriction Y}$ and $$\leqalignno{ \int_\Omega {|F|^2\,e^{-\varphi+p\psi}\over (1+|\sigma|^2e^{\psi})^{p+\varepsilon}}\,dV&\le \Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,e^{-\varphi+p\psi}\,dV, ~~~~{\it resp.}&{\rm a)}\cr \int_\Omega {|F|^2\,e^{-\varphi}\over (e^\psi+|\sigma|^2)^{p+\varepsilon}}\,dV&\le \Big(1+{(p+1)^2\over\varepsilon}\Big)\int_U|f|^2\,e^{-\varphi-(p+ \varepsilon)\psi}\,dV.&{\rm b)}\cr}$$ \endstat \begproof{} After taking convolutions with smooth kernels on pseudoconvex subdomains $\Omega_c\subset\!\subset\Omega$, we may assume $\varphi$, $\psi$ smooth. In either case a) or~b), apply Th.~7.1 to \medskip \noindent a)~~$E=\Omega\times\bbbc^r$ with the weight $e^\psi$, $L=\Omega\times\bbbc$ with the weight $e^{-\varphi+p\psi}$, and $U=\{|\sigma|^2e^\psi<1\}$. Then $$\ii\Theta(E)=-\ii d'd''\psi\otimes\Id_E\le 0,~~~~\ii\Theta(L)=\ii d'd''\varphi- p\,\ii d'd''\psi\ge p\,\ii\Theta(E).$$ \smallskip \noindent b)~~$E=\Omega\times\bbbc^r$ with the weight $e^{-\psi}$, $L=\Omega\times\bbbc$ with the weight $e^{-\varphi-(p+\varepsilon)\psi}$, and $U=\{|\sigma|^2e^{-\psi}<1\}$. Then $$\ii\Theta(E)=\ii d'd''\psi\otimes\Id_E\ge 0,~~~~\ii\Theta(L)=\ii d'd''\varphi+ (p+\varepsilon)\,\ii d'd''\psi\ge(p+\varepsilon)\,\ii\Theta(E).$$ The condition on $\Theta(L)$ is satisfied in both cases and $K_{\Omega}$ is trivial.\qed \endproof \begstat{(7.6) H\"ormander-Bombieri-Skoda theorem} Let $\Omega\subset\bbbc^n$ be a weakly pseudoconvex domain and $\varphi$ a plurisubharmonic function on $\Omega$. For every $\varepsilon>0$ and every point $z_0\in\Omega$ such that $e^{-\varphi}$ is integrable in a neighborhood of $z_0$, there exists a holomorphic function $F$ on $\Omega$ such that $F(z_0)=1$ and $$\int_\Omega{|F(z)|^2\,e^{-\varphi(z)}\over(1+|z|^2)^{n+\varepsilon}}\,dV< +\infty.$$ \endstat (Bombieri~1970) originally stated the theorem with the exponent $3n$ instead of $n+\varepsilon\,;$ the improved exponent $n+\varepsilon$ is due to (Skoda~1975). The example $\Omega=\bbbc^n$, $\varphi(z)=0$ shows that one cannot replace $\varepsilon$ by $0$. \begproof{} Apply Cor.~7.5 b) to $f\equiv 1$, $\sigma(z)=z-z_0$, $p=n$ and $\psi\equiv\log r^2$ where $U=B(z_0,r)$ is a ball such that $\int_U e^{-\varphi}\,dV<+\infty$.\qed \endproof \begstat{(7.7) Corollary} Let $\varphi$ be a plurisubharmonic function on a complex ma\-ni\-fold $X$. Let $A$ be the set of points $z\in X$ such that $e^{-\varphi}$ is not locally integrable in a neighborhood of $z$. Then $A$ is an analytic subset of $X$. \endstat \begproof{} Let $\Omega\subset X$ be an open coordinate patch isomorphic to a ball of~$\bbbc^n$, with coordinates $(z_1\ld z_n)$. Define $E\subset H^0(\Omega,\cO)$ to be the Hilbert space of holomorphic functions $f$ on $\Omega$ such that $$\int_\Omega |f(z)|^2e^{-\varphi(z)}\,dV(z)<+\infty.$$ Then $A\cap\Omega=\bigcap_{f\in E}f^{-1}(0)$. In fact, every $f$ in $E$ must obviously vanish on~$A\,;$ conversely, if $z_0\notin A$, Th.~7.6 shows that there exists $f\in E$ such that $f(z_0)\ne 0$. By Th.~II-5.5, we conclude that $A$ is analytic.\qed \endproof \titleb{8.}{Applications to Hypersurface Singularities} We first give some basic definitions and results concerning multiplicities of divisors on a complex manifold. \begstat{(8.1) Proposition} Let $X$ be a complex manifold and $\Delta=\sum\lambda_j[Z_j]$ a divisor on $X$ with real coefficients $\lambda_j\ge 0$. Let $x\in X$ and $f_j=0$, $1\le j\le N$, irreducible equations of $Z_j$ on a neighborhood $U$ of $x$. \smallskip \item{\rm a)}~~The multiplicity of $\Delta$ at $x$ is defined by $$\mu(\Delta,x)=\sum\lambda_j\,{\rm ord}_x f_j.$$ \smallskip \item{\rm b)} $\Delta$ is said to have normal crossings at a point $x\in\Supp\,\Delta$ if all hypersurfaces $Z_j$ containing $x$ are smooth at $x$ and intersect transversally, i.e.\ if the linear forms $df_j$ defining the corresponding tangent spaces $T_xZ_j$ are linearly independent at $x$. The set ${\rm nnc}(\Delta)$ of non normal crossing points is an analytic subset of $X$. \smallskip \item{\rm c)} The non-integrability locus ${\rm nil}(\Delta)$ is defined as the set of points $x\in X$ such that $\prod|f_j|^{-2\lambda_j}$ is non integrable near $x$. Then ${\rm nil}(\Delta)$ is an analytic subset of $X$ and there are inclusions $$\{x\in X\,;\,\mu(\Delta,x)\ge n\}\subset{\rm nil}(\Delta)\subset \{x\in X\,;\,\mu(\Delta,x)\ge 1\}.$$ \item{}Moreover ${\rm nil}(\Delta)\subset{\rm nnc}(\Delta)$ if all coefficients of $\Delta$ \hbox{satisfy $\lambda_j<1$}. \endstat \begproof{} b) The set ${\rm nnc}(\Delta)\cap U$ is the union of the analytic sets $$f_{j_1}=\ldots=f_{j_p}=0,~~~~df_{j_1}\wedge\ldots\wedge df_{j_p}=0,$$ for each subset $\{j_1\ld j_p\}$ of the index set $\{1\ld N\}$. Thus ${\rm nnc}(\Delta)$ is analytic. \medskip\noindent c) The analyticity of ${\rm nil}(\Delta)$ follows from Cor.~7.7 applied to the plurisubharmonic function $\varphi=\sum 2\lambda_j\, \log|f_j|$. Assume first that $\lambda_j<1$ and that $\Delta$ has normal crossings at $x$. Let $f_{j_1}(x)=\ldots=f_{j_s}(x)=0$ and \hbox{$f_j(x)\ne 0$} for $j\ne j_l$. Then, we can choose local coordinates $(w_1\ld w_n)$ on $U$ such that $w_1=f_{j_1}(z)$, $\ldots$, $w_s=f_{j_s}(z)$, and we have $$\int_{U}{d\lambda(z)\over\prod|f_j(z)|^{2\lambda_j}} \le\int_{U}{C\,d\lambda(w)\over |w_1|^{2\lambda_1}\ldots|w_s|^{2\lambda_s}}<+\infty.$$ It follows that ${\rm nil}(\Delta)\subset{\rm nnc}(\Delta)$. Let us prove now the statement relating ${\rm nil}(\Delta)$ with multiplicity sets. Near any point $x$, we have $|f_j(z)|\le C_j|z-x|^{m_j}$ with $m_j={\rm ord}_x f_j$, thus $$\prod|f_j|^{-2\lambda_j}\ge C\,|z-x|^{-2\mu(\Delta,x)}.$$ It follows that $x\in{\rm nil}(\Delta)$ as soon as $\mu(\Delta,x)\ge n$. On the other hand, we are going to prove that $\mu(\Delta,x)<1$ implies $x\notin{\rm nil}(\Delta)$, i.e.\ $\prod|f_j|^{-2\lambda_j}$ integrable near $x$. We may assume $\lambda_j$ rational; otherwise replace each $\lambda_j$ by a slightly larger rational number in such a way that $\mu(\Delta,x)<1$ is still true. Set $f=\prod f_j^{k\lambda_j}$ where $k$ is a common denominator. The result is then a consequence of the following lemma.\qed \endproof \begstat{(8.2) Lemma} If $f\in\cO_{X,x}$ is not identically $0$, there exists a neighbor\-hood $U$ of $x$ such that $\int_U|f|^{-2\lambda}\,dV$ converges for all $\lambda<1/m$, $m={\rm ord}_x f$. \endstat \begproof{} One can assume that $f$ is a Weierstrass polynomial $$f(z)=z_n^m+a_1(z')z_n^{m-1}+\cdots+a_m(z'),~~~~a_j(z')\in\cO_{n-1},~~~~ a_j(0)=0,$$ with respect to some coordinates $(z_1\ld z_n)$ centered at $x$. Let $v_j(z')$, \hbox{$1\le j\le m$,} denote the roots $z_n$ of $f(z)=0$. On a small neighborhood $U$ of $x$ we have $|v_j(z')|\le 1$. The inequality between arithmetic and geometric mean implies $$\eqalign{ \int_{\{|z_n|\le 1\}}|f(z)|^{-2\lambda}\,dx_ndy_n&=\int_{\{|z_n|\le 1\}} \prod_{1\le j\le m}|z_n-v_j(z')|^{-2\lambda}\,dx_ndy_n\cr &\le{1\over m}\int_{\{|z_n|\le 1\}}\sum_{1\le j\le m}|z_n-v_j(z') |^{-2m\lambda}\,dx_ndy_n\cr &\le\int_{\{|z_n|\le 2\}}{dx_ndy_n\over|z_n|^{2m\lambda}},\cr}$$ so the Lemma follows from the Fubini theorem.\qed \endproof Another interesting application concerns the study of multiplicities of singular points for algebraic hypersurfaces in $\bbbp^n$. Following (Waldschmidt~1975), we introduce the following definition. \begstat{(8.3) Definition} Let $S$ be a finite subset of $\bbbp^n$. For any integer $t\ge 1$, we define $\omega_t(S)$ as the minimum of the degrees of non zero homogeneous polynomials $P\in\bbbc[z_0\ld z_n]$ which vanish at order $t$ at every point of $S$, i.e.\ $D^\alpha P(w)=0$ for every $w\in S$ and every multi-index $\alpha=(\alpha_0\ld\alpha_n)$ of length $|\alpha|a$ this means that $E\cap Y=\emptyset\,)$. \endstat \begproof{} By induction on $n$, it suffices to show that $\dim(E\cap H)\le a-1$ for a generic hyperplane $H\subset\bbbp^n$. Let $E_j$ be the (finite) family of irreducible components of $E$, and $w_j\in E_j$ an arbitrary point. Then $E\cap H=\bigcup E_j\cap H$ and we have $\dim E_j\cap H<\dim E_j\le a$ as soon as $H$ avoids all points $w_j$.\qed \endproof \begproof{of Theorem 8.6.} By Bertini's lemma, there exists a linear subspace $Y\subset\bbbp^n$ of codimension $a+1$ such that ${\rm nil}(\Delta)\cap Y=\emptyset$. We consider $P$ as a section of the line bundle $\cO(D)$ over $\bbbp^n$, where $D=\deg\,P$~ (cf.\ Th.~V-15.5). There are sections $\sigma_1\ld\sigma_{a+1}$ of $\cO(1)$ such that $Y=\sigma^{-1}(0)$. We shall apply Th.~7.1 to $E=\cO(1)$ with its standard hermitian metric, and to $L=\cO(k)$ equipped with the additional weight $\varphi=\alpha\log|P|^2$. We may assume that the open set $U=\{|\sigma|<1\}$ is such that ${\rm nil}(\Delta)\cap\ovl U=\emptyset$, otherwise it suffices to multiply $\sigma$ by a large constant. This implies that the polynomial $Q=\prod P_j^{[k_j\alpha]}$ satisfies $$\int_U|Q|^2\,e^{-\varphi}\,dV= \int_U\prod|P_j|^{-2(k_j\alpha-[k_j\alpha])}\,dV<+\infty.$$ Set $\omega=ic\big(\cO(1)\big)$. We have $\ii d'd''\log|P|^2\ge-ic\big(\cO(D) \big)=-D\omega$ by the Lelong-Poincar\'e equation, thus $\ii\Theta(L_\varphi)\ge(k-\alpha D)\omega$. The desired curvature inequality $\ii\Theta(L_\varphi)\ge(a+1+\varepsilon)\ii\Theta(E)$ is satisfied if $k-\alpha D\ge (a+1+\varepsilon)$. We thus take $$k=[\alpha D]+a+2.$$ The section $f\in H^0(U,K_{\bbbp^n}\otimes L)=H^0\big(U,\cO(k-n-1)\big)$ is taken to be a multiple of $Q$ by some polynomial. This is possible provided that $$k-n-1\ge\deg\,Q~~~\Longleftrightarrow~~~ \alpha D+a+2-n-1\ge\sum[k_j\alpha]\,\deg\,P_j,$$ or equivalently, as $D=\sum k_j\,\deg\,P_j$, $$\sum(k_j\alpha-[k_j\alpha])\,\deg\,P_j\ge n-a-1.\leqno(8.10)$$ Then we get $f\in H^0(U,K_{\bbbp^n}\otimes L)$ such that $\int_U |f|^2\,e^{-\varphi}\,dV<+\infty$. Theorem 7.1 implies the existence of $F\in H^0(\bbbp^n,K_{\bbbp^n}\otimes L)$, i.e.\ of a polynomial $F$ of degree $k-n-1$, such that $$\int_{\bbbp^n}|F|^2e^{-\varphi}\,dV=\int_{\bbbp^n}{|F|^2\over|P|^{2\alpha}} \,dV<+\infty\,;$$ observe that $|\sigma|$ is bounded, for we are on a compact manifold. Near any $w\in S$, we have $|P(z)|\le C|z-w|^{t_2}$, thus \hbox{$|P(z)|^{2\alpha}\le C|z-w|^{2(t_1+n-1)}$}. This implies that the above integral can converge only if $F$ vanishes at order $\ge t_1$ at each point $w\in S$. Therefore $$\omega_{t_1}(S)\le\deg\,F=k-n-1= [\alpha D]+a+1-n\le\alpha\omega_{t_2}(S)+a+1-n,$$ which is the desired inequality. However, the above proof only works under the additional assumption (8.10). Assume on the contrary that $$\beta=\sum(k_j\alpha-[k_j\alpha])\,\deg\,P_j0$. Then for every $D''$-closed form $f$ of type $(n,k)$ with values in $Q\otimes L$ such that $\|f\|<+\infty$, there exists a $D''$-closed form $h$ of type $(n,k)$ with values in $E\otimes L$ such that $f=g\cdot h$ and $$\|h\|^2\le(1+m/\varepsilon)\,\|f\|^2.$$ \endstat The idea of the proof is essentially due to (Skoda~1978), who actually proved the special case $k=0$. The general case appeared in (Demailly~1982c). \begproof{} Let $j:S\to E$ be the inclusion morphism, $g^\star:Q\to E$ and $j^\star:E\to S$ the adjoints of $g,j$, and $$D_E = \pmatrix{D_S&-\beta^\star\cr \beta & D_Q\cr},~~ \beta\in\ci_{1,0}\big(X,\hom(S,Q)\big),~~ \beta^\star\in\ci_{0,1}\big(X,\hom(Q,S)\big),$$ the matrix of $D_E$ with respect to the orthogonal splitting $E\simeq S\oplus Q$ (cf.\ \S V-14). Then $g^\star f$ is a lifting of $f$ in $E\otimes L$. We shall try to find $h$ under the form $$h=g^\star f+ju,~~~~u\in L^2_{n,k}(X,S\otimes L).$$ As the images of $S$ and $Q$ in $E$ are orthogonal, we have $|h|^2=|f|^2+|u|^2$ at every point of $X$. On the other hand $D''_{Q\otimes L}f=0$ by hypothesis and $D''g^\star=-j\circ\beta^\star$ by V-14.3~d), hence $$D''_{E\otimes L}h=-j(\beta^\star\wedge f)+j\,D''_{S\otimes L}= j(D''_{S\otimes L}-\beta^\star\wedge f).$$ We are thus led to solve the equation $$D''_{S\otimes L}u=\beta^\star\wedge f,\leqno(9.4)$$ and for that, we apply Th.~4.5 to the $(n,k+1)$-form $\beta^\star\wedge f$. One observes now that the curvature of $S\otimes L$ can be expressed in terms of $\beta$. This remark will be used to prove: \endproof \begstat{(9.5) Lemma} $\langle A_k^{-1}(\beta^\star\wedge f), (\beta^\star\wedge f)\rangle\le(m/\varepsilon)\,|f|^2$. \endstat If the Lemma is taken for granted, Th.~4.5 yields a solution $u$ of~(9.4) in $L^2_{n,q}(X,S\otimes L)$ such that $\|u\|^2\le(m/\varepsilon)\,\|f\|^2$. As \hbox{$\|h\|^2=\|f\|^2+\|u\|^2$}, the proof of Th.~9.3 is complete.\qed \begproof{of Lemma~9.5.} Exactly as in the proof of Th.~VII-10.3, formulas (V-14.6) and (V-14.7) yield $$\ii\Theta(S)\ge_m \ii\beta^\star\wedge\beta,~~~~ \ii\Theta(\det Q)\ge\Tr_Q(\ii\beta\wedge \beta^\star)=\Tr_S(-\ii\beta^\star\wedge\beta).$$ Since $\ci_{1,1}(X,\Herm\,S)\ni\Theta:=-\ii\beta^\star\wedge\beta \ge_\Grif0$, Prop.~VII-10.1 implies $$m\,\Tr_S(-\ii\beta^\star\wedge\beta)\otimes\Id_S+\ii\beta^\star\wedge \beta\ge_m 0.$$ From the hypothesis on the curvature of $L$ we get $$\eqalign{ \ii\Theta(S\otimes L)&\ge_m \ii\Theta(S)\otimes\Id_L+(m+\varepsilon)\, \ii\Theta(\det Q)\otimes\Id_{S\otimes L}\cr &\ge_m\big(\ii\beta^\star\wedge\beta+(m+\varepsilon)\,\Tr_S(-\ii\beta^\star \wedge\beta)\otimes\Id_S\big)\otimes\Id_L\cr &\ge_m(\varepsilon/m)\,(-\ii\beta^\star\wedge\beta)\otimes\Id_S\otimes \Id_L.\cr}$$ For any $v\in\Lambda^{n,k+1}T^\star_X\otimes S\otimes L$, Lemma~VII-7.2 implies $$\langle A_{k,S\otimes L}v,v\rangle\ge(\varepsilon/m)\, \langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle,\leqno(9.6)$$ because $\rk(S\otimes L)=s$ and $m=\min\{n-k,s\}$. Let $(dz_1\ld dz_n)$ be an orthonormal basis of $T^\star_X$ at a given point $x_0\in X$ and set $$\beta=\sum_{1\le j\le n}dz_j\otimes\beta_j,~~~~\beta_j\in\hom(S,Q).$$ The adjoint of the operator~ $\beta^\star\wedge\bu=\sum d\ovl z_j\wedge \beta_j^\star\,\bu$ ~is the contraction $\beta\ort \bu$ defined by $$\beta\ort v=\sum{\partial\over\partial\ovl z_j}\ort (\beta_j v)= \sum -\ii dz_j\wedge\Lambda(\beta_j v)=-\ii\beta\wedge\Lambda v.$$ We get consequently $\langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle =|\beta\ort v|^2$ and (9.6) implies $$|\langle\beta^\star\wedge f,v\rangle|^2=|\langle f,\beta\ort v\rangle|^2\le |f|^2\,|\beta\ort v|^2\le(m/\varepsilon) \langle A_{k,S\otimes L}v,v\rangle\,|f|^2.\eqno{\square}$$ \endproof If $X$ has a plurisubharmonic exhaustion function $\psi$, we can select a convex increasing function $\chi\in\ci(\bbbr,\bbbr)$ and multiply the metric of $L$ by the weight $\exp(-\chi\circ\psi)$ in order to make the $L^2$ norm of $f$ converge. Theorem 9.3 implies therefore: \begstat{(9.7) Corollary} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler manifold, let $g:E\to Q$ be a surjective bundle morphism with $r=\rk\,E$, $q=\rk\,Q$, let \hbox{$m=\min\{n-k,r-q\}$} and let $L\to X$ be a hermitian line bundle. Suppose that $E\ge_m 0$ and $$\ii\Theta(L)-(m+\varepsilon)\,\ii\Theta(\det Q)\ge 0$$ for some $\varepsilon>0$. Then $g$ induces a surjective map $$H^k(X,K_X\otimes E\otimes L)\lra H^k(X,K_X\otimes Q\otimes L).$$ \endstat The most remarkable feature of this result is that it does not require any strict positivity assumption on the curvature (for instance $E$ can be a flat bundle). A careful examination of the proof shows that it amounts to verify that the image of the coboundary morphism $$-\beta^\star\wedge\bu~:~H^k(X,K_X\otimes Q\otimes L)\lra H^{k+1}(X,K_X\otimes S\otimes L)$$ vanishes; however the cohomology group $H^{k+1}(X,K_X\otimes S\otimes L)$ itself does not vanish in general as it would do under a strict positivity assumption (cf.\ Th.~VII-9.4). We want now to get also estimates when $Q$ is endowed with a metric given a priori, that can be distinct from the quotient metric of $E$ by $g$. Then the map $g^\star(gg^\star)^{-1}~:~Q\lra E$ is the lifting of $Q$ orthogonal to $S=\Ker g$. The quotient metric $|\bu|'$ on $Q$ is therefore defined in terms of the original metric $|\bu|$ by $$|v|^{\prime 2}=|g^\star(gg^\star)^{-1}v|^2=\langle(gg^\star)^{-1}v,v\rangle =\det(gg^\star)^{-1}\,\langle\wt{gg^\star}v,v\rangle$$ where $\wt{gg^\star}\in\End(Q)$ denotes the endomorphism of $Q$ whose matrix is the transposed of the comatrix of $gg^\star$. For every $w\in\det Q$, we find $$|w|^{\prime2}=\det(gg^\star)^{-1}\,|w|^2.$$ If $Q'$ denotes the bundle $Q$ with the quotient metric, we get $$\ii\Theta(\det Q')=\ii\Theta(\det Q)+\ii d'd''\log\det(gg^\star).$$ In order that the hypotheses of Th.~9.3 be satisfied, we are led to define a new metric $|\bu|'$ on $L$ by $|u|^{\prime 2}=|u|^2\,\big(\det(gg^\star) \big)^{-m-\varepsilon}$. Then $$\ii\Theta(L')=\ii\Theta(L)+(m+\varepsilon)\,\ii d'd''\log\det(gg^\star)\ge (m+\varepsilon)\,\ii\Theta(\det Q').$$ Theorem 9.3 applied to $(E,Q',L')$ can now be reformulated: \begstat{(9.8) Theorem} Let $X$ be a complete K\"ahler manifold equipped with a K\"ahler metric $\omega$ on~$X$, let $E\to Q$ be a surjective morphism of hermitian vector bundles and let $L\to X$ be a hermitian line bundle. Set $r=\rk\,E$, $q=\rk\,Q$ and $m=\min\{n-k,r-q\}$ and suppose $E\ge_m 0$, $$\ii\Theta(L)-(m+\varepsilon)\ii\Theta(\det Q)\ge 0$$ for some $\varepsilon>0$. Then for every $D''$-closed form $f$ of type $(n,k)$ with values in $Q\otimes L$ such that $$I=\int_X\langle\wt{gg^\star}f,f\rangle\,(\det gg^\star)^{-m-1-\varepsilon}\, dV<+\infty,$$ there exists a $D''$-closed form $h$ of type $(n,k)$ with values in $E\otimes L$ such that $f=g\cdot h$ and $$\int_X|h|^2\,(\det gg^\star)^{-m-\varepsilon}\,dV \le(1+m/\varepsilon)\,I.\eqno\square$$ \endstat Our next goal is to extend Th.~9.8 in the case when $g~:~E\lra Q$ is only {\it generically} surjective; this means that the analytic set $$Y=\{x\in X~;~g_x~:~E_x\lra Q_x~{\rm is~not~surjective~}\}$$ defined by the equation $\Lambda^q g=0$ is nowhere dense in $X$. Here $\Lambda^q g$ is a section of the bundle $\hom(\Lambda^qE,\det Q)$. \begstat{(9.9) Theorem} The existence statement and the estimates of Th.~$9.8$ remain true for a generically surjective morphism $g:E\to Q$ provided that $X$ is weakly pseudoconvex. \endstat \begproof{} Apply Th.~9.8 to each relatively compact domain $X_c\ssm Y$ (these domains are complete K\"ahler by Lemma~7.2). From a sequence of solutions on $X_c\ssm Y$ we can extract a subsequence converging weakly on $X\ssm Y$ as c tends to $+\infty$. One gets a form $h$ satisfying the estimates, such that $D''h=0$ on $X\ssm Y$ and $f=g\cdot h$. In order to see that $D''h=0$ on $X$, it suffices to apply Lemma~7.3 and to observe that $h$ has $L^2_\loc$ coefficients on $X$ by our estimates.\qed \endproof A very special but interesting case is obtained for the trivial bundles $E=\Omega\times\bbbc^r$, $Q=\Omega\times\bbbc$ over a pseudoconvex open set \hbox{$\Omega\subset\bbbc^n$}. Then the morphism $g$ is given by a $r$-tuple $(g_1\ld g_r)$ of holomorphic functions on $\Omega$. Let us take $k=0$ and $L=\Omega\times\bbbc$ with the metric given by a weight $e^{-\varphi}$. If we observe that $\wt{gg^\star}=\Id$ when $\rk\,Q=1$, Th.~9.8 applied on \hbox{$X=\Omega\ssm g^{-1}(0)$} and Lemmas~7.2, 7.3 give: \begstat{(9.10) Theorem {\rm(Skoda 1978)}} Let $\Omega$ be a complete K\"ahler open subset of $\bbbc^n$ and $\varphi$ a plurisubharmonic function on $\Omega$. Set $m=\min\{n,r-1\}$. Then for every holomorphic function $f$ on $\Omega$ such that $$I=\int_{\Omega\ssm Z}|f|^2\,|g|^{-2(m+1+\varepsilon)}e^{-\varphi}\,dV <+\infty,$$ where $Z=g^{-1}(0)$, there exist holomorphic functions $(h_1\ld h_r)$ on $\Omega$ such that $f=\sum g_jh_j$ and $$\int_{\Omega\ssm Y}|h|^2\,|g|^{-2(m+\varepsilon)}e^{-\varphi}\,dV \le(1+m/\varepsilon)I.\eqno{\square}$$ \endstat This last theorem can be used in order to obtain a quick solution of the Levi problem mentioned in \S I-4. It can be used also to prove a result of (Diederich-Pflug~1981), relating the pseudoconvexity property and the existence of complete K\"ahler metrics for domains of $\bbbc^n$. \begstat{(9.11) Theorem} Let $\Omega\subset\bbbc^n$ be an open subset. Then: \smallskip \item{\rm a)} $\Omega$ is a domain of holomorphy if and only if $\Omega$ is pseudoconvex~$;$ \smallskip \item{\rm b)} If $\omcirc=\Omega$ and if $\Omega$ has a complete K\"ahler metric $\wh\omega$, then $\Omega$ is pseudoconvex.\smallskip \endstat Note that statement b) can be false if the assumption $\omcirc=\Omega$ is omitted: in fact $\bbbc^n\ssm\{0\}$ is complete K\"ahler by Lemma~7.2, but it is not pseudoconvex if $n\ge 2$. \begproof{} b) By Th.~I-4.12, it is enough to verify that $\Omega$ is a domain of holomorphy, i.e.\ that for every connected open subset $U$ such that $U\cap\partial\Omega\ne\emptyset$ and every connected component W of $U\cap\Omega$ there exists a holomorphic function $h$ on $\Omega$ such that $h_{\restriction W}$ cannot be continued to~$U$. Since $\omcirc=\Omega$, the set $U\ssm\ovl\Omega$ is not empty. We select $a\in U\ssm\ovl\Omega$. Then the integral $$\int_\Omega|z-a|^{-2(n+\varepsilon)}\,dV(z)$$ converges. By Th.~9.10 applied to $f(z)=1$, $g_j(z)=z_j-a_j$ and $\varphi=0$, there exist holomorphic functions $h_j$ on $\Omega$ such that $\sum(z_j-a_j)\,h_j(z)=1$. This shows that at least one of the functions $h_j$ cannot be analytically continued at $a\in U$. \medskip \noindent a) Assume that $\Omega$ is pseudoconvex. Given any open connected set $U$ such that $U\cap\partial\Omega\ne\emptyset$, choose $a\in U\cap\partial\Omega$. By Th.~I-4.14~c) the function $$\varphi(z)=(n+\varepsilon)(\log(1+|z|^2)-2\log d(z,\complement\Omega)\big)$$ is plurisubharmonic on $\Omega$. Then the integral $$\int_\Omega|z-a|^{-2(n+\varepsilon)}\,e^{-\varphi(z)}\,dV(z) \le\int_\Omega(1+|z|^2)^{-n-\varepsilon}\,dV(z)$$ converges, and we conclude as for b).\qed \endproof \titleb{10.}{Application of Skoda's $L^2$ Estimates to Local Algebra} We apply here Th.~9.10 to the study of ideals in the ring $\cO_n=\bbbc\{z_1\ld z_n\}$ of germs of holomorphic functions on $(\bbbc^n,0)$. Let $\cI=(g_1\ld g_r)\ne(0)$ be an ideal of $\cO_n$. \begstat{(10.1) Definition} Let $k\in\bbbr_+$. We associate to $\cI$ the following ideals: \smallskip \item{\rm a)} the ideal $\ovl\cI^{(k)}$ of germs $u\in\cO_n$ such that $|u|\le C|g|^k$ for some constant $C\ge 0$, where $|g|^2=|g_1|^2+\cdots+|g_r|^2$. \smallskip \item{\rm b)} the ideal $\wh\cI^{(k)}$ of germs $u\in\cO_n$ such that $$\int_\Omega|u|^2\,|g|^{-2(k+\varepsilon)}\,dV<+\infty$$ on a small ball $\Omega$ centered at $0$, if $\varepsilon>0$ is small enough.\smallskip \endstat \begstat{(10.2) Proposition} For all $k,l\in\bbbr_+$ we have \smallskip \item{\rm a)} $\ovl\cI^{(k)}\subset\wh\cI^{(k)}\,;$ \smallskip \item{\rm b)} $\cI^k\subset\ovl\cI^{(k)}$ if $k\in\bbbn\,;$ \smallskip \item{\rm c)} $\ovl\cI^{(k)}.\ovl\cI^{(l)}\subset\ovl\cI^{(k+l)}\,;$ \smallskip \item{\rm d)} $\ovl\cI^{(k)}.\wh\cI^{(l)}\subset\wh\cI^{(k+l)}.$ \endstat All properties are immediate from the definitions except a) which is a consequence of Lemma~8.2. Before stating the main result, we need a simple lemma. \begstat{(10.3) Lemma} If $\cI=(g_1\ld g_r)$ and $r>n$, we can find elements $\wt g_1\ld\wt g_n\in\cI$ such that $C^{-1}|g|\le|\wt g|\le C|g|$ on a neighborhood of $0$. Each $\wt g_j$ can be taken to be a linear combination $$\wt g_j=a_j.\,g=\sum_{1\le k\le r}a_{jk}g_k,~~~~a_j\in\bbbc^r\ssm\{0\}$$ where the coefficients $([a_1]\ld[a_n])$ are chosen in the complement of a proper analytic subset of $(\bbbp^{r-1})^n$. \endstat It follows from the Lemma that the ideal $\cJ=(\wt g_1\ld\wt g_n)\subset\cI$ satisfies $\smash{\ovl\cJ}^{(k)}=\smash{\ovl\cI}^{(k)}$ and $\smash{\wh\cJ}^{(k)}=\smash{\wh\cI}^{(k)}$ for all $k$. \begproof{} Assume that $g\in\cO(\Omega)^r$. Consider the analytic subsets in $\Omega\times(\bbbp^{r-1})^n$ defined by $$\eqalign{ A&=\big\{(z,[w_1]\ld[w_n])\,;\,w_j.\,g(z)=0\big\},\cr A^\star&=\bigcup{\rm irreducible~components~of}~A~{\rm not~contained~in~} g^{-1}(0)\times(\bbbp^{r-1})^n.\cr}$$ For $z\notin g^{-1}(0)$ the fiber $A_z=\{([w_1]\ld[w_n])\,;\,w_j.\,g(z)=0\} =A^\star_z$ is a product of $n$ hyperplanes in $\bbbp^{r-1}$, hence $A\cap(\Omega\ssm g^{-1}(0))\times(\bbbp^{r-1})^n$ is a fiber bundle with base $\Omega\ssm g^{-1}(0)$ and fiber $(\bbbp^{r-2})^n$. As $A^\star$ is the closure of this set in $\Omega\times(\bbbp^{r-1})^n$, we have $$\dim A^\star=n+n(r-2)=n(r-1)=\dim(\bbbp^{r-1})^n.$$ It follows that the zero fiber $$A^\star_0=A^\star\cap\big(\{0\}\times(\bbbp^{r-1})^n\big)$$ is a proper subset of $\{0\}\times(\bbbp^{r-1})^n$. Choose $(a_1\ld a_n)\in(\bbbc^r\ssm\{0\})^n$ such that \hbox{ $(0,[a_1]\ld[a_n])$} is not in~$A^\star_0$. By an easy compactness argument the set \hbox{$A^\star\cap\big(\ovl B(0,\varepsilon)\times(\bbbp^{r-1})^n\big)$} is disjoint from the neighborhood \hbox{$B(0,\varepsilon)\times\prod[B(a_j,\varepsilon)]$} of \hbox{$(0,[a_1]\ld[a_n])$} for $\varepsilon$ small enough. For $z\in B(0,\varepsilon)$ we have \hbox{$|a_j.\,g(z)|\ge\varepsilon|g(z)|$} for some~$j$, otherwise the inequa\-lity $|a_j.\,g(z)|<\varepsilon|g(z)|$ would imply the existence of $h_j\in\bbbc^r$ with $|h_j|<\varepsilon$ and $a_j.\,g(z)=h_j.\,g(z)$. Since $g(z)\ne 0$, we would have $$(z,[a_1-h_1]\ld[a_n-h_n])\in A^\star\cap\big(B(0,\varepsilon)\times (\bbbp^{r-1})^n\big),$$ a contradiction. We obtain therefore $$\varepsilon|g(z)|\le\max|a_j.\,g(z)|\le(\max|a_j|)\,|g(z)|~~~~{\rm on}~~ B(0,\varepsilon).\eqno{\square}$$ \endproof \begstat{(10.4) Theorem {\rm(Brian\c con-Skoda~1974)}} Set $p=\min\{n-1,r-1\}$. Then \smallskip \item{\rm a)} $\wh\cI^{(k+1)}=\cI\,\wh\cI^{(k)}=\ovl\cI\,\wh\cI^{(k)}$~~ for $k\ge p$. \smallskip \item{\rm b)} $\ovl\cI^{(k+p)}\subset\wh\cI^{(k+p)}\subset\cI^k$~~ for all $k\in\bbbn$. \endstat \begproof{} a) The inclusions $\cI\,\wh\cI^{(k)}\subset\ovl\cI\,\wh\cI^{(k)} \subset\wh\cI^{(k+1)}$ are obvious thanks to Prop.~10.2, so we only have to prove that $\wh\cI^{(k+1)}\subset\cI\,\wh\cI^{(k)}$. Assume first that $r\le n$. Let $f\in\wh\cI^{(k+1)}$ be such that $$\int_\Omega|f|^2\,|g|^{-2(k+1+\varepsilon)}\,dV<+\infty.$$ For $k\ge p-1$, we can apply Th.~9.10 with $m=r-1$ and with the weight $\varphi=(k-m)\log|g|^2$. Hence $f$ can be written $f=\sum g_jh_j$ with $$\int_\Omega|h|^2\,|g|^{-2(k+\varepsilon)}\,dV<+\infty,$$ thus $h_j\in\wh\cI^{(k)}$ and $f\in\cI\,\wh\cI^{(k)}$. When $r>n$, Lemma~10.3 shows that there is an ideal $\cJ\subset\cI$ with $n$ generators such that $\smash{\wh\cJ}^{(k)}=\smash{\wh\cI}^{(k)}$. We find $$\wh\cI^{(k+1)}=\wh\cJ^{(k+1)}\subset\cJ\,\wh\cJ^{(k)}\subset \cI\,\wh\cI^{(k)}~~~~{\rm for}~~k\ge n-1.$$ \medskip \noindent b) Property a) implies inductively $\wh\cI^{(k+p)}=\cI^k\,\wh\cI^{(p)}$ for all $k\in\bbbn$. This gives in particular $\wh\cI^{(k+p)}\subset\cI^k$.\qed \endproof \begstat{(10.5) Corollary} \smallskip \item{\rm a)} The ideal $\ovl\cI$ is the integral closure of $\cI$, i.e.\ by definition the set of germs $u\in\cO_n$ which satisfy an equation $$u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^s,~~~~1\le s\le d.$$ \smallskip \item{\rm b)} Similarly, $\ovl\cI^{(k)}$ is the set of germs $u\in\cO_n$ which satisfy an equation $$u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^{]ks[},~~~~1\le s\le d,$$ where $]t[$ denotes the smallest integer $\ge t$.\smallskip \endstat As the ideal $\ovl\cI^{(k)}$ is finitely generated, property b) shows that there always exists a rational number $l\ge k$ such that $\smash{\ovl\cI}^{(l)}=\smash{\ovl\cI}^{(k)}$. \begproof{} a) If $u\in\cO_n$ satisfies a polynomial equation with coefficients $a_s\in\cI^s$, then clearly $|a_s|\le C_s\,|g|^s$ and Lemma~II-4.10 implies $|u|\le C\,|g|$. Conversely, assume that $u\in\ovl\cI$. The ring $\cO_n$ is Noetherian, so the ideal $\smash{\wh\cI}^{(p)}$ has a finite number of generators $v_1\ld v_N$. For every $j$ we have $uv_j\in\ovl\cI\,\smash{\wh\cI}^{(p)}= \cI\,\smash{\wh\cI}^{(p)}$, hence there exist elements $b_{jk}\in\cI$ such that $$uv_j=\sum_{1\le k\le N}b_{jk}v_k.$$ The matrix $(u\delta_{jk}-b_{jk})$ has the non zero vector $(v_j)$ in its kernel, thus $u$ satisfies the equation $\det(u\delta_{jk}-b_{jk})=0$, which is of the required type. \medskip \noindent b) Observe that $v_1\ld v_N$ satisfy simultaneously some integrability condition $\int_\Omega|v_j|^{-2(p+\varepsilon)}<+\infty$, thus $\smash{\wh\cI}^{(p)}=\smash{\wh\cI}^{(p+\eta)}$ for $\eta\in[0,\varepsilon[$. Let $u\in\smash{\ovl\cI}^{(k)}$. For every integer $m\in\bbbn$ we have $$u^mv_j\in\ovl\cI^{(km)}\,\wh\cI^{(p+\eta)}\subset \wh\cI^{(km+\eta+p)}.$$ If $k\notin\bbbq$, we can find $m$ such that $d(km+\varepsilon/2,\bbbz) <\varepsilon/2$, thus $km+\eta\in\bbbn$ for some $\eta\in{}]0,\varepsilon[$. If $k\in\bbbq$, we take $m$ such that $km\in\bbbn$ and $\eta=0$. Then $$u^mv_j\in\wh\cI^{(N+p)}=\cI^N\,\wh\cI^{(p)}~~~~{\rm with}~~ N=km+\eta\in\bbbn,$$ and the reasoning made in a) gives $\det(u^m\delta_{jk}-b_{jk})=0$ for some \hbox{$b_{jk}\in\cI^N$}. This is an equation of the type described in b), where the coefficients $a_s$ vanish when $s$ is not a multiple of $m$ and $a_{ms}\in\cI^{Ns}\subset\cI^{]kms[}$.\qed \endproof Let us mention that Brian\c con and Skoda's result 10.4~b) is optimal for $k=1$. Take for example $\cI=(g_1\ld g_r)$ with $g_j(z)=z_j^r$, $1\le j\le r$, and $f(z)=z_1\ldots z_r$. Then $|f|\le C|g|$ and 10.4~b) yields $f^r\in\cI\,;$ however, it is easy to verify that $f^{r-1}\notin\cI$. The theorem also gives an answer to the following conjecture made by J.~Mather. \begstat{(10.6) Corollary} Let $f\in\cO_n$ and $\cI_f=(z_1\partial f/ \partial z_1\ld z_n\partial f/\partial z_n)$. Then $f\in\ovl\cI_f$, and for every integer $k\ge 0$, $f^{k+n-1}\in\cI^k_f$. \endstat The Corollary is also optimal for $k=1\,:$ for example, one can verify that the function $f(z)=(z_1\ldots z_n)^3+z_1^{3n-1}+\ldots+z_n^{3n-1}$ is such that $f^{n-1}\notin\cI_f$. \begproof{} Set $g_j(z)=z_j\,\partial f/\partial z_j$, $1\le j\le n$. By 10.4~b), it suffices to show that $|f|\le C|g|$. For every germ of analytic curve $\bbbc\ni t\longmapsto\gamma(t)$, $\gamma\not\equiv 0$, the vanishing order of $f\circ\gamma(t)$ at $t=0$ is the same as that of $$t\,{d(f\circ\gamma)\over dt}=\sum_{1\le j\le n}t\,\gamma_j'(t)\, {\partial f\over\partial z_j}\big(\gamma(t)\big).$$ We thus obtain $$|f\circ\gamma(t)|\le C_1\,|t|~\Big|{d(f\circ\gamma)\over dt}\Big| \le C_2\,\sum_{1\le j\le n}|t\,\gamma_j'(t)|~\Big| {\partial f\over\partial z_j}\big(\gamma(t)\big)\Big| \le C_3\,|g\circ\gamma(t)|$$ and conclude by the following elementary lemma.\qed \endproof \begstat{(10.7) Lemma} Let $f,g_1\ld g_r\in\cO_n$ be germs of holomorphic functions vanishing at $0$. Then we have $|f|\le C|g|$ for some constant $C$ if and only if for every germ of analytic curve $\gamma$ through $0$ there exists a constant $C_\gamma$ such that \hbox{$|f\circ\gamma|\le C_\gamma|g\circ\gamma|$}. \endstat \begproof{} If the inequality $|f|\le C|g|$ does not hold on any neighborhood of $0$, the germ of analytic set $(A,0)\subset(\bbbc^{n+r},0)$ defined by $$g_j(z)=f(z)z_{n+j},~~~~1\le j\le r,$$ contains a sequence of points $\big(z_\nu,g_j(z_\nu)/f(z_\nu)\big)$ converging to $0$ as $\nu$ tends to $+\infty$, with $f(z_\nu)\ne 0$. Hence $(A,0)$ contains an irreducible component on which $f\not\equiv 0$ and there is a germ of curve \hbox{$\wt\gamma=(\gamma,\gamma_{n+j}): (\bbbc,0)\to(\bbbc^{n+r},0)$} contained in $(A,0)$ such that $f\circ\gamma \not\equiv 0$. We get \hbox{$g_j\circ\gamma=(f\circ\gamma)\gamma_{n+j}$}, hence \hbox{$|g\circ\gamma(t)|\le C|t|\,|f\circ\gamma(t)|$} and the inequality \hbox{$|f\circ\gamma|\le C_\gamma|g\circ\gamma|$} does not hold.\qed \endproof \titleb{11.}{Integrability of Almost Complex Structures} Let $M$ be a $\ci$ manifold of real dimension $m=2n$. An {\it almost complex structure} on $M$ is by definition an endomorphism $J\in\End(TM)$ of class $\ci$ such that $J^2=-\Id$. Then $TM$ becomes a complex vector bundle for which the scalar multiplication by $i$ is given by $J$. The pair $(M,J)$ is said to be an {\it almost complex manifold}. For such a manifold, the complexified tangent space $T_\bbbc M=\bbbc\otimes_\bbbr TM$ splits into conjugate complex subspaces $$T_\bbbc M=T^{1,0}M\oplus T^{0,1}M,~~~~\dim_\bbbc T^{1,0}M=\dim_\bbbc T^{0,1}M=n, \leqno(11.1)$$ where $T^{1,0}M$, $T^{0,1}M\subset T_\bbbc M$ are the eigenspaces of $\Id\otimes J$ corresponding to the eigenvalues $i$ and $-i$. The complexified exterior algebra $\bbbc\otimes_\bbbr\Lambda^\bu T^\star M= \Lambda^\bu T_\bbbc^\star M$ has a corresponding splitting $$\Lambda^k T^\star_\bbbc M=\bigoplus_{p+q=k}\Lambda^{p,q} T_\bbbc^\star M \leqno(11.2)$$ where we denote by definition $$\Lambda^{p,q}T^\star_\bbbc M=\Lambda^p(T^{1,0}M)^\star\otimes_\bbbc \Lambda^q(T^{0,1}M)^\star.\leqno(11.3)$$ As for complex manifolds, we let $C^s_{p,q}(M,E)$ be the space of differential forms of class $C^s$ and bidegree $(p,q)$ on $M$ with values in a complex vector bundle $E$. There is a natural antisymmetric bilinear map $$\theta~:~~\ci(M,T^{1,0}M)\times\ci(M,T^{1,0}M)\lra\ci(M,T^{0,1}M)$$ which associates to a pair $(\xi,\eta)$ of $(1,0)$-vector fields the $(0,1)$-component of the Lie bracket $[\xi,\eta]$. Since $$[\xi,f\eta]=f[\xi,\eta]+(\xi.f)\,\eta,~~~~\forall f\in\ci(M,\bbbc)$$ we see that $\theta(\xi,f\eta)=f\,\theta(\xi,\eta)$. It follows that $\theta$ is in fact a $(2,0)$-form on $M$ with values in $T^{0,1}M$. If $M$ is a {\it complex analytic manifold} and $J$ its natural almost complex structure, we have in fact $\theta=0$, because $[\partial/\partial z_j,\partial/\partial z_k]=0$, $1\le j,k\le n$, for any holomorphic local coordinate system $(z_1\ld z_n)$. \begstat{(11.4) Definition} The form $\theta\in\ci_{2,0}(M,T^{0,1}M)$ is called the torsion form of $J$. The almost complex structure $J$ is said to be integrable if $\theta=0$. \endstat \begstat{(11.5) Example} \rm If $M$ is of real dimension $m=2$, every almost complex structure is integrable, because $n=1$ and alternate $(2,0)$-forms must be zero. Assume that $M$ is a smooth oriented surface. To any Riemannian metric $g$ we can associate the endomorphism $J\in\End(TM)$ equal to the rotation of $+\pi/2$. A change of orientation changes $J$ into the conjugate structure $-J$. Conversely, if $J$ is given, $TM$ is a complex line bundle, so $M$ is oriented, and a Riemannian metric $g$ is associated to $J$ if and only if $g$ is $J$-hermitian. As a consequence, there is a one-to-one correspondence between conformal classes of Riemannian metrics on $M$ and almost complex structures corresponding to a given orientation.\qed \endstat If $(M,J)$ is an almost complex manifold and $u\in\ci_{p,q}(M,\bbbc)$, we let $d'u,~d''u$ be the components of type $(p+1,q)$ and $(p,q+1)$ in the exterior derivative $du$. Let $(\xi_1\ld\xi_n)$ be a frame of $T^{1,0}M_{\restriction\Omega}$. The torsion form $\theta$ can be written $$\theta=\sum_{1\le j\le n}\alpha_j\otimes\ovl\xi_j,~~~~ \alpha_j\in\ci_{2,0}(\Omega,\bbbc).$$ Then $\theta$ yields conjugate operators $\theta',\theta''$ on $\Lambda^\bu T_\bbbc^\star M$ such that $$\theta'u=\sum_{1\le j\le n}\alpha_j\wedge(\ovl\xi_j\ort\,u),~~~~ \theta''u=\sum_{1\le j\le n}\ovl\alpha_j\wedge(\xi_j\ort\,u).\leqno(11.6)$$ If $u$ is of bidegree $(p,q)$, then $\theta'u$ and $\theta''u$ are of bidegree $(p+2,q-1)$ and $(p-1,q+2)$. It is clear that $\theta'$, $\theta''$ are derivations, i.e.\ $$\theta'(u\wedge v)=(\theta'u)\wedge v+(-1)^{\deg\,u}u\wedge(\theta'v)$$ for all smooth forms $u,v$, and similarly for $\theta''$. \begstat{(11.7) Proposition} We have~ $d=d'+d''-\theta'-\theta''$. \endstat \begproof{} Since all operators occuring in the formula are derivations, it is sufficient to check the formula for forms of degree 0 or~1. If $u$ is of degree 0, the result is obvious because $\theta'u=\theta''u=0$ and $du$ can only have components of types $(1,0)$ or $(0,1)$. If $u$ is a $1$-form and $\xi,\eta$ are complex vector fields, we have $$du(\xi,\eta)=\xi.u(\eta)-\eta.du(\xi)-u([\xi,\eta]).$$ When $u$ is of type $(0,1)$ and $\xi,\eta$ of type $(1,0)$, we find $$(du)^{2,0}(\xi,\eta)=-u\big(\theta(\xi,\eta)\big)~$$ thus $(du)^{2,0}=-\theta'u$, and of course $(du)^{1,1}=d'u$, $(du)^{0,2}=d''u$, $\theta''u=0$ by definition. The case of a $(1,0)$-form $u$ follows by conjugation.\qed \endproof Proposition 11.7 shows that $J$ is integrable if and only if $d=d'+d''$. In this case, we infer immediately $$d^{\prime 2}=0,~~~~d'd''+d''d'=0,~~~~d^{\prime\prime 2}=0.$$ For an integrable almost complex structure, we thus have the same formalism as for a complex analytic structure, and indeed we shall prove: \begstat{(11.8) Newlander-Nirenberg theorem (1957)} Every integrable almost complex structure $J$ on $M$ is defined by a unique analytic structure. \endstat The proof we shall give follows rather closely that of (H\"ormander~1966), which was itself based on previous ideas of (Kohn~1963, 1964). A function $f\in C^1(\Omega,\bbbc)$, $\Omega\subset M$, is said to be $J$-holomorphic if $d''f=0$. Let $f_1\ld f_p\in C^1(\Omega,\bbbc)$ and let $h$ be a function of class $C^1$ on an open subset of $\bbbc^p$ containing the range of $f=(f_1\ld f_p)$. An easy computation gives $$d''(h\circ f)=\sum_{1\le j\le p}\Big({\partial h\over\partial z_j} \circ f\Big)d''f_j+\Big({\partial h\over\partial\ovl z_j}\circ f\Big) \ovl{d'f_j},\leqno(11.9)$$ in particular $h\circ f$ is $J$-holomorphic as soon as $f_1\ld f_p$ are $J$-holomorphic and $h$ holomorphic in the usual sense. Constructing a complex analytic structure on $M$ amounts to show the existence of $J$-holomorphic complex coordinates $(z_1\ld z_n)$ on a neighborhood $\Omega$ of every point $a\in M$. Formula (11.9) then shows that all coordinate changes $h:(z_k)\mapsto(w_k)$ are holomorphic in the usual sense, so that $M$ is furnished with a complex analytic atlas. The uniqueness of the analytic structure associated to $J$ is clear, since the holomorphic functions are characterized by the condition $d''f=0$. In order to show the existence, we need a lemma. \begstat{(11.10) Lemma} For every point $a\in M$ and every integer $s\ge 1$, there exist $\ci$ complex coordinates $(z_1\ld z_n)$ centered at $a$ such that $$d''z_j={\rm O}(|z|^s),~~~~1\le j\le n.$$ \endstat \begproof{} By induction on $s$. Let $(\xi^\star_1\ld\xi^\star_n)$ be a basis of $\Lambda^{1,0}T^\star_\bbbc M$. One can find complex functions $z_j$ such that $dz_j(a)=\xi^\star_j$, i.e.\ $$d'z_j(a)=\xi^\star_j,~~~~d''z_j(a)=0.$$ Then $(z_1\ld z_n)$ satisfy the conclusions of the Lemma~for $s=1$. If $(z_1\ld z_n)$ are already constructed for the integer $s$, we have a Taylor expansion $$d''z_j=\sum_{1\le k\le n}P_{jk}(z,\ovl z)\,\ovl{d'z_k}+{\rm O}(|z|^{s+1})$$ where $P_{jk}(z,w)$ is a homogeneous polynomial in $(z,w)\in\bbbc^n\times \bbbc^n$ of total degree $s$. As $J$ is integrable, we have $$\eqalign{ 0=d^{\prime\prime 2}z_j&=\sum_{1\le k,l\le n}{\partial P_{jk}\over \partial z_l}\,d''z_l\wedge\ovl{d'z_k}+{\partial P_{jk}\over \partial\ovl z_l}\,\ovl{d'z_l}\wedge\ovl{d'z_k}+{\rm O}(|z|^s)\cr &=\sum_{1\le k