% Lecture notes on $L^2$ estimates
% Ecole d'\'et\'e de Math\'ematiques "Analyse Complexe" 1996
% J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France
        
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\titlea{}{$L^2$ estimates for the $\ovlp$-operator\medskip\noindent
on complex manifolds}
\leftrunningtitle{$L^2$ estimates for the $\ovlp$-operator on complex
manifolds}
\vskip15pt\noindent
{\tbfontt Jean-Pierre Demailly}
\vskip10pt\noindent
{\bf Universit\'e de Grenoble I\newline
Laboratoire de Math\'ematiques, UMR 5582 du CNRS\newline
Institut Fourier, BP74, 38402 Saint-Martin d'H\`eres, France}
\vskip55pt

\begpet
\titled{Abstract}
The main goal of these notes is to describe a powerful differential
geometric method which yields precise existence theorems for 
solutions of equations $\smash{\ovlp u}=v$ on (pseudoconvex) complex manifolds.
The main idea is to combine Hilbert space techniques with a
geometric identity known as the Bochner-Kodaira-Nakano identity. The BKN
identity relates the complex Laplace operators $\Delta'$ and $\Delta''$
associated to $\partial$ and $\ovlp$ with a suitable curvature tensor.
The curvature tensor reflects the convexity properties of the manifold,
from the viewpoint of complex geometry. In this way, under suitable
convexity assumptions, one is able to derive existence theorems for
holomorphic functions subject to certain constraints (in the form of
$L^2$ estimates). The central ideas go back to Kodaira and Nakano
(1954) in the case of compact manifolds, and to Androtti-Vesentini and
H\"ormander (1965) in the case of open manifolds with plurisubharmonic
weights. H\"ormander's estimates can be used for instance to give a
quick solution of the Levi problem. They have many other important
applications to complex analysis, complex geometry, local algebra and
algebraic geometry$\,\ldots~$ Important variants of these
estimates have been developped in the last two decades. The first ones
are the $L^2$ estimates of Skoda (1972, 1978), which deal with the problem
of solving ``Bezout identities'' $\smash{\sum f_jg_j}=h$ when $g_j$
and $h$ are given holomorphic functions and the $f_j$'s are the
unknowns. The last ones are the $\smash{L^2}$ estimates of Ohsawa-Takegoshi
(1987), which concern the problem of extending a holomorphic function
given on a submanifold $Y\subset X$ to the whole manifold~$X$. Our
task will be to explain the main techniques leading to all three types
of $L^2$-estimates (H\"ormander, Skoda, Ohsawa-Takegoshi), and to
present a few applications.
\vskip30pt
\noindent{\bf Contents}
\smallskip
\sectref{0} \sectref{1} \sectref{2} \sectref{3} \sectref{4} \sectref{5}
\sectref{6} \sectref{7} \sectref{8} \sectref{9} \sectref{10} \sectref{11}
\sectref{12} \sectref{13} \sectref{14} \sectref{15} \sectref{16}
\endpet

\titleb{1.}{Non bounded operators on Hilbert spaces}
A few preliminary results of functional analysis are needed.
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 $v=0$. This implies $(\Dom T^\star)^\perp=0$, hence
$T^\star$ is densely defined and our discussion yields

\begstat{(1.1) Theorem {\rm(Von Neumann 1933)}} If $T:\cH_1\lra\cH_2$ is
a closed and densely defined operator, 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.}{Basic concepts of complex analysis in several variables} 
For more details on the concepts introduced here, we refer to Thierry 
Bouche's lecture notes. Let $X$ be a $n$-dimensional complex manifold
and let $(z_1,\ldots,z_n)$ be holomorphic local coordinates on some
open set $\Omega\subset X$ (we usually think of $\Omega$ as being
just an open set in $\bbbc^n$). We write $z_j=x_j+\ii y_j$ and
$$
dz_j=dx_j+\ii dy_j,\qquad d\ovl z_j=dx_j-\ii dy_j.\leqno(2.1)
$$
(Complex) differential forms over $X$ can be defined to be linear 
combinations
$$
\sum c_{\alpha_1\ldots\alpha_\ell,\beta_1\ldots\beta_m}
dx_{\alpha_1}\wedge\cdots\wedge dx_{\alpha_\ell}\wedge
dy_{\beta_1}\wedge\cdots\wedge dy_{\beta_m}
$$ 
with complex coefficients.
Since $dx_j={1\over 2}(dz_j+d\ovl z_j)$ and $dy_j={1\over 2\ii}
(dz_j-d\ovl z_j)$, we can rearrange the wedge products as products in
the complex linear forms $dz_j$ (such that $dz_j(\xi)=\xi_j$) and the
conjugate linear forms $d\ovl z_j$ (such that $d\ovl
z_j(\xi)=\ovl\xi_j$). A $(p,q)$-form is a differential form of total
degree $p+q$ with complex coefficients, which can be written as
$$
u(z)=\sum_{|I|=p,\,|J|=q}u_{IJ}(z)\,dz_I\wedge d\ovl z_J,
\leqno(2.2)
$$
where $I=(i_1\ld i_p)$ and $J=(j_1\ld j_q)$ are multiindices (arranged
in increasing order) and
$$
dz_I=dz_{i_1}\wedge\cdots\wedge dz_{i_p},\qquad
d\ovl z_J=d\ovl z_{j_1}\wedge\cdots\wedge d\ovl z_{j_q}.
$$
The vector bundle of complex valued $(p,q)$-forms over~$X$ will be
denoted by $\Lambda^{p,q}T^\star_X$. In this setting,
the differential of a $C^1$ function $f$ can be expressed as
$$
df=\sum_{1\le j\le n}{\partial f\over\partial x_j}dx_j+
{\partial f\over\partial y_j}dy_j
=\sum_{1\le j\le n}{\partial f\over\partial z_j}dz_j+
{\partial f\over\partial\ovl z_j}d\ovl z_j
$$
where
$$
{\partial f\over\partial z_j}={1\over 2}\Big(
{\partial f\over\partial x_j}-\ii{\partial f\over\partial y_j}\Big),
\qquad
{\partial f\over\partial\ovl z_j}={1\over 2}\Big(
{\partial f\over\partial x_j}+\ii{\partial f\over\partial y_j}\Big).
$$
We thus get $df=d'f+d''f$ (or $df=\partial f+\ovlp f$ in British-American
style), where
$$
d'f=\sum_{1\le j\le n}{\partial f\over\partial z_j}dz_j,\qquad\hbox{resp.}~~
d''f=\sum_{1\le j\le n}{\partial f\over\partial\ovl z_j}d\ovl z_j
$$
is $\bbbc$-linear (resp.\ conjugate $\bbbc$-linear). We say that $f$ is 
holomorphic if $df$ is $\bbbc$-linear, or, in an equivalent way, if
$d''f=0$ ({\it Cauchy-Riemann equation}). More generally, the exterior
derivative $du$ of the $(p,q)$-form $u$ is
$$
du=\sum_{|I|=p,\,|J|=q,\,1\le k\le n}
\Big({\partial u_{IJ}\over\partial z_k}dz_k+
{\partial u_{IJ}\over\partial\ovl z_k}d\ovl z_k\Big)dz_I\wedge d\ovl z_J.
$$
We may therefore write $du=d'u+d''u$ with uniquely defined forms $d'u$ 
of type $(p+1,q)$ and $d''u$ of type $(p,q+1)$, such that
$$
\leqalignno{
d'u&=\sum_{|I|=p,\,|J|=q,\,1\le k\le n}{\partial u_{IJ}\over
\partial z_k}dz_k\wedge dz_I\wedge d\ovl z_J,&(2.3')\cr
d''u&=\sum_{|I|=p,\,|J|=q,\,1\le k\le n}{\partial u_{IJ}\over
\partial\ovl z_k}d\ovl z_k\wedge dz_I\wedge d\ovl z_J.&(2.3'')\cr}
$$
The operators $d''=\ovlp$ can be viewed as linear differential operators 
acting on the bundles of complex $(p,q)$-forms (see \S$\,$4). 
As 
$$
0=d^2=(d'+d'')^2=d^{\prime 2}+(d'd''+d''d')+d^{\prime\prime 2}
$$
where each of the three components are of different types, we get the 
identities
$$
d^{\prime 2}=0,\qquad d^{\prime\prime 2}=0,\qquad d'd''+d''d'=0.
\leqno(2.4)
$$
Moreover, $d'$ and $d''$ are {\it conjugate}, i.e.,
$\ovl{d'u}=d''\ovl u$ for any $(p,q)$-form $u$ on $X$. A basic result is
the so-called Dolbeault-Grothendieck lemma, which is the complex
analogue of the Poincar\'e lemma.

\begstat{(2.5) 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 $\Omega=D(0,R)=
D(0,R_1)\times\cdots\times D(0,R_n)$ in~$\bbbc^n$.
Then there is a smooth $(0,q-1)$-form $u$ on $\Omega$ such that
$d''u=v$ on $\Omega$.
\endstat

\begproof{} We first show that a solution $u$ exists on any smaller
polydisk $D(0,r)\compact\Omega$, \hbox{$r_j<R_j$}. Let $k$ be
the smallest integer such that the monomials $d\ovl z_J$ appearing in
$v$ only involve $d\ovl z_1$, $\ldots$, $d\ovl z_k$. We prove by induction
on $k$ that the equation $d''u=v$ can be solved on the polydisk
$D(0,r)$. If $k=0$, then $v=0$ and there is nothing to prove, whilst $k=n$
is the desired result. Now, assume that the result has been settled
for $k-1$, that $v$ only involves $d\ovl z_1\ld d\ovl z_k$, and set
$$
v=d\ovl z_k\wedge f+g,\qquad f=\sum_{|J|=q-1}f_Jd\ovl z_J,\quad
g=\sum_{|J|=q}g_Jd\ovl z_J
$$ 
where $f$, $g$ only involve $d\ovl z_1$, $\ldots$, $d\ovl z_{k-1}$. 
The assumption $d''v=0$ implies
$$
d''v=-d\ovl z_k\wedge d''f+d''g=0
$$
where $d\ovl z_k\wedge d''f$ involves terms $\partial f_J/\partial\ovl z_\ell~
d\ovl z_k\wedge d\ovl z_\ell\wedge d\ovl z_J$, $\ell>k$, and $d''g$ can 
only involve one factor $d\ovl z_\ell$ with an index $\ell\ge k$. From this
we conclude that $\partial f_J/\partial z_\ell=0$ for $\ell>k$. Hence the
coefficients $f_J$ are holomorphic in $z_{k+1}\ld z_n$. Now, let us
consider the $(0,q-1)$-form
$$
F=\sum_{|J|=q-1}F_Jd\ovl z_J,~~~~F_J(z)=\big(\psi(z_k)f_J(z)\big)\star_k
\Big({1\over \pi z_k}\Big),
$$
where $\psi(z_k)$ is a cut-off function with support in $D(0,R_k)$, 
equal to 1 on some disk $D(0,r'_k)$, $r'_k\in{}]r_k,R_k[$, and
$\star_k$ denotes a partial convolution with respect to~$z_k$.
In other words,
$$
\eqalign{
F_J(z)&=\int_{w\in D(0,R_j)}\psi(w)f_J(z_1\ld z_{k-1},w,z_{k+1}\ld
z_n){1\over\pi(z_k-w)}d\lambda(w).\cr
&=\int_{w\in\bbbc}\psi(z_k-w)f_J(z_1\ld z_{k-1},z_k-w,z_{k+1}\ld
z_n){1\over\pi w}d\lambda(w).\cr}
$$
It follows from differentiation under integral sign that $F_J$ is a smooth
function on $\Omega$ which is holomorphic in all variables $z_{k+1}\ld z_n$.
Moreover, as ${1\over\pi z}$ is a fundamental
solution of ${\partial\over\partial\ovl z}$ in $\bbbc$ (that is,
${\partial\over\partial\ovl z}({1\over\pi z})=\delta_0)$, we see
that
$$
{\partial\over\partial\ovl z_k}F_J(z)=\psi(z_k)f_J(z_1\ld z_{k-1},z_k,
z_{k+1}\ld z_n),
$$
in particular ${\partial\over\partial\ovl z_k}F_J=f_J$ on some
polydisk $D(0,r')$, $r'_j\in{}]r_j,R_j[$. Therefore
$$
d''F=\sum_{|J|=q-1,\,1\le\ell\le k}{\partial f_J\over\partial\ovl z_\ell}
d\ovl z_\ell\wedge d\ovl z_J=d\ovl z_k\wedge f+g_1
$$
where $g_1$ is a $(0,q)$ form which only involves $d\ovl z_1\ld d\ovl
z_{k-1}$. Hence
$$
v_1:=v-d''F=(d\ovl z_k\wedge f+g)-(d\ovl z_k\wedge f+g_1)=g-g_1
$$
only involves $d\ovl z_1\ld d\ovl z_{k-1}$. As $v_1$ is again a $d''$-closed
form, the induction hypothesis applied on $D(0,r')$ shows
that we can find a smooth $(0,q-1)$-form $u_1$ on $D(0,r)$ such that
$d''u_1=v_1$. Therefore $v=d''(F+u_1)$ on $D(0,r)$, and we have thus
found a solution $u=F+u_1$ on $D(0,r)\compact\Omega$.

To conclude the proof, we now show by induction on $q$ that one can find
a solution $u$ defined on all of $\Omega=D(0,R)$. Set $R_\nu=(R_1-2^{-\nu}\ld
R_n-2^{-\nu})$. By what we have proved above, there exists a smooth solution
$u_\nu\in D(0,R_{(\nu)})$ of the equation $d''u_\nu=v$. Now, if $q=1$,
we get $d''(u_{\nu+1}-u_\nu)=0$ on $D(0,R_{(\nu)})$, i.e., $u_{\nu+1}-u_\nu$
is holomorphic on $D(0,R_{(\nu)})$. By looking at its Taylor expansion
at $0$, we get a polynomial $P_\nu$ (equal to the sum of all terms in
the Taylor expansion up to a certain degree) such that
$|u_{\nu+1}-u_\nu-P_\nu|\le 2^{-\nu}$ on $D(0,R_{(\nu-1)})\compact
D(0,R_{(\nu)})$. If we set $\wt u_\nu=u_\nu+P_1+\cdots+P_{\nu-1}$, then
$\wt u_\nu$ is a uniform Cauchy sequence on every compact subset
of $D(0,R)$. Since $\wt u_{\nu+1}-\wt u_\nu$ is holomorphic on
$D(0,R_{(\nu)})$, we conclude that the limit $u$ is smooth and satisfies
$d''u=d''u_\nu=v$ on $D(0,R_{(\nu)})$ for every $\nu$, QED. Now, if $q\ge 2$,
the difference $u_{\nu+1}-u_\nu$ is $d''$-closed of degree $q-1\ge 1$
on $D(0,R_{(\nu)})$. Hence, by the induction  hypothesis, we can find
a $(0,q-2)$ form $w_\nu$ on $D(0,R_{(\nu)})$ such that $u_{\nu+1}-u_\nu=
d''w_\nu$. If we replace inductively $u_{\nu+1}$ by $u_{\nu+1}-
d''(\psi_\nu w_\nu)$ where $\psi_\nu$ is a cut-off function
with support in $D(0,R_{(\nu)})$ equal to $1$ on $D(0,R_{(\nu-1)})$, we see
that we take arrange the sequence so that $u_{\nu+1}$ coincides with
$u_\nu$ on $D(0,R_{(\nu-1)})$. Hence we get a stationary sequence
converging towards a limit $u$ such that $d''u=v$.\qed
\endproof 

We now introduce the concept of cohomology group. A {\it differential complex}
is a graded module $K^\bu=\bigoplus_{q\in\bbbz}K^q$ over some (commutative)
ring~$R$, together with a differential $d:K^\bu\to K^\bu$ of degree~$1$,
that is, a $R$-linear map such that $d=d^q:K^q\to K^{q+1}$ on $K^q$
and $d^2=0$ (i.e., $d^{q+1}\circ d^q=0$ for every~$q$). One defines
the {\it cocycle} and {\it coboundary} modules to be
$$
\leqalignno{
Z^q(K^\bu)&=\Ker(d^q:K^q\to K^{q+1}),&(2.6_Z)\cr
B^q(K^\bu)&=\Im(d^{q-1}:K^{q-1}\to K^q).&(2.6_B)\cr}
$$
The assumption $d^2=0$ immediately shows that $B^q(K^\bu)\subset Z^q(K^\bu)$,
and one defines the {\it $q$-th cohomology group} of $K^\bu$ to be
$$
H^q(K^\bu)=Z^q(K^\bu)/B^q(K^\bu).\leqno(2.7)
$$
A basic example is the {\it De Rham complex} $K^q=C^\infty(X,
\Lambda^qT^\star_X)$ together with the exterior derivative~$d$, defined
whenever $X$ is a smooth differentiable manifold. Its cohomology groups
are denoted $H^q_\DR(X,\bbbr)$ (resp.\ $H^q_\DR(X,\bbbc)$ in the case of 
complex valued forms) and are called the De Rham cohomology groups of~$X$. 
Here, we will be rather concerned with the complex case. If $X$ is a 
complex $n$-dimensional manifold, we consider for each integer $p$ fixed 
the {\it Dolbeault complex} $(K^{p,\bu},d'')$ defined by
$K^{p,q}=C^\infty(X,\Lambda^{p,q}T^\star_X)$ together with the
$d''$-exterior differential; its cohomology groups $H^{p,q}(X)$
are called the Dolbeault cohomology groups of~$X$. More generally, let us
consider a holomorphic vector bundle $E\to X$. This means that we have
a collection of trivializations $E_{\restriction U_j}\simeq U_j\times\bbbc^r$,
$r=\rank E$, such that the transition matrices $g_{jk}(z)$ are holomorphic.
We consider the complex $K^{p,q}_E=C^\infty(X, \Lambda^{p,q}T^\star_X
\otimes E)$ of $E$-valued smooth $(p,q)$-forms with values in~$E$. Again,
$K^{p,q}_E$ possesses a canonical $d''$-operator. Indeed, if $u$ is
a smooth $(p,q)$-section of $E$ represented by forms
$u_j\in C^\infty(U_j,\Lambda^{p,q}T^\star_X\otimes\bbbc^r)$ over the open 
sets~$U_j$, we have the transition relation $u_j=g_{jk}u_k\,$; this
relation implies $d''u_j=g_{jk}d''u_k$ (since $d''g_{jk}=0$),
hence the collection $(d''u_j)$ defines a unique global $(p,q+1)$-section
$d''u$. By definition, the Dolbeault cohomology groups of $X$ with values
in $E$ are
$$
H^{p,q}(X,E)=H^q(K^{p,\bu}_E,d'').\leqno(2.8)
$$
An important observation is that the Dolbeault complex $K^{p,\bu}_E$
is identical to the Dolbeault complex $K^{0,\bu}_{\Lambda^pT^\star_X
\otimes E}$, thanks to the obvious equality
$$
\Lambda^{p,q}T^\star_X\otimes E=
\Lambda^{0,q}T^\star_X\otimes(\Lambda^pT^\star_X\otimes E)
$$
and the fact that $\Lambda^pT^\star_X$ is itself a holomorphic vector 
bundle. In particular, we get an equality
$$
H^{p,q}(X,E)=H^{0,q}(X,\Lambda^pT^\star_X\otimes E).\leqno(2.9)
$$
If $X=\Omega$ is an open subset of $\bbbc^n$, the bundle $\Lambda^p
T^\star_\Omega\simeq \cO_\Omega^{\oplus{n\choose n}}$ is isomorphic to a 
direct sum of ${n\choose p}$ copies of the trivial line bundle
$\cO_\Omega$, hence we simply get
$$
H^{p,q}(\Omega,E)=H^{0,q}(\Omega,E)\otimes_\bbbc\Lambda^p(\bbbc^n)^\star
=H^{0,q}(\Omega,E)^{\oplus{n\choose p}}.
$$
In this setting, the Dolbeault-Grothendieck lemma can be restated:

\begstat{(2.10) Corollary} On every polydisk $D(0,R)=
D(0,R_1)\times\cdots\times D(0,R_n)\subset\bbbc^n$, we have
\hbox{$H^{p,q}(D(0,R),\cO_{D(0,R)})=0$} for all $p\ge 0$ and $q\ge 1$.\qed
\endstat

We finally discuss some basic properties of plurisubharmonic functions. 
In complex geometry, plurisubharmonic functions play exactly the same role 
as convex functions do in real (affine) geometry. A function $\varphi:
\Omega\to[-\infty,+\infty[$ on an open subset $\Omega\subset\bbbc^n$ is 
said to be {\it plurisubharmonic} (usually abbreviated as psh) if $\varphi$
is upper semicontinuous and satisfies the mean value inequality
$$
\varphi(z_0)\le{1\over 2\pi}\int_0^{2\pi}\varphi(z_0+a\,e^{i\theta})\,d\theta
\leqno(2.11)
$$
for every $a\in\bbbc^n$ such that the closed disk $z_0+a\ovl\bbbd$ is 
contained in~$\Omega$ (here $\bbbd$ denotes the unit disk in~$\bbbc$).

\begstat{(2.12) Example} \rm Every convex function $\varphi$ on $\Omega$ 
is psh, since convexity implies continuity, and since the convexity 
inequality
$$
\varphi(z_0)\le{1\over 2}\big(\varphi(z_0+a\,e^{i\theta})+
\varphi(z_0-a\,e^{i\theta})\big)
$$
implies (2.11) by computing the average over~$\theta\in[0,\pi]$.
\endstat

Given a closed (euclidean) ball $\ovl B(z_0,r)\subset\Omega$, the spherical
mean value $(\sigma_{2n-1}r^{2n-1})^{-1}\int_{z\in S(z_0,r)}\varphi(z)\,
d\sigma(z)$ is equal to the average of the mean values computed on each 
circle $z_0+a\partial\bbbd$, when $a$ describes the sphere~$S(0,r)$.
Hence, (2.11) implies the weaker mean value inequality
$$
\varphi(z_0)\le{1\over\sigma_{2n-1}r^{2n-1}}\int_{S(z_0,r)}
\varphi(z)\,d\sigma(z)
\leqno(2.13)
$$
for every ball $\ovl B(z_0,r)\subset\Omega$, in other words, every psh
function is subharmonic (with respect to the Euclidean metric). Notice
that (2.13) still implies the apparently weaker inequality
$$
\varphi(z_0)\le{1\over v_{2n}r^{2n}}\int_{B(z_0,r)}
\varphi(z)\,dV(z)
\leqno(2.13')
$$
by averaging again over all radii in the range~$]0,r[$, with respect to the
density $2n\,r^{2n-1}dr$ (in fact, one can show that the mean value 
properties (2.13) and $(2.13')$ are equivalent). As a consequence, we get
inclusions
$$
\Conv(\Omega)\subset\Psh(\Omega)\subset\Sh(\Omega)
\leqno(2.14)
$$
where $\Conv(\Omega)$, $\Psh(\Omega)$, $\Sh(\Omega)$ are the spaces of
convex, psh and subharmonic functions, respectively. Now, if $X$ is a
complex manifold, we say that a function $\varphi:X\to[-\infty,+\infty[$
is psh if $\varphi$ is psh on every holomorphic coordinate patch~$\Omega$, 
when viewed as a function of the corresponding coordinates.
In fact, Property~2.15~j) below shows that the plurisubharmonicity
property does not depend on the choice of complex coordinates; this
contrasts with convexity or subharmonicity, which do require an additional
linear or riemannian structure, respectively.

\begstat{(2.15) Basic properties of psh functions}
\smallskip
\item{\rm a)} For any decreasing sequence of psh
functions $\varphi_k\in\Psh(X)$, the limit $\varphi=\lim \varphi_k$ is psh
on $X$.
\smallskip
\item{\rm b)} Let $(\varphi_j)_{j\in J}$ be a family of psh functions
on~$X$. Assume that $\varphi:=\sup_{j\in J}\varphi_j$ is upper semicontinuous
and locally bounded from above. Then $\varphi$ is psh on~$X$.
\smallskip
\item{\rm c)} Let $\varphi_1\ld \varphi_p\in\Psh(X)$ and
$\chi:\bbbr^p\lra\bbbr$ be a convex function such that $\chi(t_1\ld t_p)$
is increasing in each $t_j$. Then $\chi(\varphi_1\ld \varphi_p)$ is psh
on $\Omega$. In particular $\varphi_1+\cdots+\varphi_p$, 
$\max\{\varphi_1\ld \varphi_p\}$ and
$\log(e^{\varphi_1}+\cdots+e^{\varphi_p})$ are psh on~$X$.
\smallskip
\item{\rm d)} Let $f\in\cO(X)$ be a holomorphic function. Then
$\log|f|$ is psh on~$X$.
\smallskip
\item{\rm e)} Let $f_1\ld f_p\in\cO(X)$ be holomorphic functions, and let
$\gamma_1\ld\gamma_p$ be positive numbers. Then
$$
\varphi=\log\big(|f_1|^{\gamma_1}+\cdots+|f_p|^{\gamma_p}\big)
$$
is psh on~$X$.
\smallskip
\item{\rm f)} If $\mu$ is a Radon measure on some compact space $K$
and $(z,w)\mapsto\Phi(z,w)$ is an upper semicontinuous function on
$X\times K$ such that $z\mapsto\Phi(z,w)$ is psh on $X$ for
$\mu$-almost every $w\in K$, then $\varphi(z)=\int_{w\in K}
\Phi(z,w)\,d\mu(w)$ is psh on~$X$.
\smallskip
\item{\rm g)} A function $\varphi$ of class $C^2$ is psh if and only if the 
hermitian matrix of mixed derivatives $(\partial^2\varphi/\partial z_j
\partial\ovl z_k)_{1\le j,k\le n}$ is semipositive at every point.
In particular, in dimension $n=1$, a function $\varphi$ of class~$C^2$
is $($pluri$)$subharmonic if and only if $\Delta\varphi\ge 0$.
\smallskip
\item{\rm h)} A function $\varphi\in L^1_\loc(X)$ is equal $($almost
everywhere$)$ to a psh function $\varphi_0$ if and only if for 
every $a\in\bbbc^n$ we have
$$
\sum_{1\le j,k\le n}a_j\ovl a_k
{\partial^2\varphi\over\partial z_j\partial\ovl z_k}\ge 0
\qquad\hbox{$($as a positive measure$)$}
$$
when $\partial^2\varphi/\partial z_j\partial\ovl z_k$ is computed as a
distribution.
\smallskip
\item{\rm i)}  
Let $\varphi\in\Psh(\Omega)$ where $\Omega$ is an open
subset of $\bbbc^n$ and assume that $\varphi\in L^1_\loc(\Omega)$. If
$(\rho_\varepsilon)$ is a family of smoothing kernels, then
$\varphi\star\rho_\varepsilon$ is $\ci$ and psh on
$$
\Omega_\varepsilon=\big\{z\in\Omega\,;\,d(z,\complement\Omega)>\varepsilon
\big\}.
$$
Moreover, the family $(\varphi\star\rho_\varepsilon)$ is increasing in
$\varepsilon$ and $\lim_{\varepsilon\to 0}\varphi\star\rho_\varepsilon=
\varphi$.
\smallskip
\item{\rm j)} If $F:Y\to X$ is a holomorphic map from a complex manifold
$Y$ to a complex manifold $X$ and if $\varphi\in\Psh(X)$ then
$\varphi\circ F\in\Psh(Y)$.
\smallskip
\item{\rm k)}
Assume that $\Omega=\omega\oplus\ii\bbbr^n$ is a ``tube domain''
of base an open subset $\omega\subset\bbbr^n$. Let $\varphi(x+\ii y)=
\varphi(x)$ be a function depending only on $x\in\omega$. Then $z\mapsto
\varphi(z)$ is psh on $\Omega$ if and only if $x\mapsto\varphi(x)$ is
convex on~$\omega$.
\smallskip
\item{\rm l)} Let $\varphi$ be a psh function on an open subset 
$\Omega\subset\bbbc^n$. Given a point $z_0\in\Omega$, let 
$R=d(z_0,\complement\Omega)$. Then the functions
$$
\log r\mapsto \sup_{B(z_0,r)}\varphi,\qquad
\log r\mapsto {1\over\pi^nr^{2n}/n!}
  \int_{z\in B(z_0,r)}\varphi(z)\,d\lambda(z)
$$
are convex increasing functions on $]-\infty,\log R[$.
\vskip0pt
\endstat

\begproof{} a) is just a consequence of the monotone convergence theorem,
while b) follows from the obvious inequality 
$\sup\int\varphi_j\le\int\sup\varphi_j$. Now, let us prove c). The 
conclusion is clearly true if $\chi(t)=\alpha(t)=a_1t_1+\cdots+a_pt_p+b$ is
an affine function with all $a_j>0$, for the function
$$
\alpha(\varphi_1\ld \varphi_p)=a_1\varphi_1+\cdots+a_p\varphi_p+b
$$
also satisfies the mean value inequality by taking positive linear 
combinations. However, it is well known that every convex function
$\chi$ is equal to the upper envelope $\chi=\sup_{\alpha\in\cA}\alpha$
where $\cA$ is the family of all affine functions $\alpha$ such that
$\alpha\le\chi\,$; such functions $\alpha$ are necessarily increasing in 
each variable if $\chi$~is. Hence c) follows from b), and the case of
$\log(e^{\varphi_1}+\cdots+e^{\varphi_p})$ is obtained
by checking that $\chi(t_1\ld t_p)=\log(e^{t_1}+\cdots+e^{t_p})$
is a convex increasing function (exercise: check that the matrix
$(\partial^2\chi/\partial t_j\partial t_k)$ is semipositive of rank
$p-1$ at any point $t\in\bbbr^p$).

Property d) can be reduced easily to the Jensen formula in one variable:
indeed the Jensen formula tells us that the average of $\log|f|$
on a circle of radius $r$ is the sum of the value $\log|f(z_0)|$ at the center
plus a term $\sum m_j\log(r/|w_j-z_0|)\ge 0$ where $(w_j)$ are the
zeros in the disk and $m_j$ the multiplicities. Property e) is a special
case of d) when we take $\varphi_j=\gamma_j\log|f_j|$, and f) is an 
immediate consequence of the Fubini theorem. Now, the convolution
$$
\varphi\star\rho_\varepsilon(z)=\int_{B(0,\varepsilon)}
\varphi(z-w)\rho_\varepsilon(w)\,d\lambda(w)
$$
is a smooth function on $\Omega_\varepsilon$, and f) shows that it is psh;
hence the first part of i) follows. In dimension $n=1$, the proof of g)
is based on the elementary formula
$$
{1\over 2\pi}\int_0^{2\pi}\varphi(z_0+r\,e^{\ii\theta})d\theta=
\varphi(z_0)+{1\over 2\pi}\int_0^r{d\rho\over\rho}\int_{D(z_0,\rho)}
\Delta\varphi(z)\,dx\,dy.\leqno(2.16)
$$
(In fact, assuming $z_0=0$ for simplicity, the Green-Riemann formula yields
$$
\eqalign{
{1\over\rho}\int_{D(0,\rho)}\Delta\varphi(z)\,dx\,dy
&={1\over\rho}\int_{|z|=\rho}{\partial\varphi\over\partial x}dy-
{\partial\varphi\over\partial y}dx\cr
&=\int_{|z|=\rho}{\partial\varphi\over\partial x}\cos\theta\,d\theta
+{\partial\varphi\over\partial y}\sin\theta\,d\theta
={d\over d\rho}\int_0^{2\pi}\varphi(\rho e^{\ii\theta})d\theta,\cr}
$$
an we get (2.16) after an integration.) Now, if $\Delta\varphi\ge 0$, we
infer from (2.16) that the mean value inequality holds; on the other
hand, if $\Delta\varphi(z_0)<0$, the mean value inequality fails
for $r$ small, QED. In higher dimensions, the conclusion is easily
obtained by putting $\psi(w)=\varphi(z_0+aw)\,$; we then get
$$
{1\over 4}\Delta\psi(w)={\partial^2\psi\over\partial w\partial\ovl w}
=\sum_{1\le j,k\le n}a_j\ovl a_k
{\partial^2\varphi\over\partial z_j\partial\ovl z_k}(z_0+aw)
$$
and everything follows.

i) (end of proof) Notice that (2.16) implies that the circular mean value
$\int_0^{2\pi}\varphi(z_0+r\,e^{\ii\theta})d\theta$ of a $C^2$ subharmonic
function in $\Omega\subset\bbbc$ is an increasing function of $r$.
The same is true for spherical mean values of psh functions on open
sets $\Omega\subset\bbbc^n$, since we
can compute them by averaging the circular mean values with respect to
all complex directions. In particular, if $\varphi$ is of class $C^2$,
we conclude (through a use of polar coordinates) that
$$
\varphi\star\rho_\varepsilon(z)=\int_{B(0,1)}\varphi(z-\varepsilon w)
\rho_1(w)\,d\lambda(w)
$$
is an increasing function of~$\varepsilon$, for any $z\in\Omega_\varepsilon$
fixed). Since $\varphi\star\rho_\eta$ is a smooth psh function on
$\Omega_\eta$, we get that $(\varphi\star\rho_\eta)\star\rho_\varepsilon$
is increasing in~$\varepsilon$ whenever $z\in\Omega_{\varepsilon+\eta}$.
By passing to the limit when $\eta\to 0$, we see that
$\varphi\star\rho_\varepsilon(z)$ is always increasing in $\varepsilon$
(even though $\varphi$ is maybe not smooth). Since
$\varphi\star\rho_\varepsilon(z)\ge\varphi(z)$ by the mean value
inequality and $\limsup_{\varepsilon\to 0}\varphi\star\rho_\varepsilon(z)\le
\varphi(z)$ by the upper semicontinuity, we conclude that
$\lim_{\varepsilon\to 0}\varphi\star\rho_\varepsilon(z)=\varphi(z)$
everywhere.

Let us now prove h). If $\varphi_0$ is psh and $\varphi=
\varphi_0$ almost everywhere, then $\varphi\star\rho_\varepsilon=
\varphi_0\star\rho_\varepsilon$ is smooth and psh, hence
$$
\sum_{1\le j,k\le n}a_j\ovl a_k
{\partial^2\varphi\star\rho_\varepsilon\over\partial z_j\partial\ovl z_k}
=\Big(\sum_{1\le j,k\le n}a_j\ovl a_k
{\partial^2\varphi\over\partial z_j\partial\ovl z_k}\Big)
\star\rho_\varepsilon\ge 0
$$
for every $\lambda\in\bbbc^n$ and every $\varepsilon>0$. By passing to a
weak limit, we conclude that the distribution $\sum a_j\ovl a_k
{\partial^2\varphi\over\partial z_j\partial\ovl z_k}$ is a positive
measure. Conversely, if this is the case, the convolution
$\varphi\star\rho_\eta$ is psh by g).
Hence $(\varphi\star\rho_\eta)\star\rho_\varepsilon$ is an increasing
function of~$\varepsilon$, and by taking the limit as $\eta$ tends to
$0$, we see again that $\varphi\star\rho_\varepsilon$ is increasing
in $\varepsilon$. Therefore the decreasing limit
$$
\varphi_0=\lim_{k\to+\infty}\varphi\star\rho_{1/k}
$$
is psh by a), and Lebesgue's theorem shows that $\varphi_0=\varphi$
almost everywhere.

When $\varphi$ is smooth, j) follows from the formula
$$
\sum_{\ell,m}a_\ell\ovl a_m
{\partial^2(\varphi\circ F)\over\partial w_\ell\partial\ovl w_m}=
\sum_{j,k}b_j\ovl b_k
{\partial^2\varphi\over\partial z_j\partial\ovl z_k}\circ F,\qquad
b_j=\sum_\ell a_\ell{\partial F_j\over w_\ell}
$$
in suitable coordinate systems $(z_j)$ on $X$ and $(w_\ell)$ on $Y$.
In general, we conclude by considering regularizations $\varphi
\star\rho_\varepsilon$ and passing to the limit.

k) is obvious when $\varphi$ is smooth, since the convexity of
$\varphi(x)$ and the plurisubharmonicity of $\varphi(z)$ are both
characterized by the condition that $(\partial^2\varphi/\partial x_j
\partial x_k)$ is semipositive everywhere. In general, we obtain the
conclusion by using regularizations $\varphi\star\rho_\varepsilon$.

Finally, property l) follows from the following observation:
the functions
$$
\eqalign{
\sigma(w)&=\sup_{B(z_0,e^{\Re w})}\varphi=
\sup_{a\in B(0,1)}\varphi(z_0+a\,e^w),\cr
\mu(w)&={1\over\pi^ne^{2n\Re w}/n!}
  \int_{B(z_0,e^{\Re w})}\varphi(z)\,d\lambda(z)
  ={1\over\pi^n/n!}
  \int_{a\in B(0,1)}\varphi(z_0+a\,e^w)\,d\lambda(a)\cr}
$$
are psh on the half-plane $\{\Re w<\log R\}\subset\bbbc$, thanks to
j), b) and f). As they only depend on $\Re w$, they must be convex
in $\Re w$. Moreover, $\sigma(w)$ is clearly increasing with respect
to $\Re w$, and the same is true for $\mu$ by (2.16).\qed
\endproof

\begstat{(2.17) Definition} A complex $(1,1)$-form 
$$
u=\ii\sum_{1\le j,k\le n}u_{jk}dz_j\wedge d\ovl z_k
$$ 
is said to be $($semi$)$positive if the hermitian matrix $(u_{jk})$ is
$($semi$)$positive.
\endstat

Notice that $u$ is real, i.e.\ $\ovl u=u$, if and only if
$\ovl u_{jk}=u_{kj}$, i.e.\ iff the matrix is hermitian). In this
setting, a real $L^1_\loc$ function $\varphi$ is psh if and only if $\ii
d'd''\varphi\ge 0$ as a $(1,1)$-form.

\titleb{3.}{K\"ahler metrics and K\"ahler manifolds}
Let us recall that a {\it Riemannian metric} on a (real) differentiable
manifold $M$ is a positive definite symmetric bilinear form
$$
g=\sum_{1\le j,k\le n}g_{jk}(x)\,dx_j\otimes dx_k
$$
on the tangent bundle $T_M$, where $(x_1\ld x_n)$ are local coordinates
on~$M$. We usually assume that the coefficients $g_{jk}(x)$ are smooth.
Then, for any tangent vector $\xi=\sum\xi_j\partial/\partial x_j\in T_{M,x}$, 
one defines its norm with respect to~$g$ by
$$
|\xi|_g^2=\sum_{1\le j,k\le n}g_{jk}(x)\xi_j\xi_k.\leqno(3.1)
$$
If $M$ is moreover assumed to be oriented, one defines a corresponding 
volume element
$$
dV_g=\sqrt{\det(g_{jk}(x))}~dx_1\wedge dx_2\wedge\cdots\wedge dx_n
\leqno(3.2)
$$
whenever $(x_1\ld x_n)$ fit with the given orientation. It is easy to check
by the Jacobian formula that this definition of $dV_g$ is independent
of the choice of coordinates.

Now, we consider the complex case. Let $X$ be a complex $n$-dimensional
mani\-fold. A {\it hermitian metric}
on $X$ is a positive definite hermitian form of class $\ci$ on $T_X\,$;
in a coordinate system $(z_1\ld z_n)$, such a form can be written
\hbox{$h(z)=\sum_{1\le j,k\le n} h_{jk}(z)\,dz_j\otimes d\ovl z_k$,}
where $(h_{jk})$ is a positive hermitian matrix with $\ci$ coefficients.
Thanks to the hermitian condition $\smash{\ovl{h_{jk}}}=h_{kj}$, 
our form $h$ can be written as $h=g-\ii\omega$, where 
$$
\leqalignno{
h(\xi,\eta)&=\sum_{1\le j,k\le n}h_{jk}(z)\xi_j\ovl\eta_k,\cr
g(\xi,\eta)&=\Re h(\xi,\eta)={1\over 2}\sum_{1\le j,k\le n}\big(h_{jk}(z)
\xi_j\ovl\eta_k+\ovl{h_{jk}(z)}\,\ovl\xi_j\eta_k\big)&(3.3)\cr
&\phantom{{}=\Re h(\xi,\eta)}={1\over 2}\sum_{1\le j,k\le n}h_{jk}(z)\big(
\xi_j\ovl\eta_k+\eta_j\ovl\xi_k\big),\cr
\omega(\xi,\eta)&=-\Im h(\xi,\eta)
={\ii\over 2}\sum_{1\le j,k\le n}h_{jk}(z)
\big(\xi_j\ovl\eta_k-\eta_j\ovl\xi_k\big),\qquad\hbox{i.e.}\cr
\omega&=-\Im h
={\ii\over 2}\sum_{1\le j,k\le n}h_{jk}(z)
\,dz_j\wedge d\ovl z_k.&(3.4)\cr}
$$
By definition, $\omega$ is the {\it fundamental $(1,1)$-form} associated 
with $h$. It is a positive defi\-nite $(1,1)$-form (according to the general 
definition). Since $\omega$ and $h$ are ``isomorphic'' objects,
we usually do not make any difference and will think of hermitian
metrics as being positive $(1,1)$-forms.

\begstat{(3.5) Definition} \smallskip
\item{\rm a)} A hermitian manifold is a pair  $(X,\omega)$ where $\omega$
is a $\ci$ positive definite $(1,1)$-form on $X$.
\smallskip
\item{\rm b)} The metric $\omega$ is said to be k\"ahler if $d\omega=0$.
\smallskip
\item{\rm c)} $X$ is said to be a K\"ahler manifold if $X$ possesses at
least one K\"ahler metric.\smallskip
\endstat

Since $\omega$ is real, the conditions $d\omega=0$, $d'\omega=0$,
$d''\omega=0$ are all equivalent. In local coordinates we see that
$d'\omega=0$ if and only if
$${\partial h_{jk}\over\partial z_l}={\partial h_{lk}\over
\partial z_j}\quad,\quad 1\le j,k,l\le n.$$
A simple computation gives ${\ii\over 2} dz_j\wedge d\ovl z_j=dx_j\wedge dy_j$
and
$${\omega^n\over n!}
=\det(h_{jk})\bigwedge_{1\le j\le n}\Big({\ii\over 2} dz_j\wedge d\ovl z_j\Big)
=\det(h_{jk})\,dx_1\wedge dy_1\wedge\cdots\wedge dx_n\wedge dy_n,$$
where $z_n=x_n+\ii y_n$. Therefore the $(n,n)$-form
$$dV_\omega={1\over n!}\omega^n\leqno(3.6)$$
is positive with respect to the canonical orientation of~$X$.
Since $\det(g)=\det(h)^2$ by (3.3), we 
see that $dV_\omega=dV_g$ coincides with the Riemannian volume
element of~$X$.
If $X$ is compact, then $\int_X\omega^n=n!\,\Vol_\omega(X)>0$.
This simple remark already implies that compact K\"ahler
manifolds must satisfy some restrictive
topological conditions:

\begstat{(3.7) Consequence} \smallskip
\item{\rm a)} If $(X,\omega)$ is compact K\"ahler and if $\{\omega\}$
denotes the  cohomology class of $\omega$ in $H^2(X,\bbbr)$, then
$\{\omega\}^n\ne 0$.
\smallskip
\item{\rm b)} If $X$ is compact K\"ahler, then $H^{2k}(X,\bbbr)\ne 0$ for
$0\le k\le n$. In fact, $\{\omega\}^k$ is a non zero class in
$H^{2k}(X,\bbbr)$.
\vskip0pt
\endstat

\begstat{(3.8) Examples} {\rm\smallskip
\noindent{\rm a)} The most obvious example is $\bbbc^n$ together with
the standard K\"ahler metric
$$\omega={\ii\over 2}\sum_{1\le j\le n}dz_j\wedge d\ovl z_j=
\ii\sum_{1\le j\le n}dx_j\wedge dy_j.$$
The volume element $dV_\omega$ coincides with the Lebesgue measure
of $\bbbc^n\simeq\bbbr^{2n}$.
\smallskip
\noindent{\rm b)} The complex projective space $\bbbp^n$ is
K\"ahler. A natural K\"ahler metric $\omega$ on $\bbbp^n$, called
the {\it Fubini-Study metric}, is defined by
$$p^\star\omega={\ii\over 2} d'd''\log\big(|\zeta_0|^2+|\zeta_1|^2
+\cdots+|\zeta_n|^2\big)$$
where $\zeta_0,\zeta_1\ld\zeta_n$ are coordinates of $\bbbc^{n+1}$ and
where \hbox{$p:\bbbc^{n+1}\setminus\{0\}\to\bbbp^n$} is the projection.
Let $z=(\zeta_1/\zeta_0\ld\zeta_n/\zeta_0)$ be non homogeneous coordinates
on $\bbbc^n=\{\zeta_0\ne 0\}\subset\bbbp^n$. Then, since
$d'd''\log|\zeta_0|^2=0$ on $\{\zeta_0\ne 0\}$, we see that
$$
\eqalign{
\omega&={\ii\over 2}d'd''\log(1+|z|^2)=
{\ii\over 2}\Big({d'd''|z|^2\over 1+|z|^2}-
{\langle dz,z\rangle\wedge\ovl{\langle dz,z\rangle}\over(1+|z|^2)^2}
\Big),\cr
|\xi|^2_\omega&={|\xi|^2\over 1+|z|^2}-{|\langle\xi,z\rangle|^2\over
(1+|z|^2)^2}={|\xi|^2+|\xi\wedge z|^2\over (1+|z|^2)^2},\cr}
$$
thanks to Lagrange's identity $|\xi|^2|z|^2=|\langle\xi,z\rangle|^2+
|\xi\wedge z|^2$. The eigenvalues of the Fubini-Study metric with
respect to the standard Euclidean metric are $1/(1+|z|^2)^2$ in the
radial direction $\bbbc z$, and $1/(1+|z|^2)$ in the hyperplane
$(\bbbc z)^\perp$. From this we infer
$$
dV_\omega={d\lambda(z)\over(1+|z|^2)^{n+1}}.
$$
A computation shows that the global volume is $\Vol_\omega(\bbbp^n)=
\int_{\bbbp^n}dV_\omega=\pi^n/n!$.
\smallskip
\noindent{\rm c)} A {\it complex torus} is a quotient
$X=\bbbc^n/\Gamma$ by a lattice $\Gamma$ of rank~$2n$. Then $X$ is a
compact complex manifold. Any positive definite hermitian form
with constant coefficients $\omega=\ii\sum h_{jk}dz_j\wedge d\ovl z_k$
defines a K\"ahler metric on~$X$.
\smallskip
\noindent{\rm d)} Every (complex analytic) submanifold $Y$ of a
K\"ahler manifold  $(X,\omega)$ is K\"ahler with metric
$\omega_{\restriction Y}$.  Especially, all complex
submanifolds of $\bbbp^n$ are K\"ahler.
\smallskip
\noindent{\rm e)} Consider the complex surface
$$X=(\bbbc^2\setminus\{ 0\})/\Gamma$$
where $\Gamma=\{\lambda^n~;~n\in\bbbz\}$, $\lambda<1$, acts as a
group of homotheties. Since $\bbbc^2\setminus\{ 0\}$ is diffeomorphic
to  $\bbbr^\star_+\times S^3$, we have $X\simeq S^1\times S^3$. 
Therefore $H^2(X,\bbbr)=0$ by K\"unneth's formula, and
property 3.7 b) shows that $X$ is not K\"ahler. Hence there are compact
complex surfaces which are not K\"ahler.\qed}
\endstat

The following Theorem shows that a hermitian metric $\omega$ on $X$
is K\"ahler if and only if the metric $\omega$ is tangent at order $2$
to a hermitian metric with constant coefficients at every point of~$X$.

\begstat{(3.9) Theorem} Let $\omega$ be a $\ci$ positive definite $(1,1)$-form
on $X$. In order that $\omega$ be K\"ahler, it is necessary and sufficient
that to every point $x_0\in X$ corresponds a holomorphic coordinate system
$(z_1\ld z_n)$ centered at $x_0$ such that
$$\omega=\ii\sum_{1\le l,m\le n}\omega_{lm}\,dz_l\wedge d\ovl z_m,~~~~
\omega_{lm}=\delta_{lm}+O(|z|^2).$$
\endstat

\begproof{} It is clear that the existence of a Taylor expansion as above 
implies $d_{x_0}\omega=0$, so the condition is sufficient. Conversely, assume
that $\omega$ is K\"ahler. Then one can choose local holomorphic 
coordinates $(x_1\ld x_n)$ such that $(dx_1\ld dx_n)$ is
an $\omega$-orthonormal basis of $T_{X,x_0}^\star$. Therefore
$$\eqalign{
\omega&=\ii\sum_{1\le l,m\le n}\wt\omega_{lm}\,dx_l\wedge d\ovl x_m,
~~~~{\rm where}\cr
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|)=\delta_{lm}+\sum_{1\le j\le n}
(a_{jlm}x_j+a'_{jlm}\ovl x_j)+O(|x|^2).\cr}$$
Since $\omega$ is real, we have $a'_{jlm}=\ovl a_{jml}\,$; on the other
hand the K\"ahler condition $\partial\omega_{lm}/\partial x_j=
\partial\omega_{jm}/\partial x_l$ at $x_0$ implies $a_{jlm}=a_{ljm}$. Set now
$$z_m=x_m+{1\over 2}\sum_{j,l}a_{jlm}x_jx_l,~~~~1\le m\le n.$$
Then $(z_m)$ is a coordinate system at $x_0$, and
$$\eqalign{
dz_m&=dx_m+\sum_{j,l}a_{jlm}x_jdx_l,\cr
\ii\sum_mdz_m\wedge d\ovl z_m&=\ii\sum_m dx_m\wedge d\ovl x_m
+\ii\sum_{j,l,m}a_{jlm}x_j\,dx_l\wedge d\ovl x_m\cr
&\phantom{=\ii\sum_m dx_m~\wedge d\ovl x_m}
+\ii\sum_{j,l,m}\ovl a_{jlm}\ovl x_j\,dx_m\wedge d\ovl x_l+O(|x|^2)\cr
&=\ii\sum_{l,m}\wt\omega_{lm}\,dx_l\wedge\ovl dx_m+O(|x|^2)=
\omega+O(|z|^2).\cr}$$
Theorem 3.9 is proved.\qed
\endproof

\begstat{(3.10) Remark} \rm When $\omega$ is K\"ahler, one can refine the
above proof to shows that there are local coordinates $(z_1\ld z_n)$
centered at~$x_0$ such that $\omega={\ii\over 2}\sum_{lm}
\omega_{lm}dz_l\wedge d\ovl z_m$ with
$$
\omega_{lm}=\delta_{lm}-\sum_{1\le j,k\le n}c_{jklm}\,z_j\ovl z_k+O(|z|^3).
$$
The coefficients $c_{jklm}$ satisfy the symmetry relations
$$\ovl c_{jklm}=c_{kjml},~~~~c_{jklm}=c_{lkjm}=c_{jmlk}=c_{lmjk}.$$
(The $c_{jklm}$ can be interpreted as the coefficients of the Levi-Civita
curvature tensor of $(T_X,\omega)$, but we will not use this fact).
\endstat

\titleb{4.}{Differential operators on vector bundles}
We first describe some basic concepts concerning differential operators
(symbol, composition, adjunction, ellipticity), in the general setting
of vector bundles. Let $M$ be a $\ci$ differentiable manifold,
$\dim_{\bbbr}M=m$, and let $E$, $F$ be $\bbbk$-vector bundles over $M$,
with $\bbbk=\bbbr$ or $\bbbk=\bbbc$, $\rank E=r$, $\rank F=r'$.

\begstat{(4.1) Definition} A $($linear$)$ differential operator of degree
$\delta$ from $E$ to $F$ is a $\bbbk$-linear operator
$P:C^\infty(M,E)\to C^\infty(M,F)$, $u\mapsto Pu$ of the form
$$Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x),$$
where $E_{\restriction\Omega}\simeq\Omega\times\bbbk^r$,
$F_{\restriction\Omega}\simeq\Omega\times\bbbk^{r'}$ are trivialized
locally on some open chart \hbox{$\Omega\subset M$} equipped with local
coordinates $(x_1\ld x_m)$, and 
$$a_\alpha(x)=\big(a_{\alpha\lambda\mu}(x)
\big)_{1\le\lambda\le r',\,1\le\mu\le r}
$$
are $r'\times r$-matrices
with $\ci$ coefficients on $\Omega$. Here \hbox{$D^\alpha=(\partial/
\partial x_1)^{\alpha_1}\ldots(\partial/\partial x_m)^{\alpha_m}$}
as usual, and \hbox{$u=(u_\mu)_{1\le\mu\le r}$},
\hbox{$D^\alpha u=(D^\alpha u_\mu)_{1\le\mu\le r}$} are viewed as column
matrices.
\endstat

If $t\in\bbbk$ is a parameter and $f\in\ci(M,\bbbk)$, $u\in\ci(M,E)$, a
simple calculation shows that
$e^{-tf(x)}P(e^{tf(x)}u(x))$ is a polynomial of degree $\delta$ in $t$, of
the form
$$e^{-tf(x)}P(e^{tf(x)}u(x))=t^\delta\sigma_P(x,df(x))\cdot u(x)+
\hbox{\rm lower order terms $c_j(x)t^j$, $j<\delta$},$$
where $\sigma_P$ is the homogeneous polynomial map $T^\star_M\to\Hom(E,F)$
defined by
$$T^\star_{M,x}\ni\xi\mapsto\sigma_P(x,\xi)\in\Hom(E_x,F_x),\qquad
\sigma_P(x,\xi)=\sum_{|\alpha|=\delta}a_\alpha(x)\xi^\alpha.\leqno(4.2)$$
Then $\sigma_P(x,\xi)$ is smooth on $T^\star_M$ as a function of $(x,\xi)$,
and is independent of the choice of coordinates or of the trivializations
used for $E$, $F$. We say that $\sigma_P$ is {\it the principal symbol}
of~$P$. The symbol of a composition $Q\circ P$ of differential operators
is simply the product
$$
\sigma_{Q\circ P}(x,\xi)=\sigma_{Q}(x,\xi)\sigma_{P}(x,\xi),
\leqno(4.3)
$$
computed as a product of matrices.

Now, assume that $E$ is a euclidean or hermitian vector bundle.
Recall that a {\it hermitian form} $h$ on a complex vector bundle $E$
if a collection of positive definite hermi\-tian forms $h(x)$ on each 
fiber $E_x$, such that the map
$$
E\to \bbbr_+,\qquad E_x\ni\xi \mapsto |\xi |^2_h:=h(x)(\xi)
$$
is smooth. A {\it hermitian vector bundle} is a pair $(E,h)$ where
$E$ is a complex vector bundle and $h$ a hermitian metric on~$E$.
The notion of a euclidean (real) vector bundle is similar, so we
leave the reader adapt our notations to that case. We assume in
addition that $M$ is oriented and is equipped with a smooth volume
form $dV(x)=\gamma(x)dx_1\wedge\cdots dx_m$, where $\gamma(x)>0$ is a
smooth density (usually, $dV$ will be the volume element $dV_g$ of some
Riemannian metric). Then we get a Hilbert space $L^2(M,E)$ of global 
sections $u$ of~$E$ with $L^2$ coefficients, by looking at all
sections $x\mapsto u(x)\in E_x$ satisfying the $L^2$ estimate
$$\|u\|^2=\int_M|u(x)|^2\,dV(x)<+\infty.\leqno(4.4)$$
We denote the corresponding global $L^2$ inner product by
$$\Ll u,v\Gg=\int_M\langle u(x),v(x)\rangle\,dV(x),\qquad
u,v\in L^2(M,E)\leqno(4.4')$$

\begstat{(4.5) Definition} If $P:\ci(M,E)\to\ci(M,F)$
is a differential operator and both $E$, $F$ are euclidean or hermitian,
there exists a unique differential operator
$$P^\star:\ci(M,F)\to\ci(M,E),$$
called the {\it formal adjoint} of $P$, such that for all sections
$u\in\ci(M,E)$ and $v\in\ci(M,F)$ there is an identity
$$\Ll Pu,v\Gg=\Ll u,P^\star v\Gg,\qquad\hbox{\rm whenever
$\Supp u\cap\Supp v\compact M$}.$$
\endstat

\begproof{} The uniqueness is easy, using the density of smooth compactly
supported forms in $L^2(M,E)$. By a partition of unity argument, it is
enough to show the existence of $P^\star$ locally. Now, let
$Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x)$
be the expansion of $P$ with respect to trivializations of $E$, $F$ given
by orthonormal frames over some coordinate open set $\Omega\subset M$. When
\hbox{$\Supp u\cap\Supp v\compact\Omega$} an integration by parts
yields
$$\eqalign{
\Ll Pu,v\Gg&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}
a_{\alpha\lambda\mu}D^\alpha u_\mu(x)\ovl v_\lambda(x)\,\gamma(x)\,
dx_1\ld dx_m\cr
&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}(-1)^{|\alpha|}
u_\mu(x)\ovl{D^\alpha(\gamma(x)\,\ovl a_{\alpha\lambda\mu}v_\lambda(x)}
\,dx_1\ld dx_m\cr
&=\int_\Omega\langle u,\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}
D^\alpha\big(\gamma(x)\,{}^t\ovl a_\alpha v(x)\big)\rangle\,dV(x).\cr}$$
Hence we see that $P^\star$ exists and is uniquely defined by
$$P^\star v(x)=\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}D^\alpha
\big(\gamma(x)\,{}^t\ovl a_\alpha v(x)\big).\hfill\square\leqno(4.6)$$
\endproof

It follows immediately from (4.6) that the principal symbol of $P^\star$ is
$$\sigma_{P^\star}(x,\xi)=(-1)^\delta\sum_{|\alpha|=\delta}
{}^t\ovl a_\alpha\xi^\alpha=(-1)^\delta\sigma_P(x,\xi)^\star.\leqno(4.7)$$

\titled{(4.8) Hilbertian extensions of differential operators} Given a 
differential operator $P:\ci(M,E)\to\ci(M,F)$, we can extend
it as an operator $\wt P:\cD'(M,E)\to\cD'(M,F)$ by computing $Pu$ in
the sense of distributions. Especially, we get an operator
$$
P_\cH:L^2(M,E)\to L^2(M,F),
$$
which we call the maximal (Hilbertian) extension of $P$, such that
$u\in\Dom(P_\cH)$ if and only if $u\in L^2(M,E)$ and $\wt P u\in L^2(M,F)\,$;
we then set of course $P_\cH u=\wt P u$.

\begstat{(4.9) Proposition} For any differential operator $P:\ci(M,E)\to
\ci(M,F)$, its Hilbertian extension $P_\cH$ is a closed and densely
defined operator.
\endstat

\begproof{} First observe that $\Dom P_\cH$ contains the space
$\cD(M,E)$ of smooth sections with compact support in~$M$. As
$\cD(M,E)$ is dense in $L^2(M,E)$, we conclude that $P_\cH$ has a
dense domain. Now, let $(u_\nu,v_\nu)$ be a sequence in the graph of
$P_\cH$, converging towards a limit $(u,v)$ in $L^2(M,E)\times
L^2(M,F)$. Then, $u_\nu\to u$ in $L^2(M,E)$ and in particular $u_\nu$
converges weakly to $u$ in $\cD'(M,E)$. As differentiations and
multiplications by smooth functions are continuous for the weak
topology, we infer that $v_\nu=P u_\nu$ converges weakly to $\wt Pu$
in $\cD'(M,F)$. Since the weak topology is Hausdorff, we conclude that
$v=\wt P u$. Hence $u\in\Dom P_\cH$ and $v=P_\cH u$, as desired.\qed
\endproof

By the general results of \S$\,$1, we know that $P_\cH$ admits a closed
and densely defined adjoint $(P_\cH)^\star:L^2(M,F)\to L^2(M,E)$, called
the {\it Hilbert adjoint} of $P_\cH$, such that
$$
\Ll P_\cH u,v\Gg=\Ll u,(P_\cH)^\star v\Gg,\qquad
\forall u\in\Dom P_\cH,~\forall v\in\Dom (P_\cH)^\star.$$
In particular, this identity must hold true for all $u\in\cD(M,E)$,
and we conclude from this that $(P_\cH)^\star v$ coincides with the formal
adjoint $\wt{P^\star}v$ computed in the sense of of distributions. Hence,
if $(P^\star)_\cH$ is the maximal Hilbertian extension of the formal
adjoint (usually simply called the formal adjoint), we see that
$$
\Dom (P_\cH)^\star\subset \Dom (P^\star)_\cH\leqno(4.10)
$$
and that both operators $(P_\cH)^\star$ (Hilbert adjoint) and
$(P^\star)_\cH$ (formal adjoint) coincide on $\Dom(P_\cH)^\star$.
However, the domains are in general distinct, as shown by the following
simple example.

\begstat{(4.11) Example} 
{\rm Consider $M={}]0,1[$, $dV=dx$, together
with the trivial hermitian vector bundles $E=F=M\times\bbbc$, and the
differential operator
$$
P={d\over dx}:\ci(]0,1[,\bbbc)\to \ci(]0,1[,\bbbc).
$$ 
Our general formula for the formal adjoint shows that 
$$
P^\star=-{d\over dx}=-P.
$$ 
Now, the domain of $P_\cH$ consists of all $u\in L^2(]0,1[,\bbbc)$
such that $u'\in L^2(]0,1[,\bbbc)$ and is therefore nothing else 
by definition than the Sobolev space $W^1(]0,1[,\bbbc)$. However
$W^1(]0,1[,\bbbc)$ injects continuously in $C^0(]0,1[,\bbbc)$, since
$u'\in L^2(]0,1[,\bbbc)\subset L^1(]0,1[,\bbbc)$ implies that $u$ extends 
as a continuous function in $C^0([0,1],\bbbc)$. In particular, 
any $u\in\Dom P_\cH=W^1(]0,1[,\bbbc)$ can be assigned well defined 
values $u(0)$ and~$u(1)$. Now, given $u\in\Dom P_\cH$ and
$v\in\Dom(P_\cH)^\star$, (4.10) shows that
$u,v\in W^1(]0,1[,\bbbc)\subset C^0([0,1],\bbbc)$. We then get
the integration by part formula
$$
\Ll P_\cH u,v\Gg=\int_0^1 u'(x)\ovl v(x)\,dx=u(1)v(1)-u(0)v(0)-\int_0^1u(x)
v'(x)\,dx,
$$
which can be easily reduced to the case of smooth functions by using
convolution with reglarizing kernels.
As $v'\in L^2$, we conclude that the linear form $u\mapsto \Ll Pu,v\Gg$
is continuous in the $L^2$ topology if and only if $v(0)=v(1)=0$ (and in
that case we do have $(P_\cH)^\star v=-v'=(P^\star)_\cH v$). Hence
$$
\eqalignno{
\Dom(P_\cH)^\star=W^1_0(]0,1[,\bbbc):={}&\big\{v\in W^1(]0,1[,\bbbc)\,;~
v(0)=v(1)=0\big\}\cr
\subsetneq{}&W^1(]0,1[,\bbbc)=\Dom (P^\star)_\cH.&\square\cr}
$$}
\endstat

\titled{(4.12) Elliptic operators} Especially important in PDE theory
are the so-called elliptic differential operators:

\begstat{(4.13) Definition} A differential operator $P$ is said to be
elliptic if $\sigma_P(x,\xi)\in\Hom(E_x,F_x)$
is injective for every $x\in M$ and $\xi\in T^\star_{M,x}\ssm\{0\}$.
\endstat

The main result of elliptic PDE theory, which we only quote here
(see e.g.\ (H\"ormander 1963)), is

\begstat{(4.14) Theorem}
Every solution $u\in\cD'(M,E)$ of an elliptic 
equation $\wt Pu=v$ with $v\in\ci(M,F)$ is in fact smooth, i.e., 
$u\in\ci(M,E)$. In fact, if $P$ is of degree $\delta$ and $v$ is in
some Sobolev space $W^s_\loc(M,F)$, then $u\in W^s_\loc(M,E)$.
\endstat

\titleb{5.}{Operators of K\"ahler geometry and commutation identities}
In K\"ahler geometry, many linear differential operators are to be
considered, together with their commutation relations. All these
operators are $\bbbc$-linear endomorphisms acting on the (bi)graded
module $\bigoplus_{p,q\ge 0}C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)$,
where $E$ is some hermitian vector bundle over~$X$. These operators
form a bigraded algebra \hbox{$\cA=\bigoplus\cA_{r,s}$}:
an operator is called of {\it type} (or {\it bidegree}) $(r,s)$ if it maps
$$
C^\infty(X,\Lambda^{p,q}T^\star_X\otimes E)\to
C^\infty(X,\Lambda^{p+r,q+s}T^\star_X\otimes E)
\leqno(5.1)
$$
for all $p,\,q\,$; the {\it $($total$)$ degree} of such an operator is by
definition $k=r+s$, and we set $\cA_k=\bigoplus_{r+s=k}\cA_{r,s}$.
Given homogeneous operators $A$, $B$ of degrees $a$, $b$ in a
graded algebra $\cA=\bigoplus\cA_k$, their graded commutator is
defined to be
$$
[A,B]=AB-(-1)^{ab} BA.
\leqno(5.2)
$$
If $C$ is another endomorphism of degree $c$, the following {\it
Jacobi identity} holds (as a purely formal computation shows):
$$
(-1)^{ca}\big[A,[B,C]\big]+(-1)^{ab}\big[B,[C,A]\big]
+(-1)^{bc}\big[C,[A,B]\big]=0.
\leqno(5.3)
$$
To every form $\alpha\in C^\infty(X,\Lambda^{p,q}T^\star_X)$ corresponds
an endomorphism of type $(p,q)$ on $C^\infty(X,\Lambda^{\bu,\bu}
T^\star_X\otimes E)$, defined by $u\mapsto\alpha\wedge u$. We will often
use again the notation $\alpha$ for this endomorphism, i.e., we will
write $\alpha(u)=\alpha\wedge u$.

\titled{(5.4) Pointwise and global hermitian metrics on spaces of 
$(p,q)$-forms} From now on, we suppose
that $(X,\omega)$ is a K\"ahler manifold and set $n=\dim_\bbbc X$.
The underlying Riemannian metric $g$ defines a euclidean metric on
the real tangent space $T^\bbbr_X$, hence a hermitian metric on
the complexified tangent space $\bbbc\otimes_\bbbr T^\bbbr_X$. We get as
well a hermitian metric on
$$
\Hom_\bbbr(T^\bbbr_X,\bbbc)=\Hom_\bbbc(\bbbc\otimes_\bbbr T^\bbbr_X,\bbbc)
=\Lambda^{1,0}T^\star_X\oplus \Lambda^{0,1}T^\star_X.
$$
For instance, in $\bbbc^n$ with the usual euclidean metric, we have
$$
\Big|{\partial\over\partial x_j}\Big|=\Big|{\partial\over\partial y_j}\Big|=1,
\qquad
\Big|{\partial\over\partial z_j}\Big|=
\Big|{1\over 2}{\partial\over\partial x_j}-{\ii\over 2}
{\partial\over\partial y_j}\Big|={1\over\sqrt{2}}=
\Big|{\partial\over\partial\ovl z_j}\Big|,
$$
$(\partial/\partial z_j)$, $(\partial/\partial\ovl z_k)$ are orthogonal bases
of $T^{1,0}_{\bbbc^n}$, $T^{0,1}_{\bbbc^n}$ (which are themselves 
mutually orthogonal). The dual metric on 
$\Lambda^{1,0}T^\star_{\bbbc^n}\oplus \Lambda^{0,1}T^\star_{\bbbc^n}$
is such that $|dz_j|=|dx_j+\ii dy_j|=\sqrt{2}=|d\ovl z_j|$. Now, for
an arbitrary K\"ahler manifold $(X,\omega)$, we can use 
the Gram-Schmidt orthogonalization process in order to
construct on any coordinate open set $\Omega\subset X$ an
orthonormal frame $(\xi_1\ld\xi_n)$ of $T^{0,1}_{X|\Omega}$ for the
metric~$\omega$. The dual basis $(\xi_j^\star)$ defines an orthonormal
frame of $\Lambda^{1,0}T^\star_X$ for the dual metric, furthermore, 
any $(p,q)$-form can be written in a unique way
$$
u=\sum_{|I|=p,\,|J|=q}u_{IJ}\xi_I^\star\wedge\ovl\xi_J^\star.
$$
We define the (pointwise) hermitian norm of $u$ to be
$|u|^2_\omega=\sum_{I,\,J}|u_{IJ}|^2$. In this way, we get a hermitian
metric on $\Lambda^{p,q}T^\star_X$, which is actually independent of the
initial choice of the orthonormal frame~$(\xi_j)$. One can check this
by observing that the corresponding hermitian inner product satisfies 
the intrinsic property
$$
\langle u_1\wedge\cdots\wedge u_p\wedge\ovl v_1\cdots\wedge v_q,
u_1'\wedge\cdots\wedge u_p'\wedge\ovl v_1'\cdots\wedge\ovl v_q'
\rangle_\omega=\det(\langle u_j,u_k'\rangle_\omega)\,
\ovl{\det(\langle v_j,v_k'\rangle_\omega)},
$$
which characterizes the inner product in a unique way. Now, we consider 
the Hilbert space $L^2(X,\Lambda^{p,q}T^\star_X)$ of global
$(p,q)$-forms $u$ with measurable complex coefficients such that
$$
\Vert u\Vert_\omega^2=\int_X|u|^2_\omega dV_\omega<+\infty.\leqno(5.5)
$$
The corresponding global $L^2$ inner product is
$$
\Ll u,v\Gg_\omega=\int_X\langle u,v\rangle_\omega dV_\omega.
$$
Unless there are several K\"ahler metrics under consideration, we will
usually omit the subscripts in the notation of the norms or inner
products.

\titled{(5.6) Contraction of a differential form by a tangent vector}
Let $u$ be a $(p,q)$-form on $\Omega$, viewed as an antisymmetric 
$\bbbr$-linear form of degree $k=p+q$. Given a complex tangent vector 
$\xi=\sum\xi'_j\,\partial/\partial z_j+\sum\xi''_j\,\partial/\partial
\ovl z_j$, we define the {\it contraction} $\xi\ort u$ to be the 
differential form of degree $k-1=p+q-1$ such that
$$
(\xi\ort u)(\eta_1\ld\eta_{k-1})=u(\xi,\eta_1\ld\eta_{p-1})
$$
for all tangent vectors $\eta_j$. Then $(\xi,u)\longmapsto\xi\ort u$ is 
bilinear, and from the fact that $u\wedge v$ is the antisymmetrization
of $u\otimes v$ one easily sees that contraction by a tangent vector
is a {\it derivation}, i.e.\
$$
\xi\ort(u\wedge v)=(\xi\ort u)\wedge v+(-1)^{\deg u}u\wedge(\xi\ort v).
$$
From this and the obvious rule ${\partial\over\partial z_\ell}\ort dz_j=
\delta_{j\ell}$ we derive the explicit formulas
$$
\eqalign{
{\partial\over\partial z_\ell}\ort (dz_I\wedge d\ovl z_J)
&=\cases{0&if~~$\ell\notin I$,\cr
         (-1)^{s-1}dx_{I\ssm\{\ell\}}&if~~$\ell=i_s\in I$,\cr}\cr
{\partial\over\partial\ovl z_\ell}\ort(dz_I\wedge d\ovl z_J)
&=\cases{0&if~~$\ell\notin J$,\cr
         (-1)^{p+s-1}dx_{I\ssm\{\ell\}}&if~~$\ell=j_s\in J$,\cr}\cr}
$$
whenever $|I|=p$ and $|J|=q$. An easy check shows that the interior
product $\xi\ort\bu$ is the adjoint of the wedge multiplication 
$\xi^\star\wedge\bu$, where $\xi^\star=\langle\bu,\xi\rangle_\omega$
is the $(1,0)$-form associated with $\xi$, i.e.\
$$
\langle \xi\ort u,v\rangle=\langle u,\xi^\star\wedge v\rangle
$$
for any pair of forms $(u,v)$ of respective degrees $k$, $k-1$. Of course,
a similar formula also holds for global inner products $\Ll~\,,~\,\Gg$,
since we need only integrate the above pointwise formula.

\titled{(5.7) Operators of K\"ahler geometry} Here is a short list of
the operators we will have to deal with:
\smallskip
\item{\rm a)} The operators $d=d'+d''$ acting on
$C^\infty(X,\Lambda^{\bu,\bu}T^\star_X)$, which are all three of degree
$1$ ($d'$ being of bidegree $(1,0)$ and $d''$ of bidegree $(0,1)$).
\smallskip
\item{\rm b)} Their adjoints $d^\star=d^{\prime\star}+d^{\prime\prime
\star}$, computed with respect to the global $L^2$ inner product.
We have for instance
$$
\Ll d''u,v\Gg=\Ll u,d^{\prime\prime\star}v\Gg
$$
for all smooth forms $u$ of type $(p,q-1)$ and $v$ of type $(p,q)$
with $\Supp u\cap\Supp v$ compact. Hence $d'$ is of type
$(-1,0)$ and $d''$ is of type $(0,-1)$. (More generally, the adjoint
of an operator of type $(r,s)$ is of type $(-r,-s)$.)
\smallskip
\item{\rm c)} The {\it Laplace-Beltrami} operators
$$
\eqalign{
\Delta&=dd^\star+d^\star d=[d,d^\star],\cr      
\Delta'&=d'd^{\prime\star}+d^{\prime\star}d'=[d',d^{\prime\star}],\cr
\Delta''&=d''d^{\prime\prime\star}+d^{\prime\prime\star}d''
=[d'',d^{\prime\prime\star}].\cr}
$$
\smallskip
\item{\rm d)} Two other important operators are the operators $L$ of type
$(1,1)$ defined by 2
$$
Lu=\omega\wedge u,
$$
and its adjoint $\Lambda=L^\star$, which is obtained by taking the
pointwise adjoint
$$
\langle u,\Lambda v\rangle=\langle Lu,v\rangle.\eqno\square
$$

\titled{(5.8) Special case of the flat hermitian metric on
open subsets of $\bbbc^n$}
Assume that $X=\Omega\subset\bbbc^n$ is an open subset and that
$\omega$ is the standard K\"ahler metric of $\bbbc^n$, {\it multiplied
by $2$}, i.e.\ 
$$
\omega=\ii\sum_{1\le j\le n} dz_j\wedge d\ovl z_j
$$
(the reason for multiplying the standard metric by $2$ is that we get in
this way $|\partial/\partial z_j|=1$, and this allows us to avoid annoying 
constants $2$ or $1/2$ in the computations). For any form 
$u\in C^\infty(\Omega,\Lambda^{p,q}T^\star_\Omega)$ we have
$$
\eqalign{
d'u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
dz_k\wedge dz_I\wedge d\ovl z_J=\sum_{1\le k\le n}dz_k\wedge
\Big({\partial u\over\partial z_k}\Big),\cr
d''u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial\ovl z_k}
d\ovl z_k\wedge dz_I\wedge d\ovl z_J=\sum_{1\le k\le n}d\ovl z_k\wedge
\Big({\partial u\over\partial\ovl z_k}\Big),\cr}$$
where $\partial u/\partial z_k$ and $\partial u/\partial\ovl z_k$ are
the differentiations of $u$ in $z_k$, $\ovl z_k$, taken componentwise on 
each coefficient~$u_{IJ}$. From this we easily get

\begstat{(5.9) Lemma} On any open subset $\omega\subset\bbbc^n$ equipped
with the flat hermitian metric~$\omega$, we have
$$
\eqalign{
d^{\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial\ovl z_k}
{\partial\over\partial z_k}\ort (dz_I\wedge d\ovl z_J),\cr
d^{\prime\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ovl z_k}\ort (dz_I\wedge d\ovl z_J).\cr}
$$
These formulas can be written more briefly as
$$
d^{\prime\star}u=-\sum_{1\le k\le n}
{\partial\over\partial z_k}\ort\Big({\partial u\over\partial\ovl z_k}\Big),
\qquad
d^{\prime\prime\star}u=-\sum_{1\le k\le n}
{\partial\over\partial\ovl z_k}\ort\Big({\partial u\over\partial z_k}\Big).
$$
\endstat

\begproof{} The adjoint of $dz_j\wedge\bu$ is ${\partial\over\partial z_j}
\ort\bu~$. In the case of $d^{\prime\star}$, for instance, we get
$$
\eqalign{
\Ll d'u,v\Gg
&=\int_\Omega\langle\sum_{1\le k\le n}dz_k\wedge
\Big({\partial u\over\partial z_k}\Big),v\rangle\,dV
=\int_\Omega\sum_{1\le k\le n}\langle
{\partial u\over\partial z_k},{\partial\over\partial z_k}\ort v\rangle\,dV\cr
&=\int_\Omega \langle u,
-\sum_{1\le k\le n}{\partial\over\partial\ovl z_k}\Big(
{\partial\over\partial z_k}\ort v\Big)\rangle\,dV
=\int_\Omega \langle u,
-\sum_{1\le k\le n}{\partial\over\partial z_k}\ort\Big(
{\partial v\over\partial\ovl z_k}\Big)\rangle\,dV\cr}
$$
whenever $u$ (resp.\ $v$) is a $(p-1,q)$-form (resp.\ $(p,q)$-form), with
$\Supp u\cap\Supp v\compact\Omega$. The third equality is simply obtained
through an integration par parts, and amounts to observe that the formal 
adjoint of $\partial/\partial z_k$ is $-\partial/\partial\ovl z_k$.\qed
\endproof

We now prove a basic lemma due to (Akizuki and Nakano 1954).

\begstat{(5.10) Lemma} In $\bbbc^n$, we have
$[d^{\prime\prime\star},L]=\ii d'$.
\endstat

\begproof{} Using Lemma~5.9, we find
$$
\eqalign{[d^{\prime\prime\star},L]u
&=d^{\prime\prime\star}(\omega\wedge u)-\omega\wedge d^{\prime\prime\star}u\cr
&=-\sum_k {\partial\over\partial\ovl z_k}\ort\Big(
{\partial\over\partial z_k}(\omega\wedge u)\Big)+\omega\wedge\sum_k 
{\partial\over\partial\ovl z_k}\ort\Big({\partial u\over\partial z_k}\Big).\cr}
$$
Since $\omega$ has constant coefficients, we have $\displaystyle{\partial\over
\partial z_k}(\omega\wedge u)=\omega\wedge{\partial u\over\partial z_k}$
and therefore
$$\eqalign{
[d^{\prime\prime\star},L]\,u&=-\sum_k\bigg({\partial\over\partial\ovl z_k}\ort
\Big(\omega\wedge {\partial u\over\partial z_k}\Big)-\omega\wedge\Big(
{\partial\over\partial\ovl z_k}\ort{\partial u\over\partial z_k}\Big)\bigg)\cr
&=-\sum_k\Big({\partial\over\partial\ovl z_k}\ort\omega\Big)\wedge
{\partial u\over\partial z_k}\cr}$$
by the derivation property of $\ort$. Clearly
$$
{\partial\over\partial\ovl z_k}\ort\omega=
{\partial\over\partial\ovl z_k}\ort\ii\sum_{1\le j\le n}dz_j\wedge
d\ovl z_j=-\ii\,dz_k,
$$
hence
$$
[d^{\prime\prime\star},L]\,u=\ii\sum_k dz_k\wedge
 {\partial u\over\partial z_k}=\ii\,d'u.
\eqno\square
$$
\endproof

We are now ready to derive the basic commutation relations in the case of an
arbitrary K\"ahler manifold $(X,\omega)$.

\begstat{(5.11) Theorem} If $(X,\omega)$ is K\"ahler, then
$$\cmalign{&[d^{\prime\prime\star},L]&{}={}
&\ii\,d',\qquad &[d^{\prime\star},L]&{}=-{}&\ii\,d'',\cr
&[\Lambda,d'']&{}=-{}&\ii\,d^{\prime\star},\qquad
&[\Lambda,d']&{}={}&\ii\,d^{\prime\prime\star}.\phantom{\Big)}\cr}$$
\endstat

\begproof{} It is sufficient to verify the first relation, because the
second one is the conjugate of the first, and the relations of the second line
are the adjoint of those of the first line. According to Theorem~3.9, let 
$(z_j)$ be a coordinate system at a point $x_0\in X$, chosen such that
$\omega_{\ell m}=\delta_{\ell m}+O(|z|^2)$. For any $(p,q)$-forms $u,v$
with compact support in a neighborhood of $x_0$, we get
$$\Ll u,v\Gg=\int_M\Big(\sum_{I,J}u_{IJ}\ovl v_{IJ}+\sum_{I,J,K,L}a_{IJKL}\,
u_{IJ}\ovl v_{KL}\Big)\,dV,$$
with $a_{IJKL}(z)=O(|z|^2)$ at $x_0$. An integration by parts as in 
the proof of Lemma~5.9 yields
$$d^{\prime\prime\star}u=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ovl z_k}\ort(dz_I\wedge d\ovl z_J)+
\sum_{I,J,K,L}b_{IJKL}\,u_{IJ}\,dz_K\wedge d\ovl z_L,$$
where the coefficients~ $b_{IJKL}$~ are obtained by derivation of the~
$a_{IJKL}$'s. Therefore $b_{IJKL}=O(|z|)$. Since
$\partial\omega/\partial z_k=O(|z|)$, the proof of Lemma~5.10 implies
here  $[d^{\prime\prime\star},L]u=\ii d'u+O(|z|)$, in particular
both terms coincide at every given point $x_0\in X$.\qed
\endproof

\begstat{(5.12) Corollary} If $(X,\omega)$ is K\"ahler, the complex
Laplace-Beltrami operators satisfy 
$$\Delta'=\Delta''={1\over 2}\Delta.$$
\endstat

\begproof{} It will be first shown that $\Delta''=\Delta'$. We have
$$\Delta''=[d'',d^{\prime\prime\star}]=-\ii\big[d'',[\Lambda,d']\big].$$
Since $[d',d'']=0$, the Jacobi identity implies 
$$-\big[d'',[\Lambda,d']\big]+\big[d',[d'',\Lambda]\big]=0,$$
hence $\Delta''=\big[d',-\ii[d'',\Lambda]\big]=[d',d^{\prime\star}]=\Delta'$.
On the other hand
$$\Delta=[d'+d'',d^{\prime\star}+d^{\prime\prime\star}]=\Delta'+\Delta''+[d',d^{\prime\prime\star}]
+[d'',d^{\prime\star}].$$
Thus, it is enough to prove:
\endproof

\begstat{(5.13) Lemma} $[d',d^{\prime\prime\star}]=0,~[d'',d^{\prime\star}]=0$.
\endstat

\begproof{} We have $[d',d^{\prime\prime\star}]=-\ii\big[d',[\Lambda,d']\big]$
and the Jacobi identity implies
$$-\big[d',[\Lambda,d']\big]+\big[\Lambda,[d',d']\big]+
\big[d',[d',\Lambda]\big]=0,$$
hence $-2\big[d',[\Lambda,d']\big]=0$ and $[d',d^{\prime\prime\star}]=0$. 
The second relation 
$[d'',d^{\prime\star}]=0$ is the adjoint of the first.\qed
\endproof

From the above, we also get the following result, which is of fundamental
importance in Hodge theory.

\begstat{(5.14) Theorem} $\Delta$ commutes with all operators 
$d',\,d'',\,d^{\prime\star},\,d^{\prime\prime\star},\,L,\,\Lambda$.
\endstat

\begproof{} The identities $[d',\Delta']=[d^{\prime\star},\Delta']=0$,
$[d'',\Delta'']=[d^{\prime\prime\star},\Delta'']=0$ 
are immediate. Furthermore, the equality $[d',L]=d'\omega=0$ together
with the Jacobi identity implies
$$[L,\Delta']=\big[L,[d',d^{\prime\star}]\big]=
-\big[d',[d^{\prime\star},L]\big]=\ii[d',d'']=0.$$
By adjunction, we also get $[\Delta',\Lambda]=0$.\qed
\endproof

\titleb{6.}{Connections and curvature}
The goal of this section is to recall the most basic definitions of
hermitian differential geometry related to the concepts of connection,
curvature and first Chern class of a line bundle.

Let $E$ be a complex vector bundle of rank $r$ over a smooth
differentiable mani\-fold~$M$. A {\it connection} $D$ on $E$ is a
linear differential operator of order $1$
$$D:C^\infty(M,\Lambda^qT^\star_M\otimes E) \to C^\infty(M,
\Lambda^{q+1}T^\star_M\otimes E)$$
such that
$$D(f\wedge u) = df\wedge u + (-1)^{\deg\,f}f\wedge Du\leqno(6.1)$$
for all forms $f\in C^\infty(M,\Lambda^pT^\star_M)$, $u\in C^\infty(X,
\Lambda^qT^\star_M\otimes E)$. On an open set $\Omega \subset M$ where $E$
admits a trivialization $\theta:E_{\restriction\Omega }\buildo\simeq\over\lra\Omega
\times \bbbc^r$, a connection $D$ can be written
$$Du\simeq_\theta du+\Gamma \wedge u$$
where $\Gamma \in C^\infty(\Omega ,\Lambda^1T^\star_M\otimes\Hom
(\bbbc^r,\bbbc^r))$ is an arbitrary matrix of $1$-forms and $d$ acts
componentwise. It is then easy to check that
$$D^2u\simeq_\theta (d\Gamma +\Gamma \wedge \Gamma )\wedge u\quad
\hbox{on~}\Omega.$$
Since $D^2$ is a globally defined operator, there is a global $2$-form
$$\Theta(D) \in C^\infty(M,\Lambda^2T^\star_M\otimes\Hom(E,E))\leqno(6.2)$$
such that $D^2u=\Theta(D)\wedge u$ for every form $u$ with values in~$E$.

Assume now that $E$ is endowed with a $C^\infty$ hermitian metric along
the fibers and that the isomorphism $E_{\restriction\Omega }\simeq \Omega \times
\bbbc^r$ is given by a $C^\infty$ frame $(e_\lambda)$. We then have
a canonical sesquilinear pairing
$$\leqalignno{
C^\infty(M,\Lambda^pT^\star_M\otimes E)\times
C^\infty(M,\Lambda^qT^\star_M\otimes E)
&\longrightarrow C^\infty(M,\Lambda^{p+q}T^\star_M\otimes\bbbc)&(6.3)\cr
(u,v) &\longmapsto \{u,v\}\cr}$$
given by
$$\{u,v\}=\sum_{\lambda ,\mu }u_\lambda \wedge \overline v_\mu \langle
e_\lambda,e_\mu \rangle,\qquad
u=\sum u_\lambda \otimes e_\lambda,\quad 
v=\sum v_\mu \otimes e_\mu.$$
The connection $D$ is said to be {\it hermitian} if it satisfies the
additional property
$$d\{u,v\}=\{Du,v\}+(-1)^{{\rm deg~}u}\{u,Dv\}.$$
Assuming that $(e_\lambda )$ is orthonormal, one easily checks that $D$
is hermitian if and only if $\Gamma^\star=-\Gamma$. In this case
$\Theta(D)^\star=-\Theta(D)$, thus
$$\ii\Theta(D) \in C^\infty(M,\Lambda^2T^\star_M\otimes\Herm(E,E)).$$
\medskip

\begstat{(6.4) Special case} \rm For a bundle $E$ of rank 1, the
connection form $\Gamma$ of a hermitian connection $D$ can be seen as
a $1$-form with purely imaginary coefficients $\Gamma=\ii A$ ($A$ real).
Then we have $\Theta(D)=d\Gamma=\ii dA$. In particular $\ii\Theta(E)$
is a closed 2-form. The {\it First Chern class} of $E$
is defined to be the cohomology class
$$c_1(E)_\bbbr=\Big\{{\ii\over 2\pi}\Theta(D)\Big\}\in H^2_\DR(M,\bbbr).$$
The cohomology class is actually independent of the connection, since
any other connection $D_1$ differs by a global 1-form, $D_1u=Du+B\wedge u$,
so that $\Theta(D_1)=\Theta(D)+dB$. ({\it Note\/}: the normalizing factor
$2\pi$ is introduced in such a way that $c_1(E)_\bbbr$ becomes
the image in $H^2(M,\bbbr)$ of an integral class $c_1(E)\in H^2(M,\bbbz)\,$; 
see e.g.\ (Griffiths-Harris 1978) for details).\qed
\endstat

We now concentrate ourselves on the complex analytic case. If $M=X$ is a
complex manifold $X$, every connection $D$ on a complex $\ci$ vector
bundle $E$ can be split in a unique way as a
sum of a $(1,0)$ and of a $(0,1)$-connection, $D=D'+D''$. In a local
trivialization $\theta$ given by a $C^\infty$ frame, one can write 
$$\leqalignno{
D'u &\simeq_\theta d'u + \Gamma '\wedge u,&(6.5')\cr
D''u &\simeq_\theta d''u + \Gamma ''\wedge u,&(6.5'')\cr}$$
with $\Gamma =\Gamma '+\Gamma ''$. The connection is hermitian if and
only if $\Gamma '=-(\Gamma '')^\star$ in any orthonormal frame. Thus there
exists a unique hermitian connection $D$ corresponding to a prescribed
$(0,1)$ part $D''$. 

Assume now that the bundle $E$ itself has a {\it holomorphic} structure.
The unique hermitian connection for which $D''$ is the $d''$ operator
defined in \S~1 is called the {\it Chern connection} of $E$. In a local
holomorphic frame $(e_\lambda )$ of $E_{\restriction\Omega }$, the metric is given
by the hermitian matrix $H=(h_{\lambda\mu})$, $h_{\lambda\mu}
=\langle e_\lambda,e_\mu \rangle$. We have
$$\{u,v\}=\sum_{\lambda,\mu}h_{\lambda\mu}u_\lambda \wedge
\ovl v_\mu=u^\dagger\wedge H\ovl v,$$
where $u^\dagger$ is the transposed matrix of $u$, and
easy computations yield
$$\eqalign{d\{u,v\}
&=(du)^\dagger \wedge H\ovl v+(-1)^{\deg u} u^\dagger
\wedge (dH\wedge \ovl v + H\ovl{dv})\cr
&=\big(du +\ovl H^{-1}d'\ovl H \wedge u\big)^\dagger \wedge H\ovl v
+(-1)^{\deg u}u^\dagger\wedge(\ovl{dv+ \ovl H^{-1}d'\ovl H\wedge v)}\cr}$$
using the fact that $dH = d'H+\ovl{d'\ovl H}$ and
$\ovl H^\dagger=H$. Therefore the Chern connection $D$ coincides with the
hermitian connection defined by
$$\left\{\eqalign{
Du&{}\simeq_\theta du+\ovl H^{-1}d'\ovl H \wedge u,\cr
D'&{}\simeq_\theta d'+\ovl H^{-1} d'\ovl H\wedge\bu=
\ovl H^{-1} d'(\ovl H\bu),~~~~D'' = d''.\cr}\right.\leqno(6.6)$$
It is clear from this relations that $D^{\prime 2}=D^{\prime\prime 2}=0$.
Consequently $D^2$ is given by to $D^2=D'D''+D''D'$, and the curvature 
tensor $\Theta(D)$ is of type~$(1,1)$. Since $d'd''+d''d'=0$, we get
$$\eqalign{
(D'D''+D''D')u&\simeq_\theta\ovl H^{-1}d'\ovl H\wedge d''u 
+d''(\ovl H^{-1}d'\ovl H\wedge u)\cr
&=d''(\ovl H^{-1}d'\ovl H)\wedge u.\cr}$$

\begstat{(6.7) Proposition} The Chern curvature tensor $\Theta(E):=
\Theta(D)$ is such that
$$\ii\,\Theta(E) \in C^\infty(X,\Lambda^{1,1}T^\star_X\otimes\Herm(E,E)).$$
If $\theta: E_{\restriction\Omega}\to\Omega\times\bbbc^r$ is a holomorphic
trivialization and if $H$ is the hermitian matrix representing the
metric along the fibers of $E_{\restriction\Omega}$, then
$$\ii\,\Theta(E)\simeq_\theta
\ii\,d''(\ovl H^{-1} d'\ovl H)\quad\hbox{\rm on}~\Omega.\eqno\square$$
\endstat

Let $(z_1\ld z_n)$ be holomorphic coordinates on $X$ and let
$(e_\lambda)_{1\le\lambda\le r}$ be an orthonormal frame of $E$.
Writing
$$\ii\Theta(E)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}dz_j\wedge dz_k\otimes e_\lambda^\star\otimes e_\mu,$$
we can identify the curvature tensor to a hermitian form
$$\wt\Theta(E)(\xi\otimes v,\xi\otimes v)=
\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\xi_j\ovl\xi_kv_\lambda\ovl v_\mu\leqno(6.8)$$
on $T_X\otimes E$. This leads in a natural way to positivity concepts,
following definitions introduced by Kodaira (Kodaira 1953), (Nakano 1955) and
(Griffiths 1966).

\begstat{(6.9) Definition} The hermitian vector bundle $E$ is said to be
\smallskip
\item{\rm a)} positive in the sense of Nakano if
$\wt\Theta(E)(\tau,\tau)=
\sum c_{jk\lambda\mu}\tau_{j\lambda}\ovl{\tau_{k\mu}}>0$ 
for all non zero tensors $\tau=\sum
\tau_{j\lambda}\partial/\partial z_j\otimes e_\lambda\in T_X\otimes E$.
\smallskip
\item{\rm b)} positive in the sense of Griffiths if
$\wt\Theta(E)(\xi\otimes v,\xi\otimes v)>0$ for all non zero decomposable 
tensors $\xi\otimes v\in T_X\otimes E\,;$
\smallskip\noindent
Corresponding semipositivity concepts are defined by relaxing the strict
inequalities. We will write $E>_\Nak 0$, $E\ge_\Nak 0$,
$E>_\Grif 0$, $E\ge_\Grif 0$ to express that $E$ possesses a smooth
hermitian metric with the corresponding $($semi$)$positivity properties.
\endstat

\begstat{(6.10) Special case of rank 1 bundles} \rm Assume that
$E$ is a line bundle. The hermitian matrix $H=(h_{11})$ associated
to a trivialization $\theta:E_{\restriction\Omega}\simeq\Omega\times\bbbc$
is simply a positive function which we find convenient to denote by
$e^{-\varphi}$, \hbox{$\varphi\in C^\infty(\Omega,\bbbr)$}. In this case
Prop.~6.10 shows that the curvature form $\Theta(E)$ can be identified 
with the $(1,1)$-form $d'd''\varphi$, and
$$\ii\Theta(E)=\ii\,d'd''\varphi$$
is a real $(1,1)$-form. Hence $E$ is semipositive (in either Nakano or
Griffiths sense) if and only if $\varphi$ is psh, resp.\ positive definite
if and only if $\varphi$ is {\it strictly psh} $($in the sense that
$\ii\,d'd''\varphi\gg 0)$. 
\endstat

\titleb{7.}{Bochner-Kodaira-Nakano identity and inequality}

We now proceed to explain the basic ideas of the Bochner technique used
to prove existence theorems for solutions of $d''$. Let $(X,\omega)$
be a K\"ahler manifold (the assumption that $\omega$ is K\"ahler is
not absolutely necessary, but considerably simplifies the
computations). Let $(E,h)$ be a holomorphic vector bundle on~$X$,
and let $D$ be the associated Chern connection. We first prove a
lemma which will reduce the situation to the case of a trivial
vector bundle.

\begstat{(7.1) Lemma} For every point $x_0\in X$ and every
coordinate system $(z_j)_{1\le j\le n}$ at $x_0$, there exists a
holomorphic frame $(e_\lambda)_{1\le\lambda\le r}$ in a neighborhood of
$x_0$ such that
$$\langle e_\lambda(z),e_\mu(z)\rangle=\delta_{\lambda\mu}- 
\sum_{1\le j,k\le n}c_{jk\lambda\mu}\,z_j\ovl z_k+O(|z|^3)$$ 
where $(c_{jk\lambda\mu})$ are the coefficients of the Chern curvature
tensor $\Theta(E)_{x_0}$. Such a frame $(e_\lambda)$ is called a normal
coordinate frame at~$x_0$.
\endstat

\begproof{} Let $(\varepsilon_\lambda)$ be a holomorphic frame of $E$.
After replacing $(\varepsilon_\lambda)$ by suitable linear combinations
with constant coefficients, we may assume that
$\big(\varepsilon_\lambda(x_0)\big)$ is an orthonormal basis of
$E_{x_0}$.  Then the inner products  $\langle
\varepsilon_\lambda,\varepsilon_\mu\rangle$ have an expansion 
$$
\langle \varepsilon_\lambda(z),\varepsilon_\mu(z)\rangle=
\delta_{\lambda\mu}+\sum_j(a_{j\lambda\mu}\,z_j+
a'_{j\lambda\mu}\,\ovl z_j)+O(|z|^2)
$$
for some complex coefficients $a_{j\lambda\mu}$, $a'_{j\lambda\mu}$ such
that $a'_{j\lambda\mu}=\ovl a_{j\mu\lambda}$.  Set first
$$
\eta_\lambda(z)=\varepsilon_\lambda(z)-\sum_{j,\mu}a_{j\lambda\mu}\,z_j\,
\varepsilon_\mu(z).
$$
Then there are coefficients $a_{jk\lambda\mu}$, $a'_{jk\lambda\mu}$,
$a''_{jk\lambda\mu}$ such that
$$
\eqalign{\langle \eta_\lambda(z),\eta_\mu(z)\rangle
&=\delta_{\lambda\mu}+O(|z|^2)\cr
&=\delta_{\lambda\mu}+\sum_{j,k}\big(a_{jk\lambda\mu}\,z_j\ovl z_k+
a'_{jk\lambda\mu}\,z_jz_k+a''_{jk\lambda\mu}\ovl z_j\ovl z_k\big)+
O(|z|^3).}
$$ 
The holomorphic frame $(e_\lambda)$ we are looking for is 
$$
e_\lambda(z)=
\eta_\lambda(z)-\sum_{j,k,\mu}a'_{jk\lambda\mu}\,z_jz_k\,\eta_\mu(z).
$$
Since $a''_{jk\lambda\mu}=\ovl a'_{jk\mu\lambda}$, we easily find
$$
\eqalign{
\langle e_\lambda(z),e_\mu(z)\rangle&=\delta_{\lambda\mu}+
\sum_{j,k}a_{jk\lambda\mu}\,z_j\ovl z_k+O(|z|^3),\cr
d'\langle e_\lambda,e_\mu\rangle&=\{D'e_\lambda,e_\mu\}=
\sum_{j,k}a_{jk\lambda\mu}\,\ovl z_k\,dz_j+O(|z|^2),\cr
\Theta(E)\cdot e_\lambda
&=D''(D'e_\lambda)=\sum_{j,k,\mu}a_{jk\lambda\mu}\,d\ovl z_k\wedge dz_j
\otimes e_\mu+O(|z|),\cr}
$$ 
therefore $c_{jk\lambda\mu}=-a_{jk\lambda\mu}$.\qed
\endproof

\begstat{(7.2) Extended commutation relations} Let $(X,\omega)$ be
a K\"ahler manifold and let $L$ be the operators defined by
$Lu=\omega\wedge u$ and $\Lambda=L^\star$. Then
$$\eqalign{
[D^{\prime\prime\star},L]&=\ii D',\cr
[\Lambda,D'']&=-\ii D^{\prime\star},\cr}
\qquad
\eqalign{
[D^{\prime\star},L]&=-\ii D'',\cr
[\Lambda,D']&=\ii D^{\prime\prime\star}.\cr}$$
\endstat

\begproof{} Fix a point $x_0$ in X and a coordinate system $z=(z_1\ld z_n)$
centered at~$x_0$. Then Lemma~7.1 shows the existence of a
normal coordinate frame $(e_\lambda)$ at~$x_0$.
Given any section $s=\sum_\lambda\sigma_\lambda\otimes e_\lambda\in
\ci_{p,q}(X,E)$, it is easy to check that the operators $D$,
$D^{\prime\prime\star},\,\ldots~$ have Taylor expansions of the type
$$Ds=\sum_\lambda d\sigma_\lambda\otimes e_\lambda+O(|z|),~~~~
D^{\prime\prime\star}s=\sum_\lambda d^{\prime\prime\star}
\sigma_\lambda\otimes e_\lambda+O(|z|),~\ldots$$
in terms of the scalar valued operators $d$, $d^{\prime\prime\star}$,
$\ldots$.
Here the terms $O(|z|)$ depend on the curvature coefficients of $E$.
The proof of Th.~7.2 is then reduced to the case of operators
with values in the trivial bundle $X\times\bbbc$, which is granted
by Theorem~5.11.\qed
\endproof

\begstat{(7.3) Bochner-Kodaira-Nakano identity} If $(X,\omega)$ is
K\"ahler, the complex Laplace operators $\Delta'$ and $\Delta''$ acting
on $E$-valued forms satisfy the identity
$$\Delta''=\Delta'+[\ii\Theta(E),\Lambda].$$
\endstat

\begproof{} The last equality in (7.2) yields $D^{\prime\prime\star}
=-\ii[\Lambda,D']$, hence
$$
\Delta''=[D'',D^{\prime\prime\star}]=-\ii[D'',\big[\Lambda,D']\big].
$$
By the Jacobi identity we get
$$
\big[D'',[\Lambda,D']\big]=\big[\Lambda,[D',D'']]+
\big[D',[D'',\Lambda]\big]=[\Lambda,\Theta(E)]+\ii[D',D^{\prime\star}],
$$
taking into account that $[D',D'']=D^2=\Theta(E)$. The formula
follows.\qed
\endproof

Assume that $X$ is compact and that $u\in\ci(X,\Lambda^{p,q}T^\star
X\otimes E)$ is an arbitrary $(p,q)$-form. An integration by parts yields
$$
\langle\Delta'u,u\rangle=\|D'u\|^2+\|D^{\prime\star}u\|^2\ge0
$$
and similarly for $\Delta''$, hence we get the basic inequalities
$$
\leqalignno{\|D''u\|^2+\|D^{\prime\prime\star}u\|^2
&=\|D'u\|^2+\|D^{\prime\star}u\|^2+
\int_X\langle[\ii\Theta(E),\Lambda]u,u\rangle dV_\omega,\cr
\|D''u\|^2+\|D^{\prime\prime\star}u\|^2
&\ge\int_X\langle[\ii\Theta(E),\Lambda]u,u\rangle dV_\omega.&(7.4)\cr}
$$
This a priori inequality is known as the {\it Bochner-Kodaira-Nakano} 
inequality (see (Bochner 1948), (Kodaira 1953), (Nakano 1955)). Thanks to 
the general functional analysis results of \S$\,$1 (see (1.3)), this 
inequality can be used to obtain existence theorem for solutions of 
$d''$-equations.

Now, one of the main points is to compute the curvature term 
$\langle[\ii\Theta(E),\Lambda]u,u\rangle$. Unfortunately, as we will
soon see, this term turns out to be rather intricate.
Fix \hbox{$x_0\in X$} and local coordinates $(z_1\ld z_n)$ such that
$(\partial/\partial z_1\ld\partial/\partial z_n)$ is an ortho\-normal
basis of $(T_X^{1,0},\omega)$ at $x_0$. One can write
$$\eqalign{
\omega_{x_0}&=\ii\sum_{1\le j\le n}dz_j\wedge d\ovl z_j,\cr
\ii\Theta(E)_{x_0}&=\ii\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}\,dz_j\wedge d\ovl z_k
\otimes e_\lambda^\star\otimes e_\mu\cr}$$
where $(e_1\ld e_r)$ is an orthonormal basis of $E_{x_0}$. Let
$$u=\sum_{|J|=p,\,|K|=q,\,\lambda}u_{J,K,\lambda}\,dz_J\wedge d\ovl z_K
\otimes e_\lambda\in\big(\Lambda^{p,q}T^\star_X\otimes E\big)_{x_0}.$$
As $\smash{\displaystyle\Lambda=L^\star
=\Big(-\ii\sum d\ovl z_k\wedge(dz_k\wedge\bu)\Big)^\star=
\ii\sum{\partial\over\partial z_k}\ort\Big({\partial\over\partial\ovl z_k}
\ort\bu\Big)}$, a simple computation gives 
$$\eqalign{
\Lambda u&=\ii(-1)^p\sum_{J,K,\lambda,s}u_{J,K,\lambda}\,
\Big({\partial\over\partial z_s}\ort dz_J\Big)\wedge 
\Big({\partial\over\partial\ovl z_s}\ort d\ovl z_K\Big)\otimes e_\lambda,\cr
\ii\Theta(E)\wedge u&=\ii(-1)^p\sum_{j,k,\lambda,\mu,J,K}c_{jk\lambda\mu}\,
u_{J,K,\lambda}\,dz_j\wedge dz_J\wedge d\ovl z_k\wedge d\ovl z_K\otimes e_\mu,
\cr
[\ii\Theta(E),\Lambda]u&=\ii\Theta(E)\wedge(\Lambda u)-\Lambda\big(
\ii\Theta(E)\wedge u\big)\cr
&=\sum_{j,k,\lambda,\mu,J,K}
c_{jk\lambda\mu}\,u_{J,K,\lambda}\,dz_j\wedge\Big({\partial\over\partial z_k}
\ort dz_J\Big)\wedge d\ovl z_K\otimes e_\mu\cr
&+\sum_{j,k,\lambda,\mu,J,K}c_{jk\lambda\mu}\,u_{J,K,\lambda}\,dz_J\wedge d\ovl 
z_k\wedge\Big({\partial\over\partial\ovl z_j}\ort d\ovl z_K\Big)\otimes e_\mu\cr
&-\sum_{j,\lambda,\mu,J,K}c_{jj\lambda\mu}\,u_{J,K,\lambda}\,dz_J\wedge d\ovl z_K
\otimes e_\mu.\cr}$$
We extend the definition of $u_{J,K,\lambda}$ to non increasing multi-indices
$J=(j_s)$, $K=(k_s)$ by deciding that $u_{J,K,\lambda}=0$ if $J$ or $K$ 
contains identical components repeated and that $u_{J,K,\lambda}$ is 
alternate in the indices $(j_s)$, $(k_s)$. Then the above equality 
can be written
$$
\leqalignno{\langle[\ii\Theta(E),\Lambda]u,u\rangle=
&\phantom{{}+{}}\sum_{j,k,\lambda,\mu,J,S}{\,}
c_{jk\lambda\mu}\,u_{J,jS,\lambda}~\ovl{u_{J,kS,\mu}}&(7.5)\cr
&{}+\sum_{j,k,\lambda,\mu,R,K}
c_{jk\lambda\mu}\,u_{kR,K,\lambda}~\ovl{u_{jR,K,\mu}}\cr
&{}-{\,}\sum_{j,\lambda,\mu,J,K}\phantom{k}
c_{jj\lambda\mu}\,u_{J,K,\lambda}~\ovl{u_{J,K,\mu}},\cr}
$$
where the sum is extended to all indices $1\le j,k\le n$, $1\le\lambda,
\mu\le r$ and multiindices $|R|=p-1$, $|S|=q-1$ (here the notation
$u_{JK\lambda}$ is extended to non necessarily increasing multiindices by
making it alternate with respect to permutations). It is usually hard to
decide the sign of the curvature term (7.5), except in some special cases.

The easiest case is when $p=n$. Then all terms in the second summation of
(7.5) must have $j=k$ and $R=\{1\ld n\}\ssm\{j\}$, therefore the second
and third summations are equal. Formula (7.5) takes the simpler form
$$
\langle[\ii\Theta(E),\Lambda]u,u\rangle=
\sum_{j,k,\lambda,\mu,J,S}{\,}
c_{jk\lambda\mu}\,u_{J,jS,\lambda}~\ovl{u_{J,kS,\mu}},\leqno(7.6)
$$
and it follows that $[\ii\Theta(E),\Lambda]$
is positive on $(n,q)$-forms under the assumption that $E$ is 
{\it positive in the sense of Nakano} (we will see later in \S$\,$10
refined sufficient conditions).

Another tractable case is the case where $E$ is a line bundle ($r=1$).
Indeed, at each point $x\in X$, we may then choose a coordinate
system which diagonalizes simultaneously the hermitians forms
$\omega(x)$ and $\ii\Theta(E)(x)$, in such a way that
$$
\omega(x)=\ii\sum_{1\le j\le n}dz_j\wedge d\ovl z_j,\qquad
\ii\Theta(E)(x)=\ii\sum_{1\le j\le n}\gamma_jdz_j\wedge d\ovl z_j
$$
with $\gamma_1\le\ldots\le\gamma_n$. The curvature eigenvalues
$\gamma_j=\gamma_j(x)$ are then uniquely defined and depend continuously
on~$x$. With our previous notation, we have $\gamma_j=c_{jj11}$ and all
other coefficients $c_{jk\lambda\mu}$ are zero. For any $(p,q)$-form
$u=\sum u_{JK}dz_J\wedge d\ovl z_K\otimes e_1$, this gives
$$
\leqalignno{\langle[\ii\Theta(E),\Lambda]u,u\rangle
&=\sum_{|J|=p,\,|K|=q}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}\gamma_j-
\sum_{1\le j\le n}\gamma_j\Big)|u_{JK}|^2\cr
&\ge(\gamma_1+\cdots+\gamma_q-\gamma_{n-p+1}-\cdots-\gamma_n)|u|^2.
&(7.7)\cr}
$$

\titleb{8.}{$L^2$ estimates for solutions of $d''$-equations}
Our goal here is to prove a central $L^2$ existence theorem, which is
essentially due to (H\"ormander 1965, 1966), and (Andreotti-Vesentini
1965). We will only  outline the main ideas, referring e.g.\ to [
(Demailly 1982) for a more detailed exposition of the technical situation
considered here. We start with a K\"ahler manifold $(X,\omega)$ and
denote by $\delta_\omega$ the geodesic distance associated
with~$\omega$. One says that $\omega$ is {\it complete} if
$\delta_\omega$ is complete. The proof is based on the following
two observations.

\begstat{(8.1) Hopf-Rinow lemma} If $\delta_\omega$ is complete, then
all $\delta_\omega$-balls $\ovl B(x_0,r)$ are compact $($and
conversely$)$. Moreover, under this hypothesis, there
exist a sequence of compact sets $K_\nu$ with $X=\bigcup K_\nu$ and
$K_\nu\subset K_{\nu+1}^\circ$, and a sequence of cut-off functions
$\psi_\nu$ such that $|d\psi_\nu|_\omega\le 1$, $\psi_\nu=1$ on~$K_\nu$
and $\Supp\psi_\nu\subset K_{\nu+1}$.
\endstat

\begproof{$($abridged$)$.} Take the infimum $r_0$ of radii $r$ such that
the ball $\ovl B(x_0,r)$ is not compact, if any$\,$; then
$\ovl B(x_0,r_0)$ is non compact (otherwise the local compactness of $X$
would imply that some slightly larger ball is still compact). Hence,
there is a sequence of points $y_\nu\in\ovl B(x_0,r_0)$ without any
accumulation point. For each $k$, we can select (along a suitable path
from $x_0$ to $y_\nu$) a point $z_{\nu,k}$ such that
$\delta(x_0,z_{\nu,k})\le r_0-2^{-k}$ and
$\delta(z_{\nu,k},y_\nu)\le 2^{-k+1}$. Since $\ovl B(x_0,r_0-2^{-k})$
is compact, we may assume after taking subsequences and a diagonal
subsequence, that all sequences $(z_{\nu,k})_\nu$ converge to a point
$z_k\in\ovl B(x_0,r_0-2^{-k})$. One then sees that $(y_\nu)$ is a Cauchy
sequence without any accumulation point, contradiction. Now, the functions
$(\psi_\nu)$ can be defined by
$$
\psi_\nu(x)=\theta\big(3^{-\nu}d(x_0,x)\big)
$$
where $\theta:\bbbr\to\bbbr$ is a smooth function with $-0.9\le\theta'\le 0$
such that $\theta(t)=1$ for $t\le 1.1$, $\theta(t)=0$ for $t\ge 2.9$, if
we set $K_\nu=\ovl B(x_0,3^\nu)$. As the distance function
$x\mapsto \delta_\omega(x_0,x)$ is a $1$-Lipschitz function, it is
almost everywhere differentiable with a differential of norm${}\le 1$, hence
$|d\psi_\nu|\le 0.9$. It still remains to get $\psi_\nu$ smooth, and this 
can be achieved by taking suitable convolutions with regularizing 
kernels.\qed
\endproof

\begstat{(8.2) Lemma {\rm (Andreotti-Vesentini)}} Assume that $\omega$
is complete. For every measurable form $u$ on $X$ with values in
$\Lambda^{p,q}T^\star_X\otimes E$ such that
\smallskip
\item{\rm a)} $u\in L^2$ and $D''u\in L^2$, resp.\
\smallskip
\item{\rm b)} $u\in L^2$ and $D^{\prime\prime\star}u\in L^2$, resp.\
\smallskip
\item{\rm c)} $u\in L^2$, $D''u\in L^2$ and $D^{\prime\prime\star}u
\in L^2,$
\smallskip\noindent
there exists a sequence of smooth forms $u_\nu$ with compact support,
such that $u_\nu\to u$ in $L^2$ in all cases {\rm a,b,c)} and
$D''u_\nu\to D''u$ in $L^2$ in case {\rm a)}
$($resp.\ $D^{\prime\prime\star}u_\nu\to D^{\prime\prime\star}u$ in~$L^2$,
resp.\ $D''u_\nu\to D''u$ in $L^2$ and $D^{\prime\prime\star}u_\nu
\to D^{\prime\prime\star}u$ in~$L^2$ in cases {\rm b)} and {\rm c))}.
\endstat

\begproof{} Let $\psi_\nu$ be a sequence of cut-off functions as in
Lemma~8.1. If $u\in L^2$ and $D''u\in L^2$, then
$\psi_\nu u\in L^2$ and
$$
D''(\psi_\nu u)=\psi_\nu D''u+d''\psi_\nu\wedge u\in L^2.
$$
Moreover $\psi_\nu D''u\to D''u$ and $d''\psi_\nu\wedge u\to 0$
in $L^2$ by Lebesgue's bounded convergence theorem (as $\psi_\nu\to 1$
and $|d''\psi_\nu|_\omega\to 0$ pointwise, with $1$ as a
uniform bound). The above formula shows that $\psi_\nu D''=
D''(\psi_\nu\bu)-d''\psi_\nu\wedge\bu$, hence by adjunction
$$
D^{\prime\prime\star}(\psi_\nu\bu)=\psi_\nu D^{\prime\prime\star}-
(\grad\psi_\nu)^{0,1}\ort\bu,\qquad
D^{\prime\prime\star}(\psi_\nu u)=\psi_\nu D^{\prime\prime\star}u-
(\grad\psi_\nu)^{0,1}\ort u.
$$
We infer as before that $D^{\prime\prime\star}(\psi_\nu u)\to
D^{\prime\prime\star}u$ in $L^2$ as soon as $u\in L^2$ and
$D^{\prime\prime\star}u\in L^2$. We have thus been able to approximate
$u$ by the compactly supported elements $\psi_\nu u$. In order to get
smooth approximants $u_\nu$, we need only use convolution by
regularizing kernels, i.e., $u_\nu=(\psi_\nu u)\star\rho_\varepsilon$
(possibly after using a partition of unity so as to divide the support
of $\psi_\nu u$ in small pieces contained in coordinate open sets).\qed
\endproof

\begstat{(8.3) Corollary} If $\omega$ is complete, the Von Neumann
adjoint $(D''_\cH)\star$ and the hilbertian extension
$(D^{\prime\prime\star})_\cH$ of the formal adjoint coincide.
\endstat

\begproof{} The result is equivalent to proving that
$\Ll D''u,v\Gg=\Ll u,D^{\prime\prime\star}v\Gg$ whenever
$u,\,v\in L^2$ and $D''u\in L^2$, $D^{\prime\prime\star}v\in L^2$
(with these operators being computed in the sense of distributions).
However we certainly have the equality
$$
\Ll D''u_\nu,v_\nu\Gg=\Ll u_\nu,D^{\prime\prime\star}v_\nu\Gg
$$
for any smooth approximants $u_\nu,\,v_\nu$ of $u$ and $v$ (as in
Lemma~8.2 a) and b)). The desired equality is obtained by taking
the limits in $L^2$.\qed
\endproof

\begstat{(8.4) Theorem} Let $(X,\omega)$ be a K\"ahler manifold. Here
$X$ is not necessarily compact, but we assume that the metric $\omega$
is complete on~$X$. Let $E$ be a hermitian vector bundle of
rank $r$ over $X$, and assume that the curvature operator
$A=A^{p,q}_{E,\omega}=[\ii\Theta(E),\Lambda_\omega]$ is positive
definite everywhere on $\Lambda^{p,q}T^\star_X\otimes E$, for some $q\ge 1$.
Then for any form $g\in L^2(X,\Lambda^{p,q}T^\star_X\otimes E)$
satisfying $D''g=0$ and $\int_X\langle A^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in L^2(X,\Lambda^{p,q-1}T^\star_X
\otimes E)$ such that  $D''f=g$ and
$$
\int_X|f|^2\,dV_\omega\le\int_X\langle A^{-1}g,g\rangle\,dV_\omega.
$$
\endstat

\begproof{} Lemma 8.2~c) shows that the basic a priori inequality (7.4) 
extends to arbitrary forms $u$ such that $u\in L^2$, $D''u\in L^2$ and 
$D^{\prime\prime\star}u\in L^2$ . Now, consider the Hilbert space orthogonal
decomposition
$$
L^2(X,\Lambda^{p,q}T^\star_X\otimes E)=\Ker D''\oplus({\Ker D''})^\perp,
$$
observing that $\Ker D''$ is weakly (hence strongly) closed. Let
$v=v_1+v_2$ be the decomposition of a smooth form $v\in\cD^{p,q}(X,E)$
with compact support according to this decomposition ($v_1$, $v_2$ do
not have compact support in general$\,$!). Since $(\Ker
D'')^\perp\subset\Ker D^{\prime\prime\star}$ by duality and $g,\,
v_1\in\Ker D''$ by hypothesis, we get $D^{\prime\prime\star}v_2=0$ and
$$
|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle A^{-1}g,g\rangle \,dV_\omega
\int_X\langle Av_1,v_1\rangle \,dV_\omega
$$
thanks to the Cauchy-Schwarz inequality. The a priori inequality (7.4)
applied to $u=v_1$ yields
$$
\int_X\langle Av_1,v_1\rangle \,dV_\omega
\le\|D''v_1\|^2+\|D^{\prime\prime\star}v_1\|^2
=\|D^{\prime\prime\star}v_1\|^2
=\|D^{\prime\prime\star}v\|^2.
$$
Combining both inequalities, we find
$$
|\langle g,v\rangle|^2
\le\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)
\|D^{\prime\prime\star}v\|^2
$$
for every smooth $(p,q)$-form $v$ with compact support. This shows that
we have a well defined linear form
$$
w=D^{\prime\prime\star}v\longmapsto\langle v,g\rangle,\qquad
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)
\supset D^{\prime\prime\star}(\cD^{p,q}(E))\lra\bbbc
$$
on the range of $D^{\prime\prime\star}$. This linear form
is continuous in $L^2$ norm and has norm $\le C$ with
$$
C=\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)^{1/2}.
$$
By the Hahn-Banach theorem, there is an element $f\in
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)$ with $\|f\|\le C$, such that
$\langle v,g\rangle=\langle D^{\prime\prime\star}v,f\rangle$ for every
$v$, hence $D''f=g$ in the sense of distributions. The inequality
$\|f\|\le C$ is equivalent to the last estimate in the theorem.\qed
\endproof

\begstat{(8.5) Remark} \rm One can always select a solution 
$f$ satisfying the additional property that $f\in(\Ker D'')^\perp$
(otherwise, just replace f by its orthogonal projection on 
$(\Ker D'')^\perp$). This solution is clearly unique and is precisely 
the solution of minimal $L^2$ norm of the equation $D''f=g$. Since
$(\Ker D'')^\perp=\ovl{\Im D^{\prime\prime\star}}\subset \Ker
D^{\prime\prime\star}$, we see that the
minimal $L^2$ solution satisfies the additional equation
$$D^{\prime\prime\star}f=0.$$
Consequently $\Delta''f=D^{\prime\prime\star}D''f=D^{\prime\prime\star}g$. 
If $g$ is of class $\ci$, the ellipticity of the $\Delta''$-operator shows
that $f$ is automatically smooth.\qed
\endstat

The above $L^2$ existence theorem can be applied in the fairly general
context of {\it weakly $($strongly$)$ pseudoconvex} manifolds 
(these manifolds are frequently referred to as weakly (resp.\ strongly)
$1$-complete manifolds in the literature, but we feel that this 
terminology is a bit misleading).

\begstat{(8.6) Definition} A complex manifold $X$ is said to be weakly
$($resp.\ strongly$)$ pseudoconvex if there exists a smooth psh
$($resp.\ strongly psh$)$ exhaustion function $\psi$ on $X$ $(\psi$ is
said to be an exhaustion if for every $c>0$ the sublevel set
$X_c=\psi^{-1}(c)$ is relatively compact, i.e.\ $\psi(z)$ tends to
$+\infty$ when $z$ tends to ``infinity'' in~$X$, with respect to the
filter of complements of compact sets$)$.
\endstat

For example, every closed analytic submanifold of $\bbbc^N$ is strongly
pseudoconvex (take $\psi(z)=|z|^2$). Convex open subsets of $\bbbc^n$
are likewise strongly pseudoconvex (take $\psi(z)=|z|^2+(1-\gamma_a(z))^{-1}$
where $\gamma_a$ is the ``gauge function'' with center $a\in\Omega$, namely
the unique nonnegative function which is equal to $1$ on the boundary of 
$\Omega$ and linear on each half-ray through $a\,$; if $\gamma_a$ is not
smooth, one can take a small ``convolution'' $z\mapsto \int\rho_\varepsilon(w)
\gamma_{a-w}(z)d\lambda(w)$ to get a smooth function). Examples of weakly
pseudoconvex manifolds are compact manifolds (just take $\psi\equiv 0$ in that
case$\,$!), products of such with strongly pseudoconvex manifolds, closed
submanifolds of weakly pseudoconvex manifolds, etc. Now, a basic 
observation is

\begstat{(8.7) Proposition} Every weakly pseudoconvex
K\"ahler manifold $(X,\omega)$ carries a {\it complete} K\"ahler
metric~$\wh\omega$.
\endstat

\begproof{} Let $\psi\ge 0$ be a psh exhaustion function
and set
$$
\wh\omega=\omega+\ii d'd''(\chi\circ\psi)=
\omega+\ii\chi'\circ\psi~d'd''\psi+\ii\chi''\circ\psi~d'\psi\wedge d''\psi
$$
where $\chi:\bbbr\to\bbbr$ is a convex increasing function. For any function
$\rho:\bbbr\to\bbbr$, we get
$$
|d'(\rho\circ \psi)|_{\wh\omega}\le 
{|\rho'\circ\psi|\over\sqrt{\chi''\circ\psi}}.
$$
If we assume $\int_0^{+\infty}\sqrt{\chi''(t)}\,dt=+\infty$ (as is the case
e.g.\ if $\chi(t)=t^2$ or $\chi(t)=t-\log t$ for $t\ge 1$), the function
$\rho(t)=\int_0^t\sqrt{\chi''(\tau)}\,d\tau$ tends to $+\infty$ as
$t\to+\infty$, and we have $|d(\rho\circ \psi)|_{\wh\omega}\le 
\sqrt{2}\,$. By integrating along paths, this bound yields
$$
|\rho\circ\psi(x)-\rho\circ\psi(y)|\le\sqrt{2}\,
\delta_{\wh\omega}(x,y).
$$ 
It follows easily from the fact that $\psi$ is an exhaustion that all
closed geodesic balls are compact. Therefore 
$\delta_{\wh\omega}$ is complete.\qed
\endproof

If we apply the main $L^2$ existence theorem for the complete
K\"ahler metrics $\omega_\varepsilon$, we see by passing to
the limit that the theorem even applies to the non necessarily
complete metric~$\omega$. An important special consequence is the
following

\begstat{(8.8) Theorem} Let $(X,\omega)$ be a K\"ahler manifold with
$\dim X=n$ $(\omega$ is not assumed to be complete$)$. Assume that $X$
is weakly pseudoconvex. Let $E$ be a hermitian holomorphic line bundle and let
$$
\gamma_1(x)\le\ldots\le\gamma_n(x)
$$
be the curvature eigenvalues $($i.e.\ the eigenvalues of $\ii\Theta(E)$
with respect to the metric~$\omega)$ at every point. Assume that the
curvature is positive, i.e.\ $\gamma_1>0$ everywhere. Then for any form
$g\in L^2_\loc(X,\Lambda^{n,q}T^\star_X\otimes E)$, $q\ge 1$, satisfying 
\hbox{$D''g=0$} and
\hbox{$\int_X\langle(\gamma_1+\cdots+\gamma_q)^{-1}|g|^2\,dV_\omega
<+\infty$},
there exists \hbox{$f\in L^2(X,\Lambda^{n,q-1}T^\star_X\otimes E)$} such
that $D''f=g$ and
$$
\int_X|f|^2\,dV_\omega\le
\int_X(\gamma_1+\cdots+\gamma_q)^{-1}|g|^2\,dV_\omega.
$$
\endstat

\begproof{} Indeed, for $p=n$, Eormula~7.7 shows that
$$
\langle Au,u\rangle\ge(\gamma_1+\cdots+\gamma_q)|u|^2,
$$
hence $\langle A^{-1}u,u\rangle\le(\gamma_1+\cdots+\gamma_q)^{-1}|u|^2$.
The assumption that $g\in L^2_\loc(X,\Lambda^{n,q}T^\star_X\otimes E)$
instead of $g\in L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$ is not a
real problem, since we may restrict ourselves to $X_c=\{x\in X\,;\,
\psi(x)<c\}\compact X$ where $\psi$ is a psh exhaustion function
on~$X$. Then $X_c$ itself is weakly pseudoconvex (with psh
exhaustion function $\psi_c=1/(c-\psi)$), hence $X_c$ can be equipped
with a complete K\"ahler metric $\omega_{c,\varepsilon}=\omega+
\varepsilon\,\ii\,d'd''(\psi_c^2)$. For each $(c,\varepsilon)$,
Theorem~8.4 yields a solution $f_{c,\varepsilon}\in
L^2_{\omega_{c,\varepsilon}}(X_c,\Lambda^{n,q-1}T^\star_X\otimes E)$
of the equation $D''f_{c,\varepsilon}=g$ on~$X_c$, such that
$$
\int_{X_c}|f_{c,\varepsilon}|^2_{\omega_{c,\varepsilon}}\,dV_{
\omega_{c,\varepsilon}}\le
\int_{X_c}\langle(A_{E,\omega_{c,\varepsilon}}^{n,q})^{-1}
g,g\rangle_{\omega_{c,\varepsilon}}\,dV_{\omega_{c,\varepsilon}}
$$
A simple computation shows that the integral in the right hand side is
monotonic decreasing with respect to $\omega$, hence
$$
\eqalign{
\int_{X_c}\langle(A_{E,\omega_{c,\varepsilon}}^{n,q})^{-1}g,
g\rangle_{\omega_{c,\varepsilon}}\,dV_{\omega_{c,\varepsilon}}
&\le\int_{X_c}\langle(A_{E,\omega}^{n,q})^{-1}g,
g\rangle_\omega\,dV_\omega\cr
&\le\int_X(\gamma_1+\cdots+\gamma_q)^{-1}|g|^2\,dV_\omega.\cr}
$$
Therefore the solutions $f_{c,\varepsilon}$ are uniformly bounded
in $L^2$ norm on every compact subset of $X$. Since the closed
unit ball of an Hilbert space is weakly compact (and metrizable
if the Hilbert space is separable), we can extract a subsequence 
$$
f_{c_k,\varepsilon_k}\to f\in L^2_\loc
$$ 
converging weakly in $L^2$ on any compact subset $K\subset X$, for some 
$c_k\to+\infty$ and $\varepsilon_k\to 0$. By the weak continuity
of differentiations, we get again in the limit $D''f=g$. Also, for every
compact set $K\subset X$ we get
$$
\int_K|f|^2_\omega\,dV_\omega
\le\liminf_{k\to+\infty}\int_K|f_{c_k,\varepsilon_k}|^2_{\omega_{c_k,
\varepsilon_k}}\,dV_{\omega_{c_k,\varepsilon_k}}
$$
by weak $L^2_\loc$ convergence (closed balls, and more generally closed 
convex sets, are closed in the weak topology of any Banach space). 
Finally, we let $K$ increase to~$X$ and conclude that the desired 
estimate holds on all of~$X$.\qed
\endproof

An important observation is that the above theorem still applies when the
hermitian metric on $E$ is a singular metric with positive curvature in
the sense of currents.

\begstat{(8.9) Corollary} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $E$ be a
hermitian holomorphic line bundle and let $\varphi\in L^1_\loc$ be a 
weight function $($no further regularity assumption is made on~$\varphi)$.
Suppose that
$$
\ii\Theta(E)+\ii d'd''\varphi\ge\gamma\omega
$$
for some positive continuous function $\gamma>0$ on $X$. Then for
any form $g\in L^2_\loc(X,\Lambda^{n,q}T^\star_X\otimes E)$, $q\ge 1$, 
satisfying $D''g=0$ and
$\int_X\gamma^{-1}|g|^2e^{-\varphi}\,dV_\omega<+\infty$, there exists
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)$ such that $D''f=g$ and
$$
\int_X|f|^2e^{-\varphi}\,dV_\omega\le
{1\over q}\int_X\gamma^{-1}|g|^2e^{-\varphi}\,dV_\omega.
$$
\endstat

\begproof{$($sketch$)$}. The general proof is based on regularization
techniques for psh functions (see e.g.\ (Demailly 1982)). It is
technically involved essentially because the required regularization
techniques are difficult in the case of arbitrary manifolds. We will
therefore just explain the proof in the simple case when $X=\Omega$ is
a weakly pseudoconvex open set in $\bbbc^n$ with a psh exhaustion
function~$\psi$. Then the function $\varphi_\varepsilon=\varphi\star
\rho_\varepsilon$ is well defined, smooth on
$\Omega_c=\{x\in\Omega\,;\,\psi(x)<c\}$ for $\varepsilon$ small
enough. Moreover, it satisfies a lower bound of the form
$$
\ii d'd''\varphi_\varepsilon\ge
(\gamma\omega-\ii\Theta(E))\star\rho_\varepsilon\ge
\gamma_\varepsilon\omega-\ii\Theta(E)
$$
for some continuous function $\gamma_\varepsilon$ converging uniformly to
$\gamma$ on compact subsets of $\Omega$ as $\varepsilon\to 0$.
We define new hermitian metrics $h_\varepsilon$ on the line bundle $E$ by
multiplying the original metric $h$ with the weight $e^{-\varphi_\varepsilon}$,
i.e., we set $h_\varepsilon=h\,e^{-\varphi_\varepsilon}$. Then
$$
\ii\Theta_{h_\varepsilon}(E)=\ii\Theta_h(E)+\ii\,d'd''\varphi_\varepsilon\ge
\gamma_\varepsilon\omega.
$$
We thus get solutions $f_{c,\varepsilon}$ on $X_c$ such that
$$
\int_{X_c}|f_{c,\varepsilon}|^2e^{-\varphi_\varepsilon}\,dV_\omega\le
{1\over q}\int_{X_c}\gamma_\varepsilon^{-1}|g|^2e^{-\varphi_\varepsilon}
\,dV_\omega
$$
whenever $\gamma_\varepsilon>0$ on $\ovl X_c$.
As $\varphi_\varepsilon\ge\varphi$ converges to $\varphi$ monotonically, 
we conclude by extracting weak limits and applying Lebesgue's monotone 
convergence theorem as before.\qed
\endproof

The next corollaries are simple special cases of H\"ormander's
estimates which are especially convenient in the case of bounded or 
unbounded pseudoconvex domains in~$\bbbc^n$.

\begstat{(8.10) Corollary} Let $\Omega$ be a bounded $($weakly$)$
pseudoconvex open set in~$\bbbc^n$, and let $\varphi$ be a psh
function on~$\Omega$. Then for any form
$g\in L^2(\Omega,\Lambda^{p,q}T^\star_\Omega)$, $q\ge 1$,
such that $d''g=0$, there exists
$f\in L^2(\Omega,\Lambda^{p,q-1}T^\star_\Omega)$ such that 
$d''f=g$ and
$$
\int_X|f|^2e^{-\varphi}\,dV_\omega\le
{e\over 2q}(\diam\Omega)^2\int_X|g|^2e^{-\varphi}\,dV_\omega
$$
$($All norms being computed with the standard metric 
${\ii\over 2}d'd''|z|^2.)$
\endstat

\begproof{} This is a special case of Corollary~8.9 when we take 
$E=\Omega\times\bbbc$ to be the trivial bundle equipped with the 
weight function
$$
\varphi_\varepsilon(z)=\varphi(z)+\varepsilon|z-z_0|^2,\qquad
z_0\in\Omega,\quad\varepsilon=1/(\diam\Omega)^2
$$
instead of $\varphi$.
We then find $\varphi\le\varphi_\varepsilon\le\varphi+1$ and
$\ii d'd''\varphi_\varepsilon\ge 2\varepsilon\omega$
on~$\Omega$. The~$L^2$ estimate follows immediately.\qed
\endproof

\begstat{(8.11) Corollary} Let $\Omega$ be a $($weakly$)$ pseudoconvex open
set in~$\bbbc^n$, and let $\varphi$ be a psh function on~$\Omega$.
Then for any form $g\in L^2_\loc(\Omega,\Lambda^{p,q}T^\star_\Omega)$, 
$q\ge 1$, satisfying $d''g=0$ and
$\int_\Omega|g|^2(1+|z|^2)^{2-\varepsilon}e^{-\varphi}\,dV<+\infty$ for some
$\varepsilon\in{}]0,+\infty[$, there exists 
$f\in L^2_\loc(\Omega,\Lambda^{p,q-1}T^\star_\Omega)$ such that $d''f=g$ and
$$
\int_X|f|^2(1+|z|^2)^{-\varepsilon}e^{-\varphi}\,dV_\omega\le
{1\over 2\varepsilon q}
\int_X|g|^2(1+|z|^2)^{2-\varepsilon}e^{-\varphi}\,dV_\omega
$$
$($All norms being computed with the standard metric 
${\ii\over 2}d'd''|z|^2.)$
\endstat

\begproof{} The proof is essentially the same as before, except
that we take
$$
\varphi_\varepsilon(z)=\varphi(z)+\varepsilon\log(1+|z|^2)
$$
Then the computations made in example 3.8~b) gives
$$
\ii d'd''\varphi_\varepsilon\ge\varepsilon\,\ii d'd''\log(1+|z|^2)\ge
{\varepsilon\,\ii d'd''|z|^2\over(1+|z|^2)^2}
={2\varepsilon\over(1+|z|^2)^2}\,\omega.
\eqno\square
$$
\endproof


\titleb{9.}{Some applications of H\"ormander's $L^2$ estimates}

The main applications concern three principal items: vanishing theorems for
Dolbeault cohomology groups, existence and approximation theorems for 
holomorphic functions (these aspects are in fact intimately related, as we
will see).

We first list four important vanishing theorems.

\begstat{(9.1) Nakano vanishing theorem {\rm(Nakano 1955 for the
compact case, Nakano 1973 in general)}} Let $E$ we a hermitian holomorphic vector
bundle on a weakly pseudoconvex complex manifold $X$, $\dim X=n$, such
that $\ii\Theta(E)>0$ in the sense of Nakano. Then $H^{n,q}(X,E)=0$ for
all $q\ge 1$.
\endstat

\begproof{} Indeed, (7.5) implies that 
$\langle[\ii\Theta(E),\Lambda]u,u\rangle$ is positive definite. Theorem~8.4 
shows that the equation $D''f=g$ can be solved whenever
$g$ is $D''$-closed and satisfies a suitable $L^2$-condition (moreover
Remark~8.5 implies that $f$ is smooth if $g$ is smooth). In fact, we want
to solve the equation for a given smooth $D''$-closed form, whatever is its 
growth at infinity. For this, we let $\psi$ be a smooth psh exhaustion 
function on $X$ and multiply the metric of $E$ by the weight factor
$e^{-\chi\circ\psi}$ where $\chi:\bbbr\to\bbbr$ is a convex increasing
function. If the resulting curvature tensor of $E$ is denoted $\Theta_\chi(E)$,
we find
$$
\ii\Theta_\chi(E)=\ii\Theta_0(E)+\ii\,d'd''(\chi\circ\psi)
=\ii\Theta_0(E)+\ii(\chi'\circ\psi\, d'd''\psi+\chi''\circ\psi\, d'\psi\wedge
d''\psi),
$$
and both terms $d'd''\psi$ and $d'\psi\wedge d''\psi$ yield nonnegative
contributions (in the sense of Nakano) to the curvature tensor. In particular
the resulting curvature operator $A_\chi$ on $(n,q)$-forms satisfies
$A_\chi\ge A$, $A_\chi^{-1}\le A^{-1}$ and we get
$$
\int_X\langle A_\chi^{-1}g,g\rangle e^{-\chi\circ\psi}dV_\omega
\le \int_X\langle A^{-1}g,g\rangle e^{-\chi\circ\psi}dV_\omega<+\infty
$$
when $\chi$ grows quickly enough [take e.g.\ $\chi$ so that
$$
e^{-\chi(k)}\int_{\{k\le\psi\le k+1\}}\langle A^{-1}g,g\rangle dV_\omega
\le 2^{-k}
$$
for every integer~$k\ge 0$]. We then get a smooth (minimal) $L^2$ solution $f$.
This implies $H^{n,q}(X,E)=0$ for $q\ge 1$, as desired.\qed
\endproof

\begstat{(9.2) Cartan theorem B {\rm(1953)}} Let $X$ be a strongly 
pseudoconvex manifold. Then $H^{p,q}(X,E)=0$ for every holomorphic vector
bundle $E$ and every~$q\ge 1$.
\endstat

\begproof{} Fix an arbitrary hermitian metric $h$ on $E$. By the above formula
$$
\ii\Theta_\chi(E) =\ii\Theta_0(E)
+\ii(\chi'\circ\psi\, d'd''\psi+\chi''\circ\psi\, d'\psi\wedge d''\psi),
$$
we see that the curvature of $E$ can be made positive definite if
the first derivative of $\chi$ grows fast enough. We then conclude that
$H^{n,q}(X,E)=0$ for $q\ge 1$ as for Theorem~9.1. To obtain the conclusion
for $(p,q)$-forms as well, we just observe that we have a canonical duality
pairing
$$
\Lambda^kT_X\otimes \Lambda^kT^\star_X\lra\bbbc,
$$
hence a $(p,q)$-form with values in $E$ can be viewed as a section of 
$$
\Lambda^{p,q}T^\star_X\otimes E=
\Lambda^{0,q}T^\star_X\otimes \Lambda^pT^\star_X\otimes E=
\Lambda^{n,q}T^\star_X\otimes \wt E
$$
where $\wt E$ is the holomorphic vector bundle
$$
\wt E=\Lambda^nT_X\otimes \Lambda^pT^\star_X\otimes E=
\Lambda^{n-p}T_X\otimes E,
$$
through the contraction pairing
$$
\Lambda^nT_X\otimes \Lambda^pT^\star_X\buildo\simeq\over\lra\Lambda^{n-p}T_X.
$$
Moreover the Dolbeault complex $\Lambda^{p,\bu}T^\star_X\otimes E$ is
isomorphic to the Dolbeault complex $\Lambda^{n,\bu}T^\star_X\otimes
\smash{\wt E}$, hence
$$
H^{p,q}(X,E)=H^{n,q}(X,\wt E)=0.\eqno\square
$$
\endproof

\begstat{(9.3) Akizuki-Kodaira-Nakano theorem {\rm(1954, also referred
to as ``precise vanishing theorem'')}} Let $E$ be a hermitian
holomorphic line bundle on a compact complex manifold $X$, $\dim X=n$,
such that $\ii\Theta(E)>0$.  Then the Dolbeault cohomology groups
vanish in the range $p+q\ge n+1$, i.e.
$$
H^{p,q}(X,E)=0\qquad\hbox{for $p+q\ge n+1$}.
$$
\endstat

\begproof{} Since $\ii\Theta(E)$ is a closed positive $(1,1)$-form, one can
select $\omega=\ii\Theta(E)$ as the basic K\"ahler metric on $X$ (in 
particular, $X$ is automatically K\"ahler). Then the eigenvalues of
$\ii\Theta(E)$ with respect to $\omega$ are 
$$
\gamma_1=\cdots=\gamma_n=1,
$$
and (7.7) implies
$$
\langle[\ii\Theta(E),\Lambda]u,u\rangle
\ge (q-(n-p))|u|^2=(p+q-n)|u|^2.
$$
We then get the conclusion from Theorem~8.4.\qed
\endproof

\begstat{(9.4) Kodaira-Serre vanishing theorem {\rm(1953, ``unprecise 
vanishing theorem'')}} Let $E$ we a hermitian holomorphic line bundle
on a compact complex manifold $X$ such that $\ii\Theta(E)>0$. 
Then for every holomorphic vector bundle $F$, there exists an integer 
$k_0=k_0(F)$ such that
$$
H^{p,q}(X,E^{\otimes k}\otimes F)=0 \qquad
\hbox{for all $p\ge 0$, $q\ge 1$, $k\ge k_0$}.
$$
\endstat

\begproof{} If $p=n$, we reduce (9.4) to the Nakano vanishing theorem.
In fact, for any pair of hermitian holomorphic vector bundles $F$, $G$, 
the Chern connections of $F$, $G$ and $F\otimes G$ are related by
$$
D_{F\otimes G}(u\otimes v)=D_Fu\otimes v+(-1)^{\deg u}u\otimes D_G v,
$$
and this implies easily
$$
\Theta(F\otimes G)=\Theta(F)\otimes\Id_G+\Id_F\otimes G.
\leqno(9.5)
$$
In particular, as $E$ is a line bundle, we have $\Theta(E^{\otimes k})=
k\,\Theta(E)$ (with the identification $\End(E)=\bbbc$), hence
$$
\Theta(E^{\otimes k}\otimes F)=k\,\Theta(E)\otimes\Id_F+\Theta(F),
$$
again with the identifications $\End(E)=\End(E^{\otimes k})=\bbbc$.
In other words, the associated hermitian form of $T_X\otimes F$ satisfies
$$
\eqalign{
\wt\Theta(E^{\otimes k}\otimes F)(\xi\otimes v,\xi\otimes v)
&=k\,\wt\Theta(F)(\xi,\xi)|v|^2+\wt\Theta(F)(\xi\otimes v,\xi\otimes v)\cr
\wt\Theta(E^{\otimes k}\otimes F)(\tau,\tau)
&\ge k|\tau|^2+\wt\Theta(F)(\tau,\tau)\cr}
$$
for all elements $\xi\otimes v,\,\tau\in T_X\otimes F$, when
$\omega=\ii\,\Theta(E)$ is taken as the K\"ahler metric on~$X$. Hence
$E^{\otimes k}\otimes F$ is Nakano positive for $k\ge k_0$ large enough,
and we infer $H^{n,q}(X,E^{\otimes k}\otimes F)=0$ for $q\ge 1$ and
$k\ge k_0$. The case of $(p,q)$-cohomology groups is obtained by replacing
$F$ with 
$$
\wt F=\Lambda^{n-p}T_X\otimes F
$$
as in the proof of 9.2.\qed
\endproof

The next application of $L^2$ estimates is the solution of the
so-called Levi problem. In vague terms, the Levi problem asserts that
the existence of holomorphic functions on a complex manifold $X$ is
intimately related to its pseudoconvexity properties.  Complex
analysts became aware of the question with the foundational paper of
(E.E.~Levi, 1910). The final solution for domains of $\bbbc^n$ has
been finally settled in three independent papers by (Oka 1953),
(Norguet 1954) and (Bremermann, 1954). The generalization to complex
manifolds is due to (Grauert, 1958); it gives a characterization of
the so called ``Stein manifolds'', which were introduced by K.~Stein
and H.~Cartan in the early fifties.

\begstat{(9.6) Concept of holomorphic convexity} Let $X$ be a complex
manifold and let $A\subset X$ be a closed subset. The holomorphic
hull $\wh A_{\cO(X)}$ is defined to be
$$
\wh A_{\cO(X)}=\big\{x\in X\,;\;|f(x)|\le\sup_A |f|\big\}.
$$
The subset $A$ is said to be holomorphically convex $($in~$X)$ if 
$\wh A_{\cO(X)}=A$. The manifold $X$ is said to be holomorphically convex 
if $\wh K$ is compact for every compact set~$K$, or equivalently, if
$X$ can be exhausted by holomorphically convex compact sets $K_\nu$
$($we say that $X$ is ``exhausted'' by the $K_\nu$'s if $X=\bigcup K_\nu$
and $K_\nu\subset K_{\nu+1}^\circ$ for all~$\nu)$.
\endstat

Observe that $\wh A_{\cO(X)}$ is a closed subset of $X$ and that
$\wh{\kern-1pt\wh A\kern1pt}_{\cO(X)}=\wh A_{\cO(X)}$, i.e., $\wh
A_{\cO(X)}$ is holomorphically convex in~$X$. Hence, if $X$ is
holomorphically convex, we get inductively an exhausting and strictly 
increasing sequence of holomorphically convex compact sets $K_\nu$ by 
putting
$$
K_\nu=\wh{(S_\nu)}_{\cO(X)},\qquad
S_\nu=\hbox{a compact neighborhood of $K_{\nu-1}\cup L_\nu$},
$$ 
where $L_\nu$ is any exhausting sequence of compact sets.

A similar concept of ``pseudoconvex hull'' $\wh A_{P^\infty(X)}$ with
respect to the class $P^\infty(X)= \Psh(X)\cap C^\infty(X)$ exists,
namely one can set
$$
\wh A_{P^\infty(X)}=\big\{x\in X\,;\;\varphi(x)\le\sup_A\varphi\big\}.
\leqno(9.7)
$$ 
Since $\varphi=|f|^2\in P^\infty(X)$ for every $f\in\cO(X)$, the
inclusion $\wh A_{P^\infty(X)}\subset \wh A_{\cO(X)}$ always holds. It
is not hard to see that a manifold $X$ is {\it weakly pseudoconvex} if
and only if $\wh K_{P^\infty}(X)$ is compact for every compact set $K$
in~$X\,$; in fact if $\psi$ is a psh exhaustion function, then
$K_\nu=\{x\in X\,;\,\psi(x) \le\nu\}$ is an exhausting sequence of
pseudoconvex compact sets; conversely if such a sequence $K_\nu$ with
$\wh{(K_\nu)}_{P^\infty(X)}=K_\nu$ exists, we define a psh exhaustion
$\psi$ by putting
$$
\psi=\sum_{\nu=0}^{+\infty}2^\nu\chi\circ\varphi_\nu
$$
where $\chi:\bbbr\to\bbbr$ is a smooth convex increasing function such that
$\chi(t)=0$ for $t\le 0$, $\chi(t)\ge 1$ for $t\ge 1$, and 
$\varphi_\nu\in P^\infty(X)$ 
is chosen such that $\varphi_\nu\le 0$ on $K_\nu$ and $\varphi_\nu\ge 1$
on $K_{\nu+2}\ssm K_{\nu+1}^\circ$ (a finite ``regularized'' maximum
$\varphi_\nu=(\max\star\rho_\varepsilon)(\varphi_{\nu,1}\ld\varphi_{\nu,N})$
with $\varphi_{\nu,j}(x_j)>1>0>\sup_{K_\nu}\varphi_{\nu,j}$ at
sufficiently many points $x_j\in K_{\nu+2}\ssm K_{\nu+1}^\circ$ will do
by the Borel-Lebesgue lemma).

\begstat{(9.8) Stein manifolds} Let $X$ be a $n$-dimensional complex
manifold. Then $X$ is said to be a Stein manifold if it satisfies the
following two properties.
\smallskip
\item{\rm a)} $X$ is holomorphically convex\/$;$
\smallskip
\item{\rm b)} $\cO(X)$ locally separates points, in the
sense that 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

We first prove the ``easy direction'' in the Levi problem, namely
the implication
$$
\hbox{Stein${}\Rightarrow{}$strongly pseudoconvex,}
$$
which depends only on elementary considerations about psh
functions. The converse (deeper) implication $\Leftarrow$ can be
proved using $L^2$ estimates.

\begstat{(9.9) Theorem} Let $X$ be a complex manifold.
\smallskip
\item{\rm a)} If $X$ is holomorphically convex, then
$X$ is weakly pseudoconvex.
\smallskip
\item{\rm b)} If $\cO(X)$ satisfies the local separation property {\rm
9.8$\,$b)}, there exists a smooth nonnegative strictly
plurisubharmonic function $u\in\Psh(X)$.
\smallskip
\item{\rm c)} If $X$ is Stein, then $X$ is strongly pseudoconvex.
\endstat

\begproof{} a) If $X$ is holomorphically convex, we have seen that there
is an exhausting sequence $(K_\nu)$ of holomorphically convex compact 
sets of $X$. These compact sets then satisfy $\wh{(K_\nu)}_{\cO(X)}=
\wh{(K_\nu)}_{P^\infty(X)}=K_\nu$, hence $X$ is weakly pseudoconvex.
\smallskip

\noindent
b) 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
$$
u_0(z)=\cases{
v_0(z)&on $X\ssm B$,\cr
\max_\varepsilon\{v_0(z),(|z|^2+1)/3\}&on $B$.\cr}
$$ 
where $\max_\varepsilon=\max(\bu,\bu)\star\rho_\varepsilon$ is a
regularized max function in $\bbbr^2$.  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~$X$.
\smallskip

\noindent
c) Select $\psi$ as in a) and $u$ as in b). Then $\psi+u$ is a strictly
psh exhaustion function of~$X$.\qed
\endproof

Conversely, we have the following existence theorem derived from
$L^2$ estimates.

\begstat{(9.10) Theorem} Let $X$ be a strongly pseudonconvex
manifold. For every locally finite sequence $(x_\nu)$ of distinct
points of $X$ and every sequence of polynomials $P_\nu(z^{\nu})$
relative to local coordinates $z^\nu=(z^\nu_1\ld z^\nu_n)$
around~$x_\nu$ $($with given bounds \hbox{$\deg P_\nu\le m_\nu$} for the
degrees$)$, there is a global holomorphic function $f\in\cO(X)$ such
that the Taylor expansion of order $m_\nu$ of $f$ at $x_\nu$ coincides
with $P_\nu$.
\endstat

\begproof{} The main idea is to use a $L^2$ estimate with a weight assuming
a logarithmic pole at each point~$x_\nu$. Let $U_\nu$ be the open
coordinate patch where $z^\nu$ is defined and let $\theta_\nu\in\cD(U_\nu)$ 
be a cut-off function such that $\theta_\nu=1$ on a neighborhood of $x_\nu$
and $0\le\theta_\nu\le 1$. Then
$$
\varphi_0=\sum_\nu 2(n+m_\nu)\theta_\nu\log|z_nu|
$$ 
is psh in a neighborhood of $x_\nu$ and $\varphi_0$ is smooth on
$X\ssm\{x_\nu\}$.  It follows that the negative part of
$\ii\,d'd''\varphi_0$ is locally bounded below everywhere. Hence, if
$\psi$ is a strictly psh exhaustion function on~$X$, there exists a
convex increasing function $\chi_1:\bbbr\to\bbbr$ such that
$\varphi_1=\varphi_0+\chi_1\circ\psi$ is psh (with an arbitrarily
large positive preassigned Hessian form $\ii\,d'd''\varphi_1$).  Now,
the function $F=\sum\theta_\nu P_\nu$ is a smooth function which has
the prescribed Taylor expansions at all points~$x_\nu$. The idea is to
solve the equation
$$
d''u=v:=d''F
$$
where $v$ is a $(0,1)$-form with complex values, that is, a $(n,1)$-form
with values in $\Lambda^nT_X$. We assume that
$\ii\Theta(\Lambda^nT_X)+\ii\,d'd''\varphi_1\ge\omega:=\ii\,d'd''\psi$.
Then, for every convex increasing function $\chi$ and 
$\varphi=\varphi_1+\chi\circ\psi=\varphi_0+(\chi+\chi_1)\circ\psi$, 
we get a solution $u$ such that
$$
\int_X |u|^2e^{-\varphi}dV_\omega\le\int_X|v|^2e^{-\varphi}dV_\omega
=\sum_\nu\int_{U_\nu}|P_\nu|^2|d''\theta_\nu|^2e^{-2(n+m_\nu)\theta_\nu
\log|z^\nu|+\chi\circ\psi}dV_\omega,
$$
provided that the right hand side is convergent. However, each term in
the sum is convergent since  the support of $d''\theta_\nu$ avoids
a neighborhood of $x_\nu$, and the global convergence is easily guaranteed
if $\chi$ grows fast enough. As $d''$ is elliptic in bidegree $(0,0)$, 
we get a smooth solution $u$ such that
$$
\int_{U_\nu}{|u|^2\over|z^\nu|^{2(n+m_\nu)}}dV_\omega<+\infty
$$
for each~$\nu$. From this one concludes that $D^\alpha u(x_\nu)=0$
for $|\alpha|\le m_\nu$, hence $f=F-u$ is a holomorphic function on $X$
with $P_\nu$ as its Taylor expansion of order $m_\nu$ at~$x_\nu$.\qed
\endproof

\begstat{(9.11) Corollary} Every strongly pseudoconvex manifold $X$ is Stein.
Moreover, the functions in $\cO(X)$ separate any pair of distinct points
in~$X$, and for $x_0\in X$ given, there are functions $f_1\ld f_n\in\cO(X)$
such that $(f_1\ld f_n)$ is a local coordinate system at~$x_0$.
\endstat

\begproof{} Let $X$ be a complex manifold. The holomorphic convexity
property of $X$ is formally equivalent to the following assertion: for
every sequence $(x_\nu)$ in $X$, there exists a holomorphic function
$f\in\cO(X)$ such that $(f(x_\nu))$ is unbounded (the equivalence can
be seen more or less by the same argument as in 9.9~a)). By
Theorem~9.10 we need only take a function $f$ which interpolates the
values $f(x_\nu)=\nu$. The property of local separation of points
9.9~b) is also clear, as well as the stronger properties asserted in
Corollary~9.11.\qed
\endproof

We end this section by proving a general $n$-dimensional version of
the Runge theorem.

\begstat{(9.12) Runge approximation theorem} Let $E$ be a holomorphic
vector bundle on a Stein manifold $X$. Let $\varphi$ be a psh exhaustion
of $X$ and let
$$
K=K_c=\{x\in X\,;\;\varphi(x)\le c\}
$$ 
for some~$c$ $(\varphi$ need not be strictly psh$)$. Then every
holomorphic section $g$ defined on a neighborhood of $K$ is a uniform
limit on a neighborhood of $K$ of a sequence of global holomorphic
sections~$f_\nu$ of $E$ over~$X$.
\endstat

\begproof{} Fix $c_i$, $i=1,2,3,4$ such that $c<c_1<c_2<c_3<c_4$ and
$g$ is holomorphic on $\Omega_{c_4}=\{x\in X\,;\;\varphi(x)\le c_4\}$. 
Fix a cut-off function $\theta\in\cD(X)$ with
$\Supp\theta\subset\Omega_{c_4}$ and $\theta=1$ on $\Omega_{c_3}$. We
view $g\theta$ as a smooth function on~$X$ (by defining it to be $0$ 
on $X\ssm\Omega_{c_4}$), and solve the equation
$$
d''u=v:=d''(g\theta)=g\,d''\theta
$$
with a weight of the form $\varphi_\nu=\nu\varphi+\chi\circ\psi$, where 
$\psi$ is a strictly psh exhaustion function and $\chi$ a convex increasing
function such that the resulting curvature eigenvalues are${}\ge 1$. We 
then get a solution $u=u_\nu$ such that
$$
\int_X|u_\nu|^2e^{-\nu\varphi-\chi\circ\psi}dV_\omega\le
\int_X|g|^2|d''\theta|^2e^{-\nu\varphi-\chi\circ\psi}dV_\omega.
$$
As $d''\theta$ has support in $\Omega_{c_4}\ssm\Omega_{c_3}\subset\{
\varphi\ge c_3\}$, the right hand side is bounded by $C\,e^{-\nu c_3}$. 
From the $L^2$ estimate, we infer
$$
\int_{\Omega_{c_2}}|u_\nu|^2dV_\omega\le
C'e^{\nu c_2}\int_{\Omega_{c_2}}|u_\nu|^2
e^{-\nu\varphi-\chi\circ\psi}dV_\omega\le C''e^{\nu c_2}e^{-\nu c_3}
\buildu{\nu\to+\infty}\under\lra 0.
$$ 
Finally, as $d''u_\nu=g\,d''\theta=0$ on $\Omega_{c_3}$, the function
$|u_\nu|^2$ is plurisubharmonic on $\Omega_{c_2}$. By the mean value
inequality, we see that $u_\nu$ converges uniformly to $0$
on~$\Omega_{c_1}$.  Hence $f_\nu=g\theta-u_\nu\in\cO(X)$ converges
uniformly to $f$ on~$\Omega_{c_1}$.\qed
\endproof

\begstat{(9.13) Remark} \rm The assumption that $K=\{\psi\le c\}$ for
some psh exhaustion function $\psi$ is satisfied if and only if $\wh
K_{P^\infty(X)}=K$ (in which case we say that $K$ is {\it pseudoconvex}).
In fact, we have the following lemma.
\endstat

\begstat{(9.14) Lemma} Let $X$ be a weakly $($resp.\ strongly$)$ 
pseudoconvex manifold with a weakly $($strictly$)$ psh exhaustion 
function~$\psi_0$. If $K\subset X$ is a
pseudoconvex compact set, i.e., if $\wh K_{P^\infty(X)}=K$, there
exists an exhaustion function $\varphi\in P^\infty(X)$ such that
\smallskip
\item{\rm a)} $\varphi=0$ on $X\,;$
\smallskip
\item{\rm b)} $\varphi>0$ and $\varphi$ is weakly $($strictly$)$ psh on $X\ssm K$.
\endstat

\begproof{} Let $\chi:\bbbr\to\bbbr$ be a convex increasing with 
$\chi(t)=0$ for $t\le 0$ and $\chi(t)>0$ for $t>0$. For any
$x\notin K$, there is a function $\varphi_x\in P^\infty(X)$ such that
$$
\varphi_x(x)>0>\sup_K\varphi_x.
$$ 
We set $u_x=\chi(\varphi_x+\delta\psi_0)$ ($\delta>0$ small), so as
to get a nonnegative function $u_x\in P^{\infty}(X)$ such that
$u_x=0$ on $K$ and $u_x>0$ $($strictly psh$)$ on a neighborhood
$V_x$ of~$x$. Then $X\ssm K$ can be covered by countably many
such neighborhoods $V_{x_\nu}$ and we get a function $\varphi$ with
the required properties by setting
$$
\varphi=\chi(\psi_0-C)+\sum\varepsilon_\nu u_{x_\nu}
$$
for some large constant~$C$.\qed
\endproof

\begstat{(9.15) Corollary} If $X$ is strongly pseudoconvex, then
$$
\wh K_{P^\infty(X)}=\wh K_{\cO(X)}
$$
for every compact set $K\subset X$.
\endstat

\begproof{} Set $\wt K=\wh K_{P^\infty}(X)$. Then $\wt K$ is pseudoconvex
and we thus get by Lemma~9.14 a function $\varphi\in P^\infty(X)$ with
$\varphi=0$ on $K$ and $\varphi>0$ strictly psh on $X\ssm K$. 
Fix an arbitrary point $x_0\in X\ssm K$, a coordinate system
$z=(z_1\ld z_n)$ centered at $x_0$, and a cut-off function $\theta$
equal to $1$ on a neighborhood of $x_0$, with support disjoint from~$K$. 
For $0<\varepsilon\ll 1$, the function
$$
\varphi_\varepsilon(x)=\varphi(x)+\varepsilon\theta(x)\log(|z|^2+
e^{-1/\varepsilon^2})
$$
is psh everywhere and $\varphi_\varepsilon(x_0)=\varphi(x_0)-
1/\varepsilon<0$. Thus $K_\varepsilon=\{\varphi_\varepsilon\le 0\}$ is
equal to $K$ union a small neighborhood $\ovl V_\varepsilon$ of $x_0$
disjoint from~$K$. We define a holomorphic function $g$ on a
neighborhood of $K_\varepsilon$ by taking $g=0$ on a neighborhood of
$K$ and $g=1$ on a neighborhood of $\ovl V_\varepsilon$. The Runge 
approximation theorem provides a global holomorphic function $f\in\cO(X)$
such that $|f-g|\le 1/3$ on $K_\varepsilon$. We thus get $|f|\le 1/3$ on
$K$ and $|f(x_0)|\ge 2/3$. This implies that every $x_0\notin\wt K$ is not
either in the holomorphic hull of~$\wt K$, hence $\wt K$ is holomorphically
convex. From this we infer
$$
\wh K_{\cO(X)}\subset \wh{(\wt K)}_{\cO(X)}=\wt K=\wh K_{P^\infty(X)},
$$
and the opposite inclusion is clear.\qed
\endproof

\titleb{10.}{Further preliminary results of hermitian differential geometry}
In the course of the proof of Skoda's $L^2$ estimates, we will have to
deal with dual bundles and exact sequences of hermitian vector bundles. 
The following fundamental differential geometric lemma will be needed.

\begstat{(10.1) Lemma} Let $E$ be a hermitian holomorphic vector bundle
of rank $r$ on a complex $n$-dimensional manifold~$X$.
Then the Chern connections of $E$ and $E^\star$ are related by
$\Theta(E^\star)=-{}^t\Theta(E)$ where ${}^t$ denotes transposition.
In other words, the associated hermitian forms $\wt\Theta(E)$ and
$\wt\Theta(E^\star)$ are related by
$$
\eqalign{
\wt\Theta(E)(\tau,\tau)&=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\tau_{j\lambda}\ovl\tau_{k\mu},\qquad
\tau=\sum_{j,\lambda}\tau_{j,\lambda}{\partial\over\partial z_j}\otimes
e_\lambda,\cr
\wt\Theta(E^\star)(\tau,\tau)&=-\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\mu\lambda}\tau^\star_{j\lambda}\ovl\tau_{k\mu}^\star,\qquad
\tau^\star=\sum_{j,\lambda}\tau^\star_{j,\lambda}
{\partial\over\partial z_j}\otimes e^\star_\lambda.\cr}
$$
In particular $E>_\Grif 0$ if and only if $E^\star<_\Grif 0$.
\endstat

Notice that the corresponding duality statement for Nakano positivity
is wrong (because of the twist of indices, which is fortunately irrelevant
in the case of decomposable tensors).

\begproof{} The Chern connections of $E$ and $E^\star$ are related by the
Leibnitz rule
$$
d(\sigma\wedge s)=(D_{E^\star}\sigma)\wedge s+(-1)^{\deg\sigma}\sigma\wedge
D_E s
$$
whenever $s$, $\sigma$ are forms with values in $E$, $E^\star$ respectively,
and $\sigma\wedge s$ is computed using the pairing $E^\star\otimes E\to\bbbc$.
If we differentiate a second time, this yields the identity
$$
0=(D_{E^\star}^2\sigma)\wedge s+\sigma\wedge D_E^2s,
$$
which is equivalent to the formula $\Theta(E^\star)=-{}^t\Theta(E)$. All
other assertions follow.\qed
\endproof

\begstat{(10.2) Lemma} Let
$$
0\lra S\buildo j\over\lra E\buildo g\over\lra Q\lra 0
$$
be an exact sequence of holomorphic vector bundles. Assume that $E$ is 
equipped with a smooth hermitian metric, and that $S$ and $Q$ are endowed 
with the metrics (restriction-metric and quotient-metric) induced by that 
of $E$. Then 
$$
j^\star\oplus g:E\to S\oplus Q,\qquad
j\oplus g^\star:S\oplus Q\to E
\leqno(10.3)
$$ 
are $C^\infty$ isomorphisms of bundles, which are inverse of each other. 
In the $\ci$-splitting $E\simeq S\oplus Q$, the Chern connection of $E$ 
admits a matrix decomposition
$$
D_E = \pmatrix{D_S & -\beta^\star\cr \beta & D_Q\cr}
\leqno(10.4)
$$
in terms of the Chern connections of $S$ and $Q$, where
$$
\beta\in\ci\big(X,\Lambda^{1,0}T^\star_X\otimes \Hom(S,Q)\big),\qquad
\beta^\star\in\ci\big(X,\Lambda^{0,1}T^\star_X\otimes \Hom(Q,S)\big).
$$
The form $\beta$ is called the second fundamental form associated with
the exact sequence. It is uniquely defined by each of the two formulas
$$
D'_{\Hom(S,E)}j=g^\star\circ\beta,\qquad 
j\circ\beta^\star=-D''_{\Hom(Q,E)}g^\star.
\leqno(10.5)
$$
We have $D'_{\Hom(S,Q)}\beta=0$, $D''_{\Hom(Q,S)}\beta^\star=0$, and the 
curvature form of $E$ splits as
$$
\Theta(E) = \pmatrix{
\Theta(S)-\beta^\star\wedge\beta & -D'_{\Hom(Q,S)}\beta^\star\cr 
D''_{\Hom(S,Q)}\beta & \Theta(Q)-\beta\wedge\beta^\star\cr},
\leqno(10.6)
$$
and the curvature forms of $S$ and $Q$ can be expressed as
$$
\Theta(S)=\Theta(E)_{\restriction S}+\beta^\star\wedge\beta,\qquad
\Theta(Q)=\Theta(E)_{\restriction Q}+\beta\wedge\beta^\star,
\leqno(10.7)
$$
where $\Theta(E)_{\restriction S}$, $\Theta(E)_{\restriction Q}$ stand for
$j^\star\circ\Theta(E)\circ j$ and $g\circ\Theta(E)\circ g^\star$.
\endstat

\begproof{} Because of the uniqueness property of Chern connections, it
is easy to see that we have a Leibnitz formula
$$
D_F(f\wedge u)=(D_{\Hom(E,F)}f)\wedge u +(-1)^{\deg f}f\wedge D_E u
$$
whenever $u,\,f$ are forms with values in hermitian vector bundles 
$E$ and $\Hom(E,F)$ 
(where $\Hom(E,F)=E^\star\otimes F$ is equipped with the tensor product 
metric and $f\wedge u$ incorporates the evaluation mapping
$\Hom(E,F)\otimes E\to F$). In our case, given a form $u$ with values in $E$,
we write $u=ju_S+g^\star u_Q$ where $u_S=j^\star u$ and $u_Q=gu$ are the 
projections of $u$ on~$S$ and~$Q$. We then get
$$
\eqalign{
D_Eu&=D_E(ju_S+g^\star u_Q)\cr
&=(D_{\Hom(S,E)}j)\wedge u_S+ j\cdot D_Su_S+
(D_{\Hom(Q,E)}g^\star)\wedge u_Q+g^\star\cdot D_Qu_Q.\cr}
$$
Since $j$ is holomorphic as well as $j^\star\circ j=\Id_S$, we find
$D''_{Hom(S,E)}j=0$ and
$$
D''_{Hom(S,S)}\Id_S=0=D''_{\Hom(E,S)}j^\star\circ j.
$$
By taking the adjoint, we see that $j^\star\circ D'_{\Hom(S,E)}j=0$,
hence $D'_{\Hom(S,E)}j$ takes values in~$g^\star Q$ and we thus have a
unique form $\beta$ as in the Lemma such that
\hbox{$D'_{\Hom(S,E)}j=g^\star\circ\beta$}. Similarly, $g$ and 
$g\circ g^\star=\Id_Q$ are holomorphic, thus
$$
D''_{Hom(Q,Q)}\Id_Q=0=g\circ D''_{\Hom(Q,E)}g^\star
$$
and there is a form $\gamma\in
\ci\big(X,\Lambda^{0,1}T^\star_X\otimes \Hom(Q,S)\big)$ such that
\hbox{$D''_{\Hom(Q,E)}g^\star=j\circ\gamma$}. By adjunction, we get
$D'_{\Hom(E,Q)}g=\gamma^\star\circ j^\star$ and $D''_{Hom(E,Q)}g=0$ 
implies $D'_{\Hom(Q,E)}g^\star=0$. If we differentiate $g\circ j=0$ we
then get
$$
0=D'_{\Hom(E,Q)}g\circ j+g\circ D'_{\Hom(S,E)}j=
\gamma^\star\circ j^\star\circ j+g\circ g^\star\circ\beta=\gamma^\star+\beta,
$$
thus $\gamma=-\beta^\star$ and $D''_{\Hom(Q,E)}g^\star=-j\circ\beta^\star$.
Combining all this, we get
$$
\eqalign{
D_Eu&=g^\star\beta\wedge u_S+j\cdot D_Su_S-
j\beta^\star\wedge u_Q+g^\star\cdot D_Qu_Q\cr
&=j\big(D_Su_S-\beta^\star\wedge u_Q\big)
+g^\star\big(\beta\wedge u_S+D_Qu_Q\big),\cr}
$$
and the asserted matrix decomposition formula follows. By squaring the
matrix, we get
$$
D_E^2 = \pmatrix{
D_S^2-\beta^\star\wedge\beta & -D_S\circ\beta^\star-\beta^\star\circ D_Q\cr 
D_Q\circ\beta+\beta\circ D_S& D_Q^2-\beta\wedge\beta^\star\cr}.
$$
As $D_Q\circ\beta+\beta\circ D_S=D_{\Hom(S,Q)}\beta$ and
$D_S\circ\beta^\star+\beta^\star\circ D_Q=D_{\Hom(Q,S)}\beta^\star$
by the Leibnitz rule,
the curvature formulas follow (observe, since the Chern curvature form
is of type $(1,1)$, that we must have $D'_{\Hom(S,Q)}\beta=0$, 
$D''_{\Hom(Q,S)}\beta^\star=0$).\qed
\endproof

\begstat{(10.8) Corollary} Let $0\to S \to E \to Q \to 0$ be an exact
sequence of hermitian vector bundles. Then
\medskip
\item{\rm a)} $E\ge_\Grif 0~~\Longrightarrow~~Q\ge_\Grif 0,$
\smallskip
\item{\rm b)} $E\le_\Grif 0~~\Longrightarrow~~S\le_\Grif 0,$
\smallskip
\item{\rm c)} $E\le_\Nak0~~\Longrightarrow~~S\le_\Nak 0,$
\smallskip
\noindent and analogous implications hold true for strict positivity.
\endstat

\begproof{} If $\beta$ is written $\sum dz_j \otimes \beta_j$,~$\beta_j 
\in \Hom(S,Q)$, then formulas (10.7) yield
$$\eqalign{
\ii\Theta(S) &= \ii\Theta(E)_{\restriction S} - \sum dz_j \wedge d\ovl z_k
\otimes \beta^\star_k \beta_j,\cr
\ii\Theta(Q) &= \ii\Theta(E)_{\restriction Q} + \sum dz_j \wedge d\ovl z_k
\otimes \beta_j\beta^\star_k.\cr}$$
Since $\beta\cdot(\xi\otimes s)= \sum \xi_j\beta_j\cdot s$ and $\beta^\star
\cdot(\xi\otimes s)= \sum \ovl \xi_k\beta^\star_k\cdot s$ we get
$$\wt\Theta(S)(\xi\otimes s,\xi'\otimes s') = \wt\Theta(E)(\xi\otimes s,\xi'\otimes s')
- \sum_{j,k} \xi_j \ovl \xi'_k \langle \beta_j\cdot s,\beta_k\cdot s'\rangle,$$
$$\wt\Theta(S)(u,u) = \wt\Theta(E)(u,u) - |\beta\cdot u|^2,$$
$$\wt\Theta(Q)(\xi\otimes s,\xi'\otimes s') = \wt\Theta(E)(\xi\otimes s,\xi'\otimes s')
+ \sum_{j,k} \xi_j \ovl \xi'_k \langle \beta^\star_k\cdot s,\beta^\star_j\cdot 
s'\rangle,
$$
$$
\wt\Theta(Q)(\xi\otimes s,\xi\otimes s) = \wt\Theta(E)(\xi\otimes s,\xi\otimes
s) = |\beta^\star\cdot(\xi\otimes s)|^2.\eqno{\square}
$$
\endproof

Next, we need positivity properties which somehow interpolate between 
Griffiths and Nakano positivity. This leads to the concept of $m$-tensor
positivity.

\begstat{(10.9) Definition} Let $T$ and $E$ be complex vector spaces of 
dimensions $n,\,r$ respectively, and let $\Theta$ be a hermitian form on 
$T\otimes E$.
\smallskip
\item{\rm a)} A tensor $u\in T\otimes E$ is said to be of rank $m$ if $m$
is the smallest $\ge 0$ integer such that $u$ can be written
$$u = \sum_{j=1}^m \xi_j \otimes s_j,~~~~\xi_j\in T,~s_j\in E.$$
\smallskip
\item{\rm b)} $\Theta$ is said to be $m$-tensor positive $($resp. 
$m$-tensor semi-positive$)$ if
$\Theta(u,u)>0$ $($resp. $\Theta(u,u)\ge 0)$ for every tensor 
$u\in T\otimes E$ of rank $\le m$, $u\ne 0$. In this case, we write
$$
\Theta >_m 0~~~~(\hbox{\it resp.}~~\Theta \ge_m 0).
$$
\vskip-\belowdisplayskip
\endstat

We say that a hermitian vector bundle $E$ is $m$-tensor positive 
if $\wt\Theta(E)>_m 0$. Griffiths positivity corresponds to $m=1$ and 
Nakano positivity to $m\ge \min(n,r)$. Recall from (7.6) that we have
$$
\langle[\ii\Theta(E),\Lambda]u,u\rangle=\sum_{|S|=q-1}\sum_{j,k,\lambda,\mu}
c_{jk\lambda\mu}\,u_{jS,\lambda}\ovl u_{kS,\mu}
$$
for every $(n,q)$-form $u=\sum u_{K,\lambda}\,dz_1\wedge
\ldots\wedge dz_n\wedge d\ovl z_K\otimes e_\lambda$ with values in~$E$.
Since $u_{jS,\lambda}=0$ for $j\in S$, the rank of the tensor
$(u_{jS,\lambda})_{j,\lambda}\in\bbbc^n\otimes\bbbc^r$ is in fact 
$\le\min\{n-q+1,r\}$. We obtain therefore:

\begstat{(10.10) Lemma} Assume that $E\ge_m0$ $($resp.\ $E>_m0)$. Then the
hermitian operator $[\ii\Theta(E),\Lambda]$ is semipositive $($resp.\ positive 
definite$)$ on $\Lambda^{n,q}T^\star X\otimes E$ for $q\ge 1$ and 
$m\ge\min\{n-q+1,r\}.$
\endstat

The Nakano vanishing theorem can then be improved as follows.

\begstat{(10.11) Theorem} Let $X$ be a weakly pseudoconvex K\"ahler 
manifold of dimension $n$ and let $E$ a hermitian vector bundle of 
rank $r$ such that $\wt\Theta(E)>_m0$ over~$X$. Then
$$
H^{n,q}(X,E)=0~~~~\hbox{\rm for}~~q\ge 1~~\hbox{\rm and}~~
m\ge\min\{n-q+1,r\}.
$$
\endstat

We next study some important relations which exist between the various
positivity concepts. Our starting point is the following result of
(Demailly-Skoda 1979).

\begstat{(10.12) Theorem} For any hermitian vector bundle $E$, 
$$E >_\Grif 0~~\Longrightarrow~~E\otimes \det E >_\Nak 0.$$
\endstat

To prove this result, we use the fact that
$$
\Theta(\det E)=\Tr_E \Theta(E)\leqno(10.13)
$$
where $\Tr_E:\Hom(E,E)\to\bbbc$ is the trace map, together with the identity
$$
\Theta(E\otimes \det E) = \Theta(E)+\Tr_E(\Theta(E)) \otimes \Id_E,
$$
which is a special case of formula (9.5). Formula (10.13) is easily 
obtained by differentiating twice a wedge product, according to the formula
$$
D_{\Lambda^pE}(s_1\wedge\cdots s_p)=\sum_{j=1}^p(-1)^{\deg s_1+\cdots+\deg
s_{j-1}} s_1\wedge\cdots\wedge s_{j-1}\wedge D_E s_j\wedge\cdots\wedge s_p.
$$
We then get $\Theta(E\otimes \det E)=\Theta(E)+\Tr_E\Theta(E)\otimes\Id_E$.
The corresponding hermitian forms on $T_X\otimes E$ are thus related by
$$
\wt\Theta(E\otimes \det E)= \wt\Theta(E)+\Tr_E\wt\Theta(E) \otimes h,
$$
where $h$ denotes the hermitian metric on $E$ and $\Tr_E\wt\Theta(E)$
is the hermitian form on $T_X$ defined by
$$
\Tr_E \wt\Theta(E)(\xi,\xi) = \sum_{1\le \lambda\le r} \wt\Theta(E)(\xi\otimes
e_\lambda,\xi\otimes e_\lambda),\qquad\xi\in T_X,
$$
for any orthonormal frame $(e_1\ld e_r)$ of $E$. Theorem 10.12 is now a
consequence of the following simple property of hermitian forms on a tensor
product of complex vector spaces.

\begstat{(10.14) Proposition} Let $T,E$ be complex vector spaces of respective
dimensions $n,r,\,$ and $h$ a hermitian metric on $E$. Then for
every hermitian form $\Theta$ on $T\otimes E$
$$\Theta >_\Grif 0~~\Longrightarrow~~
\Theta + \Tr_E\Theta \otimes h >_\Nak 0.$$
\endstat

We first need a lemma analogous to Fourier inversion formula for discrete
Fourier transforms.

\begstat{(10.15) Lemma} Let $q$ be an integer $\ge 3$, and 
$x_\lambda,~y_\mu,~1\le \lambda,\mu \le r$, be complex numbers. 
Let $\sigma$ describe
the set $U^r_q$ of $r$-tuples of $q$-th roots of unity and put
$$x'_\sigma = \sum_{1\le \lambda\le r} x_\lambda \ovl \sigma_\lambda,~~~~
y'_\sigma = \sum_{1\le \mu\le r} y_\mu \ovl \sigma_\mu,~~~~ \sigma\in U^r_q.$$
Then for every pair $(\alpha,\beta),~1\le \alpha,\beta\le r$, the following 
identity holds:
$$q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ovl y'_\sigma \sigma_\alpha \ovl 
\sigma_\beta=\cases{
x_\alpha \ovl y_\beta\phantom{\sum_\sum}&if~~$\alpha\ne \beta,$\cr
\displaystyle\sum_{1\le \mu\le r} x_\mu \ovl y_\mu&if~~$\alpha=\beta.$\cr}$$
\endstat

\begproof{} The coefficient of $x_\lambda\ovl y_\mu$ in the summation
$q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ovl y'_\sigma \sigma_\alpha \ovl 
\sigma_\beta$ is given by
$$q^{-r} \sum_{\sigma\in U^r_q}\sigma_\alpha \ovl \sigma_\beta\ovl
\sigma_\lambda \sigma_\mu.$$
This coefficient equals 1 when the pairs $\{ \alpha,\mu\}$ and
$\{\beta,\lambda\}$ are equal (in which case $\sigma_\alpha
\ovl \sigma_\beta \ovl \sigma_\lambda \sigma_\mu = 1$ for any one of the $q^r$
elements of $U^r_q$). Hence, it is sufficient to prove that
$$\sum_{\sigma\in U^r_q} \sigma_\alpha \ovl \sigma_\beta \ovl \sigma_\lambda
\sigma_\mu = 0$$
when the pairs $\{ \alpha,\mu\}$ and $\{ \beta,\lambda\}$
are distinct.

If $\{ \alpha,\mu\} \ne \{\beta,\lambda\}$, then one of the
elements of one of the pairs does not belong to the other pair. As the four
indices $\alpha,\beta,\lambda,\mu$ play the same role, we may suppose for 
example that $\alpha\notin \{\beta,\lambda\}$. Let us apply to 
$\sigma$ the substitution $\sigma\mapsto \tau$, where $\tau$ is defined by
$$\tau_\alpha = e^{2\pi\ii/q}\sigma_\alpha,~\tau_\nu = 
\sigma_\nu \quad \hbox{\rm for}\quad \nu\ne \alpha.$$
We get
$$\sum_\sigma\sigma_\alpha\ovl\sigma_\beta\ovl\sigma_\lambda\sigma_\mu=\sum_\tau
=\cases{
\displaystyle e^{2\pi\ii/q}\sum_\sigma&if~~$\alpha\ne\mu,$\cr
\displaystyle e^{4\pi\ii/q}\sum_\sigma&if~~$\alpha=\mu,$\cr}$$
Since $q\ge 3$ by hypothesis, it follows that
$$\sum_\sigma\sigma_\alpha\ovl\sigma_\beta\ovl\sigma_\lambda\sigma_\mu=0.$$
\endproof

\begproof{of Proposition 10.14.} Let $(t_j)_{1\le j\le n}$ be a basis of
$T$, $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis of $E$ and
$\xi=\sum_j\xi_jt_j\in T$, $u=\sum_{j,\lambda}u_{j\lambda}\,t_j\otimes 
e_\lambda \in T\otimes E$. The coefficients $c_{jk\lambda\mu}$ of $\Theta$
with respect to the basis $t_j\otimes e_\lambda$ satisfy the symmetry
relation $\ovl c_{jk\lambda\mu}=c_{kj\mu\lambda}$, and we have the formulas 
$$\eqalign{\Theta(u,u)
&=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}u_{j\lambda}\ovl u_{k\mu},\cr
\Tr_E\Theta(\xi,\xi)&=\sum_{j,k,\lambda}c_{jk\lambda\lambda}\xi_j\ovl\xi_k,\cr
(\Theta+\Tr_E\Theta\otimes h)(u,u)&=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu} 
u_{j\lambda} \ovl u_{k\mu}+ c_{jk\lambda\lambda}u_{j\mu}\ovl u_{k\mu}.\cr}$$
For every $\sigma\in U^r_q$ (cf.\ Lemma~10.15), put
$$\eqalign{
u'_{j\sigma}&=\sum_{1\le\lambda\le r}u_{j\lambda}\ovl\sigma_\lambda\in\bbbc,\cr
\wh u_\sigma&=\sum_j u'_{j\sigma}t_j\in T\quad,\quad
\wh e_\sigma=\sum_\lambda\sigma_\lambda e_\lambda\in E.\cr}$$
Lemma 10.15 implies
$$\eqalign{
q^{-r} \sum_{\sigma\in U^r_q}\Theta(\wh u_\sigma\otimes\wh e_\sigma,
\wh u_\sigma\otimes\wh e_\sigma)&= q^{-r} \sum_{\sigma\in U^r_q} c_{jk\lambda\mu}
u'_{j\sigma} \ovl u'_{k\sigma}\sigma_\lambda \ovl \sigma_\mu\cr
&= \sum_{j,k,\lambda\ne \mu} c_{jk\lambda\mu} u_{j\lambda} \ovl u_{k\mu} +
\sum_{j,k,\lambda,\mu} c_{jk\lambda\lambda} u_{j\mu} \ovl u_{k\mu}.\cr}$$
The Griffiths positivity assumption shows that the left hand side is $\ge 0$,
hence
$$(\Theta + \Tr_E \Theta \otimes h)(u,u) \ge \sum_{j,k,\lambda} c_{jk\lambda\lambda} u_{j\lambda} \ovl u_{k\lambda} \ge 0$$
with strict positivity if $\Theta >_\Grif 0$ and $u\ne 0$.\qed
\endproof

We now relate Griffiths positivity to $m$-tensor positivity. The most useful
result is the following

\begstat{(10.16) Proposition} Let $T$ be a complex vector space and
$(E,h)$ a hermitian vector space of respective dimensions $n,r$ with
$r\ge 2$. Then for any hermitian form $\Theta$ on $T\otimes E$ and any
integer $m\ge 1$
$$\Theta>_\Grif0~~~\Longrightarrow~~~m\Tr_E\Theta\otimes h-\Theta>_m0.$$
\endstat

\begproof{} Let us distinguish two cases.
\medskip
\noindent a)~ $m=1$.~
Let $u\in T \otimes E$ be a tensor of rank 1. Then $u$ can be written
$u = \xi_1 \otimes e_1$ with $\xi_1\in T,~\xi_1 \ne 0$, and
$e_1 \in E,~|e_1| = 1$. Complete $e_1$ into an orthonormal basis
$(e_1\ld e_r)$ of $E$. One gets immediately
$$\eqalign{
(\Tr_E \Theta \otimes h)(u,u) &= \Tr_E \Theta (\xi_1,\xi_1) = \sum_{1\le \lambda\le r} \Theta(\xi_1\otimes e_\lambda , \xi_1 \otimes e_\lambda)\cr
&> \Theta (\xi_1\otimes e_1,\xi_1\otimes e_1) = \Theta (u,u).\cr}$$
\medskip
\noindent b)~ $m\ge 2$.~
Every tensor $u\in T\otimes E$ of $\hbox{\rm rank~} \le m$ can be written
$$u=\sum_{1\le \lambda\le q} \xi_\lambda \otimes e_\lambda\quad,
\quad \xi_\lambda \in T,$$
with $q = \min(m,r)$ and $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis
of $E$. Let $F$ be the vector subspace of $E$ generated by $(e_1\ld e_q)$
and $\Theta_F$ the restriction of $\Theta$ to $T\otimes F$. The first part
shows that
$$\Theta':= \Tr_F \Theta_F \otimes h - \Theta_F >_\Grif 0.$$
Proposition 10.14 applied to $\Theta'$ on $T\otimes F$ yields
$$\Theta'+\Tr_F \Theta'\otimes h=q\Tr_F\Theta_F\otimes h-\Theta_F>_q 0.$$
Since $u\in T\otimes F$ is of $\hbox{\rm rank~} \le q \le m$, we get
(for $u\ne 0$)
$$\eqalignno{
\Theta(u,u) = \Theta_F(u,u) &< q(\Tr_F \Theta_F \otimes h)(u,u)\cr
&=q\sum_{1\le j,\lambda\le q}\Theta(\xi_j\otimes e_\lambda,
\xi_j\otimes e_\lambda)\le m\Tr_E\Theta\otimes h(u,u).&\square\cr}$$
\endproof

Proposition 10.16 is of course also true in the semi-positive case. From these
facts, we deduce

\begstat{(10.17) Theorem} Let $E$ be a Griffiths $($semi-$)$positive bundle of 
rank $r\ge 2$. Then for any integer $m\ge 1$
$$E^\star \otimes (\det E)^m >_m 0~~~~(\hbox{\it resp.}~~\ge_m 0).$$
\endstat

\begproof{} We apply Prop.~10.16 to $\Theta = -\Theta (E^\star) =
{}^t\Theta(E)\ge_\Grif 0$ on $T_X\otimes E^\star$  and observe that
$$\Theta(\det E) = \Tr_E \Theta (E) = \Tr_{E^\star} \Theta.$$
\endproof

\begstat{(10.18) Theorem} Let $0\to S \to E \to Q \to 0$ be an exact 
sequence of hermitian vector bundles. Then for any $m\ge 1$
$$E>_m0~~~\Longrightarrow~~~S\otimes (\det Q)^m >_m 0.$$
\endstat

\begproof{} Formulas (10.7) imply
$$\ii\Theta(S) >_m \ii\beta^\star \wedge \beta\quad,\quad \ii\Theta(Q) >_m \ii\beta \wedge \beta^\star,$$
$$\ii\Theta(\det Q) = \Tr_Q(\ii\Theta(Q)) > \Tr_Q(\ii\beta \wedge \beta^\star).$$
If we write $\beta = \sum dz_j \otimes \beta_j$ as in the proof of
Corollary~10.8, then
$$\eqalign{ \Tr_Q(\ii\beta \wedge \beta^\star) &= \sum idz_j \wedge d\ovl
z_k \Tr_Q(\beta_j\beta^\star_k)\cr &=\sum idz_j \wedge d\ovl z_k
\Tr_S(\beta^\star_k\beta_j) = \Tr_S(-\ii\beta^\star \wedge \beta).\cr}$$
Furthermore, it has been already proved that $-\ii\beta^\star \wedge
\beta \ge_\Nak 0$. By Prop.~8.1 applied to the corresponding
hermitian form $\Theta$ on $T_X \otimes S$, we get 
$$
m\Tr_S(-\ii\beta^\star \wedge \beta) \otimes \Id_S + \ii\beta^\star\wedge
\beta \ge_m 0,
$$ 
and Theorem~10.18 follows.\qed
\endproof

\begstat{(10.19) Corollary} Let $X$ be a weakly pseudoconvex K\"ahler 
$n$-dimensional manifold, $E$ a holomorphic vector bundle of rank 
$r\ge 2$ and $m\ge 1$ an integer. Then
\smallskip
\item{\rm a)} $E>_\Grif0 \Rightarrow H^{n,q}(X,E\otimes \det\,E)=0$
for $q\ge 1\,;$
\smallskip
\item{\rm b)} $E>_\Grif0\Rightarrow H^{n,q}\big
(X,E^\star\otimes(\det\,E)^m\big)=0$ for $q\ge 1$ and
\hbox{$m\ge\min\{n-q+1,r\}\,;$}
\smallskip
\item{\rm c)} Let~ $0\to S\to E\to Q\to 0$ be an exact sequence of
vector bundles and $m=\min\{n-q+1,\rk\,S\}$, $q\ge 1$. If $E>_m0$ and if
$L$ is a line bundle such that $L\otimes(\det Q)^{-m}\ge 0$, then
$$H^{n,q}(X,S\otimes L)=0.$$
\endstat

\begproof{} Immediate consequence of Theorem 10.11, in combination with
10.12 for a), 10.17 for b) and 10.18 for c).\qed
\endproof

\titleb{11.}{Skoda's $L^2$ estimates for surjective bundle morphisms}
Let $(X,\omega)$ be a K\"ahler manifold, $\dim X=n$, and let $g:E\to Q$ a
holomorphic morphism of hermitian vector bundles over $X$. Assume
in the first instance that $g$ is {\it surjective}. We are interested in
conditions insuring that the induced morphisms $g:H^{n,k}(X,E)\lra 
H^{n,k}(X,Q)$ are also surjective (dealing with $(n,\bu)$ bidegrees is 
always easier, since we have to understand positivity conditions
for the curvature term). For that purpose, it is natural to consider the 
subbundle $S=\Ker g\subset E$ and the exact sequence
$$
0\lra S\buildo j\over\lra E\buildo g\over\lra Q\lra 0
\leqno(11.1)
$$
where $j:S\to E$ is the inclusion. In fact, we need a little more flexibility
to handle the curvature terms, so we take the tensor product of the exact 
sequence by a holomorphic line bundle $L$ (whose properties will be specified
later):
$$
0\lra S\otimes L\lra E\otimes L\buildo g\over\lra Q\otimes L\lra0.
\leqno(11.2)
$$

\begstat{(11.3) Theorem} Let $k$ be an integer such that $0\le k\le n$. Set
$r=\rk\,E$, $q={\rm rk Q}$, $s=\rk\,S=r-q$~ and
$$
m=\min\{n-k,s\}=\min\{n-k,r-q\}.
$$
Assume that $(X,\omega)$ possesses also a complete K\"ahler metric $\wh\omega$,
that $E\ge_m 0$, and that $L\lra X$ is a hermitian holomorphic line bundle such that
$$
\ii\Theta(L)-(m+\varepsilon)\ii\Theta(\det Q)\ge 0
$$
for some $\varepsilon>0$. Then for every $D''$-closed form $f$ of type 
$(n,k)$ with values in $Q\otimes L$ such that $\|f\|<+\infty$, 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~1982).

\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
the matrix of $D_E$ with respect to the orthogonal splitting 
$E\simeq S\oplus Q$ (cf.\ Lemma~10.2). Then $g^\star f$ is a lifting of $f$
in $E\otimes L$. We will try to find $h$ under the form
$$
h=g^\star f+ju,~~~~u\in L^2(X,\Lambda^{n,k}T^\star_X\otimes S\otimes L).
$$
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 (10.5), 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
$$
D''_{S\otimes L}u=\beta^\star\wedge f,\leqno(11.4)
$$
and for that, we apply Th.~8.4 to the $(n,k+1)$-form 
$\beta^\star\wedge f$. One now observes that the curvature of $S\otimes L$
can be expressed in terms of $\beta$. This remark will be used to prove:
\endproof

\begstat{(11.5) Lemma} Let $A_k=[\ii\Theta(S\otimes L),\Lambda]$ be the 
curvature operator acting on \hbox{$(n,k+1)$-forms}. Then 
$\langle A_k^{-1}(\beta^\star\wedge f),
(\beta^\star\wedge f)\rangle\le(m/\varepsilon)\,|f|^2$.
\endstat

If the Lemma is taken for granted, Th.~8.4 yields a solution $u$ of~(11.4)
in $L^2(X,\Lambda^{n,q}T^\star_X\otimes S\otimes L)$ such that 
$\|u\|^2\le(m/\varepsilon)\,\|f\|^2$.
As \hbox{$\|h\|^2=\|f\|^2+\|u\|^2$}, the
proof of Th.~11.3 is complete.\qed

\begproof{of Lemma~11.5.} Exactly as in the proof of Th.~10.18
formulas (10.7) 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 $\ci(X,\Lambda^{1,1}T^\star_X\otimes \Herm\,S)
\ni\Theta:=-\ii\beta^\star\wedge\beta \ge_\Grif0$, Prop.~10.16 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~10.10 implies
$$
\langle A_kv,v\rangle\ge(\varepsilon/m)\,
\langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle,\leqno(11.6)
$$
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 operator $\beta\ort \bu$ 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 (11.6) implies
$$
|\langle\beta^\star\wedge f,v\rangle|^2=|\langle f,\beta\ort v\rangle|^2\le
|f|^2\,|\beta\ort v|^2\le(m/\varepsilon)
\langle A_kv,v\rangle\,|f|^2.
$$
This is equivalent to the estimate asserted in the lemma.\qed
\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 11.3 implies 
therefore:

\begstat{(11.7) Corollary} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler 
manifold, \hbox{$g:E\to Q$} a surjective bundle morphism with
$r=\rk\,E$, $q=\rk\,Q$, and \hbox{$L\to X$} a hermitian holomorphic line 
bundle. We set \hbox{$m=\min\{n-k,r-q\}$} and assume 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
$$
H^{n,k}(X,E\otimes L)\lra H^{n,k}(X,Q\otimes L).
$$
\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
$$
-\beta^\star\wedge\bu~:~H^{n,k}(X,Q\otimes L)\lra H^{n,k+1}(X,S\otimes L)
$$
vanishes; however the cohomology group $H^{n,k+1}(X,S\otimes L)$
itself does not necessarily vanish, as it would do under a strict positivity
assumption (cf.\ Cor.~10.19~c)).

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.~11.3 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 11.3 applied to $(E,Q',L')$ can now be reformulated:

\begstat{(11.8) 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 holomorphic line bundle. Set $r=\rk\,E$,
$q=\rk\,Q$ and $m=\min\{n-k,r-q\}$, and assume that $E\ge_m 0$ and
$$
\ii\Theta(L)-(m+\varepsilon)\ii\Theta(\det Q)\ge 0
$$
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.~11.8 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 $\Hom(\Lambda^qE,\det Q)$.
The idea is to apply the above Theorem 11.8 to $X\ssm Y$. For this,
we have to know whether $X\ssm Y$ has a complete K\"ahler metric.

\begstat{(11.9) 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 $X_c=\{x\in X~;~\psi(x)<c\}$, then $X_c\ssm Y$ has a
complete K\"ahler metric for all $c\in\bbbr$.  The same conclusion
holds for $X\ssm Y$ if $(X,\omega)$ is complete and if for some
constant $C\ge 0$ we have
$\Theta_E\le_\Grif C\,\omega\otimes\langle~,~\rangle_E$ on $X$.
\endstat

\begproof{} Set $\tau=\log|\sigma|^2$. Then $d'\tau=\{D'\sigma,\sigma\}/
|\sigma|^2$ and $D''D'\sigma=D^2\sigma=\Theta(E)\sigma$, thus
$$\ii d'd''\tau=\ii{\{D'\sigma,D'\sigma\}\over|\sigma|^2}-
\ii{\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over|\sigma|^4}-
{\{\ii\Theta(E)\sigma,\sigma\}\over|\sigma|^2}.$$
For every $\xi\in T_X$, we find therefore
$$\eqalign{
H\tau(\xi)&={|\sigma|^2\,|D'\sigma\cdot \xi|^2-|\langle D'\sigma\cdot \xi,
\sigma\rangle|^2\over|\sigma|^4}-{\wt\Theta(E)(\xi\otimes\sigma,\xi\otimes
\sigma)\over|\sigma|^2}\cr
&\ge -{\wt\Theta(E)(\xi\otimes \sigma,\xi\otimes\sigma)\over|\sigma|^2}\cr}$$
by the Cauchy-Schwarz inequality. If $C$ is a bound for the coefficients of
$\wt\Theta(E)$ on the compact
subset $\ovl X_c$, we get $\ii d'd''\tau\ge -C\omega$ on
$X_c$. Let $\chi\in\ci(\bbbr,\bbbr)$ be a convex increasing function. We set
$$\wh\omega=\omega+\ii d'd''(\chi\circ\tau)=\omega+\ii\big(
\chi'\circ\tau~d'd''\tau+\chi''\circ\tau~d'\tau\wedge d''\tau\big).$$
We thus see that $\wh\omega$ is positive definite if
$\chi'\le 1/2C$, and by a computation similar to that in proposition
8.7, we check that $\wh\omega$ is complete near $Y=\tau^{-1}(-\infty)$
as soon as
$$\int_{-\infty}^0\sqrt{\chi''(t)}\,dt=+\infty.$$
One can choose for example $\chi$ such that
$\chi(t)={1\over 5C}(t-\log|t|)$ for $t\le -1$. In order to obtain a
complete K\"ahler metric on $X_c\ssm Y$, we also need the metric 
to be complete near $\partial X_c$. If $\wh\omega$ is not, such a metric
can be defined by
$$\eqalign{
\wt\omega&=\wh\omega+\ii d'd''\log(c-\psi)^{-1}=\wh\omega+{\ii d'd''\psi\over
c-\psi}+{\ii d'\psi\wedge d''\psi\over(c-\psi)^2}\cr
&\ge \ii d'\log(c-\psi)^{-1}\wedge d''\log(c-\psi)^{-1}~;\cr}$$
$\wt\omega$ is complete on $X_c\ssm\Omega$ because $\log(c-\psi)^{-1}$
tends to $+\infty$ on $\partial X_c$.\qed
\endproof

We also need another elementary lemma dealing with the extension of
partial differential equalities across analytic sets.

\begstat{(11.10) 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
$$\int_\Omega w\wedge\alpha=(-1)^{p+q}\int_\Omega v\wedge d''\alpha
\leqno(11.11)$$
for all $\alpha\in\cD(\Omega,\Lambda^{n-p,n-q}T^\star_\Omega)$. 
Set $\lambda_\varepsilon(z)=\lambda(|z_1|/\varepsilon)$ and replace 
$\alpha$ in the integral by $\lambda_\varepsilon\alpha$. Then 
$\lambda_\varepsilon\alpha\in \cD(\Omega\ssm Y,\Lambda^{n-p,n-q}
T^\star_\Omega)$ 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 (11.11) follows when $\varepsilon$ tends to $0$.\qed
\endproof

\begstat{(11.12) Theorem} The existence statement and the estimates of 
Th.~$11.8$
remain true for a generically surjective morphism $g:E\to Q$,
provided that $X$ is weakly pseudoconvex.
\endstat

\begproof{} Apply Th.~11.8 to each relatively compact domain
$X_c\ssm Y$ (these domains are complete K\"ahler by Lemma~11.9).
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~11.10 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.~11.8 applied on \hbox{$X=\Omega\ssm g^{-1}(0)$} and Lemmas~11.9,
11.10 give:

\begstat{(11.13) Theorem {\rm(Skoda 1972b)}} 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 interesting results about
domains of holomorphy in $\bbbc^n$ and the relation with
the existence of complete K\"ahler metrics. Recall that an open set
$\Omega\subset\bbbc^n$ is said to be a domain of holomorphy if
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~$U$.
It is well known that $\Omega$ is a domain of holomorphy if and only if
$\Omega$ is Stein [the latter condition being of course trivially
sufficient; see e.g.\ (H\"ormander 1966), chap.~2].

\begstat{(11.14) Theorem {\rm(Diederich-Pflug~1981)}} 
Let $\Omega\subset\bbbc^n$ be an open subset. If $\omcirc=\Omega$ 
and $\Omega$ admits a complete K\"ahler metric $\wh\omega$, then $\Omega$ 
is a domain of holomorphy.
\endstat

Note that the statement becomes false if the assumption $\omcirc=\Omega$ is
omitted: in fact $\bbbc^n\ssm\{0\}$ is complete K\"ahler by Lemma~10.9, 
but it is not a domain of holomorphy if $n\ge 2$. 

\begproof{} 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.~11.13 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 $a\in U$.\qed
\endproof

\begstat{(11.15) Remark} \rm Skoda's theorem 11.13 can also be used to 
prove the implication
$$
\hbox{$\Omega$ pseudoconvex}~\,\Longrightarrow~\,
\hbox{$\Omega$ domain of holomorphy},
$$
which is equivalent to the ``interesting implication'' in the Levi
problem (modulo the equivalence of domains of holomorphy with
Stein open sets, an easy property). In fact, if $\Omega$ is pseudoconvex,
it can be shown that the function
$z\mapsto-\log d(z,\complement\Omega)$ is psh (see again H\"ormander 1966,
chap.~2). Given any open connected set $U$ such that 
$U\cap\partial\Omega\ne\emptyset$, select $a\in U\cap\partial\Omega$. 
The weight function
$$
\varphi(z)=(n+\varepsilon)\log(1+|z|^2)-2(n+\varepsilon)
\log d(z,\complement\Omega)
$$
is psh on $\Omega$. As $|z-a|\ge d(z,\complement\Omega)$, we see that 
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 in the proof of 11.14.\qed
\endstat

\titleb{12.}{Application of Skoda's $L^2$ estimates to local algebra}
We show here how Theorem~11.13 can be applied to get deep results
concerning ideals of the local 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{(12.1) 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{(12.2) 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 the integrability of $|g|^{-\varepsilon}$
for $\varepsilon>0$ small (exercise to the reader!). Before stating 
the main result, we need a simple lemma.

\begstat{(12.3) 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 ana\-lytic 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{(12.4) Theorem {\rm(Brian\c{c}on-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.~12.2,
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.~11.13 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~12.3 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{(12.5) 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
$$
u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^{\lceil ks\rceil},~~~~1\le s\le d,
$$
where $\lceil t\rceil$ 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 the usual elementary bound 
$$
|\hbox{roots}|\le 2\max_{1\le s\le d}|a_s|^{1/s}
$$
for the roots of a monic polynomial 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
$a_{ms}\in\cI^{Ns}\subset\cI^{\lceil kms\rceil}$.\qed
\endproof

Let us mention that Brian\c{c}on and Skoda's result 12.4~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 12.4~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{(12.6) 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 $f(z)=(z_1\ldots z_n)^3+z_1^{3n-1}+\cdots+z_n^{3n-1}$ 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 12.4~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{(12.7) Curve selection 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{13.}{The Ohsawa-Takegoshi $L^2$ extension theorem}
We address here the following extension problem: let $Y$ be a complex
analytic submanifold of a complex manifold~$X\,$; given a holomorphic 
function $f$ on $Y$ satisfying suitable $L^2$ conditions on $Y$, find
a holomorphic extension $F$ of $f$ to $X$, together with a good
$L^2$ estimate for $F$ on~$X$. The first satisfactory solution
has been obtained only rather recently by (Ohsawa-Takegoshi 1987). We 
follow here a more geometric approach due to (Manivel 1993), which 
provides a generalized extension theorem in the general framework
of vector bundles. As in Ohsawa-Takegoshi's fundamental paper, the 
main idea is to use a modified Bochner-Kodaira-Nakano inequality. Such
inequalities were originally introduced in the work of 
(Donnelly-Fefferman 1983) and (Donnelly-Xavier 1984). The main a priori 
inequality we are going to use is a simplified (and slightly extended)
version of the original Ohsawa-Takegoshi a priori inequality, as
proposed recently by (Ohsawa 1995); see also (Berndtsson 1995) for
related calculations in the special case of domains in $\bbbc^n$.

\begstat{(13.1) Lemma {\rm (Ohsawa 1995)}} Let $E$ be a hermitian
vector bundle on a complex manifold $X$ equipped with a K\"ahler
metric~$\omega$. Let $\eta,\,\lambda>0$ be smooth functions
on~$X$. Then for every form $u\in\cD(X,\Lambda^{p,q} T^\star_X\otimes E)$
with compact support we have
$$ 
\eqalign{
\|(\eta^{{1\over2}}+\lambda^{{1\over2}})D^{\prime\prime\star}u\|^2&{}
+\|\eta^{{1\over2}}D''u\|^2+\|\lambda^{{1\over2}}D'u\|^2+
2\|\lambda^{-{1\over2}}d'\eta\wedge u\|^2\cr
 &\geq \Ll[\eta\,\ii\Theta(E)-\ii\,d'd''\eta-\ii\lambda^{-1}d'\eta\wedge 
d''\eta,\Lambda]u,u\Gg.\cr}
$$
\endstat

\begproof{} Let us consider the ``twisted'' Laplace-Beltrami operators
$$
\eqalign{
D'\eta D^{\prime\star} +D^{\prime\star}\eta  D' 
& =  \eta [D',D^{\prime\star}]+[D',\eta]D^{\prime\star}+
[D^{\prime\star},\eta]D' \cr
& = \eta\Delta'+(d'\eta)D^{\prime\star}-(d'\eta)^*D',\cr
D''\eta D^{\prime\prime\star} +D^{\prime\prime\star}\eta  D'' 
& = \eta [D'',D^{\prime\prime\star}]+[D'',\eta]D^{\prime\prime\star}+
[D^{\prime\prime\star},\eta]D'' \cr
& = \eta\Delta''+(d''\eta)D^{\prime\prime\star}-(d''\eta)^*D'',\cr}
$$
where $\eta$, $(d'\eta)$, $(d''\eta)$ are abbreviated notations for the 
multiplication operators $\eta\bu$, $(d'\eta)\wedge\bu$, $(d''\eta)\wedge\bu$.
By subtracting the above equalities and taking into account the 
Bochner-Kodaira-Nakano identity $\Delta''-\Delta'=[\ii\Theta(E),\Lambda]$,
we get
$$
\leqalignno{
D''\eta D^{\prime\prime\star}&{}+D^{\prime\prime\star}\eta  D'' -
D'\eta D^{\prime\star}  -D^{\prime\star}\eta  D' \cr
&{}=\eta [\ii\Theta(E),\Lambda]+(d''\eta)D^{\prime\prime\star}
-(d''\eta)^\star D''+(d'\eta)^\star D'-(d'\eta)D^{\prime\star}.
&(13.2)\cr}
$$
Moreover, the Jacobi identity yields
$$
[D'',[d'\eta,\Lambda ]]-[d'\eta,[\Lambda,D'']]+[\Lambda,[D'',d'\eta]]=0,
$$
whilst $[\Lambda,D'']=-\ii D^{\prime\star}$ by the basic commutation 
relations~7.2. A straightforward computation shows that 
$[D'',d'\eta]=-(d'd''\eta)$ and $[d'\eta,\Lambda ]=\ii(d''\eta)^\star$.
Therefore we get
$$
\ii[D'',(d''\eta)^\star]+\ii[d'\eta,D^{\prime\star}]-[\Lambda,(d'd''\eta)]=0,
$$
that is,
$$
[\ii\,d'd''\eta,\Lambda]=
[D'',(d''\eta)^\star]+[D^{\prime\star},d'\eta]=
D''(d''\eta)^\star+(d''\eta)^\star D''+
D^{\prime\star}(d'\eta)+(d'\eta)D^{\prime\star}.
$$
After adding this to (13.2), we find
$$
\eqalign{
D''\eta D^{\prime\prime\star}&{}+D^{\prime\prime\star}\eta  D'' -
D'\eta D^{\prime\star}  -D^{\prime\star}\eta  D' 
+[\ii\,d'd''\eta,\Lambda]
\cr
&{}=\eta [\ii\Theta(E),\Lambda]+(d''\eta)D^{\prime\prime\star}+
D''(d''\eta)^\star+(d'\eta)^\star D'+D^{\prime\star}(d'\eta).
\cr}
$$
We apply this identity to a form $u\in\cD(X,\Lambda^{p,q}T^\star_X\otimes E)$
and take the inner bracket with~$u$. Then
$$
\Ll (D''\eta D^{\prime\prime\star})u,u\Gg=\Ll\eta D^{\prime\prime\star}u,
D^{\prime\prime\star}u\Gg=\|\eta^{{1\over2}}D^{\prime\prime\star}u\|^2,
$$
and likewise for the other similar terms. The above equalities
imply
$$
\eqalign{
&\|\eta^{{1\over 2}}D^{\prime\prime\star}u\|^2+\|\eta^{{1\over 2}}D''u\|^2
-\|\eta^{{1\over2}}D'u\|^2-\|\eta^{{1\over2}}D^{\prime\star}u\|^2={}\cr
&\qquad\Ll[\eta\,\ii\Theta(E)-\ii\,d'd''\eta,\Lambda]u,u\Gg
+2\Re\,\Ll D^{\prime\prime\star}u,(d''\eta)^\star u\Gg
+2\Re\,\Ll D'u,d'\eta\wedge u\Gg.\cr}
$$
By neglecting the negative terms $-\|\eta^{{1\over2}}D'u\|^2
-\|\eta^{{1\over2}}D^{\prime\star}u\|^2$ and adding the squares
$$
\eqalign{
\|\lambda^{{1\over 2}}D^{\prime\prime\star}u\|^2+
2\Re\,\Ll D^{\prime\prime\star}u,(d''\eta)^\star u\Gg+
\|\lambda^{-{1\over 2}}(d''\eta)^\star u\|^2
&\ge 0,\cr
\|\lambda^{{1\over 2}}D'u\|^2+
2\Re\,\Ll D'u,d'\eta\wedge u\Gg+
\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2
&\ge0\cr}
$$
we get
$$
\eqalign{
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})
D^{\prime\prime\star}u\|^2&{}+\|\eta^{{1\over 2}}D''u\|^2
+\|\lambda^{{1\over 2}}D'u\|^2+\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2
+\|\lambda^{-{1\over 2}}(d''\eta)^\star u\|^2\cr
&\ge\Ll[\eta\,\ii\Theta(E)-\ii\,d'd''\eta,\Lambda]u,u\Gg.\cr}
$$
Finally, we use the identities
$$
\eqalign{
(d'\eta)^\star(d'\eta)&{}-(d''\eta)(d''\eta)^\star
=\ii[d''\eta,\Lambda](d'\eta)+\ii(d''\eta)[d'\eta,\Lambda]
=[\ii d''\eta\wedge d'\eta,\Lambda],\cr
\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2&{}-
\|\lambda^{-{1\over2}}(d''\eta)^\star u\|^2
=-\Ll[\ii\lambda^{-1}d'\eta\wedge d''\eta,\Lambda]u,u\Gg,\cr}
$$
The inequality asserted in Lemma~13.1 follows by adding the second identity
to our last inequality.\qed
\endproof

In the special case of $(n,q)$-forms, the forms $D'u$ and $d'\eta\wedge u$
are of bidegree $(n+1,q)$, hence the estimate takes the simpler form
$$
\|(\eta^{{1\over2}}+\lambda^{{1\over2}})D^{\prime\prime\star}u\|^2{}
+\|\eta^{{1\over2}}D''u\|^2\ge
\Ll[\eta\,\ii\Theta(E)-\ii\,d'd''\eta
-\ii\lambda^{-1}\,d'\eta\wedge d''\eta,\Lambda]u,u\Gg.\leqno(13.3)
$$

\begstat{(13.4) Proposition} Let $X$ be a complete K\"ahler manifold
equipped with a $($non ne\-ces\-sarily complete$)$ K\"ahler metric
$\omega$, and let $E$ be a hermitian vector bundle over~$X$.
Assume that there are smooth and bounded functions $\eta,\,\lambda>0$
on $X$ such that the $($hermitian$)$ curvature operator
$B=B^{n,q}_{E,\omega,\eta}=
[\eta\,\ii\Theta(E)-\ii\,d'd''\eta-\ii\lambda^{-1}d'\eta\wedge d''\eta,
\Lambda_\omega]$ is positive definite everywhere on 
$\Lambda^{n,q}T^\star_X\otimes E$, for some $q\ge 1$.
Then for every form $g\in L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$
such that $D''g=0$ and $\int_X\langle B^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{n,q-1}T^\star_X\otimes E)$ such that  $D''f=g$ and
$$
\int_X(\eta+\lambda)^{-1}
|f|^2\,dV_\omega\le 2\int_X\langle B^{-1}g,g\rangle\,dV_\omega.
$$
\endstat

\begproof{} The proof is almost identical to the proof of Theorem~8.4,
except that we use (13.4) instead of (7.4). Assume first that $\omega$ is
complete. With the same notation as in 7.4, we get for every 
$v=v_1+v_2\in(\Ker D'')\oplus(\Ker D'')^\perp$ the inequalities
$$
|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle B^{-1}g,g\rangle \,dV_\omega
\int_X\langle Bv_1,v_1\rangle \,dV_\omega,
$$
and
$$
\int_X\langle Bv_1,v_1\rangle \,dV_\omega\le
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v_1\|^2+
\|\eta^{{1\over 2}}D''v_1\|^2
=\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2
$$
provided that $v\in\Dom D^{\prime\prime\star}$. Combining both, we find
$$
|\langle g,v\rangle|^2
\le \Big(\int_X\langle B^{-1}g,g\rangle \,dV_\omega\Big)
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2.
$$
This shows the existence of an element $w\in L^2(X,\Lambda^{n,q}T^\star_X
\otimes E)$ such that 
$$
\eqalign{
\|w\|^2&\le\int_X\langle B^{-1}g,g\rangle \,dV_\omega\qquad
\hbox{and}\cr
\Ll v,g\Gg&=\Ll(\eta^{{1\over 2}}+\lambda^{{1\over2}})
D^{\prime\prime\star}v,w\Gg\qquad
\forall g\in\Dom D''\cap\Dom D^{\prime\prime\star}.\cr}
$$
As $(\eta^{1/2}+\lambda^{{1\over2}})^2\le 2(\eta+\lambda)$, it follows 
that $f=(\eta^{1/2}+\lambda^{{1\over2}})w$ satisfies
$D''f=g$ as well as the desired $L^2$ estimate. If $\omega$ is not
complete, we set $\omega_\varepsilon=\omega+\varepsilon\wh\omega$ with
some complete K\"ahler metric~$\wh\omega$. The final conclusion is
then obtained by passing to the limit and using a monotonicity argument
(the integrals are monotonic with respect to~$\varepsilon$).\qed
\endproof

\begstat{(13.5) Remark} \rm We will also need a variant of the
$L^2$-estimate, so as to obtain approximate solutions with weaker
requirements on the data$\,$: given $\delta>0$ and $g\in
L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$ such that $D''g=0$ and
$\int_X\langle (B+\delta I)^{-1}g,g\rangle\,dV_\omega<+\infty$, there
exists an approximate solution 
\hbox{$f\in L^2(X,\Lambda^{n,q-1}T^\star_X \otimes E)$} and a correcting
term $h\in L^2(X,\Lambda^{n,q}T^\star_X \otimes E)$ such that
$D''f+\delta^{1/2}h=g$ and
$$
\int_X(\eta+\lambda)^{-1}|f|^2\,dV_\omega+
\int_X|h|^2\,dV_\omega
\le 2\int_X\langle (B+\delta I)^{-1}g,g\rangle\,dV_\omega.
$$
The proof is almost unchanged, we rely instead on the estimates
$$
|\langle g,v_1\rangle|^2\le
\int_X\langle(B+\delta I)^{-1}g,g\rangle \,dV_\omega
\int_X\langle(B+\delta I)v_1,v_1\rangle \,dV_\omega,
$$
and
$$
\int_X\langle(B+\delta I)v_1,v_1\rangle \,dV_\omega\le
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2+
\delta\|v\|^2.
\eqno\qed
$$
\endstat

\begstat{(13.6) Theorem}
Let $X$ be a weakly pseudoconvex $n$-dimensional complex manifold
equipped with a K\"ahler metric $\omega$, let $L$ $($resp.\ $E)$ be
a hermitian holomorphic line bundle $($resp.\ a hermitian holomorphic 
vector bundle of rank $r$ over~$X)$, and $s$ a global holomorphic 
section of~$E$. Assume that $s$ is generically transverse to the zero
section, and let
$$
Y=\big\{x\in X\,;\;s(x)=0, \Lambda^r ds(x)\not= 0\big\},\qquad
p=\dim Y=n-r.
$$
Moreover, assume that the $(1,1)$-form 
$\ii\Theta(L)+r\,\ii\,d'd''\log|s|^2$ is semipositive and that there is
a continuous function $\alpha\ge 1$ such that the following two inequalities
hold everywhere on $X:$
\smallskip
\item{\rm a)}
$\displaystyle\ii\Theta(L)+r\,\ii\,d'd''\log|s|^2\ge\alpha^{-1}
{\{\ii\Theta(E)s,s\}\over|s|^2}\,,$
\smallskip
\item{\rm b)} $|s|\le e^{-\alpha}$.
\smallskip\noindent
Then for every smooth $D''$-closed \hbox{$(0,q)$-form $f$} over $Y$ with 
values in the line bundle $\Lambda^nT_X^\star\otimes L$ 
$($restricted to~$Y)$, such that
$\int_Y|f|^2|\Lambda^r(ds)|^{-2}dV_\omega<+\infty$, 
there  exists a $D''$-closed $(0,q)$-form $F$ over $X$ with values 
in $\Lambda^nT^\star_X\otimes L$, such that $F$ is smooth over 
$X\ssm\{s=\Lambda^r(ds)=0\}$, satisfies $F_{\restriction Y}=f$ and
$$
\int_X{|F|^2\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}\,,
$$
where $C_r$ is a numerical constant depending only on~$r$.
\endstat

Observe that the differential $ds$ (which is intrinsically defined only at 
points where $s$ vanishes) induces a vector bundle isomorphism
$ds:T_X/T_Y\to E$ along $Y$, hence a non vanishing section 
$\Lambda^r(ds)$, taking values in 
$$
\Lambda^r(T_X/T_Y)^\star\otimes \det E\subset
\Lambda^rT_X^\star\otimes \det E.
$$
The norm $|\Lambda^r(ds)|$ is computed here with respect to the metrics
on $\Lambda^rT_X^\star$ and $\det E$ induced by the K\"ahler metric
$\omega$ and by the given metric on~$E$. Also notice that if
hypothesis a) is satisfied for some $\alpha$, one can always achieve
b) by multiplying the metric of $E$ with a sufficiently small weight
$e^{-\chi\circ\psi}$ (with $\psi$ a psh exhaustion on~$X$ and $\chi$ a
convex increasing function; property a) remains valid after we multiply
the metric of $L$ by $e^{-(r+\smash{\alpha_0^{-1}})\chi\circ\psi}$,
where $\alpha_0=\inf_{x\in X}\alpha(x)$.

\begproof{} Let us first assume that the singularity set $\Sigma=
\{s=0\}\cap\{\Lambda^r(ds)=0\}$ is empty, so that $Y$ is closed and
nonsingular. We claim that there exists a smooth section
$$
F_\infty\in C^\infty(X,\Lambda^{n,q}T^\star_X\otimes L)
=C^\infty(X,\Lambda^{0,q}T^\star_X\otimes\Lambda^nT^\star_X\otimes L)
$$ 
such that
{\parindent=0cm 
\smallskip
(a) $F_\infty$ coincides with $f$ in restriction to $Y$,
\smallskip
(b) $|F_\infty|=|f|$ at every point of~$Y$,
\smallskip
(c) $D''F_\infty=0$ at every point of~$Y$.
\smallskip}

\noindent
For this, consider coordinates patches $U_j\subset X$ biholomorphic 
to polydiscs such that 
$$
U_j\cap Y=\{z\in U_j\,;\;z_1=\ldots=z_r=0\}
$$
in the corresponding
coordinates. We can find a section $\smash{\wt f}$ in
$C^\infty(X,\Lambda^{n,q}T^\star_X\otimes L)$ which achieves a) and b),
since the restriction map $(\Lambda^{0,q}T_X^*)_{\restriction Y}\to
\Lambda^{0,q}T_Y^*$ can be viewed as an orthogonal projection onto a 
$C^\infty$-subbundle of~$(\Lambda^{0,q}T_X^*)_{\restriction Y}$. 
It is enough to extend this subbundle from \hbox{$U_j\cap Y$} to $U_j$ 
(e.g.\ by extending each component of a frame), and then to extend 
$f$ globally via local smooth extensions and a partition of unity.
For any such extension~$\smash{\wt f}$ we have
$$
(D''\wt f)_{\restriction Y}=(D''\wt f_{\restriction Y})=D''f=0.
$$
It follows that we can divide $D''\wt f=\sum_{1\le \lambda\le r}
g_{j,\lambda}(z)\wedge d\ovl z_\lambda$ on $U_j\cap Y$, with suitable
smooth $(0,q)$-forms $g_{j,\lambda}$ which we also extend arbitrarily 
from $U_j\cap Y$ to~$U_j$. Then
$$
F_\infty:=\wt f - \sum_j \theta_j(z)\sum_{1\le \lambda \le r}
\ovl z_\lambda g_{j,\lambda}(z)
$$
coincides with $\wt f$ on $Y$ and satisfies (c).
Since we do not know about $F_\infty$ except in an infinitesimal
neighborhood of~$Y$, we will consider a
truncation $F_\varepsilon$ of $F_\infty$ with support in a small
tubular neighborhood $|s|<\varepsilon$ of~$Y$, and solve the equation
$D''u_\varepsilon=D''F_\varepsilon$ with the constraint that
$u_\varepsilon$ should be $0$ on~$Y$. As $\codim Y=r$, this will be
the case if we can guarantee that $|u_\varepsilon|^2|s|^{-2r}$ is
locally integrable near~$Y$. For this, we will apply Proposition 13.4
with a suitable choice of the functions $\eta$ and $\lambda$, and an
additional weight $|s|^{-2r}$ in the metric of $L$.

Let us consider the smooth strictly convex function $\chi_0:{}]-\infty,0]
\to{}]-\infty,0]$ defined by $\chi_0(t)=t-\log(1-t)$ for $t\le 0$, which
is such that \hbox{$\chi_0(t)\le t$}, \hbox{$1\le\chi_0'\le2$} and 
\hbox{$\chi_0''(t)=1/(1-t)^2$}. 
We set
$$
\sigma_\varepsilon=\log(|s|^2+\varepsilon^2),\qquad
\eta_\varepsilon=\varepsilon-\chi_0(\sigma_\varepsilon).
$$
As $|s|\le e^{-\alpha}\le e^{-1}$, we have $\sigma_\varepsilon\le 0$
for $\varepsilon$ small, and
$$
\eta_\varepsilon\ge\varepsilon-\sigma_\varepsilon
\ge\varepsilon-\log(e^{-2\alpha}+\varepsilon^2).
$$
Given a relatively compact subset $X_c=\{\psi<c\}\compact X$, we thus
have $\eta_\varepsilon\ge 2\alpha$ for $\varepsilon<\varepsilon(c)$ small
enough. Simple calculations yield
$$
\eqalign{
\ii\,d'\sigma_\varepsilon
&={\ii\{D's,s\}\over|s|^2+\varepsilon^2},\cr
\ii\,d'd''\sigma_\varepsilon
&={\ii\{D's,D's\}\over|s|^2+\varepsilon^2}-
{\ii\{D's,s\}\wedge\{s,D's\}\over(|s|^2+\varepsilon^2)^2}-
{\{\ii\Theta(E)s,s\}\over|s|^2+\varepsilon^2}\cr
&\ge{\varepsilon^2\over|s|^2}
{\ii\{D's,s\}\wedge\{s,D's\}\over(|s|^2+\varepsilon^2)^2}
-{\{\ii\Theta(E)s,s\}\over|s|^2+\varepsilon^2}\cr
&\ge{\varepsilon^2\over|s|^2}\ii d'\sigma_\varepsilon\wedge 
d''\sigma_\varepsilon-{\{\ii\Theta(E)s,s\}\over|s|^2+\varepsilon^2},\cr}
$$
thanks to Lagrange's inequality $\ii\{D's,s\}\wedge\{s,D's\}\le|s|^2
\ii\{D's,D's\}$. On the other hand, we have $d'\eta_\varepsilon=
-\chi_0'(\sigma_\varepsilon)d\sigma_\varepsilon$ with 
$1\le\chi_0'(\sigma_\varepsilon)\le 2$, hence
$$
\eqalign{
-\ii d'd''\eta_\varepsilon&=\chi_0'(\sigma_\varepsilon)\ii 
d'd''\sigma_\varepsilon+\chi_0''(\sigma_\varepsilon)\ii d'\sigma_\varepsilon
\wedge d''\sigma_\varepsilon\cr
&\ge \Big({1\over \chi_0'(\sigma_\varepsilon)}{\varepsilon^2\over |s|^2}+
{\chi_0''(\sigma_\varepsilon)\over\chi_0'(\sigma_\varepsilon)^2}\Big)
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon
-\chi_0'(\sigma_\varepsilon){\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}.\cr}
$$
We consider the original metric of $L$ multiplied by the weight $|s|^{-2r}$. 
In this way, we get a curvature form
$$
\ii\Theta_L+r\,\ii d'd''\log|s|^2
\ge{1\over 2}\chi_0'(\sigma_\varepsilon)
\alpha^{-1}{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}
$$
by hypothesis a), thanks to the semipositivity of the
left hand side and the fact that ${1\over 2}\chi_0'(\sigma_\varepsilon){1\over
|s|^2+\varepsilon^2}\le {1\over |s|^2}$. As $\eta_\varepsilon\ge 2\alpha$ 
on $X_c$ for $\varepsilon$ small, we infer
$$
\eta_\varepsilon(\ii\Theta_L+\ii d'd''\log|s|^2)
-\ii d'd''\eta_\varepsilon-
{\chi_0''(\sigma_\varepsilon)\over\chi_0'(\sigma_\varepsilon)^2}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon\ge
{\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon
$$
on~$X_c$. Hence, if $\lambda_\varepsilon=
\chi_0'(\sigma_\varepsilon)^2/\chi_0''(\sigma_\varepsilon)$,
we obtain
$$
\eqalign{
B_\varepsilon&:=\big[\eta_\varepsilon(\ii\Theta_L+\ii d'd''\log|s|^2)
-\ii d'd''\eta_\varepsilon-\lambda_\varepsilon^{-1}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon\;,\;\Lambda\big]\cr
&\phantom{:}\ge\Big[{\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}\ii 
d'\eta_\varepsilon\wedge d''\eta_\varepsilon\,,\,\Lambda\Big]
={\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}
(d''\eta_\varepsilon)(d''\eta_\varepsilon)^\star\cr}
$$
as an operator on $(n,q)$-forms (see the proof of Lemma~13.1).

Let $\theta:\bbbr\to[0,1]$ be a smooth cut-off function such 
that $\theta(t)=1$ on $]-\infty,1/2]$, $\Supp\theta\subset{}]-\infty,1[$
and $|\theta'|\le 3$. For $\varepsilon>0$ small, we consider the 
$(n,q)$-form $F_\varepsilon=\theta(\varepsilon^{-2}|s|^2)F_\infty$ and
its $D''$-derivative
$$
g_\varepsilon=D''F_\varepsilon
=(1+\varepsilon^{-2}|s|^2)\theta'(\varepsilon^{-2}|s|^2)
d''\sigma_\varepsilon\wedge F_\infty+
\theta(\varepsilon^{-2}|s|^2)D''F_\infty
$$
[as is easily seen from the equality $1+\varepsilon^{-2}|s|^2=
\varepsilon^{-2}e^{\sigma_\varepsilon}\,$].
We observe that $g_\varepsilon$ has its support contained in the tubular
neighborhood $|s|<\varepsilon\,$; moreover, as $\varepsilon\to 0$, the 
second term in the right hand side converges uniformly to $0$ on every 
compact set; it will therefore produce no contribution in the
limit. On the other hand, the first term has the same order of magnitude
as $d''\sigma_\varepsilon$ and $d''\eta_\varepsilon$, and can be controlled
in terms of~$B_\varepsilon$. In fact, for any $(n,q)$-form $u$ and any
$(n,q+1)$-form $v$ we have
$$
\eqalign{
|\langle d''\eta_\varepsilon\wedge u,v\rangle|^2
&=|\langle u,(d''\eta_\varepsilon)^\star v\rangle|^2
\le|u|^2|(d''\eta_\varepsilon)^\star v|^2
=|u|^2\langle (d''\eta_\varepsilon)(d''\eta_\varepsilon)^\star v,v
\rangle\cr
&\le {\chi_0'(\sigma_\varepsilon)|s|^2\over\varepsilon^2}
|u|^2\langle B_\varepsilon v,v\rangle.\cr}
$$
This implies
$$
\langle B_\varepsilon^{-1}(d''\eta_\varepsilon\wedge u),
(d''\eta_\varepsilon\wedge u)\rangle\le{\chi_0'(\sigma_\varepsilon)
|s|^2\over\varepsilon^2}|u|^2.
$$
The main term in $g_\varepsilon$ can be written
$$
g_\varepsilon^{(1)}:=
(1+\varepsilon^{-2}|s|^2)\theta'(\varepsilon^{-2}|s|^2)
\chi_0'(\sigma_\varepsilon)^{-1}d''\eta_\varepsilon\wedge F_\infty.
$$
On $\Supp g_\varepsilon^{(1)}\subset\{|s|<\varepsilon\}$, since
$\chi_0'(\sigma_\varepsilon)\ge 1$, we thus find
$$
\langle B_\varepsilon^{-1}g_\varepsilon^{(1)},
g_\varepsilon^{(1)}\rangle\le (1+\varepsilon^{-2}|s|^2)^2
\,\theta'(\varepsilon^{-2}|s|^2)^2|F_\infty|^2.
$$
Instead of working on $X$ itself, we will work rather on the relatively
compact subset $X_c\ssm Y_c$, where $Y_c=Y\cap X_c=Y\cap\{\psi<c\}$. We know
that $X_c\ssm Y_c$ is again complete K\"ahler by Lemma 11.9.
In this way, we avoid the singularity of the weight $|s|^{-2r}$ along~$Y$.
We find
$$
\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon^{(1)},
g_\varepsilon^{(1)}\rangle\, |s|^{-2r}dV_\omega\le
\int_{X_c\ssm Y_c}|F_\infty|^2
(1+\varepsilon^{-2}|s|^2)^2\theta'(\varepsilon^{-2}|s|^2)^2
|s|^{-2r}dV_\omega.
$$
Now, we let $\varepsilon\to 0$ and view $s$ as ``transverse local 
coordinates'' around $Y$. As $F_\infty$ coincides with $f$ on~$Y$, 
it is not hard to see that the 
right hand side converges to $c_r\int_{Y_c}|f|^2|\Lambda^r(ds)|^{-2}
dV_{Y,\omega}$ where $c_r$ is the ``universal'' constant
$$
c_r=\int_{z\in\bbbc^r,\,|z|\le 1}
(1+|z|^2)^2\theta'(|z|^2)^2{\ii^{r^2}\Lambda^r(dz)
\wedge\Lambda^r(d\ovl z)\over|z|^{2r}}<+\infty
$$
depending only on~$r$. The second term
$$
g_\varepsilon^{(2)}=\theta(\varepsilon^{-2}|s|^2)d''F_\infty
$$
in $g_\varepsilon$ satisfies $\Supp(g_\varepsilon^{(2)})\subset\{|s|
<\varepsilon\}$ and $|g_\varepsilon^{(2)}|=O(|s|)$ (just look at the 
Taylor expansion of $d''F_\infty$ near $Y$). From this we easily conclude that
$$
\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon^{(2)},
g_\varepsilon^{(2)}\rangle\, |s|^{-2r}dV_{X,\omega}=O(\varepsilon^2),
$$
provided that $B_\varepsilon$ remains locally uniformly bounded below near~$Y$
(this is the case for instance if we have strict inequalities in the
curvature assumption a)). If this holds true, we apply Proposition~8.4
on $X_c\ssm Y_c$ with the additional weight factor~$|s|^{-2r}$. Otherwise, 
we use the modified estimate stated
in Remark~8.5 in order to solve the approximate equation $D''u+
\delta^{1/2}h=g_\varepsilon$ with $\delta>0$ small. This yields sections
$u=u_{c,\varepsilon,\delta}$, $h=h_{c,\varepsilon,\delta}$ such that
$$
\eqalign{
\int_{X_c\ssm Y_c}(\eta_\varepsilon+\lambda_\varepsilon)^{-1}
|u_{c,\varepsilon,\delta}|^2|s|^{-2r}\,dV_\omega
&{}+
\int_{X_c\ssm Y_c}|h_{c,\varepsilon,\delta}|^2|s|^{-2r}\,dV_\omega\cr
&\le 2\int_{X_c\ssm Y_c}\langle (B_\varepsilon+\delta I)^{-1}g_\varepsilon,
g_\varepsilon\rangle|s|^{-2r}\,dV_\omega,\cr}
$$ 
and the right hand side is under control in all cases. The extra error
term $\delta^{1/2}h$ can be removed at the end by letting $\delta$ tend
to~$0$.  Since there is essentially no additional difficulty involved
in this process, we will assume for simplicity of exposition that we
do have the required lower bound for~$B_\varepsilon$ and the estimates
of $\smash{g_\varepsilon^{(1)}}$ and $\smash{g_\varepsilon^{(2)}}$ as
above. For $\delta=0$, the above estimate provides a solution
$u_{c,\varepsilon}$ of the equation $D''u_{c,\varepsilon}=
g_\varepsilon=D''F_\varepsilon$ on $X_c\ssm Y_c$, such that
$$
\eqalign{
\int_{X_c\ssm Y_c}(\eta_\varepsilon+\lambda_\varepsilon)^{-1}
|u_{c,\varepsilon}|^2|s|^{-2r}dV_{X,\omega}
&\le 2\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon,
g_\varepsilon\rangle\, |s|^{-2r}dV_{X,\omega}\cr
&\le 2\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
+O(\varepsilon).\cr}
$$
Here we have
$$
\eqalign{
\sigma_\varepsilon&=\log(|s|^2+\varepsilon^2)\le
\log(e^{-2\alpha}+\varepsilon^2)\le -2\alpha+O(\varepsilon^2)\le
-2+O(\varepsilon^2),\cr
\eta_\varepsilon&=\varepsilon-\chi_0(\sigma_\varepsilon)\le
(1+O(\varepsilon))\sigma_\varepsilon^2,\cr
\lambda_\varepsilon&={\chi_0'(\sigma_\varepsilon)^2\over
\chi_0''(\sigma_\varepsilon)}=(1-\sigma_\varepsilon)^2+
(1-\sigma_\varepsilon)\le(3+O(\varepsilon))\sigma_\varepsilon^2,\cr
\eta_\varepsilon+\lambda_\varepsilon&\le
(4+O(\varepsilon))\sigma_\varepsilon^2\le(4+O(\varepsilon))
\big(-\log(|s|^2+\varepsilon^2)\big)^2.\cr}
$$
As $F_\varepsilon$ is uniformly bounded with support in $\{|s|<\varepsilon\}$,
we conclude from an obvious volume estimate that
$$
\int_{X_c}{|F_\varepsilon|^2\over
(|s|^2+\varepsilon^2)^r(-\log(|s|^2+\varepsilon^2))^2}dV_{X,\omega}
\le {\Const \over (\log\varepsilon)^2}.
$$
Therefore, thanks to the usual inequality
$|t+u|^2\le (1+k)|t|^2+(1+k^{-1})|u|^2$ applied to the sum
$F_{c,\varepsilon}=\wt f_\varepsilon-u_{c,\varepsilon}$
with $k=|\log\varepsilon|$, we obtain
from our previous estimates
$$
\int_{X_c\ssm Y_c}
{|F_{c,\varepsilon}|^2\over
(|s|^2+\varepsilon^2)^r(-\log(|s|^2+\varepsilon^2))^2}dV_{X,\omega}
\le 8\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
+O(|\log\varepsilon|^{-1}).
$$
In addition to this, we have $d''F_{c,\varepsilon}=0$ by construction,
and this equation extends from $X_c\ssm Y_c$ to $X_c$ by Lemma~11.10.

If $q=0$, then $u_{c,\varepsilon}$ must be smooth also, and the
non integrability of the weight $|s|^{-2r}$ along $Y$ shows that
$u_{c,\varepsilon}$ vanishes on~$Y$, therefore
$$
F_{c,\varepsilon\restriction Y}=
F_{\varepsilon\restriction Y}=F_{\infty\restriction Y}=f.
$$
The theorem and its final estimate are thus obtained by extracting
weak limits, first as $\varepsilon\to 0$, and then as $c\to+\infty$. The
initial assumption that $\Sigma=\hbox{$\{s=\Lambda^r(ds)=0\}$}$ is empty
can be easily removed in two steps: i) the result is true if $X$ is 
Stein, since we can always find a complex hypersurface $Z$ in $X$ such 
that $\Sigma\subset\ovl Y\cap Z \subsetneq\ovl Y$, and 
then apply the extension theorem on the Stein manifold $X\ssm Z$, in 
combination with Lemma~11.10$\,$; ii) the whole procedure still works when
$\Sigma$ is nowhere dense in~$\ovl Y$ (and possibly nonempty). Indeed 
local $L^2$ extensions $\wt f_j$ still exist by step~i) applied on small 
coordinate balls~$U_j\,$; we then set $F_\infty=\sum\theta_j\wt f_j$ and
observe that $|D''F_\infty|^2|s|^{-2r}$ is locally integrable, thanks to the
estimate $\int_{\smash{U_j}}|\wt f_j|^2|s|^{-2r}(\log|s|)^{-2}dV<+\infty$ 
and the fact that $|\sum d''\theta_j\wedge\wt f_j|=O(|s|^\delta)$ for 
suitable $\delta>0$ [as follows from Hilbert's Nullstensatz applied 
to $\wt f_j-\wt f_k$ at singular points of~$\ovl Y\,]$.

When $q\ge 1$, the arguments needed to get a smooth solution involve
more delicate considerations, and we will only sketch the details. The
main difficulty lies in the presence of the weight $|s|^{-2r}$, which
creates trouble at points of~$Y$ when one tries to use an elliptic
regularity argument (see Remark~8.5). If $r=1$, however, the
subvariety $Y$ is a divisor; therefore, when we consider a $D''$~equation
with values in the line bundle $\Lambda^nT^\star_X\otimes L$,
a $L^2$ solution for the weight $|s|^{-2}$ can be interpreted as
a $L^2$ solution with values in the twisted line bundle
$\Lambda^nT^\star_X\otimes L\otimes\cO_X(-Y)$, equipped with a smooth
hermitian metric. Hence, if $r=1$, the minimal $L^2$ solution
$u_{c,\varepsilon}$ of the $D''$ equation considered earlier satisfies
the equations
$$
D''u_{c,\varepsilon}=g_\varepsilon=D''F_\varepsilon,\qquad
D^{\prime\prime\star}(|s|^{-2}u_{c,\varepsilon})=0\quad
\hbox{\rm (minimality condition)}
$$
on $X_c\ssm Y_c$. These equations can be rewritten as
$$
D''(s^{-1}u_{c,\varepsilon})=s^{-1}D''F_\varepsilon,\qquad
D^{\prime\prime\star}(s^{-1}u_{c,\varepsilon})=0.
$$
By Lemma~11.10, the latter equalities are valid on $X_c$ and not 
only on $X_c\ssm Y_c$, for $s^{-1}u_{c,\varepsilon}$ is locally $L^2$ and 
$s^{-1}D''F_\varepsilon$ is locally bounded. From this, we infer that
$F_{c,\varepsilon}=F_\varepsilon-u_{c,\varepsilon}$ satisfies
$$
\eqalign{
D''(s^{-1}F_{c,\varepsilon})
 &=D''(s^{-1}F_\varepsilon)-s^{-1}D''F_\varepsilon,\cr
D^{\prime\prime\star}(s^{-1}F_{c,\varepsilon})
 &=D^{\prime\prime\star}(s^{-1}F_\varepsilon)
 =D^{\prime\prime\star}\big(\theta(\varepsilon^{-2}|s|^2)s^{-1}F_\infty\big).
\cr}
$$
It is easy to show that $D''(s^{-1}\smash{\wt f})-s^{-1}D''\smash{\wt f}$
is independent of the choice of the smooth extension $\smash{\wt f}$ of $f$ 
(whether $\smash{\wt f}$ is $D''$-closed or not is irrelevant), and that 
it is equal to the current $D''(s^{-1})\wedge\smash{\wt f}$ with 
support in~$Y$. On the other hand, $s^{-1}F_\infty$ is locally
integrable, hence $\theta(\varepsilon^{-2}|s|^2)s^{-1}F_\infty$ converges
weakly to $0$ as $\varepsilon\to 0$. By extracting a weak limit
$F_{c,\varepsilon}\to F$ in $L^2_\loc((|s|\log|s|)^{-2})$, we easily see
that $s^{-1}F_{c,\varepsilon}\to s^{-1}F$ in the weak topology of 
distributions, therefore
$$
D''(s^{-1}F)=D''(s^{-1})\wedge\wt f,\qquad
D^{\prime\prime\star}(s^{-1}F)=0
$$ 
in the limit. In particular $s^{-1}F$ is $\Delta''$-harmonic on 
$X\ssm Y$, hence $F$ is smooth on~$X\ssm Y$. Unfortunately, the above
equations do not imply smoothness of the coefficients of $F$ all
over~$X$, but only H\"older continuity near~$Y$ (for any H\"older
exponent~$\gamma<1$). In fact, we can always choose a smooth
local extension $\smash{\wt f}$ such that $D''\smash{\wt f}=0$ and
$\nabla^{0,1}\ovl s\ort\smash{\wt f}=0$ on $Y$ (if the second
condition is not satisfied, we replace $\smash{\wt f}$ with
$\smash{\wt f}-D''(\ovl s\,h)$, where $h$ is a suitable smooth
$(n,q-1)$-form on~$X\,$; the values taken by $\smash{\wt f}$ on~$Y$ are
then uniquely defined). We find
$$
D''(s^{-1}(F-\wt f))=0,\qquad
D^{\prime\prime\star}(s^{-1}(F-\wt f))=-D^{\prime\prime\star}(s^{-1}\wt f),
$$
and the condition $\nabla^{0,1}\ovl s\ort\smash{\wt f}=0$ on $Y$ shows
that the singularity of $D^{\prime\prime\star}(s^{-1}\smash{\wt f})$ along
$Y$ is at most $O(|s|^{-1})$. Our equations yield 
$$
\Delta''(s^{-1}(F-\wt f))=-D''D^{\prime\prime\star}(s^{-1}\wt f),
$$ 
hence
$$
s^{-1}(F-\wt f)=G^{n,q}\big(D''D^{\prime\prime\star}(s^{-1}\wt f)\big)=
D''G^{n,q-1}\big(D^{\prime\prime\star}(s^{-1}\wt f)\big)\qquad\mod C^\infty
$$
where $G^{p,q}$ is a (local) Green kernel for the $\Delta''$ operator
in bidegree~$(p,q)$. As the derivatives of order $1$ of $G^{n,q-1}$
have singularity $|x-y|^{-(2n-1)}$ along the diagonal and
$D^{\prime\prime\star}(s^{-1}\smash{\wt f})=O(|s|^{-1})$, we find
$s^{-1}(F-\smash{\wt f})=O(\log|s|)$ and the H\"older continuity of
$F-\smash{\wt f}$ (hence of~$F$) follows, as well as the fact that
$F_{\restriction Y}=\smash{\wt f}_{\restriction Y}=f$. We claim that
$F$ can be corrected so as to obtain a smooth extension $\smash{\wt F}$
with $|F-\smash{\wt F}|$ small and decaying as rapidly as we wish at
infinity; hence $\smash{\wt F}$ will satisfy the desired global $L^2$
estimate. Indeed, there is a covering of $Y$ by open sets $U_j$ in $X$ such
that $f$ admits a smooth $D''$-closed extension $\smash{\wt f_j}$ on~$U_j$,
with the following additional properties: $\nabla^{0,1}\ovl s\ort\smash{
\wt f_j}=0$ on $Y\cap U_j$, and $s^{-1}(\smash{\wt f_j-\wt f_k})$ is smooth
on~$U_j\cap U_k$. Only the latter property needs to be checked. We show
by induction on $\ell\ge 1$ that $\smash{\wt f_j}$ can be chosen so that
$$
\wt f_k-\wt f_j=s\,v_{jk}+\ovl s^\ell w_{jk}
$$
with suitable smooth $(n,q)$-forms $v_{jk}$ and~$w_{jk}$. This is true when
$\ell=1$, since by uniqueness $\smash{\wt f_k-\wt f_j}$ must vanish
on~$Y\cap U_j$. Now, the $D''$-closedness implies
$$
0=s\,D''v_{jk}+\ovl s^\ell D''w_{jk}+\ell\ovl s^{\ell-1}\ovl{D's}\wedge w_{jk},
$$
and an identification of the coefficients of the Taylor expansion in $s$,
$\ovl s$ shows that $\ovl{D's}\wedge w_{jk}=0$ on~$Y\cap U_j\cap U_k$.
This implies
$$
w_{jk}=\ovl{D's}\wedge w^{(1)}_{jk}+s\,w^{(2)}_{jk}+\ovl s\,w^{(3)}_{jk}
$$
with smooth forms $w^{(1)}_{jk}$, $w^{(2)}_{jk}$, $w^{(3)}_{jk}$. The
$(n,q-1)$ form $w^{(1)}_{jk}$ is uniquely defined if we require the
additional condition $\smash{\nabla^{0,1}\ovl s}\ort w^{(1)}_{jk}=0$. 
Then $(w^{(1)}_{jk})$ satisfies the \v{C}ech cocycle condition and we can 
write $w^{(1)}_{jk}=w^{(1)}_k-w^{(1)}_j$ for some $0$-cochain $(w^{(1)}_j)$.
We conclude from these relations that $\smash{\wt f_j}-(\ell+1)^{-1}
D''(\ovl s^{\ell+1}w^{(1)}_j)$ admits a \v{C}ech $1$-coboundary
$$
\eqalign{
\wt f_k-\wt f_j-(\ell+1)^{-1}D''(\ovl s^{\ell+1}w^{(1)}_{jk})
&=\wt f_k-\wt f_j-\ovl s^\ell\ovl{D's}\wedge w^{(1)}_{jk}
\quad\mod(\ovl s^{\ell+1})\cr
&=s(v_{jk}+\ovl s^\ell w^{(2)}_{jk})\quad\mod(\ovl s^{\ell+1}),\cr}
$$
hence it satisfies the induction hypothesis at order $\ell+1$. By arranging 
the asympotic expansion up to infinite order, we
infer that $\smash{\wt f_k-\wt f_j}-s\,v_{jk}$ is flat along~$Y$, hence
that $s^{-1}(\smash{\wt f_k-\wt f_j})$ is smooth. Our claim is thus proved.
Now, the Green kernel argument shows that $F$ can be written as
$F=\smash{\wt f_j}+sD''h_j$ on~$U_j$, where
$$
h_j=G^{n,q-1}(D^{\prime\prime\star}(s^{-1}\wt f_j)) \qquad\mod C^\infty
$$
is smooth on $U_j\ssm Y$ and has its first order derivatives bounded
by $O(\log|s|)$ near~$Y$.  Furthermore, $h_k-h_j$ is smooth on
$U_j\cap U_k$. Therefore, if we select sufficiently good
approximations $h_j\star\rho_{\varepsilon_j}$ of $h_j$ and a
collection of smooth functions $(\theta_j)$ with \hbox{$\Supp\theta_j
\subset U_j$}, $0\le\theta_j\le 1$ and $\sum\theta_j=1$ near~$Y$, the
$(n,q)$-form
$$
\wt F=F-D''\Big(s\sum_j\theta_j(h_j-h_j\star\rho_{\varepsilon_j})\Big)
$$
is smooth and satisfies all our requirements.

When $r>1$, the above argument can no longer be applied directly; 
one possibility to overcome this difficulty is to blow-up $Y$
so as to deal again with the case of a divisor. We may assume that
$\Sigma=\emptyset$ (otherwise, we just replace $X_c$ with
$X_c\ssm\Sigma$, which is again complete K\"ahler). Instead of
working on $X_c\ssm Y_c$ as we did earlier, we work on the blow-up
$\wh X_c$ of $X_c$ along~$Y_c$. If $\mu:\wh X_c\to X_c$ is the blow-up
map, $\wh Y_c=\mu^{-1}(Y_c)$ the exceptional divisor and $\gamma$ a
positive constant, we equip $\wh X_c$ with the
smooth K\"ahler metric $\wh\omega_\gamma=\mu^\star\omega+\gamma\,
(\ii d'd''\log|s|^2+{\ii\over r}\Theta(L))\ge\mu^\star\omega$. Then the
minimal $L^2(\omega_\gamma)$ solution $u_{c,\varepsilon,\gamma}$ satisfies
the equations
$$
D''u_{c,\varepsilon,\gamma}=\mu^\star g_\varepsilon
=D''(\mu^\star F_\varepsilon),\qquad
D^{\prime\prime\star}_{\omega_\gamma}(|s|^{-2r}u_{c,\varepsilon,\gamma})=0
$$
on $\wh X_c\ssm\wh Y_c$, and
$\wh F_{c,\varepsilon,\gamma}=F_\varepsilon-u_{c,\varepsilon,\gamma}$
satisfies the $L^2$ estimate
$$
\int_{\wh X_c}
{|\wh F_{c,\varepsilon,\gamma}|^2\over
(|\wh s|^2+\varepsilon^2)^r(-\log(|\wh s|^2+\varepsilon^2))^2}
dV_{\wh X_c,\omega_\gamma}
\le 8\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
+{\Const\over(\log\varepsilon)^2}
$$
where $\wh s=s\circ\mu$ (one can use monotonicity with respect to metrics
and the fact that $\omega_\gamma\ge\mu^\star\omega$ to see that the
right hand side always admits the $\omega$-bound as an upper bound).
We can view $X_c$ as a submanifold of the projectivized bundle $P(E)$
of lines of~$E$, and $\cO_{\wh X_c}(-\wh Y_c)$ as the restriction to~$X_c$
of the tautological line bundle $\cO_{P(E)}(-1)$ on~$P(E)$. We thus view
$\wh s$ as a section of $\cO_{\wh Y_c}(-\wh Y_c)$ (actually, $\wh s$ is
a gene\-rator of that ideal sheaf). Since $|\wh s|^{-2r}|u_{c,\gamma,
\varepsilon}|^2$ is locally integrable by construction, we get
$$
D''({\wh s\,}^{-r}u_{c,\varepsilon,\gamma})=
{\wh s\,}^{-r}D''(\mu^\star F_\varepsilon),\qquad
D^{\prime\prime\star}_{\omega_\gamma}({\wh s\,}^{-r}u_{c,\varepsilon,\gamma})=0
$$
on $\wh X_c$. Passing to the limit as $\varepsilon,\,\gamma$ tend to $0$ and
$c$ tends to $+\infty$, we find a $(n,q)$-form $\wh F$ with $L^2_\loc$ 
coefficients on $\wh X$ such that
$$
\int_{\wh X}
{|\wh F|^2\over
|\wh s|^{2r}(-\log|\wh s|)^2}
dV_{\wh X,\omega}
\le 8\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
$$
and
$$
\eqalign{D''({\wh s\,}^{-r}\wh F)
&=D''({\wh s\,}^{-r}\mu^\star\wt f)-{\wh s\,}^{-r}D''(\mu^\star\wt f)
=D''({\wh s\,}^{-1})\wedge ({\wh s\,}^{-(r-1)}\mu^\star\wt f),\cr
D^{\prime\prime\star}_{\mu^\star\omega}({\wh s\,}^{-r}\wh F)&=0\cr}
$$
in the sense of distributions (everywhere on $\wh X$ and not only on
$\wh X\ssm\wh Y$); indeed, thanks to the equality
$$
\mu^\star(\Lambda^nT^\star_{X_c})=\Lambda^nT^\star_{\wh X_c}\otimes
\cO_{\wh X_c}(-(r-1)\wh Y_c),
$$ 
we see that $\mu^\star\wt f$ vanishes at order $r-1$ along $\wh Y$.
If we view $\wh F$ as a $(0,q)$-form with values in
$\mu^\star(\Lambda^nT^\star_X\otimes L)$ rather than as a $(n,q)$-form
on $\wh X$, we may consider philosophically that we cancel out a
factor ${\wh s\,}^{r-1}$ in the equations. This shows that we are
essentially in the same situation as in the earlier case $r=1$, except
that $D^{\prime\prime\star}_{\mu^\star\omega}$ is computed with
respect to a metric $\mu^\star\omega$ which is degenerate along~$\wh Y$.
A~finer analysis of the Green kernel of $\Delta''_{\mu^\star\omega}$
shows that $F:=\mu_\star F$ is smooth on $X\ssm Y$, that $F$ still has 
continuous coefficients near $Y$, and that $F_{\restriction Y}=f$. 
We then produce the desired solution by taking a small perturbation
of~$F$ as above. The details are rather tedious and will be left to
the reader.\qed
\endproof

\begstat{(13.7) Remarks} \rm
\smallskip
\noindent{\rm a)} When $q=0$, the estimates provided by Theorem~13.6 are
independent of the K\"ahler metric~$\omega$. In fact, if $f$ and $F$
are holomorphic sections of $\Lambda^nT^\star_X\otimes L$ over $Y$ 
(resp.~$X$), viewed as $(n,0)$-forms with values in~$L$, we can ``divide''
$f$ by $\Lambda^r(ds)\in\Lambda^r(TX/TY)^\star\otimes\det E$ to get a 
section $f/\Lambda^r(ds)$ of $\Lambda^pT^\star_Y\otimes L\otimes(\det E)^{-1}$
over~$Y$. We then find
$$
\eqalign{
|F|^2dV_{X,\omega}&=\ii^{n^2}\{F,F\},\cr
{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
&=\ii^{p^2}\{f/\Lambda^r(ds),f/\Lambda^r(ds)\},\cr}
$$
where $\{\bu,\bu\}$ is the canonical bilinear pairing described in~(6.3).
\medskip
\noindent{\rm b)} The hermitian structure on $E$ is not really used
in~depth. In fact, one only needs $E$ to be equipped with a {\it Finsler
metric}, that is, a smooth complex homogeneous function of degree~$2$
on~$E$ [or equivalently, a smooth hermitian metric on the tautological
bundle $\cO_{P(E)}(-1)$ of lines of~$E$ over the projectivized 
bundle~$P(E)$]. The section $s$ of $E$ induces a section $[s]$ of
$P(E)$ over $X\ssm s^{-1}(0)$ and a corresponding section $\wt s$ of
the pull-back line bundle $[s]^\star\cO_{P(E)}(-1)$.  A trivial check
shows that Theorem~13.6 as well as its proof extend to the case of a
Finsler metric on~$E$, if we replace everywhere $\{\ii\Theta(E)s,s\}$
by $\{\ii\Theta([s]^\star\cO_{P(E)}(-1))\wt s,\wt s\,\}$ (especially in
hypothesis 13.6~b)). A minor issue is that $|\Lambda^r(ds)|$ is 
(a priori) no longer defined, since no obvious hermitian norm exists
on~$\det E$. A posteriori, we have the following ad hoc definition of 
a metric on $(\det E)^\star$ which makes the $L^2$~estimates work as 
before: for $x\in X$ and $\xi\in\Lambda^rE^\star_x$, we set
$$
|\xi|^2_x={1\over c_r}
\int_{z\in E_x}(1+|z|^2)^2
\theta'(|z|^2)^2\,{\ii^{r^2}\xi\wedge\ovl\xi\over|z|^{2r}}
$$
where $|z|$ is the Finsler norm on~$E_x$ [the constant $c_r$ is there to 
make the result agree with the hermitian case; it is not hard to see that
this metric does not depend on the choice of~$\theta\,$].\qed
\endstat

We now present a few interesting corollaries. The first one is a
surjectivity theorem for restriction morphisms in Dolbeault cohomology.

\begstat{(13.8) Corollary}
Let $X$ be a projective algebraic manifold and $E$ a holomorphic
vector bundle of rank $r$ over~$X$, $s$ a holomorphic section of $E$
which is everywhere transverse to the zero section, $Y=s^{-1}(0)$, and
let $L$ be a holomorphic line bundle such that \hbox{$F=L^{1/r}\otimes
E^\star$} is Griffiths positive $($we just mean formally that
\hbox{${1\over r} \ii\Theta(L)\otimes\Id_E-\ii\Theta(E)>_\Grif
0)$}. Then the restriction morphism
$$
H^{0,q}(X,\Lambda^nT^\star_X\otimes L)\to H^{0,q}(Y,\Lambda^nT^\star_X
\otimes L)
$$
is surjective for every $q\ge 0$.
\endstat

\begproof{} A short computation gives
$$
\eqalign{
\ii\,d'&d''\log|s|^2=\ii\,d'\Big({\{s,D's\}\over|s|^2}\Big)\cr
&=\ii\Big({\{D's,D's\}\over|s|^2}-{\{D's,s\}\wedge \{s,D's\}\over |s|^4}+
{\{s,\Theta(E)s\}\over|s|^2}\Big)
\ge -{\{\ii\Theta(E)s,s\}\over|s|^2}\cr}
$$
thanks to Lagrange's inequality and the fact that $\Theta(E)$ is
antisymmetric. Hence, if $\delta$ is a small positive constant
such that
$$
-\ii\Theta(E)+{1\over r}\ii\Theta(L)\otimes\Id_E\ge_\Grif\delta\,
\omega\otimes\Id_E>0,
$$
we find
$$
\ii\Theta(L)+r\,\ii\,d'd''\log|s|^2\ge r\delta\,\omega.
$$
The compactness of $X$ implies $\ii\Theta(E)\le C\omega\otimes\Id_E$
for some $C>0$. Theorem~13.6 can thus be applied with $\alpha=r\delta/C$
and Corollary 13.8 follows. By remark 13.7~b), the above surjectivity
property even holds if $L^{1/r}\otimes E^\star$ is just assumed to be
ample (in the sense that the associated line bundle $\pi^\star L^{1/r}
\otimes\cO_{P(E)}(1)$ is positive on the projectivized bundle
$\pi:P(E)\to X$ of lines of~$E$).\qed
\endproof

Another interesting corollary is the following special case, 
dealing with bounded pseudoconvex domains $\Omega\compact\bbbc^n$. Even
this simple version retains highly interesting information on the behavior
of holomorphic and plurisubharmonic functions.

\begstat{(13.9) Corollary} Let $\Omega\subset\bbbc^n$ be a bounded
pseudoconvex domain, and let $Y\subset X$ be a nonsingular complex
submanifold defined by a section $s$ of some hermitian vector
bundle $E$ with {\rm bounded} curvature tensor on~$\Omega$. Assume
that $s$ is everywhere transverse to the zero section and that
$|s|\le e^{-1}$ on~$\Omega$. Then there is a constant $C>0$ $($depending 
only on $E)$, with the following property:
for every psh function $\varphi$ on $\Omega$, every holomorphic 
function $f$ on $Y$ with $\int_Y|f|^2|\Lambda^r(ds)|^{-2}e^{-\varphi}dV_Y
<+\infty$, there exists an extension $F$ of $f$ to $\Omega$ such that
$$
\int_\Omega{|F|^2\over|s|^{2r}(-\log|s|)^2}e^{-\varphi}dV_\Omega\le
C\int_Y{|f|^2\over|\Lambda^r(ds)|^2}e^{-\varphi}dV_Y.
$$
\endstat

\begproof{} We apply essentially the same idea as for the previous corollary,
in the special case when $L=\Omega\times\bbbc$ is the trivial bundle
equipped with a weight function $e^{-\varphi-A|z|^2}$. The choice of a
sufficiently large constant $A>0$ guarantees that the curvature assumption
13.6~a) is satisfied ($A$ just depends on the presupposed bound for the 
curvature tensor of~$E$).\qed
\endproof

\begstat{(13.10) Remark} \rm The special case when $Y=\{z_0\}$ is a point is 
especially interesting. In that case, we just take $s(z)=(e\diam\Omega)^{-1}
(z-z_0)$, viewed as a section of the rank $r=n$ trivial vector bundle 
$\Omega\times \bbbc^n$ with $|s|\le e^{-1}$. We take $\alpha=1$ 
and replace $|s|^{2n}(-\log|s|)^2$ in the denominator by 
$|s|^{2(n-\varepsilon)}$, using the inequality
$$
-\log|s|={1\over\varepsilon}\log|s|^{-\varepsilon}\le {1\over\varepsilon}
|s|^{-\varepsilon},\qquad\forall \varepsilon>0.
$$ 
For any given value $f_0$, we then find a holomorphic function $f$ such 
that $f(z_0)=f_0$ and
$$
\int_\Omega{|f(z)|^2\over|z-z_0|^{2(n-\varepsilon)}}e^{-\varphi(z)}dV_\Omega
\le{C_n\over\varepsilon^2(\diam\Omega)^{2(n-\varepsilon)}}|f_0|^2
e^{-\varphi(z_0)}.
$$
\endstat

\titleb{14.}{Approximation of psh functions by logarithms of\ \newline
holomorphic functions}
\titlebrunning{14. Approximation of psh functions by logarithms of
holomorphic functions}
We prove here, as an application of the Ohsawa-Takegoshi extension theorem,
that every psh function on a pseudoconvex open set $\Omega\subset\bbbc^n$
can be approximated very accurately by functions of the form
$c\log|f|$, where $c>0$ and $f$ is a holomorphic function. The main idea
is taken from (Demailly 1992). For other applications to algebraic geometry,
see (Demailly 1993b) and (Demailly-Koll\'ar~1996). Recall that 
the Lelong number of a function $\varphi\in\Psh(\Omega)$ at a point $x_0$ 
is defined to be
$$
\nu(\varphi,x_0)=\liminf_{z\to x_0}{\log\varphi(z)\over\log|z-x_0|}=
\lim_{r\to 0_+}{\sup_{B(x_0,r)}\varphi\over \log r}.
$$
In particular, if $\varphi=\log|f|$ with $f\in\cO(\Omega)$, then 
$\nu(\varphi,x_0)$ is equal to
the vanishing order $\ord_{x_0}(f)=\sup\{k\in\bbbn\,;D^{\alpha}f(x_0)=0,~
\forall|\alpha|<k\}$.

\begstat{(14.1) Theorem} Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\bbbc^n$. For every $m>0$, let
$\cH_\Omega(m\varphi)$ be the Hilbert space of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-2m\varphi}d\lambda<+\infty$ and
let $\varphi_m={1\over 2m}\log\sum|\sigma_\ell|^2$ where $(\sigma_\ell)$
is an orthonormal basis of~$\cH_\Omega(m\varphi)$. Then there are
constants $C_1,C_2>0$ independent of $m$ such that
\item{\rm a)}$\displaystyle\varphi(z)-{C_1\over m}\le
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over m}
\log{C_2\over r^n}$\hfill\break
for every $z\in\Omega$ and $r<d(z,\partial\Omega)$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$ and
\item{\rm b)}$\displaystyle\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le
\nu(\varphi,z)$~ for every $z\in\Omega$.
\endstat

\begproof{} Note that $\sum|\sigma_\ell(z)|^2$ is the square of the norm
of the evaluation linear form $f\mapsto f(z)$ on $\cH_\Omega(m\varphi)$. As
$\varphi$ is locally bounded above, the $L^2$ topology is actually stronger
than the topology of uniform convergence on compact subsets of~$\Omega$.
It follows that the series $\sum|\sigma_\ell|^2$ converges uniformly on
$\Omega$ and that its sum is real analytic. Moreover we have
$$\varphi_m(z)=\sup_{f\in B(1)}{1\over m}\log|f(z)|$$
where $B(1)$ is the unit ball of $\cH_\Omega(m\varphi)$. For
$r<d(z,\partial\Omega)$, the mean value inequality applied to the psh
function $|f|^2$ implies
$$\eqalign{
|f(z)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|\zeta-z|<r}
|f(\zeta)|^2d\lambda(\zeta)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|\zeta-z|<r}\varphi(\zeta)\Big)
\int_\Omega|f|^2e^{-2m\varphi}d\lambda.\cr}$$
If we take the supremum over all $f\in B(1)$ we get
$$
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the second inequality in a) is proved. Conversely, the
Ohsawa-Takegoshi extension theorem (estimate 13.10) applied
to the $0$-dimensional subvariety $\{z\}\subset\Omega$ shows that for any
$a\in\bbbc$ there is a holomorphic function $f$ on $\Omega$ such that
$f(z)=a$ and
$$\int_\Omega|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z)},$$
where $C_3$ only depends on $n$ and~${\rm diam}\,\Omega$. We fix $a$
such that the right hand side is~$1$. This gives the other
inequality
$$\varphi_m(z)\ge{1\over m}\log|a|=\varphi(z)-{\log C_3\over 2m}.$$
The above inequality implies $\nu(\varphi_m,z)\le\nu(\varphi,z)$. In the
opposite direction, we find
$$\sup_{|x-z|<r}\varphi_m(x)\le\sup_{|\zeta-z|<2r}\varphi(\zeta)+
{1\over m}\log{C_2\over r^n}.$$
Divide by $\log r$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a psh function over $B(x,r)$ tends to the
Lelong number at~$x$. Thus we obtain
$$
\nu(\varphi_m,x)\ge\nu(\varphi,x)-{n\over m}.\eqno\square
$$
\endproof

Theorem 14.1 implies in a straighforward manner a deep result of
(Siu 1974) on the analyticity of the Lelong number sublevel sets.

\begstat{(14.2) Corollary} Let $\varphi$ be a plurisubharmonic function
on a complex manifold~$X$. Then, for every $c>0$, the Lelong number
sublevel set
$$
E_c(\varphi)=\big\{z\in X\,;\;\nu(\varphi,z)\ge c\big\}
$$
is an analytic subset of~$X$.
\endstat

\begproof{} Since analyticity is a local property, it
is enough to consider the case of a psh function $\varphi$ on a 
pseudoconvex open set $\Omega\subset\bbbc^n$. The
inequalities obtained in 14.1~b) imply that 
$$
E_c(\varphi)=\bigcap_{m\ge m_0}E_{c-n/m}(\varphi_m).
$$
Now, it is clear that $E_c(\varphi_m)$ is the analytic set defined by the
equations $\sigma_\ell^{(\alpha)}(z)=0$ for all multi-indices $\alpha$
such that~$|\alpha|<mc$. Thus $E_c(\varphi)$ is analytic as a (countable) 
intersection of analytic sets.\qed
\endproof

\titleb{15.}{Nadel vanishing theorem}
In this final section, we prove a vanishing theorem due to (Nadel
1989), which has found recently many deep and important applications
in complex differential geometry and in algebraic geometry. It
contains as a special case the well-known Kawamata-Viehweg vanishing
theorem (Kawamata 1982, Viehweg 1982), which can be seen as an
algebraic version of the general vanishing theorem [here, the reader
is assumed to have some knowledge of sheaf theory, namely coherent
analytic sheaves, sheaf cohomology, etc]. We first introduce the
concept of {\it multiplier ideal sheaf}, following (Nadel 1989). The
main idea actually goes back to the fundamental works of (Bombieri
1970) and (Skoda 1972a).

\begstat{(15.1) Definition} Let $\varphi$ be a
psh function on an open subset $\Omega\subset X\,$; to $\varphi$ is
associated the ideal subsheaf $\cI(\varphi)\subset\cO_\Omega$ of germs of
holomorphic functions $f\in\cO_{\Omega,x}$ such that $|f|^2e^{-2\varphi}$
is integrable with respect to the Lebesgue measure in some local
coordinates near~$x$.
\endstat

The zero variety $V(\cI(\varphi))$ is thus the set of points in
a neighborhood of which $e^{-2\varphi}$ is non integrable. Of course,
such points occur only if $\varphi$ has logarithmic poles. This is made
precise as follows.

\begstat{(15.2) Definition} A psh function $\varphi$ is said to have a
logarithmic pole of coefficient $\gamma$ at a point $x\in X$ if the Lelong
number
$$\nu(\varphi,x):=\liminf_{z\to x}{\varphi(z)\over\log|z-x|}$$
is non zero and if $\nu(\varphi,x)=\gamma$.
\endstat

\begstat{(15.3) Lemma {\rm (Skoda 1972a)}} Let $\varphi$ be a psh function
on an open set $\Omega$ and let $x\in\Omega$.
\smallskip
\item{\rm a)} If $\nu(\varphi,x)<1$, then $e^{-2\varphi}$ is integrable in
a neighborhood of~$x$, in particular $\cI(\varphi)_x=\cO_{\Omega,x}$.
\smallskip
\item{\rm b)} If $\nu(\varphi,x)\ge n+s$ for some integer $s\ge 0$, then
$e^{-2\varphi}\ge C|z-x|^{-2n-2s}$ in a neighborhood of $x$ and
$\cI(\varphi)_x\subset\gm_{\Omega,x}^{s+1}$, where $\gm_{\Omega,x}$ is the
maximal ideal of $\cO_{\Omega,x}$.
\smallskip
\item{\rm c)} The zero variety $V(\cI(\varphi))$ of $\cI(\varphi)$
satisfies 
$$E_n(\varphi)\subset V(\cI(\varphi))\subset E_1(\varphi)$$
where $E_c(\varphi)=\{x\in X\,;\,\nu(\varphi,x)\ge c\}$ is the $c$-sublevel
set of Lelong numbers of~$\varphi$.
\vskip0pt
\endstat

\begproof{} a) We use the following well-known facts about Lelong numbers, see
(Lelong 1968) or (Demailly 1993a): set $d^c={1\over2\ii\pi}(d'-d'')$, so that
$dd^c={\ii\over\pi}d'd''$, and put $\Theta=dd^c\varphi\,$; we then have the
equality
$$
\nu(\Theta,x,r)={1\over 2^{n-1}r^{2n-2}}\int_{B(x,r)}
\Theta\wedge(dd^c|z|^2)^{n-1}=\int_{B(x,r)}\Theta\wedge(dd^c\log|z-x|)^{n-1}\,;
$$
it follows from the last integral that $\nu(\Theta,x,r)$ is an
increasing function of~$r\,$; the limit $\nu(\Theta,x):=\lim_{r\to 0_+}
\nu(\Theta,x,r)$ is equal to $\nu(\varphi,x)$. Now, let $\chi$ be
a cut-off function will support in a small ball $B(x,r)$, equal to $1$
in $B(x,r/2)$. As $(dd^c\log|z|)^n=\delta_0$, we get
$$\eqalign{\varphi(z)
&=\int_{B(x,r)}\chi(\zeta)\varphi(\zeta)(dd^c\log|\zeta-z|)^n\cr
&=\int_{B(x,r)}dd^c(\chi(\zeta)\varphi(\zeta))\wedge
\log|\zeta-z|(dd^c\log|\zeta-z|)^{n-1}\cr}$$
for $z\in B(x,r/2)$. Expanding $dd^c(\chi\varphi)$ and observing that
$d\chi=dd^c\chi=0$ on $B(x,r/2)$, we find
$$\varphi(z)=\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge
\log|\zeta-z|(dd^c\log|\zeta-z|)^{n-1}+\hbox{smooth terms}$$
on $B(x,r/2)$. Fix $r$ so small that
$$\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge(dd^c\log|\zeta-x|)^{n-1}
\le\nu(\Theta,x,r)<1.$$
By continuity, there exists $\delta,\varepsilon>0$ such that
$$I(z):=\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge(dd^c\log|\zeta-z|)^{n-1}
\le 1-\delta$$
for all $z\in B(x,\varepsilon)$. Applying Jensen's convexity inequality to
the probability measure
$$d\mu_z(\zeta)=I(z)^{-1}\chi(\zeta)\Theta(\zeta)\wedge
(dd^c\log|\zeta-z|)^{n-1},
$$
we find
$$\eqalign{
-\varphi(z)&=\int_{B(x,r)}I(z)\log|\zeta-z|^{-1}\,d\mu_z(\zeta)+O(1)\quad
\Longrightarrow\cr
e^{-2\varphi(z)}&\le C\int_{B(x,r)}|\zeta-z|^{-2I(z)}\,d\mu_z(\zeta).\cr}$$
As
$$d\mu_z(\zeta)\le C_1|\zeta-z|^{-(2n-2)}\Theta(\zeta)\wedge
(dd^c|\zeta|^2)^{n-1}=C_2|\zeta-z|^{-(2n-2)}d\sigma_\Theta(\zeta),$$
we get
$$e^{-2\varphi(z)}\le C_3\int_{B(x,r)}|\zeta-z|^{-2(1-\delta)-(2n-2)}
d\sigma_\Theta(\zeta),$$
and the Fubini theorem implies that $e^{-2\varphi(z)}$ is integrable on a
neighborhood of~$x$.
\smallskip

\noindent b) If $\nu(\varphi,x)=\gamma$, the convexity properties of
psh functions, namely, the convexity of $\log r\mapsto\sup_{|z-x|=r}
\varphi(z)$ implies that
$$\varphi(z)\le\gamma\log|z-x|/r_0+M,$$
where $M$ is the supremum on $B(x,r_0)$. Hence there exists a constant
$C>0$ such that $e^{-2\varphi(z)}\ge C|z-x|^{-2\gamma}$ in a neighborhood
of $x$. The desired result follows from the identity
$$\int_{B(0,r_0)}{\big|\sum a_\alpha z^\alpha\big|^2\over|z|^{2\gamma}}dV(z)=
{\rm Const}\int_0^{r_0}\Big(\sum |a_\alpha|^2r^{2|\alpha|}\Big)
r^{2n-1-2\gamma}\,dr,$$
which is an easy consequence of Parseval's formula. In fact, if $\gamma$
has integral part $[\gamma]=n+s$, the integral converges if and only if
$a_\alpha=0$ for $|\alpha|\le s$.
\smallskip
\noindent c) is just a simple formal consequence of a) and b).\qed
\endproof

\begstat{(15.4) Proposition {\rm(Nadel 1989)}} For any psh function
$\varphi$ on~$\Omega\subset X$, the sheaf
$\cI(\varphi)$ is a coherent sheaf of ideals over $\Omega$.
\endstat

\begproof{} Since the result is local, we may assume that $\Omega$ is the
unit ball in $\bbbc^n$. Let $E$ be the set of all holomorphic functions $f$ 
on $\Omega$ such that $\int_\Omega|f|^2e^{-2\varphi}\,d\lambda<+\infty$.
By the strong noetherian property of coherent sheaves, the set $E$
generates a coherent ideal sheaf $\cJ\subset\cO_\Omega$. It is clear that
$\cJ\subset\cI(\varphi)$; in order to prove the equality,
we need only check that $\cJ_x+\cI(\varphi)_x\cap
\gm^{s+1}_{\Omega,x}=\cI(\varphi)_x$ for every integer $s$, in
view of the Krull lemma. Let $f\in\cI(\varphi)_x$ be defined 
in a neighborhood $V$ of $x$ and let $\theta$ be a cut-off function
with support in $V$ such that $\theta=1$ in a neighborhood of $x$.
We solve the equation $d''u=g:=d''(\theta f)$ by means of H\"ormander's
$L^2$ estimates~8.9, where $E$ is the trivial line bundle
$\Omega\times\bbbc$ equipped with the strictly psh weight 
$$\wt\varphi(z)=\varphi(z)+(n+s)\log|z-x|+|z|^2.$$ 
We get a solution $u$ such that
$\int_\Omega |u|^2e^{-2\varphi}|z-x|^{-2(n+s)}d\lambda<\infty$, thus
$F=\theta f-u$ is holomorphic, $F\in E$ and 
$f_x-F_x=u_x\in\cI(\varphi)_x\cap\gm_{\Omega,x}^{s+1}$.
This proves our contention.\qed
\endproof

The multiplier ideal sheaves satisfy the following basic
fonctoriality property with respect to direct images of sheaves
by modifications.

\begstat{(15.5) Proposition} Let $\mu:X'\to X$ be a modification
of non singular complex manifolds $($i.e.\ a proper generically 1:1
holomorphic map$)$, and let $\varphi$ be a psh function on~$X$. Then 
$$\mu_\star\big(\cO(K_{X'})\otimes\cI(\varphi\circ\mu)\big)=
\cO(K_X)\otimes\cI(\varphi).$$
\endstat

\begproof{} Let $n=\dim X=\dim X'$ and let $S\subset X$ be an analytic
set such that $\mu:X'\ssm S'\to X\ssm S$ is a biholomorphism. By
definition of multiplier ideal sheaves, $\cO(K_X)\otimes\cI(\varphi)$ is
just the sheaf of holomorphic $n$-forms $f$ on open sets $U\subset X$
such that
$\ii^{n^2}f\wedge\ovl f\,e^{-2\varphi}\in L^1_\loc(U)$. Since
$\varphi$ is locally bounded from above, we may even consider forms $f$
which are a priori defined only on $U\ssm S$, because $f$ will be
in $L^2_\loc(U)$ and therefore will automatically extend through~$S$.
The change of variable formula yields
$$
\int_U\ii^{n^2}f\wedge\ovl f\,e^{-2\varphi}=
\int_{\mu^{-1}(U)}\ii^{n^2}\mu^\star f\wedge\ovl{\mu^\star f}
\,e^{-2\varphi\circ\mu},
$$
hence $f\in\Gamma(U,\cO(K_X)\otimes\cI(\varphi))$ if and only if
$\mu^\star f\in\Gamma(\mu^{-1}(U),\cO(K_{X'})\otimes\cI(\varphi\circ
\mu))$. Proposition~15.5 is proved.\qed
\endproof

\begstat{(15.6) Remark} \rm If $\varphi$ has ``analytic singularities''
the computation of $\cI(\varphi)$ can be reduced to a purely algebraic problem.

The first observation is that $\cI(\varphi)$ can be computed easily if
$\varphi$ has the form $\varphi=\sum\alpha_j\log|g_j|$ where
$D_j=g^{-1}_j(0)$ are nonsingular irreducible divisors with normal
crossings. Then $\cI(\varphi)$ is the sheaf of functions $h$ on open
sets $U\subset X$ such that
$$\int_U|h|^2\prod|g_j|^{-2\alpha_j}dV<+\infty.$$
Since locally the $g_j$ can be taken to be coordinate functions from a
local coordinate system $(z_1\ld z_n)$, the condition is that
$h$ is divisible by $\prod g_j^{m_j}$ where $m_j-\alpha_j>-1$ for
each $j$, i.e.\ $m_j\ge\lfloor\alpha_j\rfloor$ (integer part). Hence
$$\cI(\varphi)=\cO(-\lfloor D\rfloor)=\cO(-\sum\lfloor\alpha_j\rfloor D_j)$$
where $\lfloor D\rfloor$ denotes the integral part of the $\bbbq$-divisor
$D=\sum\alpha_jD_j$.

Now, consider the general case of analytic singularities, i.e., the case
of a psh function such that
$$
\varphi={\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)+O(1)
$$
near the poles. Let $\cJ$ be the (coherent) integrally closed ideal
sheaf of holomorphic functions $h$ such that $|h|\le C\exp(\varphi/\alpha)$.
In this case, the computation is made as follows.
First, one computes a smooth modification $\mu:\wt X\to X$ of $X$ such
that $\mu^\star\cJ$ is an invertible sheaf $\cO(-D)$ associated with
a normal crossing divisor $D=\sum\lambda_jD_j$, where $(D_j)$ are the
components of the exceptional divisor of $\wt X$ (take the blow-up $X'$
of $X$ with respect to the ideal $\cJ$ so that the pull-back of $\cJ$
to $X'$ becomes an invertible sheaf $\cO(-D')$, then blow up again by
the Hironaka desingularization theorem (Hironaka 1964) to make $X'$ 
smooth and $D'$ have normal crossings).
Now, we have $K_{\wt X}=\mu^\star K_X+R$ where $R=\sum\rho_j
D_j$ is the zero divisor of the Jacobian function $J_\mu$ of the blow-up
map. By the direct image formula~15.5, we get
$$\cI(\varphi)=\mu_\star\big(\cO(K_{\wt X}-\mu^\star K_X)\otimes
\cI(\varphi\circ\mu)\big)=\mu_\star\big(\cO(R)\otimes
\cI(\varphi\circ\mu)\big).$$
Now, $(f_j\circ\mu)$ are generators of the ideal $\cO(-D)$, hence
$$\varphi\circ\mu\sim\alpha\sum\lambda_j\log|g_j|$$
where $g_j$ are local generators of $\cO(-D_j)$. We are thus reduced to
computing multiplier ideal sheaves in the case where the poles are
given by a $\bbbq$-divisor with normal crossings $\sum\alpha\lambda_jD_j$.
We obtain $\cI(\varphi\circ\mu)=\cO(-\sum\lfloor\alpha\lambda_j\rfloor D_j)$,
hence
$$\cI(\varphi)=\mu_\star\cO_{\wt X}\big(\sum(\rho_j-\lfloor\alpha\lambda_j
\rfloor)D_j\big).\eqno\square$$
\endstat

\begstat{(15.7) Exercise} \rm Compute the multiplier ideal sheaf
$\cI(\varphi)$ associated with
$\varphi=\log(|z_1|^{\alpha_1}+\ldots+|z_p|^{\alpha_p})$
for arbitrary real numbers $\alpha_j>0$.\newline
{\it Hint:} using Parseval's formula and polar coordinates $z_j=r_j
e^{\ii\theta_j}$, show that the problem is equivalent to determining for
which $p$-tuples $(\beta_1\ld\beta_p)\in\bbbn^p$ the integral
$$\int_{[0,1]^p}{r_1^{2\beta_1}\ldots r_p^{2\beta_p}\,r_1dr_1\ldots r_pdr_p
\over r_1^{2\alpha_1}+\ldots+r_p^{2\alpha_p}}
=\int_{[0,1]^p}{t_1^{(\beta_1+1)/\alpha_1}\ldots t_p^{(\beta_p+1)/\alpha_p}
\over t_1+\ldots+t_p}{dt_1\over t_1}\ldots{dt_p\over t_p}$$
is convergent. Conclude from this that $\cI(\varphi)$ is generated by
the monomials $z_1^{\beta_1}\ldots z_p^{\beta_p}$ such that
$\sum(\beta_p+1)/\alpha_p>1$. (This exercise shows that the analytic
definition of $\cI(\varphi)$ is sometimes also quite convenient for
computations).\qed
\endstat

Let $F$ be a line bundle over $X$ with a singular metric $h$ of
curvature current $\Theta_h(F)$. If $e^{-2\varphi}$ is the weight
representing the metric in an open set $\Omega\subset X$, the ideal
sheaf $\cI(\varphi)$ is independent of the choice of the trivialization
and so it is the restriction to $\Omega$ of a global coherent sheaf
$\cI(h)$ on~$X$. We will sometimes still write $\cI(h)=\cI(\varphi)$ by
abuse of notation. In this context, we have the following fundamental
vanishing theorem, which is probably one of the most central results of
analytic and algebraic geometry (especially, it contains the
Kawamata-Viehweg vanishing theorem as a special case).

\begstat{(15.8) Nadel vanishing theorem {\rm(Nadel 1989, Demailly 1993b)}} 
Let $(X,\omega)$ be a K\"ahler weakly pseudoconvex manifold, and let $L$ be a
holomorphic line bundle over $X$ equipped with a singular hermitian metric
$h$ of weight~$e^{-2\varphi}$. Assume that $\ii\Theta_h(L)\ge\varepsilon
\omega$ for some continuous positive function $\varepsilon$ on~$X$. Then
$$
H^q\big(X,\cO(K_X\otimes L)\otimes\cI(h)\big)=0\qquad
\hbox{for all $q\ge 1$.}
$$
\endstat

\begproof{} Let $\cL^q$ be the sheaf of germs of $(n,q)$-forms 
$u$ with values in $L$ and with measurable coefficients, such that
both $|u|^2e^{-2\varphi}$ and $|d''u|^2e^{-2\varphi}$ are locally
integrable. The $d''$ operator defines a complex of sheaves
$(\cL^\bu,d'')$ which is a resolution of the sheaf
$\cO(K_X\otimes L)\otimes\cI(\varphi)$: indeed, the kernel of $d''$ in degree
$0$ consists of all germs of holomorphic $n$-forms with values in $L$
which satisfy the integrability condition; hence the coefficient
function lies in $\cI(\varphi)$; the exactness in degree $q\ge 1$
follows from Corollary~8.9 applied on arbitrary small balls.
Each sheaf $\cL^q$ is a ${\cal C}^\infty$-module, so $\cL^\bu$ 
is a resolution by acyclic sheaves. Let $\psi$ be a smooth psh
exhaustion function on~$X$. Let us apply Corollary~8.9 globally on~$X$,
with the original metric of $L$ multiplied by the factor
$e^{-\chi\circ\psi}$, where $\chi$ is a convex increasing function of
arbitrary fast growth at infinity. This factor can be used to ensure
the convergence of integrals at infinity. By Corollary~8.9, we conclude
that $H^q\big(\Gamma(X,\cL^\bu)\big)=0$ for $q\ge 1$. The theorem
follows.\qed
\endproof 

\begstat{(15.9) Corollary} Let $(X,\omega)$, $L$ and $\varphi$ be as in
Theorem~15.8 and let $x_1\ld x_N$ be isolated points in the zero
variety~$V(\cI(\varphi))$. Then there is a surjective map
$$H^0(X,K_X\otimes L)\lraww\bigoplus_{1\le j\le N}\cO(K_X\otimes L)_{x_j}
\otimes\big(\cO_X/\cI(\varphi)\big)_{x_j}.$$
\endstat

\begproof{} Consider the long exact sequence of cohomology associated
to the short exact sequence
$0\to\cI(\varphi)\to\cO_X\to\cO_X/\cI(\varphi)\to 0$ twisted by
$\cO(K_X\otimes L)$, and apply Theorem~15.8 to obtain the vanishing of
the first $H^1$ group. The asserted surjectivity property follows.\qed
\endproof

\begstat{(15.10) Corollary} Let $(X,\omega)$, $L$ and $\varphi$ be as in
Theorem~15.8 and suppose that the weight function $\varphi$ is such that
$\nu(\varphi,x)\ge n+s$ at some point $x\in X$ which is an isolated point
of $E_1(\varphi)$. Then $H^0(X,K_X\otimes L)$ generates all $s$-jets at~$x$.
\endstat

\begproof{} The assumption is that $\nu(\varphi,y)<1$ for $y$ near $x$,
$y\ne x$. By Skoda's lemma 15.3~b), we conclude that $e^{-2\varphi}$ is
integrable at all such points~$y$, hence $\cI(\varphi)_y=\cO_{X,y}$, whilst
$\cI(\varphi)_x\subset\gm_{X,x}^{s+1}$ by 15.3~a). Corollary 15.10 is thus
a special case of 15.9.\qed
\endproof

The philosophy of these results (which can be seen as generalizations
of the H\"ormander-Bombieri-Skoda theorem (Bombieri 1970), (Skoda
1972a,~1975) is that the problem of constructing holomorphic sections
of $K_X\otimes L$ can be solved by constructing suitable hermitian
metrics on $L$ such that the weight $\varphi$ has isolated poles at
given points~$x_j$.

\begstat{(15.11) Exercise} \rm Assume that $X$ is compact and that $L$ is
a positive line bundle on~$X$. Let $\{x_1\ld x_N\}$ be a finite set.
Show that there are constants $a,b\ge 0$ depending only on $L$ and $N$
such that $H^0(X,L^{\otimes m})$ generates jets of any order $s$ at all 
points $x_j$ for $m\ge as+b$.\newline
{\it Hint:} Apply Corollary~15.9 to $L'=K_X^{-1}\otimes L^{\otimes m}$, 
with a singular metric on $L$ of the form $h=h_0e^{-\varepsilon\psi}$,
where $h_0$ is smooth of positive curvature, $\varepsilon>0$ small and
$\psi(z)\sim\log|z-x_j|$ in a neighborhood of $x_j$.

Recall that a line bundle $L$ is said to be {\it very ample} if the
sections of $H^0(X,L)$ generate any pair of $L_x\oplus L_y$ for
distinct points $x\ne y$ in $X$, as well as $1$-jets of $L$ at any
point $x\in X$. The line bundle $L$ is said to be {\it ample} if some
positive multiple $L^{\otimes m}$ is very ample. Then derive the
Kodaira embedding theorem:
\endstat

\begstat{(15.12) Theorem {\rm(Kodaira 1954)}} If $L$ is a line bundle
on a compact complex manifold, then $L$ is ample if and only if $L$ is
positive.
\endstat

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\ref Wells, R.O. (1980)& Differential analysis on complex manifolds,&
Graduate Texts in Math.~{\bf 65}, 2nd edition, Springer-Verlag, Berlin (1980)&

}
%%\bigskip\noindent
\vfill\eject

\noindent
Jean-Pierre Demailly\newline
Universit\'e de Grenoble I\newline
Institut Fourier, BP74\newline
38400 Saint-Martin d'H\`eres, France\newline
{\tt demailly@fourier.ujf-grenoble.fr}

\bigskip\noindent
(version of September 17, 1996, printed on \today, \timeofday)

\end

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The direct image $\mu_\star\smash{\wh F_{c,\varepsilon,\gamma}}= 
F_\varepsilon-\mu_\star u_{c,\varepsilon,\gamma}$ is a
$D''$-closed locally integrable $(0,q)$-form on $X_c$ with values in
$\Lambda^nT^\star_X\otimes L$ which, in some sense, should have $f$ as
its restriction to~$Y_c$. The difficulty is that $\mu_\star\smash{\wh
F_{c,\varepsilon,\gamma}}$ need not be smooth at points of~$Y_c$. This
difficulty can be overcome by correcting $\mu_\star \smash{\wh
F_{c,\varepsilon,\gamma}}$ and chasing through the commutative diagram
$$
\eqalign{
H^{0,q}\big(X_c,\Lambda^nT^\star_X\otimes L\big)&{}\lra
H^{0,q}\big(Y_c,(\Lambda^nT^\star_X\otimes L)_{\restriction Y_c}\big)\cr
\hfill\mu^\star\Big\downarrow\kern1.8cm&\kern2.7cm\Big\downarrow
\mu_{\restriction\wh Y_c}^\star\hfill\cr
H^{0,q}\big(\wh X_c,\mu^\star(\Lambda^nT^\star_X\otimes L)\big)&{}\lra
H^{0,q}\big(\wh Y_c,\mu^\star(\Lambda^nT^\star_X\otimes L)_{\restriction
\wh Y_c}\big),\cr}
$$
in which the horizontal arrows are restriction morphisms. Both vertical 
arrows $\mu^\star$ and $\mu_{\restriction\wh Y_c}^\star$ are isomorphisms, 
since $R^q\mu_\star\cO_{\wh X_c}=0$ and
$R^q(\mu_{\restriction\wh Y_c})_\star\cO_{\wh Y_c}=0$ for all~$q\ge 1$.
We must show that the $L^2$ estimate can still be achieved through these
isomorphisms. Let us take a smooth form $F_{c,\varepsilon,\gamma}$ on~$X_c$
such that $F_{c,\varepsilon,\gamma}$ is in the same cohomology class as
$\mu_\star\smash{\wh F_{c,\varepsilon,\gamma}}$. Thanks to the isomorphisms,
we have
$$
\mu^\star F_{c,\varepsilon,\gamma}=\wh F_{c,\varepsilon,\gamma}+
D''\wh h_{c,\varepsilon,\gamma}
$$
for some smooth $(0,q-1)$-form $\wh h_{c,\varepsilon,\gamma}$ on~$\wh X_c$.
By taking the restriction to $\smash{\wh Y_c}$ we get
$$
\mu^\star F_{c,\varepsilon,\gamma\restriction\wh Y_c}=
\mu_{\restriction\wh Y_c}^\star f
+D''\wh h_{c,\varepsilon,\gamma\restriction\wh Y_c}.
$$
As $\mu_{\restriction\wh Y_c}^\star$ is an isomorphism in cohomology, we infer
the existence of a smooth $(0,q-1)$-form $k_{c,\varepsilon,\gamma}$ on $Y_c$
such that 
$$
F_{c,\varepsilon,\gamma\restriction\wh Y_c}=f+D''k_{c,\varepsilon,\gamma}.
$$
By taking a smooth extension $\ovl k_{c,\varepsilon,\gamma}$ of
$k_{c,\varepsilon,\gamma}$ to $X_c$ and replacing
$F_{c,\varepsilon,\gamma}$ with \hbox{$F_{c,\varepsilon,\gamma}-
D''\ovl k_{c,\varepsilon,\gamma}$}, we can even assume that
$F_{c,\varepsilon,\gamma\restriction Y_c}=f$. Now, we find
$$
D''\wh h_{c,\varepsilon,\gamma\restriction\wh Y_c}=
\mu_{\restriction\wh Y_c}^\star f-\mu_{\restriction\wh Y_c}^\star f=0,
$$
hence, by the extension property in bidegree $(n,q-1)$, there is a
$D''$-closed $(0,q-1)$-form $\smash{\wh\ell}_{c,\varepsilon,\gamma}$ on 
$\wh X_c$ which extends $\smash{\wh h}_{c,\varepsilon,\gamma\restriction
\wh Y_c}$. By replacing $\smash{\wh h}_{c,\varepsilon,\gamma}$ with 
$\smash{\wh h_{c,\varepsilon,\gamma}-\wh\ell_{c,\varepsilon,\gamma}}$, we 
can assume in addition to the above that 
$\smash{\wh h}_{c,\varepsilon,\gamma\restriction Y_c}=0$. By correcting
again $\smash{\wh h}_{c,\varepsilon,\gamma}$ with
the $D''$ of a $(0,q-2)$-form which vanishes along $\smash{\wh Y_c}$, we can
finally assume that $\smash{\wh h}_{c,\varepsilon,\gamma}$ vanishes at each
point of~$\smash{\wh Y_c}$ (and not only in restriction to the tangent space).



