\input broccod.tex
\input ams.tex
\vsize=20cm
\parindent=0.5cm

\def\nb{{\Bbb N}}
\def\zb{{\Bbb Z}}
\def\qb{{\Bbb Q}}
\def\rb{{\Bbb R}}
\def\cb{{\Bbb C}}
\def\pb{{\Bbb P}}

\def\fr{{\cal F}}
\def\gr{{\cal G}}
\def\hr{{\cal H}}
\def\lr{{\cal L}}
\def\oc{{\cal O}}

\def\ov{\overline}
\def\ci{{C^\infty}}
\def\la{\longrightarrow}
\def\ld{,\,\ldots\,,}
\def\dbar{\smash{\overline\partial}}
\def\wt{\widetilde}
\def\pav#1|{\vskip15pt{\bf #1.}\vskip8pt}
\def\pointir{.--- }
\def\th#1|#2\endth{\vskip3pt{\bf#1}\pointir {\sl#2}}
\def\proof{{\sl Proof}\pointir}
\def\endpr{~\hbox{$\square$}}

\def\article#1|#2|#3|#4|#5|#6|#7|
{{\leftskip=7mm\noindent
\hangindent=2mm\hangafter=1
\llap{[#1]\hskip1.35em}{\petcap #2}\pointir {\sl #3}, {\rm #4}, 
\nobreak{\bf #5} ({\oldstyle #6}), \nobreak #7.\par}}

\def\book#1|#2|#3|#4|#5|
{{\leftskip=7mm\noindent
\hangindent=2mm\hangafter=1
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\def\miscel#1|#2|#3|#4|
{{\leftskip=7mm\noindent
\hangindent=2mm\hangafter=1
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\centerline{\bf TRANSCENDENTAL PROOF OF A GENERALIZED}
\centerline{\bf KAWAMATA-VIEHWEG VANISHING THEOREM}

\vskip 0.5cm
{\noindent\hangindent4cm\hangafter-1
{\sl Jean-Pierre DEMAILLY

}}

{\noindent\hangindent4cm\hangafter-4{\sl
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
Laboratoire associ\'e au C.N.R.S.
n\kern-2pt{\rm\char'27}188,\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\vskip20pt
{\hfill Dedicated to Professor Leon Ehrenpreis on his sixtieth birthday}
\vskip30pt

{\bf Abstract}\pointir 
{\sl Let $\lr$ be a holomorphic line bundle over a projective algebraic
manifold $X$~.  It is shown that the differential geometric technique of
Bochner-Kodaira-Nakano and the $L^2$ estimates for $\dbar$ yield a very
elementary proof of the Kawamata-Viehweg theorem [7,13,15]: if $\lr$ is
numerically effective, then $H^q(X,\lr^{-1})=0$ for $q<s$~, where $s$ is
the largest integer such that $c_1(\lr)^s\ne 0$~.  More generally, our
method implies a vanishing result when $\lr$ is tensorized with an
effective $\bf Q$--divisor which may have non normal crossings, under a
natural integrability hypothesis for the divisor.} \vskip 30pt

\pav 1.  Statement of results|
Recall that a line bundle $\lr$ over a projective algebraic manifold
$X$ is said to be {\sl numerically effective} (nef) if
$c_1(\lr)_{\restriction\Gamma}\ge 0$ for every curve $\Gamma$ in $X$~. 
Then, it is known [10] that $c_1(\lr)^d_{\restriction Y}\ge 0$ for any
subvariety $Y\subset X$ of dimension $d$~.  On the other hand, the
well-known Nakai-Moishezon criterion says that a line bundle $\hr$ is
ample if and only if $\smash{c_1(\hr)^d_{\restriction Y}}>0$ for any
subvariety $Y$ of dimension $d$~.  If $\hr$ is a given ample line
bundle, it follows easily that a line bundle $\lr$ is nef if and only if
$\lr^m\otimes\hr$ is ample for every integer $m\ge 0$~. 

Let $\fr$ be a line bundle over $X$ and $Z(V_m)$ the set of common zeros
of all sections in $V_m=H^0(X,\fr^m)$~.  The {\sl Kodaira dimension}
$\kappa(\fr)$ is the maximum when $m$ runs over all positive integers 
of the generic rank of the analytic morphism $\Phi_m:X\setminus
Z(V_m)\la P(V_m^\star)$ which maps a point $x\in X\setminus Z(V_m)$ to
the hyperplane of sections $\sigma\in V_m$ such that $\sigma(x)=0$~.  As
usual, we set $\kappa(\fr)=-\infty$ if $V_m=\{0\}$ for all $m$~. 
The following definition (related to the concept of log-terminal 
singularities) gives a way of measuring how singular is a divisor.

\th Definition|We say that a divisor $D=\sum\alpha_j D_j$ with rational
coefficients $\alpha_j\in\qb$ is integrable at a point $x_0\in X$ 
if the function $\prod|g_j|^{-2\alpha_j}$ equal to the product of 
local generators $g_j$ of the ideal of $D_j$ at $x_0$
is integrable on a neighborhood of $x_0$~.\endth

When $D$ has normal crossings, the $g_j$'s can be taken to be
coordinates at $x_0$~; thus $D$ is integrable if and only if
$\alpha_j<1$ for all $j$~. When $D$ is effective and has arbitrary
singularities, a sufficient (but rather restrictive) condition for
integrability of $D$ at $x_0$ is that the multiplicity (or Lelong number) 
$$m(D,x_0)=\sum\alpha_j\,m(D_j,x_0)$$ 
be $<1$~; indeed, if $\alpha_j=p_j/q\ge 0$ and if $\prod g_j(z)^{p_j}$
is written as a Weierstrass polynomial $\prod_{1\le k\le p}(z_n-r_k(z'))$
of degree $p=\sum p_j\,m(D_j,x_0)$ in $z_n$~, the inequality between 
arithmetic and geometric mean shows that
$$\prod|g_j|^{-2p_j/q}=\Big|\prod\big(z_n-r_k(z')\big)\Big|^{-2/q}
\le p^{-1}\sum|z_n-r_k(z')|^{-2p/q}$$
is integrable as soon as $q>p$~.

If neither $D$ has normal crossings nor $m(D,x_0)<1$~, the integrability
condition can be checked by means of a sequence of blowing-ups which
lift $D$ into a divisor with normal crossings (this is always possible
by Hironaka [11]).  Let $\varphi:\smash{\wt X}\la X$ be the blowing-up of
$X$ along a submanifold $Y$~, ${\rm codim}\,Y=r$~, and $\smash{\wt Y}
\subset\smash{\wt X}$ the exceptional divisor. On a coordinate
neighborhood $U$ of a point $x_0\in X$~, we have
$$\int_U\prod|g_j(z)|^{-2\alpha_j}d\lambda(z)=
\int_{\varphi^{-1}(U)}\prod\big|g_j(\varphi(\zeta))\big|^{-2\alpha_j}
|J\varphi(\zeta)|^2\,d\lambda(\zeta)$$
where $d\lambda$ is the Lebesgue measure. The divisor of zeros of
the Jacobian $J\varphi$ is $(r-1)\smash{\wt Y}$~, thus $D$ is
integrable on $U$ if and only if $\smash{D^{(1)}}=\varphi^\star D-(r-1)
\smash{\wt Y}$ is integrable on $\varphi^{-1}(U)\subset\smash{\wt X}$~. 
At the end we get a divisor $\smash{D^{(N)}}$ with normal crossings and 
$D$ is integrable if and only if all coefficients of $\smash{D^{(N)}}$ are 
$<1$~.\break  A consequence of this is that integrability is an open condition~:
if $E$ is an arbitrary effective divisor and if $D$ is integrable at $x_0$
(resp. on $X$), then $D+p^{-1}E$ is again integrable at $x_0$ (resp. on
$X$) for $p$ large enough. Observe also that the set of points $B$ where
$D$ is non integrable is an analytic subset of $X$ (the image in $X$ of
the union of all components of $\smash{D^{(N)}}$ which have a 
coefficient $\ge 1$).

\th Main theorem|Let $\fr$ be a line bundle over a projective
manifold $X$~. Assume that some positive power $\fr^k$
can be written $\fr^k=\lr\otimes\oc(D)$ with $\lr$ nef and with an 
effective divisor $D$ such that $k^{-1}D$ is integrable on $X\setminus B$~.
Let $s$ be the largest integer such that $c_1(\lr)^s\ne 0$~. Then
$$H^q(X,\fr^{-1})=0~~~~{\sl for}~~q<\min\{\max\{s,\kappa(\fr)\},
{\rm codim}\,B\}~.$$\endth

The special case $D=0$ and $\fr=\lr$ gives in particular
$H^q(X,\lr^{-1})=0$ for $\lr$ nef and $q<s$~: this is the original
Kawamata-Viehweg vanishing theorem ([13,15]). The above theorem has also
been proved by Kawamata [13] in some important cases and it can be deduced
formally from the method of Esnault-Viehweg [7], although their main
result (theorem 2.13) is not stated in such a wide generality
(they assume $D$ to be a normal crossing divisor and $\kappa(\fr)=n$). 
The proof given here is an expanded and improved version of
the one given in our note [5]. It is elementary in the sense that it avoids
some rather sophisticated tools of algebraic geometry that were
used in the previous proofs. Our proof involves only one non trivial
ingredient --- the $L^2$-estimates for $\dbar$
deduced from the Bochner-Kodaira-Nakano curvature inequality [1] ---
combined with standard slicing arguments and a simple trick of
Kawamata [13] which reduces the situation to the case when 
$\lr$ is ample.

Thus, it now appears that the original arguments of Akizuki-Nakano~[1]
were in fact good enough to prove the Kawamata-Viehweg theorem and the
above strong refinement~! In the same vein, we shall give at the end a 
new differential-geometric proof of the Bogomolov-Sommese vanishing 
theorem. Since our approach rests heavily on methods of partial 
differential equations to which Leon Ehrenpreis contributed so much, 
it is a pleasure for us to dedicate this work to him on the occasion of 
his sixtieth birthday.

\pav 2. $L^2$ estimates for $\dbar$ on line bundles|

Let $X$ be a complex manifold admitting a complete K\"ahler metric $\omega$
and $\gr$ a holomorphic line bundle over $X$ equipped with a hermitian metric
of class $\ci$. If $d\sigma$ denotes the K\"ahler volume form and
$\lambda_1(x)\ge\ldots\ge\lambda_n(x)$ are the eigenvalues of the curvature 
form $c(\gr)$ at any point $x\in X$~, the well-known Bochner-Kodaira-Nakano
inequality gives
$$||\dbar u||^2+||\dbar^\star u||^2\ge
\int_X(\lambda_1+\cdots+\lambda_q-\lambda_{p+1}-\cdots-\lambda_n)|u|^2
d\sigma\leqno(\star)$$
for every $(p,q)$--form $u$ of class $\ci$ with values in $\gr$ and
compact support~.
 
From this we readily infer the Kodaira-Akizuki-Nakano vanishing
theorem~[1]. Indeed, let $\gr$ be an ample line bundle on a compact 
manifold $X$~; then there exists a $C^\infty$ metric with positive 
curvature on $\gr$~. If we take
$\omega=c(\gr)$~, then $\lambda_j(x)=1$ for all $j$ and $x$~, hence
$$||\dbar u||^2+||\dbar^\star u||^2\ge (p+q-n)||u||^2~.$$ 
For $p+q>n$~, it follows that every $(p,q)$--form that is harmonic 
(\ie such that $\dbar u=\dbar^\star u=0$) must vanish. Hodge theory implies
the desired result:
$$\gr~~{\rm ample}\Longrightarrow
H^q(X,\Omega^p_X\otimes\gr)=0~~~~{\rm for}~~p+q>n~.$$ 
By Serre duality, we also infer that $H^q(X,\Omega^p_X\otimes\gr^{-1})=0$
for \hbox{$p+q<n$.} A natural problem is to relax the ampleness or positivity
hypothesis.  Historically, the first results proved in this direction are
contained in the following:

\th Proposition|Assume $X$ is a compact connected K\"ahler
manifold and that $c(\gr)$ is semi-positive everywhere.
\hfill\break\llap{\hbox{$\phantom{T^{T^{T^T}}}$}}{\rm(i)~
(Grauert-Riemenschneider [9])} If $c(\gr)$
has rank at least equal to $s$ at some point $x_0\in X$ then
$$H^q(X,\gr^{-1})=0~~~~{\rm for}~~q<s~.$$
\llap{\hbox{$\phantom{T^{T^{T^T}}}$}}{\rm(ii)~ (Girbau [8])} 
If $c(\gr)$ has rank at least equal to $s$ at every point, then
$$H^q(X,\Omega^p_X\otimes\gr^{-1})=0~~~~{\rm for}~~p+q<s~.$$\endth

{\sl Sketch of proof}\pointir First observe that the ordered sequence
of eigenvalues of\break
$c(\gr^{-1})=-c(\gr)$ is $-\lambda_n\le\ldots\le-\lambda_1\le 0$~.

(i) For $(0,q)$--forms with values in $\gr^{-1}$~, inequality $(\star)$ gives
$$||\dbar u||^2+||\dbar^\star u||^2\ge
\int_X(\lambda_1+\cdots+\lambda_{n-q})|u|^2d\sigma~.$$
As $\lambda_{n-s+1}(x_0)>0$~, we see that any harmonic $(0,q)$--form $u$
with $q<s$ must vanish identically on a neighborhood of $x_0$~. Hence
$u$ vanishes identically on $X$ (by Aronszajn' lemma on solutions of
elliptic equations).

(ii) In this case, we replace the K\"ahler metric $\omega$ by
$\omega_\varepsilon=\varepsilon\omega+c(\gr)$ with $\varepsilon>0$ very
small. Then all eigenvalues $\lambda_{j,\varepsilon}$ with respect to
$\omega_\varepsilon$ are in $[0,1[$ and $\lambda_{n-s+1,\varepsilon}\ld 
\lambda_{n,\varepsilon}>0$ become very close to $1$~,
say $\ge 1-\delta$~. For a $(p,q)$--form with values in $\gr^{-1}$~,
inequality $(\star)$ gives
$$\eqalign{
||\dbar u||^2+||\dbar^\star u||^2
&\ge\int_X(\lambda_1+\cdots+\lambda_{n-q}
          -\lambda_{n-p+1}-\cdots-\lambda_n)|u|^2d\sigma\cr
&\ge\big((s-q)(1-\delta)-p\big)||u||^2~,\cr}$$
and the Girbau vanishing result follows.\endpr

The vanishing result in (ii) is no longer true under the weaker hypothesis
of (i), even if $c(\gr)$ is assumed to be of rank $n$ on a dense open set~:
in fact, an example of C.P. Ramanujam [14] shows that for $X$ equal
to the blow-up of $\pb^n$ at one point and $\gr$ equal to the pull-back
of $\oc(1)$ over $X$~, then $H^1(X,\Omega^1_X \otimes\gr^{-1})\ne 0$~.
The failure of the proof when $p>0$ is due to the fact that the sum 
$\lambda_{n-p+1}+\cdots+\lambda_n$ may become larger than 
$\lambda_1+\cdots+\lambda_{n-q}$ at points where the rank of 
$c(\gr)$ drops below its generic value $s$~. The easiest way of 
avoiding this difficulty is to restrict ourselves to the case
$H^q(X,\gr^{-1})$ with $p=0$~, or, by Serre duality, to the dual case
$H^q(X,K_X\otimes\gr)$ with $p=n$~.

Another draw-back of the above proposition is that semi-positivity
cannot be translated easily in purely algebraic terms~: it is clear that
$c(\gr)\ge 0$ implies $\gr$ nef, but the converse implication is
false.  Our desire is to formulate vanishing theorems in
such a way that the hypotheses involve only numerical conditions on the
Chern class or easily understood conditions on divisors of sections. 
Then the difficulty in the analytic proof is to find hermitian metrics
with the required (semi)-positivity properties.  As we will see below,
this can be carried out in a simple way if we allow {\sl singular 
metrics}.  If $\gr_{\restriction\Omega}\simeq\Omega\times\cb$
is a local trivialization of $\gr$~, the metric is defined by a
positive function on $\Omega$ that will be denoted $e^{-\varphi}$~.
The only requirement we make on $\varphi$ is that 
$\varphi\in L^1_{\rm loc}(\Omega)$~.
Then the curvature form $c(\gr)=i\partial\dbar\varphi$~, computed
in the sense of distributions, is a real $(1,1)$--current. The
main analytic tool that will be used is the following consequence of
H\"ormander's $L^2$--estimates for $\dbar$ [12].

\th Theorem|Let $(X,\omega)$ be a complete K\"ahler manifold
and $\gr$ a line bundle over $X$ with a singular metric such that
$c(\gr)\ge\varepsilon\omega$ in the sense of currents. Then
for every $(n,q)$--form $v$ with values in $\gr$ such that $\dbar v=0$ 
and $\int_X|v|^2d\sigma<+\infty$~, there exists a $(n,q-1)$--form $u$ such that
$$\dbar u=v~~~~{\rm and}~~~~
\int_X|u|^2d\sigma\le(q\varepsilon)^{-1}\int_X|v|^2d\sigma~.$$\endth

\proof If the metric is smooth, $(\star)$ implies
$||\dbar u||^2+||\dbar^\star u||^2\ge q\varepsilon||u||^2$ for all
smooth $(n,q)$--forms, and the result follows by the usual Hilbert space
method. The hypothesis on the curvature implies that the weight
functions $\varphi$ are plurisubharmonic. The general case follows
from the smoothing procedure for plurisubharmonic functions
on K\"ahler manifolds developed in [4] (th\'eor\`emes 5.1 et 9.1).
Since the arguments are rather technical, we give here a simple proof
when $X$ is algebraic. In this case, there is an affine open subset 
$\Omega\subset X$ such that $\gr_{\restriction\Omega}$ is trivial.
Then the hermitian metric is given by a global plurisubharmonic
function $\varphi$ on $\Omega$~. As $\Omega$ is Stein, one can find an
increasing sequence of Stein open sets $\Omega_j\subset\!\subset\Omega$
and plurisubharmonic functions $\varphi_j\in C^\infty(\Omega_j)$ decreasing
to $\varphi$~, such that $i\partial\dbar\varphi_j\ge\varepsilon_j\omega$
with $\lim\varepsilon_j=\varepsilon$ (embed $\Omega$ in some $\cb^N$~, 
use a holomorphic retraction $\rho$ of a neighborhood of $\Omega$ onto 
$\Omega$ and approximate $\varphi\circ\rho$ by convolution). 
By the standard H\"ormander
$L^2$--estimates, there is a sequence of $(n,q-1)$--forms $u_j$ on
$\Omega_j$ such that $\dbar u_j=v$ and
$$\int_{\Omega_j}|u_j|^2e^{-\varphi_j}d\sigma\le(q\varepsilon_j)^{-1}
\int_{\Omega_j}|v|^2e^{-\varphi_j}d\sigma\le(q\varepsilon_j)^{-1}
\int_\Omega|v|^2e^{-\varphi}d\sigma~.$$
Then the sequence $u_j$ is uniformly bounded in $L^2$ norm on compact 
subsets of $\Omega$ ($\varphi_j$ is uniformly locally bounded from above),
so there is a weak limit $u$ satisfying $\dbar u=v$ on $\Omega$ and 
$\int_X\smash{|u|_\gr^2}d\sigma<\infty$ 
(observe that $X\setminus\Omega$ has measure
zero). This estimate implies that $u$ has coefficients in $L^2_{\rm loc}$~.
Then, by easy estimates (see e.g. [4], lemme 6.9), the relation $\dbar u=v$ 
extends to $X$~.\endpr

\th Corollary|Let $\gr$ be a line bundle with a singular hermitian 
metric. Assume that $c(\gr)\ge\varepsilon\omega$ and that the metric 
(i.e. the weight $e^{-\varphi}$) is integrable near all but
finitely many points of $X$~. Then
$$H^q(X,K_X\otimes\gr)=0~~{\rm for}~~q>0~,~~~~
H^q(X,\gr^{-1})=0~~{\rm for}~~q<n~.$$\endth

\proof By Serre duality, we only have to consider the case of $H^q(X,
K_X\otimes\gr)$~. Let $x_1\ld x_m$ be the points where the metric is not
integrable and $v$ a smooth $(n,q)$--form with values in $\gr$ such that
$\dbar v=0$~. Let $u_j$ be a smooth solution of $\dbar u_j=v$ in
a neighborhood of $V_j$ of $x_j$ and $\psi_j$ a cut-off function
with support in $V_j$ such that $\psi_j(x_j)=1$~. Then 
$v'=v-\sum\dbar(\psi_ju_j)$ satisfies $\dbar v'=0$ and 
$\smash{\int_X|v'|_\gr^2d\sigma }<+\infty$ because the metric is 
integrable except at the $x_j$'s and $v'$ vanishes in a neighbor\-hood
of $x_j$~. By the above theorem, there is a $L^2$ solution $u'$
of $\dbar u'=v'$~, hence $u=u'+\sum\psi_ju_j$ is a solution
of $\dbar u=v$~. It is well-known that the existence of a
$L^2_{\rm loc}$ solution implies the existence of a smooth one,
whence the corollary.\endpr

\pav 3. Proof of the main theorem|

The proof consists of three steps. The first two are easy
algebraic reductions and the third one is a direct application
of the above corollary. We set $n=\dim_\cb X$~.

$\bullet$ {\bf First step~:} {\sl reduction to the case when
$\min\{\max\{s,\kappa(\fr)\},{\rm codim}\,B\}=n$~.}

This step is achieved by induction on $n$ and a use of suitable
hyperplane sections. Suppose $n>\min\{\max\{s,\kappa(\fr)\},{\rm codim}\,B\}$
and choose a very ample line bundle $\hr$ on $X$~. Take a generic
section of $\hr$ admitting a non singular zero divisor $A$ and consider
the exact sequence
$$0\la \fr^{-1}\otimes\oc(-A)\la \fr^{-1}\la \fr^{-1}_{\restriction A}\la 0~,$$
together with the associated long exact sequence of cohomology. Then all
groups $H^q(X,\fr^{-1}\otimes\oc(-A))$ vanish for $q<n$ if $\hr=\oc(A)$ 
is ample enough (if not, replace $\hr$ by some power). The vanishing of 
$H^q(X,\fr^{-1})$ is then equivalent to the vanishing of 
$H^q(A,\smash{\fr^{-1}_{\restriction A}})$~, so we have to choose 
$A$ in such a way that the hypotheses still hold for 
$\fr_{\restriction A}$ and $\smash{\fr^k_{\restriction A}}=
\lr_{\restriction A}\otimes\oc(D_{\restriction A})$~.
The Fubini theorem applied to the slices of functions
$\prod|g_j|^{-2\alpha_j}$ associated to $k^{-1}D$ shows that
$D_{\restriction A}$ is integrable on $A\setminus (B\cap A)$ for
generic $A$~; in addition, we select $A$ such that
${\rm codim}_A B\cap A={\rm codim}_X B$~. Now if we choose $A$
transversal to the fiber of $\Phi_m:X\setminus Z(V_m)\la P(V_m^\star)$~,
at a point where $\Phi_m$ has maximal rank $\kappa(\fr)$~, we see that
$\Phi_{m\restriction A}$ has the same rank as $\Phi_m$ if $\kappa(\fr)<n$
and has rank $n-1$ otherwise. Hence 
$\kappa(\fr_{\restriction A})\ge\min\{\kappa(\fr),n-1\}$~.
Finally $\lr_{\restriction A}$ is nef and~:

\th Lemma 1|For $r=\min\{s,n-1\}$ we have~ 
$c_1(\lr_{\restriction A})^r\ne 0$~.\endth

Indeed the line bundle $\lr^m\otimes\hr$ is ample, so
$\big(c_1(\lr)+(1/m)c_1(\hr)\big)^r$ is the cohomology class of a
positive rational cycle $Z_m\ge 0$ of codimension $r$~, obtained by
intersecting $r$ generic divisors of some power of $\lr^m\otimes\hr$~. By
extracting a weak limit as $m$ tends to infinity, we see that $c_1(\lr)^r$
is represented by a closed positive current $T$ on $X$~, and $T\ne 0$
by hypothesis. The class $\{A\}=c_1(\hr)$ is represented by a
smooth positive $(1,1)$--form, thus if $i_A$ is the injection of $A$ 
in $X$ we have
$$(i_A)_\star\big(c_1(\lr)^r_{\restriction A}\big)=c_1(\lr)^r\wedge\{A\}
=\{T\}\wedge c_1(\hr)\ne 0~,$$
because $T\wedge c(\hr)$ is a non zero positive current.\endpr

Now the induction hypothesis shows that 
$$H^q(A,\fr_{\restriction A})=0~~~~{\rm for}~~
q<\min\{\max\{r,\kappa(\fr_{\restriction A}\},{\rm codim}_AB\cap A\}~.$$
By the above choice of the divisor $A$~, the upper bound on $q$ is larger
than\break $\min\{\max\{s,\kappa(\fr)\},{\rm codim}\,B\}$~. 
Next, we need the following standard lemma on numerically effective
line bundles. 

\th Lemma 2|If $\lr$ is nef, then for every line bundle $\gr$ we have
$$\eqalign{
h^q(X,\lr^m\otimes \gr)&\le Cm^{n-1}~~~{\sl for}~~q\ge 1~,\cr
h^0(X,\lr^m\otimes \gr)&=c_1(\lr)^n\,m^n/n!+{\rm O}(m^{n-1})~.\cr}$$
\endth

The first is a consequence of the exact sequence
$$H^{q-1}(A,\lr^m\otimes \gr\otimes\oc(A)_{\restriction A})\la 
H^q(X,\lr^m\otimes \gr)\la H^q(X,\lr^m\otimes\gr\otimes\oc(A))~;$$
choose $A$ so ample that $K_X^{-1}\otimes\gr\otimes\oc(A)$ is ample~;
then the last group vanishes by the Akizuki-Kodaira-Nakano vanishing 
theorem and the first one has rank $\le Cm^{n-1}$ since $\dim A=n-1$~.
The second line results from the fact that the Euler-Poincar\'e
characteristic $\chi(X,\lr^m\otimes\gr)$ is a polynomial in $m$ of
leading term $c_1(\lr)^n\,m^n/n!$~.\endpr

{\sl Remark}\pointir Lemma 2 shows that the Kodaira dimension $\kappa(\lr)$
is equal to $n$ if and only if $c_1(\lr)^n>0$~. In the case $s<n$~, 
we always have $\kappa(\lr)\le s$~, for there exists a subvariety
$Y\subset X$ of dimension $d=\kappa(\lr)$ such that 
$\kappa(\lr_{\restriction Y})=d$ and lemma 2 gives
$c_1(\lr)^d_{\restriction Y}>0$~. However, it may happen that
$\kappa(\lr)<s$~: take $\lr\la X=X_1\times X_2$ equal to
the tensor product of an ample line bundle on $X_1$ and of a 
hermitian flat line bundle on $X_2$ with non trivial powers~;
then $\kappa(\lr)=-\infty$ and $s=\dim X_1$~. The same example shows
that the Kodaira dimension may increase by restriction to a
subvariety (take $Y=X_1\times\{{\rm point}\}$).\endpr\vskip5pt

$\bullet$ {\bf Second step~:} {\sl reduction to the case when $\lr$
is ample}\pointir By the first step, we may assume ${\rm codim}\,B=n$
and $\max\{s,\kappa(\fr)\}=n$~. The injection $\lr\hookrightarrow
\lr\otimes\oc(D)=\fr^k$ implies $\kappa(\fr)\ge\kappa(\lr)$~, hence 
the case $s=n$ implies also 
$\kappa(\fr)=n$ by the above remark. We may therefore assume
$\kappa(\fr)=n$~, whence $\limsup~m^{-n}\,h^0(X,\fr^m)>0$~.
Let $\hr=\oc(A)$ be an ample line bundle. The exact sequence
$$0\la H^0(X,\fr^m\otimes\oc(-A))\la H^0(X,\fr^m)\la H^0(A,
\fr^m_{\restriction A})$$
where $h^0(A,\fr^m_{\restriction A})\le Cm^{n-1}$ shows that
$\fr^m\otimes\hr^{-1}$ has non trivial sections for $m$ large enough.
If $E$ is the divisor of such a section, we get
$\fr^m=\hr\otimes\oc(E)$~. After raising $\fr^k=\lr\otimes\oc(D)$
to the power $p$ and multiplying by $\fr^m$~, we obtain~:
$$\fr^{kp+m}=\lr^p\otimes\hr\otimes\oc(pD+E)=\lr'\otimes\oc(D')~,$$
with $\lr'=\lr^p\otimes\hr$ and $D'=pD+E$~. Moreover $\lr'$ is ample
and for $p$ large enough $(kp+m)^{-1}D'\le k^{-1}D+(kp+m)^{-1}E$ is
integrable.\vskip5pt

$\bullet$ {\bf Third step~:} {\sl Proof of the main theorem when
$\lr$ is ample}\pointir We choose on $\lr$ a smooth hermitian
metric with positive curvature and set $\omega=c(\lr)$~.
On the other hand, $\oc(D)$ can be equipped with a
natural singular metric~: if $s$ is a local section of $\oc(D)$~,
viewed as a meromorphic function with poles along $D$~, we define
$|s|^2=s\ov s\in[0,+\infty]$~. If $g=\prod g_j^{p_j}$ is a local generator
of the ideal of $D=\sum p_j D_j$~, then $g^{-1}$ is a non vanishing section
of $\oc(D)$ which may be used to get a trivialization of $\oc(D)$~.
The plurisubharmonic weight corresponding to the metric is
$e^{-\varphi}$ with $\varphi=\log|g|^2$~. By the Lelong-Poincar\'e
equation, the curvature of this metric is ${1\over 2\pi}c(\oc(D))=[D]=$ 
current of integration over $D$~; in particular $c(\oc(D))\ge 0$ in the 
sense of currents. The induced metric on $\fr$ has curvature
$$c(\fr)=k^{-1}\big(c(\lr)+c(\oc(D))\big)=k^{-1}\omega+k^{-1}2\pi[D]
\ge k^{-1}\omega~.$$
{\sl Caution}~: in spite of the fact that our singular metric has 
strongly positive curvature, it may happen that $\fr$ is not ample 
nor even nef when $D$ itself is not nef (e.g. when $D$ is an
exceptional divisor).

Now the weight associated to $\fr$ is equal to $\exp(-k^{-1}
(\varphi+\psi))$ where $\psi$ is the (smooth) weight of $\lr$~,
so it is integrable if and only if $\exp(-k^{-1}\varphi)=|g|^{-2/k}$
is integrable. This assumption holds on $X\setminus B$ by
hypothesis, and here $B$ is finite (codim$\,B=n$). The corollary in
\S 2 implies $H^q(X,\fr^{-1})=0$ for $q<n$~.\endpr

{\sl Remark}\pointir If the linear system $|D|$ is not reduced to the
divisor $D$~, the integrability requirement can be weakened~.  Let
$\smash{D^{(1)}=D,\,D^{(2)}\ld D^{(N)}}$ be the zero divisors of
sections in a basis $s=(s^{(1)}\ld s^{(N)})$ of $H^0(X,\oc(D))$~.  Then
we take the quotient metric on $\oc(D)$ induced by the generically
surjective homomorphism $s:\oc^N\la\oc(D)$ and the trivial metric on
$\oc^N$~.  If $e$ is a local holomorphic frame of $\oc(D)$~, we may write
$s^{(\ell)}=g^{(\ell)}e$ where $g^{(\ell)}$ is a local generator of
${\cal I}_{D^{(\ell)}}$ and the quotient metric is given by the weight
$e^{-\varphi}$ with $\varphi=\log(|g^{(1)}|^2+\cdots+|g^{(N)}|^2)$~. 
The integrability condition is now weakened into the integrability of
$\big(|g^{(1)}|^2+\cdots+|g^{(N)}|^2\big)^{-2/k}$ instead of
$|g^{(1)}|^{-2/k}$~.\eject

\pav 4. New proof of the Bogomolov-Sommese vanishing theorem|
 
By similar ideas, we can obtain a new simple proof of this 
theorem, without using Deligne's degeneracy criterion
for the Hodge spectral sequence.

\th Theorem {\rm ([2], [15])}|Let $(X,\omega)$ be a compact K\"ahler
manifold and let $\gr$ be a line bundle over $X$~. Then
$$H^0(X,\Omega^p_X\otimes\gr^{-1})=0~~~~{\rm for}~~p<\kappa(\gr)~.$$
\endth

\proof Let $\Phi_m:X\setminus Z\la\pb^N$ be a morphism of
generic rank $d=\kappa(\gr)$ associated to $V_m=H^0(X,\gr^m)$~. Then
$\gr^m\simeq\Phi_m^\star\oc(1)$ and we endow $\gr$ with the induced
metric, which is singular along $Z$~. If we set $V_m=\langle
s^{(1)}\ld s^{(N)}\rangle$
and use the same notations as in the last remark, the singular metric
on $\gr$ (resp. $\gr^{-1}$) is defined by the weight $e^{-\varphi}$ 
(resp. by $e^\varphi$) with
$$\varphi={1\over m}\log(|g^{(1)}|^2+\cdots+|g^{(N)}|^2)~.$$
It is clear that the curvature form
$c(\gr)=i\partial\dbar\varphi=\Phi_m^\star c(\oc(1))$ is a positive
current, equal to a smooth form of generic rank $\kappa(\gr)$ on
$X\setminus Z$~. If $u$ is a holomorphic $p$--form with values in
$\gr^{-1}$~, the curvature inequality $(\star)$ gives~:
$$0=\int_X|\dbar u|^2d\sigma\ge
\int_{X\setminus Z}(\lambda_{p+1}+\cdots+\lambda_n)|u|^2d\sigma$$
where $\lambda_1\ge\ldots\ge\lambda_n\ge 0$ are the eigenvalues of
$c(\gr)$~. This can be checked by using a sequence of smooth weights
$\varphi_j$ converging to $\varphi$ with $\lambda_i(\varphi_j)\ge -$ 
constant converging to $\lambda_i=\lambda_i(\varphi)$ and 
$e^{\varphi_j}$ converging uniformly to $e^\varphi$ (cf. [4]). 
As $\lambda_d>0$ almost everywhere on $X\setminus Z$~, we conclude 
that $u=0$ for $p<d$~.\endpr
\vfill\eject

\centerline{\bf References}
\vskip15pt
\eightpoint
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\end

