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\centerline{\bftwelve Singular hermitian metrics on positive line bundles}
\vskip10pt
{\noindent\hangindent4cm\hangafter-1
{\bf 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

}}
\vskip24pt
{\bf Abstract}\pointir {\sl The notion of a singular hermitian metric
on a holomorphic line bundle is introduced as a tool for the study of
various algebraic questions.  One of the main interests of such metrics
is the corresponding $L^2$ vanishing theorem for $\op$ cohomology,
which gives a useful criterion for the existence of sections.  In this
context, numerically effective line bundles and line bundles with
maximum Kodaira dimension are characterized by means of positivity
properties of the curvature in the sense of currents.  The coefficients
of isolated logarithmic poles of a plurisubharmonic singular metric are
shown to have a simple interpretation in terms of the constant
$\varepsilon$ of Seshadri's ampleness criterion.  Finally, we use
singular metrics and approximations of the curvature current to prove 
a new asymptotic estimate for the dimension of cohomology groups with 
values in high multiples $\oc(kL)$ of a line bundle $L$ with maximum
Kodaira dimension.}
\vskip6pt

\titre 1. Introduction|

Our purpose is to show that several important concepts of algebraic
geo\-metry have a nice interpretation in differential geometric terms, 
once we admit hermitian metrics with singularities, and especially
plurisubharmonic weights with {\sl logarithmic poles}. 

A {\sl singular (hermitian) metric} on a line bundle $L$ is simply a
hermitian metric which is given in any trivialization by a weight
function $e^{-\varphi}$ such that $\varphi$ is locally integrable.  We
then have a well-defined {\sl curvature current}
$c(L)={i\over\pi}\partial\op \varphi$ and the case when $c(L)\ge 0$ as
a current is especially interesting.  One of the main reasons for this
is the basic $L^2$ existence theorem of H\"ormander-Andreotti-Vesentini
for solutions of $\op$ equations with plurisubharmonic weights.  With
relatively few efforts, the $L^2$ theory gives strong vanishing theorems 
(of Kawamata-Viehweg type) and existence results for sections of the
adjoint line bundle $K_X+L\,$; here $K_X$ denotes the canonical line 
bundle of the base manifold~$X$, and an additive notation is used for 
the group ${\rm Pic}(X)=H^1(X,{\cal O}^\star)$. These techniques can 
also be applied in combination with the Calabi-Yau theorem to obtain
explicit numerical criteria for very ample line bundles; we refer to
[De~90] for results in this direction.

On a projective algebraic manifold $X$, the real vector space generated
by the Neron-Severi group $H^2(X,\zb)\cap H^{1,1}(X)$ contains
two canonical closed convex cones $\Gamma_+\supset\Gamma_a$, which
are generated by cohomology classes of effective or ample divisors,
respectively. For a line bundle $L$ on $X$, we show that $L$ has a
singular metric with positive curvature current if and only if
$c_1(L)\in\Gamma_+$. We also give similar differential geometric 
descriptions of the line bundles for which $c_1(L)$ belongs to
$\Gamma_a$ (numerically effective line bundles), or to the open 
cone $\Gamma_+^\circ$ (line bundles with $\kappa(L)=n$), or else to
$\Gamma_+^\circ\cap\Gamma_a$ (big and nef line bundles).

The well-known Seshadri ampleness criterion asserts that a line 
bundle $L$ on a projective manifold $X$ is ample if and only if
there is a constant $\varepsilon>0$ such that $L\cdot C\ge
\varepsilon\,m(C)$ for every curve $C\subset X$, where $m(C)$ is the
maximum of the multiplicity of the singular points of~$C$.
We show that the optimal constant $\varepsilon(L)$ is precisely
equal to the supremum of coefficients $\gamma$ for which a 
plurisubharmonic weight on $L$ may have an isolated logarithmic pole of
slope $\gamma$ at any point. This result is then refined by introducing
``local" Seshadri constants $\varepsilon(L,x)$ which measure 
ampleness along curves passing though a fixed point $x$.

Finally we use approximation techniques for singular metrics,
combined with the general holomorphic Morse inequalities of [De~85],
to obtain an asymptotic upper bound for the dimensions of
cohomology groups $H^q(X,kL)$ when $L$ is a line bundle of maximum
Kodaira dimension $\kappa(L)=n$ and $k$ tends to $+\infty$.  If some
multiple of $L$ is written as $mL\simeq\oc(A+D)$ where $A$ (resp. $D$)
is an ample (resp. effective) divisor, the upper bound is expressed in
a simple way in terms of the first Chern class $c_1(L)$, the 
multiplicities of the singular points of $D$ and the curvature of 
the tangent bundle $TX$.

Several results of the present article have been worked out while
the author was visiting the Tata Institute in Bombay in August~1989.
The author wishes to thank this institution for its hospitality,
and especially Prof. M.S.~Narasimhan, R.R~Simha and S.~Subramanian
for stimulating discussions.

\titre 2. Notion of singular hermitian metrics|

Let $L$ be a holomorphic line bundle over a complex manifold $X$.
We are mostly interested in the case of a compact manifold, but this 
restriction is irrelevant in the present section.

\tha Definition 2.1|A singular $($hermitian$)$ metric on $L$ is a metric
which is given in any trivialization $\theta:L_{\restriction\Omega}
\buildo\simeq\over\longrightarrow\Omega\times\cb$ by
$$\|\xi\|=|\theta(\xi)|\,e^{-\varphi (x)},~~~~x\in \Omega ,~\xi\in L_x$$
where $\varphi\in L^1_\loc(\Omega)$ is an arbitrary function, called the
weight of the metric with respect to the trivialization $\theta$.\fintha

If $\theta':L_{\restriction\Omega'}\la\Omega'\times\cb$ 
is another trivialization, $\varphi'$ the associated weight and
$g\in\oc^\star(\Omega\cap\Omega')$ the transition function, then
$\theta'(\xi)=g(x)\,\theta(\xi)$ for $\xi\in L_x$, and so
$\varphi'=\varphi+\log|g|$ on $\Omega\cap\Omega'$. The curvature form 
of $L$ is then given by the closed $(1,1)$-current $c(L)=
{i\over\pi}\partial\op\varphi$ on $\Omega$, if we compute formally
$c(L)={i\over 2\pi}D^2$ as in the smooth case$\,$; $c(L)$ is, of course, a 
global current on $X$ which is independent of the choice of trivializations. 
Our assumption $\varphi\in L^1_\loc(\Omega)$ guarantees that $c(L)$ exists 
in the sense of distribution theory. Then the De Rham cohomology
class of $c(L)$ is the image of the first Chern class $c_1(L)\in
H^2(X,\zb)$ in $H^2_{DR}(X,\rb)$ (De~Rham cohomology can be computed 
either by means of smooth differential forms or by means of currents). 
Before going further, we discuss two basic examples. 

\remarque Example 2.2|Let $D=\sum\alpha_jD_j$ be a divisor with 
coefficients $\alpha_j\in\zb$ and let $L=\oc(D)$ be the associated
invertible sheaf of meromorphic functions $f$ such that
${\rm div}(f)+D\ge 0\,$; the corresponding line bundle can be
equipped with the singular metric defined by $||f||=|f|$. If $g_j$ is a
generator of the ideal of $D_j$ on an open set $\Omega\subset X$ then 
$\theta(f)=f\prod\smash{g_j^{\alpha_j}}$ defines a trivialization of $\oc(D)$
over $\Omega$, thus our singular metric is associated to the weight
$\varphi=\sum\alpha_j\log|g_j|$. By the Lelong-Poincar\'e equation,
we find
$$c\big(\oc(D)\big)={i\over\pi}\partial\op\varphi=[D],\leqno(2.3)$$
where $[D]=\sum\alpha_j[D_j]$ denotes the current of integration over $D$
(cf. [Le~57] and [Le~69]).

\remarque Example 2.4|Assume that $\sigma_1\ld \sigma_N$ are non zero 
holomorphic sections of $L$. Then we can define a natural (possibly
singular) hermitian metric 
on $L^\star$ by
$$||\xi^\star||^2=\sum_{1\le j\le n}\big|\xi^\star.\sigma_j(x)\big|^2
~~~~\hbox{\rm for}~~\xi^\star\in L^\star_x.$$
The dual metric on $L$ is given by
$$||\xi||^2={|\theta(\xi)|^2\over|\theta(\sigma_1(x))|^2+\cdots+
|\theta(\sigma_N(x))|^2}$$
with respect to any trivialization $\theta$. The associated weight function is
thus given by $\varphi(x)=\log\big(\sum_{1\le j\le N}|\theta(\sigma_j(x))|^2
\big){}^{1/2}$. In this case $\varphi$ is a plurisubharmonic function, so
$c(L)$ is a (closed) positive current.\finpr\vskip5pt

It is worth observing that the weight functions $\varphi$ have logarithmic
poles in both examples. In the second case, the set of poles is the
base locus $\bigcap\sigma_j^{-1}(0)$ of the linear system generated
by the sections $\sigma_1\ld \sigma_N$. 
In the sequel, we always suppose that the curvature current $c(L)$ is 
positive, i.e. that the weight functions $\varphi$ are plurisubharmonic.

\tha Definition 2.5|A singular metric on $L$ with positive curvature 
current $c(L)\ge 0$ 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 $\nu(\varphi,x)=\gamma$.\fintha

For the basic properties of Lelong numbers, we refer to [Le~69],
[Siu~74] and [De~87]. It is well known that $\nu(\varphi,x)$ is always 
equal to the Lelong number of the associated current 
$T={i\over\pi}\partial\op\varphi$, defined by
$\nu(T,x)=\lim_{r\to 0_+}\nu(T,x,r)$ with
$$\nu(T,x,r)={1\over(2\pi r^2)^{n-1}}\int_{B(x,r)}T(z)
\wedge(i\partial\op|z|^2)^{n-1}.\leqno(2.6)$$
Finally, for every $c>0$, we consider the sublevel sets
$$E_c(T)=\big\{x\in X\,;\,\nu(T,x)\ge c\big\}.\leqno(2.7)$$
By a theorem of [Siu~74], $E_c(T)$ is a (closed) analytic subset 
of~$X$. If $T={i\over\pi}\partial\op\varphi$ on an open set
$\Omega\subset X$, we denote accordingly $E_c(\varphi)=E_c(T)\cap\Omega$.
The following simple lemma is very useful in this context.

\tha Lemma 2.8|If $\varphi$ is a plurisubharmonic function on $X$,
then $e^{-2\varphi}$ is integrable in a neighborhood of $x$ as soon
as $\nu(\varphi,x)<1$ and non integrable as soon as $\nu(\varphi,x)\ge n$.
\fintha

\remarque Proof|If $\nu(\varphi,x)=\gamma$, the usual convexity
properties of plurisubharmonic functions show that 
$$\varphi(z)\le\gamma\log|z-x|+O(1)~~~~\hbox{\rm near}~~x,\leqno(2.9)$$ 
thus $e^{-2\varphi(z)}\ge C|z-x|^{-2\gamma}$ is non integrable as soon
as $\gamma\ge n$. For a proof that $e^{-2\varphi}$ is integrable when 
$\nu(\varphi,x)<1$, we refer to Skoda [Sk~72]. Both bounds are best 
possible as the examples $\varphi(z)=(n-\varepsilon)\log|z|$ and 
$\varphi(z)=\log|z_1|$ in $\cb^n$ easily show; in the first case
$\nu(\varphi,0)=n-\varepsilon$ but $e^{-2\varphi}$ is integrable at $0\,$;
in the second case $\nu(\varphi,0)=1$ but 
$e^{-2\varphi}$ is non integrable at $0\,$.\finpr

\titre 3. $L^2$ vanishing theorem and criterion for the existence of sections|

One of the main reasons for which singular metrics are especially
interesting is the powerful existence theorem of
H\"ormander for solutions of equations $\op u=v$.

\tha Theorem 3.1|Suppose that $X$ is a Stein or compact projective
manifold equipped with a K\"ahler metric $\omega$.  Let $L$ be a 
holomorphic line bundle equipped with a singular metric associated to
plurisubharmonic weight functions $\varphi$ such that 
$c(L)\ge\varepsilon\,\omega$ for some $\varepsilon>0$. For every
$q\ge 1$ and every $(n,q)$ form $v$ with values in $L$ such that 
$\op v=0$ and $\int_X|v|^2e^{-2\varphi}dV_\omega<+\infty$, there is a 
$(n,q-1)$-form $u$ with values in $L$ such that $\op u=v$ and
$$\int_X|u|^2e^{-2\varphi}dV_\omega\le{1\over2\pi q\varepsilon}
\int_X|v|^2e^{-2\varphi}dV_\omega.$$\fintha

Here $dV_\omega$ stands for the K\"ahler volume element $\omega^n/n!$
and $|u|^2e^{-2\varphi}$ denotes somewhat abusively the pointwise norm of
$u(z)$ at each point $z\in X$, although $\varphi$ is only defined on
an open set in $X$. The constant $2\pi$ comes from the fact that we have
included $2\pi$ in the definition of $c(L)$.

\remarque Proof|The result is standard when $X$ is Stein and $L$ is the
trivial bundle (see [AV~65] and [H\"o~66]); the proof can then be
reduced to the case of a smooth metric, because any plurisubharmonic
function is the limit of a decreasing sequence of smooth
plurisubharmonic functions.  In general, there exists a
hypersurface $H\subset X$ such that $X\ssm H$ is Stein and $L$
is trivial over $X\ssm H$.  We then solve the equation $\op
u=v$ over $X\ssm H$ and observe that the solution extends to
$X$ thanks to the $L^2$ estimate (cf. [De~82],
lemma~6.9).\finpr\vskip8pt

From this general theorem, we can easily derive an abstract vanishing
theorem for the adjoint line bundle $K_X+L$ and a criterion for the 
existence of sections of $K_X+L$.

\tha Corollary 3.2|Let $L$ be a line bundle with a singular hermitian 
metric. Assume that $c(L)\ge\varepsilon\,\omega$ and that the metric 
(i.e. the weight $e^{-2\varphi}$) is integrable near all but
finitely many points of $X$~. Then~
$H^q(X,K_X+L)=0$ for $q\ge 1$.\fintha

\remarque Proof|Let $x_1\ld x_m$ be the points where the metric is not
integrable and let $v$ be a smooth $(n,q)$--form with values in $L$ such 
that $\op v=0$~. Let $u_j$ be a smooth solution of $\op u_j=v$ 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 
$v'=v-\sum\op(\psi_ju_j)$ satisfies $\op v'=0$ and 
$\int_X|v'|^2e^{-2\varphi}d\sigma<+\infty$, because $e^{-2\varphi}$ is 
integrable except at the $x_j$'s and $v'$ vanishes in a neighborhood
of $x_j$~. By theorem~3.1, there is a $L^2$ solution $u'$
of $\op u'=v'$~, hence $u=u'+\sum\psi_ju_j$ is a solution
of $\op u=v$~. It is well-known that the existence of a
$L^2_{\rm loc}$ solution implies the existence of a smooth one,
whence the corollary.\finpr

\tha Corollary 3.3|Let $L$ be a line bundle with a singular metric such
that $c(L)\ge\varepsilon\,\omega$, $\varepsilon>0$.  If 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
$H^0(X,K_X+L)$ generates all $s$-jets~at~$x$.\fintha

\remarque Proof|The proof is a straightforward adaptation of the
H\"ormander-Bombieri-Skoda technique ([Bo~70], [Sk~75]). 
We have $e^{-2\varphi(z)}\ge C|z-x|^{-2(n+s)}$ near $x$ by (2.9),
and in particular $e^{-2\varphi }$ is non integrable near $x$. 
Since $x$ is supposed to be isolated in $E_1(\varphi)$, we infer from 
lemma~2.8 that $e^{-2\varphi }$ is locally integrable on a small punctured
neighborhood $V\ssm \{x\}$. Let $P(z)$ be an arbitrary polynomial of
degree $\le s$ in given analytic coordinates $(z_1\ld z_n)$ on $V$. Fix
a smooth cut-off function $\chi $ with compact support in $V$ such that
$\chi =1$ near $x$ and a non vanishing local section $h\in
H^0(V,K_X+L)$. Then $v=P\op\chi \otimes h$ can be viewed as a
$\op$-closed $(n,1)$-form on $X$ with values in $L$, such that
$\int_X|v|^2e^{-2\varphi }dV_\omega  < +\infty$~; indeed $v$ is smooth,
has compact support in $V$ and vanishes near $x$. The solution $u$ of
$\op u = v$ is smooth and we have $|u(z)|=o(|z-x|^s)$ near $x$ 
thanks to the $L^2$ estimate. Therefore
$$f=\chi Ph-u\in H^0(X,K_X+L)$$
has the prescribed $s$-jet $Ph$ at $x$.\finpr

\titre 4. Numerical cones associated to positive line bundles|

We suppose here that $X$ is a projective algebraic manifold and denote
$n=\dim X$. It is well known that an integral cohomology class in
$H^2(X,\zb)$ is the first Chern class of a holomorphic (or algebraic)
line bundle if and only if this class is of type $(1,1)$. Hence the
so-called {\sl Neron-Severi} group 
$$NS(X)=H^2(X,\zb)\cap H^{1,1}(X)\subset H^2(X,\rb)$$
is the set of cohomology classes of algebraic line bundles (or of 
integral divisors).

\tha Definition 4.1|A holomorphic line bundle $L$ over $X$ is said to be
numerically effective, nef for short, if $L\cdot C=\int_C c_1(L)\ge 0$
for every curve $C\subset X$, and in this case $L$ is said to be big
if $L^n = \int_X c_1(L)^n>0$.\fintha

If $L$ is nef, it is well known that $L^p\cdot Y=\int_Y c_1(L)^p\ge 0$
for any $p$-dimensional subvariety $Y\subset X$ (see e.g. [Ha~70]).
The Nakai-Moishezon ampleness criterion then shows that $L+A$ is ample
as soon as $A$ is ample. In fact, if $A$ is ample, it is easy to see 
that $L$ is nef if and only if $kL+A$ is ample for all integers $k\ge 1$.
We are going to describe a simple dictionary relating these concepts to
similar concepts in the context of differential geometry.

Let $NS_\rb(X)$ be the real vector space $NS(X)\otimes\rb\subset
H^2(X,\rb)$ and let $\Gamma_+\subset NS_\rb(X)$, resp.
$\Gamma_a\subset\Gamma_+$, be the closed convex cone generated by 
cohomology classes of effective (resp. ample) divisors $D\,$;  denote
by $\Gamma^\circ_+$ (resp. $\Gamma^\circ_a$) the interior of $\Gamma_+$
(resp. $\Gamma_a$). We will call $\Gamma_+$ and $\Gamma^\circ_a$
respectively the {\sl effective cone} and the {\sl ample cone} of~$X$.
Finally, recall that the {\sl Kodaira dimension} $\kappa(L)$ is the 
supremum of the rank of the canonical maps
$$\Phi_m:X\ssm Z_m\la P(V_m^\star),~~~~x\longmapsto H_x=\{\sigma\in
V_m\,;\,\sigma(x)=0\},~~~~m\ge 1$$
with $V_m=H^0(X,mL)$ and $Z_m=\bigcap_{\sigma\in V_m}\sigma^{-1}(0)={}$base
locus of $V_m$. If $V_m=\{0\}$ for all $m\ge 1$, we set $\kappa(L)=
-\infty$. Then we have $h^0(X,mL)\le O(m^{\kappa(L)})$ for $m\ge 1$,
and $\kappa(L)$ is the smallest constant for which this estimate holds.

\tha Proposition 4.2|If $L$ is a holomorphic line bundle on $X$, $\omega$ 
a K\"ahler metric and $\varepsilon>0$, we have the following equivalent
properties:
$$\leqalignno{
&\qquad c_1(L)\in\Gamma_+\Llra L~\hbox{\rm has a singular metric with}~
c(L)\ge 0\,;&\hbox{\rm(a)}\cr
&\qquad c_1(L)\in\Gamma^\circ_+\Llra\exists\varepsilon,~L~
\hbox{\rm has a singular metric with}~
c(L)\ge \varepsilon\,\omega\Llra \kappa(L)=n\,;&\hbox{\rm(b)}\cr
&\qquad c_1(L)\in\Gamma_a\Llra\forall\varepsilon,~L~
\hbox{\rm has a smooth metric with}~
c(L)\ge-\varepsilon\,\omega\Llra L~\hbox{is nef}\,;&\hbox{\rm(c)}\cr
&\qquad c_1(L)\in\Gamma^\circ_a\Llra\exists\varepsilon,~L~
\hbox{\rm has a smooth metric with}~
c(L)\ge \varepsilon\,\omega\Llra L~\hbox{is ample}.&\hbox{\rm(d)}\cr}$$\fintha

\remarque Proof|It is well known that $L$ is ample if and only if $L$ 
has a smooth metric with positive definite curvature, and this
gives the last equivalence in (4.2d).

(4.2a) Suppose that $c_1(L)\in\Gamma_+$. By definition
$c_1(L)$ is a limit of coho\-mo\-logy classes of effective real
divisors $D_k=\sum\lambda_{j,k}D_{j,k}$. Then $[D_k]$ is a sequence of
closed positive currents, with a uniform bound of the mass
$\int_X [D_k]\wedge\omega^{n-1}$, for this integral converges to
$\int_X c_1(L)\wedge\omega^{n-1}$. By weak compactness, there is a
subsequence $[D_{k_\nu}]$ converging to some closed positive current
$T$ of bidegree $(1,1)$, such that the cohomology class of $T$ is
equal to $c_1(L)$. Therefore, if $L$ is equipped with an arbitrary smooth
metric, we find $T=c(L)+{i\over\pi}\partial\op\psi$ for some function
$\psi\in L^1(X)$, and $T$ is the curvature current of the singular metric
obtained by multiplication of the original smooth metric by $e^{-\psi}$.
Conversely, if $L$ has a singular metric with $c(L)\ge 0$,
we fix a point $x_0\in X$ such that the associated weight satisfies
$\nu(\varphi_L,x_0)=0$. Let $\psi_0$ be a smooth function
on $X\ssm\{x_0\}$ which is equal to $n\log|z-x_0|$ (in some
coordinates) near $x_0$, and let $A$ be a fixed ample line bundle,
equipped with a smooth metric of positive curvature.
For $m_0$ large enough, we have $m_0c(A)+{i\over\pi}\partial\op\psi_0
\ge\omega$ and the tensor product metric on $kL+m_0A$
multiplied by $e^{-2\psi_0}$ is associated to a weight $\varphi_k=
k\varphi_L+m_0\varphi_A+\psi_0$ such that
$${i\over\pi}\partial\op\varphi_k=
k\,c(L)+m_0c(A)+{i\over\pi}\partial\op\psi_0\ge k\,c(L)+\omega\ge
\omega~~~~~\forall k\ge 1.$$
Moreover $\nu(\varphi_k,x_0)=n$, whereas $\nu(\varphi_k,x)=
k\,\nu(\varphi_L,x)<1$ for $x\ne x_0$ near $x_0$. We infer from corollary~3.3
that $kL+m_0A+K_X$ has non zero sections for all $k\ge 1$. Let $D_k$ be the
divisor of any such section and $\{D_k\}$ its cohomology class. Then
$$c_1(L)={1\over k}\Big(\{D_k\}-m_0c_1(A)-c_1(K_X)\Big)=
\lim_{k\to +\infty}{1\over k}\{D_k\}$$
and therefore $c_1(L)\in\Gamma_+$.

(4.2b) Without loss of generality, we may suppose that the cohomology
class $\{\omega\}$ is integral, i.e. $\{\omega\}\in NS(X)$. The first 
equivalence in (4.2b) is then an immediate consequence of (4.2a), since
a class $\{\alpha\}$ is in the interior $\Gamma^\circ_+$ if and only if
$\{\alpha-\varepsilon\,\omega\}\in\Gamma_+$ for $\varepsilon$ small enough
($\{\omega\}$ is obviously an interior point). If
$c(L)\ge\varepsilon\,\omega$, we can construct as above a singular
metric $\varphi_k$ on $kL-K_X$ such that
${i\over\pi}\partial\op\varphi_k\ge \omega$ for $k\ge k_0$,
$\nu(\varphi_k,x_0)=n+1$ and $\nu(\varphi_k,x)<1$ for $x\ne x_0$
near $x_0$. Then corollary 3.3 shows that $kL$ has sections with
arbitrary $1$-jets at $x_0$, hence $\kappa(L)=n$. Conversely, if
$\kappa(L)=n$, then $h^0(X,kL)\ge ck^n$ for $k\ge k_0$ and $c>0$. 
Let $A$ be a smooth ample divisor. The exact cohomology sequence
$$0\la H^0(X,kL-A)\la H^0(X,kL)\la H^0(A,kL_{\restriction A})$$
where $h^0(A,kL_{\restriction A})=O(k^{n-1})$ shows that
$kL-A$ has non zero sections for $k$ large. If $D$ is the divisor of
such a section, then $kL\simeq\oc(A+D)$. If we select a smooth metric
on $A$ such that $c(A)\ge\varepsilon_0\omega$ and the singular metric 
on $\oc(D)$ described in example~2.2, then $c(L)={1\over k}\big(c(A)+[D]\big)
\ge(\varepsilon_0/k)\,\omega$, as desired.

(4.2c) If $c_1(L)\in\Gamma_a$, then $L$ is nef, because the
condition that a cohomology class of type $(1,1)$ has nonnegative
integrals over curves is preserved through convex combinations
and limits. Conversely, when $L$ is nef, $kL+A$ is ample
as soon as $A$ is ample; thus $c_1(kL+A)$ is the cohomology
class of an ample rational divisor $D_k$ and
$c_1(L)=\lim_{k\to+\infty}k^{-1}\{D_k\}\in\Gamma_a$. Moreover, arbitrary
choices of smooth metrics with positive curvature on $A$ and $kL+A$
yield a smooth metric on $L$ such that
$$c(L)={1\over k}\big(c(kL+A)-c(A)\big)\ge -{1\over k}c(A)\,;$$
in this way the negative part can be made smaller than $\varepsilon\,\omega$
for any $\varepsilon>0$ if we take $k$ large enough. Finally, if 
$c(L)\ge-\varepsilon\,\omega$ for every $\varepsilon>0$, then $L\cdot C
\ge-\varepsilon\int_C\omega$ and we conclude that $L\cdot C\ge 0$,
thus $L$ is nef.

(4.2d) Only the first equivalence remains to be checked: this is
an immediate consequence of (4.2c) and the fact that
$\{\omega\}\in\Gamma_a^\circ$.\finpr\vskip8pt

\tha Corollary 4.3|If $L$ is nef, then $\kappa(L)=n$ if and only if
$L^n>0$. Moreover, the following properties are equivalent:

\noindent{\rm(a)}~~$L$ is nef and big$\,$;

\noindent{\rm(b)}~~$c_1(L)\in\Gamma_a\cap\Gamma_+^\circ\,$;

\noindent{\rm(c)}~~$\,$for every $\delta>0$, $L$ has a singular
metric such that $c(L)\ge\varepsilon\,\omega$ for some $\varepsilon>0$\break
\phantom{\rm(c)~~$\,$}and such that $\,\max_{x\in X}\nu(\varphi,x)\le\delta$. 

\noindent The metrics obtained in {\rm(c)}
can be chosen to be smooth on the complement of a fixed divisor $D$, 
with logarithmic poles along $D$.\fintha

\remarque Proof|The first statement is well-known and obtained as follows:
if $L$ is nef, the Hilbert polynomial of $\chi(X,kL)$ has leading 
coefficient $L^n/n!\ge 0\,$; by the Kodaira-Nakano vanishing theorem, 
we have an exact sequence
$$H^{j-1}\big(A,(kL+A)_{\restriction A}\big)
\la H^j(X,kL)\la H^j(X,kL+A)=0,~~~~j\ge 1$$
whenever $A$ is a smooth divisor chosen sufficiently ample so that 
$A-K_X$ is ample; thus $h^j(X,kL)=O(k^{n-1})$ for $j\ge 1$ and 
$$h^0(X,kL)=(L^n/n!)\,k^n+O(k^{n-1}),$$
therefore $L$ is big if and only if $\kappa(L)=n$. The equivalence of (a) 
and (b) follows immediately from this and from (4.2b), (4.2c).

We also observe that $H^0(X,kL-A)$ is the kernel of the restriction 
morphism $H^0(X,kL)\to H^0(A,kL_{\restriction A})$ in which the target 
has dimension $O(k^{n-1})$. If $L^n>0$, we infer $H^0(X,kL-A)\ne 0$ 
for $k$ large, so there is an effective divisor $D$ such that $kL\simeq
\oc(A+D)$. Now, $pL+A$ is ample for every $p\ge 0$, so $pL+A$ has a 
smooth metric
with $c(pL+A)\ge\varepsilon_p\omega$ and the isomorphism $(k+p)L\simeq pL+A+D$
gives a metric on $L$ such that
$$c(L)=(k+p)^{-1}c(pL+A)+(k+p)^{-1}[D]\ge (k+p)^{-1}\varepsilon_p\,\omega.$$
Observe that the singular part $(p+k)^{-1}[D]$ can be chosen as small 
as desired by taking $p$ large, so $\max_X\nu(\varphi,x)\le(k+p)^{-1}
\max_D\nu(D,x)$ can be made arbitrarily small. Hence (a) implies (c).

Finally, if property (c) holds, the regularization theorem of 
[De~91] applied to $T=c(L)$ shows that $L$ has smooth metrics such that 
the regularized curvature form $T_\varepsilon$ has arbitrary small
negative part. Hence $L$ is nef by (4.2c) and $\kappa(L)=n$ by 
(4.2b). Therefore (c) implies (a).\finpr

\titre 5. The Kawamata-Viehweg vanishing theorem|

To illustrate the strength of theorem 3.1, we give below a very
simple derivation of the Kawamata-Viehweg vanishing theorem
[Ka~82], [Vi~82]. Only the case of maximum Kodaira dimension will be 
treated here (the general case can be easily deduced by a slicing
argument and an induction on $\dim X$, cf. [De~89]).

\tha Definition 5.1|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 $\prod|g_j|^{-2\alpha_j}$ associated to 
local generators $g_j$ of the ideal of $D_j$ at $x_0$
is integrable on a neighborhood of $x_0$~.\fintha

Observe that $\prod|g_j|^{-2\alpha_j}=e^{-2\varphi}$ where $\varphi$ 
is the weight function of the natural singular metric on $\oc(D)$
described in example~2.2. 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, lemma~2.8 shows that a sufficient 
condition for the integrability of $D$ at $x_0$ is that the 
multiplicity (or Lelong number) 
$$\nu(D,x_0)=\sum\alpha_j\,\nu(D_j,x_0)$$ 
be $<1$. If neither $D$ has normal crossings nor $\nu(D,x_0)<1$~, 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 [Hi~64]). Taking into account the
jacobian divisor $J$ of the blow-up morphism $\pi$, we get at the end
a divisor $D'=\pi^\star D-J$ with normal crossings which is integrable
if and only if $D$ is integrable.
A consequence of this is that integrability is an open condition~:
if $E$ is an arbitrary effective divisor and if $D$ is integrable at $x_0$, 
then $D+p^{-1}E$ is again integrable at $x_0$ for $p$ large enough. 
With these definitions, we have:

\tha Theorem 5.2 {\rm (Kawamata-Viehweg)}|Let $L$ be a line bundle 
over a projective manifold $X$ with $\kappa(L)=n$. Assume that some 
positive multiple $mL$ can be written $mL=\oc(F+D)$ where $F$ is a 
nef line bundle and $D$ an effective divisor such that $m^{-1}D$ is 
integrable on $X\ssm\{\hbox{\sl finite~set}\}$. Then
$$H^q(X,K_X+L)=0~~~~{\sl for}~~q\ge 1.$$\fintha

\remarque Proof|By the proof of (4.2b), there is an ample divisor $A$
and an effective divisor $E$ such that $kL\simeq\oc(A+E)$. Then
$(pm+k)L\simeq\oc(pF+A+pD+E)$ where $pF+A$ is ample and
$$(pm+k)^{-1}(pD+E)\le m^{-1}D+(pm+k)^{-1}E$$
is integrable on $X\ssm\{$finite set$\}$ for $p\ge 1$ large enough 
(integrability is an open 
condition). If we select a smooth metric on $pF+A$ with positive curvature
$\omega_p=c(pF+A)$ and take the singular metric on $\oc(pD+E)$ described in
example~2.2, we find a singular metric on $L$ such that
$$c(L)=(pk+m)^{-1}\omega_p+(pk+m)^{-1}(p[D]+[E]),$$
and the associated weight $e^{-2\varphi}$ is locally integrable on
$X\ssm\{$finite set$\}$. Hence we can apply corollary~3.2 to conclude
that $H^q(X,K_X+L)=0$ for $q\ge 1$.\finpr

\titre 6. Seshadri constants of nef line bundles|

Let $L$ be a nef line bundle over a projective algebraic 
manifold~$X$. To every point $x\in X$, we attach the number
$$\varepsilon (L,x) = \inf_{C\ni x}~{L\cdot C\over \nu(C,x)}\leqno(6.1)$$
where the infimum is taken over all irreducible curves $C$ passing
through $x$ and $\nu(C,x)$ is the multiplicity of $C$ at $x$.
The infimum
$$\varepsilon (L) =\inf_{x\in X}~\varepsilon (L,x) = \inf_C~{L\cdot C
\over \nu(C)}~~~~\hbox{\rm where}~~\nu(C)=\max_{x\in C}~\nu(C,x)\leqno(6.1')$$
will be called the {\sl Seshadri constant} of $L$. It is well known
that $L$ is ample if and only if $\varepsilon(L)>0$ (Seshadri's criterion
[Ha~70]). For two nef line bundles $L_1,L_2$ we have
$\varepsilon(L_1+L_2)\ge \varepsilon(L_1)+\varepsilon(L_2)$,
in particular this shows again that $L_1+L_2$ is ample if 
$L_1$ or $L_2$ is ample.

If $L$ is a nef line bundle, we are especially interested in singular
metrics with isolated logarithmic poles: we say that a logarithmic pole 
$x$ of the weight $\varphi$ is {\sl isolated} if $\varphi$
is finite and continuous on $V\ssm \{x\}$ for some neighborhood $V$ of~$x$
and we define 
$$\gamma(L,x)=\sup\left.\cases{\hbox{\rm $\gamma\in\rb_+$ such that 
$L$ has a singular metric with $ic(L){\ge}0$}\cr
\hbox{\rm and with an isolated log pole of coefficient 
$\gamma$ at $x$}\cr}\right\};\leqno(6.2)$$
if there are no such metrics, we set $\gamma(L,x)=0$.

The numbers $\varepsilon(L,x)$ and $\gamma(L,x)$ will be seen to 
carry a lot of useful information about the global sections of 
$L$ and its multiples $kL$. To make this precise,
we first introduce some further definitions. Let
$s(L,x)$ be the largest integer $s\in\nb$ such that the global
sections in $H^0(X,L)$ generate all $s$-jets $J^s_xL = \oc_x(L)/
\mc_x^{s+1}\oc_x(L)$. If $L_x$ is not generated, i.e. if all
sections of $L$ vanish at $x$, we set $s(L,x)=-\infty$. We also
introduce the limit value
$$\sigma (L,x) = \limsup_{k\to +\infty} {1\over k} s(kL,x) =
\sup_{k\in\nb^\star} {1\over k} s(kL,x)\leqno(6.3)$$
if $s(kL,x)\ne-\infty$ for some $k$, and $\sigma(L,x)=0$ otherwise.
The limsup is actually equal to the sup thanks to the superadditivity
property 
$$s(L_1+L_2,x) \ge s(L_1,x) + s(L_2,x).$$
The limsup is in fact a limit as soon as $kL$ spans at $x$
for $k\ge k_0$, e.g. when $L$ is ample.

\tha Theorem 6.4|Let $L$ be a nef line bundle over $X$. For
every point $x\in X$ we have
$$\varepsilon(L,x)\ge\gamma(L,x)\ge\sigma(L,x).$$
If $L$ is ample, the equality holds for every $x\in X$. If $L$ is nef and big,
the equality holds outside any divisor $D$ prescribed by corollary 
{\rm 4.3}.\fintha

\remarque Proof|Fix a point $x\in X$ and a coordinate system $(z_1\ld z_n)$
centered at $x$. If $s=s(kL,x)$, then $H^0(X,kL)$ 
generates all
$s$-jets at $x$ and we can find holomorphic sections $f_1\ld f_N$ whose
$s$-jets are all monomials $z^\alpha$, $|\alpha|=s$. We define a global
singular metric on $L$ by
$$|\xi | = \bigg( \sum_{1\le j\le N} |f_j(z)\cdot \xi
^{-k}|^2\bigg)^{-1/2k},~\xi \in L_z\leqno(6.5)$$
associated to the weight function $\varphi(z)={1\over 2k}\log\sum\big|
\theta(f_j(z))\big|^2$ in any trivialization $L_{\restriction\Omega}\simeq 
\Omega\times\cb$. Then $\varphi $ has an isolated logarithmic pole of
coefficient $s/k$ at $x$, thus
$$\gamma (L,x) \ge {1\over k} s(kL,x)$$
and in the limit we get $\gamma(L,x)\ge\sigma(L,x)$.

Now, suppose that $L$ has a singular metric with an isolated log pole of
coefficient $\ge\gamma$ at $x$. Set $c(L)={i\over\pi}\partial\op\varphi$ 
on a neighborhood $\Omega$ of $x$ and let $C$ be an
irreducible curve passing through $x$. Then all weight functions
associated to the metric of $L$ must be locally integrable along $C$
(since $\varphi $ has an isolated pole at $x$). We infer
$$L\cdot C=\int_C c(L)\ge\int_{C\cap\Omega}
{i\over\pi}\partial\op\varphi\ge\gamma\,\nu(C,x)$$
because the last integral is larger than the Lelong number of the
current $[C]$ with respect to the weight $\varphi $ (cf. [De~87]) and we
may apply the comparison theorem with the ordinary Lelong number
associated to the weight $\log|z-x|$. Therefore
$$\varepsilon (L,x) = \inf {L\cdot C\over \nu(C,x)} \ge \sup \gamma
=\gamma (L,x).$$

Finally, we show that $\sigma(L,x)\ge\varepsilon(L,x)$ when $L$ is
ample. This is done essentially by same arguments as in the proof 
of Seshadri's criterion, as explained in [Ha~70]. Consider 
the blow-up $\pi:\smash{\wt X}\to X$ 
at point $x$, the exceptional divisor $E=\pi^{-1}(x)$ and the 
line bundles $F_{p,q} =\oc(p\,\pi^\star L-q\,E)$ over $\smash{\wt X}$, 
where $p,q>0$. Recall that $\oc(-E)_{\restriction E}$ is the canonical 
line bundle $\oc_E(1)$ over $E\simeq\pb^{n-1}$, in particular we have
$E^n=\oc_E(-1)^{n-1}=(-1)^{n-1}$. For any irreducible curve
$\smash{\wt C}\subset\smash{\wt X}$, either $\smash{\wt C}\subset E$ and
$$F_{p,q}\cdot \wt C = \oc(-q\,E)\cdot \wt C = q\,\oc_E(1)\cdot\wt C=
q\deg\wt C$$
or $\pi (\wt C) = C$ is a curve and
$$F_{p,q}\cdot \wt C=p\,L\cdot C - q\,\nu(C,x) \ge \big(p-q/\varepsilon
(L,x)\big)L\cdot C.$$
Thus $F_{p,q}$ is nef provided that $p\ge q/\varepsilon (L,x)$.
Since $F_{p,q}$ is ample when $p/q$ is large, a simple interpolation 
argument shows that $F_{p,q}$ is ample for $p>q/\varepsilon(L,x)$. 
In that case, the Kodaira-Serre vanishing theorem gives
$$H^1(\wt X,k\,F_{p,q}) = H^1\big(\wt X,\oc(kp\,\pi^\star L-kq\,E)\big)=0$$
for $k$ large. Hence we get a surjective map
$$H^0(\wt X,kp\,\pi^\star L) \lraww  H^0\Big(\wt X,
\oc(kp\,\pi^\star L)\otimes(\oc/\oc(-kq\,E)\big)\Big)\simeq
J_x^{kq-1}(kp\,L),$$
that is, $H^0(X,kp\,L)$ generates all $(kq-1)$ jets at $x$.
Therefore $p>q/\varepsilon (L,x)$ implies $s(kp\,L,x) \ge kq-1$
for $k$ large, so $\sigma (L,x)\ge q/p$. At the limit we get
$\sigma(L,x)\ge\varepsilon(L,x)$.

Assume now that $L$ is nef and big and that $\varepsilon(L,x)>0$. By the 
proof of lemma~4.3, there exist an integer $k_0\ge 1$ and effective divisors
$A,D$ such that $k_0L\simeq A+D$ where $A$ is ample. Then $a\,\pi^\star A-E$ 
is  ample for $a$ large. Hence there are
integers $a,b>0$ such that $a\,\pi^\star A-b\,E-K_{\wt X}$
is ample. When $F_{p,q}$ is nef, the sum with any positive multiple 
$k\,F_{p,q}$ is still ample and the Akizuki-Nakano vanishing theorem gives
$$H^1(\wt X,k\,F_{p,q}+a\,\pi^\star A-b\,E)
=H^1\big(\wt X,(kp+k_0a)\,\pi^\star L-a\,\pi^\star D-(kq+b)E\big)=0$$
when we substitute $A=k_0L-D$. As above, this implies that we have a 
surjective map
$$H^0\big(X,(kp+k_0a)\,L-a\,D\big)\lraww
J^{kq+b-1}_x\big((kp+k_0a)\,L-a\,D\big)$$
when $p\ge q/\varepsilon(L,x)$. Since $\oc(-aD)\subset\oc$, we infer
$s\big((kp+k_0a)L,x\big)\ge kq+b-1$ at every point $x\in X\ssm D$
and at the limit $\sigma(L,x)\ge\varepsilon(L,x)$.\finpr

\remarque Remark 6.6|Suppose that $L$ is ample. The same arguments
show that if $\pi :\wt X \to X$ is the blow-up at two points
$x,y$ and if $E_x+E_y$ is the exceptional divisor,
then $F_{p,q} =p\,\pi^\star L-q\,E_x-E_y$ is
ample for $p>q/\varepsilon (L,x) + 1/\varepsilon (L,y)$. In that  case,
$H^0(X,kp\,L)$ generates $J^{kq-1}_x(kp\,L)\oplus
J^{k-1}_y(kp\,L)$ for $k$ large. Take $p>q/\varepsilon
(L,x)+1/\varepsilon (L)$ and let $y$ run over $X\ssm  \{x\}$.
For $k$ large, we obtain sections $f_j\in H^0(X,kp\,L)$ whose
jets at $x$ are all monomials $z^\alpha $, $|\alpha |=kq-1$, and with
no other common zeros. Moreover, formula (6.5) produces a metric on $L$
which is smooth and has positive definite curvature on $X\ssm 
\{x\}$, and which has a log pole of coefficient $(kq-1)/kp$ at $x$.
Therefore the supremum $\gamma (L,x) =\sup\{\gamma \}$ is always
achieved by metrics that are smooth and have positive definite curvature
on $X\ssm \{x\}$.

\remarque Remark 6.7|If $Y$ is a $p$-dimensional algebraic subset of $X$
passing through $x$, then
$$L^p\cdot Y\ge \varepsilon (L,x)^p \nu(Y,x)$$
where $L^p\cdot Y=\int_Yc_1(L)^p$ and $\nu(Y,x)$ is the multiplicity of
$Y$ at $x$ (equal by Thie's theorem [Th~67] to the Lelong number of the
integration current $[Y]$). If $L$ is ample, we can take a metric on $L$ 
which is smooth on $X\ssm \{x\}$ and defined on a neighborhood $\Omega $ 
of $x$ by a weight function $\varphi $ with a log pole of coefficient 
$\gamma $ at $x$. By the comparison theorem for Lelong numbers, we get
$$L^p\cdot Y \ge \int_{Y\cap \Omega } \Big({i\over \pi }
\partial\op \varphi \Big)^p \ge \gamma^p\nu(Y,x)$$
and $\gamma $ can be chosen arbitrarily close to $\varepsilon
(L,x)$. If $L$ is nef, we apply the inequality to
$k\,L+M$ with $M$ ample and take the limit as $k\to
+\infty$.\finpr

\medskip
The numbers $\varepsilon (L,x)$ and Seshadri's constant $\varepsilon
(L)=\inf \varepsilon (L,x)$ are especially interesting because they
provide effective results concerning the existence of sections of 
$K_X+L$. The following proposition illustrates this observation.

\tha Proposition 6.8|Let $L$ be a big nef line bundle over $X$. 

\noindent{\rm(a)}~~If $\varepsilon (L,x) > n+s$, then $H^0(X,K_X+L)$
generates all $s$-jets at $x$. 

\noindent{\rm(b)}~~If $\varepsilon (L) >2n$, then $K_X+L$ is very
ample.\fintha

\remarque Proof|By the proof of theorem~6.4, the line bundle 
$\pi^\star L-q\,E$ is nef for $q\le\varepsilon (L,x)$. Moreover,
its $n$-th self intersection is equal to $L^n+(-q)^nE^n=L^n-q^n$ and
as $L^n\ge\varepsilon(L,x)^n$ by remark 3.5, we see that
$\pi^\star L-q\,E$ is big for $q<\varepsilon (L,x)$. 
The Kawamata-Viehweg vanishing theorem~5.2 then gives
$$H^1(\wt X,K_{\wt X}+\pi^\star L-q\,E)=
H^1\big(\wt X,\pi^\star K_X+\pi^\star L-(q-n+1)E\big)=0,$$
since $K_{\wt X}=\pi^\star K_X+(n-1)E$. Thus we get a
surjective map
$$\cmalign{
H^0\big(\wt X,\pi^\star&K_X+\pi^\star L)&\lraww
H^0\big(\wt X,\pi^\star\oc(K_X&+ L)\otimes
\oc/\oc(-(q-n+1)E)\big)\cr
&\|&&\|\cr
\hfill H^0(X,&K_X+L)&\lraww\hfill J^{q-n}_x(&K_X+L)\cr}$$
provided that $\varepsilon (L,x)>q$. The first statement is proved. To
show that $K_X+L$ is very ample, we blow up at two points $x,y$. The
line bundle $\pi^\star L-n\,E_x-n\,E_y$ is ample for $1/\varepsilon
(L,x) + 1/\varepsilon (L,y) < 1/n$, a sufficient condition for this is
$\varepsilon (L)>2n$. Then we see that
$$H^0(X,K_X+L) \longrightarrow  (K_X+L)_x \oplus
(K_X+L)_y$$
is also surjective.\finpr\vskip8pt

These results are related to a conjecture of Fujita [Fu~88], asserting that
if $L$ is an ample line bundle, then $K_X+mL$ is spanned for
$m\ge n+1$ and very ample for $m\ge n+2$. The answer is positive for 
surfaces, thanks to I.~Reider's numerical criterion (a deep
extension of Bombieri's work [Bo~73] on pluricanonical
embeddings of surfaces of general type). Our paper [De~90] describes
a new method which gives partial results in the higher dimensional 
case. By proposition 6.8 above, we know that $K_X+mL$
generates $s$-jets for $m>(n+s)/\varepsilon(L)$ and is very ample
for $m>2n/\varepsilon(L)$. It is easy to see for example that 
$\varepsilon(L)\ge 1$ for any flag manifold, in which the ample cone 
has a very simple structure. In general, unfortunately, it seems to 
be a rather hard problem to compute the Seshadri constant $\varepsilon(L)$, 
even in the case of surfaces. An answer to the following question would be
urgently needed.

\tha Question 6.9|Given a projective algebraic manifold $X$, is there
always a universal lower bound for $\varepsilon(L)$ 
when $L$ runs over all ample line bundles of $X$~? In this case, 
is it possible to compute explicitly such a lower bound in terms of 
geometric invariants of $X$~?\fintha

\titre 7. Asymptotic estimates of cohomology groups|

Let $X$ be a compact K\"ahler manifold, $E$ a holomorphic vector bundle of
rank $r$ and $L$ a line bundle over~$X$. If $L$ is equipped with a smooth 
metric of curvature form $c(L)$, we define the $q$-index set of $L$ to be
the open subset
$$X(q,L)=\left\{x\in X~;~c(L)_x \hbox{~has~}
{\displaystyle q\atop\displaystyle n-q}~
{\hbox{negative eigenvalues}\atop\hbox{positive eigenvalues}}\right\},
~~~~0\le q\le n.\leqno(7.1)$$
It is shown in [De~85] that the cohomology groups
$H^q\big(X,E\otimes\oc(kL)\big)$ satisfy the asymptotic ``Morse inequalities"
$$h^q\big(X,E\otimes\oc(kL)\big)\le
r{k^n\over n!}\int_{X(q,L)} (-1)^q \big(c(L)\big)^n + o(k^n)~~~~
\hbox{\rm as}~~k\to+\infty.\leqno(7.2)$$
One difficulty in the application of this result is that the curvature
integral is in general quite uneasy to compute, since it is neither a
topological nor a holomorphic invariant. However, when $c_1(L)\in\Gamma_+$,
the results of [De~91] allow us to measure the distance of $L$ to the
ample cone $\Gamma_a$. In that case, a use of singular metrics combined with
the approximation theorem of [De~91] produces smooth metrics on $L$ for which
an explicit bound of the negative part of the curvature is known.
It follows that (7.2) gives an explicit upper bound of the cohomology
groups of $E\otimes\oc(kL)$ in terms of a polynomial in the first Chern
class~$c_1(L)$ (related techniques have already been used
in [Su] in a slightly different context). To state the precise result,
we need the notion of nefness for a real $(1,1)$ cohomology class
which does not necessarily belong to $NS_\rb(X)$: we say that $\{u\}\in
H^{1,1}(X)$ is nef if $\{u\}$ belongs to the closure of the convex cone
generated by classes of K\"ahler forms $\big($the so-called {\sl
K\"ahler cone} of $H^{1,1}(X)\,\big)$.

\tha Theorem 7.3|Suppose that there is a nef cohomology class $\{u\}$ in
$H^{1,1}(X)$ such that $c_1\big(\oc_{TX}(1)\big)+\pi^\star\{u\}$ is nef
over the hyperplane bundle $P(T^\star X)$. Suppose 
moreover that $L$ is equipped with a singular metric such that
$T=c(L)\ge 0$. For $p=1,2\ld n,n+1$ set
$$b_p=\inf\{c>0\,;\,\codim E_c(T)\ge p\},$$
with $b_{n+1}=\max_{x\in X}\nu(T,x)$. Then for any holomorphic
vector bundle $E$ of rank $r$ over~$X$ we have
$$h^q\big(X,E\otimes\oc(kL)\big)\le A_qr\,k^n+o(k^n)$$
where $A_q$ is the cup product
$$A_q={1\over q!\,(n-q)!}\big(b_{n-q+1}\{u\}\big)^q\cdot
\big(c_1(L)+b_{n-q+1}\{u\}\big)^{n-q}$$
in $H^{2n}(X,\rb)$, identified to a positive number.\fintha

\remarque Remark 7.4|When $X$ is projective algebraic and $\kappa(L)=n$,
the proof of (4.2b) shows  that $mL\simeq\oc(A+D)$ with $A$ ample and $D$
effective, for some $m\ge 1$. Then we can choose a singular metric on $L$
such that $T=c(L)=\omega+m^{-1}[D]$, 
where $\omega=m^{-1}c(A)$ is a K\"ahler metric. As $\nu(T,x)=m^{-1}\nu(D,x)$
at each point, the constants $b_j$ of theorem~7.3 are obtained
by counting the multiplicities of the singular points of $D\,$; 
for example, if $D$ only has isolated singularities, then $b_1=0$,
$b_2=\ldots=b_n=1/m$. Observe moreover that the nefness
assumption on $\oc_{TX}(1)$ is satisfied with $\{u\}=c_1(G)$ if $G$ is a nef
$\qb$-divisor such that $\oc(TX)\otimes\oc(G)$ is nef, e.g. if
$\oc(S^mTX)\otimes\oc(mG)$ is spanned by sections for some $m\ge 1$.

\remarque Proof of theorem {\rm 7.3}|By definition, we have
$0=b_1\le b_2\le\ldots\le
b_n\le b_{n+1}$, and for $c\in{}]b_p,b_{p+1}]$,
$E_c(T)$ has codimension $\ge p$ with some component(s) of codimension~$p$
exactly. Let $\omega$ be a fixed K\"ahler metric on $X$. By adding
$\varepsilon\omega$ to $u$ if necessary, we may assume that $u\ge 0$
and that $\oc_{TX}(1)$ has a smooth hermitian metric such that
$c\big(\oc_{TX}(1)\big)+\pi^\star u\ge 0$.

Under this assumption, the main approximation theorem of [De~91] shows
that the metric of $L$ can be approximated by a sequence of smooth metrics
such that the associated curvature forms $T_j$ satisfy the uniform lower bound 
$$T_j\ge -\lambda_j(x)\,u(x)-\varepsilon_j\,
\omega(x)\leqno(7.5)$$
where ${\lim\!\downarrow\,}_{j\to+\infty}\varepsilon_j=0$ and 
$(\lambda_j)_{j>0}$ is a decreasing sequence of
continuous functions on $X$ such that $\lim_{j\to+\infty}
\lambda_j(x)=\nu(T,x)$ at each point.

Estimate (7.2) cannot be used directly with $T=c(L)$ because wedge
products of currents do not make sense in general. Therefore, we
replace $c(L)$ by its approximations $T_j$ and try to find
an upper bound for the limit. 

\tha Lemma 7.6|Let $U_j=X(q,T_j)$ be the $q$-index set 
associated to $T_j$ and let $c$ be a positive number. On the open 
set $\Omega_{c,j}=\{x\in X\,;\,\lambda_j(x)<c\}$
we have
$$(-1)^q\oneb_{U_j}T_j^n\le{n!\over q!\,(n-q)!}
\big(c\,u+\varepsilon_j\,\omega\big)^q\wedge\big(T_j+c\,u
+\varepsilon_j\,\omega\big)^{n-q}.$$\fintha

\remarque Proof|Write $v=c\,u+\varepsilon_j\,\omega>0$ and let 
$\alpha_{1,j}\le\ldots
\le\alpha_{n,j}$ be the eigenvalues of $T_j$ with
respect to $v$ at each point. Then $T_j^n=
\alpha_{1,j}\ldots\alpha_{n,j}\,v^n$ and
$$v^q\wedge(T_j+v)^{n-q}={q!\,(n-q)!\over n!}
\sum_{1\le i_1<\ldots<i_{n-q}\le n}(1+\alpha_{i_1,j})\ldots
(1+\alpha_{i_{n-q},j})\,v^n.$$
On $\Omega_{c,j}$ we get $T_j\ge -v$ 
by inequality (7.5), thus $\alpha_{i,j}\ge -1$; moreover, we 
have $\alpha_1\le\ldots\le\alpha_q<0$ and $0<\alpha_{q+1}\le\ldots\le
\alpha_n$ on $U_j$. On $\Omega_{c,j}$ we thus find
$$0\le(-1)^q\oneb_{U_j}\alpha_{1,j}\ldots
\alpha_{n,j}\le
\oneb_{U_j}\alpha_{q+1,j}\ldots\alpha_{n,j}\le
(1+\alpha_{q+1,j})\ldots(1+\alpha_{n,j}),$$
therefore $(-1)^q\oneb_{U_j}T_j^n\le
\big(n!/q!\,(n-q)!\big)\,v^q\wedge(T_j+v)^{n-q}$.\finpr\vskip8pt

\remarque End of the proof of theorem {\rm 7.3}|Set $\Lambda=
\max_X\lambda_1(x)$. By lemma~7.6 applied with an arbitrary
$c>\Lambda$ we have
$$(-1)^q\oneb_{U_j}T_j^n\le{n!\over q!(n-q)!}
(\Lambda\,u+\varepsilon_1\omega)^q\wedge(T_j+\Lambda\,u+\varepsilon_1
\omega)^{n-q}~~~~\hbox{\rm on}~~X.$$
Then estimate (7.2) and lemma 7.6 again imply
$$\leqalignno{
h^q\big(X,E\otimes\oc(kL)\big)
&\le r{k^n\over n!}\int_X(-1)^q\oneb_{U_j}T_j^n+o(k^n)\cr
&\le{r\,k^n\over q!\,(n-q)!}\Big(\int_{\Omega_{c,j}}
(c\,u+\varepsilon_j\,\omega)^q\wedge(T_j+c\,u+\varepsilon_j\,\omega)^{n-q}\cr
&+\int_{X\ssm\Omega_{c,j}}
(\Lambda\,u+\varepsilon_1\omega)^q\wedge(T_j+\Lambda\,u+\varepsilon_1
\omega)^{n-q}\Big)+o(k^n).&(7.7)\cr}$$
Since $\lambda_j(x)$ decreases to $\nu(T,x)$ as 
$j\to+\infty$, the set $X\ssm\Omega_{c,j}$ decreases
to $E_c(T)$. Now, $T_j+\Lambda\,u+\varepsilon_1\omega$ is a closed
positive $(1,1)$-form belonging to a fixed cohomology class, so the mass of
any wedge power $(T_j+\Lambda\,u+\varepsilon_1\omega)^p$ with respect
to $\omega$ is constant. By weak compactness, there is a subsequence
$(j_\nu)$ such that
$(T_{j_\nu}+\Lambda\,u+\varepsilon_1\omega)^p$ converges 
weakly to a closed positive current $\Theta_p$ of bidegree $(p,p)$,
for each $p=1\ld n$.
For $c>b_{p+1}$, we have $\codim E_c(T)\ge p+1$, hence
$\oneb_{E_c(T)}\Theta_p=0$. It follows that the integral
over $X\ssm\Omega_{c,j}$ in (7.7) converges to $0$ when 
$c>b_{n-q+1}$. For the same reason the integral over 
$\Omega_{c,j}$ converges to the same limit as
its value over $X$: observe that 
$(T_j+c\,u+\varepsilon_j\,\omega)^{n-q}$ can be
expressed in terms of powers of $u,\omega$ and of the positive forms
$(T_j+\Lambda\,u+\varepsilon_1\omega)^p$ with $p\le n-q\,$; thus
the limit is a linear combination with smooth coefficients of the 
currents $\Theta_p$, which carry no mass on $E_c(T)$. In the limit,
we obtain
$$h^q\big(X,E\otimes\oc(kL)\big)\le
{r\,k^n\over q!\,(n-q)!}(c\{u\})^q\cdot\big(c_1(L)+c\{u\}\big)^{n-q}+o(k^n),$$
and since this is true for every $c>b_{n-q+1}$, theorem~7.3
follows.\finpr\vfill\eject

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\vskip10pt
\tenpoint\noindent(August 17, 1990,~ revised on December 21, 1990)

\end
