%% ICM94 Z\"urich
%% Plain-TeX file of J.-P. Demailly's 45mn Lecture
%% New version 09/08/94q
%% $L^2$ methods and effective results in Algebraic Geometry

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%% Main Text

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\noindent
{\fourteenpointbf $L^2$-Methods and Effective} 
\medskip
\noindent
{\fourteenpointbf 
Results in Algebraic Geometry}
\titlerunning{$L^2$-Methods and Effective Results in Algebraic Geometry}
\vskip20pt

\noindent
{\twelvebf Jean-Pierre Demailly}
\vskip20pt

\noindent{\twelverm Universit\'e de Grenoble I, Institut Fourier}
\smallskip
\noindent{\twelverm URA 188 du CNRS, BP74, F-38402 Saint-Martin d'H\`eres, 
France}
\vskip50pt

{\eightpoint
\noindent{\bf Abstract.} 
One important problem arising in algebraic geometry is
the computation of effective bounds for the degree of embeddings in
a projective space of given algebraic varieties. This problem is intimately
related to the following question: Given a positive (or ample) line bundle
$L$ on a projective manifold~$X$, can one compute explicitly an integer
$m_0$ such that $mL$ is very ample for $m\ge m_0\,$? It turns out that the
answer is much easier to obtain in the case of adjoint line bundles
$2(K_X+mL)$, for which universal values of $m_0$ exist. We indicate
here how such bounds can be derived by a combination of powerful
analytic methods: theory of positive currents and plurisubharmonic
functions (Lelong), $L^2$ estimates for $\dbar$ (Andreotti-Vesentini,
H\"ormander, Bombieri, Skoda), Nadel vanishing theorem,
Aubin-Calabi-Yau theorem, and holomorphic Morse inequalities.
}
\vskip20pt

\section{Basic concepts of hermitian differential geometry}

Let $X$ be a complex manifold of dimension $n$ and let $F$ be a $C^\infty$
complex vector bundle of rank $r$ over $X$. A~{\it connection} $D$ on $F$ is a
linear differential operator $D$ acting on spaces
$C^\infty(X,\Lambda^{p,q}T^\star_X\otimes F)$ of $F$-valued differential
forms, increasing the degree by $1$ and satisfying Leibnitz' rule
$$D(f\wedge u)= df\wedge u + (-1)^{\deg\,f}f\wedge Du$$
for all forms $f\in C^\infty(X,\Lambda^{a,b}T^\star_X)$,
$u\in C^\infty(X,\Lambda^{p,q}T^\star_X\otimes F)$. As usual, we split
$D=D'+D''$ into its $(1,0)$ and $(0,1)$ parts, where
$$D'+D'':C^\infty(X,\Lambda^{p,q}T^\star_X\otimes F)\lra
C^\infty(X,\Lambda^{p+1,q}T^\star_X\otimes F)\oplus
C^\infty(X,\Lambda^{p,q+1}T^\star_X\otimes F).$$
With respect to a trivialization $\tau:F_{\restriction\Omega }\buildo\simeq
\over\lra\Omega\times\bC^r$, a connection $D$ can be written
$Du\simeq_\tau du+\Gamma \wedge u$, where $\Gamma=\Gamma'+\Gamma''$ is an
arbitrary $(r\times r)$-matrix of $1$-forms and $d$ acts componentwise.
A standard computation shows that $D^2u\simeq_\tau\Theta(D)\wedge u$,
where $\Theta(D)=d\Gamma+\Gamma\wedge\Gamma$ is a global $2$-form on $X$
with values in $\Hom(F,F)$. This form is called the {\it curvature tensor}
of $F$. In the important case of rank $1$ bundles, $\Theta(F)=d\Gamma$
is a $d$-closed form with complex values; it is well known that the De
Rham cohomology class of $\theta(F):={\ii\over2\pi}\Theta(F)=
{\ii\over2\pi}D^2$ is the image in De Rham cohomology of the first
Chern class $c_1(F)\in H^2(M,\bZ)$. For any line bundles $F_1\ld F_p$
on $X$ and any compact $p$-dimensional analytic set $Y$ in~$X$, we set
$$F_1\cdot\ldots\cdot F_p\cdot Y=\int_Yc_1(F_1)\wedge\ldots\wedge c_1(F_p).$$
If $F$ is equipped with a $C^\infty$ hermitian metric~$h$, the connection
$D$ is said to be {\it compatible with $h$} if 
$$d\langle u,v\rangle_h=\langle Du,v\rangle_h+\langle u,Dv\rangle_h$$
for all smooth sections $u$, $v$ of $F$. This is equivalent to the
antisymmetry condition $\Gamma^\star=-\Gamma$ (in a unitary frame),
i.e.\ $\Gamma''=-\Gamma^{\prime\star}$. In particular, a compatible
connection $D$ is uniquely determined by its $(0,1)$-component $D''$.
If $F$ has a {\it holomorphic structure}, we precisely have a canonical
$(0,1)$-connection $D''=\dbar$ obtained by letting $\dbar$ act
componentwise. Hence, there exists a unique $(1,0)$-connection $D'$
that makes \hbox{$D=D'+\dbar$} compatible with the hermitian metric.
This connection is called the {\it Chern connection}. Let $(e_\lambda)$
be a local holomorphic frame of $F_{\restriction\Omega }$ and let
\hbox{$H=(h_{\lambda\mu})$}, $h_{\lambda\mu}=\langle e_\lambda,
e_\mu\rangle$ be the hermitian matrix representing the metric. Standard
computations show that the Chern connection and curvature are given~by
$$D'\simeq_\tau\partial+\ol H^{-1}\partial\ol H\wedge\bu=
\ol H^{-1} \partial(\ol H\bu),\qquad
\Theta(F)=\dbar(\ol H^{-1}\partial\ol H).$$
In the special case where $F$ has rank $1$, it is convenient to
write the unique coefficient $H=h_{11}$ of the hermitian metric in the
form $H=e^{-2\varphi}$. The function $\varphi$ is called the {\it weight}
of the metric in the local coordinate patch $\Omega$. We then find
$\Theta(F)=2\partial\dbar\varphi$. It is important to observe that this
formula still makes sense in the context of distribution theory if
$\varphi$ is just an arbitrary $L^1_{\loc}$ function. As we shall see later,
the case of logarithmic poles is very important for the applications.

\claim{Definition.} A singular hermitian metric on a line
bundle $F$ is a metric given in any trivialization
$\tau:F_{\restriction\Omega}
\buildo\simeq\over\longrightarrow\Omega\times\bC$ by
$$\|\xi\|=|\tau(\xi)|\,e^{-\varphi(x)},\qquad x\in\Omega,~\xi\in F_x\,,$$
where $\varphi\in L^1_\loc(\Omega)$ is an arbitrary function. The
associated curvature current is
$$\theta(F)={\ii\over\pi}\partial\dbar\varphi.$$
\endclaim


\noindent The {\it Lelong-Poincar\'e} equation states that
${\ii\over\pi}\partial\dbar\log|f|=[D_f]$, where $f$ is a holomorphic
or meromorphic function  and $[D_f]$ is the current of integration over
the divisor of $f$. More generally, we have
$${\ii\over\pi}\partial\dbar\log\|\sigma\|=[D_\sigma]-\theta(F)$$
for every section $\sigma\in H^0(X,F)$, as follows from the equality
$\|\sigma\|=|f|e^{-\varphi}$, if $f=\tau(\sigma)$. As a consequence, the
De Rham cohomology class of $[D_\sigma]$ coincides with the first Chern
class $c_1(F)_\bR\in H^2_\DR(X,\bR)$.

\section{Positivity and ampleness}

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 $F$.
Let the curvature tensor of $F$ be
$$\Theta(F)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}dz_j\wedge dz_k\otimes e_\lambda^\star\otimes e_\mu.$$
Clearly, this tensor can be identified with a hermitian form on
$T_X\otimes F$, namely
$$\wt\Theta(F)(t)=\sum c_{jk\lambda\mu}t_{j\lambda}\ol{t_{k\mu}},\qquad
t=\sum t_{j\lambda}{\partial\over\partial z_j}\otimes e_\lambda\in
T_X\otimes F.$$

\claim{Definition} {\rm(Kodaira, Nakano, Griffiths){\bf .}} A holomorphic
vector bundle $F$~is
\item{$\bullet$} positive in the sense of Nakano if
$\wt\Theta(F)(t)>0$ for all nonzero tensors $t\in T_X\otimes F$;
\item{$\bullet$} positive in the sense of Griffiths if
$\wt\Theta(F)(\xi\otimes v)>0$ for all nonzero decomposable tensors
$\xi\otimes v\in T_X\otimes F$.
\medskip

\noindent%
In particular, a holomorphic line bundle $F$ is positive if and only if
its weights $\varphi$ are strictly plurisubharmonic $($psh$)$, i.e.\
if $(\partial^2\varphi/\partial z_j\dbar z_k)$ is positive definite.
\endclaim

\noindent%
\claim{Example.} \rm Let $D=\sum\alpha_jD_j$ be a divisor with 
coefficients $\alpha_j\in\bZ$ and let $F=\cO(D)$ be the associated
invertible sheaf of meromorphic functions $u$ such that
${\rm div}(u)+D\ge 0\,$; the corresponding line bundle can be
equipped with the singular metric defined by $\|u\|=|u|$. If $g_j$ is a
generator of the ideal of $D_j$ on an open set $\Omega\subset X$, then 
$\tau(u)=u\prod\smash{g_j^{\alpha_j}}$
defines a trivialization of $\cO(D)$ over $\Omega$; thus, our singular metric
is associated with the weight $\varphi=\sum\alpha_j\log|g_j|$. By the
Lelong-Poincar\'e equation, we find
$${\ii\over\pi}\Theta\big(\cO(D)\big)={\ii\over\pi}\partial\dbar
\varphi=[D]\ge 0,$$
where $[D]=\sum\alpha_j[D_j]$ denotes the current of integration over~$D$.
\endclaim

\noindent%
\claim{Example.} \rm Assume that $\sigma_1$, $\ldots$, $\sigma_N$ are
nonzero holomorphic sections of $F$. Then we can define a natural
(possibly singular) hermitian metric on $F$ by
$$
\|\xi\|^2={|\tau(\xi)|^2\over\sum_{1\le j\le N}|\tau(\sigma_j(x))|^2}
$$
with respect to any trivialization~$\tau$. The associated weight function is
$\varphi(x)=\log\big(\sum|\tau(\sigma_j(x))|^2
\big){}^{1/2}$. In this case $\varphi$ is a psh function;
thus, $\ii\Theta(F)$ is a closed positive current. Let us denote by
$\Sigma$ the linear system defined by $\sigma_1\ld \sigma_N$ and by
$B_\Sigma=\bigcap\sigma_j^{-1}(0)$ its base locus. Let
$$
\Phi_\Sigma:X\setminus B_\Sigma\to\bP^{N-1},\qquad
x\mapsto(\sigma_1(x):\sigma_2(x):\ldots:\sigma_N(x))
$$
be the associated map. Then
$\theta(F)={\ii\over 2\pi}\log(|\sigma_1|^2+\cdots+|\sigma_N|^2)$
is the pullback over $X\setminus B_\Sigma$ of the Fubini-Study metric
$\omega_\FS$ on $\bP^{N-1}$.
\endclaim

\claim{Definition.} A holomorphic line bundle $F$ over a compact complex
manifold $X$ is
\item{$\bullet$} very ample, if the map $\Phi_{|F|}:X\to\bP^{N-1}$ defined by
the complete linear system $|F|=P(H^0(X,F))$ is a regular embedding
$($this means in particular that $B_{|F|}=\emptyset)$;
\item{$\bullet$} ample, if $mF$ is very ample for some positive integer $m$.
\endclaim

\noindent Here we used an additive notation for
$\Pic(X)=H^1(X,\cO^\star)$, i.e.\ $mF=F^{\otimes m}$. By Example~(2.2),
every ample line bundle $F$ has a smooth hermitian metric with positive
definite curvature form; indeed, if $\Phi_{|mF|}$ is an embedding, then
we get a positive definite curvature form $\theta(F^{\otimes
m})=\Phi_{|mF|}^\star(\omega_\FS)$ and we need only extract the $m$th
root of this metric to get the desired smooth metric on~$F$. The converse
is also true:

\claim{Kodaira embedding theorem (1954).} A line bundle $F$ is ample if
and only if $F$ can be equipped with a smooth hermitian metric of positive
curvature.
\endclaim

\noindent In this context, Fujita [Fuj87] has raised the following important
conjecture.

\claim{Conjecture (Fujita, 1987).} If $L$ is an ample line bundle on a
projective $n$-fold~$X$, then $K_X+(n+1)L$ is globally generated and
$K_X+(n+2)L$ is very ample.
\endclaim

\noindent Here $K_X=\Lambda^n T^\star_X$ is the canonical bundle. The
example of curves shows that $K_X$ is needed to get a uniform answer
(if $L$ is a bundle of degree $1$ on a curve, then in general $mL$ does
not have any nonzero section unless $m\ge g={}$ genus). Also, the
example of projective spaces show that Fujita's bounds would be
optimal, because $K_{\bP^n}=\cO(-n-1)$.

Such questions have attracted a lot of attention in recent years.
First, the case of surfaces has been completely settled by Reider;
in [Rei88] he obtains a very sharp criterion for global generation and
very ampleness of line bundles in dimension~$2$. In higher dimensions, let
us mention [Dem90,$\,$93,$\,$94,$\,$95] and the works of Fujita
[Fuj87,$\,$94], Koll\'ar [Kol93], Ein-Lazarsfeld [EL92,$\,$93],
Lazarsfeld [Laz93], and Siu [Siu93,$\,$94]. Our goal is to describe a
few powerful analytic methods that are useful in this context.

\section{Bochner technique and vanishing theorems}

Let $X$ be a compact complex $n$-fold equipped with a
{\it K\"ahler metric}, namely a positive $(1,1)$-form
$\omega=\ii\sum\omega_{jk}dz_j\wedge d\ol z_k$ with $d\omega=0$.
Let $F$ be a holomorphic vector bundle on $X$ equipped with a hermitian
metric, and let
$$\Delta'=D'D^{\prime\star}+D^{\prime\star}D',\qquad
\Delta''=D''D^{\prime\prime\star}+D^{\prime\prime\star}D'',$$
be the complex Laplace operators associated with the Chern connection~$D$.
Here the adjoints $D^{\prime\star}$, $D^{\prime\prime\star}$ are the formal
adjoints computed with respect to the $L^2$ norm $\|u\|^2=\int_X
|u(x)|^2\;dV_\omega(x)$, where $|u|$ is the pointwise hermitian norm and
$dV_\omega=\omega^n/n!$ is the volume form. The fundamental results of Hodge
theory imply isomorphisms
$$H^q(X,\Omega^p_X\otimes F)=H^{p,q}_{\dbar}(X,F)\simeq\cH^{p,q}(X,F)$$
between sheaf cohomology groups, Dolbeault $\dbar$-cohomology groups, and the
space $\cH^{p,q}$ of harmonic $(p,q)$-forms $\Delta''u=0$.
The next fundamental fact is an identity originally used by Bochner to
prove vanishing results for Betti numbers. Slightly later, the identity
was extended to the complex situation by Kodaira and Nakano.

\claim{Bochner-Kodaira-Nakano formula (1954).} For all
$u=\sum u_{J,K,\lambda}dz_I\wedge d\ol z_J\otimes e_\lambda$
of class $C^\infty$ and type $(p,q)$, we have
$$\Delta''u=\Delta'u+A^{p,q}_{F,\omega} u,$$
where $A^{p,q}_{F,\omega}$ is the hermitian endomorphism such that
$\langle A^{p,q}_{F,\omega}u,u\rangle={}$
$$\sum c_{jk\lambda\mu}\,u_{J,jS,\lambda}\,\ol{u_{J,kS,\mu}}+
\sum c_{jk\lambda\mu}\,u_{kR,K,\lambda}\,\ol{u_{jR,K,\mu}}-
\sum c_{jj\lambda\mu}\,u_{J,K,\lambda}\,\ol{u_{J,K,\mu}},$$
and the summations are extended to all relevant indices $1\le j,k\le n$,
$1\le\lambda,\mu\le r$, and all relevant multiindices $|J|=p$, $|K|=q$,
$|R|=p-1$, $|S|=q-1$.
\endclaim

\noindent As $\langle\Delta'u,u\rangle=\|D'u\|^2+\|D^{\prime\star}u\|^2\ge0$
the Bochner-Kodaira-Nakano formula implies
$$\langle\Delta''u,u\rangle\ge\int_X\langle A^{p,q}_{F,\omega}u,u\rangle\;
dV_\omega.$$
If $A^{p,q}_{F,\omega}$ is positive definite, every $(p,q)$-harmonic form
has to vanish and we conclude that $H^q(X,\Omega^p_X\otimes F)=0$.
In the special case of rank $1$ bundles, we can take at each point $x\in X$
simultaneous diagonalizations
$$\omega(x)=\ii\sum dz_j\wedge d\ol z_j,\qquad
\Theta(F)(x)=\ii\sum\gamma_j(x)dz_j\wedge d\ol z_j,$$
where $\gamma_1(x)\le\cdots\le\gamma_n(x)$ are the curvature eigenvalues.
Then $c_{jj\lambda\mu}=\gamma_j$ and
$$\langle A^{p,q}_{F,\omega}u,u\rangle
=\sum_{J,K}\Big(\sum_{j\in K}\gamma_j-\sum_{j\notin J}\gamma_j\Big)|u_{JK}|^2
\ge(\gamma_1+\cdots+\gamma_q-\gamma_{n-p+1}-\cdots-\gamma_n)|u|^2.$$
Assume now that $\ii\Theta(F)$ is positive. The choice $\omega=\ii\Theta(F)$
yields $\gamma_j=1$ for $j=1,2\ld n$ and
$\langle A^{p,q}_{F,\omega}u,u\rangle=(p+q-n)|u|^2$. From this, we immediately
infer:

\claim{Akizuki-Kodaira-Nakano vanishing theorem (1954).}
If $F$ is a positive line bundle on a compact complex manifold~$X$, then
$$H^{p,q}(X,F)=H^q(X,\Omega^p_X\otimes F)=0\qquad\hbox{\it for $p+q\ge n+1$}.
$$
\endclaim

\noindent The above vanishing result is optimal. Unfortunately, it cannot be
extended to semipositive or numerically effective line bundles of bidegrees
$(p,q)$ with $p<n$, as shown by a counterexample of Ramanujam [Ram74].

\section{H\"ormander's $L^2$ estimates and existence theorems}

The basic existence theorem is the following result, which is essentially
due to H\"ormander [H\"o65] and, in a more geometric setting, to
Andreotti-Vesentini [AV65].

\claim{Theorem.} Let $(X,\omega)$ be a complete K\"ahler manifold.
Let $F$ be a hermitian vector bundle of
rank $r$ over $X$, and assume that $A=A^{p,q}_{F,\omega}$ is positive
definite everywhere on $\Lambda^{p,q}T^\star_X\otimes F$, $q\ge 1$.
Then for any form $g\in L^2(X,\Lambda^{p,q}T^\star_X\otimes F)$ with
$$D''g=0\quad\hbox{\it and}\quad
\int_X\langle(A^{p,q}_{F,\omega})^{-1}g,g\rangle\,dV_\omega<+\infty,$$
there exists a $(p,q-1)$-form $f$ such that $D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X\langle(A^{p,q}_{F,\omega})^{-1}g,g
\rangle\,dV_\omega.$$
\endclaim

\noindent The proof can be ultimately reduced to a simple duality argument
for unbounded operators on a Hilbert space, based on the a priori inequality
$$\|D''u\|^2+\|D^{\prime\prime\star}u\|^2\ge
\int_X\langle A^{p,q}_{F,\omega}u,u\rangle \;dV_\omega.$$
The above $L^2$ existence theorem can be applied in the fairly general
context of {\it weakly pseudoconvex} manifolds (i.e.\ manifolds
possessing a weakly psh exhaustion function), thanks to the fact that
every weakly pseudoconvex K\"ahler manifold $(X,\omega)$ carries a {\it
complete} K\"ahler metric. In particular, the existence theorem can be
applied on compact manifolds, pseudoconvex open sets in $\bC^n$,
Stein manifolds, etc. By regularization arguments, the existence
theorem also applies when $F$ is a line bundle and the hermitian metric is
a singular metric with positive curvature in the sense of currents. In
fact, the solutions obtained with the regularized metrics have weak
$L^2$ limits satisfying the desired estimates. Especially, we get the
following more tractable version in the case $p=n$.

\claim{Corollary.} Let $(X,\omega)$ be a K\"ahler weakly pseudoconvex
complex manifold of dimension~$n$. Let $F$ be a holomorphic line bundle
on~$X$, equipped with a singular metric whose local
weights $\varphi\in L^1_\loc$ satisfy
$\ii\Theta(F)=2\ii \partial\dbar\varphi\ge\varepsilon\omega$
for some $\varepsilon>0$. For every $g\in L^2(X,\Lambda^{n,q}
T^\star_X\otimes F)$ with $D''g=0$, there exists
\hbox{$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$}
such that $D''f=g$ and
$$\int_X|f|^2e^{-2\varphi}\,dV_\omega\le{1\over q\varepsilon}\int_X
|g|^2e^{-2\varphi}\,dV_\omega.
$$
\endclaim

\noindent This result leads in a natural way to the concept of {\it
multiplier ideal sheaves}, according to Nadel [Nad89]. The basic
idea was already implicit in the work of Bombieri [Bom70]
and Skoda [Sk72].

\claim{Multiplier ideal sheaves.} Let $\varphi$ be a psh function on an
open subset $\Omega\subset X$. We define $\cI(\varphi)\subset\cO_X$ to
be the sheaf of germs $f\in\cO_{\Omega,x}$ such that
$|f|^2e^{-2\varphi}$ is integrable on a small neighborhood $V$ of $x$ with
respect to the Lebesgue measure.
\endclaim

\claim{Main property {\rm([Nad89], [Dem93])}.} The ideal sheaf
$\cI(\varphi)\subset\cO_X$ is a coherent analytic sheaf. Its zero variety
$V(\cI(\varphi))$ is the set of points in a neighborhood of which
$e^{-2\varphi}$ is nonintegrable.
\endclaim

\noindent A basic observation is that the zero variety $V(\cI(\varphi))$
is closed related to the sublevel sets of Lelong numbers of~$\varphi$.

\claim{Definition.} The Lelong number of a psh function $\varphi$
at a point $x\in X$ is the limit
$\nu(\varphi,x):=\liminf_{z\to x}\varphi(z)/\log|z-x|$.
The function $\varphi$ is said to have a logarithmic pole of coefficient
$\gamma$ if $\gamma=\nu(\varphi,x)>0$.
\endclaim

\claim{Lemma {\rm ([Sk72])}.} Let $\varphi$ be psh on $\Omega$
and let $x\in\Omega$.
\item{$\bullet$} If $\nu(\varphi,x)<1$, then $e^{-2\varphi}$ is integrable 
near~$x$ $\Rightarrow$ $\cI(\varphi)_x=\cO_{\Omega,x}$.
\item{$\bullet$} If $\nu(\varphi,x)\ge n+s$, $s\in\bN$, then
$e^{-2\varphi}\ge C|z-x|^{-2n-2s}$ near $x$ and
$\cI(\varphi)_x\subset\gm_{\Omega,x}^{s+1}$.
\endclaim

\claim{Simple algebraic case.} {\rm Let
$\varphi=\sum\alpha_j\log|g_j|$, $\alpha_j\in\bQ^+$, be associated
with a normal crossing $\bQ$-divisor $D=\sum\alpha_jD_j$,
$D_j=g_j^{-1}(0)$. An easy computation gives
$$\cI(\varphi)=\cO(-\sum\lfloor\alpha_j\rfloor D_j)=\cO(-\lfloor D\rfloor),$$
where $\lfloor\alpha_j\rfloor={}$ the integral part of $\alpha_j$. If the
assumption on normal crossings is omitted, a desingularization of $D$ has to
be used in combination with the following fonctoriality property for direct
images.}
\endclaim

\claim{Basic fonctoriality property.} Let $\mu:X'\to X$ be a
modification $($i.e.\ a proper generically $1:1$ holomorphic map$)$, and let
$\varphi$ be a psh function on~$X$. Then 
$$\mu_\star\big(\cO(K_{X'})\otimes\cI(\varphi\circ\mu)\big)
=\cO(K_X)\otimes\cI(\varphi).$$
\endclaim

\noindent Let us now consider the case of general algebraic singularities
$$\varphi\sim{\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)$$
with $\alpha\in\bQ^+$ and $f_j$ holomorphic on an open set
$\Omega\subset X$. By Hironaka's theorem, there exists a smooth modification
$\mu:\wt X\to X$ of $X$ such that $\mu^\star(f_1\ld f_N)$ is an
invertible sheaf $\cO(-D)$ associated with a normal crossing divisor
$D=\sum\lambda_jD_j$. Then
$$\cI(\varphi)=\mu_\star\cO_{\wt X}\big(\sum(\rho_j-\lfloor\alpha\lambda_j
\rfloor)D_j\big),$$
where $R=\sum\rho_jD_j$ is the zero divisor of the jacobian $J_\mu$ of
the blow-up map. In this context, we get the following important vanishing
theorem, which can be seen as a generalization of the Kawamata-Viehweg
vanishing theorem (see [Kaw82], [Vie82], [EV86]).

\claim{Nadel vanishing theorem ([Nad89], [Dem93]).} Let
$(X,\omega)$ be a K\"ahler weakly pseudoconvex manifold, and let $F$ be a
holomorphic line bundle over $X$ equipped with a singular hermitian metric
of weight~$\varphi$. Assume that $\ii\Theta(F)\ge\varepsilon\omega$ for some
continuous positive function $\varepsilon$ on~$X$. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(\varphi)\big)=0\qquad\hbox{\it
for all $q\ge 1$.}$$
\endclaim

\noindent{\it Proof.} In virtue of H\"ormander's $L^2$ estimates
applied on small balls, the $\dbar$-complex of $L^2_\loc$ $(n,q)$-forms
is a (fine) resolution of the sheaf \hbox{$\cO(K_X+F)\otimes\cI(\varphi)$}.
The global $L^2$ cohomology is also zero by the $L^2$ estimates applied
globally on~$X$.\qed

\claim{Corollary.} Let $x_1\ld x_N$ be isolated points in the zero
variety~$V(\cI(\varphi))$. Then there is a surjective map
$$H^0(X,K_X+F)\lraww\bigoplus_{1\le j\le N}\big(\cO(K_X+L)\otimes
\cO_X/\cI(\varphi)\big)_{x_j}.$$
In particular, if the weight function $\varphi$ is such that
$\nu(\varphi,x)\ge n+s$ at some point $x\in X$ and $\nu(\varphi,y)<1$ at
nearby points, then $H^0(X,K_X+F)$ generates all~$s$-jets at~$x$.
\endclaim

\claim{Remark.} {\rm It is an easy exercise (left to the reader!) to show
that Corollary~(4.10) implies the Kodaira embedding theorem.}
\endclaim

\section{Numerical criteria for very ample line bundles}

The simplest approach to this problem is a recent technique due to [Siu94],
which rests merely on Nadel's vanishing theorem and the Riemann-Roch formula.
We formulate here a slightly improved version (see also [Dem94, 95]).

\claim{Theorem.} Let $L$ be an ample line bundle on a
projective $n$-fold~$X$. Let $x_j\in X$ and $s_j\in\bN$
be given, $1\le j\le N$. For
$$m\ge m_0=2+\sum_{1\le j\le N}{3n+2s_j-1\choose n}$$
$H^0(X,2K_X+mL)$ generates simultaneously jets of order $s_j$ at all
points~$x_j$. In~particular, $2K_X+mL$ is very ample for
$m\ge 2+{3n+1\choose n}$.
\endclaim

\noindent{\it Proof.} By a result of Fujita, $K_X+mL$ is ample for
$m\ge m_0$ (in fact Fujita has shown that $K_X+mL$ is nef for
$m\ge m+1$ and ample for $m\ge n+2$). The idea is to use a recursion
procedure for the construction of psh weights $(\varphi_\nu)_{\nu\ge 1}$
on $K_X+m_0L$ such that
\item{$(\alpha)$} the curvature of $K_X+m_0L$ is positive definite:
$\ii\partial\dbar\varphi_\nu\ge\varepsilon_\nu\omega$ for some
$\varepsilon_\nu>0$, where $\omega$ is the K\"ahler metric;
\item{$(\beta)$} $\nu(\varphi_\nu,x_j)\ge n+s_j$ for all $j$;
\item{$(\gamma)$} $\cI(\varphi_{\nu+1})\supsetneq\cI(\varphi_\nu)$
whenever $\dim V(\cI(\varphi_\nu))>0$.
\medskip

\noindent%
Indeed, Nadel's vanishing theorem implies
$$H^q(X,\cO(2K_X+mL)\otimes\cO/\cI(\varphi_\nu))=0\qquad
\hbox{\rm for $m\ge m_0$ and $q\ge 1$.}$$
Hence, $h^0=\chi$ is large for some $m\in[m_0,2m_0-1]$, and the existence
of a section $\sigma$ vanishing at order $2(n+s_j)$ at all points $x_j$
follows by the Riemann-Roch formula and an elementary count of dimensions.
We then set inductively
$$\varphi_{\nu+1}=\log(e^{\varphi_\nu}+e^{(1-m/2m_0)\psi}|\sigma|^{1/2}),$$
where $\psi$ is a weight for a smooth metric of positive definite curvature
on~$L$. Condition $(\gamma)$ guarantees that the process stops after a
finite number of steps.\qed

One weak point of the above result is that large multiples of $L$
are required. Instead, we would like to find conditions on $L$ implying
that $2(K_X+L)$ is very ample. For this, we need a convenient measurement
of how large $L$ is.

\claim{Definition.} Let $L$ be a numerically effective line bundle, i.e.\
a line bundle such that $L^p\cdot Y\ge 0$ for all $p$-dimensional
subvarieties~$Y$. For every $S\subset X$, we set
$$\mu(L,S)=\min_{Y\cap S\ne\emptyset}(L^p\cdot Y)^{1/p},$$
where $Y$ runs over all $p$-dimensional subvarieties intersecting~$S$. The
main pro\-perties of this invariant are\/$:$
\item{$\bullet$} Linearity\/$:$ $\forall k\ge 0$,~ $\mu(kL,S)=k\,\mu(L,S)\,;$
\item{$\bullet$} Nakai-Mo\u\i{}shezon criterion\/$:$ $L$ is ample if
and only if $\mu(L,X)>0$.
\endclaim

\claim{Theorem ([Dem93]).} Let $s,m\in\bN$, $s\ge 1$, $m\ge 2$.
If $L$ is ample and satisfies
$$(m-1)\,\mu(L,X)\ge 6(n+s)^n-s,$$
then $2K_X+mL$ generates $s$-jets.
Moreover, the result still holds with $6(n+s)^n$ replaced by $12n^n$ if
$s=1$; in particular, $2K_X+12n^nL$ is always very ample.
\endclaim

\noindent{\it Proof.} By Corollary (4.10), the main point is to
construct psh weights $\varphi$ that achieve the desired ideals
$\cI(\varphi)_{x_j}$ for the jets. This is done by solving a complex
Monge-Amp\`ere equation
$$\left(\omega+{\ii\over\pi}\partial\dbar\varphi\right)=f,\qquad
\omega=\theta(L),$$
where $f$ is a linear combination of Dirac measures $\delta_{x_j}$ and
of a uniform density with respect to~$\omega^n$. The solution $\varphi$
does exist by the Aubin-Calabi-Yau theorem, but in general, the poles of
$\varphi$ are not isolated. Hence, the Lelong numbers have to be estimated
precisely: this is indeed possible by means of intersection inequalities for
positive currents. We refer to [Dem93,$\,$94] for details.\qed

\section{Holomorphic Morse inequalities}

The starting point is the following differential geometric asymptotic
inequality, in which $X({\le}q,L)$ denotes the set of points $x\in X$ at
which $\theta_h(L)(x)$ has at most $q$ negative eigenvalues. The proof
is obtained by a careful study of the spectrum of the complex Laplace
operator $\Delta''$. See [Dem85,$\,$91] for details. 

\claim{Strong Morse inequalities ([Dem85]).} Let $X$ be a compact
complex $n$-fold and $(L,h)$ a hermitian line bundle. Then, as $k\to+\infty$,
$$\sum_{0\le j\le q}(-1)^{q-j}h^j\big(X,kL)\le 
{k^n\over n!}\int_{X(\le q,L)}(-1)^q\big(\theta_h(L)\big)^n+o(k^n).$$
\endclaim

\claim{Special case (algebraic version).} Let $L=F-G$, where $F$ and $G$
are numerically effective. Then for all $q=0,1,\ldots,n$,
$$\sum_{0\le j\le q}(-1)^{q-j}h^j(X,kL)\le
{k^n\over n!}\sum_{0\le j\le q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j
+o(k^n).$$
In particular, for $q=1$ we get
$$h^0(X,kL)-h^1(X,kL)\ge{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n).$$
\endclaim

\claim{Corollary.} If $F,G$ are nef and $F^n>0$, then $k(mF-G)$
has sections as soon as $m>n F^{n-1}\cdot G/F^n$ and $k\gg 0$.
\endclaim

\claim{Corollary.} If $F,G$ are nef and $F^n>0$, then
$H^0(X,K_X+mF-G)\ne 0$ for some $m\le n F^{n-1}\cdot G/F^n+n+1$.
\endclaim

\noindent {\it Proof.} Set $m_0:=\lfloor nF^{n-1}\cdot G/F^n\rfloor+1$.
By Corollary (6.3), $m_0F-G$ has a psh weight $\varphi$
with $\ii\partial\dbar\varphi\gg 0$; thus,
$H^q(X,\cO(K_X+mF-G)\otimes\cI(\varphi))=0$
for $q\ge 1$ and $m\ge m_0$. The Hilbert polynomial is thus equal to
$$h^0(X,\cO(K_X+mF-G)\otimes\cI(\varphi))\ge 0,$$
and it must be nonzero for some $m\in[m_0,m_0+n]$
because there are at most $n$ roots.\qed

A similar proof yields

\claim{Corollary.} If $F,G$ are nef with $F^n>0$, and $Y$ is a
$p$-dimensional subvariety, then $H^0(Y,\omega_Y\otimes\cO_Y(mF-G))\ne 0$
for some $m\le pF^{p-1}\cdot G\cdot Y/F^p\cdot Y+p+1$,
where $\omega_Y$ is the $L^2$ dualizing sheaf of~$Y$.
\endclaim

\noindent A proof by backward induction on $\dim Y$ then yields the
following effective version of the big Matsusaka theorem ([Mat72],
[KoM83]), improving Siu's result [Siu93].

\claim{Theorem ([Siu93], [Dem94, 95])} Let $F$ and
$G$ be nef line bundles on a projective $n$-fold~$X$. Assume that $F$
is ample and set $H=\lambda_n(K_X+(n+2)F)$ with $\lambda_2=1$ and
$\lambda_n={3n+1\choose n}-2n$ for $n\ge 3$. Then $mF-G$ is very ample for
$$m\ge(2n)^{(3^{n-1}-1)/2}{(F^{n-1}\cdot(G+H))^{(3^{n-1}+1)/2}
(F^{n-1}\cdot H)^{3^{n-2}(n/2-3/4)-1/4}\over(F^n)^{3^{n-2}(n/2-1/4)+1/4}}.$$
In particular $mF$ is very ample for
$$m\ge C_n~(F^n)^{3^{n-2}}\left(n+2+{F^{n-1}\cdot K_X\over F^n}
\right)^{3^{n-2}(n/2+3/4)+1/4}$$
with $C_n=(2n)^{(3^{n-1}-1)/2}(\lambda_n)^{3^{n-2}(n/2+3/4)+1/4}$.
\endclaim
\vskip30pt

\noindent
{\twelvebf References}
\bigskip

\bibitem{AV65}&A.\ Andreotti {\rm and} E.\ Vesentini,& {\it Carleman
   estimates for the Laplace-Beltra\-mi equation in complex
   manifolds},& Publ.\ Math.\ I.H.E.S. {\bf25} (1965) 81--130&

\bibitem{Bom70}&E.\ Bombieri,& {\it Algebraic values of meromorphic
   maps},& Invent.\ Math. {\bf10} (1970), 267--287; Addendum,
   Invent.\ Math. {\bf 11} (1970), 163--166&

\bibitem{Dem85}&J.-P.\ Demailly,& {\it Champs magn\'etiques et
   in\'egalit\'es de Morse pour la $d''$-coho\-mo\-logie},& Ann.\
   Inst.\ Fourier (Grenoble) {\bf 35} (1985) 189--229&

\bibitem{Dem89}&J.-P.\ Demailly,& {\it Une g\'en\'eralisation du
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   Sci.\ Paris S\'erie I Math. {\bf 309} (1989), 123--126&

\bibitem{Dem90}&J.-P.\ Demailly,& {\it Singular hermitian metrics on
   positive line bundles},& in: Proc.\ Complex Algebraic Varieties,
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 \bibitem{Dem91}&J.-P.\ Demailly,& {\it Holomorphic Morse
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