%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Analytic approach of the minimal model program
% and of the abundance conjectures
%
% Lecture notes by
% Jean-Pierre Demailly (Universit\'e de Grenoble I, Institut Fourier)
%
% Park City Mathematical Institute, July 2008
% PCMI Graduate Summer School held at Park City in July 2008 :
% Analytic and Algebraic Geometry: Common Problems-Different Methods
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\tableofcontents

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% The title of the lecture series.  Modify and uncomment.
\LectureSeries[Analytic approach of MMP and of abundance]%
{Analytic approach of the minimal model program and of
the abundance conjectures \author{Jean-Pierre Demailly}}

% The short-title is used in headers.  If your real title is short
% enough, this option is not needed.

% Author information.  Address and email is required.  
\address{Universit\'e de Grenoble I, Institut Fourier}
\curraddr{}
\email{demailly@fourier.ujf-grenoble.fr}
\date{July 31, 2009}
%%\thanks{}
%%\dedicatory{}

% Introductory material.	

\section*{Introduction}

The main purpose of these notes is to describe analytic techniques
which are useful to study questions such as linear series, multiplier
ideals and vanishing theorems for algebraic vector bundles. One
century after the ground-breaking work of Riemann on geometric aspects
of function theory, the general progress achieved in differential
geometry and global analysis on manifolds resulted into major advances
in the theory of algebraic and analytic varieties of arbitrary
dimension. One central unifying concept is the concept of positivity,
which can ve viewed either in algebraic terms (positivity of divisors
and algebraic cycles), or in more analytic terms (plurisubharmonicity,
hermitian connections with positive curvature). In this direction, one
of the most basic result is Kodaira's vanishing theorem for positive
vector bundles (1953-54), which is a deep consequence of the Bochner
technique and of the theory of harmonic forms initiated by W.V.D.~Hodge
during the 1940's. This method quickly led Kodaira to the well-known
embedding theorem for projective varieties, a far reaching extension
of Riemann's characterization of abelian varieties. Further
refinements of the Bochner technique led ten years later to the theory
of $L^2$ estimates for the Cauchy-Riemann operator, (J.J.~Kohn [Koh63,
64], Andreotti-Vesentini [AV65], [H\"or65]).  Not only vanishing
theorems can be proved of reproved in that manner, but perhaps more 
importantly, extremely precise information of a quantitative nature
is obtained about solutions of $\dbar$-equations, their zeroes, poles
and growth at infinity.

What makes the theory extremely flexible is the possibility to
formulate existence theorems with a wide assortment of different $L^2$ norms,
namely norms of the form $\int_X|f|^2e^{-2\varphi}$ where $\varphi$ is
a plurisubharmonic or strictly plurisubharmonic function on the
given manifold or variety~$X$. Here, the weight $\varphi$ need not be
smooth, and it is on the contrary extremely important to allow weights
which have logarithmic poles of the form
$\varphi(z)=c\log\sum|g_j|^2$, where $c>0$ and $(g_j)$ is a collection
of holomorphic functions possessing a common zero zet $Z\subset X$.
Following Nadel [Nad89],
one defines the {\it multiplier ideal sheaf} $\cI(\varphi)$ to be the
sheaf of germs of holomorphic functions $f$ such that
$|f|^2e^{-2\varphi}$ is locally summable.  Then $\cI(\varphi)$ is a
coherent algebraic sheaf over $X$ and $H^q(X,K_X\otimes
L\otimes\cI(\varphi))=0$ for all $q\ge 1$ if the curvature of $L$ is
positive as a current. This important result can be seen as a
generalization of the Kawamata-Viehweg vanishing theorem ([Kaw82],
[Vie82]), which is one of the cornerstones of higher dimensional
algebraic geometry, especially in relation with Mori's 
minimal model program.

In the dictionary between analytic geometry and algebraic geometry, the
ideal $\cI(\varphi)$ plays a very important role, since it directly converts
an analytic object into an algebraic one, and, simultaneously, takes care
of the singularities in a very efficient way. Another analytic tool used to
deal with singularities is the theory of positive currents introduced by
Lelong [Lel57]. Currents can be seen as generalizations of algebraic cycles,
and many classical results of intersection theory still apply to currents.
The concept of Lelong number of a current is the analytic analogue of the
concept of multiplicity of a germ of algebraic variety. Intersections
of cycles correspond to wedge products of currents (whenever these products
are defined).

Another very important result is the $L^2$ extension theorem by
Ohsawa-Takegoshi [OT87, Ohs88] (see also Manivel [Man93]). The main
statement is that every $L^2$ section $f$ of a suitably positive line
bundle defined on a subavariety $Y\subset X$ can be extended to a
$L^2$ section $\tilde f$ defined over the whole of~$X$. The positivity
condition can be understood in terms of the canonical sheaf and normal
bundle to the subvariety.  The extension theorem turns out to have an
incredible amount of important consequences: among them, let us
mention for instance Siu's theorem [Siu74] on the analyticity of
Lelong numbers, Skoda's division theorem for ideals of holomorphic functions,
a basic approximation theorem of closed positive
$(1,1)$-currents by divisors, the subadditivity property
$\cI(\varphi+\psi)\subset\cI(\varphi)\cI(\psi)$ of multiplier ideals
[DEL00], the restriction formula
$\cI(\varphi_{|Y})\subset\cI(\varphi)_{|Y}$, $\ldots$~. A suitable
combination of these results can be used to reprove Fujita's result
[Fuj94] on approximate Zariski decomposition, as detailed in
section~10. 

In section~11, we show how subadditivity can be used to derive 
an ``equisingular'' approximation theorem for (almost)
plurisubharmonic functions: any
such function can be approximated by a sequence of (almost) plurisubharmonic 
functions which are smooth outside an analytic set, and which define
the same multiplier ideal sheaves. From this, we derive a generalized version
of the {\it hard Lefschetz theorem} for cohomology with values in a
pseudo-effective line bundle; namely, the Lefschetz map is surjective
when the cohomology groups are twisted by the relevant multiplier ideal 
sheaves.

Section 12 explains the proof of Siu's theorem
on the invariance of plurigenera, according to a beautiful
approach developped by Mihai P\u{a}un [Pau07]. The proofs consists
of an iterative process based on the Ohsawa-Takegoshi theorem,
and a very clever limiting argument for currents.

Sections 13 to 15 are devoted to the study of positive cones in
K\"ahler or projective geometry. Recent ``algebro-analytic'' characterizations
of the K\"ahler cone ([DP04]) and the pseudo-effective cone of divisors
([BDPP04]) are explained in detail. This leads to a discussion
of the important concepts of volume and mobile intersections,
following S.Boucksom's PhD work [Bou02]. As a consequence, we show that
a projective algebraic manifold has a pseudo-effective canonical line bundle
if and only if it is not uniruled.

Section 16 presents some important ideas of H.~Tsuji, later refined
by Berndtsson and P\u{a}un, concerning the so-called
``super-canonical metrics'', and their
interpretation in terms of the invariance of plurigenera and of
the abundance conjecture. As the concluding section, we state
P\u{a}un's version of the Shokurov-Hacon-McKernan-Siu non vanishing
theorem and give an account of the very recent approach of the proof
of the finiteness of the canonical ring by Birkar-P\u{a}un [BiP09], based
on the ideas of Hacon-McKernan and Siu.

I would like to thank the organizers of the Graduate Summer School on
Analytic and Algebraic Geometry held at the Park City Mathematical
Institute in July 2008 for their invitation to give a series of
lectures, and thus for the opportunity of publishing these notes.

\section{Preliminary material}
\subsection*{\csect.A. Dolbeault cohomology and sheaf cohomology}
Let $X$ be a $\bC$-analytic manifold of dimension $n$. We denote by
$\Lambda^{p,q}T^\star_X$ the bundle of differential forms of bidegree
$(p,q)$ on $X$, i.e., differential forms which can be written as
$$u=\sum_{|I|=p,\,|J|=q}u_{I,J}dz_I\wedge d\overline z_J.$$
Here $(z_1,\ldots,z_n)$ denote arbitrary local holomorphic coordinates,
\hbox{$I=(i_1,\ldots,i_p)$}, \hbox{$J=(j_1,\ldots,j_q)$} are multiindices
(increasing sequences of integers in the range\break $[1,\ldots,n]$, of lengths
$|I|=p$, $|J|=q$), and
$$dz_I:=dz_{i_1}\wedge\ldots\wedge dz_{i_p},\qquad
d\overline z_J:=d\overline z_{j_1}\wedge\ldots\wedge d\overline z_{j_q}.$$
Let $\cE^{p,q}$ be the sheaf of germs of complex valued differential
$(p,q)$-forms with $C^\infty$ coefficients. Recall that the exterior
derivative $d$ splits as $d=d'+d''$ where
$$\eqalign{
d'u&=\sum_{|I|=p,\,|J|=q,1\le k\le n}{\partial u_{I,J}
\over\partial z_k}dz_k\wedge dz_I\wedge d\overline z_J,\cr
d''u&=\sum_{|I|=p,\,|J|=q,1\le k\le n}{\partial u_{I,J}
\over\partial\overline z_k}d\overline z_k\wedge dz_I\wedge d\overline z_J\cr}$$
are of type $(p+1,q)$, $(p,q+1)$ respectively. The well-known
Dolbeault-Grothendieck lemma asserts that any $d''$-closed
form of type $(p,q)$ with $q>0$ is locally $d''$-exact (this is the
analogue for $d''$ of the usual Poincar\'e lemma for~$d$, see e.g.\
[H\"or66]). In other words, the complex of sheaves
$(\cE^{p,\bu},d'')$ is exact in degree $q>0$; in degree $q=0$, $\Ker d''$
is the sheaf $\Omega^p_X$ of germs of holomorphic forms of degree $p$
on~$X$.

More generally, if $F$ is a holomorphic vector bundle
of rank $r$ over~$X$, there is a natural $d''$ operator acting
on the space $C^\infty(X,\Lambda^{p,q}T^\star_X\otimes F)$ of smooth
$(p,q)$-forms with values in~$F$; if $s=\sum_{1\le\lambda\le r}s_\lambda
e_\lambda$ is a $(p,q)$-form expressed in terms of a local holomorphic
frame of $F$, we simply define $d''s:=\sum d''s_\lambda\otimes e_\lambda$,
observing that the holomorphic transition matrices involved in changes
of holomorphic frames do not affect the computation of~$d''$.
It is then clear that the Dolbeault-Grothendieck lemma still
holds for $F$-valued forms. For every integer $p=0,1,\ldots,n$, the
{\it Dolbeault Cohomology} groups $H^{p,q}(X,F)$ are defined to be the
cohomology groups of the complex of global $(p,q)$ forms (graded by $q$):
$$H^{p,q}(X,F)=H^q\big(C^\infty(X,\Lambda^{p,\bu}T^\star_X\otimes F)\big).
\leqno(1.1)$$
Now, let us recall the following fundamental result from sheaf theory
(De Rham-Weil isomorphism theorem): let $(\cL^{\bu},d)$ be a resolution
of a sheaf $\cA$ by acyclic sheaves, i.e.\ a complex of sheaves
$(\cL^\bu,\delta)$ such that there is an exact sequence of sheaves
$$
0\longrightarrow\cA\buildo\longrightarrow^j \cL^0\buildo
\longrightarrow^{\delta^0}\cL^1\longrightarrow\cdots
\longrightarrow\cL^q\buildo \longrightarrow^{\delta^q}
\cL^{q+1}\longrightarrow\cdots~,
$$
and $H^s(X,\cL^q)=0$ for all $q\ge 0$ and $s\ge 1$. Then there
is a functorial isomorphism 
$$
H^q\big(\Gamma(X,\cL^\bu)\big)\longrightarrow H^q(X,\cA).\leqno(1.2)
$$
We apply this to the following situation: let $\cE(F)^{p,q}$ be the
sheaf of germs of $C^\infty$ sections of $\Lambda^{p,q}T^\star_X\otimes
F$. Then $(\cE(F)^{p,\bu},d'')$ is a resolution of the locally free
$\cO_X$-module $\Omega^p_X\otimes\cO(F)$ (Dolbeault-Grothendieck
lemma), and the sheaves $\cE(F)^{p,q}$ are acyclic as modules over the
soft sheaf of rings~$\cC^\infty$. Hence by (1.2) we get

\claim{(1.3) Dolbeault Isomorphism Theorem {\rm (1953)}} For every
holomorphic vector bundle $F$ on~$X$, there is a canonical isomorphism
$$H^{p,q}(X,F)\simeq H^q(X,\Omega^p_X\otimes\cO(F)).\eqno\square$$
\endclaim
%
If $X$ is projective algebraic and $F$ is an algebraic vector bundle,
Serre's GAGA theorem [Ser56] shows that the algebraic sheaf cohomology
group $H^q(X,\Omega^p_X\otimes\cO(F))$ computed with algebraic sections
over Zariski open sets is actually isomorphic to the analytic
cohomology group. These results are the most basic tools to attack
algebraic problems via analytic methods. Another important tool
is the theory of plurisubharmonic functions and positive
currents originated by K.~Oka and P.~Lelong in the decades 1940-1960.

\subsection*{\csect.B. Plurisubharmonic functions}
Plurisubharmonic functions have been introduced independently
by Lelong and Oka in the study of holomorphic convexity. We refer to
[Lel67, 69] for more details.

\claim{(1.4) Definition} A function $u:\Omega\longrightarrow[-\infty,+\infty[$
defined on an open subset $\Omega\subset\bC^n$ is said to be plurisubharmonic
$($psh for short$)$ if
\vskip3pt
\plainitem{\rm (a)} $u$ is upper semicontinuous~$;$
\vskip3pt
\plainitem{\rm (b)} for every complex line $L\subset\bC^n$, 
$u_{\restriction\Omega\cap L}$ is subharmonic on $\Omega\cap L$, that is,
for all $a\in\Omega$ and $\xi\in\bC^n$ with $|\xi|<d(a,\complement\Omega)$,
the function $u$ satisfies the mean value inequality
$$u(a)\le{1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta.$$
\vskip0pt\noindent
The set of psh functions on $\Omega$ is denoted by $\Psh(\Omega)$.
\endclaim
%
We list below the most basic properties of psh functions.
They all follow easily from the definition.

\claim{(1.5) Basic properties} \rm
\vskip3pt
\plainitem{\rm (a)} Every function $u\in\Psh(\Omega)$ is subharmonic, namely it
satisfies the mean value inequality on euclidean balls or spheres:
$$u(a)\le{1\over\pi^nr^{2n}/n!}\int_{B(a,r)}u(z)\,d\lambda(z)$$
for every $a\in\Omega$ and $r<d(a,\complement\Omega)$. Either $u\equiv-\infty$
or $u\in L^1_\loc$ on every connected component of~$\Omega$.
\vskip3pt
\plainitem{\rm (b)} For any decreasing sequence of psh
functions $u_k\in\Psh(\Omega)$, the limit $u=\lim u_k$ is psh
on $\Omega$.
\vskip3pt
\plainitem{\rm (c)} Let $u\in\Psh(\Omega)$ be such that $u\not\equiv-\infty$ 
on every connected component of~$\Omega$. If $(\rho_\varepsilon)$ is a 
family of smoothing kernels, then $u\star\rho_\varepsilon$ is $C^\infty$ and 
psh on
$$\Omega_\varepsilon=\big\{x\in\Omega\,;\,d(x,\complement\Omega)>\varepsilon
\big\},$$
the family $(u\star\rho_\varepsilon)$ is increasing in $\varepsilon$ and 
$\lim_{\varepsilon\to 0}u\star\rho_\varepsilon=u$.
\vskip3pt
\plainitem{\rm (d)} Let $u_1,\ldots,u_p\in\Psh(\Omega)$ and
$\chi:\bR^p\longrightarrow\bR$ be a convex function such that 
$\chi(t_1,\ldots,t_p)$ is increasing in each $t_j$. Then
$\chi(u_1,\ldots,u_p)$ is psh
on $\Omega$. In particular~ $u_1+\cdots+u_p$, $\max\{u_1,\ldots,u_p\}$, 
$\log(e^{u_1}+\cdots+e^{u_p})$ are psh on~$\Omega$.\qed
\vskip0pt
\endclaim
%
\claim{(1.6) Lemma} A function $u\in C^2(\Omega,\bR)$ is psh on
$\Omega$ if and only if the hermitian form 
$$Hu(a)(\xi)=\sum_{1\le j,k\le n}
\partial^2 u/\partial z_j\partial\overline z_k(a)\,\xi_j\overline \xi_k$$
is semi-positive at every point $a\in\Omega$.
\endclaim
%
\begproof{Proof.} This is an easy consequence of the following standard formula
$${1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta - u(a)=
{2\over\pi}\int_0^1{dt\over t}\int_{|\zeta|<t}Hu(a+\zeta\xi)(\xi)\,
d\lambda(\zeta),$$
where $d\lambda$ is the Lebesgue measure on $\bC$.
Lemma~1.6 is a strong evidence that plurisubharmonicity is the natural
complex analogue of linear convexity.\qed
\endproof
%
For non smooth functions, a similar characterization of plurisubharmonicity
can be obtained by means of a regularization process.

\claim{(1.7) Theorem} If $u\in\Psh(\Omega)$, $u\not\equiv-\infty$ on every
connected component of $\Omega$, then for all $\xi\in\bC^n$
$$Hu(\xi)=\sum_{1\le j,k\le n}{\partial^2 u\over\partial z_j\partial\overline z_k}
\,\xi_j\overline\xi_k\in\cD'(\Omega)$$
is a positive measure. Conversely, if $v\in\cD'(\Omega)$ is such that 
$Hv(\xi)$ is a positive measure for every $\xi\in\bC^n$, there exists a 
unique function $u\in\Psh(\Omega)$ which is locally integrable on $\Omega$
and such that $v$ is the distribution associated to $u$.\qed
\endclaim
%
In order to get a better geometric insight of this notion, we assume more
generally that $u$ is a function on a complex $n$-dimensional manifold 
$X$. If $\Phi:X\to Y$ is a holomorphic mapping and if $v\in C^2(Y,\bR)$, we
have $\hbox{$d'd''(v\circ \Phi)$}=\Phi^\star d'd''v$, hence
$$H(v\circ \Phi)(a,\xi)=Hv\big(\Phi(a),\Phi'(a).\xi\big).$$
In particular $Hu$, viewed as a hermitian form on $T_X$, does not depend
on the choice of coordinates $(z_1,\ldots,z_n)$. Therefore, the notion of
psh function makes sense on any complex manifold. More generally,
we have

\claim{(1.8) Proposition} If $\Phi:X\longrightarrow Y$ is a holomorphic map and
$v\in\Psh(Y)$, then $v\circ \Phi\in\Psh(X)$.\qed
\endclaim
%
\claim{(1.9) Example} \rm It is a standard fact that $\log|z|$ is psh
(i.e.\ subharmonic) on $\bC$. Thus $\log|f|\in\Psh(X)$ for every
holomorphic function $f\in H^0(X,\cO_X)$. More generally 
$$\log\big(|f_1|^{\alpha_1}+\cdots+|f_q|^{\alpha_q}\big)\in\Psh(X)$$
for every $f_j\in H^0(X,\cO_X)$ and $\alpha_j\ge 0$ (apply Property~1.5$\,$d
with $u_j=\alpha_j\,\log|f_j|$). We will be especially interested in the
singularities obtained at points of the zero variety $f_1=\ldots=f_q=0$,
when the $\alpha_j$ are rational numbers.\qed
\endclaim
%
\claim{(1.10) Definition} A psh function $u\in\Psh(X)$ will be said to
have analytic singularities if $u$ can be written locally as
$$u={\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)+v,$$
where $\alpha\in\bR_+$, $v$ is a locally bounded function and the $f_j$
are holomorphic functions. If $X$ is algebraic, we say that $u$ has
algebraic singularities if $u$ can be written as above on sufficiently
small Zariski open sets, with $\alpha\in\bQ_+$ and $f_j$ algebraic.
\endclaim
%
We then introduce the ideal $\cJ=\cJ(u/\alpha)$ of germs of holomorphic
functions $h$ such that $|h|\le Ce^{u/\alpha}$ for some constant~$C$, i.e.
$$|h|\le C\big(|f_1|+\cdots+|f_N|\big).$$
This is a globally defined ideal sheaf on $X$, locally equal to the
integral closure $\overline\cI$ of the ideal sheaf $\cI=(f_1,\ldots,f_N)$, 
thus $\cJ$ is coherent on~$X$. If $(g_1,\ldots,g_{N'})$ are local 
generators of $\cJ$, we still have
$$u={\alpha\over 2}\log\big(|g_1|^2+\cdots+|g_{N'}|^2\big)+O(1).$$
If $X$ is projective algebraic and $u$ has analytic singularities with
$\alpha\in\bQ_+$, then $u$ automatically has algebraic singularities.
From an algebraic point of view, the singularities of $u$ are in 1:1
correspondence with the ``algebraic data'' $(\cJ,\alpha)$. Later
on, we will see another important method for associating an ideal sheaf
to a psh function.

\claim{(1.11) Exercise} \rm Show that the above definition of the integral
closure of an ideal $\cI$ is equivalent to the following more algebraic
definition: $\overline\cI$ consists of all germs $h$ satisfying
an integral equation
$$h^d+a_1h^{d-1}+\ldots+a_{d-1}h+a_d=0,\qquad a_k\in\cI^k.$$
{\it Hint}. One inclusion is clear. To prove the other inclusion, consider
the normalization of the blow-up of $X$ along the (non necessarily reduced)
zero variety $V(\cI)$.\qed
\endclaim
%
\subsection*{\csect.C. Positive currents}
The reader can consult [Fed69] for a more thorough treatment of
current theory. Let us first recall a few basic definitions. A {\it
current} of degree $q$ on an oriented differentiable manifold $M$ is
simply a differential $q$-form $\Theta$ with distribution coefficients.
The space of currents of degree $q$ over $M$ will be denoted by
$\cD^{\prime q}(M)$. Alternatively, a current of degree $q$ can be seen
as an element $\Theta$ in the dual space $\cD'_p(M):=\big(\cD^p(M)\big)'$ of
the space $\cD^p(M)$ of smooth differential forms of degree $p=\dim M-q$ with
compact support; the duality pairing is given by
$$\langle \Theta,\alpha\rangle=\int_M \Theta\wedge\alpha,~~~~
\alpha\in\cD^p(M).\leqno(1.12)$$
A basic example is the {\it current of integration} $[S]$ over a compact
oriented submanifold $S$ of $M\,$:
$$\langle[S],\alpha\rangle=\int_S\alpha,~~~~{\rm deg}\alpha=p=\dim_\bR S.
\leqno(1.13)$$
Then $[S]$ is a current with measure coefficients, and
Stokes' formula shows that $d[S]=(-1)^{q-1}[\partial S]$, in particular
$d[S]=0$ if $S$ has no boundary. Because of this example, the integer $p$
is said to be the dimension of~$\Theta$ when $\Theta\in\cD_p'(M)$. The
current $\Theta$ is said to be {\it closed} if $d\Theta=0$.

On a complex manifold $X$, we have similar notions of bidegree and
bidimension; as in the real case, we denote by
$$\cD^{\prime p,q}(X)=\cD'_{n-p,n-q}(X),\qquad n=\dim X,$$
the space of currents of bidegree $(p,q)$ and bidimension $(n-p,n-q)$ on~$X$.
According to [Lel57], a current $\Theta$
of bidimension $(p,p)$ is said to be {\it $($weakly$)$ positive} if
for every choice of smooth $(1,0)$-forms $\alpha_1,\ldots,\alpha_p$ on~$X$
the distribution
$$\Theta\wedge\ii\alpha_1\wedge\overline\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\overline\alpha_p~~~~\hbox{\rm is a positive measure.}
\leqno(1.14)$$

\claim{(1.15) Exercise} \rm If $\Theta$ is positive, show that the
coefficients $\Theta_{I,J}$ of $\Theta$ are complex measures, and that,
up to constants, they are dominated by the trace measure
$$\sigma_\Theta=\Theta\wedge{1\over p!}\beta^p=2^{-p}\sum\Theta_{I,I},\qquad
\beta={\ii\over 2}d'd''|z|^2={\ii\over 2}\sum_{1\le j\le n}dz_j\wedge
d\overline z_j,$$
which is a positive measure.\newline
{\it Hint}. Observe that $\sum\Theta_{I,I}$ is invariant by unitary changes
of coordinates and that the $(p,p)$-forms
$\ii\alpha_1\wedge\overline\alpha_1\wedge\ldots\wedge\ii\alpha_p\wedge\overline\alpha_p$
generate $\Lambda^{p,p}T^\star_{\bC^n}$ as a $\bC$-vector space.\qed
\endclaim
%
A current $\Theta=\ii\sum_{1\le j,k\le n}\Theta_{jk}dz_j\wedge dz_k$
of bidegree $(1,1)$ is easily seen to be positive if and only if the
complex measure $\sum\lambda_j\overline\lambda_k \Theta_{jk}$ is a positive
measure for every $n$-tuple $(\lambda_1,\ldots,\lambda_n)\in\bC^n$.

\claim{(1.16) Example} \rm If $u$ is a (not identically $-\infty$)
psh function on $X$, we can associate with $u$ a (closed) positive
current $\Theta=\ii\partial\overline\partial u$ of bidegree~$(1,1)$. Conversely,
every closed positive current of bidegree $(1,1)$ can be written under
this form on any open subset $\Omega\subset X$ such that
$H^2_{DR}(\Omega,\bR)=H^1(\Omega,\cO)=0$, e.g.\ on small coordinate
balls (exercise to the reader).\qed
\endclaim
%
It is not difficult to show that a product $\Theta_1\wedge\ldots\wedge
\Theta_q$ of positive currents of bidegree $(1,1)$ is positive whenever
the product is well defined (this is certainly the case if all
$\Theta_j$ but one at most are smooth; much finer conditions will be
discussed in Section~2).

We now discuss another very important example of closed positive current.
In fact, with every closed analytic set $A\subset X$ of pure dimension $p$
is associated a current of integration
$$\langle [A],\alpha\rangle=\int_{A_\reg}\alpha,~~~~\alpha\in\cD^{p,p}(X),
\leqno(1.17)$$
obtained by integrating over the regular points of~$A$. In order to show
that (1.17) is a correct definition of a current on~$X$, one must show that
$A_\reg$ has locally finite area in a neighborhood of $A_\sing$.
This result, due to [Lel57] is shown as follows. Suppose that $0$
is a singular point of $A$. By the local parametrization theorem for
analytic sets, there is a linear change of coordinates on $\bC^n$
such that all projections
$$\pi_I:(z_1,\ldots,z_n)\mapsto (z_{i_1},\ldots,z_{i_p})$$
define a finite ramified covering of the intersection $A\cap\Delta$ with
a small polydisk $\Delta$ in $\bC^n$ onto a small
polydisk $\Delta_I$ in~$\bC^p$. Let $n_I$ be the sheet number. Then the
\hbox{$p$-dimensional} area of $A\cap\Delta$ is bounded above by the sum
of the areas of its projections counted with multiplicities, i.e.
$$\hbox{\rm Area}(A\cap\Delta)\le\sum n_I\hbox{\rm Vol}(\Delta_I).$$
The fact that $[A]$ is positive is also easy. In fact
$$\ii\alpha_1\wedge\overline\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\overline\alpha_p=|\det(\alpha_{jk})|^2\,
\ii w_1\wedge\overline w_1\wedge\ldots\wedge
\ii w_p\wedge\overline w_p$$
if $\alpha_j=\sum\alpha_{jk}dw_k$ in terms of local coordinates
$(w_1,\ldots,w_p)$ on $A_\reg$. This shows that all such forms are $\ge 0$
in the canonical orientation defined by
$\ii w_1\wedge\overline w_1\wedge\ldots\wedge\ii w_p\wedge\overline w_p$.
More importantly, Lelong [Lel57] has shown that $[A]$ is $d$-closed in~$X$,
even at points of $A_\sing$. This last result can be seen today as a
consequence of the Skoda-El Mir extension theorem. For this we need the
following definition: a {\it complete pluripolar} set is a set $E$ such
that there is an open covering $(\Omega_j)$ of $X$ and psh functions $u_j$ on
$\Omega_j$ with $E\cap\Omega_j=u_j^{-1}(-\infty)$. Any (closed) analytic
set is of course complete pluripolar (take $u_j$ as in Example~1.9). 

\claim{(1.18) Theorem {\rm(Skoda [Sko82], El Mir [EM84], Sibony [Sib85])}}
Let $E$ be a closed complete pluripolar set in $X$,
and let $\Theta$ be a closed positive current on $X\ssm E$ such that the
coefficients $\Theta_{I,J}$ of $\Theta$ are measures with locally finite
mass near~$E$. Then the trivial extension $\widetilde \Theta$ obtained by extending
the measures $\Theta_{I,J}$ by $0$ on $E$ is still closed on~$X$.
\endclaim
%
Lelong's result $d[A]=0$ is obtained by applying the Skoda-El Mir
theorem to $\Theta=[A_\reg]$ on $X\ssm A_\sing$. 

\begproof{Proof of Theorem 1.18.} The statement is local on $X$, so we may
work on a small open set $\Omega$ such that $E\cap\Omega=v^{-1}(-\infty)$,
$v\in\Psh(\Omega)$. Let $\chi:\bR\to\bR$ be a convex increasing function
such that $\chi(t)=0$ for $t\le -1$ and $\chi(0)=1$. By shrinking $\Omega$
and putting $v_k=\chi(k^{-1}v\star\rho_{\varepsilon_k})$ with
$\varepsilon_k\to 0$ fast, we get a sequence of functions
$v_k\in\Psh(\Omega)\cap C^\infty(\Omega)$ such that $0\le v_k\le 1$, $v_k=0$ in
a neighborhood of $E\cap\Omega$ and $\lim v_k(x)=1$ at every point of
$\Omega\ssm E$. Let $\theta\in C^\infty([0,1])$ be a
function such that $\theta=0$ on $[0,1/3]$, $\theta=1$ on $[2/3,1]$ and 
$0\le\theta\le 1$. Then $\theta\circ v_k=0$ near $E\cap\Omega$ and 
$\theta\circ v_k\to 1$ on $\Omega\ssm E$. Therefore 
$\tilde\Theta=\lim_{k\to+\infty}(\theta\circ v_k)\Theta$ and
$$d'\tilde\Theta=\lim_{k\to+\infty}\Theta\wedge d'(\theta\circ v_k)$$
in the weak topology of currents. It is therefore sufficient to verify that 
\hbox{$\Theta\wedge d'(\theta\circ v_k)$} converges weakly to $0$
(note that $d''\tilde \Theta$ is conjugate to $d'\tilde \Theta$, 
thus $d''\tilde \Theta$ will also vanish).

Assume first that $\Theta\in\cD^{\prime n-1,n-1}(X)$. Then
$\Theta\wedge d'(\theta\circ v_k)\in\cD^{\prime n,n-1}(\Omega)$, and we have
to show that
$$\langle \Theta\wedge d'(\theta\circ v_k),\overline\alpha\rangle=\langle\Theta,
\theta'(v_k)d'v_k\wedge\overline\alpha\rangle
\buildu\longrightarrow_{k\to+\infty} 0,~~~~\forall\alpha\in\cD^{1,0}(\Omega).$$
As $\gamma\mapsto\langle\Theta,\ii\gamma\wedge\overline\gamma\rangle$ is a
non-negative hermitian form on $\cD^{1,0}(\Omega)$, the Cauchy-Schwarz 
inequality yields
$$\big|\langle \Theta,\ii\beta\wedge\overline\gamma\rangle\big|^2\le
\langle \Theta,\ii\beta\wedge\overline\beta\rangle~
\langle \Theta,\ii\gamma\wedge\overline\gamma\rangle,~~~~
\forall\beta,\gamma\in\cD^{1,0}(\Omega).$$
Let $\psi\in\cD(\Omega)$, $0\le\psi\le 1$, be equal to $1$ in a neighborhood
of ${\rm Supp}\,\alpha$. We find
$$\big|\langle \Theta,\theta'(v_k)d'v_k\wedge\overline\alpha\rangle\big|^2\le
\langle \Theta,\psi \ii d'v_k\wedge d''v_k\rangle~
\langle \Theta,\theta'(v_k)^2 \ii\alpha\wedge\overline\alpha\rangle.$$
By hypothesis $\int_{\Omega\ssm E}\Theta\wedge \ii\alpha\wedge\overline\alpha<
+\infty$ and $\theta'(v_k)$ converges everywhere to $0$ on $\Omega$, thus
$\langle \Theta,\theta'(v_k)^2 \ii\alpha\wedge\overline\alpha\rangle$ converges
to $0$ by Lebesgue's dominated convergence theorem. On the other hand
$$\eqalign{
&\ii d'd''v_k^2=2v_k\,\ii d'd''v_k+2\ii d'v_k\wedge d''v_k\ge 2\ii d'v_k
\wedge d''v_k,\cr
&2\langle\Theta,\psi \ii d'v_k\wedge d''v_k\rangle\le\langle\Theta,\psi
\ii d'd''v_k^2\rangle.\cr}$$
As $\psi\in\cD(\Omega)$, $v_k=0$ near $E$ and $d\Theta=0$ on $\Omega\ssm E$,
an integration by parts yields
$$\langle \Theta,\psi \ii d'd''v_k^2\rangle=\langle \Theta,v_k^2
\ii d'd''\psi\rangle\le C\int_{\Omega\ssm E}\|\Theta\|<+\infty$$
where $C$ is a bound for the coefficients of $\ii d'd''\psi$. Thus 
$\langle \Theta,\psi \ii d'v_k\wedge d''v_k\rangle$ is bounded, 
and the proof is complete when $\Theta\in\cD^{\prime n-1,n-1}$.

In the general case $\Theta\in\cD^{\prime p,p}$, $p<n$, we simply apply the
result already proved to all positive currents $\Theta\wedge\gamma\in
\cD^{\prime n-1,n-1}$ where
$\gamma=\ii\gamma_1\wedge\overline\gamma_1\wedge\ldots\wedge\ii\gamma_{n-p-1,}
\wedge\overline\gamma_{n-p-1}$ runs over a basis of forms of
$\Lambda^{n-p-1,n-p-1}T^\star_\Omega$ with constant coefficients.
Then we get $d(\tilde \Theta\wedge\gamma)=d\tilde \Theta\wedge\gamma=0$ for
all such $\gamma$, hence $d\tilde \Theta=0$.\qed
\endproof
%
\claim{(1.19) Corollary} Let $\Theta$ be a closed positive current on
$X$ and let $E$ be a complete pluripolar set. Then $\bOne_E \Theta$
and $\bOne_{X\ssm E}\Theta$ are closed positive currents. In fact,
\hbox{$\widetilde\Theta=\bOne_{X\ssm E}\Theta$} is the trivial extension of
$\Theta_{\restriction X\ssm E}$ to~$X$, and
$\bOne_E\Theta=\Theta-\widetilde\Theta$.\qed
\endclaim
%
As mentioned above, any current $\Theta=\ii d'd''u$ associated with a psh
function $u$ is a closed positive $(1,1)$-current. In the special
case $u=\log|f|$ where $f\in H^0(X,\cO_X)$ is a non zero holomorphic
function, we have the important

\vbox{\claim{(1.20) Lelong-Poincar\'e equation}
Let $f\in H^0(X,\cO_X)$ be a non zero holomorphic function,
$Z_f=\sum m_jZ_j$, $m_j\in\bN$, the zero divisor of~$f$ and
$[Z_f]=\sum m_j[Z_j]$ the associated current of integration. Then
$${\ii\over\pi}\partial\overline\partial\log|f|=[Z_f].$$
\endclaim\removelastskip}

\begproof{Proof {\rm (Sketch)}.} 
It is clear that $\ii d'd''\log|f|=0$ in a neighborhood
of every point $x\notin\Supp(Z_f)=\bigcup Z_j$, so it is enough to check
the equation in a neighborhood of every point of $\Supp(Z_f)$. Let $A$ be
the set of singular points of $\Supp(Z_f)$, i.e.\ the union of the
pairwise intersections $Z_j\cap Z_k$ and of the singular loci
$Z_{j,\sing}$; we thus have $\dim A\le n-2$. In a neighborhood of any
point $x\in\Supp(Z_f)\ssm A$ there are local coordinates $(z_1,\ldots,z_n)$
such that $f(z)=z_1^{m_j}$ where $m_j$ is the multiplicity of $f$ along
the component $Z_j$ which contains~$x$ and $z_1=0$ is an equation for
$Z_j$ near~$x$. Hence
$${\ii\over\pi}d'd''\log|f|=m_j{\ii\over\pi}d'd''\log|z_1|=m_j[Z_j]$$
in a neighborhood of $x$, as desired (the identity comes from the
standard formula ${\ii\over\pi}d'd''\log|z|={}$ Dirac measure $\delta_0$
in $\bC$). This shows that the equation holds on $X\ssm A$. Hence
the difference ${\ii\over\pi}d'd''\log|f|-[Z_f]$ is a closed current of
degree $2$ with measure coefficients, whose support is contained in~$A$.
By Exercise~1.21, this current must be $0$, for $A$ has too small
dimension to carry its support ($A$ is stratified by submanifolds of real
codimension $\ge 4$).\qed
\endproof
%
\claim{(1.21) Exercise} \rm Let $\Theta$ be a current of degree $q$
on a real manifold~$M$, such that both $\Theta$ and $d\Theta$ have
measure coefficients (``normal current''). Suppose that $\Supp \Theta$
is contained in a real submanifold $A$ with $\codim_\bR A>q$. Show
that $\Theta=0$.\newline
{\it Hint:} Let $m=\dim_\bR M$ and let $(x_1,\ldots,x_m)$ be a coordinate
system in a neighborhood $\Omega$ of a point $a\in A$ such that
$A\cap\Omega=\{x_1=\ldots=x_k=0\}$, $k>q$. Observe that
$x_j\Theta=x_jd\Theta=0$ for $1\le j\le k$, thanks to the hypothesis on
supports and on the normality of~$\Theta$, hence $dx_j\wedge
\Theta=d(x_j\Theta)-x_jd\Theta=0$, $1\le j\le k$. Infer from this that
all coefficients in $\Theta=\sum_{|I|=q}\Theta_Idx_I$ vanish.\qed
\endclaim
%
We now recall a few basic facts of slicing theory (the reader will
profitably consult [Fed69] and [Siu74] for further developments).
Let \hbox{$\sigma:M\to M'$} be a submersion of smooth differentiable
manifolds and let $\Theta$ be a {\it locally flat} current on~$M$,
that is, a current which can be written locally as $\Theta=U+dV$
where $U$, $V$ have $L^1_\loc$ coefficients. It is a standard fact
(see Federer) that every current $\Theta$ such that both $\Theta$ and
$d\Theta$ have measure coefficients is locally flat; in particular,
closed positive currents are locally flat. Then, for almost every
$x'\in M'$, there is a well defined {\it slice} $\Theta_{x'}$,
which is the current on the fiber $\sigma^{-1}(x')$ defined by
$$\Theta_{x'}=U_{\restriction\sigma^{-1}(x')}+
  dV_{\restriction\sigma^{-1}(x')}.$$
The restrictions of $U$, $V$ to the fibers exist for almost all $x'$ by
the Fubini theorem. The slices $\Theta_{x'}$ are currents on the fibers with
the same degree as~$\Theta$ (thus of dimension $\dim \Theta-\dim
\hbox{(fibers)}$). Of course, every slice $\Theta_{x'}$ coincides with
the usual restriction of $\Theta$ to the fiber if $\Theta$ has smooth
coefficients. By using a regularization $\Theta_\varepsilon=\Theta\star
\rho_\varepsilon$, it is easy to show that the slices of a
closed positive current are again closed and positive: in fact
$U_{\varepsilon,x'}$ and $V_{\varepsilon,x'}$ converge to $U_{x'}$ and
$V_{x'}$ in $L^1_{\rm loc}(\sigma^{-1}(x'))$, thus
$\Theta_{\varepsilon,x'}$ converges weakly to $\Theta_{x'}$ for almost
every~$x'$. Now, the basic slicing formula is
$$\int_M \Theta\wedge\alpha\wedge\sigma^\star\beta=
\int_{x'\in M'}\Big(\int_{x''\in\sigma^{-1}(x')}\Theta_{x'}(x'')\wedge
\alpha_{\restriction\sigma^{-1}(x')}(x'')\Big)\beta(x')\leqno(1.22)$$
for every smooth form $\alpha$ on $M$ and $\beta$ on $M'$, such that
$\alpha$ has compact support and $\deg\alpha=\dim M-\dim M'-\deg \Theta$,
$\deg\beta=\dim M'$. This is an easy consequence of the usual Fubini
theorem applied to $U$ and $V$ in the decomposition $\Theta=U+dV$, if we
identify locally $\sigma$ with a projection map
$M=M'\times M''\to M'$, $x=(x',x'')\mapsto x'$, and use a partitition
of unity on the support of $\alpha$.

To conclude this section, we discuss De Rham and Dolbeault cohomology theory
in the context of currents. A basic observation is that the Poincar\'e and
Dolbeault-Grothendieck lemmas still hold for currents. Namely, if
$(\cD^{\prime q},d)$ and $(\cD'(F)^{p,q},d'')$ denote the complex of sheaves
of degree $q$ currents (resp.\ of $(p,q)$-currents with values in a
holomorphic vector bundle~$F$), we still have De Rham and Dolbeault sheaf
resolutions
$$0\to\bR\to\cD^{\prime \bu},\qquad 0\to\Omega^p_X\otimes\cO(F)\to
\cD'(F)^{p,\bu}.$$
Hence we get canonical isomorphisms
$$\leqalignno{
H^q_\DR(M,\bR)&=H^q\big((\Gamma(M,\cD^{\prime\bu}),d)\big),&(1.23)\cr
H^{p,q}(X,F)&=H^q\big((\Gamma(X,\cD'(F)^{p,\bu}),d'')\big).&{}\cr}$$
In other words, we can attach a cohomology class
$\{\Theta\}\in H^q_\DR(M,\bR)$ to any closed current $\Theta$ of degree~$q$,
resp.\ a cohomology class $\{\Theta\}\in H^{p,q}(X,F)$ to any $d''$-closed
current of bidegree~$(p,q)$. Replacing if necessary every current by a
smooth representative in the same cohomology class, we see that there is a
well defined cup product given by the wedge product of differential forms
$$\eqalign{
H^{q_1}(M,\bR)\times\ldots\times H^{q_m}(M,\bR)&\longrightarrow
H^{q_1+\ldots+q_m}(M,\bR),\cr
(\{\Theta_1\},\ldots,\{\Theta_1\})&\longmapsto\{\Theta_1\}\wedge\ldots\wedge
\{\Theta_m\}.\cr}$$
In particular, if $M$ is a compact oriented variety and
$q_1+\ldots+q_m=\dim M$, there is a well defined intersection number
$$\{\Theta_1\}\cdot\{\Theta_2\}\cdot\,\cdots\,\cdot
\{\Theta_m\}=\int_M\{\Theta_1\}\wedge\ldots\wedge\{\Theta_m\}.$$
However, as we will see in the next section, the pointwise product
\hbox{$\Theta_1\wedge\ldots\wedge\Theta_m$} need not exist in general.

\section{Lelong numbers and intersection theory}
\subsection*{\csect.A. Multiplication of currents and Monge-Amp\`ere operators}
Let $X$ be a $n$-dimensional complex manifold. We set
$$d^c={1\over2\ii\pi}(d'-d'').$$
It follows in particular that $d^c$ is a real operator, i.e. $\overline{d^cu}=
d^c\overline u$, and that $dd^c={\ii\over\pi}d'd''$. Although not quite standard,
the $1/2\ii\pi$ normalization is very convenient for many purposes, since
we may then forget the factor $\pi$ or $2\pi$ almost everywhere (e.g.\ in
the Lelong-Poincar\'e equation (1.20)).

Let $u$ be a psh function and let $\Theta$ be a closed
positive current on~$X$. Our desire is to define the wedge product
$dd^cu\wedge\Theta$ even when neither $u$ nor $\Theta$ are smooth.
In general, this product does not make sense because $dd^cu$ and $\Theta$
have measure coefficients and measures cannot be multiplied; see
Kiselman [Kis84] for interesting counterexamples. Even in the algebraic
setting considered here, multiplication of currents is not always possible:
suppose e.g.\ that $\Theta=[D]$ is the exceptional divisor of a blow-up
in a surface; then $D\cdot D=-1$ cannot be the cohomology class of a
closed positive current $[D]^2$. Assume however that
$u$ is a {\it locally bounded} psh function. Then the
current $u\Theta$ is well defined since $u$ is a locally bounded Borel
function and $\Theta$ has measure coefficients. According to
Bedford-Taylor [BT82] we define
$$dd^cu\wedge\Theta=dd^c(u\Theta)$$
where $dd^c(~~)$ is taken in the sense of distribution theory.

\claim{(2.1) Proposition} If $u$ is a locally bounded psh function, the
wedge product $dd^cu\wedge\Theta$ is again a closed positive current.
\endclaim
%
\begproof{Proof.} The result is local. Use a convolution $u_\nu=u\star\rho_{1/\nu}$
to get a decreasing sequence of smooth psh functions converging to~$u$.
Then write
$$dd^c(u\Theta)=\lim_{\nu\to+\infty}dd^c(u_\nu\Theta)=
dd^cu_\nu\wedge\Theta$$
as a weak limit of closed positive currents. Observe that $u_\nu\Theta$
converges weakly to $u\Theta$ by Lebesgue's monotone convergence theorem.\qed
\endproof
%
More generally, if $u_1,\ldots,u_m$ are locally bounded
psh functions, we can define
$$dd^cu_1\wedge\ldots\wedge dd^cu_m\wedge\Theta=
dd^c\big(u_1dd^cu_2\wedge\ldots\wedge dd^cu_m\wedge\Theta\big)$$
by induction on~$m$. Chern, Levine and Nirenberg [CLN69] noticed
the following useful inequality. Define the {\it mass} of a current
$\Theta$ on a compact set $K$ to be
$$||\Theta||_K=\int_K\sum_{I,J}|\Theta_{I,J}|$$
whenever $K$ is contained in a coordinate patch and
$\Theta=\sum\Theta_{I,J}dz_I\wedge d\overline z_J$. Up to seminorm equivalence,
this does not depend on the choice of coordinates. If $K$ is not contained
in a coordinate patch, we use a partition of unity to define a suitable
seminorm $||\Theta||_K$. If $\Theta\ge 0$, Exercise~1.15 shows that the mass
is controlled by the trace measure, i.e.\
$||\Theta||_K\le C\int_K\Theta\wedge\beta^p$.

\claim{(2.2.4) Chern-Levine-Nirenberg inequality}
For all compact subsets $K,L$ of $X$ with $L\subset K^\circ$, there exists
a constant $C_{K,L}\ge0$ such that
$$||dd^cu_1\wedge\ldots\wedge dd^cu_m\wedge\Theta||_L\le C_{K,L}~
||u_1||_{L^\infty(K)}\ldots||u_m||_{L^\infty(K)}\,||\Theta||_K$$
\endclaim
%
\begproof{Proof.} By induction, it is sufficient to prove the result for
$m=1$ and $u_1=u$. There is a covering of $L$ by a family of open balls
$B'_j\compact B_j\subset K$ contained in coordinate patches of~$X$. Let
$(p,p)$ be the bidimension of $\Theta$, let $\beta={\ii\over 2}d'd''|z|^2$,
and let $\chi\in\cD(B_j)$ be equal to $1$ on $\smash{\overline B'_j}$. Then
$$||dd^cu\wedge\Theta||_{L\cap\overline B'_j}\le C\int_{\overline B'_j}dd^cu\wedge
\Theta\wedge\beta^{p-1}\le C\int_{B_j}\chi\,dd^c u\wedge\Theta\wedge
\beta^{p-1}.$$
As $\Theta$ and $\beta$ are closed, an integration by parts yields
$$||dd^cu\wedge\Theta||_{L\cap\overline B'_j}\le C\int_{B_j}u\,\Theta\wedge
dd^c\chi\wedge\beta^{p-1}\le C'||u||_{L^\infty(K)}||\Theta||_K$$
where $C'$ is equal to $C$ multiplied by a bound for the coefficients
of the smooth form $dd^c\chi\wedge\beta^{p-1}$.\qed
\endproof
%
Various examples (cf.\ [Kis84]) show however that products of
$(1,1)$-currents $dd^cu_j$ cannot be defined in a reasonable way for
arbitrary psh functions~$u_j$. However, functions $u_j$
with $-\infty$ poles can be admitted if the polar sets are sufficiently
small.

\claim{(2.3) Proposition} Let $u$ be a psh function
on $X$, and let $\Theta$ be a closed positive current
of bidimension $(p,p)$. Suppose that $u$ is locally bounded on $X\ssm A$,
where $A$ is an analytic subset of $X$ of dimension $<p$ at each point.
Then $dd^cu\wedge\Theta$ can be defined in such a way that 
$dd^cu\wedge\Theta=\lim_{\nu\to+\infty}dd^cu_\nu\wedge\Theta$
in the weak topology of currents, for any decreasing sequence
$(u_\nu)_{\nu\ge 0}$ of psh functions converging to~$u$.
\endclaim
%
\begproof{Proof.} When $u$ is locally bounded everywhere, we have 
$\,\lim u_\nu\,\Theta=u\,\Theta$ by the monotone convergence 
theorem and the result follows from the continuity of $dd^c$ with
respect to the weak topology.

First assume that $A$ is discrete. Since our results are local, we 
may suppose that $X$ is a ball $B(0,R)\subset\bC^n$ and that $A=\{0\}$. 
For every $s\le 0$, the function $u^{\gge s}=\max(u,s)$
is locally bounded on $X$, so the product $\Theta\wedge 
dd^cu^{\gge s}$ is well defined. For $|s|$ large, the function
$\smash{u^{\gge s}}$ differs from $u$ only in a small neighborhood 
of the origin, at which $u$ may have a $-\infty$ pole. Let $\gamma$
be a $(p-1,p-1)$-form with constant coefficients and set
$s(r)=\liminf_{|z|\to r-0}u(z)$. By Stokes' formula, we see that
the integral
$$I(s):=\int_{B(0,r)}dd^cu^{\gge s}\wedge\Theta\wedge\gamma\leqno(2.4)$$
does not depend on $s$ when $s<s(r)$, for the difference $I(s)-I(s')$
of two such integrals involves the $dd^c$ of a current
$(u^{\gge s}-u^{\gge s'})\wedge\Theta\wedge\gamma$ with compact support
in $B(0,r)$. Taking $\gamma=(dd^c|z|^2)^{p-1}$, we see that the current
$dd^cu\wedge\Theta$ has finite mass on $B(0,r)\ssm\{0\}$
and we can define $\langle\bOne_{\{0\}}(dd^cu\wedge\Theta),
\gamma\rangle$ to be the limit of the integrals $(2.4)$ as $r$ tends to
zero and $s<s(r)$. In this case, the weak convergence statement is easily
deduced from the locally bounded case discussed above.
 
In the case where $0<\dim A<p$, we use a slicing technique to reduce the 
situation to the discrete case. Set $q=p-1$. There are linear
coordinates $(z_1,\ldots,z_n)$ centered at any point of $A$,
such that $0$ is an isolated point of $A\cap\big(\{0\}\times\bC^{n-q}\big)$.
Then there are small balls $B'=B(0,r')$ in $\bC^q$, $B''=B(0,r'')$ in
$\bC^{n-q}$ such that $A\cap(\smash{\overline B}'\times\partial B'')=
\emptyset$, and the projection map
$$\pi:\bC^n\to\bC^q,~~~~z=(z_1,\ldots,z_n)\mapsto z'=(z_1,\ldots,z_q)$$
defines a finite proper mapping $A\cap(B'\times B'')\to B'$.
These properties are preserved if we slightly change the direction of
projection. Take sufficiently many projections $\pi_m$ associated
to coordinate systems $(z^m_1,\ldots,z^m_n)$, \hbox{$1\le m\le N$}, in such
a way that the family of $(q,q)$-forms
$$i\,dz^m_1\wedge d\overline z^m_1\wedge\ldots\wedge i\,dz^m_q\wedge d\overline z^m_q$$
defines a basis of the space of $(q,q)$-forms. Expressing any
compactly supported smooth $(q,q)$-form in such a basis, we see
that we need only define
$$\leqalignno{
&\int_{B'\times B''}dd^cu\wedge\Theta\wedge f(z',z'')\,
\ii\,dz_1\wedge d\overline z_1\wedge\ldots\wedge\ii\,dz_q\wedge d\overline z_q=&(2.5)\cr
&\int_{B'}\Big\{\int_{B''}f(z',\bu)\,
dd^cu(z',\bu)\wedge\Theta(z',\bu)\Big\}
\ii\,dz_1\wedge d\overline z_1\wedge\ldots\wedge\ii\,dz_q\wedge d\overline z_q\cr}$$
where $f$ is a test function with compact support in $B'\times B''$,
and $\Theta(z',\bu)$ denotes the slice of $\Theta$ on the fiber
$\{z'\}\times B''$ of the projection \hbox{$\pi:\bC^n\to\bC^q$}.
Each integral $\int_{B''}$ in the right hand side of (2.5) makes sense
since the slices $(\{z'\}\times B'')\cap A$ are discrete.
Moreover, the double integral $\int_{B'}\int_{B''}$ is convergent.
Indeed, observe that $u$ is bounded on any compact cylinder
$$K_{\delta,\varepsilon}=\smash{\overline B}\big((1-\delta)r'\big)\times
\Big(\smash{\overline B}(r'')\ssm\smash{\overline B}\big((1-\varepsilon)r''\big)\Big)$$
disjoint from $A$. Take $\varepsilon\ll\delta\ll 1$ so small that
$${\rm Supp}\,f\subset\smash{\overline B}\big((1-\delta)r'\big)\times
\smash{\overline B}\big((1-\varepsilon)r''\big).$$
For all $z'\in\smash{\overline B}((1-\delta)r')$, the proof of the
Chern-Levine-Nirenberg inequality (2.2) with a cut-off function $\chi(z'')$
equal to $1$ on $B((1-\varepsilon)r'')$ and with support in
$B((1-\varepsilon/2)r'')$ shows that
$$\eqalign{
&\int_{B((1-\varepsilon)r'')}dd^cu(z',\bu)\wedge\Theta(z',\bu)\cr
&\qquad\quad{}\le C_\varepsilon ||u||_{L^\infty(K_{\delta,\varepsilon})}
\int_{z''\in B((1-\varepsilon/2)r'')}
\Theta(z',z'')\wedge dd^c|z''|^2.\cr}$$
This implies that the double integral is convergent. Now replace $u$
everywhere by $u_\nu$ and observe that $\lim_{\nu\to+\infty}\int_{B''}$
is the expected integral for every $z'$ such that $\Theta(z',\bu)$ exists
(apply the discrete case already proven). Moreover, the
Chern-Levine-Nirenberg inequality yields uniform bounds for all
functions $u_\nu$, hence Lebes\-gue's dominated convergence theorem can
be applied to $\int_{B'}$. We conclude from this that the sequence of
integrals (2.5) converges when \hbox{$u_\nu\downarrow u$}, as
expected.\qed
\endproof
%
\claim{(2.6) Remark} \rm In the above proof, the fact that $A$ is an
analytic set does not play an essential role. The main point is just
that the slices $(\{z'\}\times B'')\cap A$ consist of isolated points
for generic choices of coordinates $(z',z'')$. In fact, the proof even
works if the slices are totally discontinuous, in particular if they are
of zero Hausdorff measure~$\cH_1$. It follows that Proposition~2.3 still
holds whenever $A$ is a closed set such that $\cH_{2p-1}(A)=0$.\qed
\endclaim
%
\subsection*{\csect.B. Lelong numbers}
The concept of Lelong number is an analytic analogue of the algebraic
notion of multiplicity. It is a very useful technique to extend
results of the intersection theory of algebraic cycles to currents.
Lelong numbers have been introduced for the first time by Lelong in [Lel57].
See also [Lel69], [Siu74], [Dem82a, 85a, 87] for further developments.

Let us first recall a few definitions.
Let $\Theta$ be a closed positive current of bidimension $(p,p)$
on a coordinate open set $\Omega\subset\bC^n$ of a complex manifold~$X$.
The Lelong number of $\Theta$ at a point $x\in\Omega$ is defined to be
the limit
$$\nu(\Theta,x)=\lim_{r\to 0{\scriptscriptstyle+}}\nu(\Theta,x,r),\qquad
\hbox{where}~~
\nu(\Theta,x,r)={\sigma_\Theta(B(x,r))\over\pi^p r^{2p}/p!}$$
measures the ratio of the area of $\Theta$ in the ball $B(x,r)$ to the 
area of the ball of radius $r$ in $\bC^p$. As $\sigma_\Theta=\Theta\wedge
{1\over p!}(\pi dd^c|z|^2)^p$ by 1.15, we also get
$$\nu(\Theta,x,r)={1\over r^{2p}}\int_{B(x,r)}\Theta(z)\wedge
(dd^c|z|^2)^p.\leqno(2.7)$$
The main results concerning Lelong numbers are summarized in the following
theorems, due respectively to Lelong, Thie and Siu.

\claim{(2.8) Theorem {\rm([Lel57])}}
\vskip3pt
\plainitem{\rm (a)}  For every positive current $\Theta$, the
ratio $\nu(\Theta,x,r)$ is a nonnegative increasing function of $r$, in
particular the limit $\nu(\Theta,x)$ as $r\to0{\scriptstyle+}$
always exists.
\vskip3pt
\plainitem{\rm (b)} If $\Theta=dd^cu$ is the bidegree $(1,1)$-current associated
with a psh function~$u$, then
$$\nu(\Theta,x)=\sup\big\{\gamma\ge 0\,;\,u(z)\le\gamma\log|z-x|+O(1)
~~\hbox{at $x$}\big\}.$$
In particular, if $u=\log|f|$ with $f\in H^0(X,\cO_X)$ and
$\Theta=dd^cu=[Z_f]$, we have
$$\nu([Z_f],x)=\ord_x(f)=\max\{m\in\bN\,;\,D^\alpha f(x)=0,\,|\alpha|<m\}.$$
\vskip0pt
\endclaim
%
\claim{(2.9) Theorem {\rm([Thi67])}} In the case where $\Theta$
is a current of integration $[A]$ over an analytic subvariety $A$, the
Lelong number $\nu([A],x)$ coincides with the multiplicity of $A$ at $x$
$($defined e.g.\ as the sheet number in the ramified covering obtained by
taking a generic linear projection of the germ $(A,x)$ onto a
$p$-dimensional linear subspace through $x$ in any coordinate
patch~$\Omega)$.
\endclaim
%
\claim{(2.10) Theorem {\rm([Siu74])}} Let $\Theta$ be a closed positive
current of bidimension $(p,p)$ on the complex manifold~$X$.
\vskip3pt
\plainitem{\rm (a)} The Lelong number $\nu(\Theta,x)$ is invariant by holomorphic
changes of local coordinates.
\vskip3pt
\plainitem{\rm (b)} For every $c>0$, the set
$E_c(\Theta)=\big\{x\in X\,;\,\nu(\Theta,x)\ge c\big\}$
is a closed analytic subset of~$X$ of dimension $\le p$.\vskip0pt
\endclaim
%
The most important result is 2.10~b), which is a deep application of
H\"ormander $L^2$ estimates (see Section~5). The earlier proofs of all
other results were rather intricate in spite of their rather simple nature.
We reproduce below a sketch of elementary arguments based on the use of
a more general and more flexible notion of Lelong number introduced in
[Dem87]. Let $\varphi$ be a continuous psh function
with an isolated $-\infty$ pole at $x$, e.g.\ a function of the form
$\varphi(z)=\log\sum_{1\le j\le N}|g_j(z)|^{\gamma_j}$, $\gamma_j>0$,
where $(g_1,\ldots,g_N)$ is an ideal of germs of holomorphic functions in 
$\cO_x$ with $g^{-1}(0)=\{x\}$. The {\it generalized
Lelong number} $\nu(\Theta,\varphi)$ of $\Theta$ with respect to the
weight $\varphi$ is simply defined to be the mass of the measure
$\Theta\wedge(dd^c\varphi)^p$ carried by the point $x$
(the measure $\Theta\wedge(dd^c\varphi)^p$ is always well defined thanks
to Proposition~2.3). This number can also
be seen as the limit $\nu(\Theta,\varphi)=\lim_{t\to-\infty}
\nu(\Theta,\varphi,t)$, where
$$\nu(\Theta,\varphi,t)=\int_{\varphi(z)<t}\Theta\wedge
(dd^c\varphi)^p.\leqno(2.11)$$
The relation with our earlier definition of Lelong numbers (as well as part
a) of Theorem~2.8) comes from the identity
$$\nu(\Theta,x,r)=\nu(\Theta,\varphi,\log r),~~~~
\varphi(z)=\log|z-x|,\leqno(2.12)$$
in particular $\nu(\Theta,x)=\nu(\Theta,\log|\bu-x|)$. This equality is
in turn a consequence of the following general formula, applied to
$\chi(t)=e^{2t}$ and $t=\log r\,$:
$$\int_{\varphi(z)<t}\Theta\wedge(dd^c\chi\circ\varphi)^p=
\chi'(t-0)^p\int_{\varphi(z)<t}\Theta\wedge(dd^c\varphi)^p,
\leqno(2.13)$$
where $\chi$ is an arbitrary convex increasing function. To prove the
formula, we use a regularization and thus suppose that $\Theta$,
$\varphi$ and $\chi$ are smooth, and that $t$ is a non critical value of
$\varphi$. Then Stokes' formula shows that the integrals on the left and
right hand side of (2.13) are equal respectively to
$$\int_{\varphi(z)=t}\Theta\wedge(dd^c\chi\circ\varphi\big)^{p-1}
\wedge d^c(\chi\circ\varphi),~~~~
\int_{\varphi(z)=t}\Theta\wedge\big(dd^c\varphi\big)^{p-1}
\wedge d^c\varphi,$$
and the differential form of bidegree $(p-1,p)$ appearing in the integrand
of the first integral is equal to
$(\chi'\circ\varphi)^p\,(dd^c\varphi)^{p-1}\wedge d^c\varphi$.
The expected formula follows. Part (b) of Theorem 2.8 is a consequence of
the Jensen-Lelong formula, whose proof is left as an exercise to the reader.

\claim{(2.14) Jensen-Lelong formula} Let $u$ be any psh function 
on~$X$. Then $u$ is integrable with respect to the measure
$\mu_r=(dd^c\varphi)^{n-1}\wedge d^c\varphi$ supported by the
pseudo-sphere $\{\varphi(z)=r\}$ and
$$\mu_r(u)=\int_{\{\varphi<r\}}u(dd^c\varphi)^n+
\int_{-\infty}^r\nu(dd^cu,\varphi,t)\,dt.\eqno\square$$
\endclaim
%
In our case, we set $\varphi(z)=\log|z-x|$. Then $(dd^c\varphi)^n=\delta_x$
and $\mu_r$ is just the unitary invariant mean value measure on the
sphere $S(x,e^r)$. For $r<r_0$, Formula 2.14 implies
$$\mu_r(u)-\mu_{r_0}(u)=\int_{r_0}^r\nu(dd^cu,x,t)\sim
(r-r_0)\nu(dd^cu,x)\qquad\hbox{as $r\to-\infty$}.$$
From this, using the Harnack inequality for subharmonic functions, we get
$$\liminf_{z\to x}{u(z)\over\log|z-x|}=
\lim_{r\to-\infty}{\mu_r(u)\over r}=\nu(dd^cu,x).$$
These equalities imply statement 2.8~b).

Next, we show that the Lelong numbers $\nu(T,\varphi)$ only
depend on the asymptotic behaviour of $\varphi$ near the polar set
$\varphi^{-1}(-\infty)$. In a precise way:

\claim{(2.15) Comparison theorem} Let $\Theta$ be a closed positive
current on~$X$, and let $\varphi,\psi:X\to[-\infty,+\infty[$
be continuous psh functions with isolated poles at some point
$x\in X$. Assume that
$$\ell:=\limsup_{z\to x}{\psi(z)\over\varphi(z)}<+\infty.$$
Then $\nu(\Theta,\psi)\le \ell^p\nu(\Theta,\varphi)$, and the equality holds
if $\ell=\lim\psi/\varphi$.
\endclaim
%
\begproof{Proof.} (2.12) shows that
$\nu(\Theta,\lambda\varphi)=\lambda^p\nu(\Theta,\varphi)$
for every positive constant $\lambda$. It is thus sufficient to verify the
inequality $\nu(\Theta,\psi)\le\nu(\Theta,\varphi)$ under the hypothesis 
$\limsup\psi/\varphi<1$. For any $c>0$, consider the psh function 
$$u_c=\max(\psi-c,\varphi).$$
Fix $r\ll 0$. For $c>0$ large enough, we have $u_c=\varphi$ on a
neighborhood of $\varphi^{-1}(r)$ and Stokes' formula gives
$$\nu(\Theta,\varphi,r)=\nu(\Theta,u_c,r)\ge\nu(\Theta,u_c).$$
On the other hand, the hypothesis $\limsup\psi/\varphi<1$ implies that
there exists $t_0<0$ such that $u_c=\psi-c$ on $\{u_c<t_0\}$. We thus get 
$$\nu(\Theta,u_c)=\nu(\Theta,\psi-c)=\nu(\Theta,\psi),$$
hence $\nu(\Theta,\psi)\le\nu(\Theta,\varphi)$. The equality case is obtained
by reversing the roles of $\varphi$ and $\psi$ and observing that
$\lim\varphi/\psi=1/l$.\qed
\endproof
%
Part (a) of Theorem~2.10 follows immediately from 2.15 by considering
the weights $\varphi(z)=\log|\tau(z)-\tau(x)|$, $\psi(z)=\log
|\tau'(z)-\tau'(x)|$ associated to coordinates systems
$\tau(z)=(z_1,\ldots,z_n)$, $\tau'(z)=(z'_1,\ldots,z'_n)$ in a neighborhood
of~$x$. Another application is a direct simple proof of Thie's Theorem
2.9 when $\Theta=[A]$ is the current of integration over an
analytic set $A\subset X$ of pure dimension~$p$. For this, we have to
observe that Theorem 2.15 still holds provided that $x$ is an isolated
point in $\Supp(\Theta)\cap\varphi^{-1}(-\infty)$ and
$\Supp(\Theta)\cap\psi^{-1}(-\infty)$ (even though $x$ is not isolated in
$\varphi^{-1}(-\infty)$ or $\psi^{-1}(-\infty)$), under the weaker assumption
that $\limsup_{\Supp(\Theta)\ni z\to x}\psi(z)/\varphi(z)=\ell$. The reason
for this is that all integrals involve currents supported on $\Supp(\Theta)$.
Now, by a generic choice of local coordinates \hbox{$z'=(z_1,\ldots,z_p)$}
and \hbox{$z''=(z_{p+1},\ldots,z_n)$} on $(X,x)$, the germ $(A,x)$ is contained
in a cone \hbox{$|z''|\le C|z'|$}. If
$B'\subset\bC^p$ is a ball of center $0$ and radius $r'$ small,
and \hbox{$B''\subset\bC^{n-p}$} is the ball of center $0$ and
radius~$r''=Cr'$, the projection
      $$\pr:A\cap(B'\times B'')\longrightarrow B'$$
is a ramified covering with finite sheet number $m$.
When $z\in A$ tends to $x=0$, the functions
$$\varphi(z)=\log|z|=\log(|z'|^2+|z''|^2)^{1/2},~~~~\psi(z)=\log|z'|.$$
satisfy $\lim_{z\to x}\psi(z)/\varphi(z)=1$. Hence Theorem~2.15 implies
      $$\nu([A],x)=\nu([A],\varphi)=\nu([A],\psi).$$
Now, Formula 2.13 with $\chi(t)=e^{2t}$ yields
$$\eqalign{\nu([A],\psi,\log t)
&=t^{-2p}\int_{\{\psi<\log t\}}[A]\wedge\Big({1\over 2}dd^ce^{2\psi}\Big)^p\cr
&=t^{-2p}\int_{A\cap\{|z'|<t\}}\Big({1\over 2}\pr^\star dd^c|z'|^2\Big)^p\cr
&=m\,t^{-2p}\int_{\bC^p\cap\{|z'|<t\}}\Big({1\over 2}dd^c|z'|^2\Big)^p=m,
\cr}$$
hence $\nu([A],\psi)=m$.  Here, we have used the fact that pr is an \'etale
covering with $m$ sheets over the complement of the ramification locus
$S\subset B'$, and the fact that $S$ is of zero Lebesgue measure in~$B'$.

\claim{(2.16) Proposition} Under the assumptions of Proposition 2.3,
we have
$$\nu(dd^cu\wedge\Theta,x)\ge\nu(u,x)\,\nu(\Theta,x)$$
at every point $x\in X$.
\endclaim
%
\begproof{Proof.} Assume that $X=B(0,r)$ and $x=0$. By definition
$$\nu(dd^c u\wedge\Theta,x)=
\lim_{r\to 0}\int_{|z|\le r}dd^cu\wedge\Theta\wedge(dd^c\log|z|)^{p-1}.$$
Set $\gamma=\nu(u,x)$ and
$$u_\nu(z)=\max\big(u(z),(\gamma-\varepsilon)\log|z|-\nu\big)$$
with $0<\varepsilon<\gamma$ (if $\gamma=0$, there is nothing to prove).
Then $u_\nu$ decreases to $u$ and
$$\int_{|z|\le r}\kern-7pt dd^c u\wedge\Theta\wedge(dd^c\log|z|)^{p-1}\ge
\limsup_{\nu\to+\infty}\int_{|z|\le r}\kern-7pt dd^cu_\nu\wedge
\Theta\wedge(dd^c\log|z|)^{p-1}$$
by the weak convergence of $dd^cu_\nu\wedge\Theta$; here 
$(dd^c\log|z|)^{p-1}$ is not
smooth on $\overline B(0,r)$, but the integrals remain unchanged if we replace
$\log|z|$ by $\chi(\log|z|/r)$ with a smooth convex function $\chi$ such
that $\chi(t)=t$ for $t\ge -1$ and $\chi(t)=0$ for $t\le -2$. Now, we have
$u(z)\le\gamma\log|z|+C$ near $0$, so $u_\nu(z)$ coincides with
$(\gamma-\varepsilon)\log|z|-\nu$ on a small ball $B(0,r_\nu)\subset
B(0,r)$ and we infer
$$\eqalign{
\int_{|z|\le r}dd^c u_\nu\wedge\Theta\wedge 
(dd^c\log|z|)^{p-1}&\ge(\gamma-\varepsilon)
\int_{|z|\le r_\nu}\Theta\wedge(dd^c\log|z|)^p\cr
&\ge(\gamma-\varepsilon)\nu(\Theta,x).\cr}$$
As $r\in{}]0,R[$ and $\varepsilon\in{}]0,\gamma[$ were arbitrary, the
desired inequality follows.\qed
\endproof
%
We will later need an important decomposition formula of [Siu74].
We start with the following lemma.

\claim{(2.17) Lemma} If $\Theta$ is a closed
positive current of bidimension $(p,p)$ and $Z$ is an irreducible
analytic set in $X$, we set
$$m_Z=\inf\{x\in Z\,;\,\nu(\Theta,x)\}.$$
\plainitem{\rm (a)} There is a countable family of proper analytic subsets
$(Z'_j)$ of $Z$ such that $\nu(\Theta,x)=m_Z$ for all
$x\in Z\ssm\bigcup Z'_j$. We say that $m_Z$ is the generic Lelong number
of $\Theta$ along $Z$.
\vskip3pt
\plainitem{\rm (b)} If $\dim Z=p$, then $\Theta\ge m_Z[Z]$ and
$\bOne_Z\Theta=m_Z[Z]$.\vskip0pt
\endclaim
%
\begproof{Proof.} (a) By definition of $m_Z$ and $E_c(\Theta)$, we
have $\nu(\Theta,x)\ge m_Z$ for every $x\in Z$ and 
$$\nu(\Theta,x)=m_Z~~~~\hbox{\rm on}~~
Z\ssm\bigcup_{c\in\bQ,\,c>m_Z}Z\cap E_c(\Theta).$$
However, for $c>m_Z$, the intersection $Z\cap E_c(\Theta)$ is a proper
analytic subset of~$A$.

\noindent (b) Left as an exercise to the reader. It is enough to prove
that $\Theta\ge m_Z[Z_\reg]$ at regular points of $Z$, so one may assume
that $Z$ is a $p$-dimensional linear subspace in~$\bC^n$. Show that
the measure $(\Theta-m_Z[Z])\wedge(dd^c|z|^2)^p$ has nonnegative mass
on every ball $|z-a|<r$ with center $a\in Z$. Conclude by using 
arbitrary affine changes of coordinates that $\Theta-m_Z[Z]\ge 0$.\qed
\endproof
%
\claim{(2.18) Decomposition formula {\rm([Siu74])}} Let $\Theta$
be a closed positive current of bidimension~$(p,p)$. Then
$\Theta$ can be written as a convergent series of
closed positive currents
$$\Theta=\sum_{k=1}^{+\infty}\lambda_k\,[Z_k]+R,$$
where $[Z_k]$ is a current of integration over an irreducible
analytic set of dimension~$p$, and $R$ is a residual current 
with the property that $\dim E_c(R)<p$ for every $c>0$.
This decomposition is locally and globally unique: the sets $Z_k$
are precisely the $p$-dimensional components occurring in the
upperlevel sets $E_c(\Theta)$, and $\lambda_k=\min_{x\in Z_k}\nu(\Theta,x)$ is
the generic Lelong number of $\Theta$ along $Z_k$.
\endclaim
%
\begproof{Proof of uniqueness.} If $\Theta$ has such a decomposition, the
$p$-dimensional components of $E_c(\Theta)$ are $(Z_j)_{\lambda_j\ge c}$,
for $\nu(\Theta,x)=\sum\lambda_j\nu([Z_j],x)+\nu(R,x)$ is non zero
only on $\bigcup Z_j\cup\bigcup E_c(R)$, and is equal to
$\lambda_j$ generically on $Z_j$ $\big($more precisely,
$\nu(\Theta,x)=\lambda_j$ at every regular
point of $Z_j$ which does not belong to any intersection $Z_j\cup Z_k$,
$k\ne j$ or to $\bigcup E_c(R)\big)$. In particular $Z_j$ and
$\lambda_j$ are unique.
\endproof
%
\begproof{Proof of existence.} Let $(Z_j)_{j\ge 1}$ be the countable 
collection of
$p$-dimensional components occurring in one of the sets $E_c(\Theta)$,
$c\in\bQ_+^\star$, and let $\lambda_j>0$ be the generic Lelong
number of $\Theta$ along $Z_j$. Then Lemma~2.17 shows by induction on $N$
that $R_N=\Theta-\sum_{1\le j\le N}\lambda_j[Z_j]$ is positive. As $R_N$
is a decreasing sequence, there must be a
limit $R=\lim_{N\to+\infty}R_N$ in the weak topology. Thus we have
the asserted decomposition. By construction, $R$ has zero generic
Lelong number along $Z_j$, so $\dim E_c(R)<p$ for every $c>0$.\qed
\endproof
%
It is very important to note that some components of lower dimension
can actually occur in $E_c(R)$, but they cannot be subtracted because
$R$ has bidimension $(p,p)$. A typical case is the case of a
bidimension \hbox{$(n-1,n-1)$} current $\Theta=dd^cu$ with
$u=\log(|f_j|^{\gamma_1}+\ldots|f_N|^{\gamma_N})$ and
$f_j\in H^0(X,\cO_X)$. In general $\bigcup E_c(\Theta)=\bigcap f_j^{-1}(0)$
has dimension~$<n-1$. 

\claim{(2.19) Corollary} Let $\Theta_j=dd^c u_j$, $1\le j\le p$,
be closed positive \hbox{$(1,1)$-currents} 
on a complex manifold $X$. Suppose that there are analytic sets 
$A_2\supset\ldots\supset A_p$ in $X$ with
$\codim A_j\ge j$ at every point such that each $u_j$, $j\ge 2$, 
is locally bounded on $X\ssm A_j$. Let $\{A_{p,k}\}_{k\ge 1}$ be the
irreducible components of $A_p$ of codimension $p$ exactly and let
$\nu_{j,k}=\min_{x\in A_{p,k}}\nu(\Theta_j,x)$ be the generic Lelong number
of $\Theta_j$ along $A_{p,k}$. Then $\Theta_1\wedge\ldots\wedge \Theta_p$
is well-defined and
$$\Theta_1\wedge\ldots\wedge \Theta_p\ge\sum_{k=1}^{+\infty}\nu_{1,k}\ldots
\nu_{p,k}\,[A_{p,k}].$$
\endclaim
%
\begproof{Proof.} By induction on $p$, Proposition~2.3 shows that 
$\Theta_1\wedge\ldots\wedge \Theta_p$ is well defined. Moreover, 
Proposition~2.16 implies 
$$\nu(\Theta_1\wedge\ldots\wedge \Theta_p,x)\ge
\nu(\Theta_1,x)\ldots\nu(\Theta_p,x)\ge\nu_{1,k}\ldots\nu_{p,k}$$
at every point $x\in A_{p,k}$. The desired inequality is then a consequence
of Siu's decomposition theorem.\qed
\endproof
%
\section{Hermitian vector bundles, connections and curvature}
The goal of this section is to recall the most basic definitions of
hemitian differential geometry related to the concepts of connection,
curvature and first Chern class of a line bundle.

Let $F$ be a complex vector bundle of rank $r$ over a smooth
differentiable manifold~$M$. A {\it connection} $D$ on $F$ is a
linear differential operator of order $1$
$$D:C^\infty(M,\Lambda^qT^\star_M\otimes F) \to C^\infty(M,
\Lambda^{q+1}T^\star_M\otimes F)$$
such that
$$D(f\wedge u) = df\wedge u + (-1)^{\deg\,f}f\wedge Du\leqno(3.1)$$
for all forms $f\in C^\infty(M,\Lambda^pT^\star_M)$, $u\in C^\infty(X,
\Lambda^qT^\star_M\otimes F)$. On an open set $\Omega \subset M$ where $F$
admits a trivialization $\theta:F_{|\Omega }\buildo\longrightarrow^\simeq \Omega
\times \bC^r$, a connection $D$ can be written
$$Du\simeq_\theta du+\Gamma \wedge u$$
where $\Gamma \in C^\infty(\Omega ,\Lambda^1T^\star_M\otimes\Hom
(\bC^r,\bC^r))$ is an arbitrary matrix of $1$-forms and $d$ acts
componentwise (the coefficents of $\Gamma$ are called the 
{\it Christoffel symbols} of the connection). It is then easy to check that
$$D^2u\simeq_\theta (d\Gamma +\Gamma \wedge \Gamma )\wedge u\quad
\hbox{on~}\Omega.$$
Since $D^2$ is a globally defined operator, there is a global $2$-form
$$\Theta_D \in C^\infty(M,\Lambda^2T^\star_M\otimes\Hom(F,F))\leqno(3.2)$$
such that $D^2u=\Theta_D\wedge u$ for every form $u$ with values in~$F$.

Assume now that $F$ is endowed with a $C^\infty$ hermitian metric $h$ along
the fibers and that the isomorphism $F_{|\Omega }\simeq \Omega \times
\bC^r$ is given by a $C^\infty$ frame $(e_\lambda)$. We then have
a canonical sesquilinear pairing
$$\leqalignno{
C^\infty(M,\Lambda^pT^\star_M\otimes F)\times
C^\infty(M,\Lambda^qT^\star_M\otimes F)
&\longrightarrow C^\infty(M,\Lambda^{p+q}T^\star_M\otimes\bC)&(3.3)\cr
(u,v) &\longmapsto \{u,v\}_h\cr}$$
given by
$$\{u,v\}_h=\sum_{\lambda ,\mu }u_\lambda \wedge \overline v_\mu \langle
e_\lambda,e_\mu \rangle_h,\qquad
u=\sum u_\lambda \otimes e_\lambda,\quad 
v=\sum v_\mu \otimes e_\mu.$$
The connection $D$ is said to be {\it hermitian} (with respect to $h$) if 
it satisfies the additional property
$$d\{u,v\}_h=\{Du,v\}_h+(-1)^{{\rm deg~}u}\{u,Dv\}_h.$$
Assuming that $(e_\lambda )$ is orthonormal, one easily checks that $D$
is hermitian if and only if $\Gamma^\star=-\Gamma$. In this case
$\Theta_D^\star=-\Theta_D$, thus
$$\ii\Theta_D \in C^\infty(M,\Lambda^2T^\star_M\otimes\Herm(F,F)).$$
\medskip

\claim{(3.4) Special case} \rm For a bundle $F$ of rank 1, the
connection form $\Gamma$ of a hermitian connection $D$ can be seen as
a $1$-form with purely imaginary coefficients $\Gamma=\ii A$ ($A$ real).
Then we have $\Theta_D=d\Gamma=\ii dA$. In particular $\ii\Theta_F$
is a closed 2-form. The {\it first Chern class} of $F$
is defined to be the cohomology class
$$c_1(F)_\bR=\Big\{{\ii\over 2\pi}\Theta_F\Big\}\in H^2_\DR(M,\bR).$$
The cohomology class is actually independent of the connection, since
any other connection $D_1$ differs by a global 1-form, $D_1u=Du+B\wedge u$,
so that $\Theta_{D_1}=\Theta_D+dB$. It is well-known that $c_1(F)_\bR$
is the image in $H^2(M,\bR)$ of an integral class $c_1(F)\in
H^2(M,\bZ)\,$; by using the exponential exact sequence
$$0\to\bZ\to\cE\to\cE^\star\to 0,$$
$c_1(F)$ can be defined in \v Cech cohomology theory as the image
by the coboundary map $H^1(M,\cE^\star)\to H^2(M,\bZ)$ of the cocycle
$\{g_{jk}\}\in H^1(M,\cE^\star)$ defining $F\,$; see e.g.\ [GrH78] for
details.\qed
\endclaim
%
We now concentrate ourselves on the complex analytic case. If $M=X$ is a
complex manifold $X$, every connection $D$ on a complex $C^\infty$ vector
bundle $F$ can be splitted in a unique way as a
sum of a $(1,0)$ and of a $(0,1)$-connection, $D=D'+D''$. In a local
trivialization $\theta$ given by a $C^\infty$ frame, one can write 
$$\leqalignno{
D'u &\simeq_\theta d'u + \Gamma '\wedge u,&(3.5')\cr
D''u &\simeq_\theta d''u + \Gamma ''\wedge u,&(3.5'')\cr}$$
with $\Gamma =\Gamma '+\Gamma ''$. The connection is hermitian if and
only if $\Gamma '=-(\Gamma '')^\star$ in any orthonormal frame. Thus there
exists a unique hermitian connection $D$ corresponding to a prescribed
$(0,1)$ part $D''$. 

Assume now that the bundle $F$ itself has a {\it holomorphic} structure, and
is equipped with a hermitian metric~$h$.
The unique hermitian connection for which $D''$ is the $d''$ operator
defined in \S~1 is called the {\it Chern connection} of $F$. In a local
holomorphic frame $(e_\lambda )$ of $E_{|\Omega }$, the metric is given
by the hermitian matrix $H=(h_{\lambda\mu})$, $h_{\lambda\mu}
=\langle e_\lambda,e_\mu \rangle$. We have
$$\{u,v\}_h=\sum_{\lambda,\mu}h_{\lambda\mu}u_\lambda \wedge
\overline v_\mu=u^\dagger\wedge H\overline v,$$
where $u^\dagger$ is the transposed matrix of $u$, and
easy computations yield
$$\eqalign{d\{u,v\}_h
&=(du)^\dagger \wedge H\overline v+(-1)^{\deg u} u^\dagger
\wedge (dH\wedge \overline v + H\overline{dv})\cr
&=\big(du +\overline H^{-1}d'\overline H \wedge u\big)^\dagger \wedge H\overline v
+(-1)^{\deg u}u^\dagger\wedge(\overline{dv+ \overline H^{-1}d'\overline H\wedge v)}\cr}$$
using the fact that $dH = d'H+\overline{d'\overline H}$ and
$\overline H^\dagger=H$. Therefore the Chern connection $D$ coincides with the
hermitian connection defined by
$$\left\{\eqalign{
Du&{}\simeq_\theta du+\overline H^{-1}d'\overline H \wedge u,\cr
D'&{}\simeq_\theta d'+\overline H^{-1} d'\overline H\wedge\bu=
\overline H^{-1} d'(\overline H\bu),~~~~D'' = d''.\cr}\right.\leqno(3.6)$$
It is clear from this relations that $D^{\prime 2}=D^{\prime\prime 2}=0$.
Consequently $D^2$ is given by to $D^2=D'D''+D''D'$, and the curvature 
tensor $\Theta_D$ is of type~$(1,1)$. Since $d'd''+d''d'=0$, we get
$$\eqalign{
(D'D''+D''D')u&\simeq_\theta\overline H^{-1}d'\overline H\wedge d''u 
+d''(\overline H^{-1}d'\overline H\wedge u)\cr
&=d''(\overline H^{-1}d'\overline H)\wedge u.\cr}$$

\claim{(3.7) Proposition} The Chern curvature tensor $\Theta_{F,h}:=
\Theta_D$ of $(F,h)$ is such that
$$\ii\,\Theta_{F,h} \in C^\infty(X,\Lambda^{1,1}T^\star_X\otimes\Herm(F,F)).$$
If $\theta: F_{\restriction\Omega}\to\Omega\times\bC^r$ is a holomorphic
trivialization and if $H$ is the hermitian matrix representing the
metric along the fibers of $F_{\restriction\Omega}$, then
$$\ii\,\Theta_{F,h}\simeq_\theta
\ii\,d''(\overline H^{-1} d'\overline H)\quad\hbox{\rm on}~\Omega.\eqno\square$$
\endclaim
%
Let $(z_1,\ldots,z_n)$ be holomorphic coordinates on $X$ and let
$(e_\lambda)_{1\le\lambda\le r}$ be an orthonormal frame of $F$.
Writing
$$\ii\Theta_{F,h}=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}dz_j\wedge dz_k\otimes e_\lambda^\star\otimes e_\mu,$$
we can identify the curvature tensor to a hermitian form
$$\widetilde\Theta_{F,h}(\xi\otimes v)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\xi_j\overline\xi_kv_\lambda\overline v_\mu\leqno(3.8)$$
on $T_X\otimes F$. This leads in a natural way to positivity concepts,
following definitions introduced by Kodaira [Kod53], Nakano [Nak55] and
Griffiths [Gri69].

\claim{(3.9) Definition} The hermitian vector bundle $(F,h)$ is said to be
\vskip3pt
\plainitem{\rm (a)} positive in the sense of Nakano if
$\widetilde\Theta_{F,h}(\tau)>0$ for all non zero tensors $\tau=\sum
\tau_{j\lambda}\partial/\partial z_j\otimes e_\lambda\in T_X\otimes F$.
\vskip3pt
\plainitem{\rm (b)} positive in the sense of Griffiths if
$\widetilde\Theta_{F,h}(\xi\otimes v)>0$ for all non zero decomposable tensors
$\xi\otimes v\in T_X\otimes F\,;$
\vskip3pt\noindent
Corresponding semipositivity concepts are defined by relaxing the strict
inequalities.
\endclaim
%
\claim{(3.10) Special case of rank 1 bundles} \rm Assume that
$F$ is a line bundle. The hermitian matrix $H=(h_{11})$ associated
to a trivialization $\theta:F_{\restriction\Omega}\simeq\Omega\times\bC$
is simply a positive function. It is often convenient to denote it as
an exponential, namely $e^{-2\varphi}$ 
(and also sometimes $e^{-\varphi}$ simply, if we do not want to stress that $H$
is a quadratic form), with  \hbox{$\varphi\in C^\infty(\Omega,\bR)$}. 
In this case
the curvature form $\Theta_{F,h}$ can be identified to the $(1,1)$-form
$d'd''\varphi$, and
$${\ii\over 2\pi}\Theta_{F,h}={\ii\over\pi}d'd''\varphi=dd^c\varphi$$
is a real $(1,1)$-form. Hence $F$ is semi-positive (in either the Nakano or
Griffiths sense) if and only if $\varphi$ is psh, resp.\ positive if
and only if $\varphi$ is {\it strictly psh}. In this setting, the
Lelong-Poincar\'e equation can be generalized as follows: let
$\sigma\in H^0(X,F)$ be a non zero holomorphic section. Then
$$dd^c\log\|\sigma\|_h=[Z_\sigma]-{\ii\over 2\pi}\Theta_{F,h}.\leqno(3.11)$$
Formula (3.11) is immediate if we write $\|\sigma\|=|\theta(\sigma)|
e^{-\varphi}$ and if we apply (1.20) to the holomorphic function
$f=\theta(\sigma)$. As we shall see later, it is very important for
the applications to consider also singular hermitian metrics.
\endclaim
%
\claim{(3.12) Definition} A singular $($hermitian$)$ metric $h$ on a line
bundle $F$ is a metric which is given in any trivialization
$\theta:F_{\restriction\Omega}
\buildo \longrightarrow^\simeq \Omega\times\bC$ by
$$\|\xi\|_h=|\theta(\xi)|\,e^{-\varphi(x)},~~~~x\in\Omega,~\xi\in F_x$$
where $\varphi\in L^1_\loc(\Omega)$ is an arbitrary function, called the
weight of the metric with respect to the trivialization $\theta$.
\endclaim
%
If $\theta':F_{\restriction\Omega'}\longrightarrow\Omega'\times\bC$ 
is another trivialization, $\varphi'$ the associated weight and
$g\in\cO^\star(\Omega\cap\Omega')$ the transition function, then
$\theta'(\xi)=g(x)\,\theta(\xi)$ for $\xi\in F_x$, and so
$\varphi'=\varphi+\log|g|$ on $\Omega\cap\Omega'$. The curvature form 
of $F$ is then given formally by the closed $(1,1)$-current
${\ii\over 2\pi}\Theta_{F,h}=dd^c\varphi$ on $\Omega\,$; our assumption
$\varphi\in L^1_\loc(\Omega)$ guarantees that $\Theta_{F,h}$ exists in
the sense of distribution theory. As in the smooth case, ${\ii\over 2\pi}
\Theta_{F,h}$ is globally defined on $X$ and independent of the choice of
trivializations, and its De Rham cohomology class is the image of the first
Chern class $c_1(F)\in H^2(X,\bZ)$ in $H^2_{DR}(X,\bR)$.
Before going further, we discuss two basic examples. 

\claim{(3.13) 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 
$\theta(u)=u\prod\smash{g_j^{\alpha_j}}$ defines a trivialization of
$\cO(D)$ 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
$${\ii\over 2\pi}\Theta_{\cO(D)}=dd^c\varphi=[D],$$
where $[D]=\sum\alpha_j[D_j]$ denotes the current of integration
over~$D$.\qed
\endclaim
%
\claim{(3.14) Example} \rm Assume that $\sigma_0,\sigma_1,\ldots,\sigma_N$ 
are non zero 
holomorphic sections of $F$. Then we can define a natural (possibly
singular) hermitian metric $h^*$ on $F^\star$ by
$$\|\xi^\star\|_{h^*}^2=\sum_{0\le j\le N}\big|\xi^\star.\sigma_j(x)\big|^2
~~~~\hbox{\rm for}~~\xi^\star\in F^\star_x.$$
The dual metric $h$ on $F$ is given by
$$\|\xi\|^2_h={|\theta(\xi)|^2\over|\theta(\sigma_0(x))|^2+
|\theta(\sigma_1(x))|^2+\ldots+|\theta(\sigma_N(x))|^2}$$
with respect to any trivialization $\theta$. The associated weight function is
thus given by 
$$
\varphi(x)=\log\Big(\sum_{0\le j\le N}|\theta(\sigma_j(x))|^2
\Big){}^{1/2}.
$$
In this case $\varphi$ is a psh function, thus
$\ii\Theta_{F,h}$ is a closed positive current. Let us denote by $\Sigma$
the linear system defined by $\sigma_0,\ldots,\sigma_N$ and by
$B_\Sigma=\bigcap\sigma_j^{-1}(0)$ its base locus. We have a meromorphic
map
$$\Phi_\Sigma:X\ssm B_\Sigma\to\bP^N,\qquad
x\mapsto(\sigma_0(x):\sigma_1(x):\sigma_2(x):\ldots:\sigma_N(x)).$$
Then ${\ii\over 2\pi}\Theta_{F,h}$ is equal to the pull-back over $X\ssm
B_\Sigma$ of the Fubini-Study metric $\omega_\FS={\ii\over 2\pi}
\log(|z_0|^2+|z_1|^2+\ldots+ |z_N|^2)$ of $\bP^N$ by $\Phi_\Sigma$.\qed
\endclaim
%
\claim{(3.15) Ample and very ample line bundles} A holomorphic line
bundle $F$ over a compact complex manifold $X$ is said to be
\vskip3pt
\plainitem{\rm (a)} very ample if the map $\Phi_{|F|}:X\to\bP^N$ associated
to the complete linear system $|F|=P(H^0(X,F))$ is a regular embedding
$($by this we mean in particular that the base locus is empty, i.e.\
$B_{|F|}=\emptyset)$.
\vskip3pt
\plainitem{\rm (b)} ample if some multiple $mF$, $m>0$, is very ample.
\endclaim
%
Here we use an additive notation for $\Pic(X)=H^1(X,\cO^\star)$, hence
the symbol $mF$ denotes the line bundle $F^{\otimes m}$. By
Example~3.14, every ample line bundle $F$ has a smooth hermitian metric
with positive definite curvature form; indeed, if the linear system
$|mF|$ gives an embedding in projective space, then we get a smooth
hermitian metric on $F^{\otimes m}$, and the $m$-th root yields a
metric $h$ on $F$ such that ${\ii\over 2\pi}\Theta_{F,h}={1\over m}
\Phi_{|mF|}^\star\omega_\FS$. Conversely, the Kodaira embedding theorem
[Kod54] tells us that every positive line bundle $F$ is ample (see
Exercise~5.14 for a straightforward analytic proof of the Kodaira
embedding theorem).

\section{Bochner technique and vanishing theorems}

\subsection*{\csect.A. Laplace-Beltrami operators and Hodge theory}
We first recall briefly a few basic facts of Hodge theory. Assume for
the moment that $M$ is a differentiable manifold equipped with a Riemannian
metric $g=\sum g_{ij}dx_i\otimes dx_j$ and that
$(F,h)$ is a hermitian vector bundle over $M$. Given a $q$-form $u$ 
on $M$ with values in $F$, we consider the global $L^2$ norm
$$\|u\|^2 = \int_M |u(x)|^2dV_g(x)$$
where $|u|$ is the pointwise hermitian norm and $dV_g$ is the
Riemannian volume form (we omit the dependence on the metrics in the 
notation, but we should really put $|u(x)|_{g,h}$ and 
$\|u\|_{g,h}$ here). The Laplace-Beltrami operator associated to the
connection $D$ is by definition
$$\Delta =DD^\star+D^\star D$$
where
$$D^\star:C^\infty(M,\Lambda^qT^\star_M\otimes F)\to
C^\infty(M,\Lambda^{q-1}T^\star_M\otimes F)$$
is the (formal) adjoint of $D$ with respect to the $L^2$ inner product.
Assume that $M$ is {\it compact}. Since
$$\Delta:C^\infty(M,\Lambda^qT^\star_M\otimes F)\to
C^\infty(M,\Lambda^qT^\star_M\otimes F)$$
is a self-adjoint elliptic operator in each degree, standard results of
PDE theory show that there is an orthogonal decomposition
$$C^\infty(M,\Lambda^qT^\star_M\otimes F)=\cH^q(M,F)\oplus\Im\Delta$$
where $\cH^q(M,F)=\Ker\Delta$ is the space of harmonic forms of degree~$q$;
$\cH^q(M,F)$ is a finite dimensional space.
Assume moreover that the connection $D$ is {\it integrable}, i.e.\ that
$D^2=0$. It is then easy to check that there is an orthogonal direct sum
$$\Im\Delta=\Im D\oplus\Im D^\star,$$
indeed $\langle Du,D^\star v\rangle=\langle D^2u,v\rangle=0$ for all $u,v$.
Hence we get an orthogonal decomposition
$$C^\infty(M,\Lambda^qT^\star_M\otimes F)=\cH^q(M,F)\oplus\Im D\oplus\Im D^\star,$$
and $\Ker\Delta$ is precisely equal to $\cH^q(M,F)\oplus\Im D$. Especially,
the $q$-th cohomology group $\Ker\Delta/\Im\Delta$ is isomorphic to
$\cH^q(M,F)$. All this can be applied for example in the case of the
De Rham groups $H^q_\DR(M,\bC)$, taking $F$ to be the trivial bundle
$F=M\times\bC$ (notice, however, that a nontrivial bundle $F$
usually does not admit any integrable connection):

\claim{(4.1) Hodge Fundamental Theorem} If $M$ is a compact Riemannian
manifold, there is an isomorphism
$$H^q_\DR(M,\bC)\simeq\cH^q(M,\bC)$$
from De Rham cohomology groups onto spaces of harmonic forms.\qed
\endclaim
%
A rather important consequence of the Hodge fundamental theorem is a proof
of the {\it Poincar\'e duality theorem}. Assume that the Riemannian manifold
$(M,g)$ is oriented. Then there
is a (conjugate linear) Hodge star operator
$$\star:\Lambda^qT^\star_M\otimes\bC\to\Lambda^{m-q}T^\star_M\otimes\bC,
\qquad m=\dim_\bR M$$
defined by $u\wedge\star v=\langle u,v\rangle dV_g$ for any two complex
valued $q$-forms $u$, $v$. A standard computation shows
that $\star$ commutes with $\Delta$, hence $\star u$ is harmonic if and only
if $u$ is. This implies that the natural pairing
$$H^q_\DR(M,\bC)\times H^{m-q}_\DR(M,\bC),\qquad
(\{u\},\{v\})\mapsto\int_M u\wedge v\leqno(4.2)$$
is a nondegenerate duality, the dual of a class $\{u\}$ represented by a
harmonic form being $\{\star u\}$.

\subsection*{\csect.B. Serre duality theorem}
Let us now suppose that $X$ is a compact complex manifold
equipped with a hermitian metric $\omega=\sum\omega_{jk}dz_j\wedge d\overline
z_k$. Let $F$ be a holomorphic vector bundle on $X$ equipped with a
hermitian metric, and let $D=D'+D''$ be its Chern curvature form.
All that we said above for the Laplace-Beltrami operator $\Delta$
still applies to the complex Laplace operators
$$\Delta'=D'D^{\prime\star}+D^{\prime\star}D',\qquad
\Delta''=D''D^{\prime\prime\star}+D^{\prime\prime\star}D'',$$
with the great advantage that we always have 
$D^{\prime 2}=D^{\prime\prime 2}=0$. Especially, if $X$ is a compact
complex manifold, there are isomorphisms
$$H^{p,q}(X,F)\simeq\cH^{p,q}(X,F)\leqno(4.3)$$
between Dolbeault cohomology groups $H^{p,q}(X,F)$ and spaces
$\cH^{p,q}(X,F)$ of $\Delta''$-harmonic forms of bidegree $(p,q)$ with
values in~$F$. Now, there is a generalized Hodge star operator
$$\star:\Lambda^{p,q}T^\star_X\otimes F\to
\Lambda^{n-p,n-q}T^\star_X\otimes F^\star,\qquad n=\dim_\bC X,$$
such that $u\wedge\star v=\langle u,v\rangle dV_g$,
for any two $F$-valued $(p,q)$-forms, when the wedge product
$u\wedge\star v$ is combined with the pairing $F\times F^\star\to\bC$.
This leads to the {\it Serre duality theorem} [Ser55]: the bilinear pairing
$$H^{p,q}(X,F)\times H^{n-p,n-q}(X,F^\star),\qquad
(\{u\},\{v\})\mapsto\int_X u\wedge v\leqno(4.4)$$
is a nondegenerate duality. Combining this with the Dolbeault isomorphism,
we may restate the result in the form of the duality formula
$$H^q(X,\Omega^p_X\otimes\cO(F))^\star\simeq H^{n-q}(X,
\Omega^{n-p}_X\otimes\cO(F^\star)).\leqno(4.4')$$

\subsection*{\csect.C. Bochner-Kodaira-Nakano identity on K\"ahler manifolds}
We now proceed to explain the basic ideas of the Bochner technique used
to prove vanishing theorems. Great simplifications occur in the computations
if the hermitian metric on $X$ is supposed to be {\it K\"ahler}, i.e.\
if the associated {\it fundamental $(1,1)$-form}
$$\omega=\ii\sum\omega_{jk}dz_j\wedge d\overline z_k$$
satisfies $d\omega=0$. It can be easily shown that $\omega$ is K\"ahler
if and only if there are holomorphic coordinates $(z_1,\ldots,z_n)$ centered
at any point $x_0\in X$ such that the matrix of coefficients $(\omega_{jk})$
is tangent to identity at order $2$, i.e.\ 
$$\omega_{jk}(z)=\delta_{jk}+O(|z|^2)\quad\hbox{at $x_0$}.$$
It follows that all order 1 operators $D$, $D'$, $D''$ and their
adjoints $D^\star$, $D^{\prime\star}$, $D^{\prime\prime\star}$ admit at $x_0$
the same expansion as the analogous operators obtained when all hermitian
metrics on $X$ or $F$ are constant. From this, the basic commutation
relations of K\"ahler geometry can be checked. If $A,B$ are differential
operators acting on the algebra $C^\infty(X,\Lambda^{\bu,\bu}T^\star_X\otimes
F)$, their graded commutator (or graded Lie bracket) is defined by
$$[A,B]= AB - (-1)^{ab} BA$$
where $a,b$ are the degrees of $A$ and $B$ respectively. If $C$ is another
endomorphism of degree $c$, the following purely formal {\it Jacobi identity}
holds:
$$(-1)^{ca} \big[A,[B,C]\big] + (-1)^{ab}\big[B,[C,A]\big]
+ (-1)^{bc} \big[C,[A,B]\big] = 0.$$

\claim{(4.5) Basic commutation relations} Let $(X,\omega)$ be a K\"ahler
manifold and let $L$ be the operators defined by $Lu=\omega\wedge u$ and
$\Lambda=L^\star$. Then
$$\eqalign{
[D^{\prime\prime\star},L]&=\ii D',\cr
[\Lambda,D'']&=-\ii D^{\prime\star},\cr}
\qquad
\eqalign{
[D^{\prime\star},L]&=-\ii D'',\cr
[\Lambda,D']&=\ii D^{\prime\prime\star}.\cr}$$
\endclaim
%
\begproof{Proof {\rm(sketch)}.} The first step is to check the identity
$[d^{\prime\prime\star},L]=\ii d'$ for constant metrics on $X=\bC^n$ and
$F=X\times\bC$, by a brute force calculation. All three other identities
follow by taking conjugates or adjoints. The case of variable metrics follows
by looking at Taylor expansions up to order~1.\qed
\endproof
%
\claim{(4.6) Bochner-Kodaira-Nakano identity} If $(X,\omega)$ is
K\"ahler, the complex Laplace operators $\Delta'$ and $\Delta''$ acting
on $F$-valued forms satisfy the identity
$$\Delta''=\Delta'+[\ii\Theta_{F,h},\Lambda].$$
\endclaim
%
\begproof{Proof.} The last equality in (4.5) yields $D^{\prime\prime\star}
=-\ii[\Lambda,D']$, hence
$$\Delta''=[D'',\delta'']=-\ii[D'',\big[\Lambda,D']\big].$$
By the Jacobi identity we get
$$\big[D'',[\Lambda,D']\big]=\big[\Lambda,[D',D'']]+\big[D',[D'',\Lambda]\big]
=[\Lambda,\Theta_{F,h}]+\ii[D',D^{\prime\star}],$$
taking into account that $[D',D'']=D^2=\Theta_{F,h}$. The formula follows.\qed
\endproof
%
\subsection*{\csect.D. Vanishing theorems}
Assume that $X$ is compact and that $u\in C^\infty(X,\Lambda^{p,q}T^\star
X\otimes F)$ is an arbitrary $(p,q)$-form. An integration by parts yields
$$\langle\Delta'u,u\rangle=\|D'u\|^2+\|D^{\prime\star}u\|^2\ge0$$
and similarly for $\Delta''$, hence we get the basic a priori inequality
$$\|D''u\|^2+\|D^{\prime\prime\star}u\|^2\ge\int_X
\langle[\ii\Theta_{F,h},\Lambda]u,u\rangle dV_\omega.\leqno(4.7)$$
This inequality is known as the {\it Bochner-Kodaira-Nakano} inequality
(see [Boc48], [Kod53], [Nak55]). When $u$ is $\Delta''$-harmonic, we get
$$\int_X\big(\langle[\ii\Theta_{F,h},\Lambda]u,u\rangle+
\langle T_\omega u,u\rangle\big)dV\le 0.$$
If the hermitian operator $[\ii\Theta_{F,h},\Lambda]$ acting on
$\Lambda^{p,q}T^\star_X\otimes F$ is positive on each fiber, we infer
that $u$ must be zero, hence
$$H^{p,q}(X,F)=\cH^{p,q}(X,F)=0$$
by Hodge theory. The main point is thus to compute the curvature form
$\Theta_{F,h}$ and find sufficient conditions under which the operator
$[\ii\Theta_{F,h},\Lambda]$ is positive definite. Elementary (but somewhat
tedious) calculations yield the following formulae: if the curvature of
$F$ is written as in (3.8) and 
$u=\sum u_{J,K,\lambda}dz_I\wedge d\overline z_J\otimes e_\lambda$,
$|J|=p$, $|K|=q$, $1\le\lambda\le r$ is a $(p,q)$-form with values in~$F$,
then
$$\leqalignno{\langle[\ii\Theta_{F,h},\Lambda]u,u\rangle=
&\phantom{{}+{}}\sum_{j,k,\lambda,\mu,J,S}{\,}
c_{jk\lambda\mu}\,u_{J,jS,\lambda}~\overline{u_{J,kS,\mu}}&(4.8)\cr
&{}+\sum_{j,k,\lambda,\mu,R,K}
c_{jk\lambda\mu}\,u_{kR,K,\lambda}~\overline{u_{jR,K,\mu}}\cr
&{}-{\,}\sum_{j,\lambda,\mu,J,K}\phantom{k}
c_{jj\lambda\mu}\,u_{J,K,\lambda}~\overline{u_{J,K,\mu}},\cr}$$
where the sum is extended to all indices $1\le j,k\le n$, $1\le\lambda,
\mu\le r$ and multiindices $|R|=p-1$, $|S|=q-1$ (here the notation
$u_{JK\lambda}$ is extended to non necessarily increasing multiindices by
making it alternate with respect to permutations). It is usually hard to
decide the sign of the curvature term (4.8), except in some special cases.

The easiest case is when $p=n$. Then all terms in the second summation of
(4.8) must have $j=k$ and $R=\{1,\ldots,n\}\ssm\{j\}$, therefore the second
and third summations are equal. It follows that $[\ii\Theta_{F,h},\Lambda]$ is
positive on $(n,q)$-forms under the assumption that $F$ is positive in
the sense of Nakano. In this case $X$ is automatically K\"ahler since
$$\omega=\Tr_F(\ii\Theta_{F,h})=\ii\sum_{j,k,\lambda}c_{jk\lambda\lambda}dz_j
\wedge d\overline z_k=\ii\Theta_{\det F,h}$$
is a K\"ahler metric.

\claim{(4.9) Nakano vanishing theorem {\rm ([Nak55])}} Let $X$ be a
compact complex manifold and let $F$ be a Nakano positive vector
bundle on~$X$. Then
$$H^{n,q}(X,F)=H^q(X,K_X\otimes F)=0\qquad\hbox{for every $q\ge 1$.}
\eqno\square$$
\endclaim
%
Another tractable case is the case where $F$ is a line bundle ($r=1$).
Indeed, at each point $x\in X$, we may then choose a coordinate
system which diagonalizes simultaneously the hermitians forms
$\omega(x)$ and $\ii\Theta_{F,h}(x)$, in such a way that
$$\omega(x)=\ii\sum_{1\le j\le n}dz_j\wedge d\overline z_j,\qquad
\ii\Theta_{F,h}(x)=\ii\sum_{1\le j\le n}\gamma_jdz_j\wedge d\overline z_j$$
with $\gamma_1\le\ldots\le\gamma_n$. The curvature eigenvalues
$\gamma_j=\gamma_j(x)$ are then uniquely defined and depend continuously
on~$x$. With our previous notation, we have $\gamma_j=c_{jj11}$ and all
other coefficients $c_{jk\lambda\mu}$ are zero. For any $(p,q)$-form
$u=\sum u_{JK}dz_J\wedge d\overline z_K\otimes e_1$, this gives
$$\leqalignno{\langle[\ii\Theta_{F,h},\Lambda]u,u\rangle
&=\sum_{|J|=p,\,|K|=q}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}\gamma_j-
\sum_{1\le j\le n}\gamma_j\Big)|u_{JK}|^2\cr
&\ge(\gamma_1+\ldots+\gamma_q-\gamma_{n-p+1}-\ldots-\gamma_n)|u|^2.
&(4.10)\cr}$$
Assume that $\ii\Theta_{F,h}$ is positive. It is then natural to make the
special choice $\omega=\ii\Theta_{F,h}$ for the K\"ahler metric. Then
$\gamma_j=1$ for $j=1,2,\ldots,n$ and we obtain
$\langle[\ii\Theta_{F,h},\Lambda]u,u\rangle=(p+q-n)|u|^2$.
As a consequence:

\claim{(4.11) Akizuki-Kodaira-Nakano vanishing theorem {\rm ([AN54])}}
If $F$ is a positive line bundle on a compact complex manifold~$X$, then
$$H^{p,q}(X,F)=H^q(X,\Omega^p_X\otimes F)=0\qquad\hbox{for}~~p+q\ge n+1.
\eqno\square$$
\endclaim
%
More generally, if $F$ is a Griffiths positive (or ample) vector bundle
of rank~$r\ge 1$, Le Potier [LP75] proved that $H^{p,q}(X,F)=0$ for
$p+q\ge n+r$. The proof is not a direct consequence of the Bochner
technique. A rather easy proof has been found by M.~Schneider [Sch74],
using the Leray spectral sequence associated to the projectivized bundle
projection $\bP(F)\to X$, using the following more or less standard 
notation.

\claim{(4.12) Notation} \rm If $V$ is a complex vector space (resp.\
complex vector bundle), we let $P(V)$ be
the projective space (resp.\ bundle) of {\it lines} of $V$,
and $\bP(V)=P(V^*)$ be the projective space (resp.\ bundle) 
of {\it hyperplanes} of~$V$.
\endclaim
%
\claim{(4.13) Exercise} \rm It is important for various applications
to obtain vanishing theorems which are also valid in the case of semi-positive
line bundles. The easiest case is the following result of Girbau [Gir76]:
let $(X,\omega)$ be compact K\"ahler; assume that $F$ is a line bundle and
that $\ii\Theta_{F,h}\ge 0$ has at least $n-k$ positive eigenvalues at each
point, for some integer~$k\ge 0$; show that $H^{p,q}(X,F)=0$ for $p+q\ge
n+k+1$.\newline
{\it Hint:}\/ use the K\"ahler metric $\omega_\varepsilon=
\ii\Theta_{F,h}+\varepsilon\omega$ with $\varepsilon>0$ small.
\vskip3pt
A stronger and more natural ``algebraic version'' of this result has been
obtained by Sommese [Som78]: define $F$ to be $k$-ample if some multiple
$mF$ is such that the canonical map
$$\Phi_{|mF|}:X\ssm B_{|mF|}\to\bP^{N-1}$$
has at most $k$-dimensional fibers and $\dim B_{|mF|}\le k$. If
$X$ is projective and $F$ is $k$-ample, show that $H^{p,q}(X,F)=0$ for
$p+q\ge n+k+1$.\newline
{\it Hint:}\/ prove the dual result $H^{p,q}(X,F^{-1})=0$ for $p+q\le n-k-1$
by induction on~$k$. First show that $F$ 0-ample $\Rightarrow$ $F$ positive;
then use hyperplane sections $Y\subset X$ to prove the induction step,
thanks to the exact sequences
$$\eqalignno{
&0\longrightarrow\Omega^p_X\otimes F^{-1}\otimes\cO(-Y)\longrightarrow\Omega^p_X\otimes F^{-1}
\longrightarrow\big(\Omega^p_X\otimes F^{-1}\big)_{\restriction Y}\longrightarrow 0,&\cr
&0\longrightarrow\Omega^{p-1}_Y\otimes F^{-1}_{\restriction Y}\longrightarrow
\big(\Omega^p_X\otimes F^{-1}\big)_{\restriction Y}\longrightarrow
\Omega^p_Y\otimes F^{-1}_{\restriction Y}\longrightarrow 0.&\square\cr}$$
\endclaim
%
\section{$L^2$ estimates and existence theorems}

\subsection*{\csect.A. Basic $L^2$ existence theorems}
The starting point is the following $L^2$ existence theorem, which is
essentially due to H\"ormander [H\"or65, 66], and Andreotti-Vesentini
[AV65], following fundamental work by Kohn [Koh63, 64]. We will only  
present the strategy and the main ideas and tools, referring e.g.\ to [Dem82b]
for a more detailed exposition of the technical situation considered here.

\claim{(5.1) Theorem} Let $(X,\omega)$ be a K\"ahler manifold. Here $X$
is not necessarily compact, but we assume that the geodesic distance
$\delta_\omega$ is complete on~$X$. Let $F$ be a hermitian vector bundle of
rank $r$ over $X$, and assume that the curvature operator
$A=A^{p,q}_{F,h,\omega}=[\ii\Theta_{F,h},\Lambda_\omega]$ is positive
definite everywhere on $\Lambda^{p,q}T^\star_X\otimes F$, $q\ge 1$.
Then for any form $g\in L^2(X,\Lambda^{p,q}T^\star_X\otimes F)$
satisfying $D''g=0$ and $\int_X\langle A^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ such that  $D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X\langle A^{-1}g,g\rangle\,dV_\omega.$$
\endclaim
%
\begproof{Proof.} The assumption that $\delta_\omega$ is complete implies the
existence of cut-off functions $\psi_\nu$ with arbitrarily large compact
support such that $|d\psi_\nu|\le 1$ (take $\psi_\nu$ to be a function of
the distance $x\mapsto \delta_\omega(x_0,x)$, which is an almost
everywhere differentiable 1-Lipschitz function, and regularize if
necessary). From this, it follows that very form $u\in
L^2(X,\Lambda^{p,q}T^\star_X\otimes F)$ such that $D''u\in L^2$ and
$D^{\prime\prime\star}u\in L^2$ in the sense of distribution theory is a
limit of a sequence of smooth forms $u_\nu$ with compact support, in such
a way that $u_\nu\to u$, $D''u_\nu\to D''u$ and $D^{\prime\prime\star}u_\nu
\to D^{\prime\prime\star}u$ in~$L^2$ (just take $u_\nu$ to be a
regularization of $\psi_\nu u$). As a consequence, the basic a priori
inequality (4.7) extends to arbitrary forms $u$ such that
$u,~D''u,D^{\prime\prime\star}u\in L^2$ . Now, consider the Hilbert
space orthogonal decomposition
$$L^2(X,\Lambda^{p,q}T^\star_X\otimes F)=\Ker
D''\oplus({\Ker D''})^\perp,$$
observing that $\Ker D''$ is weakly (hence strongly) closed. Let $v=v_1+v_2$
be the decomposition of a smooth form $v\in\cD^{p,q}(X,F)$ with compact
support according to this decomposition ($v_1$, $v_2$ do not have compact
support in general$\,$!).
Since $(\Ker D'')^\perp\subset\Ker D^{\prime\prime\star}$ by duality
and $g,v_1\in\Ker D''$ by hypothesis, we get
$D^{\prime\prime\star}v_2=0$ and
$$|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle A^{-1}g,g\rangle \,dV_\omega
\int_X\langle Av_1,v_1\rangle \,dV_\omega$$
thanks to the Cauchy-Schwarz inequality. The a priori inequality (4.7)
applied to $u=v_1$ yields
$$\int_X\langle Av_1,v_1\rangle \,dV_\omega\le
\|D''v_1\|^2+\|D^{\prime\prime\star}v_1\|^2=\|D^{\prime\prime\star}v_1\|^2=
\|D^{\prime\prime\star}v\|^2.$$
Combining both inequalities, we find
$$|\langle g,v\rangle|^2\le\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)
\|D^{\prime\prime\star}v\|^2$$
for every smooth $(p,q)$-form $v$ with compact support. This shows that we
have a well defined linear form
$$w=D^{\prime\prime\star}v\longmapsto\langle v,g\rangle,\qquad
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)
\supset D^{\prime\prime\star}(\cD^{p,q}(F))\longrightarrow\bC$$
on the range of $D^{\prime\prime\star}$. This linear form
is continuous in $L^2$ norm and has norm $\le C$ with
$$C=\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)^{1/2}.$$
By the Hahn-Banach theorem, there is an element
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ with $||f||\le C$, such that
$\langle v,g\rangle=\langle D^{\prime\prime\star}v,f\rangle$ for every $v$,
hence $D''f=g$ in the sense of distributions. The inequality $||f||\le C$
is equivalent to the last estimate in the theorem.\qed
\endproof
%
The above $L^2$ existence theorem can be applied in the fairly general
context of {\it weakly pseudoconvex} manifolds. By this, we mean a complex
manifold $X$ such that there exists a smooth psh exhaustion function
$\psi$ on $X$ ($\psi$ is said to be an exhaustion if for every $c>0$ the
upperlevel set $X_c=\psi^{-1}(c)$ is relatively compact, i.e.\ $\psi(z)$
tends to $+\infty$ when $z$ is taken outside larger and larger compact
subsets of~$X$). In particular, every compact complex manifold $X$ is
weakly pseudoconvex (take $\psi=0$), as well as every Stein
manifold, e.g.\ affine algebraic submanifolds of $\bC^N$ (take
$\psi(z)=|z|^2$), open balls $X=B(z_0,r)$ $\big($take $\psi(z)=
1/(r-|z-z_0|^2)\big)$, convex open subsets, etc. Now, a basic observation
is that every weakly pseudoconvex K\"ahler manifold $(X,\omega)$ carries a
{\it complete} K\"ahler metric: let $\psi\ge 0$ be a psh exhaustion function
and set
$$\omega_\varepsilon=\omega+\varepsilon\,\ii d'd''\psi^2=
\omega+2\varepsilon(2\ii\psi d'd''\psi+\ii d'\psi\wedge d''\psi).$$
Then $|d\psi|_{\omega_\varepsilon}\le 1/\varepsilon$ and
$|\psi(x)-\psi(y)|\le\varepsilon^{-1}\delta_{\omega_\varepsilon}(x,y)$. 
It follows easily from this estimate that the geodesic balls are relatively
compact, hence $\delta_{\omega_\varepsilon}$ is complete
for every $\varepsilon>0$. Therefore, the $L^2$ existence theorem can be
applied to each K\"ahler metric $\omega_\varepsilon$, and by passing to
the limit it can even be applied to the non necessarily complete
metric~$\omega$. An important special case is the following

\claim{(5.2) Theorem} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $F$ be a hermitian
line bundle and let
$$\gamma_1(x)\le\ldots\le\gamma_n(x)$$
be the curvature eigenvalues $($i.e.\ the eigenvalues of $\ii\Theta_{F,h}$
with respect to the metric~$\omega)$ at every point. Assume that the
curvature is positive, i.e.\ $\gamma_1>0$ everywhere. Then for any form
$g\in L^2(X,\Lambda^{n,q}T^\star_X\otimes F)$ satisfying $D''g=0$ and
\hbox{$\int_X\langle(\gamma_1+\ldots+\gamma_q)^{-1}|g|^2\,dV_\omega<+\infty$},
there exists \hbox{$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$} such that
$D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X
(\gamma_1+\ldots+\gamma_q)^{-1}|g|^2\,dV_\omega.$$
\endclaim
%
\begproof{Proof.} Indeed, for $p=n$, Formula~4.10 shows that
$$\langle Au,u\rangle\ge(\gamma_1+\ldots+\gamma_q)|u|^2,$$
hence $\langle A^{-1}u,u\rangle\ge(\gamma_1+\ldots+\gamma_q)^{-1}|u|^2$.\qed
\endproof
%
An important observation is that the above theorem still applies when the
hermitian metric on $F$ is a singular metric with positive curvature in
the sense of currents. In fact, by standard
regularization techniques (convolution of psh functions by smoothing
kernels), the metric can be made smooth and the solutions obtained by
(5.1) or (5.2) for the smooth metrics have limits satisfying the desired
estimates. Especially, we get the following

\claim{(5.3) Corollary} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $F$ be a
holomorphic line bundle equipped with a singular metric whose local
weights are denoted $\varphi\in L^1_\loc$, i.e.\ $H=E^{-\varphi}$. 
Suppose that
$$\ii\Theta_{F,h}=\ii d'd''\varphi\ge\varepsilon\omega$$
for some $\varepsilon>0$. Then for any form $g\in L^2(X,\Lambda^{n,q}
T^\star_X\otimes F)$ satisfying $D''g=0$, there exists
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ such that  $D''f=g$ and
$$\int_X|f|^2e^{-\varphi}\,dV_\omega\le{1\over q\varepsilon}\int_X
|g|^2e^{-\varphi}\,dV_\omega.\eqno\square$$
\endclaim
%
Here we denoted somewhat incorrectly the metric by $|f|^2e^{-\varphi}$,
as if the weight $\varphi$ was globally defined on~$X$ (of course, this
is so only if $F$ is globally trivial). We will use this notation anyway,
because it clearly describes the dependence of the $L^2$ norm on the psh
weights.

\subsection*{\csect.B. Multiplier ideal sheaves and Nadel vanishing theorem}
We now introduce the concept of {\it multiplier ideal sheaf},
following A.~Nadel [Nad89]. The main idea actually goes back to the
fundamental works of Bombieri [Bom70] and H.~Skoda [Sko72a].

\claim{(5.4) Definition} Let $\varphi$ be a
psh function on an open subset $\Omega\subset X\,$; to $\varphi$ is
associated the ideal subsheaf $\cI(\varphi)\subset\cO_\Omega$ of germs of
holomorphic functions $f\in\cO_{\Omega,x}$ such that $|f|^2e^{-2\varphi}$
is integrable with respect to the Lebesgue measure in some local
coordinates near~$x$.
\endclaim
%
The zero variety $V(\cI(\varphi))$ is thus the set of points in
a neighborhood of which $e^{-2\varphi}$ is non integrable. Of course,
such points occur only if $\varphi$ has logarithmic poles. This is made
precise as follows.

\claim{(5.5) Definition} A psh function $\varphi$ is said to have a
logarithmic pole of coefficient $\gamma$ at a point $x\in X$ if the Lelong
number
$$\nu(\varphi,x):=\liminf_{z\to x}{\varphi(z)\over\log|z-x|}$$
is non zero and if $\nu(\varphi,x)=\gamma$.
\endclaim
%
\claim{(5.6) Lemma {\rm (Skoda [Sko72a])}} Let $\varphi$ be a psh function
on an open set $\Omega$ and let $x\in\Omega$.
\vskip3pt
\plainitem{\rm (a)} If $\nu(\varphi,x)<1$, then $e^{-2\varphi}$ is integrable in
a neighborhood of~$x$, in particular $\cI(\varphi)_x=\cO_{\Omega,x}$.
\vskip3pt
\plainitem{\rm (b)} If $\nu(\varphi,x)\ge n+s$ for some integer $s\ge 0$, then
$e^{-2\varphi}\ge C|z-x|^{-2n-2s}$ in a neighborhood of $x$ and
$\cI(\varphi)_x\subset\gm_{\Omega,x}^{s+1}$, where $\gm_{\Omega,x}$ is the
maximal ideal of $\cO_{\Omega,x}$.
\vskip3pt
\plainitem{\rm (c)} The zero variety $V(\cI(\varphi))$ of $\cI(\varphi)$
satisfies 
$$E_n(\varphi)\subset V(\cI(\varphi))\subset E_1(\varphi)$$
where $E_c(\varphi)=\{x\in X\,;\,\nu(\varphi,x)\ge c\}$ is the $c$-upperlevel
set of Lelong numbers of~$\varphi$.
\vskip0pt
\endclaim
%
\begproof{Proof.} (a) Set $\Theta=dd^c\varphi$ and $\gamma=\nu(\Theta,x)=
\nu(\varphi,x)$. Let $\chi$ be a cut-off function with support in a small
ball $B(x,r)$, equal to $1$ in $B(x,r/2)$. As $(dd^c\log|z|)^n=\delta_0$,
we get
$$\eqalign{\varphi(z)
&=\int_{B(x,r)}\chi(\zeta)\varphi(\zeta)(dd^c\log|\zeta-z|)^n\cr
&=\int_{B(x,r)}dd^c(\chi(\zeta)\varphi(\zeta))\wedge
\log|\zeta-z|(dd^c\log|\zeta-z|)^{n-1}\cr}$$
for $z\in B(x,r/2)$. Expanding $dd^c(\chi\varphi)$ and observing that
$d\chi=dd^c\chi=0$ on $B(x,r/2)$, we find
$$\varphi(z)=\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge
\log|\zeta-z|(dd^c\log|\zeta-z|)^{n-1}+\hbox{smooth terms}$$
on $B(x,r/2)$. Fix $r$ so small that
$$\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge(dd^c\log|\zeta-x|)^{n-1}
\le\nu(\Theta,x,r)<1.$$
By continuity, there exists $\delta,\varepsilon>0$ such that
$$I(z):=\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge(dd^c\log|\zeta-z|)^{n-1}
\le 1-\delta$$
for all $z\in B(x,\varepsilon)$. Applying Jensen's convexity inequality to
the probability measure
$$d\mu_z(\zeta)=I(z)^{-1}\chi(\zeta)\Theta(\zeta)\wedge
(dd^c\log|\zeta-z|)^{n-1},
$$
we find
$$\eqalign{
-\varphi(z)&=\int_{B(x,r)}I(z)\log|\zeta-z|^{-1}\,d\mu_z(\zeta)+O(1)\quad
\Longrightarrow\cr
e^{-2\varphi(z)}&\le C\int_{B(x,r)}|\zeta-z|^{-2I(z)}\,d\mu_z(\zeta).\cr}$$
As
$$d\mu_z(\zeta)\le C_1|\zeta-z|^{-(2n-2)}\Theta(\zeta)\wedge
(dd^c|\zeta|^2)^{n-1}=C_2|\zeta-z|^{-(2n-2)}d\sigma_\Theta(\zeta),$$
we get
$$e^{-2\varphi(z)}\le C_3\int_{B(x,r)}|\zeta-z|^{-2(1-\delta)-(2n-2)}
d\sigma_\Theta(\zeta),$$
and the Fubini theorem implies that $e^{-2\varphi(z)}$ is integrable on a
neighborhood of~$x$.
\vskip3pt

\noindent (b) If $\nu(\varphi,x)=\gamma$, the convexity properties of
psh functions, namely, the convexity of $\log r\mapsto\sup_{|z-x|=r}
\varphi(z)$ implies that
$$\varphi(z)\le\gamma\log|z-x|/r_0+M,$$
where $M$ is the supremum on $B(x,r_0)$. Hence there exists a constant
$C>0$ such that $e^{-2\varphi(z)}\ge C|z-x|^{-2\gamma}$ in a neighborhood
of $x$. The desired result follows from the identity
$$\int_{B(0,r_0)}{\big|\sum a_\alpha z^\alpha\big|^2\over|z|^{2\gamma}}dV(z)=
{\rm Const}\int_0^{r_0}\Big(\sum |a_\alpha|^2r^{2|\alpha|}\Big)
r^{2n-1-2\gamma}\,dr,$$
which is an easy consequence of Parseval's formula. In fact, if $\gamma$
has integral part $[\gamma]=n+s$, the integral converges if and only if
$a_\alpha=0$ for $|\alpha|\le s$.
\vskip3pt
\noindent (c) is just a simple formal consequence of (a) and (b).\qed
\endproof
%
\claim{(5.7) Proposition {\rm([Nad89])}} For any psh function
$\varphi$ on~$\Omega\subset X$, the sheaf
$\cI(\varphi)$ is a coherent sheaf of ideals over~$\Omega$. Moreover,
if $\Omega$ is a bounded Stein open set, the sheaf $\cI(\varphi)$ is
generated by any Hilbert basis of the $L^2$~space $\cH^2(\Omega,\varphi)$
of holomorphic functions $f$ 
on $\Omega$ such that $\int_\Omega|f|^2e^{-2\varphi}\,d\lambda<+\infty$.
\endclaim
%
\begproof{Proof.} Since the result is local, we may assume that $\Omega$ is a
bounded pseudoconvex open set in $\bC^n$. 
By the strong noetherian property of coherent sheaves, the family of
sheaves generated by finite subsets of $\cH^2(\Omega,\varphi)$ has
a maximal element on each compact subset of $\Omega$, hence
$\cH^2(\Omega,\varphi)$ 
generates a coherent ideal sheaf $\cJ\subset\cO_\Omega$. It is clear that
$\cJ\subset\cI(\varphi)$; in order to prove the equality,
we need only check that $\cJ_x+\cI(\varphi)_x\cap
\gm^{s+1}_{\Omega,x}=\cI(\varphi)_x$ for every integer $s$, in
view of the Krull lemma. Let $f\in\cI(\varphi)_x$ be defined 
in a neighborhood $V$ of $x$ and let $\theta$ be a cut-off function
with support in $V$ such that $\theta=1$ in a neighborhood of $x$.
We solve the equation $d''u=g:=d''(\theta f)$ by means of H\"ormander's
$L^2$ estimates~5.3, where $F$ is the trivial line bundle
$\Omega\times\bC$ equipped with the strictly psh weight 
$$\widetilde\varphi(z)=\varphi(z)+(n+s)\log|z-x|+|z|^2.$$ 
We get a solution $u$ such that
$\int_\Omega |u|^2e^{-2\varphi}|z-x|^{-2(n+s)}d\lambda<\infty$, thus
$F=\theta f-u$ is holomorphic, $F\in \cH^2(\Omega,\varphi)$ and 
$f_x-F_x=u_x\in\cI(\varphi)_x\cap\gm_{\Omega,x}^{s+1}$.
This proves the coherence. Now, $\cJ$ is generated by any Hilbert basis
of $\cH^2(\Omega,\varphi)$, because it is well-known that the space of 
sections of any coherent sheaf is a Fr\'echet space, therefore
closed under local $L^2$ convergence.\qed
\endproof
%
The multiplier ideal sheaves satisfy the following basic
functoriality property with respect to direct images of sheaves
by modifications.

\claim{(5.8) Proposition} Let $\mu:X'\to X$ be a modification
of non singular complex manifolds $($i.e.\ a proper generically 1:1
holomorphic map$)$, and let $\varphi$ be a psh function on~$X$. Then 
$$\mu_\star\big(\cO(K_{X'})\otimes\cI(\varphi\circ\mu)\big)=
\cO(K_X)\otimes\cI(\varphi).$$
\endclaim
%
\begproof{Proof.} Let $n=\dim X=\dim X'$ and let $S\subset X$ be an analytic
set such that $\mu:X'\ssm S'\to X\ssm S$ is a biholomorphism. By
definition of multiplier ideal sheaves, $\cO(K_X)\otimes\cI(\varphi)$ is
just the sheaf of holomorphic $n$-forms $f$ on open sets $U\subset X$
such that
$\ii^{n^2}f\wedge\overline f\,e^{-2\varphi}\in L^1_\loc(U)$. Since
$\varphi$ is locally bounded from above, we may even consider forms $f$
which are a priori defined only on $U\ssm S$, because $f$ will be
in $L^2_\loc(U)$ and therefore will automatically extend through~$S$.
The change of variable formula yields
$$\int_U\ii^{n^2}f\wedge\overline f\,e^{-2\varphi}=
\int_{\mu^{-1}(U)}\ii^{n^2}\mu^\star f\wedge\overline{\mu^\star f}
\,e^{-2\varphi\circ\mu},$$
hence $f\in\Gamma(U,\cO(K_X)\otimes\cI(\varphi))$ iff
$\mu^\star f\in\Gamma(\mu^{-1}(U),\cO(K_{X'})\otimes\cI(\varphi\circ
\mu))$. Proposition~5.8 is proved.\qed
\endproof
%
\claim{(5.9) Remark} \rm If $\varphi$ has analytic singularities
(according to Definition~1.10), the computation of $\cI(\varphi)$
can be reduced to a purely algebraic problem.

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

Now, consider the general case of analytic singularities and suppose
that $\varphi\sim{\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)$
near the poles. By the remarks after Definition~1.10, we may assume
that the $(f_j)$ are generators of the integrally closed ideal
sheaf $\cJ=\cJ(\varphi/\alpha)$, defined as the sheaf of holomorphic
functions $h$ such that $|h|\le C\exp(\varphi/\alpha)$. In this case,
the computation is made as follows (see also L.~Bonavero's work [Bon93],
where similar ideas are used in connection with ``singular'' holomorphic
Morse inequalities).

First, one computes a smooth modification $\mu:\widetilde X\to X$ of $X$ such
that $\mu^\star\cJ$ is an invertible sheaf $\cO(-D)$ associated with
a normal crossing divisor $D=\sum\lambda_jD_j$, where $(D_j)$ are the
components of the exceptional divisor of $\widetilde X$ (take the blow-up $X'$
of $X$ with respect to the ideal $\cJ$ so that the pull-back of $\cJ$
to $X'$ becomes an invertible sheaf $\cO(-D')$, then blow up again by
Hironaka [Hir64] to make $X'$ smooth and $D'$ have normal crossings).
Now, we have $K_{\widetilde X}=\mu^\star K_X+R$ where $R=\sum\rho_j
D_j$ is the zero divisor of the Jacobian function $J_\mu$ of the blow-up
map. By the direct image formula~5.8, we get
$$\cI(\varphi)=\mu_\star\big(\cO(K_{\widetilde X}-\mu^\star K_X)\otimes
\cI(\varphi\circ\mu)\big)=\mu_\star\big(\cO(R)\otimes
\cI(\varphi\circ\mu)\big).$$
Now, $(f_j\circ\mu)$ are generators of the ideal $\cO(-D)$, hence
$$\varphi\circ\mu\sim\alpha\sum\lambda_j\log|g_j|$$
where $g_j$ are local generators of $\cO(-D_j)$. We are thus reduced to
computing multiplier ideal sheaves in the case where the poles are
given by a $\bQ$-divisor with normal crossings $\sum\alpha\lambda_jD_j$.
We obtain $\cI(\varphi\circ\mu)=\cO(-\sum\lfloor\alpha\lambda_j\rfloor D_j)$,
hence
$$\cI(\varphi)=\mu_\star\cO_{\widetilde X}\big(\sum(\rho_j-\lfloor\alpha\lambda_j
\rfloor)D_j\big).\eqno\square$$
\endclaim
%
\claim{(5.10) Exercise} \rm Compute the multiplier ideal sheaf
$\cI(\varphi)$ associated with
$\varphi=\log(|z_1|^{\alpha_1}+\ldots+|z_p|^{\alpha_p})$
for arbitrary real numbers $\alpha_j>0$.\newline
{\it Hint:} using Parseval's formula and polar coordinates $z_j=r_j
e^{\ii\theta_j}$, show that the problem is equivalent to determining for
which $p$-tuples $(\beta_1,\ldots,\beta_p)\in\bN^p$ the integral
$$\int_{[0,1]^p}{r_1^{2\beta_1}\ldots r_p^{2\beta_p}\,r_1dr_1\ldots r_pdr_p
\over r_1^{2\alpha_1}+\ldots+r_p^{2\alpha_p}}
=\int_{[0,1]^p}{t_1^{(\beta_1+1)/\alpha_1}\ldots t_p^{(\beta_p+1)/\alpha_p}
\over t_1+\ldots+t_p}{dt_1\over t_1}\ldots{dt_p\over t_p}$$
is convergent. Conclude from this that $\cI(\varphi)$ is generated by
the monomials\break $\smash{z_1^{\beta_1}\ldots z_p^{\beta_p}}$ such that
$\sum(\beta_p+1)/\alpha_p>1$. (This exercise shows that the analytic
definition of $\cI(\varphi)$ is sometimes also quite convenient for
computations).\qed
\endclaim
%
Let $F$ be a line bundle over $X$ with a singular metric $h$ of
curvature current $\Theta_{F,h}$. If $\varphi$ is the weight
representing the metric in an open set $\Omega\subset X$, the ideal
sheaf $\cI(\varphi)$ is independent of the choice of the trivialization
and so it is the restriction to $\Omega$ of a global coherent sheaf
$\cI(h)$ on~$X$. We will sometimes still write $\cI(h)=\cI(\varphi)$ by
abuse of notation. In this context, we have the following fundamental
vanishing theorem, which is probably one of the most central results of
analytic and algebraic geometry (as we will see later, it contains the
Kawamata-Viehweg vanishing theorem as a special case).

\claim{(5.11) Nadel vanishing theorem {\rm([Nad89], [Dem93b])}} 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
$h$ of weight~$\varphi$. Assume that $\ii\Theta_{F,h}\ge\varepsilon\omega$
for some continuous positive function $\varepsilon$ on~$X$. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(h)\big)=0\qquad
\hbox{for all $q\ge 1$.}$$
\endclaim
%
\begproof{Proof.} Let $\cL^q$ be the sheaf of germs of $(n,q)$-forms 
$u$ with values in $F$ and with measurable coefficients, such that
both $|u|^2e^{-2\varphi}$ and $|d''u|^2e^{-2\varphi}$ are locally
integrable. The $d''$ operator defines a complex of sheaves
$(\cL^\bu,d'')$ which is a resolution of the sheaf
$\cO(K_X+F)\otimes\cI(\varphi)$: indeed, the kernel of $d''$ in degree
$0$ consists of all germs of holomorphic $n$-forms with values in $F$
which satisfy the integrability condition; hence the coefficient
function lies in $\cI(\varphi)$; the exactness in degree $q\ge 1$
follows from Corollary~5.3 applied on arbitrary small balls.
Each sheaf $\cL^q$ is a $\cC^\infty$-module, so $\cL^\bu$ 
is a resolution by acyclic sheaves. Let $\psi$ be a smooth psh
exhaustion function on~$X$. Let us apply Corollary~5.3 globally on~$X$,
with the original metric of $F$ multiplied by the factor
$e^{-\chi\circ\psi}$, where $\chi$ is a convex increasing function of
arbitrary fast growth at infinity. This factor can be used to ensure
the convergence of integrals at infinity. By Corollary~5.3, we conclude
that $H^q\big(\Gamma(X,\cL^\bu)\big)=0$ for $q\ge 1$. The theorem
follows.\qed
\endproof 

\claim{(5.12) Corollary} Let $(X,\omega)$, $F$ and $\varphi$ be as in
Theorem~5.11 and let $x_1,\ldots,x_N$ be isolated points in the zero
variety~$V(\cI(\varphi))$. Then there is a surjective map
$$H^0(X,K_X+F)\dlraw\bigoplus_{1\le j\le N}\cO(K_X+L)_{x_j}\otimes
\big(\cO_X/\cI(\varphi)\big)_{x_j}.$$
\endclaim
%
\begproof{Proof.} Consider the long exact sequence of cohomology associated to the
short exact sequence $0\to\cI(\varphi)\to\cO_X\to\cO_X/\cI(\varphi)\to 0$
twisted by $\cO(K_X+F)$, and apply Theorem~5.11 to obtain the vanishing of
the first $H^1$ group. The asserted surjectivity property follows.\qed
\endproof
%
\claim{(5.13) Corollary} Let $(X,\omega)$, $F$ and $\varphi$ be as in
Theorem~5.11 and suppose that the weight function $\varphi$ is such that
$\nu(\varphi,x)\ge n+s$ at some point $x\in X$ which is an isolated point
of $E_1(\varphi)$. Then $H^0(X,K_X+F)$ generates all $s$-jets at~$x$.
\endclaim
%
\begproof{Proof.} The assumption is that $\nu(\varphi,y)<1$ for $y$ near $x$,
$y\ne x$. By Skoda's lemma 5.6~b), we conclude that $e^{-2\varphi}$ is
integrable at all such points~$y$, hence $\cI(\varphi)_y=\cO_{X,y}$, whilst
$\cI(\varphi)_x\subset\gm_{X,x}^{s+1}$ by 5.6~a). Corollary 5.13 is thus
a special case of 5.12.\qed
\endproof
%
The philosophy of these results (which can be seen as generalizations
of the H\"or\-man\-der-Bombieri-Skoda theorem [Bom70], [Sko72a,~75]) is that
the problem of constructing holomorphic sections of $K_X+F$ can be
solved by constructing suitable hermitian metrics on $F$ such that the
weight $\varphi$ has isolated poles at given points~$x_j$.

\claim{(5.14) Exercise} \rm Assume that $X$ is compact and that $L$ is
a positive line bundle on~$X$. Let $\{x_1,\ldots,x_N\}$ be a finite set.
Show that there are constants $a,b\ge 0$ depending only on $L$ and $N$
such that $H^0(X,mL)$ generates jets of any order $s$ at all points $x_j$
for $m\ge as+b$.\newline
{\it Hint:} Apply Corollary~5.12 to $F=-K_X+mL$, with a singular
metric on $L$ of the form $h=h_0e^{-\varepsilon\psi}$, where $h_0$ is
smooth of positive curvature, $\varepsilon>0$ small and
$\psi(z)\sim\log|z-x_j|$ in a neighborhood of $x_j$.\newline
Derive the Kodaira embedding theorem from the above result:
\endclaim
%
\claim{(5.15) Theorem {\rm(Kodaira embedding theorem)}} If $L$ is a 
line bundle on a compact
complex manifold, then $L$ is ample if and only if $L$ is positive.\qed
\endclaim
%
\claim{(5.16) Exercise {\rm (solution of the Levi problem)}} \rm Show
that the following two properties are equivalent.
\vskip3pt
\plainitem{\rm (a)} $X$ is strongly pseudoconvex, i.e.\ $X$ admits a strongly psh
exhaustion function.
\vskip3pt
\plainitem{\rm (b)} $X$ is Stein, i.e.\ the global holomorphic functions
$H^0(X,\cO_X)$ separate points and yield local coordinates at any point,
and $X$ is holomorphically convex $($this means that for any discrete
sequence $z_\nu$ there is a function $f\in H^0(X,\cO_X)$ such that
$|f(z_\nu)|\to\infty)$.\qed
\vskip0pt
\endclaim
%
\claim{(5.17) Remark} \rm As long as forms of bidegree $(n,q)$ are
considered, the $L^2$ estimates can be extended to complex spaces with
arbitrary singularities. In fact, if $X$ is a complex space and $\varphi$
is a psh weight function on~$X$, we may still define a sheaf $K_X(\varphi)$
on~$X$, such that the sections on an open set $U$ are the holomorphic
$n$-forms $f$ on the regular part $U\cap X_\reg$, satisfying the
integrability condition $\ii^{n^2}f\wedge\overline f\,e^{-2\varphi}\in
L^1_\loc(U)$. In this setting, the functoriality property 5.8 becomes
$$\mu_\star\big(K_{X'}(\varphi\circ\mu)\big)=K_X(\varphi)$$
for arbitrary complex spaces $X$, $X'$ such that $\mu:X'\to X$
is a modification. If $X$ is nonsingular we have $K_X(\varphi)=
\cO(K_X)\otimes\cI(\varphi)$, however, if $X$ is
singular, the symbols $K_X$ and $\cI(\varphi)$ must not be
dissociated. The statement of the Nadel vanishing theorem becomes
$H^q(X,\cO(F)\otimes K_X(\varphi))=0$ for $q\ge 1$, under the same
assumptions ($X$ K\"ahler and weakly pseudoconvex, curvature
$\ge\varepsilon\omega$). The proof can be obtained by restricting
everything to $X_\reg$. Although in general $X_\reg$ is not weakly
pseudoconvex (e.g.\ in case $\codim X_\sing\ge 2$), $X_\reg$ is
always K\"ahler complete (the complement of a proper analytic subset in
a K\"ahler weakly pseudoconvex space is complete K\"ahler, see
e.g.\ [Dem82b]). As a consequence, Nadel's vanishing theorem is
essentially insensitive to the presence of singularities.\qed
\endclaim
%
\section{Numerically effective and pseudo-effective line bundles}

\subsection*{\csect.A. Pseudo-effective line bundles and metrics with 
minimal singularities}
The concept of pseudo-effectivity is quite general and makes sense 
on an arbitrary compact complex manifold~$X$ (no projective or 
K\"ahler assumption is needed). In this general context, it is better to 
work with $\ddbar$-cohomology classes instead of De Rham cohomology classes: 
we define
$$H_{\ddbar}^{p,q}(X)=\big\{d\hbox{-closed $(p,q)$-forms}\}/
\big\{\ddbar\hbox{-exact $(p,q)$-forms}\}.\leqno(6.1)$$
By means of the Fr\"olicher spectral sequence, it is easily shown
that these cohomology groups are finite dimensional and can be computed
either with spaces of smooth forms or with currents. In both cases,
the quotient topology of $\smash{H_{\ddbar}^{p,q}(X)}$ induced by
the Fr\'echet topology of smooth forms or by the weak topology of
currents is Hausdorff. Clearly $H_{\ddbar}^\bu(X)$ is a bigraded
algebra. This algebra can be shown to be isomorphic to the usual 
De Rham cohomology
algebra $H^\bu(X,\bC)$ if $X$ is K\"ahler or more generally if $X$ is
in the Fujiki class $\cC$ of manifolds bimeromorphic to K\"ahler
manifolds.

\claim{(6.2) Definition} Let $L$ we a holomorphic line bundle on a compact
complex manifold $X$.
we say that $L$ pseudo-effective if $c_1(L)\in H_{\ddbar}^{1,1}(X)$ is 
the cohomology class of some
closed positive current~$T$, i.e.\ if $L$ can be equipped with a singular
hermitian metric $h$ with $T={\ii\over 2\pi}\Theta_{L,h}\ge 0$ as a current.
\endclaim
%
The locus where $h$ has singularities turns out to be extremely important.
The following definition was introduced in [DPS00].

\claim{(6.3) Definition} Let $L$ be a pseudo-effective line bundle
on a compact complex manifold~$X$. Consider two hermitian metrics 
$h_1$, $h_2$ on $L$ with curvature $i\Theta_{L,h_j}\ge 0$
in the sense of currents.
\vskip3pt
\plainitem{\rm(i)} We will write $h_1\preccurlyeq h_2$, and say that $h_1$ 
is less singular than $h_2$, if there exists a constant $C>0$ such that
$h_1\le C h_2$.
\vskip3pt
\plainitem{\rm(ii)} We will write $h_1\sim h_2$, and say that $h_1$, $h_2$ are
equivalent with respect to singularities, if there exists a constant
$C>0$ such that $C^{-1}h_2\le h_1\le C h_2$.
\vskip0pt
\endclaim
%
\noindent
Of course $h_1\preccurlyeq h_2$ if and only if the associated weights
in suitable trivializations locally satisfy $\varphi_2\le \varphi_1+C$.
This implies in particular $\nu(\varphi_1,x)\le\nu(\varphi_2,x)$ at each
point. The above definition is motivated by the following observation.

\claim{(6.4) Theorem} For every pseudo-effective line bundle $L$ over
a compact complex manifold $X$, there exists up to equivalence of
singularities a unique class of hermitian metrics $h$ with minimal
singularities such that $i\Theta_{L,h}\ge 0$.
\endclaim
%
\begproof{Proof.} The proof is almost trivial. We fix once for all a smooth
metric $h_\infty$ (whose curvature is of random sign and signature),
and we write singular metrics of $L$ under the form $h=h_\infty e^{-\psi}$. 
The condition $i\Theta_{L,h}\ge 0$ is equivalent to 
${i\over2\pi}\ddbar\psi\ge -u$ where $u={i\over 2\pi}\Theta_{L,h_\infty}$. This
condition implies that $\psi$ is plurisubharmonic up to the addition
of the weight $\varphi_\infty$ of $h_\infty$, and therefore locally
bounded from above.  Since we are concerned with metrics only up to
equivalence of singularities, it is always possible to adjust
$\psi$ by a constant in such a way that $\sup_X\psi=0$. We now set
$$
h_{\min}=h_\infty e^{-\psi_{\min}},\qquad \psi_{\min}(x)=\sup_\psi\psi(x)
$$
where the supremum is extended to all functions $\psi$ such that
$\sup_X\psi=0$ and ${i\over2\pi}\ddbar\psi\ge -u$. By standard
results on plurisubharmonic functions (see Lelong [Lel69]), 
$\psi_{\min}$ still satisfies ${i\over2\pi}\ddbar\psi_{\min}\ge -u$
(i.e.\ the weight $\varphi_\infty+\psi_{\min}$ of $h_{\min}$ is
plurisubharmonic), and $h_{\min}$ is obviously the metric with
minimal singularities that we were looking for. [In~principle one
should take the upper semicontinuous regularization $\psi_{\min}^*$
of $\psi_{\min}$ to really get a plurisubharmonic weight, but since 
$\psi_{\min}^*$ also participates to the upper envelope, we obtain
here $\psi_{\min}=\psi_{\min}^*$ automatically].
\qed
\endproof
%
\claim{(6.5) Remark} \rm In general, the supremum 
$\psi=\sup_{j\in I}\psi_j$ of a 
locally dominated family of plurisubharmonic functions $\psi_j$ is 
not plurisubharmonic strictly speaking, but its ``upper semi-continuous
regularization'' $\psi^*(z)=\limsup_{\zeta\to z}\psi(\zeta)$ is
plurisubharmonic and coincides almost everywhere with $\psi$, with
$\psi^*\ge\psi$. However, in the context of (6.5), $\psi^*$ still
satisfies $\psi^*\le 0$ and ${i\over2\pi}\ddbar\psi\ge -u$, hence
$\psi^*$ participates to the upper envelope. As a consequence, we have
$\psi^*\le \psi$ and thus $\psi=\psi^*$ is indeed plurisubharmonic. 
Under a strict
positivity assumption, namely if $L$ is a big line bundle (i.e.\ the
curvature can be taken to be strictly positive in the sense of
currents, see 6.12) and (6.14$\,$f)), then $h_{\min}$ can be shown
to possess some regularity properties. The reader may consult
[BmD09] for a rather general (but certainly non trivial) proof that 
$\psi_{\min}$ possesses locally bounded
second derivatives $\partial^2\psi_{\min}/\partial z_j\partial\overline z_k$
outside an analytic set $Z\subset X\,$; in other words, $i\Theta_{L,h_{\min}}$
has locally bounded coefficients on~$X\ssm Z$.
\endclaim
%
\claim{(6.6) Definition} Let $L$ be a pseudo-effective line bundle.
If $h$ is a singular hermitian metric such 
that $i\Theta_{L,h}\ge 0$ and
$$
H^0(X,mL\otimes\cI(h^{\otimes m}))\simeq H^0(X,mL)\qquad
\hbox{for all $m\ge 0$},
$$
we say that $h$ is an {\it analytic Zariski decomposition} of~$L$. 
\endclaim
%
In other words, we require that $h$ has singularities so mild that
the vanishing conditions prescribed by the multiplier ideal sheaves
$\cI(h^{\otimes m})$ do not kill any sections of $L$ and its multiples.

\claim{(6.7) Exercise} {\rm A special case is when there is an 
isomorphism $pL=A+E$ where $A$ and $E$ are effective divisors such 
that $H^0(X,mpL)=H^0(X,mA)$
for all $m$ and $\cO(A)$ is generated by sections. Then $A$ possesses
a smooth hermitian metric~$h_A$, and this metric defines a singular hermitian
metric $h$ on $L$ with poles ${1\over p}E$ and curvature 
${1\over p}\Theta_{A,h_A}+{1\over p}[E]$. Show that this metric $h$
is an analytic Zariski decomposition.\newline
{\it Note}: when $X$ projective and there is a decomposition $pL=A+E$
with $A$ nef (see (6.9) below), $E$ effective and
$H^0(X,mpL)=H^0(X,mA)$ for all $m$, one says that this is an 
{\it algebraic Zariski decomposition} of $L$. It can be shown that Zariski
decompositions exist in dimension $2$, but in higher dimension one can see
that they do not exist.}
\endclaim
%
\claim{(6.8) Theorem} The metric $h_{\min}$ with minimal singularities
provides an analytic Zariski decomposition.
\endclaim
%
It follows that an analytic Zariski decomposition always exists (while
algebraic decompositions do not exist in general, especially in
dimension $3$ and more.)

\begproof{Proof.} Let $\sigma\in H^0(X,mL)$ be any section.
Then we get a singular metric
$h$ on $L$ by putting $|\xi|_h=|\xi/\sigma(x)^{1/m}|$ for $\xi\in L_x$, and
it is clear that $|\sigma|_{h^m}=1$ for this metric. Hence
$\sigma\in H^0(X,mL\otimes\cI(h^{\otimes m}))$, and a fortiori
$\sigma\in H^0(X,mL\otimes\cI(h^{\otimes m}_{\min}))$ since $h_{\min}$ is
less singular than~$h$.\qed
\endproof
%
\subsection*{\csect.B. Nef line bundles}
Many problems of algebraic geometry (e.g.\ problems of classification of
algebraic surfaces or higher dimensional varieties) lead in a natural
way to the study of line bundles satisfying semipositivity conditions.
It turns out that semipositivity in the sense of curvature (at least,
as far as smooth metrics are considered) is not a very satisfactory
notion. A more flexible notion perfectly suitable for algebraic purposes
is the notion of {\it numerical effectivity}. The goal of this section
is to give a few fundamental algebraic definitions and to discuss their
differential geometric counterparts. We first suppose that $X$ is a
projective algebraic manifold, \hbox{$\dim X=n$}.

\claim{(6.9) Definition} A holomorphic line bundle $L$ over a projective
manifold $X$ is said to be numerically effective, nef for short, if
$L\cdot C=\int_C c_1(L)\ge 0$ for every curve $C\subset X$.
\endclaim
%
If $L$ is nef, it can be shown that $L^p\cdot Y=\int_Y c_1(L)^p\ge 0$
for any $p$-dimensional subvariety $Y\subset X$ (see e.g.\ [Har70]).
In relation to this, let us recall the Nakai-Moishezon ampleness
criterion: a line bundle $L$ is ample if and only if $L^p\cdot Y>0$ for
every $p$-dimensional subvariety~$Y$. From this, we easily infer

\claim{(6.10) Proposition} Let $L$ be a line bundle on a projective
algebraic manifold~$X$, on which an ample line bundle $A$ and a hermitian
metric $\omega$ are given. The following properties are equivalent:
\vskip3pt
\plainitem{\rm (a)} $L$ is nef$\,;$
\vskip3pt
\plainitem{\rm (b)} for any integer $k\ge 1$, the line bundle $kL+A$ is ample$\,;$
\vskip3pt
\plainitem{\rm (c)} for every $\varepsilon>0$, there is a smooth metric
$h_\varepsilon$ on $L$ such that $\ii\Theta_{L,h_\varepsilon}\ge
-\varepsilon\omega$.
\vskip0pt
\endclaim
%
\begproof{Proof.} (a) $\Rightarrow$ (b). If $L$ is nef and $A$ is ample then
clearly $kL+A$ satisfies the Nakai-Moishezon criterion, hence
$kL+A$ is ample.
\vskip3pt
\noindent (b) $\Rightarrow$ (c). Condition (c) is independent of the choice
of the hermitian metric, so we may select a metric $h_A$ on $A$ with
positive curvature and set $\omega=\ii\Theta_{A,h_A}$. If $kL+A$ is ample,
this bundle has a metric $h_{kL+A}$ of positive curvature. Then the
metric $h_L=(h_{kL+A}\otimes h_A^{-1})^{1/k}$ has curvature
$$\ii\Theta_{L,h_L}={1\over k}\big(\ii\Theta(kL+A)-\ii\Theta_A\big)\ge
-{1\over k}\ii\Theta_{A,h_A}\,;$$
in this way the negative part can be made smaller than $\varepsilon\,\omega$
by taking $k$ large enough.
\vskip3pt
\noindent (c) $\Rightarrow$ (a). Under hypothesis (c), we get
$L\cdot C=\int_C{\ii\over2\pi}\Theta_{L,h_\varepsilon}\ge-
{\varepsilon\over2\pi}\int_C\omega$ for every curve $C$ and every
$\varepsilon>0$, hence $L\cdot C\ge 0$ and $L$ is nef.\qed
\endproof
%
Let now $X$ be an arbitrary compact complex manifold. Since there need
not exist any curve in $X$, Property 6.10~c) is simply taken as a definition
of nefness ([DPS94]):

\claim{(6.11) Definition} A line bundle $L$ on a compact complex
manifold $X$ is said to be nef if for every $\varepsilon>0$, there is a
smooth hermitian metric $h_\varepsilon$ on $L$ such that
$\ii\Theta_{L,h_\varepsilon}\ge -\varepsilon\omega$.
\endclaim
%
In general, it is not possible to extract a smooth limit $h_0$ such that
$\ii\Theta_{L,h_0}\ge 0$. The following simple example is given in
[DPS94] (Example~1.7). Let $E$ be a non trivial extension
$0\to\cO\to E\to\cO\to 0$ over an elliptic curve $C$ and let $X=\bP(E)$
(with notation as in (4.12)) be the corresponding ruled surface 
over~$C$. Then $L=\cO_{\bP(E)}(1)$ is
nef but does not admit any smooth metric of nonnegative curvature.
This example answers negatively a question raised by Fujita [Fuj83].

Let us now introduce the important concept of {\it Kodaira-Iitaka dimension}
of a line bundle.

\claim{(6.12) Definition} If $L$ is a line bundle, the Kodaira-Iitaka
dimension $\kappa(L)$ is the supremum of the rank of the canonical maps
$$\Phi_m:X\ssm B_m\longrightarrow \bP(V_m),~~~~x\longmapsto H_x=\{\sigma\in
V_m\,;\,\sigma(x)=0\},~~~~m\ge 1$$
with $V_m=H^0(X,mL)$ and $B_m=\bigcap_{\sigma\in V_m}\sigma^{-1}(0)={}$base
locus of $V_m$. In case $V_m=\{0\}$ for all $m\ge 1$, we set $\kappa(L)=
-\infty$.\vskip5pt
\noindent
A line bundle is said to be {\rm big} if $\kappa(L)=\dim X$.
\endclaim
%
The following lemma is well-known (the proof is a rather elementary
consequence of the Schwarz lemma).

\claim{(6.13) Serre-Siegel lemma {\rm([Ser54], [Sie55])}} Let $L$ be any
line bundle on a compact complex manifold. Then we have
$$h^0(X,mL)\le O(m^{\kappa(L)})\qquad\hbox{for $m\ge 1$,}$$
and $\kappa(L)$ is the smallest constant for which this estimate holds.\qed
\endclaim
%
\subsection*{\csect.C. Pseudoeffective line bundles and positive cones}
We now discuss the various concepts of positive cones in the space of
numerical classes of line bundles, and establish a simple dictionary
relating these concepts to corresponding concepts in the context of
differential geometry.

Let us recall that an integral cohomology class in
$H^2(X,\bZ)$ is the first Chern class of a holomorphic (or algebraic)
line bundle if and only if it lies in the
{\it Neron-Severi} group 
$$\NS(X)=\Ker\big(H^2(X,\bZ)\to H^2(X,\cO_X)\big)$$
(this fact is just an elementary consequence of the exponential exact
sequence\break $0\to\bZ\to\cO\to\cO^\star\to 0$). If $X$ is compact
K\"ahler, as we will suppose from now on in this section,
this is the same as saying that the class is of type $(1,1)$
with respect to Hodge decomposition.

Let $\NS_\bR(X)$ be the real vector space $\NS(X)\otimes\bR\subset
H^2(X,\bR)$. We define four convex cones
$$\eqalign{
&\Amp(X)\subset \Eff(X)\subset \NS_\bR(X),\cr
&\Nef(X)\subset \Psef(X)\subset\NS_\bR(X)\cr}$$
which are, respectively, the {\it convex cones} generated by Chern classes
$c_1(L)$ of ample and effective line bundles, resp.\ the {\it closure of
the convex cones} generated by numerically effective and pseudo-effective
line bundles; we say that $L$ is effective if $mL$ has a section for some
$m>0$, i.e.\ if $\cO(mL)\simeq\cO(D)$ for some effective divisor~$D$.

For each of the ample, effective, nef and pseudo-effective
cones, the first Chern class $c_1(L)$ of a line bundle $L$ lies in the
cone if and only if $L$ has the corresponding property (for $\Psef(X)$
use the fact that the space of positive currents of mass $1$ is
weakly compact; the case of all other cones is obvious).

\claim{(6.14) Proposition} Let $(X,\omega)$ be a compact K\"ahler manifold.
The numerical cones satisfy the following properties.
\vskip3pt
\plainitem{\rm (a)} $\Amp(X)=\Amp(X)^\circ\subset \Nef(X)^\circ$,~~
$\Nef(X)\subset \Psef(X)$.
\vskip3pt
\plainitem{\rm (b)} If moreover $X$ is projective algebraic, we have
$\Amp(X)=\Nef(X)^\circ$ $($therefore $\overline{\Amp(X)}=\Nef(X))$,
and $\overline{\Eff(X)}=\Psef(X)$.
\vskip3pt\noindent
If $L$ is a line bundle on $X$ and $h$ denotes a hermitian metric on $L$,
the following properties are equivalent:
\vskip3pt
\plainitem{\rm (c)} $c_1(L)\in \Amp(X)$
$\Leftrightarrow$ $\exists\varepsilon>0$, $\exists h$ smooth such that
$\ii\Theta_{L,h}\ge\varepsilon\omega$.
\vskip3pt
\plainitem{\rm (d)} $c_1(L)\in \Nef(X)$
$\Leftrightarrow$ $\forall\varepsilon>0$, $\exists h$ smooth such that
$\ii\Theta_{L,h}\ge-\varepsilon\omega$.
\vskip3pt
\plainitem{\rm (e)} $c_1(L)\in \Psef(X)$
$\Leftrightarrow$ $\exists h$ possibly singular such that
$\ii\Theta_{L,h}\ge 0$.
\vskip3pt
\plainitem{\rm (f)} If moreover $X$ is projective algebraic, then\newline
$c_1(L)\in \Psef(X)^\circ$~$\Leftrightarrow\kappa(L)=\dim X$\newline
\phantom{$c_1(L)\in \Psef(X)^\circ$}~$\Leftrightarrow\exists\varepsilon>0$,
$\exists h$ possibly singular such that
$\ii\Theta_{L,h}\ge\varepsilon\omega$.
\vskip0pt
\endclaim
%
\begproof{Proof.} (c) and (d) are already known and (e) is a definition.
\vskip3pt\noindent
a) The ample cone $\Amp(X)$ is always open by definition and contained
in $\Nef(X)$, so the first inclusion is obvious ($\Amp(X)$ is
of course empty if $X$ is not projective algebraic). Let us now prove
that $\Nef(X)\subset \Psef(X)$. Let $L$ be a line bundle with
$c_1(L)\in \Nef(X)$. Then for every $\varepsilon>0$, there is
a current $T_\varepsilon={\ii\over 2\pi}\Theta_{L,h_\varepsilon}\ge
-\varepsilon\omega$. Then $T_\varepsilon+\varepsilon\omega$ is
a closed positive current and the family is uniformly bounded
in mass for $\varepsilon\in{}]0,1]$, since
$$\int_X(T_\varepsilon+\varepsilon\omega)\wedge\omega^{n-1}=
\int_X c_1(L)\wedge\omega^{n-1}+\varepsilon\int_X\omega^n.$$
By weak compactness, some subsequence converges to a weak limit
$T\ge 0$ and $T\in c_1(L)$ (the cohomology class $\{T\}$ of a current
is easily shown to depend continuously on $T$ with respect to the weak
topology; use e.g.\ Poincar\'e duality to check this).
\vskip3pt\noindent
b) If $X$ is projective, the equality $\Amp(X)=\Nef(X)^\circ$
is a simple consequence of 6.10~b) and of the fact that ampleness (or
positivity) is an open property. It remains to show that
$\Psef(X)\subset\overline{\Eff(X)}$. Let $L$ be a line bundle with
$c_1(L)\in \Psef(X)$ and let $h_L$ be a singular hermitian on $L$
such that $T={\ii\over 2\pi}\Theta_{L,h_L}\ge 0$. Fix a point $x_0\in X$
such that the Lelong number of $T$ at $x_0$ is zero, and take a sufficiently
positive line bundle $A$ (replacing $A$ by a multiple if necessary), such
that $A-K_X$ has a singular metric $h_{A-K_X}$ of curvature
$\ge\varepsilon\omega$ and such that $h_{A-K_X}$ is smooth on
$X\ssm\{x_0\}$ and has an isolated logarithmic pole of Lelong number $\ge n$
at~$x_0$. Then apply Corollary 5.13 to $F=mL+A-K_X$ equipped with
the metric $h_L^{\otimes m}\otimes h_{A-K_X}$. Since the weight
$\varphi$ of this metric has a Lelong number $\ge n$ at $x_0$ and
a Lelong number equal to the Lelong number of $T={\ii\over2\pi}\Theta_{L,h_L}$
at nearby points, $\limsup_{x\to x_0}\nu(T,x)=\nu(T,x_0)=0$,
Corollary 5.13 implies that $H^0(X,K_X+F)=H^0(X,mL+A)$ has a section
which does not vanish at~$x_0$. Hence there is an effective divisor $D_m$
such that $\cO(mL+A)=\cO(D_m)$ and $c_1(L)={1\over m}\{D_m\}-{1\over m}
c_1(A)=\lim{1\over m}\{D_m\}$ is in $\overline{\Eff(X)}$.\qed
\vskip3pt

\noindent
f) Fix a nonsingular ample divisor $A$. If $c_1(L)\in \Psef(X)^\circ$, there
is an integer $m>0$ such that $c_1(L)-{1\over m}c_1(A)$ is still effective,
hence for $m,p$ large we have $mpL-pA=D+F$ with an effective divisor $D$
and a numerically trivial line bundle $F$. This implies
$\cO(kmpL)=\cO(kpA+kD+kF)\supset\cO(kpA+kF)$, hence $h^0(X,kmpL)\ge h^0(X,kpA+
kF)\sim (kp)^nA^n/n!$ by the Riemann-Roch formula. Therefore $\kappa(L)=n$.

If $\kappa(L)=n$, then $h^0(X,kL)\ge ck^n$ for $k\ge k_0$ and $c>0$. 
The exact cohomology sequence
$$0\longrightarrow H^0(X,kL-A)\longrightarrow H^0(X,kL)\longrightarrow H^0(A,kL_{\restriction A})$$
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 $kL\simeq\cO(A+D)$. Select a smooth metric $h_A$
on $A$ such that ${\ii\over2\pi}\Theta_{A,h_A}\ge\varepsilon_0\omega$ for
some $\varepsilon_0>0$, and take the singular metric 
on $\cO(D)$ with weight function $\varphi_D=\sum\alpha_j\log|g_j|$
described in Example~3.13. Then the metric with weight $\varphi_L=
{1\over k}(\varphi_A+\varphi_D)$ on $L$ yields
$${\ii\over 2\pi}\Theta_{L,h_L}={1\over k}
\Big({\ii\over2\pi}\Theta_{A,h_A}+[D]\Big)\ge(\varepsilon_0/k)\,\omega,$$
as desired.

Finally, the curvature condition $\ii\Theta_{L,h}\ge\varepsilon\omega$ in the
sense of currents yields by definition $c_1(L)\in \Psef(X)^\circ$.\qed
\endproof
%
Before going further, we need a lemma.

\claim{(6.15) Lemma} Let $X$ be a compact K\"ahler $n$-dimensional
manifold, let $L$ be a nef line bundle on~$X$, and let $E$ be an
arbitrary holomorphic vector bundle. Then
$h^q(X,\cO(E)\otimes\cO(kL))=o(k^n)$
as $k\to+\infty$, for every $q\ge1$. If $X$ is projective algebraic, the
following more precise bound holds:
$$h^q(X,\cO(E)\otimes\cO(kL))=O(k^{n-q}),\qquad\forall q\ge0.$$
\endclaim
%
\begproof{Proof.} The K\"ahler case will be proved in Section~12, as a
consequence of the holomorphic Morse inequalities. In the projective
algebraic case, we proceed by induction on $n=\dim X$. If $n=1$ the result
is clear, as well as if $q=0$. Now let $A$ be a nonsingular ample divisor
such that $E\otimes\cO(A-K_X)$ is Nakano positive. Then the Nakano
vanishing theorem applied to the vector bundle $F=E\otimes\cO(kL+A-K_X)$
shows that $H^q(X,\cO(E)\otimes\cO(kL+A))=0$ for all $q\ge 1$.
The exact sequence
$$0\to\cO(kL)\to\cO(kL+A)\to\cO(kL+A)_{\restriction A}\to0$$
twisted by $E$ implies
$$H^q(X,\cO(E)\otimes\cO(kL))\simeq H^{q-1}(A,\cO(E_{\restriction A}
\otimes\cO(kL+A)_{\restriction A}),$$
and we easily conclude by induction since $\dim A=n-1$. Observe that the
argument does not work any more if $X$ is not algebraic. It seems to be
unknown whether the $O(k^{n-q})$ bound still holds in that case.\qed
\endproof
%
\claim{(6.16) Corollary} If $L$ is nef, then $L$ is big $($i.e.\
$\kappa(L)=n)$ if and only if $L^n>0$. Moreover, if $L$ is nef and big,
then for every $\delta>0$, $L$ has a singular metric $h=e^{-2\varphi}$
such that $\,\max_{x\in X}\nu(\varphi,x)\le\delta$ and
$\ii\Theta_{L,h}\ge\varepsilon\,\omega$ for some $\varepsilon>0$.
The metric $h$ can be chosen to be smooth on the complement of a fixed
divisor $D$, with logarithmic poles along $D$.
\endclaim
%
\begproof{Proof.} By Lemma~6.15 and the Riemann-Roch formula, we have
$h^0(X,kL)=\chi(X,kL)+o(k^n)=k^nL^n/n!+o(k^n)$, whence the first
statement. If $L$ is big, the proof made in (6.14~f) shows that
there is a singular metric $h_1$ on $L$ such that
$${\ii\over 2\pi}\Theta_{L,h_1}={1\over k}
\Big({\ii\over2\pi}\Theta_{A,h_A}+[D]\Big)$$
with a positive line bundle $A$ and an effective divisor $D$.
Now, for every $\varepsilon>0$, there is a smooth metric $h_\varepsilon$
on $L$ such that ${\ii\over2\pi}\Theta_{L,h_\varepsilon}\ge-\varepsilon
\omega$, where $\omega={\ii\over2\pi}\Theta_{A,h_A}$. The convex combination
of metrics $h'_\varepsilon=h_1^{k\varepsilon}h_\varepsilon^{1-k\varepsilon}$
is a singular metric with poles along $D$ which satisfies
$${\ii\over2\pi}\Theta_{L,h'_\varepsilon}\ge \varepsilon(\omega+[D])-
(1-k\varepsilon)\varepsilon\omega\ge k\varepsilon^2\omega.$$
Its Lelong numbers are $\varepsilon\nu(D,x)$ and they can be made
smaller than $\delta$ by choosing $\varepsilon>0$ small.\qed
\endproof
%
We still need a few elementary facts about the numerical dimension of nef
line bundles.

\claim{(6.17) Definition} Let $L$ be a nef line bundle on a compact
K\"ahler manifold~$X$. One defines the numerical dimension of $L$ to be
$$\num(L)=\max\big\{k=0,\ldots,n\,;\,c_1(L)^k\ne0~\hbox{in}~
H^{2k}(X,\bR)\big\}.$$
\endclaim
%
By Corollary 6.16, we have $\kappa(L)=n$ if and only if $\num(L)=n$.
In general, we merely have an inequality.

\claim{(6.18) Proposition} If $L$ is a nef line bundle on a compact
K\"ahler manifold, then $\kappa(L)\le\num(L)$.
\endclaim
%
\begproof{Proof.} By induction on $n=\dim X$. If $\num(L)=n$ or $\kappa(L)=n$
the result is true, so we may assume $r:=\kappa(L)\le n-1$ and $k:=\num(L)\le
n-1$. Fix $m>0$ so that $\Phi=\Phi_{|mL|}$ has generic rank $r$.
Select a nonsingular ample divisor $A$ in $X$ such that the restriction of
$\Phi_{|mL|}$ to $A$ still has rank $r$ (for this, just take $A$
passing through a point $x\notin B_{|mL|}$ at which $\rank(d\Phi_x)=r<n$,
in such a way that the tangent linear map $d\Phi_{x\restriction T_{A,x}}$
still has rank~$r$). Then $\kappa(L_{\restriction A})\ge r=\kappa(L)$
(we just have an equality because there might exist sections in
$H^0(A,mL_{\restriction A})$ which do not extend to~$X$). On the other hand,
we claim that $\num(L_{\restriction A})=k=\num(L)$. The inequality
$\num(L_{\restriction A})\ge\num(L)$ is clear. Conversely, if we set
$\omega={\ii\over2\pi}\Theta_{A,h_A}>0$, the cohomology class $c_1(L)^k$ can
be represented by a closed positive current of bidegree~$(k,k)$
$$T=\lim_{\varepsilon\to0}\Big({\ii\over 2\pi}\Theta_{L,h_\varepsilon}+
\varepsilon\omega\Big)^k$$
after passing to some subsequence (there is a uniform bound for the mass
thanks to the K\"ahler assumption, taking wedge products with
$\omega^{n-k}$). The current $T$ must be non zero since $c_1(L)^k\ne0$ by
definition of $k=\num(L)$. Then $\{[A]\}=\{\omega\}$ as cohomology classes,
and
$$\int_A c_1(L_{\restriction A})^k\wedge\omega^{n-1-k}=
\int_X c_1(L)^k\wedge[A]\wedge\omega^{n-1-k}=
\int_X T\wedge\omega^{n-k}>0.$$
This implies $\num(L_{\restriction A})\ge k$, as desired. The induction
hypothesis with $X$ replaced by $A$ yields
$$\kappa(L)\le\kappa(L_{\restriction A})\le
\num(L_{\restriction A})\le\num(L).\eqno\square$$
\endproof
%
\claim{(6.19) Remark} \rm It may happen that $\kappa(L)<\num(L)$: take e.g.
$$L\to X=X_1\times X_2$$
equal to the total tensor product of an ample line bundle $L_1$ on a
projective manifold $X_1$ and of a unitary flat line bundle $L_2$ on an
elliptic curve $X_2$ given by a representation $\pi_1(X_2)\to U(1)$ such
that no multiple $kL_2$ with $k\ne0$ is trivial. Then
$H^0(X,kL)=H^0(X_1,kL_1)\otimes H^0(X_2,kL_2)=0$ for $k>0$, and thus
$\kappa(L)=-\infty$. However $c_1(L)=\pr_1^\star c_1(L_1)$ has
numerical dimension equal to $\dim X_1$. The same example shows
that the Kodaira dimension may increase by restriction to a
subvariety (if $Y=X_1\times\{{\rm point}\}$, then
$\kappa(L_{\restriction Y})=\dim Y$).\qed
\endclaim
%

\subsection*{\csect.D. The Kawamata-Viehweg vanishing theorem}
We now derive an algebraic version of the Nadel vanishing theorem in the
context of nef line bundles. This algebraic vanishing theorem has been
obtained independently by Kawamata [Kaw82] and Viehweg [Vie82], who both
reduced it to the Kodaira-Nakano vanishing theorem by cyclic covering
constructions. Since then, a number of other proofs have been given,
one based on connections with logarithmic singularities [EV86], another
on Hodge theory for twisted coefficient systems [Kol85], a third one
on the Bochner technique [Dem89] (see also [EV92] for a general survey,
and [Eno93] for an extension to the compact K\"ahler case). Since the
result is best expressed in terms of multiplier ideal sheaves (avoiding
then any unnecessary desingularization in the statement), we feel that
the direct approach via Nadel's vanishing theorem is probably the most
natural one.

If $D=\sum\alpha_jD_j\ge 0$ is an effective $\bQ$-divisor, we define
the {\it multiplier ideal sheaf} $\cI(D)$ to be equal to $\cI(\varphi)$
where $\varphi=\sum\alpha_j|g_j|$ is the corresponding psh function
defined by generators $g_j$ of $\cO(-D_j)\,$; as we saw in Remark 5.9,
the computation of $\cI(D)$ can be made algebraically by using
desingularizations $\mu:\widetilde X\to X$ such that $\mu^\star D$ becomes a
divisor with normal crossings on~$\widetilde X$.

\claim{(6.20) Kawamata-Viehweg vanishing theorem} Let $X$ be a projective
algebraic manifold and let $F$ be a line bundle over $X$ such that some
positive multiple $mF$ can be written $mF=L+D$ where $L$ is a 
nef line bundle and $D$ an effective divisor. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(m^{-1}D)\big)=0~~~~
{\it for}~~q>n-\num(L).$$
\endclaim
%
\claim{(6.21) Special case} If $F$ is a nef line bundle, then
$$H^q\big(X,\cO(K_X+F)\big)=0~~~~{\it for}~~q>n-\num(F).$$
\endclaim
%
\begproof{Proof of Theorem 6.20.} First suppose that $\num(L)=n$, i.e.\ that
$L$ is big. By the proof of 6.13~f), there is a singular hermitian
metric $h_0$ on $L$ such that the corresponding weight $\varphi_0$
has algebraic singularities and
$$\ii\Theta_{L,h_0}=2\ii d'd''\varphi_0\ge\varepsilon_0\omega$$
for some $\varepsilon_0>0$. On the other hand, since $L$ is nef, there
are metrics given by weights $\varphi_\varepsilon$ such that
${\ii\over 2\pi}\Theta_{L,h_\varepsilon}\ge -\varepsilon\omega$ for
every $\varepsilon>0$, $\omega$ being a K\"ahler metric. Let
$\varphi_D=\sum\alpha_j\log|g_j|$ be the weight of the singular metric on
$\cO(D)$ described in Example~3.13. We define a singular metric on $F$ by
$$\varphi_F={1\over m}
\big((1-\delta)\varphi_{L,\varepsilon}+\delta\varphi_{L,0}+\varphi_D\big)$$
with $\varepsilon\ll\delta\ll 1$, $\delta$ rational. Then $\varphi_F$
has algebraic singularities, and by taking $\delta$ small enough we
find $\cI(\varphi_F)=\cI({1\over m}\varphi_D)=\cI({1\over m}D)$.
In fact, $\cI(\varphi_F)$ can be computed by taking integer parts
of $\bQ$-divisors (as explained in Remark~5.9), and adding
$\delta\varphi_{L,0}$ does not change the integer part of the rational
numbers involved when $\delta$ is small. Now
$$\eqalign{dd^c\varphi_F
&={1\over m}\big((1-\delta)dd^c\varphi_{L,\varepsilon}+
\delta dd^c\varphi_{L,0}+dd^c\varphi_D\big)\cr
&\ge{1\over m}\big(-(1-\delta)\varepsilon\omega+
\delta\varepsilon_0\omega+[D]\ge{\delta\varepsilon\over m}\omega,\cr}$$
if we choose $\varepsilon\le\delta\varepsilon_0$. Nadel's theorem thus
implies the desired vanishing result for all $q\ge 1$.

Now, if $\num(L)<n$, we use hyperplane sections and argue by induction
on $n=\dim X$. Since the sheaf $\cO(K_X)\otimes\cI(m^{-1}D)$ behaves
functorially with respect to modifications (and since the $L^2$
cohomology complex is ``the same'' upstairs and downstairs),
we may assume after blowing-up that $D$ is a divisor with normal
crossings. By Remark~5.9, the multiplier ideal sheaf
$\cI(m^{-1}D)=\cO(-\lfloor m^{-1}D\rfloor)$ is locally free.
By Serre duality, the expected vanishing is equivalent to
$$H^q(X,\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor))=0\qquad
\hbox{\rm for}~q<\num(L).$$
Then select a nonsingular ample divisor $A$ such that $A$ meets all
components $D_j$ transversally. Select $A$ positive enough so that
$\cO(A+F-\lfloor m^{-1}D\rfloor)$ is ample. Then
$H^q(X,\cO(-A-F)\otimes\cO(\lfloor m^{-1}D\rfloor))=0$ for $q<n$
by Kodaira vanishing, and the exact sequence
$0\to\cO_X(-A)\to\cO_X\to(i_A)_\star\cO_A\to 0$ twisted by
$\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor)$ yields an isomorphism
$$H^q(X,\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor))\simeq
H^q(A,\cO(-F_{\restriction A})\otimes\cO(\lfloor m^{-1}
D_{\restriction A}\rfloor).$$
The proof of 6.18 showed that $\num(L_{\restriction A})=\num(L)$, hence
the induction hypothesis implies that the cohomology group on $A$ on
the right hand side is zero for \hbox{$q<\num(L)$}.\qed
\endproof
%
\subsection*{\csect.E. A uniform global generation property due to 
Y.T.~Siu}
Let $X$ be a projective manifold, and $(L,h)$ a pseudo-effective
line bundle.
The ``uniform global generation property'' of shows in some sense that the
tensor product sheaf $L\otimes\cI(h)$ has a uniform positivity property,
for any singular hermitian metric $h$ with nonnegative curvature on~$L$.

\claim{(6.22) Theorem {\rm Y.T.~Siu, ([Siu98])}} Let $X$ be a 
projective manifold. There exists an ample line bundle $G$ on $X$
such that for every pseudo-effective line bundle $(L,h)$, the sheaf
$\cO(G+L)\otimes\cI(h)$ is generated by its global sections. In fact, $G$
can be chosen as follows: pick any very ample line bundle $A$, and take
$G$ such that $G-(K_X+nA)$ is ample, e.g.\ $G=K_X+(n+1)A$.
\endclaim
%
\begproof{Proof.} Let $\varphi$ be the weight of the metric $h$ on a small
neighborhood of a point~$z_0\in X$. Assume that we have a local section 
$u$ of $\cO(G+L)\otimes\cI(h)$ on a coordinate open ball $B=B(z_0,\delta)$, 
such that
$$
\int_B|u(z)|^2e^{-2\varphi(z)}|z-z_0|^{-2(n+\varepsilon)}dV(z)<+\infty.
$$
Then Skoda's division theorem [Sko72b] (see also Corollary 8.21 below)
implies $u(z)=\sum(z_j-z_{j,0})v_j(z)$
with
$$
\int_B|v_j(z)|^2e^{-2\varphi(z)}|z-z_0|^{-2(n-1+\varepsilon)}dV(z)<+\infty,
$$
in particular $u_{z_0}\in\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$. Select
a very ample line bundle $A$ on~$X$. We take a basis
$\sigma=(\sigma_j)$ of sections of $H^0(X,G\otimes\gm_{X,z_0})$ and
multiply the metric $h$ of $G$ by the factor $|\sigma|^{-2(n+\varepsilon)}$.
The weight of the above metric has singularity $(n+\varepsilon)\log|z-z_0|^2$
at~$z_0$, and its curvature is
$$
i\Theta_G+(n+\varepsilon)i\ddbar\log|\sigma|^2\ge
i\Theta_G-(n+\varepsilon)\Theta_A.
\leqno(6.23)
$$
Now, let $f$ be a local section in $H^0(B,\cO(G+L)\otimes\cI(h))$ on 
$B=B(z_0,\delta)$, $\delta$ small. We solve the global $\dbar$ equation 
$$
\dbar u =\dbar(\theta f)\qquad\hbox{on $X$}
$$ 
with a cut-off function $\theta$ supported near $z_0$ and with the 
weight associated with our above choice of metric on $G+L$. 
Thanks to Nadel's Theorem 5.11, the solution exists if the metric of 
$G+L-K_X$ has positive curvature. As $i\Theta_{L,h}\ge 0$ in the sense 
of currents, $(6.23)$ shows that a sufficient condition is $G-K_X-nA>0$
(provided that $\varepsilon$ is small enough).
We then find a smooth solution $u$ such that $u_{z_0}\in 
\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$, hence
$$
F:=\theta f -u \in H^0(X,\cO(G+L)\otimes\cI(h))
$$ 
is a global section differing from $f$ by a germ in 
$\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$. Nakayama's lemma implies 
that $H^0(X,\cO(G+L)\otimes\cI(h))$ generates the stalks of
$\cO(G+L)\otimes\cI(h)$.
\endproof
%
\section{Holomorphic Morse inequalities}
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 $h$ of curvature form $\Theta_{L,h}$, we define the $q$-index set of $L$
to be the open subset
$$X(q,L)=\left\{x\in X~;~\ii\Theta_{L,h}(x) \hbox{~has~}
{\displaystyle q\atop\displaystyle n-q}~
{\hbox{negative eigenvalues}\atop\hbox{positive eigenvalues}}\right\}
\leqno(7.1)$$
for $0\le q\le n$. Hence $X$ admits a partition $X=\Delta\cup
\bigcup_q X(q,L)$ where $\Delta=\{\hbox{$x\in X$}\,;\,\det(\Theta_{L,h}(x))=0\}$
is the degeneracy set. We also introduce
$$X(\le q,L)=\bigcup_{0\le j\le q}X(j,L).\leqno(7.1')$$

\claim{(7.2) Morse inequalities {\rm ([Dem85b])}} For any hermitian holomorphic
line bundle $L,h)$ and any holomorphic vector bundle $E$ over a compact 
complex manifold~$X$, the cohomology groups
$H^q\big(X,E\otimes\cO(kL)\big)$ satisfy the following asymptotic
inequalities as $k\to+\infty$~:
\vskip3pt
\plainitem{\rm (a)} Weak Morse inequalities
$$h^q\big(X,E\otimes\cO(kL)\big)\le
r{k^n\over n!}\int_{X(q,L)} (-1)^q\Big({\ii\over2\pi}\Theta_{L,h}\Big)^n+o(k^n).
$$
\vskip3pt
\plainitem{\rm (b)} Strong Morse inequalities
$$
\sum_{0\le j\le q}(-1)^{q-j}h^j\big(X,E\otimes\cO(kL)\big)
\le r{k^n\over n!}\int_{X(\le q,L)}(-1)^q\Big({\ii\over2\pi}\Theta_{L,h}\Big)^n
+o(k^n).$$
\endclaim
%
The proof is based on the spectral theory of the complex Laplace operator, 
using either a localization procedure or, alternatively, a heat kernel 
technique.
These inequalities are a useful complement to the Riemann-Roch formula
when information is needed about individual cohomology groups, and not
just about the Euler-Poincar\'e characteristic.

One difficulty in the application of these inequalities is that the
curvature integral is in general quite uneasy to compute, since it is
neither a topological nor an algebraic invariant. However, the Morse
inequalities can be reformulated in a more algebraic setting in which
only algebraic invariants are involved. We give here two such
reformulations.

\claim{(7.3) Theorem} Let $L=F-G$ be a holomorphic line bundle over a
compact K\"ahler manifold~$X$, where $F$ and $G$ are numerically effective
line bundles. Then for every $q=0,1,\ldots,n=\dim X$, there is an asymptotic
strong Morse inequality
$$\sum_{0\le j\le q}(-1)^{q-j}h^j(X,kL)\le{k^n\over n!}
\sum_{0\le j\le q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j+o(k^n).$$
\endclaim
%
\begproof{Proof.} By adding $\varepsilon$ times a K\"ahler metric $\omega$ to the
curvature forms of $F$ and $G$, $\varepsilon>0$ one can write ${\ii\over
2\pi}\Theta_L=\theta_{F,\varepsilon}-\theta_{G,\varepsilon}$ where
$\theta_{F,\varepsilon}={\ii\over 2\pi}\Theta_F+\varepsilon\omega$ and
$\theta_{G,\varepsilon}={\ii\over 2\pi}\Theta_G+\varepsilon\omega$ are
positive definite. Let $\lambda_1\ge\ldots\ge\lambda_n>0$ be the
eigenvalues of $\theta_{G,\varepsilon}$ with respect to $\theta_{F,\varepsilon}$.
Then the eigenvalues of ${\ii\over 2\pi}\Theta_L$ with respect
to $\theta_{F,\varepsilon}$ are the real numbers $1-\lambda_j$ and the
set $X(\le q,L)$ is the set $\{\lambda_{q+1}<1\}$ of points $x\in X$ such
that $\lambda_{q+1}(x)<1$. The strong Morse inequalities yield 
$$\sum_{0\le j\le q}(-1)^{q-j}h^j(X,kL)\le{k^n\over n!}
\int_{\{\lambda_{q+1}<1\}}\!\!\!(-1)^q\!\!
\prod_{1\le j\le n}(1-\lambda_j)\theta_{F,\varepsilon}^n+o(k^n).$$
On the other hand we have
$${n\choose j}\theta_{F,\varepsilon}^{n-j}\wedge\theta_{G,\varepsilon}^j=
\sigma_n^j(\lambda)\,\theta_{F,\varepsilon}^n,$$
where $\sigma_n^j(\lambda)$ is the $j$-th elementary symmetric function in
$\lambda_1,\ldots,\lambda_n\,$, hence
$$\sum_{0\le j\le q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j=
\lim_{\varepsilon\to 0}\int_X\sum_{0\le j\le q}(-1)^{q-j}
\sigma_n^j(\lambda)\,\theta_{F,\varepsilon}^n.$$
Thus, to prove the Lemma, we only have to check that
$$\sum_{0\le j\le n}(-1)^{q-j}\sigma_n^j(\lambda)-
\bOne_{\{\lambda_{q+1}<1\}}(-1)^q\prod_{1\le j\le n}(1-\lambda_j)\ge 0$$
for all $\lambda_1\ge\ldots\ge\lambda_n\ge 0$, where $\bOne_
{\{\ldots\}}$ denotes the characteristic function of a set.
This is easily done by induction on~$n$ (just split apart the parameter
$\lambda_n$ and write $\sigma_n^j(\lambda)=
\sigma_{n-1}^j(\lambda)+\sigma_{n-1}^{j-1}(\lambda)\,\lambda_n$).\qed
\endproof
%
In the case $q=1$, we get an especially interesting lower bound (this
bound has been observed and used by S.~Trapani [Tra95] in a similar
context).

\claim{(7.4) Consequence}
$h^0(X,kL)-h^1(X,kL)\ge{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n)$.\newline
Therefore some multiple $kL$ has a section as soon as $F^n-nF^{n-1}\cdot
G>0$.
\endclaim
%
\claim{(7.5) Remark} \rm The weaker inequality
$$h^0(X,kL)\ge{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n)$$
is easy to prove if $X$ is projective algebraic. Indeed, by adding a
small ample $\bQ$-divisor to $F$ and $G$, we may assume that
$F$, $G$ are ample. Let $m_0G$ be very ample and let $k'$ be the smallest
integer $\ge k/m_0$. Then $h^0(X,kL)\ge h^0(X,kF-k'm_0G)$. We select
$k'$ smooth members $G_j$, $1\le j\le k'$ in the linear system $|m_0G|$
and use the exact sequence
$$0\to H^0(X,kF-\sum G_j)\to H^0(X,kF)\to\bigoplus H^0(G_j,kF_{|G_j}).$$
Kodaira's vanishing theorem yields $H^q(X,kF)=0$ and $H^q(G_j,kF_{|G_j})=0$
for $q\ge 1$ and $k\ge k_0$. By the exact sequence combined with
Riemann-Roch, we get
$$\eqalign{
h^0(X,kL)&\ge h^0(X,kF-\sum G_j)\cr
&\ge{k^n\over n!}F^n-O(k^{n-1})-
\sum\Big({k^{n-1}\over(n-1)!}F^{n-1}\cdot G_j-O(k^{n-2})\Big)\cr
&\ge{k^n\over n!}\Big(F^n-n{k'm_0\over k}F^{n-1}\cdot G\Big)-O(k^{n-1})\cr
&\ge{k^n\over n!}\Big(F^n-n\,F^{n-1}\cdot G\Big)-O(k^{n-1}).\cr}$$
(This simple proof is due to F.~Catanese.)\qed
\endclaim
%
\claim{(7.6) Corollary} Suppose that $F$ and $G$ are nef and that $F$ is
big. Some multiple of $mF-G$ has a section as soon as
$$m>n\,{F^{n-1}\cdot G\over F^n}.$$
\endclaim
%
In the last condition, the factor $n$ is sharp: this is easily seen by taking
$X=\bP_1^n$ and $F=\cO(a,\ldots,a)$ and $G=\cO(b_1,\ldots,b_n)$ over
$\bP_1^n\,$; the condition of the Corollary is then
$m>\sum b_j/a$, whereas $k(mF-G)$ has a section if and only if
$m\ge\sup b_j/a$; this shows that we cannot replace $n$ by
$n(1-\varepsilon)$.\qed

\section{The Ohsawa-Takegoshi $L^2$ extension theorem}

The Ohsawa-Takegoshi theorem addresses the following extension problem:
let $Y$ be a complex analytic submanifold of a complex manifold~$X\,$;
given a holomorphic function $f$ on $Y$ satisfying suitable $L^2$
conditions on $Y$, find a holomorphic extension $F$ of $f$ to $X$,
together with a good $L^2$ estimate for $F$ on~$X$. The first
satisfactory solution has been obtained in the fundamental papers
[OT87, Ohs88].  We follow here a more geometric
approach due to Manivel [Man93], which provides a generalized
extension theorem in the general framework of vector bundles. As in
Ohsawa-Takegoshi's fundamental paper, the main idea is to use a
modified Bochner-Kodaira-Nakano inequality. Such inequalities were
originally introduced in the work of Donnelly-Fefferman [DF83] and
Donnelly-Xavier [DX84]. 

\subsection*{\csect.A. The basic a priori inequality}
The main a priori inequality we are going to
use is a simplified (and slightly extended) version of the original
Ohsawa-Takegoshi a priori inequality, along the lines proposed by Ohsawa
[Ohs95].

\claim{(8.1) Lemma {\rm (Ohsawa [Ohs95])}} Let $E$ be a hermitian
vector bundle on a complex manifold $X$ equipped with a K\"ahler
metric~$\omega$. Let $\eta,\,\lambda>0$ be smooth functions
on~$X$. Then for every form $u\in\cD(X,\Lambda^{p,q} T^\star_X\otimes E)$
with compact support we have
$$ 
\eqalign{
\|(\eta^{{1\over2}}+\lambda^{{1\over2}})D^{\prime\prime\star}u\|^2&{}
+\|\eta^{{1\over2}}D''u\|^2+\|\lambda^{{1\over2}}D'u\|^2+
2\|\lambda^{-{1\over2}}d'\eta\wedge u\|^2\cr
 &\ge \Ll[\eta\,\ii\Theta_E-\ii\,d'd''\eta-\ii\lambda^{-1}d'\eta\wedge 
d''\eta,\Lambda]u,u\Gg.\cr}
$$
\endclaim
%
\begproof{Proof.} Let us consider the ``twisted'' Laplace-Beltrami operators
$$
\eqalign{
D'\eta D^{\prime\star} +D^{\prime\star}\eta  D' 
& =  \eta [D',D^{\prime\star}]+[D',\eta]D^{\prime\star}+
[D^{\prime\star},\eta]D' \cr
& = \eta\Delta'+(d'\eta)D^{\prime\star}-(d'\eta)^*D',\cr
D''\eta D^{\prime\prime\star} +D^{\prime\prime\star}\eta  D'' 
& = \eta [D'',D^{\prime\prime\star}]+[D'',\eta]D^{\prime\prime\star}+
[D^{\prime\prime\star},\eta]D'' \cr
& = \eta\Delta''+(d''\eta)D^{\prime\prime\star}-(d''\eta)^*D'',\cr}
$$
where $\eta$, $(d'\eta)$, $(d''\eta)$ are abbreviated notations for the 
multiplication operators $\eta\bu$, $(d'\eta)\wedge\bu$, $(d''\eta)\wedge\bu$.
By subtracting the above equalities and taking into account the 
Bochner-Kodaira-Nakano identity $\Delta''-\Delta'=[\ii\Theta_E,\Lambda]$,
we get
$$
\leqalignno{
D''\eta D^{\prime\prime\star}&{}+D^{\prime\prime\star}\eta  D'' -
D'\eta D^{\prime\star}  -D^{\prime\star}\eta  D' \cr
&{}=\eta [\ii\Theta_E,\Lambda]+(d''\eta)D^{\prime\prime\star}
-(d''\eta)^\star D''+(d'\eta)^\star D'-(d'\eta)D^{\prime\star}.
&(8.2)\cr}
$$
Moreover, the Jacobi identity yields
$$
[D'',[d'\eta,\Lambda ]]-[d'\eta,[\Lambda,D'']]+[\Lambda,[D'',d'\eta]]=0,
$$
whilst $[\Lambda,D'']=-\ii D^{\prime\star}$ by the basic commutation 
relations~4.5. A straightforward computation shows that 
$[D'',d'\eta]=-(d'd''\eta)$ and $[d'\eta,\Lambda ]=\ii(d''\eta)^\star$.
Therefore we get
$$
\ii[D'',(d''\eta)^\star]+\ii[d'\eta,D^{\prime\star}]-[\Lambda,(d'd''\eta)]=0,
$$
that is,
$$
[\ii\,d'd''\eta,\Lambda]=
[D'',(d''\eta)^\star]+[D^{\prime\star},d'\eta]=
D''(d''\eta)^\star+(d''\eta)^\star D''+
D^{\prime\star}(d'\eta)+(d'\eta)D^{\prime\star}.
$$
After adding this to (8.2), we find
$$
\eqalign{
D''\eta D^{\prime\prime\star}&{}+D^{\prime\prime\star}\eta  D'' -
D'\eta D^{\prime\star}  -D^{\prime\star}\eta  D' 
+[\ii\,d'd''\eta,\Lambda]
\cr
&{}=\eta [\ii\Theta_E,\Lambda]+(d''\eta)D^{\prime\prime\star}+
D''(d''\eta)^\star+(d'\eta)^\star D'+D^{\prime\star}(d'\eta).
\cr}
$$
We apply this identity to a form $u\in\cD(X,\Lambda^{p,q}T^\star_X\otimes E)$
and take the inner bracket with~$u$. Then
$$
\Ll (D''\eta D^{\prime\prime\star})u,u\Gg=\Ll\eta D^{\prime\prime\star}u,
D^{\prime\prime\star}u\Gg=\|\eta^{{1\over2}}D^{\prime\prime\star}u\|^2,
$$
and likewise for the other similar terms. The above equalities
imply
$$
\eqalign{
&\|\eta^{{1\over 2}}D^{\prime\prime\star}u\|^2+\|\eta^{{1\over 2}}D''u\|^2
-\|\eta^{{1\over2}}D'u\|^2-\|\eta^{{1\over2}}D^{\prime\star}u\|^2={}\cr
&\qquad\Ll[\eta\,\ii\Theta_E-\ii\,d'd''\eta,\Lambda]u,u\Gg
+2\Re\,\Ll D^{\prime\prime\star}u,(d''\eta)^\star u\Gg
+2\Re\,\Ll D'u,d'\eta\wedge u\Gg.\cr}
$$
By neglecting the negative terms $-\|\eta^{{1\over2}}D'u\|^2
-\|\eta^{{1\over2}}D^{\prime\star}u\|^2$ and adding the squares
$$
\eqalign{
\|\lambda^{{1\over 2}}D^{\prime\prime\star}u\|^2+
2\Re\,\Ll D^{\prime\prime\star}u,(d''\eta)^\star u\Gg+
\|\lambda^{-{1\over 2}}(d''\eta)^\star u\|^2
&\ge 0,\cr
\|\lambda^{{1\over 2}}D'u\|^2+
2\Re\,\Ll D'u,d'\eta\wedge u\Gg+
\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2
&\ge0\cr}
$$
we get
$$
\eqalign{
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})
D^{\prime\prime\star}u\|^2&{}+\|\eta^{{1\over 2}}D''u\|^2
+\|\lambda^{{1\over 2}}D'u\|^2+\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2
+\|\lambda^{-{1\over 2}}(d''\eta)^\star u\|^2\cr
&\ge\Ll[\eta\,\ii\Theta_E-\ii\,d'd''\eta,\Lambda]u,u\Gg.\cr}
$$
Finally, we use the identities
$$
\eqalign{
(d'\eta)^\star(d'\eta)&{}-(d''\eta)(d''\eta)^\star
=\ii[d''\eta,\Lambda](d'\eta)+\ii(d''\eta)[d'\eta,\Lambda]
=[\ii d''\eta\wedge d'\eta,\Lambda],\cr
\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2&{}-
\|\lambda^{-{1\over2}}(d''\eta)^\star u\|^2
=-\Ll[\ii\lambda^{-1}d'\eta\wedge d''\eta,\Lambda]u,u\Gg,\cr}
$$
The inequality asserted in Lemma~8.1 follows by adding the second identity
to our last inequality.\qed
\endproof
%
In the special case of $(n,q)$-forms, the forms $D'u$ and $d'\eta\wedge u$
are of bidegree $(n+1,q)$, hence the estimate takes the simpler form
$$
\|(\eta^{{1\over2}}+\lambda^{{1\over2}})D^{\prime\prime\star}u\|^2{}
+\|\eta^{{1\over2}}D''u\|^2\ge
\Ll[\eta\,\ii\Theta_E-\ii\,d'd''\eta
-\ii\lambda^{-1}\,d'\eta\wedge d''\eta,\Lambda]u,u\Gg.
\leqno(8.3)
$$

\subsection*{\csect.B. Abstract $L^2$ existence theorem for solutions of
$\dbar$-equations}
Using standard arguments from functional analysis -- actually just basic
properties of Hilbert spaces along the lines already explained in
section~5 -- the a priori inequality (8.3) implies
a very strong $L^2$ existence theorem for solutions of $\dbar$-equations.

\claim{(8.4) Proposition} Let $X$ be a complete K\"ahler manifold
equipped with a $($non ne\-ces\-sarily complete$)$ K\"ahler metric
$\omega$, and let $E$ be a hermitian vector bundle over~$X$.
Assume that there are smooth and bounded functions $\eta,\,\lambda>0$
on $X$ such that the $($hermitian$)$ curvature operator
$B=B^{n,q}_{E,\omega,\eta}=
[\eta\,\ii\Theta_E-\ii\,d'd''\eta-\ii\lambda^{-1}d'\eta\wedge d''\eta,
\Lambda_\omega]$ is positive definite everywhere on 
$\Lambda^{n,q}T^\star_X\otimes E$, for some $q\ge 1$.
Then for every form $g\in L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$
such that $D''g=0$ and $\int_X\langle B^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{n,q-1}T^\star_X\otimes E)$ such that  $D''f=g$ and
$$
\int_X(\eta+\lambda)^{-1}
|f|^2\,dV_\omega\le 2\int_X\langle B^{-1}g,g\rangle\,dV_\omega.
$$
\endclaim
%
\begproof{Proof.} The proof is almost identical to the proof of standard
$L^2$ estimates for $\dbar$ (see Theorem~5.1),
except that we use (8.3) instead of (4.7). Assume first that $\omega$ is
complete. With the same notation as in 7.4, we get for every 
$v=v_1+v_2\in(\Ker D'')\oplus(\Ker D'')^\perp$ the inequalities
$$
|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle B^{-1}g,g\rangle \,dV_\omega
\int_X\langle Bv_1,v_1\rangle \,dV_\omega,
$$
and
$$
\int_X\langle Bv_1,v_1\rangle \,dV_\omega\le
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v_1\|^2+
\|\eta^{{1\over 2}}D''v_1\|^2
=\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2
$$
provided that $v\in\Dom D^{\prime\prime\star}$. Combining both, we find
$$
|\langle g,v\rangle|^2
\le \Big(\int_X\langle B^{-1}g,g\rangle \,dV_\omega\Big)
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2.
$$
This shows the existence of an element $w\in L^2(X,\Lambda^{n,q}T^\star_X
\otimes E)$ such that 
$$
\eqalign{
\|w\|^2&\le\int_X\langle B^{-1}g,g\rangle \,dV_\omega\qquad
\hbox{and}\cr
\Ll v,g\Gg&=\Ll(\eta^{{1\over 2}}+\lambda^{{1\over2}})
D^{\prime\prime\star}v,w\Gg\qquad
\forall g\in\Dom D''\cap\Dom D^{\prime\prime\star}.\cr}
$$
As $(\eta^{1/2}+\lambda^{{1\over2}})^2\le 2(\eta+\lambda)$, it follows 
that $f=(\eta^{1/2}+\lambda^{{1\over2}})w$ satisfies
$D''f=g$ as well as the desired $L^2$ estimate. If $\omega$ is not
complete, we set $\omega_\varepsilon=\omega+\varepsilon\widehat\omega$ with
some complete K\"ahler metric~$\widehat\omega$. The final conclusion is
then obtained by passing to the limit and using a monotonicity argument
(the integrals are monotonic with respect to~$\varepsilon$).\qed
\endproof
%
\claim{(8.5) Remark} \rm We will also need a variant of the
$L^2$-estimate, so as to obtain approximate solutions with weaker
requirements on the data$\,$: given $\delta>0$ and $g\in
L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$ such that $D''g=0$ and
$\int_X\langle (B+\delta I)^{-1}g,g\rangle\,dV_\omega<+\infty$, there
exists an approximate solution 
\hbox{$f\in L^2(X,\Lambda^{n,q-1}T^\star_X \otimes E)$} and a correcting
term $h\in L^2(X,\Lambda^{n,q}T^\star_X \otimes E)$ such that
$D''f+\delta^{1/2}h=g$ and
$$
\int_X(\eta+\lambda)^{-1}|f|^2\,dV_\omega+
\int_X|h|^2\,dV_\omega
\le 2\int_X\langle (B+\delta I)^{-1}g,g\rangle\,dV_\omega.
$$
The proof is almost unchanged, we rely instead on the estimates
$$
|\langle g,v_1\rangle|^2\le
\int_X\langle(B+\delta I)^{-1}g,g\rangle \,dV_\omega
\int_X\langle(B+\delta I)v_1,v_1\rangle \,dV_\omega,
$$
and
$$
\int_X\langle(B+\delta I)v_1,v_1\rangle \,dV_\omega\le
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2+
\delta\|v\|^2.
\eqno\qed
$$
\endclaim
%
\subsection*{\csect.C. The $L^2$ extension theorem}
According to a concept already widely used in section~5, a 
(non necessarily compact) complex manifold will be said to be
weakly pseudoconvex if it possesses a smooth weakly plurisubharmonic
exhaustion function.

\claim{(8.6) Theorem}
Let $X$ be a weakly pseudoconvex complex $n$-dimensional manifold possessing
a K\"ahler metric $\omega$, and let $L$ $($resp.\ $E)$
be a hermitian holomorphic line bundle $($resp.\ a hermitian holomorphic 
vector bundle of rank $r$ over~$X)$, and $s$ a global holomorphic 
section of~$E$. Assume that $s$ is generically transverse to the zero
section, and let
$$
Y=\big\{x\in X\,;\;s(x)=0, \Lambda^r ds(x)\not= 0\big\},\qquad
p=\dim Y=n-r.
$$
Moreover, assume that the $(1,1)$-form 
$\ii\Theta_L+r\,\ii\,d'd''\log|s|^2$ is semi-positive and that there is
a continuous function $\alpha\ge 1$ such that the following two inequalities
hold everywhere on $X:$
\vskip3pt
\plainitem{\rm (a)}
$\displaystyle\ii\Theta_L+r\,\ii\,d'd''\log|s|^2\ge\alpha^{-1}
{\{\ii\Theta_Es,s\}\over|s|^2}\,,$
\vskip3pt
\plainitem{\rm (b)} $|s|\le e^{-\alpha}$.
\vskip3pt\noindent
Then for every smooth $D''$-closed \hbox{$(0,q)$-form $f$} over $Y$ with 
values in the line bundle $\Lambda^nT_X^\star\otimes L$ 
$($restricted to~$Y)$, such that
$\int_Y|f|^2|\Lambda^r(ds)|^{-2}dV_\omega<+\infty$, 
there  exists a $D''$-closed $(0,q)$-form $F$ over $X$ with values 
in $\Lambda^nT^\star_X\otimes L$, such that $F$ is smooth over 
$X\ssm\{s=\Lambda^r(ds)=0\}$, satisfies $F_{\restriction Y}=f$ and
$$
\int_X{|F|^2\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}\,,
$$
where $C_r$ is a numerical constant depending only on~$r$.
\endclaim
%
Observe that the differential $ds$ (which is intrinsically defined only at 
points where $s$ vanishes) induces a vector bundle isomorphism
$ds:T_X/T_Y\to E$ along $Y$, hence a non vanishing section 
$\Lambda^r(ds)$, taking values in 
$$
\Lambda^r(T_X/T_Y)^\star\otimes \det E\subset
\Lambda^rT_X^\star\otimes \det E.
$$
The norm $|\Lambda^r(ds)|$ is computed here with respect to the metrics
on $\Lambda^rT_X^\star$ and $\det E$ induced by the K\"ahler metric
$\omega$ and by the given metric on~$E$. Also notice that if
hypothesis (a) is satisfied for some $\alpha$, one can always achieve
b) by multiplying the metric of $E$ with a sufficiently small weight
$e^{-\chi\circ\psi}$ (with $\psi$ a psh exhaustion on~$X$ and $\chi$ a
convex increasing function; property (a) remains valid after we multiply
the metric of $L$ by $e^{-(r+\smash{\alpha_0^{-1}})\chi\circ\psi}$,
where $\alpha_0=\inf_{x\in X}\alpha(x)$.

\begproof{Proof.} Let us first assume that the singularity set $\Sigma=
\{s=0\}\cap\{\Lambda^r(ds)=0\}$ is empty, so that $Y$ is closed and
nonsingular. We claim that there exists a smooth section
$$
F_\infty\in C^\infty(X,\Lambda^{n,q}T^\star_X\otimes L)
=C^\infty(X,\Lambda^{0,q}T^\star_X\otimes\Lambda^nT^\star_X\otimes L)
$$ 
such that
{\parindent=0cm 
\vskip3pt
(a) $F_\infty$ coincides with $f$ in restriction to $Y$,
\vskip3pt
(b) $|F_\infty|=|f|$ at every point of~$Y$,
\vskip3pt
(c) $D''F_\infty=0$ at every point of~$Y$.
\vskip3pt}

\noindent
For this, consider coordinates patches $U_j\subset X$ biholomorphic 
to polydiscs such that 
$$
U_j\cap Y=\{z\in U_j\,;\;z_1=\ldots=z_r=0\}
$$
in the corresponding
coordinates. We can certainly find a section $\smash{\widetilde f}\in
C^\infty(X,\Lambda^{n,q}T^\star_X\otimes L)$ which achieves (a) and (b),
since the restriction map $(\Lambda^{0,q}T_X^*)_{\restriction Y}\to
\Lambda^{0,q}T_Y^*$ can be viewed as an orthogonal projection onto a 
$C^\infty$-subbundle of~$(\Lambda^{0,q}T_X^*)_{\restriction Y}$. 
It is enough to extend this subbundle from \hbox{$U_j\cap Y$} to $U_j$ 
(e.g.\ by extending each component of a frame), and then to extend 
$f$ globally via local smooth extensions and a partition of unity.
For any such extension~$\smash{\widetilde f}$ we have
$$
(D''\widetilde f)_{\restriction Y}=(D''\widetilde f_{\restriction Y})=D''f=0.
$$
It follows that we can divide $D''\widetilde f=\sum_{1\le \lambda\le r}
g_{j,\lambda}(z)\wedge d\overline z_\lambda$ on $U_j\cap Y$, with suitable
smooth $(0,q)$-forms $g_{j,\lambda}$ which we also extend arbitrarily 
from $U_j\cap Y$ to~$U_j$. Then
$$
F_\infty:=\widetilde f - \sum_j \theta_j(z)\sum_{1\le \lambda \le r}
\overline z_\lambda g_{j,\lambda}(z)
$$
coincides with $\widetilde f$ on $Y$ and satisfies (c).
Since we do not know about $F_\infty$ except in an infinitesimal
neighborhood of~$Y$, we will consider a
truncation $F_\varepsilon$ of $F_\infty$ with support in a small
tubular neighborhood $|s|<\varepsilon$ of~$Y$, and solve the equation
$D''u_\varepsilon=D''F_\varepsilon$ with the constraint that
$u_\varepsilon$ should be $0$ on~$Y$. As $\codim Y=r$, this will be
the case if we can guarantee that $|u_\varepsilon|^2|s|^{-2r}$ is
locally integrable near~$Y$. For this, we will apply Proposition 8.4
with a suitable choice of the functions $\eta$ and $\lambda$, and an
additional weight $|s|^{-2r}$ in the metric of $L$.

Let us consider the smooth strictly convex function $\chi_0:{}]-\infty,0]
\to{}]-\infty,0]$ defined by $\chi_0(t)=t-\log(1-t)$ for $t\le 0$, which
is such that \hbox{$\chi_0(t)\le t$}, \hbox{$1\le\chi_0'\le2$} and 
\hbox{$\chi_0''(t)=1/(1-t)^2$}. 
We set
$$
\sigma_\varepsilon=\log(|s|^2+\varepsilon^2),\qquad
\eta_\varepsilon=\varepsilon-\chi_0(\sigma_\varepsilon).
$$
As $|s|\le e^{-\alpha}\le e^{-1}$, we have $\sigma_\varepsilon\le 0$
for $\varepsilon$ small, and
$$
\eta_\varepsilon\ge\varepsilon-\sigma_\varepsilon
\ge\varepsilon-\log(e^{-2\alpha}+\varepsilon^2).
$$
Given a relatively compact subset $X_c=\{\psi<c\}\compact X$, we thus
have $\eta_\varepsilon\ge 2\alpha$ for $\varepsilon<\varepsilon(c)$ small
enough. Simple calculations yield
$$
\eqalign{
\ii\,d'\sigma_\varepsilon
&={\ii\{D's,s\}\over|s|^2+\varepsilon^2},\cr
\ii\,d'd''\sigma_\varepsilon
&={\ii\{D's,D's\}\over|s|^2+\varepsilon^2}-
{\ii\{D's,s\}\wedge\{s,D's\}\over(|s|^2+\varepsilon^2)^2}-
{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}\cr
&\ge{\varepsilon^2\over|s|^2}
{\ii\{D's,s\}\wedge\{s,D's\}\over(|s|^2+\varepsilon^2)^2}
-{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}\cr
&\ge{\varepsilon^2\over|s|^2}\ii d'\sigma_\varepsilon\wedge 
d''\sigma_\varepsilon-{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2},\cr}
$$
thanks to Lagrange's inequality $\ii\{D's,s\}\wedge\{s,D's\}\le|s|^2
\ii\{D's,D's\}$. On the other hand, we have $d'\eta_\varepsilon=
-\chi_0'(\sigma_\varepsilon)d\sigma_\varepsilon$ with 
$1\le\chi_0'(\sigma_\varepsilon)\le 2$, hence
$$
\eqalign{
-\ii d'd''\eta_\varepsilon&=\chi_0'(\sigma_\varepsilon)\ii 
d'd''\sigma_\varepsilon+\chi_0''(\sigma_\varepsilon)\ii d'\sigma_\varepsilon
\wedge d''\sigma_\varepsilon\cr
&\ge \Big({1\over \chi_0'(\sigma_\varepsilon)}{\varepsilon^2\over |s|^2}+
{\chi_0''(\sigma_\varepsilon)\over\chi_0'(\sigma_\varepsilon)^2}\Big)
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon
-\chi_0'(\sigma_\varepsilon){\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}.\cr}
$$
We consider the original metric of $L$ multiplied by the weight $|s|^{-2r}$. 
In this way, we get a curvature form
$$
\ii\Theta_L+r\,\ii d'd''\log|s|^2
\ge{1\over 2}\chi_0'(\sigma_\varepsilon)
\alpha^{-1}{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}
$$
by hypothesis (a), thanks to the semipositivity of the
left hand side and the fact that ${1\over 2}\chi_0'(\sigma_\varepsilon){1\over
|s|^2+\varepsilon^2}\le {1\over |s|^2}$. As $\eta_\varepsilon\ge 2\alpha$ 
on $X_c$ for $\varepsilon$ small, we infer
$$
\eta_\varepsilon(\ii\Theta_L+\ii d'd''\log|s|^2)
-\ii d'd''\eta_\varepsilon-
{\chi_0''(\sigma_\varepsilon)\over\chi_0'(\sigma_\varepsilon)^2}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon\ge
{\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon
$$
on~$X_c$. Hence, if $\lambda_\varepsilon=
\chi_0'(\sigma_\varepsilon)^2/\chi_0''(\sigma_\varepsilon)$,
we obtain
$$
\eqalign{
B_\varepsilon&:=\big[\eta_\varepsilon(\ii\Theta_L+\ii d'd''\log|s|^2)
-\ii d'd''\eta_\varepsilon-\lambda_\varepsilon^{-1}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon\;,\;\Lambda\big]\cr
&\phantom{:}\ge\Big[{\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}\ii 
d'\eta_\varepsilon\wedge d''\eta_\varepsilon\,,\,\Lambda\Big]
={\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}
(d''\eta_\varepsilon)(d''\eta_\varepsilon)^\star\cr}
$$
as an operator on $(n,q)$-forms (see the proof of Lemma~8.1).

Let $\theta:\bR\to[0,1]$ be a smooth cut-off function such 
that $\theta(t)=1$ on $]-\infty,1/2]$, $\Supp\theta\subset{}]-\infty,1[$
and $|\theta'|\le 3$. For $\varepsilon>0$ small, we consider the 
$(n,q)$-form $F_\varepsilon=\theta(\varepsilon^{-2}|s|^2)F_\infty$ and
its $D''$-derivative
$$
g_\varepsilon=D''F_\varepsilon
=(1+\varepsilon^{-2}|s|^2)\theta'(\varepsilon^{-2}|s|^2)
d''\sigma_\varepsilon\wedge F_\infty+
\theta(\varepsilon^{-2}|s|^2)D''F_\infty
$$
[as is easily seen from the equality $1+\varepsilon^{-2}|s|^2=
\varepsilon^{-2}e^{\sigma_\varepsilon}\,$].
We observe that $g_\varepsilon$ has its support contained in the tubular
neighborhood $|s|<\varepsilon\,$; moreover, as $\varepsilon\to 0$, the 
second term in the right hand side converges uniformly to $0$ on every 
compact set; it will therefore produce no contribution in the
limit. On the other hand, the first term has the same order of magnitude
as $d''\sigma_\varepsilon$ and $d''\eta_\varepsilon$, and can be controlled
in terms of~$B_\varepsilon$. In fact, for any $(n,q)$-form $u$ and any
$(n,q+1)$-form $v$ we have
$$
\eqalign{
|\langle d''\eta_\varepsilon\wedge u,v\rangle|^2
&=|\langle u,(d''\eta_\varepsilon)^\star v\rangle|^2
\le|u|^2|(d''\eta_\varepsilon)^\star v|^2
=|u|^2\langle (d''\eta_\varepsilon)(d''\eta_\varepsilon)^\star v,v
\rangle\cr
&\le {\chi_0'(\sigma_\varepsilon)|s|^2\over\varepsilon^2}
|u|^2\langle B_\varepsilon v,v\rangle.\cr}
$$
This implies
$$
\langle B_\varepsilon^{-1}(d''\eta_\varepsilon\wedge u),
(d''\eta_\varepsilon\wedge u)\rangle\le{\chi_0'(\sigma_\varepsilon)
|s|^2\over\varepsilon^2}|u|^2.
$$
The main term in $g_\varepsilon$ can be written
$$
g_\varepsilon^{(1)}:=
(1+\varepsilon^{-2}|s|^2)\theta'(\varepsilon^{-2}|s|^2)
\chi_0'(\sigma_\varepsilon)^{-1}d''\eta_\varepsilon\wedge F_\infty.
$$
On $\Supp g_\varepsilon^{(1)}\subset\{|s|<\varepsilon\}$, since
$\chi_0'(\sigma_\varepsilon)\ge 1$, we thus find
$$
\langle B_\varepsilon^{-1}g_\varepsilon^{(1)},
g_\varepsilon^{(1)}\rangle\le (1+\varepsilon^{-2}|s|^2)^2
\,\theta'(\varepsilon^{-2}|s|^2)^2|F_\infty|^2.
$$
Instead of working on $X$ itself, we will work rather on the relatively
compact subset $X_c\ssm Y_c$, where $Y_c=Y\cap X_c=Y\cap\{\psi<c\}$. We know
that $X_c\ssm Y_c$ is again complete K\"ahler by a standard Lemma
(see [Dem82b], Th.~1.5).
In this way, we avoid the singularity of the weight $|s|^{-2r}$ along~$Y$.
We find
$$
\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon^{(1)},
g_\varepsilon^{(1)}\rangle\, |s|^{-2r}dV_\omega\le
\int_{X_c\ssm Y_c}|F_\infty|^2
(1+\varepsilon^{-2}|s|^2)^2\theta'(\varepsilon^{-2}|s|^2)^2
|s|^{-2r}dV_\omega.
$$
Now, we let $\varepsilon\to 0$ and view $s$ as ``transverse local 
coordinates'' around $Y$. As $F_\infty$ coincides with $f$ on~$Y$, 
it is not hard to see that the 
right hand side converges to $c_r\int_{Y_c}|f|^2|\Lambda^r(ds)|^{-2}
dV_{Y,\omega}$ where $c_r$ is the ``universal'' constant
$$
c_r=\int_{z\in\bC^r,\,|z|\le 1}
(1+|z|^2)^2\theta'(|z|^2)^2{\ii^{r^2}\Lambda^r(dz)
\wedge\Lambda^r(d\overline z)\over|z|^{2r}}<+\infty
$$
depending only on~$r$. The second term
$$
g_\varepsilon^{(2)}=\theta(\varepsilon^{-2}|s|^2)d''F_\infty
$$
in $g_\varepsilon$ satisfies $\Supp(g_\varepsilon^{(2)})\subset\{|s|
<\varepsilon\}$ and $|g_\varepsilon^{(2)}|=O(|s|)$ (just look at the 
Taylor expansion of $d''F_\infty$ near $Y$). From this we easily conclude that
$$
\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon^{(2)},
g_\varepsilon^{(2)}\rangle\, |s|^{-2r}dV_{X,\omega}=O(\varepsilon^2),
$$
provided that $B_\varepsilon$ remains locally uniformly bounded below near~$Y$
(this is the case for instance if we have strict inequalities in the
curvature assumption (a)). If this holds true, we apply Proposition~8.4
on $X_c\ssm Y_c$ with the additional weight factor~$|s|^{-2r}$. Otherwise, 
we use the modified estimate stated
in Remark~8.5 in order to solve the approximate equation $D''u+
\delta^{1/2}h=g_\varepsilon$ with $\delta>0$ small. This yields sections
$u=u_{c,\varepsilon,\delta}$, $h=h_{c,\varepsilon,\delta}$ such that
$$
\eqalign{
\int_{X_c\ssm Y_c}(\eta_\varepsilon+\lambda_\varepsilon)^{-1}
|u_{c,\varepsilon,\delta}|^2|s|^{-2r}\,dV_\omega
&{}+
\int_{X_c\ssm Y_c}|h_{c,\varepsilon,\delta}|^2|s|^{-2r}\,dV_\omega\cr
&\le 2\int_{X_c\ssm Y_c}\langle (B_\varepsilon+\delta I)^{-1}g_\varepsilon,
g_\varepsilon\rangle|s|^{-2r}\,dV_\omega,\cr}
$$ 
and the right hand side is under control in all cases. The extra error
term $\delta^{1/2}h$ can be removed at the end by letting $\delta$ tend
to~$0$.  Since there is essentially no additional difficulty involved
in this process, we will assume for simplicity of exposition that we
do have the required lower bound for~$B_\varepsilon$ and the estimates
of $\smash{g_\varepsilon^{(1)}}$ and $\smash{g_\varepsilon^{(2)}}$ as
above. For $\delta=0$, the above estimate provides a solution
$u_{c,\varepsilon}$ of the equation $D''u_{c,\varepsilon}=
g_\varepsilon=D''F_\varepsilon$ on $X_c\ssm Y_c$, such that
$$
\eqalign{
\int_{X_c\ssm Y_c}(\eta_\varepsilon+\lambda_\varepsilon)^{-1}
|u_{c,\varepsilon}|^2|s|^{-2r}dV_{X,\omega}
&\le 2\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon,
g_\varepsilon\rangle\, |s|^{-2r}dV_{X,\omega}\cr
&\le 2\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
+O(\varepsilon).\cr}
$$
Here we have
$$
\eqalign{
\sigma_\varepsilon&=\log(|s|^2+\varepsilon^2)\le
\log(e^{-2\alpha}+\varepsilon^2)\le -2\alpha+O(\varepsilon^2)\le
-2+O(\varepsilon^2),\cr
\eta_\varepsilon&=\varepsilon-\chi_0(\sigma_\varepsilon)\le
(1+O(\varepsilon))\sigma_\varepsilon^2,\cr
\lambda_\varepsilon&={\chi_0'(\sigma_\varepsilon)^2\over
\chi_0''(\sigma_\varepsilon)}=(1-\sigma_\varepsilon)^2+
(1-\sigma_\varepsilon)\le(3+O(\varepsilon))\sigma_\varepsilon^2,\cr
\eta_\varepsilon+\lambda_\varepsilon&\le
(4+O(\varepsilon))\sigma_\varepsilon^2\le(4+O(\varepsilon))
\big(-\log(|s|^2+\varepsilon^2)\big)^2.\cr}
$$
As $F_\varepsilon$ is uniformly bounded with support in $\{|s|<\varepsilon\}$,
we conclude from an obvious volume estimate that
$$
\int_{X_c}{|F_\varepsilon|^2\over
(|s|^2+\varepsilon^2)^r(-\log(|s|^2+\varepsilon^2))^2}dV_{X,\omega}
\le {\Const \over (\log\varepsilon)^2}.
$$
Therefore, thanks to the usual inequality
$|t+u|^2\le (1+k)|t|^2+(1+k^{-1})|u|^2$ applied to the sum
$F_{c,\varepsilon}=\widetilde f_\varepsilon-u_{c,\varepsilon}$
with $k=|\log\varepsilon|$, we obtain
from our previous estimates
$$
\int_{X_c\ssm Y_c}
{|F_{c,\varepsilon}|^2\over
(|s|^2+\varepsilon^2)^r(-\log(|s|^2+\varepsilon^2))^2}dV_{X,\omega}
\le 8\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
+O(|\log\varepsilon|^{-1}).
$$
In addition to this, we have $d''F_{c,\varepsilon}=0$ by construction,
and this equation can be seen to extend from $X_c\ssm Y_c$ to $X_c$ by 
the $L^2$ estimate ([Dem82b], Lemma 6.9).

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

When $q\ge 1$, the arguments needed to get a smooth solution involve
more delicate considerations, and we will skip the details, which are
extremely technical and not very enlightening.
\endproof
%
\claim{(8.7) Remarks} \rm
\vskip3pt
\noindent{\rm (a)} When $q=0$, the estimates provided by Theorem~8.6 are
independent of the K\"ahler metric~$\omega$. In fact, if $f$ and $F$
are holomorphic sections of $\Lambda^nT^\star_X\otimes L$ over $Y$ 
(resp.~$X$), viewed as $(n,0)$-forms with values in~$L$, we can ``divide''
$f$ by $\Lambda^r(ds)\in\Lambda^r(TX/TY)^\star\otimes\det E$ to get a 
section $f/\Lambda^r(ds)$ of $\Lambda^pT^\star_Y\otimes L\otimes(\det E)^{-1}$
over~$Y$. We then find
$$
\eqalign{
|F|^2dV_{X,\omega}&=\ii^{n^2}\{F,F\},\cr
{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
&=\ii^{p^2}\{f/\Lambda^r(ds),f/\Lambda^r(ds)\},\cr}
$$
where $\{\bu,\bu\}$ is the canonical bilinear pairing described in~(3.3).
\medskip
\noindent{\rm (b)} The hermitian structure on $E$ is not really used
in~depth. In fact, one only needs $E$ to be equipped with a {\it Finsler
metric}, that is, a smooth complex homogeneous function of degree~$2$
on~$E$ [or equivalently, a smooth hermitian metric on the tautological
bundle $\cO_{P(E)}(-1)$ of lines of~$E$ over the projectivized 
bundle~$P(E)$, see (4.12)]. The section $s$ of $E$ induces a section $[s]$ of
$P(E)$ over $X\ssm s^{-1}(0)$ and a corresponding section $\widetilde s$ of
the pull-back line bundle $[s]^\star\cO_{P(E)}(-1)$.  A trivial check
shows that Theorem~8.6 as well as its proof extend to the case of a
Finsler metric on~$E$, if we replace everywhere $\{\ii\Theta_Es,s\}$
by $\{\ii\Theta([s]^\star\cO_{P(E)}(-1))\widetilde s,\widetilde s\,\}$ (especially in
hypothesis 8.6~b)). A minor issue is that $|\Lambda^r(ds)|$ is 
(a priori) no longer defined, since no obvious hermitian norm exists
on~$\det E$. A posteriori, we have the following ad hoc definition of 
a metric on $(\det E)^\star$ which makes the $L^2$~estimates work as 
before: for $x\in X$ and $\xi\in\Lambda^rE^\star_x$, we set
$$
|\xi|^2_x={1\over c_r}
\int_{z\in E_x}(1+|z|^2)^2
\theta'(|z|^2)^2\,{\ii^{r^2}\xi\wedge\overline\xi\over|z|^{2r}}
$$
where $|z|$ is the Finsler norm on~$E_x$ [the constant $c_r$ is there to 
make the result agree with the hermitian case; it is not hard to see that
this metric does not depend on the choice of~$\theta\,$].\qed
\endclaim
%
We now present a few interesting corollaries. The first one is a
surjectivity theorem for restriction morphisms in Dolbeault cohomology.

\claim{(8.8) Corollary}
Let $X$ be a projective algebraic manifold and $E$ a holomorphic
vector bundle of rank $r$ over~$X$, $s$ a holomorphic section of $E$
which is everywhere transverse to the zero section, $Y=s^{-1}(0)$, and
let $L$ be a holomorphic line bundle such that \hbox{$F=L^{1/r}\otimes
E^\star$} is Griffiths positive $($we just mean formally that
\hbox{${1\over r} \ii\Theta_L\otimes\Id_E-\ii\Theta_E>_\Grif
0)$}. Then the restriction morphism
$$
H^{0,q}(X,\Lambda^nT^\star_X\otimes L)\to H^{0,q}(Y,\Lambda^nT^\star_X
\otimes L)
$$
is surjective for every $q\ge 0$.
\endclaim
%
\begproof{Proof.} A short computation gives
$$
\eqalign{
\ii\,d'&d''\log|s|^2=\ii\,d'\Big({\{s,D's\}\over|s|^2}\Big)\cr
&=\ii\Big({\{D's,D's\}\over|s|^2}-{\{D's,s\}\wedge \{s,D's\}\over |s|^4}+
{\{s,\Theta_Es\}\over|s|^2}\Big)
\ge -{\{\ii\Theta_Es,s\}\over|s|^2}\cr}
$$
thanks to Lagrange's inequality and the fact that $\Theta_E$ is
antisymmetric. Hence, if $\delta$ is a small positive constant
such that
$$
-\ii\Theta_E+{1\over r}\ii\Theta_L\otimes\Id_E\ge_\Grif\delta\,
\omega\otimes\Id_E>0,
$$
we find
$$
\ii\Theta_L+r\,\ii\,d'd''\log|s|^2\ge r\delta\,\omega.
$$
The compactness of $X$ implies $\ii\Theta_E\le C\omega\otimes\Id_E$
for some $C>0$. Theorem~8.6 can thus be applied with $\alpha=r\delta/C$
and Corollary 8.8 follows. By remark 8.7~b), the above surjectivity
property even holds if $L^{1/r}\otimes E^\star$ is just assumed to be
ample (in the sense that the associated line bundle $\pi^\star L^{1/r}
\otimes\cO_{P(E)}(1)$ is positive on the projectivized bundle
$\pi:P(E)\to X$ of lines of~$E$).\qed
\endproof
%
Another interesting corollary is the following special case, 
dealing with bounded pseudoconvex domains $\Omega\compact\bC^n$. Even
this simple version retains highly interesting information on the behavior
of holomorphic and plurisubharmonic functions.

\claim{(8.9) Corollary} Let $\Omega\subset\bC^n$ be a bounded
pseudoconvex domain, and let $Y\subset X$ be a nonsingular complex
submanifold defined by a section $s$ of some hermitian vector
bundle $E$ with {\rm bounded} curvature tensor on~$\Omega$. Assume
that $s$ is everywhere transverse to the zero section and that
$|s|\le e^{-1}$ on~$\Omega$. Then there is a constant $C>0$ $($depending 
only on $E)$, with the following property:
for every psh function $\varphi$ on $\Omega$, every holomorphic 
function $f$ on $Y$ with $\int_Y|f|^2|\Lambda^r(ds)|^{-2}e^{-\varphi}dV_Y
<+\infty$, there exists an extension $F$ of $f$ to $\Omega$ such that
$$
\int_\Omega{|F|^2\over|s|^{2r}(-\log|s|)^2}e^{-\varphi}dV_\Omega\le
C\int_Y{|f|^2\over|\Lambda^r(ds)|^2}e^{-\varphi}dV_Y.
$$
\endclaim
%
\begproof{Proof.} We apply essentially the same idea as for the previous corollary,
in the special case when $L=\Omega\times\bC$ is the trivial bundle
equipped with a weight function $e^{-\varphi-A|z|^2}$. The choice of a
sufficiently large constant $A>0$ guarantees that the curvature assumption
8.6~a) is satisfied ($A$ just depends on the presupposed bound for the 
curvature tensor of~$E$).\qed
\endproof
%
\claim{(8.10) Remark} \rm The special case when $Y=\{z_0\}$ is a point is 
especially interesting. In that case, we just take $s(z)=(e\diam\Omega)^{-1}
(z-z_0)$, viewed as a section of the rank $r=n$ trivial vector bundle 
$\Omega\times \bC^n$ with $|s|\le e^{-1}$. We take $\alpha=1$ 
and replace $|s|^{2n}(-\log|s|)^2$ in the denominator by 
$|s|^{2(n-\varepsilon)}$, using the inequality
$$
-\log|s|={1\over\varepsilon}\log|s|^{-\varepsilon}\le {1\over\varepsilon}
|s|^{-\varepsilon},\qquad\forall \varepsilon>0.
$$ 
For any given value $f_0$, we then find a holomorphic function $f$ such 
that $f(z_0)=f_0$ and
$$
\int_\Omega{|f(z)|^2\over|z-z_0|^{2(n-\varepsilon)}}e^{-\varphi(z)}dV_\Omega
\le{C_n\over\varepsilon^2(\diam\Omega)^{2(n-\varepsilon)}}|f_0|^2
e^{-\varphi(z_0)}.
$$
\endclaim
%
\subsection*{\csect.D. Skoda's division theorem for ideals of holomorphic functions}
Following a strategy inpired by T.~Ohsawa [Ohs02, Ohs04], we give here 
a version of Skoda's division theorem for ideals of holomorphic functions, 
by reducing it to an extension problem. Our approach uses Manivel's version
of the extension theorem presented above, and leads to results very close
to those of Skoda [Sko80], albeit somewhat weaker.

Let $(X,\omega)$ be a K\"ahler manifold, $\dim X=n$, and let $g:E\to Q$ a
holomorphic morphism of hermitian vector bundles over $X$. Assume
for a moment that $g$ is {\it everywhere surjective}. Given a holomorphic line
bundle $L\to X$, we are interested in
conditions insuring that the induced morphism $g:H^0(X,K_X\otimes E\otimes L)
\to H^0(X,K_X\otimes Q\otimes L)$ is also surjective (as is observed
frequently in algebraic geometry, it will be easier to twist by an 
adjoint line bundle $K_X\otimes L$
than by $L$ alone). For that purpose, it is natural to consider the 
subbundle $S=\Ker g\subset E$ and the exact sequence
$$
0\longrightarrow S\buildo \longrightarrow^j E\buildo \longrightarrow^g Q\longrightarrow 0
\leqno(8.11)
$$
where $j:S\to E$ is the inclusion, as well as the dual exact sequence
$$
0\longrightarrow Q^*\buildo \longrightarrow^{g^*} E^*\buildo \longrightarrow^{j^*} S^*\longrightarrow 0,
\leqno(8.11')
$$
which we will twist by suitable line bundles. The main idea
of [Ohs02, Ohs04] is that finding a lifting of a section by $g$
is essentially equivalent to extending the related section on
$\cY=P(Q^*)=\bP(Q)$ to \hbox{$\cX=P(E^*)=\bP(E)$}, using the
obvious embedding $\cY\subset\cX$ of the projectivized bundles.
In fact, if $r_S=r_E-r_Q$ are 
the respective ranks of our vector bundles, we have the classical formula
$$
K_\cX=K_{\bP(E)}=\pi^*(K_X\otimes\det E)\otimes\cO_{\bP(E)}(-r_S)
\leqno(8.12)
$$
where $\pi:\bP(E)\to X$ is the canonical projection. Therefore,
since $E$ coincides with the direct image sheaf $\pi_*\cO_{\bP(E)}(1)$,
a section of $H^0(X,K_X\otimes E\otimes L)$ can also be seen as a section
of 
$$
H^0(\cX,K_\cX\otimes \cO_\cX(r_S+1)\otimes \pi^*(L\otimes\det E^{-1})).
\leqno(8.13)
$$
Now, since $\cO_{\cX}(1)_{\restriction\ cY}=\cO_{\cY}(1)=\cO_{\bP(Q)}(1)$, the
lifting problem is equivalent to extending to $\cX$ a section of the
line bundle $(K_\cX\otimes\cL)_{\restriction\cY}$ where 
$\cL=\cO_\cX(r_S+1)\otimes 
\pi^*(L\otimes\det E^{-1})$. As a submanifold, $\cY$ is the zero locus 
of the bundle morphism
$$
\cO_{\bP(E)}(-1)\hookrightarrow \pi^* E^* \to \pi^*(E^*/Q^*)=\pi^* S^*,
$$
hence it is the (transverse) zero locus of a naturally defined section
$$
s\in H^0(\cX,\cE)\qquad\hbox{where}\quad
\cE:=\pi^*S^*\otimes\cO_{\bP(E)}(1).\leqno(8.14)
$$
Let us assume that $E$ is endowed with a smooth hermitian metric $h$ such 
that $\Theta_{E,h}$ is Griffiths semi-positive. We equip $Q$ with the quotient
metric and $S$, $\cO_{\bP(E)}(1)$, $\det E$, $\cE$ (...) with the induced 
metrics. 
A sufficient curvature condition needed to apply the Ohsawa-Takegoshi-Manivel
extension theorem is
$$
\ii\Theta_{\cL}+r_S\ii d'd''\log|s|^2\ge \varepsilon
{\{\ii\Theta_\cE s,s\}\over |s|^2}
$$
for $\varepsilon>0$ small enough (i.e.\ in some range 
$\varepsilon\in[0,\varepsilon_0]$, $\varepsilon_0\le 1$). Since 
$\ii d'd''\log|s|^2\ge -\ii\Theta_{\cO(\bP(E)}(1)
-{\{\ii\Theta_{\pi^*S^*}s,s\}\over |s|^2}$, we obtain the 
sufficient condition
$$
\pi^*\ii\Theta_{L\otimes \det E^{-1}}+(1-\varepsilon)\ii\Theta_{\cO_{\bP(E)}(1)}-
(r_S+\varepsilon){\{\ii\Theta_{\pi^*S^*}s,s\}\over |s|^2}\ge 0,\qquad
\varepsilon\in[0,\varepsilon_0].
\leqno(8.15)
$$
The assumption that $E$ is Griffiths semi-positive implies 
$\ii\Theta_{\det E}\ge 0$, $\ii\Theta_{\cO_{\bP(E)}(1)}\ge 0$ and also
$$
{\{\ii\Theta_{\pi^*S^*}s,s\}\over |s|^2}\le \ii\Theta_{\det Q}.\leqno(8.16)
$$
In fact this is equivalent to proving that $S\otimes \det Q$ is 
Griffiths semi-positive, but we have in fact
$S\otimes \det Q=S\otimes\det S^{-1}\otimes\det E=\Lambda^{r_S-1}S^*\otimes
\det E$, which is a quotient of $\Lambda^{r_S-1}E^*\otimes \det E=
\Lambda^{r_E-r_S+1}E\ge 0$. This shows that (8.15) is implied by the
simpler condition
$$
\ii\Theta_L\ge\ii\Theta_{\det E}+(r_S+\varepsilon_0)\ii\Theta_{\det Q},
\leqno(8.17)
$$
in particular $L=\det E\otimes (\det Q)^k$, $k>r_S$, satisfies the curvature 
condition. We derive from there:

\claim{(8.18) Theorem} Assume that $(X,\omega)$ is a K\"ahler manifold 
possessing a complete 
K\"ahler metric $\widehat\omega$, and let \hbox{$g:E\to Q$} be a surjective morphism
of holomorphic vector bundles, where $(E,h_E)$ is a Griffiths semi-positive
hermitian bundle. Consider a hermitian holomorphic line bundle 
$(L,h_L)$ such that
$$
\ii\Theta_L-(r_S+\varepsilon)\ii\Theta_{\det Q}
-\ii\Theta_{\det E}\ge 0,\qquad r_S=r_E-r_Q,~~
\varepsilon>0.
$$
Then for every $L^2$ holomorphic section $f\in H^0(X,K_X\otimes Q\otimes L)$ 
there exists a $L^2$ holomorphic section
\hbox{$h\in H^0(X,K_X\otimes E\otimes L)$}
such that $f=g\cdot h$ and  $\|h\|^2\le C_{n,r_E,\varepsilon}\|f\|^2$.
\endclaim
%
\begproof{Proof.} We apply Theorem 8.6 with respect to the data $(\cX,\cY,\cE,\cL)$
and $\alpha=\varepsilon^{-1}$, $r=r_S$. Since
$|s|\le 1$, we have to multiply $s$ by $\delta=\exp(-1/\varepsilon)$ to
enforce hypothesis 8.6~b). This affects the final estimate only as far
as the term $\log|s|$ is concerned, since both $|s|^{2r}$ and
$|\Lambda^r(ds)|^2=1$ are multiplied by $\delta^{2r}$. Finally, we apply
Fubini's theorem to reduce integrals over $\cX$ or $\cY$ to integrals
over $X$, observing that all fibers of $\cX=\bP(E)\to X$ are isometric 
and therefore
produce the same fiber integral. Theorem 8.18 follows. By exercising
a little more care in the estimates, one sees that the constant
$C_{n,r_E,\varepsilon}$ is actually bounded by $C_{n,r_E}\varepsilon^{-2}$, 
where the $\varepsilon^{-2}$ comes from the term $(-\log|s|)^2$, after 
$s$ has been multiplied by $\exp(-1/\varepsilon)$.\qed
\endproof
%
Skoda's original method is slightly more accurate. It shows that one
can take $C_{n,r_E,\varepsilon}=\varepsilon^{-1}$, and, more importantly,
replaces the curvature hypothesis by the weaker one
$$
\ii\Theta_L-(k+\varepsilon)\ii\Theta_{\det Q}
-\ii\Theta_{\det E}\ge 0,\qquad k=\min(r_S,n),~~r_S=r_E-r_Q
\leqno(8.19)
$$
for some $\varepsilon>0$, which does not seem so easy to obtain with
the present method. It is however possible to get estimates also when
$Q$ is endowed with a metric given a priori, that can be distinct from
the quotient metric of $E$ by $g$.  Then the map
$g^\star(gg^\star)^{-1}~:~Q\longrightarrow E$ is the lifting of $Q$
orthogonal to $S=\Ker g$. The quotient metric $|\bu|'$ on $Q$ is
therefore defined in terms of the original metric $|\bu|$ by
$$
|v|^{\prime 2}=|g^\star(gg^\star)^{-1}v|^2=\langle(gg^\star)^{-1}v,v\rangle
=\det(gg^\star)^{-1}\,\langle\widetilde{gg^\star}v,v\rangle
$$
where $\widetilde{gg^\star}\in\End(Q)$ denotes the endomorphism of $Q$
whose matrix is the transposed comatrix of $gg^\star$. For every
$w\in\det Q$, we find
$$
|w|^{\prime2}=\det(gg^\star)^{-1}\,|w|^2.
$$
If $Q'$ denotes the bundle $Q$ with the quotient metric, we get
$$
\ii\Theta_{\det Q'}=\ii\Theta_{\det Q}+\ii d'd''\log\det(gg^\star).
$$
In order that the hypotheses of Theorem~8.18 be satisfied, we are led to define
a new metric $|\bu|'$ on $L$ by $|u|^{\prime 2}=|u|^2\,\big(\det(gg^\star)
\big)^{-m-\varepsilon}$. Then
$$
\ii\Theta_{L'}=\ii\Theta_L+(m+\varepsilon)\,\ii d'd''\log\det(gg^\star)\ge
(m+\varepsilon)\,\ii\Theta_{\det Q'}.
$$
Theorem 8.18 applied to $(E,Q',L')$ can now be reformulated:

\claim{(8.20) Theorem} Let $X$ be a weakly pseudoconvex
manifold equipped with a K\"ahler metric $\omega$, let $E\to Q$ be
a generically surjective morphism of hermitian vector bundles with
$E$ Griffiths semi-positive, and let
$L\to X$ be a hermitian holomorphic line bundle. Assume that
$$
\ii\Theta_L-(r_S+\varepsilon)\ii\Theta_{\det Q}
-\ii\Theta_{\det E}\ge 0,\qquad r_S=r_E-r_Q,~~
\varepsilon>0.
$$
Then for every holomorphic section $f$ of $K_X\otimes Q\otimes L$
such that 
$$
I=\int_X\langle\widetilde{gg^\star}f,f\rangle\,(\det gg^\star)^{-r_S-1-\varepsilon}\,
dV<+\infty,
$$
there exists a holomorphic section of $K_X\otimes E\otimes L$ such that 
$f=g\cdot h$ and 
$$
\int_X|h|^2\,(\det gg^\star)^{-r_S-\varepsilon}\,dV
\le C_{n,r_E,\varepsilon}\,I.
$$
In case $Q$ is of rank $1$, the estimate reduces to
$$
\int_X|h|^2\,|g|^{-2r_S-2\varepsilon}\,dV
\le C_{n,r_E,\varepsilon}\,
\int_X|f|^2\,|g|^{-2(r_S+1)-2\varepsilon}\,dV.
$$
\endclaim
%
\begproof{Proof.} if $Z\subset X$ is the analytic locus where $g:E\to Q$ is
not surjective and $X_c=\{\psi<c\}$ is an exhaustion of $X$ by weakly
pseudoconvex relatively compact open subsets, we exploit here the fact
that $X_c\ssm Z$ carries a complete metric (see [Dem82b]). It is
easy to see that the $L^2$ conditions forces a section defined a priori
only on $X\ssm Z$ to extend to~$X$.\qed
\endproof
%
The special case where $E=\cO_\Omega^{\oplus p}$ and $Q=\cO_\Omega$ are trivial
bundles over a weakly pseudocovex open set $\Omega\subset\bC^n$ is already
a quite substantial theorem, which goes back to [Sko72b]. In this case, we 
take $L$ to be the hermitian line bundle $(\cO_\Omega, e^{-\varphi})$ associated
with an arbitrary plurisubharmonic function $\varphi$ on $\Omega$.

\claim{(8.21) Corollary {\rm (Skoda's division theorem)}} 
Let $f,\,g_1,\ldots, g_p$ be holomorphic functions
on a weakly pseudoconvex open set $\Omega\subset\bC^n$ such that
$$
\int_\Omega|f|^2|g|^{-2(p+1)-2\varepsilon}e^{-\varphi}dV<+\infty
$$
for some plurisubharmonic function~$\varphi$. Then there exist
holomorphic functions $h_j$, $1\le j\le p$, such that $f=\sum g_jh_j$
on $\Omega$, and
$$
\int_X|h|^2\,|g|^{-2(p-1)-2\varepsilon}e^{-\varphi}\,dV
\le C_{n,p,\varepsilon}\,
\int_X|f|^2\,|g|^{-2p-2\varepsilon}e^{-\varphi}\,dV.
$$
\endclaim
%

\section{Approximation of closed positive (1,1)-currents by divisors}

\subsection*{\csect.A. Local approximation theorem through Bergman kernels}
We prove here, as an application of the Ohsawa-Takegoshi extension theorem,
that every psh function on a pseudoconvex open set $\Omega\subset\bC^n$
can be approximated very accurately by functions of the form
$c\log|f|$, where $c>0$ and $f$ is a holomorphic function. The main idea
is taken from [Dem92]. For other applications to algebraic geometry,
see [Dem93b] and Demailly-Koll\'ar [DK01]. Recall that 
the Lelong number of a function $\varphi\in\Psh(\Omega)$ at a point $x_0$ 
is defined to be
$$
\nu(\varphi,x_0)=\liminf_{z\to x_0}{\varphi(z)\over\log|z-x_0|}=
\lim_{r\to 0_+}{\sup_{B(x_0,r)}\varphi\over \log r}.\leqno(9.1)
$$
In particular, if $\varphi=\log|f|$ with $f\in\cO(\Omega)$, then 
$\nu(\varphi,x_0)$ is equal to
the vanishing order 
$$\ord_{x_0}(f)=\sup\{k\in\bN\,;D^{\alpha}f(x_0)=0,~
\forall|\alpha|<k\}.$$

\claim{(9.2) Theorem} Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\bC^n$. For every $m>0$, let
$\cH_\Omega(m\varphi)$ be the Hilbert space of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-2m\varphi}d\lambda<+\infty$ and
let $\varphi_m={1\over 2m}\log\sum|\sigma_\ell|^2$ where $(\sigma_\ell)$
is an orthonormal basis of~$\cH_\Omega(m\varphi)$. Then there are
constants $C_1,C_2>0$ independent of $m$ such that
{\plainitemindent=7mm
\plainitem{\rm (a)}$\displaystyle\varphi(z)-{C_1\over m}\le
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over m}
\log{C_2\over r^n}$\newline
for every $z\in\Omega$ and $r<d(z,\partial\Omega)$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$ and
\plainitem{\rm (b)}$\displaystyle\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le
\nu(\varphi,z)$~ for every $z\in\Omega$.}
\endclaim
%
\begproof{Proof.} (a) Note that $\sum|\sigma_\ell(z)|^2$ is the square of the norm
of the evaluation linear form $\ev_z:f\mapsto f(z)$ on $\cH_\Omega(m\varphi)$,
since $\sigma_\ell(z)=\ev_z(\sigma_\ell)$ is the $\ell$-th coordinate of $\ev_z$
in the orthonormal basis $(\sigma_\ell)$. In other words, we have
$$\sum|\sigma_\ell(z)|^2=\sup_{f\in B(1)}|f(z)|^2$$
where $B(1)$ is the unit ball of $\cH_\Omega(m\varphi)$ (The sum
is called the {\it Bergman kernel} associated with $\cH_\Omega(m\varphi)$.)
As $\varphi$ is locally bounded from above, the $L^2$ topology is 
actually stronger
than the topology of uniform convergence on compact subsets of~$\Omega$.
It follows that the series $\sum|\sigma_\ell|^2$ converges uniformly on
$\Omega$ and that its sum is real analytic. Moreover, by what we just 
explained, we have
$$\varphi_m(z)=\sup_{f\in B(1)}{1\over m}\log|f(z)|.$$
For $z_0\in\Omega$ and $r<d(z_0,\partial\Omega)$, the mean value
inequality applied to the psh function $|f|^2$ implies
$$\eqalign{
|f(z_0)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|z-z-0|<r}
|f(z)|^2d\lambda(z)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|z-z_0|<r}\varphi(z)\Big)
\int_\Omega|f|^2e^{-2m\varphi}d\lambda.\cr}$$
If we take the supremum over all $f\in B(1)$ we get
$$
\varphi_m(z_0)\le\sup_{|z-z_0|<r}\varphi(z)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the second inequality in (a) is proved -- as we see, this is an easy
consequence of the mean value inequality. Conversely, the
Ohsawa-Takegoshi extension theorem (Corollary 8.9) applied
to the $0$-dimensional subvariety $\{z_0\}\subset\Omega$ shows that for any
$a\in\bC$ there is a holomorphic function $f$ on $\Omega$ such that
$f(z_0)=a$ and
$$\int_\Omega|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z_0)},$$
where $C_3$ only depends on $n$ and~${\rm diam}\,\Omega$. We fix $a$
such that the right hand side is~$1$. Then $\Vert f\Vert \le 1$ and so
we get
$$\varphi_m(z_0)\ge{1\over m}\log|f(z_0)|=
{1\over m}\log|a|=\varphi(z)-{\log C_3\over 2m}.$$
The inequalities given in (a) are thus proved. Taking $r=1/m$, we find
that $\lim_{m\to+\infty}\sup_{|\zeta-z|<1/m}\varphi(\zeta)=\varphi(z)$
by the upper semicontinuity of $\varphi$, and therefore
$\lim\varphi_m(z)=\varphi(z)$, since $\lim{1\over m}\log (C_2m^n)=0$.
\medskip

\noindent
(b) The above estimates imply
$$
\sup_{|z-z_0|<r}\varphi(z)-{C_1\over  m}\le
\sup_{|z-z_0|<r}\varphi_m(z)\le\sup_{|z-z_0|<2r}\varphi(z)+
{1\over m}\log{C_2\over r^n}.$$
After dividing by $\log r<0$ when $r\to 0$, we infer
$$
{\sup_{|z-z_0|<2r}\varphi(z)+{1\over m}\log{C_2\over r^n}\over \log r}\le
{\sup_{|z-z_0|<r}\varphi_m(z)\over \log r}\le
{\sup_{|z-z_0|<r}\varphi(z)-{C_1\over  m}\over\log r},
$$
and from this and definition (9.1), it follows immediately that 
$$
\nu(\varphi,x)-{n\over m}\le\nu(\varphi_m,z)\le\nu(\varphi,z).
\eqno\square
$$
\endproof
%
Theorem 9.2 implies in a straightforward manner the deep result of
[Siu74] on the analyticity of the Lelong number upperlevel sets.

\claim{(9.3) Corollary {\rm[Siu74])}} Let $\varphi$ be a plurisubharmonic
function on a complex manifold~$X$. Then, for every $c>0$, the Lelong number
upperlevel set
$$
E_c(\varphi)=\big\{z\in X\,;\;\nu(\varphi,z)\ge c\big\}
$$
is an analytic subset of~$X$.
\endclaim
%
\begproof{Proof.} Since analyticity is a local property, it
is enough to consider the case of a psh function $\varphi$ on a 
pseudoconvex open set $\Omega\subset\bC^n$. The
inequalities obtained in 9.2~b) imply that 
$$
E_c(\varphi)=\bigcap_{m\ge m_0}E_{c-n/m}(\varphi_m).
$$
Now, it is clear that $E_c(\varphi_m)$ is the analytic set defined by the
equations $\sigma_\ell^{(\alpha)}(z)=0$ for all multi-indices $\alpha$
such that~$|\alpha|<mc$. Thus $E_c(\varphi)$ is analytic as a (countable) 
intersection of analytic sets.\qed
\endproof
%
\claim{(9.4) Remark} {\rm It can be easily shown that the Lelong numbers
of any closed positive $(p,p)$-current coincide (at least locally) with the
Lelong numbers of a suitable plurisubharmonic potential $\varphi$
(see Skoda [Sko72a]). Hence Siu's theorem also holds true for the
Lelong number upperlevel sets $E_c(T)$ of any closed positive 
$(p,p)$-current~$T$.}
\endclaim
%
\subsection*{\csect.B. Global approximation of closed (1,1)-currents
on a compact complex manifold}
We take here $X$ to be an arbitrary compact complex manifold (no K\"ahler
assumption is needed). Now, let $T$ be a closed $(1,1)$-current
on~$X$. We assume that $T$ is {\it almost positive}, i.e.\ that there exists
a $(1,1)$-form $\gamma$ with continuous coefficients such
that~$T\ge\gamma\;$; the case of positive currents ($\gamma=0$) is of course
the most important.

\claim{(9.5) Lemma} There exists a smooth closed $(1,1)$-form $\alpha$ 
representing the same~$\ddbar$-
cohomology class as $T$ and an {\rm almost psh} function $\varphi$
on $X$ such that \hbox{$T=\alpha+{i\over\pi}\ddbar\varphi$}. $($We say that
a function $\varphi$ is almost psh if its complex Hessian is bounded below
by a $(1,1)$-form with locally bounded coefficients, that is, if
$i\ddbar\varphi$ is almost positive$)$.
\endclaim
%
\begproof{Proof.} Select
an open covering $(U_j)$ of $X$ by coordinate balls such that 
$T=\smash{{i\over\pi}}
\ddbar\varphi_j$ over $U_j$, and construct a global function
$\varphi=\sum\theta_j \varphi_j$ by means of a partition of unity $(\theta_j)$
subordinate to~$U_j$. Now, we observe that $\varphi-\varphi_k$ is smooth 
on $U_k$  because all differences $\varphi_j-\varphi_k$ are smooth in 
the intersections
$U_j\cap U_k$ and $\varphi-\varphi_k=\sum\theta_j(\varphi_j-\varphi_k)$).
Therefore $\alpha:=T-{i\over\pi}\ddbar\varphi$ is smoth.\qed
\endproof
%
By replacing $T$ with $T-\alpha$ and $\gamma$ with
$\gamma-\alpha$, we can assume without loss of
generality that $\{T\}=0$, i.e. that $T={i\over\pi}\ddbar\varphi$
with an almost psh function~$\varphi$ on~$X$ such that
${i\over\pi}\ddbar\varphi\ge\gamma$.

Our goal is to approximate $T$ in the weak topology by currents 
$T_m={i\over\pi}\ddbar\varphi_m$ such their potentials $\varphi_m$
have analytic singularities in the sense of Definition 1.10, more
precisely, defined on a neighborhood $V_{x_0}$ of any point $x_0\in X$ 
in the form 
$\varphi_m(z)=c_m\log\sum_j|\sigma_{j,m}|^2+O(1)$,
where $c_m>0$ and the $\sigma_{j,m}$ are holomorphic functions on $V_{x_0}$.

We select a finite covering $(W_\nu)$ of $X$ with open
coordinate charts. Given $\delta>0$, we take in each $W_\nu$ a
maximal family of points with (coordinate) distance to the boundary
$\ge 3\delta$ and mutual distance $\ge\delta/2$. In this way, we get for 
$\delta>0$ small a finite covering of $X$ by open balls $U'_j$ of
radius~$\delta$ (actually every point is even at 
\hbox{distance${}\le\delta/2$} of
one of the centers, otherwise the family of points would not be maximal), 
such that the concentric ball $U_j$ of radius
$2\delta$ is relatively compact in the corresponding chart~$W_\nu$. Let
$\tau_j:U_j\longrightarrow B(a_j,2\delta)$ be the isomorphism given by the
coordinates of~$W_\nu$. Let $\varepsilon(\delta)$ be a modulus
of continuity for $\gamma$ on the sets $U_j$, such that
$\lim_{\delta\to 0}\varepsilon(\delta)=0$ and 
$\gamma_x-\gamma_{x'}\le{1\over 2}\varepsilon(\delta)\,\omega_x$ for all
$x,x'\in U_j$. We denote by $\gamma_j$ the $(1,1)$-form with constant
coefficients on $B(a_j,2\delta)$ such that $\tau_j^\star\gamma_j$
coincides with $\gamma-\varepsilon(\delta)\,\omega$ at
$\tau_j^{-1}(a_j)$. Then we have
$$0\le\gamma-\tau_j^\star\gamma_j\le 2\varepsilon(\delta)\,\omega~~~~
\hbox{\rm on}~~U'_j\leqno(9.6)$$
for $\delta>0$ small. We set $\varphi_j=\varphi\circ\tau_j^{-1}$ on
$B(a_j,2\delta)$ and let $q_j$ be the homogeneous quadratic
function in $z-a_j$ such that ${i\over\pi}\ddbar q_j=
\gamma_j$ on $B(a_j,2\delta)$. Finally, we set
$$\psi_j(z)=\varphi_j(z)-q_j(z)~~~~\hbox{\rm on}~B(a_j,2\delta).
\leqno(9.7)$$
Then $\psi_j$ is plurisubharmonic, since
$${i\over\pi}\ddbar(\psi_j\circ\tau_j)=T-\tau_j^\star\gamma_j
\ge\gamma-\tau_j^\star\gamma_j\ge 0.$$
We let $U'_j\compact U''_j\compact U_j$ be 
concentric balls of radii $\delta$, $1.5\,\delta$, $2\delta$ respectively.
On each open set $U_j$ the function
$\psi_j:=\varphi-q_j\circ\tau_j$ defined
in (9.7) is plurisubharmonic, so Theorem~(9.2) applied
with $\Omega=U_j$ produces functions
$$\psi_{j,m}={1\over 2m}\log\sum_\ell|\sigma_{j,\ell}|^2,~~~~
(\sigma_{j,\ell})=\hbox{\rm basis of}~\cH_{U_j}(m\psi_j).\leqno(9.8)$$
These functions approximate $\psi_j$ as $m$ tends to $+\infty$ and 
satisfy the inequalities
$$\psi_j(x)-{C_1\over m}\le\psi_{j,m}(x)\le\sup_{|\zeta-x|<r}
\psi_j(\zeta)+{1\over m}\log{C_2\over r^n}.\leqno(9.9)$$
The functions $\psi_{j,m}+q_j\circ\tau_j$ on $U_j$ then have
to be glued together by a partition of unity technique. For this,
we rely on the following ``discrepancy'' lemma, estimating the variation
of the approximating functions on overlapping balls.

\claim{(9.10) Lemma} There are constants $C_{j,k}$ 
independent of $m$ and $\delta$ such that the almost psh functions
$w_{j,m}=2m(\psi_{j,m}+q_j\circ\tau_j)$, i.e.\
$$w_{j,m}(x)=2m\,q_j\circ\tau_j(x)+
\log\sum_\ell\big|\sigma_{j,\ell}(x)\big|^2,~~~~
x\in U''_j,$$
satisfy
$$|w_{j,m}-w_{k,m}|\le C_{j,k}\big(\log\delta^{-1}+
m\varepsilon(\delta)\delta^2\big)
~~~~\hbox{\rm on}~~U''_j\cap U''_k.$$
\endclaim
%
\begproof{Proof.} The details will be left as an exercise to the reader.
The main idea is the following: for any holomorphic function 
$f_j\in \cH_{U_j}(m\psi_j)$,
a $\dbar$ equation $\dbar u = \dbar(\theta f_j)$ can be solved on $U_k$, 
where $\theta$ is a cut-off function with support in $U''_j\cap U''_k$,
on a ball of radius${}<\delta/4$, 
equal to $1$ on the ball of radius $\delta/8$ centered at a given 
point $x_0\in U''_j\cap U''_k$. We apply
the $L^2$ estimate with respect to the weight 
\hbox{$(n+1)\log|x-x_0|^2+2m\psi_k$}, where the first term is picked up so as 
to force the solution $u$ to vanish at $x_0$, in such a way that
$F_k=u-\theta f_j$ is holomorphic and $F_k(x_0)=f_j(x_0)$. 
The discrepancy between the weights on $U''_j$ and $U''_k$~is
$$
\psi_j(x)-\psi_k(x)=-\big(q_j\circ\tau_j(x)-q_k\circ\tau_k(x)\big)
$$
and the $\ddbar$ of this difference is $O(\varepsilon(\delta))$, so it
is easy to correct the discrepancy up to a $O(\varepsilon(\delta)\delta^2)$
term by multiplying our functions by an invertible holomorphic
function~$G_{jk}$.
In this way, we get a uniform $L^2$ control on the $L^2$ norm of the
solution $f_k=G_{jk}F_k=G_{jk}(u-\theta f_j)$ of the form
$$\int_{U_k}|f_k|^2e^{-2m\psi_k}\le C_{j,k}\delta^{-2n-4}
e^{m O(\varepsilon(\delta)\delta^2)} \int_{U_j}|f_j|^2e^{-2m\psi_j}.$$
The required estimate follows, using the fact that
$$
e^{2m\psi_{j,m}(x)}=\sum_\ell|\sigma_{j,\ell}(x)|^2=
\sup_{f\in\cH_{U_j}(m\psi_j),\,\Vert f\Vert\le 1}|f(x)|^2~~~~\hbox{on $U_j$},
$$
and the analogous equality on $U_k$.\qed
\endproof
%
Now, the actual glueing of our almost psh functions is performed
using the following elementary partition of unity calculation.

\claim{(9.11) Lemma} Let $U'_j\compact U''_j$ be locally finite 
open coverings
of a complex manifold $X$ by relatively compact open sets, and let
$\theta_j$ be smooth nonnegative functions with support in $U''_j$,
such that $\theta_j\le 1$ on $U''_j$ and $\theta_j=1$ on $U'_j$.
Let $A_j\ge 0$ be such that
$$i(\theta_j\ddbar\theta_j-\partial\theta_j\wedge\dbar\theta_j)\ge
-A_j\omega~~~~\hbox{\rm on}~~U''_j\ssm U'_j$$
for some positive $(1,1)$-form~$\omega$.
Finally, let $w_j$ be almost psh functions on $U_j$ with
the property that $i\partial\smash{\dbar}w_j\ge\gamma$
for some real $(1,1)$-form $\gamma$ on~$M$, and let $C_j$ be constants
such that
$$w_j(x)\le C_j+\sup_{k\ne j,\,U'_k\ni x}w_k(x)~~~~\hbox{\rm on}~~
U''_j\ssm U'_j.$$
Then the function $w=\log\big(\sum\theta_j^2 e^{w_j}\big)$
is almost psh and satisfies
$$i\ddbar w\ge\gamma-2\Big(\sum_j\bOne_{U''_j\ssm U'_j}A_je^{C_j}\Big)
\omega.$$
\endclaim
%
\begproof{Proof.} If we set $\alpha_j=\theta_j\partial w_j+2\partial\theta_j$,
a straightforward computation shows that
$$\eqalign{
\partial w&={\sum(\theta_j^2\partial w_j+2\theta_j\partial\theta_j)e^{w_j}
\over\sum\theta_j^2e^{w_j}}={\sum\theta_je^{w_j}\alpha_j
\over\sum\theta_j^2e^{w_j}},\cr
\ddbar w&={\sum\!\big(\alpha_j\wedge\overline\alpha_j{+}\theta_j^2\ddbar w_j
{+}2\theta_j\ddbar\theta_j{-}2\partial\theta_j{\wedge}\dbar\theta_j\big)
e^{w_j}\over\sum\theta_j^2e^{w_j}}
-{\sum_{j,k}\theta_je^{w_j}\theta_ke^{w_k}\alpha_j{\wedge}
\overline\alpha_k\over\big(\sum\theta_j^2e^{w_j}\big)^2}\cr
&={\sum_{j<k}\!\big|\theta_j\alpha_k{-}\theta_k\alpha_j\big|^2e^{w_j}e^{w_k}
\over\big(\sum\theta_j^2e^{w_j}\big)^2}{+}{\sum\theta_j^2e^{w_j}\ddbar w_j
\over\sum\theta_j^2e^{w_j}}{+}{\sum\!\big(2\theta_j\ddbar\theta_j{-}
2\partial\theta_j{\wedge}\dbar\theta_j\big)e^{w_j}\over\sum\theta_j^2e^{w_j}}\cr}$$
by using the Legendre identity. The first term in the last line is nonnegative
and the second one is $\ge\gamma$. In the third term, if $x$ is in the support
of $\theta_j\ddbar\theta_j-\partial\theta_j\wedge\dbar\theta_j$, then
$x\in U''_j\ssm U'_j$ and so $w_j(x)\le C_j+w_k(x)$ for some $k\ne j$
with $U'_k\ni x$ and $\theta_k(x)=1$. This gives
$$i{\sum\big(
2\theta_j\ddbar\theta_j-2\partial\theta_j\wedge\dbar\theta_j\big)e^{w_j}
\over\sum\theta_j^2e^{w_j}}\ge
-2\sum_j\bOne_{U''_j\ssm U'_j}e^{C_j}A_j\omega.$$
The expected lower bound follows.\qed
\endproof
%
We apply Lemma~(9.11) to functions $\widetilde w_{j,m}$ which are just
slight modifications of the functions $w_{j,m}=2m(\psi_{j,m}+q_j\circ\tau_j)$
occurring in (9.10)~:
$$\eqalign{
\widetilde w_{j,m}(x)
&=w_{j,m}(x)+2m\Big({C_1\over m}+
C_3\varepsilon(\delta)(\delta^2/2-|\tau_j(x)|^2)\Big)\cr
&=2m\Big(\psi_{j,m}(x)+q_j\circ\tau_j(x)+{C_1\over m}+
C_3\varepsilon(\delta)(\delta^2/2-|\tau_j(x)|^2)\Big)\cr}$$
where $x\mapsto z=\tau_j(x)$ is a local coordinate identifying $U_j$
to $B(0,2\delta)$, $C_1$ is the constant occurring in (9.9) and $C_3$ is 
a sufficiently large constant. 
It is easy to see that we can take $A_j=C_4\delta^{-2}$ in Lemma~(9.11).
We have
$$
\widetilde w_{j,m}\ge w_{j,m}+2C_1+m{C_3\over 2}\varepsilon(\delta)\delta^2
~~~~\hbox{on $B(x_j,\delta/2)\subset U'_j$},
$$
since $|\tau_j(x)|\le\delta/2$ on $B(x_j,\delta/2)$, while
$$
\widetilde w_{j,m}\le w_{j,m}+2C_1-mC_3\varepsilon(\delta)\delta^2
~~~~\hbox{on $U''_j\ssm U'_j$}.
$$
For $m\ge m_0(\delta)=(\log\delta^{-1}/(\varepsilon(\delta)\delta^2)$, 
Lemma (9.10) implies $|w_{j,m}-w_{k,m}|\le C_5m\varepsilon(\delta)\delta^2$ on
$U''_j\cap U''_k$. Hence, for $C_3$ large enough, we get
$$\widetilde w_{j,m}(x)\le \sup_{k\ne j,\,B(x_k,\delta/2)\ni x}w_{k,m}(x)
\le \sup_{k\ne j,\,U'_k\ni x}w_{k,m}(x)~~~~\hbox{\rm on}~~
U''_j\ssm U'_j,$$
and we can take $C_j=0$ in the hypotheses of Lemma~(9.11). 
The associated function $w=\log\big(\sum\theta_j^2e^{\widetilde w_{j,m}}\big)$ 
is given by
$$w=\log\sum_j\theta_j^2\exp\Big(2m\big(\psi_{j,m}+q_j\circ\tau_j+
{C_1\over m}+C_3\varepsilon(\delta)(\delta^2/2-|\tau_j|^2)\big)
\Big).$$
If we define $\varphi_m={1\over 2m}w$, we get
$$
\varphi_m(x):={1\over 2m}w(x)\ge \psi_{j,m}(x)+
q_j\circ\tau_j(x)+{C_1\over m}+{C_3\over 4}\varepsilon(\delta)\delta^2
>\varphi(x)
$$
in view of (9.9), by picking an index $j$ such that $x\in B(x_j,\delta/2)$.
In the opposite
direction, the maximum number $N$ of overlapping
balls~$U_j$ does not depend on $\delta$, and we thus get
$$w\le\log N+2m\Big(
\max_j\big\{\psi_{j,m}(x)+q_j\circ\tau_j(x)\big\}+
{C_1\over m}+{C_3\over 2}\varepsilon(\delta)\delta^2\Big).$$
By definition of $\psi_j$ we have $\sup_{|\zeta-x|<r}\psi_j(\zeta)\le
\sup_{|\zeta-x|<r}\varphi(\zeta)-q_j\circ\tau_j(x)+C_5r$ thanks to
the uniform Lipschitz continuity of $q_j\circ\tau_j$, thus by 
(9.9) again we find
$$\varphi_m(x)\le
{\log N\over 2m}+\sup_{|\zeta-x|<r}\varphi(\zeta)+
{C_1\over m}+{1\over m}\log{C_2\over r^n}+
{C_3\over 2}\varepsilon(\delta)\delta^2+C_5 r$$
By taking for instance $r=1/m$ and $\delta=\delta_m\to 0$, we see that 
$\varphi_m$ converges to $\varphi$.
On the other hand $(9.6)$ implies
${i\over\pi}\ddbar q_j\circ\tau_j(x)=\tau_j^\star\gamma_j\ge
\gamma-2\varepsilon(\delta)\omega$, thus
$${i\over\pi}\ddbar\widetilde w_{j,m}\ge 
2m\big(\gamma-C_6\varepsilon(\delta)\omega\big).$$
Lemma~(9.11) then produces the lower bound
$${i\over\pi}\ddbar w
\ge 2m\big(\gamma-C_6\varepsilon(\delta)\omega\big)-C_7\delta^{-2}\omega,$$
whence
$${i\over\pi}\ddbar\varphi_m\ge\gamma-C_8\varepsilon(\delta)\omega$$
for $m\ge m_0(\delta)=(\log\delta^{-1})/(\varepsilon(\delta)\delta^2)$.
We can fix $\delta=\delta_m$ to be the smallest value of $\delta>0$ such that
$m_0(\delta)\le m$, then $\delta_m\to 0$ and we have
obtained a sequence of quasi psh functions $\varphi_m$ satisfying 
the following properties.

\claim{(9.12) Theorem} 
Let $\varphi$ be an almost psh function on a compact complex
manifold $X$ such that ${i\over\pi}\ddbar\varphi\ge\gamma$ for some
continuous $(1,1)$-form $\gamma$. Then there is a sequence of
almost psh functions $\varphi_m$ such that $\varphi_m$ has the 
same singularities
as a logarithm of a sum of squares of holomorphic functions and
a decreasing sequence $\varepsilon_m>0$ converging to $0$ such that
{\plainitemindent=7mm
\plainitem{\rm (i)}$\displaystyle\varphi(x)<\varphi_m(x)\le\sup_{|\zeta-x|<r}
\varphi(\zeta)+C\Big({|\log r|\over m}+r+\varepsilon_m\Big)$\newline
with respect to coordinate open sets covering $X$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1(X)$ and
\plainitem{\rm (ii)}$\displaystyle\nu(\varphi,x)-{n\over m}\le\nu(\varphi_m,x)\le
\nu(\varphi,x)$~ for every $x\in X\,;$
\plainitem{\rm (iii)}$\displaystyle{i\over\pi}\ddbar\varphi_m\ge
\gamma-\varepsilon_m\omega$.}
\endclaim
%
In particular, we can apply this to an arbitrary positive or 
almost positive closed
$(1,1)$-current $T=\alpha+{i\over\pi}\ddbar\varphi$.

\claim{(9.13) Corollary} 
Let $T$ be an almost positive  closed
$(1,1)$-current on a compact complex
manifold $X$ such that $T\ge\gamma$ for some
continuous $(1,1)$-form $\gamma$. Then there is a sequence of
currents $T_m$ whose local potentials have the 
same singularities as $1/m$ times a logarithm of a sum of squares 
of holomorphic functions and
a decreasing sequence $\varepsilon_m>0$ converging to $0$ such that
{\plainitemindent=7mm
\plainitem{\rm (i)} $T_m$ converges weakly to $T$,
\plainitem{\rm (ii)}$\displaystyle\nu(T,x)-{n\over m}\le\nu(T_m,x)\le
\nu(T,x)$~ for every $x\in X\,;$
\plainitem{\rm (iii)}$T_m\ge\gamma-\varepsilon_m\omega$.
\vskip0pt
}
\noindent
We say that our currents $T_m$ are approximations of $T$ possessing
logarithmic poles.
\endclaim
%
By using blow-ups of $X$, the structure of the currents $T_m$ can be
better understood.  In fact, consider the coherent ideals $\cJ_m$
generated locally by the holomorphic functions $(\sigma^{(k)}_{j,m})$
on $U_k$ in the local approximations
$$
\varphi_{k,m}={1\over 2m}\log\sum_j|\sigma^{(k)}_{j,m}|^2+O(1)
$$
of the potential $\varphi$ of~$T$ on~$U_k$. These ideals are in fact
globally defined, because the local ideals
$\cJ_m^{(k)}=(\sigma^{(k)}_{j,m})$ are integrally closed, and they
coincide on the intersections $U_k\cap U_\ell$ as they have the same
order of vanishing by the proof of Lemma~(13,10).  By Hironaka
[Hir64], we can find a composition of blow-ups with smooth centers
$\mu_m:\widetilde X_m\to X$ such that $\mu_m^*\cJ_m$ is an invertible ideal
sheaf associated with a normal crossing divisor $D_m$. Now, we can
write
$$
\mu_m^*\varphi_{k,m}=\varphi_{k,m}\circ\mu_m=
{1\over m}\log|s_{D_m}|+\widetilde\varphi_{k,m}
$$
where $s_{D_m}$ is the canonical section of $\cO(-D_m)$ and 
$\widetilde\varphi_{k,m}$ is a smooth potential. This implies
$$
\mu_m^*T_m={1\over m}[D_m]+\beta_m\leqno(9.14)
$$
where $[D_m]$ is the current of integration over $D_m$ and $\beta_m$
is a smooth closed $(1,1)$-form which satisfies the lower bound
$\beta_m\ge\mu_m^*(\gamma-\varepsilon_m\omega)$. (Recall that
the pull-back of a closed $(1,1)$-current by a holomorphic map $f$
is always well-defined, by taking a local plurisubharmonic potential 
$\varphi$ such that $T=i\ddbar\varphi$ and writing 
$f^*T=i\ddbar(\varphi\circ f)$). In the remainder of
this section, we derive from this a rather important geometric consequence,
first appeared in~[DP04]). We need two related definitions.

\claim{(9.15) Definition} A K\"ahler current on a compact
complex space $X$ is a closed positive current $T$ of bidegree $(1,1)$
which satisfies $T\ge\varepsilon\omega$ for some $\varepsilon>0$ and
some smooth positive hermitian form $\omega$ on $X$.
\endclaim
%
\claim{(9.16) Definition} A compact complex manifold is said to be
in the {\rm Fujiki class $\cC$)} if it is bimeromorphic to a K\"ahler manifold
$($or equivalently, using Hironaka's desingularization theorem, if 
it admits a proper K\"ahler modification$)$.
\endclaim
%
\claim{(9.17) Theorem} A compact complex manifold $X$ 
is bimeromorphic to a K\"ahler manifold $($i.e.\ $X\in\cC)$
if and only if it admits a K\"ahler current.
\endclaim
%
\begproof{Proof.} If $X$ is bimeromorphic to a K\"ahler manifold $Y$, 
Hironaka's desingularization theorem implies that there exists
a blow-up $\widetilde Y$ of $Y$ (obtained by a sequence of blow-ups with
smooth centers) such that the bimeromorphic map from
$Y$ to $X$ can be resolved into a modification $\mu:\widetilde Y\to X$. Then
$\widetilde Y$ is K\"ahler and the push-forward $T=\mu_*\widetilde\omega$
of a K\"ahler form $\widetilde\omega$ on $\widetilde Y$ provides a K\"ahler 
current on~$X$. In fact, if $\omega$ is a smooth hermitian form on~$X$,
there is a constant $C$ such that $\mu^*\omega\le C\widetilde\omega$ (by compactness
of $\widetilde Y$), hence
$$
T=\mu_*\widetilde\omega\ge \mu_*(C^{-1}\mu^*\omega)=C^{-1}\omega.
$$
Conversely, assume that $X$ admits a K\"ahler current $T\ge\varepsilon\omega$. 
By Theorem 9.13~(iii), there exists a K\"ahler current 
$\widetilde T=T_m\ge{\varepsilon\over 2}\omega$ (with $m\gg 1$ so large 
that $\varepsilon_m\le\varepsilon/2$) in 
the same $\ddbar$-cohomology class as~$T$, possessing logarithmic poles. 
Observation (9.14) implies the existence of a composition of blow-ups
$\mu:\widetilde X\to X$ such that
$$
\mu^*\widetilde T=[\widetilde D]+\widetilde\beta\qquad\hbox{on $\widetilde X$},
$$
where $\widetilde D$ is a $\bQ$-divisor with normal crossings and $\widetilde\beta$
a smooth closed $(1,1)$-form such that $\widetilde\beta\ge{\varepsilon\over 2}
\mu^*\omega$. In particular $\widetilde\beta$ is positive outside the exceptional
locus of~$\mu$. This is not enough yet to produce a K\"ahler
form on $\widetilde X$, but we are not very far. Suppose that $\widetilde X$ 
is obtained as a tower of blow-ups
$$
\widetilde X=X_N\to X_{N-1}\to\cdots \to X_1\to X_0=X,
$$
where $X_{j+1}$ is the blow-up of $X_j$ along a smooth center $Y_j\subset X_j$.
Denote by $E_{j+1}\subset X_{j+1}$ the exceptional divisor, and let
$\mu_j:X_{j+1}\to X_j$ be the blow-up map. Now, we use the following
simple
\endproof
%
\claim{(9.18) Lemma} For every K\"ahler current 
$T_j$ on $X_j$, there exists $\varepsilon_{j+1}>0$ and a smooth form $u_{j+1}$
in the $\ddbar$-cohomology class of $[E_{j+1}]$ such that
$$
T_{j+1}=\mu_j^\star T_j-\varepsilon_{j+1}u_{j+1}
$$ 
is a K\"ahler current on~$X_{j+1}$.
\endclaim
%
\begproof{Proof.} The line bundle
$\cO(-E_{j+1})|E_{j+1}$ is equal to $\cO_{P(N_j)}(1)$ where $N_j$
is the normal bundle to $Y_j$ in $X_j$. Pick an arbitrary smooth hermitian 
metric on $N_j$, use this metric to get an induced Fubini-Study metric on
$\cO_{P(N_j)}(1)$, and finally extend this metric as a smooth hermitian 
metric on the line bundle $\cO(-E_{j+1})$. Such a metric has positive
curvature along tangent vectors of $X_{j+1}$ which are tangent to the 
fibers of $E_{j+1}=P(N_j)\to Y_j$. Assume furthermore that 
$T_j\ge\delta_j\omega_j$ for some hermitian form $\omega_j$ on $X_j$
and a suitable $0<\delta_j\ll 1$. Then
$$
\mu_j^\star T_j-\varepsilon_{j+1}u_{j+1}\ge
\delta_j\mu_j^\star\omega_j-\varepsilon_{j+1}u_{j+1}
$$
where $\mu_j^*\omega_j$ is semi-positive on $X_{j+1}$, positive definite on 
$X_{j+1}\ssm E_{j+1}$, and also positive definite on tangent vectors of 
$T_{X_{j+1}|E_{j+1}}$ which are not tangent to the fibers of $E_{j+1}\to Y_j$.
The statement is then easily proved by taking $\varepsilon_{j+1}\ll\delta_j$ 
and by using an elementary compactness argument on the unit sphere bundle 
of~$T_{X_{j+1}}$ associated with any given hermitian metric.\qed
\endproof
%
\noindent
{\it End of proof of Theorem 9.17.}
If $\widetilde u_j$ is the pull-back of $u_j$ to the final blow-up
$\widetilde X$, we conclude inductively that $\mu^\star
\widetilde T-\sum\varepsilon_j\widetilde u_j$ is a K\"ahler current. Therefore
the smooth form
$$
\widetilde\omega:=\widetilde\beta-\sum\varepsilon_j\widetilde u_j
=\mu^\star \widetilde T-\sum\varepsilon_j\widetilde u_j-[\widetilde D]
$$
is K\"ahler and we see that $\widetilde X$ is a K\"ahler manifold.\qed
\medskip

\claim{(9.19) Remark} {\rm A special case of Theorem (9.16) is the following
characterization of Moishezon varieties (i.e.\ manifolds which are
bimeromorphic to projective algebraic varieties or, equivalently, whose
algebraic dimension is equal to their complex dimension): 
\vskip3pt
\noindent
{\it A compact complex manifold $X$ is Moishezon if and only if $X$ 
possesses a K\"ahler current $T$ such that the De Rham cohomology class
$\{T\}$ is rational, i.e.\ $\{T\}\in H^2(X,\bQ)$}. 
\vskip3pt
\noindent
In fact, in the above proof, we get an integral current $T$ if we take
the push forward $T=\mu_*\widetilde\omega$ of an integral ample class 
$\{\widetilde\omega\}$ on $Y$, where $\mu:Y\to X$ is a projective model 
of~$Y$. Conversely, if $\{T\}$ is rational, we can take
the $\varepsilon_j's$ to be rational in Lemma~3.5. This produces
at the end a K\"ahler metric $\widetilde\omega$ with rational De Rham cohomology
class on $\widetilde X$. Therefore $\widetilde X$ is projective by the Kodaira
embedding theorem. This result was already observed in [JS93]
(see also [Bon93, Bon98] for a more general perspective based
on a singular version of holomorphic Morse inequalities).}
\endclaim
%
\subsection*{\csect.C. Global approximation by divisors}
We now translate our previous approximation theorems into a more 
algebro-geometric setting. Namely, we assume that $T$ is a closed
positive $(1,1)$-current which belongs to the first Chern class
$c_1(L)$ of a holomorphic line bundle~$L$, and we assume here $X$
to be algebraic (i.e.\ projective or at the very least Moishezon).

Our goal is to show that $T$ can be approximated by divisors which
have roughly the same Lelong numbers as~$T$. The existence of weak
approximations by divisors has already been proved in [Lel72] for
currents defined on a pseudoconvex open set $\Omega\subset\bC^n$
with $H^2(\Omega,\bR)=0$, and in [Dem92, 93b] in the situation considered
here (cf.\ also [Dem82b], although the argument given there is somewhat
incorrect). We take the opportunity to present here a slightly simpler
derivation. 

Let $X$ be
a projective manifold and $L$ a line bundle over~$X$. A singular hermitian
metric $h$ on $L$ is a metric such that the weight function $\varphi$ of
$h$ is $L^1_\loc$ in any local trivialization (such that
$L_{|U}\simeq U\times\bC$ and $\Vert\xi\Vert_h=|\xi|e^{-\varphi(x)}$,
$\xi\in L_x\simeq\bC$). The curvature of $L$ can then be computed in
the sense of distributions
$$
T={\ii\over 2\pi}\Theta_{L,h}={\ii\over \pi}\ddbar\varphi,
$$
and $L$ is said to be pseudo-effective if $L$ admits a singular hermitian
metric $h$ such that the curvature current $T={\ii\over 2\pi}\Theta_{L,h}$
is semi-positive [The weight functions $\varphi$ of $L$ are thus
plurisubharmonic]. In what follows, we sometimes use an additive 
notation for $\Pic(X)$, i.e.\ $kL$ is meant for the line 
bundle $L^{\otimes k}$.

We will also make use of the concept of complex singularity exponent, 
following e.g.\ [Var82, 83], [ArGV85] and [DK01].
A quasi-plurisubharmonic (quasi-psh) function is by definition a
function $\varphi$ which is locally equal to the sum of a psh function
and of a smooth function, or equivalently, a locally integrable function
$\varphi$ such that $i\ddbar\varphi$ is locally bounded below by
$-C\omega$ where $\omega$ is a hermitian metric and $C$ a constant.

\claim{(9.20) Definition} If $K$ is a compact subset of $X$ and $\varphi$
is a quasi-psh function defined near~$K$, we define
\vskip3pt
\plainitem{\rm(a)} the
complex singularity exponent $c_K(\varphi)$ to be the supremum of all
positive numbers $c$ such that  $e^{-2c\varphi}$ is
integrable in a neighborhood of every point $z_0\in K$, with respect to 
the Lebesgue measure in holomorphic coordinates centered at~$z_0$. 
In particular $c_K(\varphi)=\inf_{z_0\in K}(\varphi)$.
\vskip3pt
\plainitem{\rm(b)} The concept is easily extended to hermitian metrics 
$h=e^{-2\varphi}$ by putting $c_K(h)=c_K(\varphi)$, to holomorphic
functions $f$ by $c_K(f)=c_K(\log|f|)$, to coherent ideals
$\cJ=(g_1,\ldots,g_N)$ by $c_K(\cJ)=c_K(\varphi)$ where
$\varphi={1\over 2}\log\sum|g_j|^2$. Also for an effective
$\bR$-divisor $D$, we put $c_K(D)=c_K(\log|\sigma_D|)$ where
$\sigma_D$ is the canonical section.\vskip0pt
\endclaim
%
\noindent
The main technical result of this section can be stated as follows, in the
case of big line bundles (cf.\ Proposition (6.14$\,$f)).

\claim{(9.21) Theorem} Let $L$ be a line bundle on a compact complex manifold
$X$ possessing a singular hermitian metric $h$ with $\Theta_{L,h}\ge 
\varepsilon\omega$ for some $\varepsilon>0$ and some smooth positive definite
hermitian $(1,1)$-form $\omega$ on~$X$. For every real number $m>0$, consider
the space $\cH_m=H^0(X,L^{\otimes m}\otimes\cI(h^m))$ of holomorphic sections 
$\sigma$ of $L^{\otimes m}$ on $X$ such that 
$$
\int_X|\sigma|_{h^m}^2dV_\omega
=\int_X|\sigma|^2e^{-2m\varphi}dV_\omega<+\infty,
$$
where $dV_\omega={1\over m!}\omega^m$ is the hermitian volume form.
Then for $m\gg 1$, $\cH_m$ is a non zero finite dimensional Hilbert space 
and we consider the closed positive $(1,1)$-current
$$
T_m={i\over \pi}\ddbar\Big({1\over 2m}\log\sum_k|g_{m,k}|^2\Big)
={i\over \pi}\ddbar\Big({1\over 2m}\log\sum_k|g_{m,k}|_h^2\Big)
+\Theta_{L,h}
$$
where $(g_{m,k})_{1\le k\le N(m)}$ is an orthonormal basis of~$\cH_m$. 
Then$\;:$
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} For every trivialization  $L_{|U}\simeq U\times\bC$
on a cordinate open set $U$ of $X$ and every compact set $K\subset U$,
there are constants $C_1,C_2>0$ independent of $m$ and $\varphi$ such that
$$
\varphi(z)-{C_1\over m}\le
\psi_m(z):={1\over 2m}\log\sum_k|g_{m,k}(z)|^2
\le\sup_{|x-z|<r}\varphi(x)+{1\over m}\log{C_2\over r^n}
$$
for every $z\in K$ and $r\le{1\over 2}d(K,\partial U)$. In particular,
$\psi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$, hence $T_m$ converges weakly 
to $T=\Theta_{L,h}$.
\vskip3pt
\plainitem{\rm (ii)} The Lelong numbers $\nu(T,z)=\nu(\varphi,z)$ and 
$\nu(T_m,z)=\nu(\psi_m,z)$ are related by
$$
\nu(T,z)-{n\over m}\le\nu(T_m,z)\le
\nu(T,z)\quad\hbox{ for every $z\in X$.}
$$
\plainitem{\rm (iii)} For every compact set $K\subset X$, the complex singularity
exponents of the metrics given locally by $h=e^{-2\varphi}$ and 
$h_m=e^{-2\psi_m}$ satisfy
$$
c_K(h)^{-1}-{1\over m}\le c_K(h_m)^{-1}\le c_K(h)^{-1}.
$$}
\endclaim
%
\begproof{Proof.} The major part of the proof is a variation
of the arguments already explained in section 9.A.\vskip3pt

\noindent
(i) We note that $\sum|g_{m,k}(z)|^2$ is the square of the norm of 
the evaluation linear form $\sigma\mapsto \sigma(z)$ on $\cH_m$, hence
$$
\psi_m(z)=\sup_{\sigma\in B(1)}{1\over m}\log|\sigma(z)|
$$
where $B(1)$ is the unit ball of $\cH_m$. For
$r\le {1\over 2}d(K,\partial\Omega)$, the mean value inequality 
applied to the plurisubharmonic function $|\sigma|^2$ implies
$$
\eqalign{
|\sigma(z)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|x-z|<r}
|\sigma(x)|^2d\lambda(x)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|x-z|<r}\varphi(x)\Big)
\int_\Omega|\sigma|^2e^{-2m\varphi}d\lambda.\cr}
$$
If we take the supremum over all $\sigma\in B(1)$ we get
$$
\psi_m(z)\le\sup_{|x-z|<r}\varphi(x)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the right hand inequality in (i) is proved. Conversely, the
Ohsawa-Takegoshi extension theorem [OhT87], [Ohs88] applied to 
the $0$-dimensional
subvariety $\{z\}\subset U$ shows that for any $a\in\bC$ there is
a holomorphic function $f$ on $U$ such that $f(z)=a$ and
$$
\int_U|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z)},
$$
where $C_3$ only depends on $n$ and~$\diam U$. Now, provided $a$ remains in
a compact set $K\subset U$, we can use a cut-off function $\theta$ with
support in $U$ and equal to $1$ in a neighborhood of $a$, and solve the
$\dbar$-equation $\dbar g=\dbar(\theta f)$ in the $L^2$ space associated
with the
weight $2m\varphi+2(n+1)|\log|z-a|$, that is, the singular hermitian
metric $h(z)^m|z-a|^{-2(n+1)}$ on~$L^{\otimes m}$. For this, we apply
the standard Andreotti-Vesentini-H\"ormander $L^2$ estimates (see e.g.\ 
[Dem82b] for the required version). This is possible for $m\ge m_0$ 
thanks to the 
hypothesis that $\Theta_{L,h}\ge \varepsilon\omega>0$, even if $X$ 
is non K\"ahler ($X$ is in any event a Moishezon variety from our
assumptions). The bound $m_0$ depends only on $\varepsilon$ and the geometry
of a finite covering of $X$ by compact sets $K_j\subset U_j$, where
$U_j$ are coordinate balls (say); it is independent of the point $a$ and 
even of the metric~$h$. It follows that $g(a)=0$ and therefore
$\sigma=\theta f-g$ is a holomorphic section of $L^{\otimes m}$ such that
$$
\int_X|\sigma|^2_{h^m}dV_\omega=
\int_X|\sigma|^2e^{-2m\varphi}dV_\omega\le C_4
\int_U|f|^2e^{-2m\varphi}dV_\omega\le C_5
|a|^2e^{-2m\varphi(z)},
$$
in particular $\sigma\in \cH_m=H^0(X,L^{\otimes m}\otimes\cI(h^m))$.
We fix $a$ such that the right hand side is~$1$. This gives
the inequality
$$
\psi_m(z)\ge{1\over m}\log|a|=\varphi(z)-{\log C_5\over 2m}
$$
which is the left hand part of statement (i).
\medskip

\noindent
(ii) The first inequality in (i) implies $\nu(\psi_m,z)\le\nu(\varphi,z)$. 
In the opposite direction, we find
$$
\sup_{|x-z|<r}\psi_m(x)\le\sup_{|x-z|<2r}\varphi(x)+
{1\over m}\log{C_2\over r^n}.
$$
Divide by $\log r<0$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a psh function over $B(x,r)$ tends to the
Lelong number at~$x$. Thus we obtain
$$
\nu(\psi_m,x)\ge\nu(\varphi,x)-{n\over m}.
$$
(iii) Again, the first inequality (in (i) immediately yields $h_m\le C_6 h$,
hence $c_K(h_m)\ge c_K(h)$. For the converse inequality, since we
have $c_{\,\cup K_j}(h)=\min_j c_{K_j}(h)$, we can assume without loss 
of generality that $K$ is contained in a trivializing open patch $U$ of~$L$.
Let us take $c<c_K(\psi_m)$. Then, by definition, if 
$V\subset X$ is a sufficiently small open neighborhood of $K$, 
the H\"older inequality
for the conjugate exponents $p=1+mc^{-1}$ and $q=1+m^{-1}c$
implies, thanks to equality ${1\over p}={c\over mq}$,
$$
\leqalignno{
\int_V & e^{-2(m/p)\varphi}dV_\omega
=\int_V\Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2e^{-2m\varphi}\Big)^{1/p}
  \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/mq}dV_\omega\cr
&\le
\left(\int_X\sum_{1\le k\le N(m)}|g_{m,k}|^2e^{-2m\varphi}dV_\omega\right)^{1/p}
\left(\int_V \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/m}dV_\omega
\right)^{1/q}\cr
&= N(m)^{1/p}
\left(\int_V \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/m}dV_\omega
\right)^{1/q}<+\infty.
\cr}
$$
From this we infer $c_K(h)\ge m/p$, i.e., $c_K(h)^{-1}\le p/m=1/m+c^{-1}$.
As $c<c_K(\psi_m)$ was arbitrary, we get $c_K(h)^{-1}\le 1/m+c_K(h_m)^{-1}$
and the inequalities of (iii) are proved.\qed
\endproof
%
\claim{(9.22) Remark} {\rm The proof would also work, with a few
modifications, when $X$ is a Stein manifold and $L$ is an arbitrary 
holomorphic line bundle.}
\endclaim
%
\claim{(9.23) Corollary} Let $L\to X$ be a holomorphic line bundle
and $T={\ii\over 2\pi}\Theta_{L,h}$ the curvature current of some singular
hermitian metric $h$ on $L$.
\vskip3pt
{\plainitemindent=7mm
\plainitem{\rm (i)} If $L$ is big and $\Theta_{L,h}\ge \varepsilon\omega>0$, 
there exists a sequence of holomorphic sections $h_s\in H^0(X,q_sL)$ with
\hbox{$\lim q_s=+\infty$} such that the $\bQ$-divisors $D_s=\smash{1\over q_s}
\div(h_s)$ satisfy $T=\lim D_s$ in the weak topology and
\hbox{$\,\sup_{x\in X}|\nu(D_s,x)-\nu(T,x)|\to 0$} as $s\to+\infty$.
\plainitem{\rm (ii)} If $L$ is just pseudo-effective and $\Theta_{L,h}\ge 0$, 
for any ample line bundle $A$, there exists a sequence of non zero sections
$h_s\in H^0(X,p_sA+q_sL)$ with \hbox{$p_s,q_s>0$}, $\lim q_s=+\infty$
and $\lim p_s/q_s=0$, such that the divisors $D_s=\smash{1\over
q_s}\div(h_s)$ satisfy $T=\lim D_s$ in the weak topology and
$\,\sup_{x\in X}|\nu(D_s,x)-\nu(T,x)|\to 0$ as \hbox{$s\to+\infty$}.
\vskip0pt}
\endclaim
%
\begproof{Proof.} Part (ii) is a rather straightforward consequence of part (i)
applied to $mL+A$ and $T_m={1\over m}\Theta_{mL+A,h^mh_A}=T+
{1\over m}\Theta_{A,h_A}\to T$  when $m$ tends to infinity.
Therefore, it suffices to prove (i).
\vskip3pt
\noindent
(i) By Theorem (9.20), we can find sections
$g_1,\ldots,g_N\in H^0(X,mL)$ (omitting the index $m$ for simplicity of
notation), such that
$$
T_m={i\over \pi}\ddbar\Big({1\over 2m}\log\sum_{1\le j\le N}|g_j|_h^2\Big)
+\Theta_{L,h}=
{i\over \pi}\ddbar\Big({1\over 2m}\log\sum_{1\le j\le N}|g_j|^2\Big)
$$
converges weakly to $T$ and satisfies $\nu(T,x)-n/m\le\nu(T_m,x)\le\nu(T,x)$.
In fact, since the number $N$ of sections grows at most as $O(m^n)$, 
we can replace $\sum_{1\le j\le N}|g_j|^2$ by $\max_{1\le j\le N}|g_j|^2$,
as the difference of the potentials tends uniformly to $0$ with the
help of the renormalizing constant ${1\over 2m}$. Hence, we can use
instead the approximating currents
$$
\widetilde T_m={i\over \pi}\ddbar u_m,\qquad
u_m={1\over m}\log\max_{1\le j\le N}|g_j|.
$$
Now, as $L$ is big, by the proof of (6.7f) we can write
$k_0L=A+D$ where $A$ is an ample divisor and $D$ is an effective divisor,
for some $k_0>0$. By enlarging $k_0$, we can even assume that $A$ is very
ample. Let $\sigma_D$ be the canonical section of $D$ and let
$h_1,\ldots,h_N$ be sections of $H^0(X,A)$. We get a section of
$H^0(X,(m\ell+k_0)L)$ by considering
$$
u_{\ell,m}=(g_1^\ell h_1+\ldots+g_N^\ell h_N)\sigma_D
$$
By enlarging $N$ if necessary and putting e.g.\ $g_j=g_N$ for $j>N$, 
we can assume
that the sections $h_j$ generate all $1$-jets of sections of $A$ at every
point (actually, one can always achieve this with $n+1$ sections only,
so this is not really a big demand). Then, for almost every 
$N$-tuple $(h_1,\ldots,h_N)$, Lemma~9.24 below
and the weak continuity of $\partial\dbar$ imply that
$$
\Delta_{\ell,m}={1\over \ell m}{i\over\pi}\partial\dbar\log|u_{\ell ,m}|=
{1\over \ell m}\,{\rm div}(u_{\ell ,m})
$$
converges weakly to $\widetilde T_m={i\over\pi}\ddbar u_m$ as $\ell$
tends to $+\infty$, and that
$$
\nu(T_m,x)\le\nu\Big({1\over \ell m}\Delta_{\ell ,m},x\Big)\le\nu(T,x)+
{\mu+1\over \ell m},
$$
where $\mu=\max_{x\in X}\ord_x(\sigma_D)$.
This, together with the first step, implies the proposition for some
subsequence $D_s=\Delta_{\ell(s),s}$, $\ell(s)\gg s\gg 1$. We even obtain 
the more explicit inequality
$$
\nu(T,x)-{n\over m}\le\nu\Big({1\over \ell m}\Delta_{\ell,m},x\Big)\le
\nu(T,x)+{\mu+1\over\ell m}.\qquad\qed
$$
\endproof
%
\claim{(9.24) Lemma} Let $\Omega$ be an open subset in $\bC^n$ and let
$g_1,\ldots,g_N\in H^0(\Omega,\cO_\Omega)$ be non zero functions. 
Let $S\subset H^0(\Omega,\cO_\Omega)$ be a finite dimensional subspace
whose elements generate all $1$-jets at any point of $\Omega$. 
Finally, set $u=\log\max_j|g_j|$ and
$$u_\ell =g_1^\ell h_1+\ldots+g_N^\ell h_N,~~~~h_j\in S\ssm\{0\}.$$
Then for all $(h_1,\ldots,h_N)$ in $(S\ssm\{0\})^N$ except a set of measure~$0$,
the sequence ${1\over \ell }\log|u_\ell |$ converges to $u$ in 
$L^1_{\rm loc}(\Omega)$ and
$$\nu(u,x)\le\nu\Big({1\over \ell }\log|u_\ell |\Big)\le\nu(u,x)+
{1\over \ell },~~~~\forall x\in X,~~\forall \ell \ge 1.$$
\endclaim
%
\begproof{Proof.} The sequence ${1\over \ell }\log|u_\ell |$ is locally uniformly
bounded above and we have
$$\lim_{\ell \to+\infty}{1\over \ell }\log\big|u_\ell (z)\big|=u(z)$$
at every point $z$ where all absolute values $|g_j(z)|$ are distinct
and all $h_j(z)$ are nonzero. This is a set of full measure in $\Omega$
because the sets $\{|g_j|^2=|g_l|^2,\,j\ne l\}$ and $\{h_j=0\}$ are real
analytic and thus of zero measure (without loss of generality, we may
assume that $\Omega$ is connected and that the $g_j$'s are not pairwise
proportional). The well-known uniform integrability properties of 
plurisubharmonic functions then show that ${1\over \ell }\log|u_\ell |$ converges
to $u$ in $L^1_{\rm loc}(\Omega)$. It is easy to see that $\nu(u,x)$ is the
minimum of the vanishing orders ${\rm ord}_x(g_j)$, hence
$$\nu(\log|u_\ell |,x)={\rm ord}_x(u_\ell )\ge \ell \,\nu(u,x).$$
In the opposite direction, consider the set $\cE_\ell $ of all $(N+1)$-tuples
$$(x,h_1,\ldots,h_N)\in\Omega\times S^N$$
for which $\nu(\log|u_\ell |,x)\ge \ell \,\nu(u,x)+2$. Then $\cE_\ell $ is a
constructible set in $\Omega\times S^N$: it has a locally finite
stratification by analytic sets, since
$$\cE_\ell =\bigcup_{s\ge 0}\Big(\bigcup_{j,\,|\alpha|=s}
\big\{x\,;\,D^\alpha g_j(x)\ne 0\big\}\times S^N\Big)\cap
\bigcap_{|\beta|\le \ell s+1}\big\{(x,(h_j))\,;\,D^\beta u_\ell (x)=0\big\}.$$
The fiber $\cE_\ell \cap(\{x\}\times S^N)$ over a point $x\in\Omega$
where $\nu(u,x)=\min{\rm ord}_x(g_j)=s$ is the vector space of $N$-tuples
$(h_j)\in S^N$ satisfying the equations $D^\beta
\big(\sum g_j^\ell h_j(x)\big)=0$, $|\beta|\le \ell s+1$. However, if
${\rm ord}_x(g_j)=s$, the linear map
$$(0,\ldots,0,h_j,0,\ldots,0)\longmapsto
\big(D^\beta(g_j^\ell h_j(x))\big)_{|\beta|\le \ell s+1}$$
has rank $n+1$, because it factorizes into an injective map
$J^1_xh_j\mapsto J^{\ell s+1}_x(g_j^\ell h_j)$. It follows that the fiber
$\cE_\ell \cap(\{x\}\times S^N)$ has codimension at least $n+1$.
Therefore
$$\dim \cE_\ell \le\dim(\Omega\times S^N)-(n+1)=\dim S^N-1$$
and the projection of $\cE_\ell $ on $S^N$ has measure zero by Sard's
theorem. By definition of $\cE_\ell $, any choice of $(h_1,\ldots,h_N)\in
S^N\ssm\bigcup_{\ell \ge 1} {\rm pr}(\cE_\ell )$ produces functions $u_\ell $
such that  $\nu(\log|u_\ell |,x)\le \ell \,\nu(u,x)+1$ on~$\Omega$.\qed
\endproof
%
\claim{(9.25) Exercise} {\rm When $L$ is ample and 
$h$ is a smooth metric with $T={i\over2\pi}\Theta_{L,h}>0$, show that 
the approximating divisors can be taken smooth (and thus irreducible
if $X$ is connected).\newline
{\it Hint.} In the above proof of Corollary (9.23), the sections $g_j$ 
have no common zeroes and one can take $\sigma_D=1$. Moreover, a smooth
divisor $\Delta$ in an ample linear system is always connected, otherwise
two disjoint parts $\Delta'$, $\Delta''$ would be big and nef and
$\Delta'\cdot\Delta''=0$ would contradict the Hovanskii-Teissier
inequality when $X$ is connected.}
\endclaim
%
\claim{(9.26) Corollary} On a projective manifold $X$, effective 
$\bQ$-divisors are dense in the weak topology in the cone
$P^{1,1}_{\NS}(X)$ of closed positive $(1,1)$-currents $T$ whose cohomology
class $\{T\}$ belongs to the Neron-Severi space $\NS_\bR(X)$.
\endclaim
%
\begproof{Proof.}We may add $\varepsilon$ times a K\"ahler metric $\omega$ so
as to get $T+\varepsilon\omega>0$, and then perturb by a small
combination $\sum\delta_j\alpha_j$ of classes ${\alpha_j}$ in a
$\bZ$-basis of $\NS(X)$ so that $\Theta=T+\varepsilon\omega+\sum
\delta_j\alpha_j\ge{\varepsilon\over 2}\omega$ and $\{\Theta\}\in H^2(X,\bQ)$.
Then $\Theta$ can be approximated by $\bQ$-divisors by Corollary (9.23),
and the conclusion follows.\qed
\endproof
%
\claim{(9.27) Comments} {\rm We can rephrase the above results by
saying that the cone of closed positive currents $P^{1,1}_{\NS}(X)$ is a
completion of the cone of effective $\bQ$-divisors. A considerable
advantage of using currents is that the cone of currents is locally
compact in the weak topology, namely the section of the cone
consisting of currents $T$ of mass $\int_XT\wedge\omega^{n-1}=1$ is
compact. This provides a very strong tool for the study of the
asymptotic behaviour of linear systems, as required for instance in
the Minimal Model Program of Kawamata-Mori-Shokurov.  One should be
aware, however, that the cone of currents is really huge and contains
objects which are very far from being algebraic in any reasonable sense.
This occurs very frequently in the realm of complex dynamics. For instance,
if $P_m(z,c)$ denotes the $m$-th iterate of the quadratic polynomial
$z\mapsto z^2+c$, then $P_m(z,c)$ defines a polynomial of degree $2^m$
on $\bC^2$, and the sequence of $\bQ$-divisors $D_m={1\over
m}{i\over\pi} \ddbar\log|P_m(z,c)|$ which have mass $1$ on
$\bC^2\subset\bP_\bC^2$ can be shown to converge to a closed positive
current $T$ of mass $1$ on~$\bP_\bC^2$. The support of this current
$T$ is extremely complicated~: its slices $c=c_0$ are the Julia sets
$J_c$ of the quadratic polynomial $z\mapsto z^2+c$, and the slice
$z=0$ is the famous Mandelbrot set $M$.  Therefore, in general, limits
of divisors in asymptotic linear systems may exhibit a fractal
behavior.}
\endclaim
%
\subsection*{\csect.D. Singularity exponents and log canonical thresholds}
The goal of this section to relate ``log canonical thresholds'' with
the $\alpha$ invariant introduced by G.~Tian [Tia87] for the study of
the existence of K\"ahler-Einstein metrics. The approximation
technique of closed positive $(1,1)$-currents introduced above can be
used to show that the $\alpha$ invariant actually coincides with the
log canonical threshold (see also [DK01], [JK01], [BGK05], [Dem08]). 

Usually, in these applications, only the case of the anticanonical
line bundle $L=-K_X$ is considered. Here we will consider more generally
the case of an arbitrary line bundle $L$ (or $\bQ$-line bundle $L$) on a 
complex  manifold $X$, with some additional restrictions which will
be stated later. We introduce a generalized version of Tian's invariant 
$\alpha$, as defined in [Tia87] (see also~[Siu88]).

\claim{(9.28) Definition} Assume that $X$ is a compact manifold and that
$L$ is a pseudo-effective line bundle, i.e.\ $L$ admits a singular
hermitian metric $h_0$ with $\Theta_{L,h_0}\ge 0$. If $K$ is a compact 
subset of $X$, we put
$$
\alpha_K(L)=\inf_{\{h,\,\Theta_{L,h}\ge 0\}}c_K(h)
$$
where $h$ runs over all singular hermitian metrics on $L$ such that
$\Theta_{L,h}\ge 0$.
\endclaim
%
In algebraic geometry, it is more usual to look instead at linear systems
defined by a family of linearly independent sections 
$\sigma_0,\sigma_1,\ldots\sigma_N\in 
H^0(X,L^{\otimes m})$. We denote by $\Sigma$ the vector subspace generated by
these sections and by 
$$
|\Sigma|:=P(\Sigma)\subset|mL|:=P(H^0(X,L^{\otimes m}))
$$
the corresponding linear system. Such an $(N+1)$-tuple of sections 
$\sigma=(\sigma_j)_{0\le j\le N}$ defines a
singular hermitian metric $h$ on $L$ by putting in any trivialization
$$
\vert \xi\vert^2_h
={|\xi|^2\over\big(\sum_j|\sigma_j(z)|^2\big)^{1/m}}
={|\xi|^2\over|\sigma(z)|^{2/m}},\qquad \xi\in L_z,
$$
hence $h(z)=|\sigma(z)|^{-2/m}$ with
$\varphi(z)={1\over m}\log|\sigma(z)|={1\over 2m}\log\sum_j|\sigma_j(z)|^2$
as the associated weight function. Therefore, we are interested in the
number $c_K(|\sigma|^{-2/m})$. In the case of a single section 
$\sigma_0$ (corresponding to a one-point linear system), this is the 
same as the log canonical threshold $\lct_K(X,{1\over m}D)=c_K({1\over m}D)$ 
of the associated divisor $D$, in the notation
of~Section~1 of [CS08]. We will also use the formal notation
$c_K({1\over m}|\Sigma|)$ in the case of a higher dimensional
linear system $|\Sigma|\subset|mL|$. The main result of this section is

\claim{(9.29) Theorem} Let $L$ be a big line bundle on a compact complex
manifold $X$. Then for every compact set $K$ in $X$ we have
$$
\alpha_K(L)=\inf_{\{h,\,\Theta_{L,h}\ge 0\}} c_K(h) =\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}
c_K\Big({1\over m}D\Big).
$$
\endclaim
%
\begproof{Proof.}Observe that the inequality 
$$
\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}c_K\Big({1\over m}D\Big)
\ge \inf_{\{h,\,\Theta_{L,h}\ge 0\}} c_K(h)
$$
is trivial, since any divisor $D\in|mL|$ gives rise to a singular
hermitian metric~$h$.

The converse inequality will follow from the
approximation techniques discussed above. Given a big line bundle $L$ 
on~$X$, there exists a modification
$\mu:\widetilde X\to X$ of $X$ such that $\widetilde X$ is projective and 
$\mu^*L=\cO(A+E)$ where $A$ is an ample divisor and $E$ an effective
divisor with rational coefficients. By pushing forward by $\mu$ a smooth 
metric $h_A$ with positive curvature on $A$, we get a singular hermitian 
metric $h_1$ on $L$ such that 
$\Theta_{L,h_1}\ge\mu_*\Theta_{A,h_A}\ge \varepsilon\omega$ on~$X$. Then
for any $\delta>0$ and any singular hermitian metric $h$ on $L$ with
$\Theta_{L,h}\ge 0$, the interpolated metric 
$h_\delta=h_1^\delta h^{1-\delta}$ satisfies
$\Theta_{L,h_\delta}\ge\delta\varepsilon\omega$. Since $h_1$ is bounded
away from $0$, it follows that $c_K(h)\ge (1-\delta)c_K(h_\delta)$
by monotonicity. By theorem (9.21, iii) applied to $h_\delta$, we infer
$$
c_K(h_\delta)=\lim_{m\to+\infty} c_K(h_{\delta,m}),
$$
and we also have
$$
c_K(h_{\delta,m})\ge c_K\Big({1\over m}D_{\delta,m}\Big)
$$
for any divisor $D_{\delta,m}$ associated with a section $\sigma\in
H^0(X,L^{\otimes m}\otimes\cI(h_\delta^m))$, since the metric $h_{\delta,m}$
is given by $h_{\delta,m}=(\sum_k|g_{m,k}|^2)^{-1/m}$ for an orthornormal basis
of such sections. This clearly implies
$$
c_K(h)\ge\liminf_{\delta\to 0}\;\liminf_{m\to+\infty}\;
c_K\Big({1\over m}D_{\delta,m}\Big)\ge
\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}
c_K\Big({1\over m}D\Big).\eqno\square
$$
\endproof
%
In the applications, it is frequent to have a finite or compact group 
$G$ of automorphisms of $X$ and to look at $G$-invariant objects, namely
$G$-equivariant metrics on $G$-equivariant line bundles~$L\;$; in the case
of a reductive algebraic group $G$ we simply consider a compact real form
$G^\bR$ instead of $G$ itself.

One then gets an $\alpha$ invariant $\alpha_{K,G}(L)$ by
looking only at $G$-equivariant metrics in
Definition~(9.28). All contructions made are then
$G$-equivariant, especially $\cH_m\subset |mL|$ is a $G$-invariant
linear system. For every $G$-invariant compact set $K$ in $X$, we thus
infer
$$
\alpha_{K,G}(L)
:=\inf_{\{h\;G{\mathrm{-equiv.}},\,\Theta_{L,h}\ge 0\}}c_K(h)
=\inf_{m\in\bZ_{>0}}~~\inf_{|\Sigma|\subset|mL|,~\Sigma^G=\Sigma}~~
c_K\Big({1\over m}|\Sigma|\Big).
\leqno(9.30)
$$
When $G$ is a finite group, one can pick for $m$ large enough a
$G$-invariant divisor $D_{\delta,m}$ associated with a
$G$-invariant section $\sigma$, possibly after multiplying $m$ by
the order of $G$. One then gets the slightly simpler equality
$$
\alpha_{K,G}(L)
:=\inf_{\{h\;G{\mathrm{-equiv.}},\,\Theta_{L,h}\ge 0\}}c_K(h)
=\inf_{m\in\bZ_{>0}}~~\inf_{D\in|mL|^G}~~c_K\Big({1\over m}D\Big).
\leqno(9.31)
$$
In a similar manner, one can work on an orbifold $X$ rather than on a non
singular variety. The $L^2$ techniques work in this setting with almost
no change ($L^2$ estimates are essentially insensitive  to
singularities,  since one can just use an orbifold metric on
the open set of regular points).

The main interest of Tian's invariant $\alpha_{X,G}$ (and of the related
concept of log canonical threshold) is that it provides a neat criterion 
for the existence of K\"ahler-Einstein metrics for Fano manifolds
(see [Tia87], [Siu88], [Nad89], [DK01]).

\claim{(9.32) Theorem} Let $X$ be a Fano manifold, i.e.\ a projective
manifold with $-K_X$ ample. Assume that $X$ admits a compact group
of automorphisms $G$ such that $\alpha_{X,G}(K_X)>n/(n+1)$. Then
$X$ possesses a $G$-invariant K\"ahler-Einstein metric.
\endclaim
%
We will not give here the details  of the proof, which rely on very
delicate $C^k$-estimates (successively for $k=0,1,2,\ldots$) for 
the Monge-Amp\`ere 
operator. In fine, the required estimates can be shown to depend only on 
the boundedness of the integral
$\int_X e^{-2\gamma\varphi}$ for a suitable constant 
$\gamma\in{}]{n\over n+1},1]$, where $\varphi$ is the potential of 
the K\"ahler metric $\omega\in c_1(X)$ (also viewed as the weight 
of a hermitian metric on $K_X$). Now, one can restrict the estimate
to $G$-invariant weights $\varphi$, and this translates into the
sufficient condition (9.32). The approach explained in [DK01] 
simplifies the analysis developped in earlier works by proving first
a general semi-continuity theorem which implies the desired a priori
bound under the assumption of Theorem~9.32. The semi-continuity
theorem states as

\claim{(9.33) Theorem {\rm([DK01])}} Let $K$ be a compact set in a
complex manifold $X$. Then the map $\varphi\mapsto c_K(\varphi)^{-1}$
is upper semi-continuous with respect to the weak $({}=L^1_\loc$)
topology on the space of plurisubharmonic functions. Moreover, if
$\gamma<c_K(\varphi)$, then
$\int_K|e^{-2\gamma\psi}-e^{-2\gamma\varphi}|$ converges to $0$ when
$\psi$ converges to $\varphi$ in the weak topology.
\endclaim
%
\medskip
{\it Sketch of proof.} We will content ourselves by explaining the main 
points. It is convenient to observe (by a quite easy integration argument
suggested to us by J.~McNeal) 
that $c_K(\varphi)$ can be calculated by estimating the Lebesgue volume
$\mu_U(\{\varphi<\log r\}$ of tubular neighborhoods as $r\to 0$~:
$$
c_K(\varphi)=\sup\big\{c\ge 0\,;\,r^{-2c}\mu_U(\{\varphi<\log r\})~~
\hbox{is bounded as $r\to 0$, for some $U\supset K$}\big\}.
\leqno(9.34)
$$
The first step is the following important monotonicity result,
which is a straightforward consequence of the $L^2$ extension theorem.

\claim{(9.35) Proposition} Let $\varphi$ be a quasi-psh function on
a complex manifold~$X$, and let $Y\subset X$ be a complex submanifold such
that $\varphi_{|Y}\not\equiv-\infty$ on every connected component of~$Y$.
Then, if $K$ is a compact subset of $Y$, we have
$$
c_K(\varphi_{|Y})\le c_K(\varphi).
$$
$($Here, of course, $c_K(\varphi)$ is computed on $X$, i.e., by means of
neighborhoods of $K$ in~$X)$.
\endclaim
%
\noindent We need only proving monotonicity for $c_{z_0}(\varphi_{|Y})$ when
$z_0$ is a point of $Y$. This is done by just extending the holomorphic
function $f(z)=1$ on $B(z_0,r)\cap Y$ with respect to the weight
$e^{-2\gamma\varphi}$ whenever $\gamma<c_{z_0}(\varphi_{|Y})$.

\claim{(9.36) Proposition} Let $X$, $Y$ be complex manifolds of 
respective dimensions $n$,~$m$, let $\cI\subset\cO_X$, $\cJ\subset\cO_Y$
be coherent ideals, and let $K\subset X$, $L\subset Y$ be compact
sets. Put $\cI\oplus\cJ:=\pr_1^\star\cI + \pr_2^\star\cJ\subset
\cO_{X\times Y}$. Then
$$
c_{K\times L}(\cI\oplus\cJ)=c_K(\cI)+c_L(\cJ).
$$
\endclaim
%
\begproof{Proof.} It is enough to show that
$c_{(x,y)}(\cI\oplus\cJ)=c_x(\cI)+c_y(\cJ)$ at every point $(x,y)\in
X\times Y$. Without loss of generality, we may assume that
\hbox{$X\subset\bC^n$}, \hbox{$Y\subset\bC^m$} are open sets and
$(x,y)=(0,0)$. Let $g=(g_1,\,\ldots\,,g_p)$, resp.\
$h=(h_1,\,\ldots\,,h_q)$, be systems of generators of $\cI$ (resp.\ $\cJ$)
on a neighborhood of~$0$. Set
$$
\varphi=\log\sum|g_j|,\qquad\psi=\log\sum|h_k|.
$$
Then $\cI\oplus\cJ$ is generated by the $p+q$-tuple of functions
$$
g\oplus h=(g_1(x),\,\ldots\,g_p(x),h_1(y),\,\ldots\,,h_q(y))
$$
and the corresponding psh function $\Phi(x,y)=\log\big(\sum|g_j(x)|+
\sum|h_k(y)|\big)$ has the same behavior along the poles as
$\Phi'(x,y)=\max(\varphi(x),\psi(y))$ (up to a term $O(1)\le\log 2$).
Now, for sufficiently small neighborhoods $U$, $V$ of $0$, we have
$$
\mu_{U\times V}\big(\big\{\max(\varphi(x),\psi(y))<\log r\big\}\big)
=\mu_U\big(\{\varphi<\log r\}\times\mu_V(\{\psi<\log r\}\big),
$$
and one can derive from this that
$$
C_1r^{2(c+c')}\le
\mu_{U\times V}\big(\big\{\max(\varphi(x),\psi(y))<\log r\big\}\big)
\le C_2r^{2(c+c')}\,|\log r|^{n-1+m-1}
$$
with $c=c_0(\varphi)=c_0(\cI)$ and $c'=c_0(\psi)=c_0(\cJ)$. We infer 
$$
c_{(0,0)}(\cI\oplus\cJ)=c+c'=c_0(\cI)+c_0(\cJ).
\eqno\square
$$
\endproof

\claim{(9.37) Proposition} Let $f$, $g$ be holomorphic on a complex 
manifold~$X$. Then, for every $x\in X$,
$$
c_x(f+g)\leq c_x(f)+c_x(g).
$$
More generally, if $\cI$ and $\cJ$ are coherent ideals, then
$$
c_x(\cI+\cJ)\leq c_x(\cI)+c_x(\cJ).
$$
\endclaim
%
\begproof{Proof.} Let $\Delta$ be the diagonal in $X\times X$. Then $\cI+\cJ$
can be seen as the restriction of $\cI\oplus\cJ$ to~$\Delta$. Hence
Prop.~9.35 and 9.36 combined imply
$$
c_x(\cI+\cJ)=c_{(x,x)}((\cI\oplus\cJ)_{|\Delta})\le
c_{(x,x)}(\cI\oplus\cJ)=c_x(\cI)+c_x(\cJ).
$$
Since $(f+g)\subset(f)+(g)$, we get
$$
c_x(f+g)\le c_x((f)+(g))\le c_x(f)+c_x(g).\eqno\square
$$
\endproof

\noindent
Now we can explain in rough terms the strategy of proof of Theorem~9.33. 
We start by approximating psh singularities with analytic singularities,
using theorem 9.21. By the argument of Corollary~9.23, we can even
reduce ourselves to the case of invertible ideals $(f)$ near $z_0=0$,
and look at what happens when we have a uniformly convergent sequence
$f_\nu\to f$. In this case, we use the Taylor expansion of $f$ at $0$ to
write $f=p_N+s_N$ where $p_N$ is a polynomial of degree $N$ and 
$s_N(z)=O(|z|^{N+1})$. Clearly $c_0(s_N)\le n/(N+1)$, and from this
we infer $|c_0(f)-c_0(P_N)|\le n/(N+1)$ by 9.37. Similarly, we get 
the uniform estimate $|c_0(f_\nu)-c_0(P_{\nu,N})|\le n/(N+1)$
for all indices~$\nu$. This means that the proof of the semi-continuity 
theorem is reduced to handling the situation of a finite dimensional space
of polynomials. This case is well-known -- one can apply 
Hironaka's desingularization theorem, in a relative version involving
the coefficients of our polynomials as parameters. The conclusion is
obtained by putting together carefully all required uniform estimates
(which involve a lot of $L^2$ estimates).\qed

\section{Subadditivity of multiplier ideals and Fujita's
approximate Zariski~decomposition theorem}
\sectionrunning{Subadditivity of multiplier ideals and Fujita approximation}

We first notice the following basic restriction formula for multiplier
ideals, which is just a rephrasing of the Ohsawa-Takegoshi extension 
theorem.

\claim{(10.1) Restriction formula} Let $\varphi$ be a
plurisubharmonic function on a complex mani\-fold $X$, and let $Y\subset
X$ be a submanifold. Then
$$\cI(\varphi_{|Y})\subset\cI(\varphi)_{|Y}.$$
\endclaim
%
Thus, in some sense, the singularities of $\varphi$ can only get worse
if we restrict to a submanifold (if the restriction of $\varphi$ to some
connected component of $Y$ is identically $-\infty$, we agree that the 
corresponding multiplier ideal sheaf is zero). The proof is straightforward
and just amounts to extending locally a germ of function $f$ on $Y$ near 
a point $y_0\in Y$ to a function $\tilde f$ on a small Stein neighborhood 
of $y_0$ in $X$, which is possible by the Ohsawa-Takegoshi extension theorem. 
As a direct consequence, we get:

\claim{(10.2) Subadditivity Theorem}
\plainitem{\rm(i)} Let $X_1$, $X_2$  be complex manifolds,
$\pi_i:X_1\times X_2\to X_i$, $i=1,2$ the projections, and
let $\varphi_i$ be a plurisubharmonic function on~$X_i$.
Then
$$
\cI(\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2) = \pi_1^\star
\cI(\varphi_1) \cdot  \pi_2^\star\cI(\varphi_2).
$$
\plainitem{\rm(ii)} Let $X$ be a complex manifold and let
$\varphi$, $\psi$ be plurisubharmonic functions on $X$.
Then
$$
\cI(\varphi + \psi) \subset \cI(\varphi) \cdot \cI(\psi)
$$
\endclaim
%
\begproof{Proof.} (i) Let us fix two relatively
compact Stein open subsets $U_1\subset X_1$, 
$U_2\subset X_2$. Then $\cH^2(U_1\times U_2,
\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2, \pi_1^\star
dV_1\otimes \pi_2^\star dV_2)$ is the Hilbert tensor
product of $\cH^2(U_1,\varphi_1,dV_1)$ and
$\cH^2(U_2,\varphi_2,dV_2)$, and admits $(f'_k\boxtimes
f''_l)$ as a Hilbert  basis, where $(f'_k)$ and
$(f''_l)$ are respective Hilbert bases. Since
$\cI(\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2)_{|U_1\times
U_2}$ is generated  as an $\cO_{U_1\times U_2}$ module
by the $(f'_k\boxtimes f''_l)$ (Proposition 5.7), we  conclude that (i)
holds true.
\medskip

\noindent
(ii) We apply (i) to $X_1=X_2=X$ and the restriction
formula to $Y={}$diagonal of $X\times X$. Then
$$
\eqalign{
\cI(\varphi+\psi)
&=\cI\big((\varphi\circ \pi_1 + \psi\circ \pi_2)_{|Y}\big)
\subset \cI\big(\varphi\circ \pi_1 + \psi\circ \pi_2\big)_{|Y}\cr
&=\Big(\pi_1^\star\cI(\varphi)\otimes \pi_2^\star\cI(\psi)\Big)_{|Y}
=\cI(\varphi)\cdot\cI(\psi).\cr}
$$
\endproof
%
\claim{(10.3) Proposition} Let $f:X\to Y$ be an arbirary holomorphic 
map, and let $\varphi$ be a plurisubharmonic function on~$Y$. Then
$\cI(\varphi\circ f)\subset f^\star\cI(\varphi)$.
\endclaim
%
\begproof{Proof.} Let 
$$
\Gamma_f=\{(x,f(x)\,;\,x\in X\}\subset X\times Y
$$ 
be the graph of $f$, and let $\pi_X:X\times Y\to X$, $\pi_Y:X\times
Y\to Y$ be the natural projections. Then we can view $\varphi\circ f$
as the restriction of $\varphi\circ\pi_Y$ to $\Gamma_f$, as $\pi_X$ is
a biholomorphism from $\Gamma_f$ to $X$. Hence the restriction formula
implies
$$
\cI(\varphi\circ f)=\cI\big((\varphi\circ\pi_Y)_{|\Gamma_f}\big)
\subset\cI(\varphi\circ\pi_Y)_{|\Gamma_f}
=\big(\pi_Y^\star\cI(\varphi)\big)_{|\Gamma_f}=f^\star\cI(\varphi).
\eqno\square
$$
\endproof
%
As an application of subadditivity, we now reprove a result of Fujita
[Fuj93], relating the growth of sections of multiples of a line bundle
to the Chern numbers of its ``largest nef part''. Fujita's original
proof is by contradiction, using the Hodge index theorem and
intersection inequalities. The present method arose in
the course of joint work with R.~Lazarsfeld~[Laz99].

Let $X$ be a projective $n$-dimensional algebraic variety and $L$
a line bundle over~$X$. We define the {\it volume} of $L$ to be
$$
\Vol(L)=\limsup_{k\to+\infty}{n!\over k^n}h^0(X,kL)\in[0,+\infty[.
$$
In view of Definition 6.12 and of the Serre-Siegel Lemma 6.13, the 
line bundle is {\it big} if and only if $\Vol(L)>0$. If $L$ is ample, 
we have $h^q(X,kL)=0$ for $q\ge 1$ and $k\gg 1$ by the Kodaira-Serre
vanishing theorem, hence 
$$
h^0(X,kL)\sim\chi(X,kL)\sim{L^n\over n!}k^n
$$ 
by the Riemann-Roch formula.  Thus $\Vol(L)=L^n\,({}=c_1(L)^n)$ if $L$ is
ample. This is still true if $L$ is nef (numerically effective), i.e.\
if $L\cdot C\ge 0$ for every effective curve~$C$. In fact, one can show 
that $h^q(X,kL)=O(k^{n-q})$ in that case. The following
well-known proposition characterizes big line bundles.

\claim{(10.4) Proposition} The line bundle $L$ is big if and only if there
is a multiple $m_0L$ such that $m_0L=E+A$, where $E$ is an effective divisor
and $A$ an ample divisor.
\endclaim
%
\begproof{Proof.} If the condition is satisfied, the decomposition 
$km_0L=kE+kA$ gives
rise to an injection $H^0(X,kA)\hookrightarrow H^0(X,km_0L)$, thus
$\Vol(L)\ge m_0^{-n}\Vol(A)>0$. Conversely, assume that $L$ is big, and take 
$A$ to be a very ample nonsingular divisor in $X$. The exact sequence
$$
0\longrightarrow \cO_X(kL-A)\longrightarrow \cO_X(kL)\longrightarrow \cO_A(kL_{|A})\longrightarrow 0
$$
gives rise to a cohomology exact sequence
$$
0\to H^0(X,kL-A) \longrightarrow H^0(X,kL) \longrightarrow H^0(A,kL_{|A}), 
$$
and $h^0(A,kL_{|A})=O(k^{n-1})$ since $\dim A=n-1$. Now, the assumption that
$L$ is big implies that $h^0(X,kL)> ck^n$ for infinitely many $k$, hence
$H^0(X,m_0L-A)\ne 0$ for some large integer $m_0$. If $E$ is the divisor 
of a section in $H^0(X,m_0L-A)$, we find $m_0L-A=E$, as required.\qed
\endproof
%
\claim{(10.5) Lemma} Let $G$ be an arbitrary line bundle. For every
$\varepsilon>0$, there exists a positive integer $m$ and a sequence
$\ell_\nu\uparrow+\infty$ such that
$$
h^0\big(X,\ell_\nu(mL-G)\big)\ge {\ell_\nu^mm^n\over n!}
\big(\Vol(L)-\varepsilon\big),
$$
in other words, $\Vol(mL-G)\ge m^n(\Vol(L)-\varepsilon)$ for $m$ large enough.
\endclaim
%
\begproof{Proof.} Clearly, $\Vol(mL-G)\ge \Vol(mL-(G+E))$ for every effective divisor
$E$. We can take $E$ so large that $G+E$ is very ample, and we are thus 
reduced to the case where $G$ is very ample by replacing $G$ with $G+E$.
By definition of $\Vol(L)$, there exists a sequence $k_\nu\uparrow+\infty$
such that
$$
h^0(X,k_\nu L)\ge {k_\nu^n\over n!}\Big(\Vol(L)-{\varepsilon\over 2}\Big).
$$
We take $m\gg 1$ (to be precisely chosen later), and $\ell_\nu=
\big[{k_\nu\over m}\big]$, so that $k_\nu=\ell_\nu m+r_\nu$,
$0\le r_\nu<m$. Then
$$
\ell_\nu(mL-G)=k_\nu L-(r_\nu L+\ell_\nu G).
$$
Fix a constant $a\in\bN$ such that $aG-L$ is an effective divisor. Then
$r_\nu L\le maG$ (with respect to the cone of effective divisors), hence
$$
h^0\big(X,\ell_\nu(mL-G)\big)\ge
h^0\big(X,k_\nu L-(\ell_\nu+am)G\big).
$$
We select a smooth divisor $D$ in the very ample linear system
$|G|$. By looking at global sections associated with the exact
sequences of sheaves
$$
0\to\cO(-(j+1)D)\otimes\cO(k_\nu L)\to\cO(-jD)\otimes\cO(k_\nu L)
\to\cO_D(k_\nu L-jD)\to 0,
$$
$0\le j<s$, we infer inductively that
$$
\eqalign{
h^0(X,k_\nu L-sD)
&\ge h^0(X,k_\nu L)-\sum_{0\le j< s}h^0\big(D,\cO_D(k_\nu L-jD)\big)\cr
&\ge h^0(X,k_\nu L)- s\,h^0\big(D,k_\nu L_{|D})\cr
&\ge {k_\nu^n\over n!}\Big(\Vol(L)-{\varepsilon\over 2}\Big) -s\,Ck_\nu^{n-1}
\cr}
$$
where $C$ depends only on $L$ and~$G$. Hence, by putting $s=\ell_\nu+am$,
we get
$$
\eqalign{
h^0\big(X,\ell_\nu(mL-G)\big) 
&\ge {k_\nu^n\over n!}\Big(\Vol(L)-{\varepsilon\over 2}\Big) 
-C(\ell_\nu+am)k_\nu^{n-1}\cr
&\ge{\ell_\nu^nm^n\over n!}\Big(\Vol(L)-{\varepsilon\over 2}\Big)
-C(\ell_\nu+am)(\ell_\nu+1)^{n-1}m^{n-1}\cr}
$$
and the desired conclusion follows by taking $\ell_\nu\gg m\gg 1$.\qed
\endproof
%
We are now ready to prove Fujita's decomposition theorem, as reproved
in [DEL00].

\claim{(10.6) Theorem (Fujita)} Let $L$ be a big line bundle. Then for 
every $\varepsilon>0$, there exists a modification $\mu:\widetilde X\to X$ and
a decomposition $\mu^\star L=E+A$, where $E$ is an effective $\bQ$-divisor
and $A$ an ample $\bQ$-divisor, such that $A^n>\Vol(L)-\varepsilon$.
\endclaim
%
\claim{(10.7) Remark} {\rm Of course, if $\mu^\star L= E+A$ with $E$ 
effective and $A$ nef, we get an injection
$$
H^0(\widetilde X,kA)\hookrightarrow H^0(\widetilde X, kE+kA)=H^0(\widetilde X, k\mu^\star L)=
H^0(X,kL)
$$
for every integer $k$ which is a multiple of the denominator of $E$, hence
$A^n\le \Vol(L)$.}
\endclaim
%
\claim{(10.8) Remark} {\rm Once Theorem 10.6 is proved, the same kind
of argument easily shows that
$$
\Vol(L)=\lim_{k\to +\infty}{n!\over k^n}h^0(X,kL),
$$
because the formula is true for every ample line bundle $A$.}
\endclaim
%
\begproof{Proof of Theorem 10.6.} It is enough to prove the theorem with $A$
being a big and nef divisor. In fact, Proposition 10.4 then shows that we can
write $A=E'+A'$ where $E'$ is an effective $\bQ$-divisor and $A'$ an
ample $\bQ$-divisor, hence
$$
E+A=E+\varepsilon E'+(1-\varepsilon)A+\varepsilon A'
$$
where $A''=(1-\varepsilon)A+\varepsilon A'$ is ample and the intersection
number $A^{\prime\prime n}$ approaches $A^n$ as closely as we want.
Let $G$ be as in Theorem (6.22) (Siu's theorem on uniform global
generation). Lemma 10.5 implies that
$\Vol(mL-G)>m^n(\Vol(L)-\varepsilon)$ for $m$ large. By Theorem~(6.8) on
the existence of analytic Zariski decomposition, there
exists a hermitian metric $h_m$ of weight $\varphi_m$ on $mL-G$ such that
$$
H^0\big(X,\ell(mL-G)\big)=H^0\big(X,\ell(mL-G)\otimes\cI(\ell\varphi_m)\big)
$$
for every $\ell\ge 0$. We take a smooth modification $\mu:\widetilde X\to X$ such
that 
$$
\mu^\star\cI(\varphi_m)=\cO_{\widetilde X}(-E)
$$
is an invertible ideal sheaf in $\cO_{\widetilde X}$. This is possible by
taking the blow-up of $X$ with respect to the ideal $\cI(\varphi_m)$
and by resolving singularities (Hironaka [Hir64]). Theorem~6.22
applied to $L'=mL-G$
implies that $\cO(mL)\otimes\cI(\varphi_m)$ is generated by its
global sections, hence its pull-back $\cO(m\,\mu^\star L-E)$ is 
also generated. This implies
$$
m\,\mu^\star L=E+A
$$ 
where $E$ is an effective divisor and $A$ is a nef (semi-ample)
divisor in~$\widetilde X$. We find
$$
\eqalign{
H^0(\widetilde X,\ell A)
  &=H^0\big(\widetilde X,\ell(m\,\mu^\star L-E)\big)\cr
  &\supset H^0\big(\widetilde X,\mu^\star\big(\cO(\ell mL)\otimes
       \cI(\varphi_m)^\ell\big)\big)\cr
  &\supset H^0\big(\widetilde X,\mu^\star\big(\cO(\ell mL)\otimes
       \cI(\ell\varphi_m)\big)\big),\cr}
$$
thanks to the subadditivity property of multiplier ideals.
Moreover, the direct image 
$\mu_\star\mu^\star\cI(\ell\varphi_m)$ coincides with the integral
closure of $\cI(\ell\varphi_m)$, hence with $\cI(\ell\varphi_m)$,
because a multiplier ideal sheaf is always integrally closed.
From this we infer
$$
\eqalign{
H^0(\widetilde X,\ell A)
  &\supset H^0\big(X,\cO(\ell mL)\otimes \cI(\ell\varphi_m)\big)\cr
  &\supset H^0\big(X,\cO(\ell(mL-G))\otimes \cI(\ell\varphi_m)\big)\cr
  &= H^0\big(X,\cO(\ell(mL-G))\big).\cr}
$$
By Lemma 10.5, we find
$$
h^0(\widetilde X,\ell A)\ge{\ell^n\over n!}m^n\big(\Vol(L)-\varepsilon\big)
$$
for infinitely many $\ell$, therefore $\Vol(A)=A^n\ge m^n(\Vol(L)-\varepsilon)$.
Theorem 10.6 is proved, up to a minor change of notation
$E\mapsto{1\over m}E$, $A\mapsto{1\over m}A$.\qed
\endproof
%
We conclude by using Fujita's theorem to establish a geometric
interpretation of the volume $\Vol(L)$. Suppose as above that $X$ is a
smooth projective variety of dimension $n$, and that $L$ is a big line
bundle on $X$. Given a large integer $k \gg 0$, denote by $B_k
\subset X$ the base-locus of the linear system $| kL |$. The
{\it moving self-intersection number} $(kL)^{[n]}$ of
$|kL|$ is defined by choosing $n$ general divisors $D_1, \dots , D_n
\in |kL|$ and putting
$$( kL)^{[n]} = \# \Big( D_1 \cap \ldots
\cap D_n \cap ( X - B_k)  \Big).
$$
In other words, we simply count the number of intersection points
away from the base locus of $n$ general divisors in the linear system
$|kL|$. This notion arises for example in Matsusaka's proof of his
``big theorem". We show that the volume $\Vol(L)$ of $L$ 
measures the rate of growth with respect to $k$ of these moving 
self-intersection numbers:

\claim{(10.9) Proposition} One has
$$\Vol(L) = \limsup_{k \to \infty} { (kL)^{[n]} \over k^n}.$$
\endclaim
%
\begproof{Proof.} We start by interpreting 
$(kL)^{[n]}$ geometrically. Let $\mu_k: X_k \longrightarrow X$
be a modification of $|kL|$ such that $\mu_k^\star |kL| = |V_k| +  F_k $, 
where
$$ P_k := \mu_k^\star (kL) - F_k$$  
is generated by sections, and
$H^0(X,\cO_X(kL)) = V_k = H^0(X_k,\cO_{X_k}(P_k))$, so that $B_k =
\mu_k(F_k)$. Then evidently
$(kL)^{[n]}$ counts the number of intersection points
of $n$ general divisors in $P_k$, and consequently
$$(kL)^{[n]} = (P_k)^n .$$  
Since then $P_k$ is big (and nef) for $k\gg 0$, 
we have  $\Vol(P_k)=(P_k)^n$. Also, $\Vol(kL) \ge \Vol(P_k)$ 
since $P_k$ embeds in $\mu_k^\star(k L)$.  Hence
$$
\Vol(kL) \ge  ( kL)^{[n]}\qquad \forall k \gg 0.
$$
On the other hand, an easy argument in the spirit of Lemma (10.5)
shows that $\Vol(kL) = k^n \cdot \Vol(L)$
(cf.\ also [ELN96], Lemma 3.4), and so we conclude that
$$
\Vol(L) \ge { (kL)^{[n]} \over k^n}. \leqno(10.10)
$$
for every $k \gg 0$. 

For the reverse inequality we use Fujita's theorem. Fix
$\varepsilon > 0$, and consider the  decomposition $\mu^\star
L = A + E$ on $\mu : \widetilde X \longrightarrow X$ constructed in
Fujita's theorem. Let $k$ be any positive integer
such that $kA$ is integral and globally generated. By taking a
common resolution we can assume that $X_k$ dominates
$\widetilde X$, and hence we can write 
$$
\mu_k^\star  kL \sim  A_k + E_k
$$
with $A_k$ globally generated and 
$$
(A_k)^n\ge k^n \cdot ( \Vol(L) - \varepsilon ).
$$  
But then $A_k$ embeds in $P_k$ and both $\cO(A_k)$ and $\cO(P_k)$
are globally generated, consequently 
$$
(A_k)^n \le (P_k)^n =  ( kL)^{[n]}.
$$
Therefore
$$
{(kL)^{[n]}\over k^n} \ge  \Vol(L) - \varepsilon. \leqno(10.11)
$$
But (10.11) holds for any sufficiently large and divisible $k$, and in 
view of (10.10) the Proposition follows.\qed
\endproof

\endsectionrunning
%
\section{Hard Lefschetz theorem with multiplier ideal sheaves}
\subsection*{\csect.A. Main statement}
The goal of this section is to prove the following surjectivity theorem,
which can be seen as an extension of the hard Lefschetz theorem. We
closely follow the exposition of [DPS00].

\claim{(11.1) Theorem} Let $(L,h)$ be a pseudo-effective line bundle on a
compact K\"ahler manifold $(X,\omega)$, of dimension $n$, let 
$\Theta_{L,h}\ge 0$ be its curvature current and $\cI(h)$ the 
associated multiplier ideal sheaf. Then, the wedge multiplication 
operator $\omega^q\wedge\bu$ induces a surjective morphism
$$
\Phi^q_{\omega,h}:
H^0(X,\Omega_X^{n-q}\otimes L\otimes\cI(h))\longrightarrow
H^q(X,\Omega_X^n\otimes L\otimes\cI(h)).
$$
\endclaim
%
\noindent
The special case when $L$ is nef is due to Takegoshi [Take97]. An even
more special case is when $L$ is semi-positive, i.e.\ possesses a
smooth metric with semi-positive curvature.
In that case the multiple ideal sheaf $\cI(h)$ coincides with $\cO_X$
and we get the following consequence already observed by Mourougane
[Mou99].

\claim{(11.2) Corollary} Let $(L,h)$ be a semi-positive line bundle
on a compact K\"ahler mani\-fold $(X,\omega)$ of dimension~$n$. Then, the wedge
multiplication operator $\omega^q\wedge\bu$ induces a surjective morphism
$$
\Phi^q_\omega:H^0(X,\Omega_X^{n-q}\otimes L)\longrightarrow
H^q(X,\Omega_X^n\otimes L).
$$
\endclaim
%
It should be observed that although all objects involved in Theorem (11.1)
are algebraic when $X$ is a projective manifold, there are no known
algebraic proof of the statement; it is not even clear how to define
algebraically $\cI(h)$ for the case when $h=h_{min}$ is a metric
with minimal singularity. However, even in the special circumstance
when $L$ is nef, the multiplier
ideal sheaf is crucially needed (see section 11.E for a counterexample).

The proof of Theorem (11.1) is based on the Bochner formula, combined
with a use of harmonic forms with values in the hermitian line bundle
$(L,h)$. The method can be applied only after $h$ has been made smooth
at least in the complement of an analytic set. However, we have to
accept singularities even in the regularized metrics because only a
very small incompressible loss of positivity is acceptable in the
Bochner estimate (by the results of [Dem92], singularities can only be
removed at the expense of a fixed loss of positivity). Also, we need
the multiplier ideal sheaves to be preserved by the smoothing process.
This is possible thanks to a suitable ``equisingular'' regularization
process.

\subsection*{\csect.B. Equisingular approximations of quasi plurisubharmonic functions}
Let $\varphi$ be a quasi-psh function.
We say that $\varphi$ has logarithmic poles if $\varphi$ is locally
bounded outside an analytic set $A$ and has singularities of the form
$$\varphi(z)=c\log\sum_k|g_k|^2+O(1)$$
with $c>0$ and $g_k$ holomorphic, on a neighborhood of every point
of~$A$. Our goal is to show the following

\claim{(11.3) Theorem} Let $T=\alpha+i\ddbar\varphi$ be a closed
$(1,1)$-current on a compact hermitian manifold $(X,\omega)$, where
$\alpha$ is a smooth closed $(1,1)$-form and $\varphi$ a quasi-psh
function. Let $\gamma$ be a continuous real $(1,1)$-form such that
$T\ge\gamma$. Then one can write
\hbox{$\varphi=\lim_{\nu\to+\infty}\varphi_\nu$} where
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (a)} $\varphi_\nu$ is smooth in the complement $X\ssm Z_\nu$
of an analytic set $Z_\nu\subset X\,;$
\vskip3pt
\plainitem{\rm (b)} $(\varphi_\nu)$ is a decreasing sequence, and $Z_\nu\subset
Z_{\nu+1}$ for all~$\nu\,;$
\vskip3pt
\plainitem{\rm (c)} $\int_X(e^{-2\varphi}-e^{-2\varphi_\nu})dV_\omega$
is finite for every $\nu$ and converges to $0$ as $\nu\to+\infty\,;$
\vskip3pt
\plainitem{\rm (d)} $\cI(\varphi_\nu)=\cI(\varphi)$ for all $\nu$
$($``equisingularity''$)\,;$
\vskip3pt
\plainitem{\rm (e)} $T_\nu=\alpha+i\ddbar\varphi_\nu$ satisfies $T_\nu\ge
\gamma-\varepsilon_\nu\omega$, where $\lim_{\nu\to+\infty}\varepsilon_\nu=0$.
\vskip0pt}
\endclaim
%
\claim{(11.4) Remark} {\rm It would be interesting to know whether the
$\varphi_\nu$ can be taken to have logarithmic poles along $Z_\nu$.
Unfortunately, the proof given below destroys this property in the
last step. Getting it to hold true seems to be more or less
equivalent to proving the semi-continuity property
$$\lim_{\varepsilon\to 0_+}
\cI((1+\varepsilon)\varphi)=\cI(\varphi).$$
Actually, this can be checked in dimensions $1$ and $2$, but is unknown in
higher dimensions (and probably quite hard to establish).}
\endclaim
%

\begproof{Proof of Theorem 11.3}. Clearly, by replacing $T$ with
$T-\alpha$ and $\gamma$ with $\gamma-\alpha$, we may assume that
$\alpha=0$ and $T=i\ddbar\varphi\ge \gamma$. We divide the proof in
four steps.
\medskip

\noindent
{\it Step 1. Approximation by quasi-psh functions with logarithmic
poles.}

\noindent
By [Dem92], there is a decreasing sequence $(\psi_\nu)$ of quasi-psh
functions with logarithmic poles such that $\varphi=\lim\psi_\nu$ and
$i\ddbar\psi_\nu\ge\gamma- \varepsilon_\nu\omega$. We need a little bit
more information on those functions, hence we first recall the main
techniques used for the construction of~$(\psi_\nu)$. For $\varepsilon>0$
given, fix a covering of $X$ by open balls $B_j=\{|z^{(j)}|<r_j\}$ with
coordinates $z^{(j)}=(z^{(j)}_1,\ldots, z^{(j)}_n)$, such that
$$
0\le\gamma+c_j\,i\ddbar|z^{(j)}|^2\le \varepsilon\omega
\qquad\hbox{on}\quad B_j,
\leqno(11.5)
$$
for some real number~$c_j$. This is possible by selecting coordinates
in which $\gamma$ is diagonalized at the center of the ball, and by
taking the radii $r_j>0$ small enough (thanks to the fact that $\gamma$
is continuous). We may assume that these coordinates come from a finite
sample of coordinates patches covering $X$, on which we perform suitable
linear coordinate changes (by invertible matrices lying in some
compact subset of the complex linear group). By taking additional balls,
we may also assume that $X=\bigcup B''_j$ where
$$
B''_j\compact B'_j\compact B_j
$$
are concentric balls $B'_j=\{|z^{(j)}|<r'_j=r_j/2\}$,
$B''_j=\{|z^{(j)}|<r''_j=r_j/4\}$. We define
$$
\psi_{\varepsilon,\nu,j}=
{1\over 2\nu}\log\sum_{k\in\bN}|f_{\nu,j,k}|^2-c_j|z^{(j)}|^2
\qquad\hbox{on}\quad B_j,
\leqno(11.6)
$$
where $(f_{\nu,j,k})_{k\in\bN}$ is an orthonormal basis of the Hilbert
space $\cH_{\nu,j}$ of holomorphic functions on $B_j$ with finite
$L^2$ norm
$$
\Vert u\Vert^2 = \int_{B_j} |u|^2 e^{-2\nu(\varphi+c_j|z^{(j)}|^2)}
d\lambda(z^{(j)}).
$$
(The dependence of $\psi_{\varepsilon,\nu,j}$ on $\varepsilon$ is through
the choice of the open covering~$(B_j)$). Observe that the choice of
$c_j$ in (11.5) guarantees that $\varphi+c_j|z^{(j)}|^2$ is
plurisubharmonic on $B_j$, and notice also that
$$
\sum_{k\in\bN}|f_{\nu,j,k}(z)|^2=\sup_{f\in\cH_{\nu,j},\,\Vert f\Vert\le 1}
|f(z)|^2
\leqno(11.7)
$$
is the square of the norm of the continuous linear form $\cH_{\nu,j}\to\bC$,
$f\mapsto f(z)$. We claim that there exist
constants $C_i$, $i=1,2,\ldots$ depending only on $X$ and $\gamma$
(thus independent of $\varepsilon$ and $\nu$), such that the following
uniform estimates hold:
$$
\leqalignno{\qquad\qquad
&i\ddbar\psi_{\varepsilon,\nu,j}\ge -c_j\,i\ddbar|z^{(j)}|^2\ge\gamma-
\varepsilon\omega\qquad\hbox{on}~~B'_j~~(B'_j\compact B_j),
&(11.8)\cr
&\varphi(z)\le\psi_{\varepsilon,\nu,j}(z)\le\sup_{|\zeta-z|\le r}\varphi(\zeta)
+{n\over\nu}\log{C_1\over r}+C_2r^2
\qquad\hbox{$\forall z\in B'_j$,~ $r<r_j-r'_j$,}
&(11.9)\cr
&|\psi_{\varepsilon,\nu,j}-\psi_{\varepsilon,\nu,k}|\le {C_3\over\nu}
+C_4\varepsilon\big(\min(r_j,r_k)\big)^2\qquad
\hbox{on}~~B'_j\cap B'_k.
&(11.10)\cr}
$$
Actually, the Hessian estimate (11.8) is obvious from (11.5) and (11.6).
As in the proof of ([Dem92], Prop.~3.1), (11.9) results from the
Ohsawa-Takegoshi $L^2$ extension theorem (left hand inequality)
and from the mean value inequality (right hand inequa\-lity).
Finally, as in ([Dem92], Lemma~3.6 and Lemma~4.6), (11.10) is a
consequence of H\"ormander's $L^2$ estimates. We briefly sketch the idea.
Assume that the balls $B_j$ are small enough, so that the
coordinates $z^{(j)}$ are still defined on a neighborhood of all
balls $\overline B_k$ which intersect~$B_j$ (these coordinates can be
taken to be linear transforms of coordinates belonging to a fixed
finite set of coordinate patches covering~$X$, selected once for all).
Fix a point
$z_0\in B'_j\cap B'_k$. By (11.6) and (11.7), we have
$$
\psi_{\varepsilon,\nu,j}(z_0)={1\over \nu}\log|f(z_0)|-c_j|z^{(j)}|^2
$$
for some holomorphic function $f$ on $B_j$ with $\Vert f\Vert=1$. We
consider the weight function
$$
\Phi(z)=2\nu(\varphi(z)+c_k|z^{(k)}|^2)+2n\log|z^{(k)}-z^{(k)}_0|,
$$
on both $B_j$ and $B_k$. The trouble is that a priori we have to deal with
different weights, hence a comparison of weights is needed. By the Taylor
formula applied at $z_0$, we get
$$
\Big|c_k|z^{(k)}-z^{(k)}_0|^2 - c_j|z^{(j)}-z^{(j)}_0|^2\Big|\le
C\varepsilon\big(\min(r_j,r_k)\big)^2\qquad\hbox{on $B_j\cap B_k$}
$$
[the only nonzero term of degree 2 has type $(1,1)$ and its Hessian satisfies
$$
-\varepsilon\omega\le i\ddbar(c_k|z^{(k)}|^2-c_j|z^{(j)}|^2)\le
\varepsilon\omega
$$
by (11.5); we may suppose $r_j\ll\varepsilon$ so that the terms of order
$3$ and more are negligible]. By writing
$|z^{(j)}|^2=|z^{(j)}-z^{(j)}_0|^2+|z^{(j)}_0|^2+
2\Re\langle z^{(j)}-z^{(j)}_0,z^{(j)}_0\rangle$, we obtain
$$\eqalign{
c_k|z^{(k)}|^2-c_j|z^{(j)}|^2
&=2c_k\Re\langle z^{(k)}-z^{(k)}_0,z^{(k)}_0\rangle
-2c_j\Re\langle z^{(j)}-z^{(j)}_0,z^{(j)}_0\rangle\cr
&\qquad{}+c_k|z^{(k)}_0|^2-c_j|z^{(j)}_0|^2
\pm C\varepsilon(\min(r_j,r_k))^2.\cr}
$$
We use a cut-off function $\theta$ equal to $1$ in a neighborhood of
$z_0$ and with support in $B_j\cap B_k$; as $z_0\in B'_j\cap B'_k$, the
function $\theta$ can be taken to have its derivatives uniformly bounded
when $z_0$ varies. We solve the equation
$\dbar u=\dbar(\theta f e^{\nu g})$ on $B_k$, where $g$ is the holomorphic
function
$$
g(z)=c_k\langle z^{(k)}-z^{(k)}_0,z^{(k)}_0\rangle
-c_j\langle z^{(j)}-z^{(j)}_0,z^{(j)}_0\rangle.
$$
Thanks to H\"ormander's $L^2$ estimates [H\"or66], the $L^2$ solution
for the weight $\Phi$ yields a holomorphic function
$f'=\theta fe^{\nu g}-u$ on $B_k$ such that $f'(z_0)=f(z_0)$ and
$$
\eqalign{
&\int_{B_k}|f'|^2e^{-2\nu(\varphi+c_k|z^{(k)}|^2)}
d\lambda(z^{(k)})
\le C'\int_{B_j\cap B_k} |f|^2 |e^{\nu g}|^2e^{-2\nu(\varphi+c_k|z^{(k)}|^2)}
d\lambda(z^{(k)})\le{}\cr
&
C'\exp\!\Big(\!2\nu\big(c_k|z^{(k)}_0|^2-c_j|z^{(j)}_0|^2
+C\varepsilon(\min(r_j,r_k))^2\big)\!\Big)\!
\int_{B_j}\kern-1pt|f|^2 e^{-2\nu(\varphi+c_j|z^{(j)}|^2)}
d\lambda(z^{(j)}).\cr}
$$
Let us take the supremum of ${1\over\nu}\log|f(z_0)|={1\over\nu}\log|f'(z_0)|$
over all $f$ with $\Vert f\Vert\le 1$. By the definition of
$\psi_{\varepsilon,\nu,k}$ ((11.6) and (11.7)) and the bound on
$\Vert f'\Vert$, we find
$$
\psi_{\varepsilon,\nu,k}(z_0)\le \psi_{\nu,j}(z_0)+{\log C'\over 2\nu}
+C\varepsilon(\min(r_j,r_k))^2,
$$
whence (11.10) by symmetry. Assume that $\nu$ is so large that
$C_3/\nu<C_4\varepsilon(\inf_j r_j)^2$.
We ``glue'' all functions $\psi_{\varepsilon,\nu,j}$
into a function $\psi_{\varepsilon,\nu}$ globally defined on~$X$,
and for this we set
$$
\psi_{\varepsilon,\nu}(z)=\sup_{j,~B'_j\ni z}
\Big(\psi_{\varepsilon,\nu,j}(z)+
12\,C_4\varepsilon(r^{\prime 2}_j-|z^{(j)}|^2)\Big)
\qquad\hbox{on}~~X.
$$
Every point of $X$ belongs to some ball $B''_k$, and for such a point
we get
$$
12\,C_4\varepsilon(r^{\prime 2}_k-|z^{(k)}|^2)\ge
12\,C_4\varepsilon(r^{\prime 2}_k-r^{\prime\prime2}_k)>2C_4r_k^2>
{C_3\over\nu}+C_4\varepsilon(\min(r_j,r_k))^2.
$$
This, together with (11.10), implies that in $\psi_{\varepsilon,\nu}(z)$
the supremum is never reached for indices $j$ such that
$z\in \partial B'_j$, hence $\psi_{\varepsilon,\nu}$ is well defined
and continuous, and by standard properties of upper envelopes of
(quasi)-plurisubharmonic functions we get
$$
i\ddbar\psi_{\varepsilon,\nu}\ge \gamma-C_5\varepsilon\omega
\leqno(11.11)
$$
for $\nu\ge\nu_0(\varepsilon)$ large enough. By inequality (11.9)
applied with $r=e^{-\sqrt{\nu}}$, we see that
$\lim_{\nu\to+\infty}\psi_{\varepsilon,\nu}(z) = \varphi(z)$.
At this point, the difficulty is to show that
$\psi_{\varepsilon,\nu}$ is decreasing with $\nu$ -- this may not be 
formally true, but we will see at Step 3 that this is essentially true.
Another difficulty is that we must simultaneously let $\varepsilon$
go to $0$, forcing us to change the covering as we want the error
to get smaller and smaller in (11.11).
\medskip

\noindent{\it Step 2. A comparison of integrals.}

\noindent We claim that
$$
I:=\int_X\big(e^{-2\varphi}-e^{-2\max(\varphi,{\ell\over\ell-1}
\psi_{\nu,\varepsilon})+a}\big)dV_\omega<+\infty
\leqno(11.12)
$$
for every $\ell\in{}]1,\nu]$ and $a\in\bR$. In fact
$$
\eqalign{
I&\le\int_{\{\varphi<{\ell\over\ell-1}\psi_{\varepsilon,\nu}+a\}}
e^{-2\varphi}dV_\omega
=\int_{\{\varphi<{\ell\over\ell-1}\psi_{\varepsilon,\nu}\}+a}
e^{2(\ell-1)\varphi-2\ell\varphi}dV_\omega\cr
&\le e^{2(\ell-1)a}\int_Xe^{2\ell(\psi_{\varepsilon,\nu}-\varphi)}dV_\omega
\le C\Big(\int_Xe^{2\nu(\psi_{\varepsilon,\nu}-\varphi)}
dV_\omega\Big)^{\ell\over\nu}\cr}
$$
by H\"older's inequality. In order to show that these integrals are finite,
it is enough, by the definition and properties of the functions
$\psi_{\varepsilon,\nu}$ and $\psi_{\varepsilon,\nu,j}$, to prove that
$$
\int_{B'_j}e^{2\nu\psi_{\varepsilon,\nu,j}-2\nu\varphi}d\lambda=
\int_{B'_j}\Big(\sum_{k=0}^{+\infty}|f_{\nu,j,k}|^2\Big)
e^{-2\nu\varphi}d\lambda<+\infty.
$$
By the strong Noetherian property of coherent ideal sheaves
(see e.g.\ [GR84]), we know that the sequence of ideal sheaves
generated by the holomorphic functions
$(f_{\nu,j,k}(z)\overline{f_{\nu,j,k}(\overline w)})_{k\le k_0}$ on
$B_j\times B_j$ is locally statio\-nary as $k_0$ increases, hence
independant of $k_0$ on $B'_j\times B'_j\compact B_j\times B_j$ for
$k_0$ large enough. As the sum of the series $\sum_k f_{\nu,j,k}(z)\overline
{f_{\nu,j,k}(\overline w)}$ is bounded by
$$\Big(\sum_k |f_{\nu,j,k}(z)|^2 \sum_k|f_{\nu,j,k}(\overline w)|^2
\Big)^{1/2}$$
and thus uniformly covergent on every compact subset of $B_j\times B_j$,
and as the space of sections of a coherent ideal sheaf is closed
under the topology of uniform convergence on compact subsets, we infer
from the Noetherian property that the holomorphic function
$\sum_{k=0}^{+\infty}f_{\nu,j,k}(z)\overline{f_{\nu,j,k}(\overline w)}$
is a section of the coherent ideal sheaf generated by
$(f_{\nu,j,k}(z)\overline{f_{\nu,j,k}(\overline w)})_{k\le k_0}$ over
$B'_j\times B'_j$, for $k_0$ large enough. Hence, by restricting
to the conjugate diagonal $w=\overline z$, we get
$$
\sum_{k=0}^{+\infty}|f_{\nu,j,k}(z)|^2\le C
\sum_{k=0}^{k_0}|f_{\nu,j,k}(z)|^2\qquad\hbox{on}~~B'_j.
$$
This implies
$$
\int_{B'_j}\Big(\sum_{k=0}^{+\infty}|f_{\nu,j,k}|^2\Big)e^{-2\varphi}d\lambda
\le C
\int_{B'_j}\Big(\sum_{k=0}^{k_0}|f_{\nu,j,k}|^2\Big)e^{-2\varphi}d\lambda
= C(k_0+1).
$$
Property (11.12) is proved.\medskip

\noindent{\it Step 3. Subadditivity of the approximating
sequence $\psi_{\varepsilon,\nu}$.}

\noindent
We want to compare $\psi_{\varepsilon,\nu_1+\nu_2}$ and
$\psi_{\varepsilon,\nu_1}$, $\psi_{\varepsilon,\nu_2}$
for every pair of indices~$\nu_1$, $\nu_2$, first when the functions are
associated with the same covering $X=\bigcup B_j$. Consider a function
$f\in\cH_{\nu_1+\nu_2,j}$ with
$$
\int_{B_j}|f(z)|^2e^{-2(\nu_1+\nu_2)\varphi_j(z)}d\lambda(z)\le 1,\qquad
\varphi_j(z)=\varphi(z)+c_j|z^{(j)}|^2.
$$
We may view $f$ as a function $\hat f(z,z)$ defined on the diagonal
$\Delta$ of $B_j\times B_j$. Consider the Hilbert space of holomorphic
functions $u$ on $B_j\times B_j$ such that
$$
\int_{B_j\times B_j}|u(z,w)|^2
e^{-2\nu_1\varphi_j(z)-2\nu_2\varphi_j(w)}d\lambda(z)d\lambda(w)<+\infty.
$$
By the Ohsawa-Takegoshi $L^2$ extension theorem [OT87], there exists a
function $\widetilde f(z,w)$ on $B_j\times B_j$ such that $\widetilde f(z,z)=f(z)$ and
$$
\eqalign{
\int_{B_j\times B_j}|\widetilde f(z,w)|^2
& e^{-2\nu_1\varphi_j(z)-2\nu_2\varphi_j(w)}d\lambda(z)d\lambda(w)\cr
&\qquad{}
\le C_7\int_{B_j}|f(z)|^2e^{-2(\nu_1+\nu_2)\varphi_j(z)}d\lambda(z)=C_7,\cr}
$$
where the constant $C_7$ only depends on the dimension $n$
(it is actually independent of the radius $r_j$ if say $0<r_j\le 1$).
As the Hilbert
space under consideration on $B_j\times B_j$ is the completed tensor
product $\cH_{\nu_1,j}\mathop{\widehat\otimes}\cH_{\nu_2,j}$, we infer that
$$
\widetilde f(z,w)=\sum_{k_1,k_2}c_{k_1,k_2}f_{\nu_1,j,k_1}(z)
f_{\nu_2,j,k_2}(w)
$$
with $\sum_{k_1,k_2}|c_{k_1,k_2}|^2\le C_7$. By restricting to the
diagonal, we obtain
$$
|f(z)|^2=|\widetilde f(z,z)|^2\le\sum_{k_1,k_2}|c_{k_1,k_2}|^2
\sum_{k_1}|f_{\nu_1,j,k_1}(z)|^2\sum_{k_2}|f_{\nu_2,j,k_2}(z)|^2.
$$
From (11.5) and (11.6), we get
$$
\psi_{\varepsilon,\nu_1+\nu_2,j}\le {\log C_7\over\nu_1+\nu_2}+
{\nu_1\over\nu_1+\nu_2}\psi_{\varepsilon,\nu_1,j}+{\nu_2\over\nu_1+\nu_2}
\psi_{\varepsilon,\nu_2,j},
$$
in particular
$$
\psi_{\varepsilon,2^\nu,j}\le \psi_{\varepsilon,2^{\nu-1},j}+{C_8\over 2^\nu},
$$
and we see that $\psi_{\varepsilon,2^\nu}+C_8 2^{-\nu}$ is a decreasing
sequence. By Step~2 and Lebesgue's monotone convergence theorem,
we infer that for every $\varepsilon,\delta>0$ and $a\le a_0\ll 0$ fixed,
the integral
$$
I_{\varepsilon,\delta,\nu}=\int_X
\Big(e^{-2\varphi}-e^{-2\max(\varphi,(1+\delta)(\psi_{2^\nu,\varepsilon}+a
))}\Big)dV_\omega
$$
converges to $0$ as $\nu$ tends to~$+\infty$ (take
$\ell={1\over\delta}+1$ and $2^\nu>\ell$ and $a_0$ such that
$\delta\sup_X\varphi+a_0\le 0$; we do not have monotonicity strictly speaking
but need only replace $a$ by $a+C_8 2^{-\nu}$ to get it, thereby slightly
enlarging the integral).
\medskip

\noindent
{\it Step 4. Selection of a suitable upper envelope.}

\noindent For the simplicity of notation, we assume here that
$\sup_X\varphi=0$ (possibly after subtracting a constant), hence we can take
$a_0=0$ in the above. We may even further assume that all our
functions $\psi_{\varepsilon,\nu}$ are nonpositive. By Step~3,
for each $\delta=\varepsilon=2^{-k}$, we can select an index
$\nu=p(k)$ such that
$$
I_{2^{-k},2^{-k},p(k)}=
\int_X\Big(e^{-2\varphi}-e^{-2\max(\varphi,(1+2^{-k})\psi_{2^{-k},2^{p(k)}})}
\Big)dV_\omega\le 2^{-k}
\leqno(11.13)
$$
By construction, we have an estimate $i\ddbar\psi_{2^{-k},2^{p(k)}}\ge
\gamma-C_52^{-k}\omega$, and the functions $\psi_{2^{-k},2^{p(k)}}$ are
quasi-psh with logarithmic poles. Our estimates (especially (11.9))
imply that $\lim_{k\to+\infty}\psi_{2^{-k},2^{p(k)}}(z)=\varphi(z)$
as soon as $2^{-p(k)}\log\big(1/\inf_j r_j(k)\big)\to 0$ (notice that
the $r_j$'s now depend on $\varepsilon=2^{-k}$). We set
$$
\varphi_\nu(z)=\sup_{k\ge\nu}(1+2^{-k})\psi_{2^{-k},2^{p(k)}}(z).
\leqno(11.14)
$$
By construction $(\varphi_\nu)$ is a decreasing sequence and satisfies the
estimates
$$\varphi_\nu\ge
\max\big(\varphi,(1+2^{-\nu})\psi_{2^{-\nu},2^{p(\nu)}}\big),
\qquad
i\ddbar\varphi_\nu\ge \gamma-C_52^{-\nu}\omega.$$
Inequality (11.13) implies that
$$
\int_X(e^{-2\varphi}-e^{-2\varphi_\nu})dV_\omega\le
\sum_{k=\nu}^{+\infty}2^{-k}=2^{1-\nu}.
$$
Finally, if $Z_\nu$ is the set of poles of $\psi_{2^{-\nu},2^{p(\nu)}}$, then
$Z_\nu\subset Z_{\nu+1}$ and $\varphi_\nu$ is continuous on~$X\ssm Z_\nu$.
The reason is that in a neighborhood of every point
\hbox{$z_0\in X\ssm Z_\nu$},
the term \hbox{$(1+2^{-k})\psi_{2^{-k},2^{p(k)}}$} contributes to
$\varphi_\nu$ only when it is larger than
$(1+2^{-\nu})\psi_{2^{-\nu},2^{p(\nu)}}$.
Hence, by the almost-monotonicity, the relevant terms of the sup
in (11.14) are squeezed
between $(1+2^{-\nu})\psi_{2^{-\nu},2^{p(\nu)}}$ and
$(1+2^{-k})(\psi_{2^{-\nu},2^{p(\nu)}}+C_82^{-\nu})$, and therefore
there is uniform convergence in a neighborhood of~$z_0$.
Finally, condition (c) implies that
$$
\int_U|f|^2(e^{-2\varphi}-e^{-2\varphi_\nu})dV_\omega<+\infty
$$
for every germ of holomorphic function $f\in\cO(U)$ at a point~$x\in X$.
Therefore both integrals $\int_U|f|^2e^{-2\varphi}dV_\omega$ and
$\int_U|f|^2e^{-2\varphi_\nu}dV_\omega$ are simultaneously convergent
or divergent, i.e.\ $\cI(\varphi)=\cI(\varphi_\nu)$. Theorem 11.3 is
proved, except that $\varphi_\nu$ is possibly just continuous instead of
being smooth. This can be arranged by Richberg's regularization theorem
[Ri68], at the expense of an arbitrary small loss in the Hessian form.\qed
\endproof
%
\claim{(11.15) Remark} {\rm By a very slight variation of the proof, we
can strengthen condition (c) and obtain that for every~$t>0$
$$
\int_X(e^{-2t\varphi}-e^{-2t\varphi_\nu})dV_\omega
$$
is finite for $\nu$ large enough and converges to $0$ as $\nu\to+\infty$.
This implies that the sequence of multiplier ideals
$\cI(t\varphi_\nu)$ is a stationary decreasing
sequence, with \hbox{$\cI(t\varphi_\nu)=\cI(t\varphi)$} for $\nu$ large.}
\endclaim
%
\subsection*{\csect.C. A Bochner type inequality}
Let $(L,h)$ be a smooth hermitian line bundle on a (non necessarily
compact) K\"ahler manifold $(Y,\omega)$.  We denote by
$|~~|=|~~|_{\omega,h}$ the pointwise hermitian norm on
\hbox{$\Lambda^{p,q}T^\star_Y\otimes L$} associated with $\omega$ and $h$,
and by $\Vert~~\Vert=\Vert~~\Vert_{\omega,h}$ the global $L^2$ norm
$$
\Vert u\Vert^2 = \int_Y |u|^2 dV_\omega \qquad
\hbox{where}\quad  dV_\omega={\omega^n\over n!}
$$
We consider the $\dbar$ operator acting on $(p,q)$-forms with values in
$L$, its adjoint $\dbar_h^\star$ with respect to $h$ and the complex
Laplace-Beltrami operator $\Delta''_h=\dbar\dbar_h^\star+\dbar_h^\star\dbar$.
Let $v$ be a smooth $(n-q,0)$-form with compact support
in~$Y$. Then $u=\omega^q\wedge v$ satisfies
$$
\Vert\dbar u\Vert^2+\Vert\dbar^\star_h u\Vert^2=
\Vert\dbar v\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)|u_{IJ}|^2
\leqno(11.16)
$$
where $\lambda_1\le\ldots\le\lambda_n$ are the curvature
eigenvalues of $\Theta_{L,h}$ expressed in an orthonormal frame
$(\partial/\partial z_1,\ldots,\partial/\partial z_n)$ (at some fixed
point $x_0\in Y$), in such a way that
$$
\omega_{x_0}=i\sum_{1\le j\le n}dz_j\wedge d\overline z_j,\qquad
(\Theta_{L,h})_{x_0}=i\ddbar\varphi_{x_0}=
i\sum_{1\le j\le n} \lambda_jdz_j\wedge d\overline z_j.
$$
The proof of (11.16) proceeds by checking that
$$
(\dbar^\star_\varphi\,\dbar+\dbar\,\dbar^\star_\varphi)(v\wedge\omega^q)-
(\dbar^\star_\varphi\,\dbar v)\wedge\omega^q=
q\,i\ddbar\varphi\wedge\omega^{q-1}\wedge v,
\leqno(11.17)
$$
taking the inner product with $u=\omega^q\wedge v$ and integrating by
parts in the left hand side. In order to check (11.16), we use the
identity $\dbar^\star_\varphi=e^\varphi\dbar^\star(e^{-\varphi}\bu)
=\dbar^\star+\nabla^{0,1}\varphi\ort\bu$~. Let us work in
a local trivialization of $L$ such that $\varphi(x_0)=0$ and
$\nabla\varphi(x_0)=0$. At~$x_0$ we then find
$$
\eqalign{
(\dbar^\star_\varphi\,\dbar+\dbar\,\dbar^\star_\varphi)(\omega^q\wedge v)&{}-
\omega^q\wedge(\dbar^\star_\varphi\,\dbar v)={}\cr
&\big[(\dbar^\star\,\dbar+\dbar\,\dbar^\star)(\omega^q\wedge v)-
\omega^q\wedge(\dbar^\star\,\dbar v)\big]+\dbar(\nabla^{0,1}\varphi\ort(
\omega^q\wedge v)).\cr}
$$
However, the term $[\,\ldots\,]$ corresponds to the case of a trivial vector
bundle and it is well known in that case that $[\Delta'',
\omega^q\wedge\bu]=0$, hence $[\,\ldots\,]=0$. On the other hand
$$\nabla^{0,1}\varphi\ort(\omega^q\wedge v)=
q(\nabla^{0,1}\varphi\ort\omega)\wedge \omega^{q-1}\wedge v=
-q\,i\partial\varphi\wedge \omega^{q-1}\wedge v,$$
and so
$$
(\dbar^\star_\varphi\,\dbar+\dbar\,\dbar^\star_\varphi)(\omega^q\wedge v)-
\omega^q\wedge(\dbar^\star_\varphi\,\dbar v)=q\,i\ddbar\varphi
\wedge \omega^{q-1}\wedge v.$$
Our formula is thus proved when $v$ is smooth and compactly
supported. In general, we have:

\claim{(11.18) Proposition} Let $(Y,\omega)$ be a {\rm complete} K\"ahler
manifold and $(L,h)$ a smooth hermitian line bundle such that the
curvature possesses a uniform lower bound $\Theta_{L,h}\ge -C\omega$.
For every measurable $(n-q,0)$-form $v$ with $L^2$ coefficients and
values in $L$ such that $u=\omega^q\wedge v$ has differentials
$\dbar u$, $\dbar^\star u$ also in $L^2$, we have
$$
\Vert\dbar u\Vert^2+\Vert\dbar^\star_h u\Vert^2=
\Vert\dbar v\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)|u_{IJ}|^2
$$
$($here, all differentials are computed in the sense of distributions$)$.
\endclaim
%
\begproof{Proof.} Since $(Y,\omega)$ is assumed to be complete,
there exists a sequence of smooth forms $v_\nu$ with compact support in $Y$
(obtained by truncating $v$ and taking the convolution with a regularizing
kernel) such that $v_\nu\to v$ in $L^2$ and such that
$u_\nu=\omega^q\wedge v_\nu$ satisfies $u_\nu\to u$, $\dbar u_\nu\to\dbar u$,
$\dbar^\star u_\nu\to\dbar^\star u$ in~$L^2$. By the curvature assumption,
the final integral in the right hand side of (11.16) must be under control
(i.e.\ the integrand becomes nonnegative if we add a term $C\Vert u\Vert^2$
on both sides, $C\gg 0$). We thus get the equality by passing to the limit
and using Lebesgue's monotone convergence theorem.\qed
\endproof
%
\subsection*{\csect.D. Proof of Theorem (11.1)}
To fix the ideas, we first indicate the proof in the much simpler case
when $(L,h)$ is hermitian semi-positive, and then treat the general
case.\medskip

\noindent
{\bf (11.19) Special case.} $(L,h)$ is (smooth) hermitian semi-positive.
\vskip3pt
\noindent
Let $\{\beta\}\in H^q(X,\Omega^n_X\otimes L)$ be an arbitrary cohomology class.
By standard $L^2$ Hodge theory, $\{\beta\}$ can be represented by a smooth
harmonic $(0,q)$-form $\beta$ with values in $\Omega^n_X\otimes L$. We can
also view $\beta$ as a $(n,q)$-form with values in $L$. The pointwise
Lefschetz isomorphism produces a unique $(n-q,0)$-form $\alpha$ such
that $\beta=\omega^q\wedge\alpha$. Proposition 11.18 then yields
$$
\Vert\dbar\alpha\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)
|\alpha_{IJ}|^2=\Vert\dbar\beta\Vert^2+\Vert\dbar^\star_h \beta\Vert^2=0,
$$
and the curvature eigenvalues $\lambda_j$ are nonnegative by our assumption.
Hence $\dbar\alpha=0$ and $\{\alpha\}\in H^0(X,\Omega^{n-q}_X\otimes L)$
is mapped to $\{\beta\}$ by $\Phi^q_{\omega,h}=\omega^q\wedge\bu~$.
\medskip

\noindent
{\bf (11.20) General case.}
\vskip3pt
\noindent
There are several difficulties. The first difficulty is that the
metric $h$ is no longer smooth and we cannot directly represent
cohomology classes by harmonic forms. We circumvent this problem by
smoothing the metric on an (analytic) Zariski open subset and by 
avoiding the remaining poles on the complement. However, some careful 
estimates have to be made in order to take the error terms into account.

Fix $\varepsilon=\varepsilon_\nu$ and let $h_\varepsilon=h_{\varepsilon_\nu}$
be an approximation of~$h$, such that $h_\varepsilon$ is smooth on
$X\ssm Z_\varepsilon$ ($Z_\varepsilon$ being an analytic subset of $X$),
$\Theta_{L,h_\varepsilon}\ge -\varepsilon\omega$,
$h_\varepsilon\le h$ and $\cI(h_\varepsilon)=\cI(h)$. This is possible
by Theorem~11.3. Now, we can find a family
$$
\omega_{\varepsilon,\delta}=\omega+\delta(i\ddbar \psi_\varepsilon+\omega),
\qquad \delta>0
$$
of {\it complete K\"ahler} metrics on $X\ssm Z_\varepsilon$, where
$\psi_\varepsilon$ is a quasi-psh function on $X$ with
$\psi_\varepsilon=-\infty$ on $Z_\varepsilon$,
$\psi_\varepsilon$ on $X\ssm Z_\varepsilon$ and $i\ddbar
\psi_\varepsilon+\omega\ge 0$ (see e.g.\ [Dem82b], Th\'eor\`eme 1.5).
By construction, $\omega_{\varepsilon,\delta}\ge\omega$ and
$\lim_{\delta\to 0}\omega_{\varepsilon,\delta}=\omega$.
We look at the $L^2$ Dolbeault complex $K^\bu_{\varepsilon,\delta}$
of $(n,\bu)$-forms on $X\ssm Z_\varepsilon$, where the $L^2$ norms are
induced by $\omega_{\varepsilon,\delta}$ on differential forms and by
$h_\varepsilon$ on elements in~$L$. Specifically
$$
K^q_{\varepsilon,\delta}=\Big\{u{:}X\ssm Z_\varepsilon{\to}\Lambda^{n,q}
T^\star_X\otimes L;\int_{X\ssm Z_\varepsilon}\kern-15pt
(|u|^2_{\Lambda^{n,q}\omega_{\varepsilon,\delta}
\otimes h_\varepsilon}+|\dbar u|^2_{\Lambda^{n,q+1}\omega_{\varepsilon,\delta}
\otimes h_\varepsilon})dV_{\omega_{\varepsilon,\delta}}<\infty\Big\}.
$$
Let $\cK^q_{\varepsilon,\delta}$ be the corresponding sheaf of germs
of locally $L^2$ sections on $X$ (the local $L^2$ condition
should hold on $X$, not only on $X\ssm Z_\varepsilon\,$!). Then,
for all $\varepsilon>0$ and $\delta\ge 0$,
$(\cK^q_{\varepsilon,\delta},\dbar)$ is a resolution of the sheaf
$\Omega^n_X\otimes L\otimes\cI(h_\varepsilon)=
\Omega^n_X\otimes L\otimes\cI(h)$. This is
because $L^2$ estimates hold locally on small Stein open sets, and the
$L^2$ condition on $X\ssm Z_\varepsilon$ forces holomorphic sections
to extend across~$Z_\varepsilon$ ([Dem82b], Lemme 6.9).

Let $\{\beta\}\in H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$ be a
cohomology class represented by a smooth form with values in
$\Omega^n_X\otimes L\otimes\cI(h)$ (one can use a \v Cech cocycle
and convert it to an element in the $C^\infty$ Dolbeault complex by
means of a partition of unity, thanks to the usual De Rham-Weil
isomorphism). Then
$$
\Vert\beta\Vert_{\varepsilon,\delta}^2\le \Vert\beta\Vert^2=
\int_X|\beta|^2_{\Lambda^{n,q}\omega\otimes h}dV_\omega<+\infty.
$$
The reason is that $|\beta|^2_{\Lambda^{n,q}\omega\otimes h}dV_\omega$
decreases as $\omega$ increases. This is just an easy calculation,
shown by comparing two metrics $\omega$, $\omega'$ which are
expressed in diagonal form in suitable coordinates; the norm
$|\beta|^2_{\Lambda^{n,q}\omega\otimes h}$ turns out to decrease
faster than the volume $dV_\omega$ increases; see e.g.\ [Dem82b], Lemme 3.2;
a special case is $q=0$, then $|\beta|^2_{\Lambda^{n,q}\omega\otimes h}
dV_\omega=i^{n^2}\beta\wedge\overline\beta$ with the identification
$L\otimes\overline L\simeq\bC$ given by the metric $h$, hence the
integrand is even independent of $\omega$ in that case. 

By the proof of the De Rham-Weil isomorphism, the map
$\alpha\mapsto\{\alpha\}$ from the cocycle space
$Z^q(\cK^\bu_{\varepsilon,\delta})$ equipped with its $L^2$ topology,
into $H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$ equipped with its
finite vector space topology, is continuous.  Also, Banach's open
mapping theorem implies that the coboundary space
$B^q(\cK^\bu_{\varepsilon,\delta})$ is closed in
$Z^q(\cK^\bu_{\varepsilon,\delta})$. This is true for all $\delta\ge
0$ (the limit case $\delta=0$ yields the strongest $L^2$ topology in
bidegree $(n,q)$). Now, $\beta$ is a $\dbar$-closed form in the Hilbert space
defined by $\omega_{\varepsilon,\delta}$ on $X\ssm Z_\varepsilon$, so there
is a $\omega_{\varepsilon,\delta}$-harmonic form
$u_{\varepsilon,\delta}$ in the same cohomology class
as $\beta$, such that
$$
\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}\le
\Vert\beta\Vert_{\varepsilon,\delta}.
$$
\claim{(11.21) Remark} {\rm
The existence of a harmonic representative holds true only for $\delta>0$, 
because we need to have a complete K\"ahler metric on $X\ssm Z_\varepsilon$.
The trick of employing $\omega_{\varepsilon,\delta}$ instead of a fixed
metric $\omega$, however, is not needed when $Z_\varepsilon$ is (or can 
be taken to be) empty. This is the case if $(L,h)$ is such that 
$\cI(h)=\cO_X$ and $L$ is nef. Indeed, in that case, from the very 
definition of nefness, it is easy to prove that we can take the 
$\varphi_\nu$'s to be everywhere smooth in Theorem~11.3. However,
we will see in \S~11.E that multiplier ideal sheaves are needed even
in case $L$ is nef, when $\cI(h)\ne\cO_X$.}
\endclaim
%
Let $v_{\varepsilon,\delta}$ be the unique $(n-q,0)$-form such that
$u_{\varepsilon,\delta}=v_{\varepsilon,\delta}\wedge
\omega_{\varepsilon,\delta}^q$ ($v_{\varepsilon,\delta}$ exists by the
pointwise Lefschetz isomorphism). Then
$$
\Vert v_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}=
\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}\le
\Vert\beta\Vert_{\varepsilon,\delta}\le\Vert\beta\Vert.
$$
As $\sum_{j\in J}\lambda_j\ge -q\varepsilon$ by the assumption on
$\Theta_{L,h_\varepsilon}$, the Bochner formula yields
$$
\Vert\dbar v_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}^2\le
q\varepsilon\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}^2
\le q\varepsilon\Vert\beta\Vert^2.
$$
These uniform bounds imply that there are subsequences $u_{\varepsilon,
\delta_\nu}$ and $v_{\varepsilon,\delta_\nu}$ with $\delta_\nu\to 0$,
possessing weak-$L^2$ limits $u_\varepsilon=
\lim_{\nu\to+\infty}u_{\varepsilon,\delta_\nu}$
and $v_\varepsilon=\lim_{\nu\to+\infty}v_{\varepsilon,\delta_\nu}$.
The limit $u_\varepsilon=
\lim_{\nu\to+\infty}u_{\varepsilon,\delta_\nu}$ is with respect to
$L^2(\omega)=L^2(\omega_{\varepsilon,0})$. To check this, notice that
in bidegree $(n-q,0)$, the space $L^2(\omega)$ has the weakest topology
of all spaces $L^2(\omega_{\varepsilon,\delta})$; indeed, an easy calculation
as in ([Dem82b], Lemme 3.2) yields
$$
|f|^2_{\Lambda^{n-q,0}\omega\otimes h}dV_\omega\le
|f|^2_{\Lambda^{n-q,0}\omega_{\varepsilon,\delta}\otimes h}
dV_{\omega_{\varepsilon,\delta}}\qquad
\hbox{if $f$ is of type $(n-q,0)$}.
$$
On the other hand, the limit
$v_\varepsilon=\lim_{\nu\to+\infty}v_{\varepsilon,\delta_\nu}$
takes place in all spaces $L^2(\omega_{\varepsilon,\delta})$, $\delta>0$,
since the topology gets stronger and stronger as $\delta\downarrow 0$
[$\,$possibly not in $L^2(\omega)$, though, because in bidegree $(n,q)$
the topology of $L^2(\omega)$ might be strictly stronger than that 
of all spaces $L^2(\omega_{\varepsilon,\delta})\,$].
The above estimates yield
$$
\eqalign{
&\Vert v_\varepsilon\Vert^2_{\varepsilon,0}=
\int_X|v_\varepsilon|^2_{\Lambda^{n-q,0}\omega\otimes h_\varepsilon}
dV_\omega\le\Vert\beta\Vert^2,\cr
&\Vert \dbar v_\varepsilon\Vert^2_{\varepsilon,0}\le q\varepsilon
\Vert\beta\Vert^2_{\varepsilon,0},\cr
\noalign{\vskip3pt}
&u_\varepsilon=\omega^q\wedge v_\varepsilon\equiv\beta
\qquad\hbox{in}~~H^q(X,\Omega^n_X\otimes L\otimes\cI(h_\varepsilon)).\cr}
$$
Again, by arguing in a given Hilbert space $L^2(h_{\varepsilon_0})$,
we find $L^2$ convergent subsequences $u_\varepsilon\to u$,
$v_\varepsilon\to v$ as $\varepsilon\to 0$, and in this way get
$\dbar v=0$ and
$$
\eqalignno{
&\Vert v\Vert^2\le \Vert \beta\Vert^2,\cr
&u=\omega^q\wedge v\equiv \beta
\qquad\hbox{in}~~H^q(X,\Omega^n_X\otimes L\otimes\cI(h)).\cr}
$$
Theorem 11.1 is proved. Notice that the equisingularity property
$\cI(h_\varepsilon)=\cI(h)$ is crucial in the above proof,
otherwise we could not infer that $u\equiv \beta$ from the fact
that $u_\varepsilon\equiv \beta$. This is true only because
all cohomology classes $\{u_\varepsilon\}$ lie in the same fixed 
cohomology group $H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$, whose
topology is induced by the topology of $L^2(\omega)$ on $\dbar$-closed
forms (e.g.\ through the De Rham-Weil isomorphism).\qed

\subsection*{\csect.E. A counterexample}
In view of Corollary 11.2, one might wonder whether the morphism
$\Phi^q_\omega$ would not still be surjective when $L$ is a nef
vector bundle. We will show that this is unfortunately not so,
even in the case of algebraic surfaces.

Let $B$ be an elliptic curve and let $V$ be the rank $2$ vector bundle 
over $B$ which is defined as the (unique) non split
extension
$$0 \to \cO_B \to V \to \cO_B \to 0.$$
In particular, the bundle $V$ is numerically flat, i.e.\ $c_1(V)=0$,
$c_2(V)=0$. We consider the ruled surface $X=\bP(V)$. On that surface
there is a unique section $C=\bP(\cO_B) \subset X$ with $C^2=0$ and
$$\cO_X (C)= \cO_{\bP (V)} (1)$$
is a nef line bundle. It is easy to see that
$$h^0 (X,\cO_{\bP (V)} (m))= h^0 (B,S^m V)=1$$
for all $m \in \bN$ (otherwise we would have $mC=aC+M$ where
$aC$ is the fixed part of the linear system $|mC|$ and $M\ne 0$ the
moving part, thus $M^2\ge 0$ and $C\cdot M>0$, contradiction).
We claim that
$$h^0 (X,\Omega_X^1 (kC))=2$$
for all $k \ge 2.$ This follows by tensoring the exact sequence
$$
0 \to \Omega^1_{X \vert C} \to \Omega_X^1 \to \pi^* \Omega^1_{C}
\simeq \cO_{C} \to 0
$$
by $\cO_X (kC)$ and observing that
$$\Omega^1_{X \vert C}= K_X=\cO_X (-2C).$$
From this, we get
$$
0 \to H^0(X,\cO_X((k-2)C)) \to H^0(X,\Omega_X^1\cO(kC))
\to H^0(X,\cO_X(kC))
$$
where $h^0 (X,\cO_X ((k-2)C))=h^0 (X,\cO_X (kC))=1$ for
all $k\ge 2$. Moreover, the last arrow is surjective because we
can multiply a section of $H^0(X,\cO_X(kC))$ by a nonzero section
in $H^0(X,\pi^*\Omega_B^1)$ to get a preimage. Our claim follows.
We now consider the diagram
$$
\plainmatrix{
H^0 (X,\Omega_X^1 (2C)) & \buildo\vlra^{\textstyle \wedge \omega} &
H^1 (X,K_X (2C)) \cr
\noalign{\vskip4pt}
\simeq \Big\downarrow & & \Big\downarrow \varphi \cr
H^0 (X,\Omega_X^1 (3C)) &
\build\vlra^{\textstyle\wedge \omega}_{\textstyle\psi} & H^1(X,K_X (3C)).\cr
}
$$
Since $K_X (2C) \simeq \cO_X$ and $K_X (3C) \simeq \cO_X(C),$ the
cohomology sequence of
$$
0 \to K_X (2C) \to K_X (3C) \to K_X (3C) \vert C \simeq \cO_{C}
\to 0
$$
immediately implies $\varphi=0$ (notice that $h^1 (X,K_X (2C))=h^1 (X,K_X
(3C))=1,$ since $h^1 (B,\cO_B)=h^1 (B,V)=1)$, and
$h^2(X,K_X(2C))=h^2(B,\cO_B)=0$). Therefore the diagram
implies $\psi=0$, and we get:

\claim{(11.22) Proposition} $L=\cO_{\bP(V)}(3)$ is a counterample to
$(11.2)$ in the nef case.\qed
\endclaim
%
\noindent
By Corollary (11.2), we infer that $\cO_X(3)$ cannot be hermitian
semi-positive and we thus again obtain -- by a quite different method --
the result of [DPS94], example~1.7.

\claim{(11.23) Corollary} Let $B$ be an elliptic curve, $V$ the vector
bundle given by the unique non-split extension
$$0 \to \cO_B \to V\to \cO_B \to 0.$$
Let $X=\bP(V)$. Then $L=\cO_X(1)$ is nef but not hermitian
semi-positive $($nor does any multiple, e.g.\ the
anticanonical line bundle $-K_X=\cO_X(-2)$ is nef but not
semi-positive$)$.
\endclaim
%
\section{Invariance of plurigenera of projective varieties}

The goal of this section is to give a proof of the following 
fundamental result on the invariance of plurigenera, which has been
proved by Y.T.~Siu [Siu98] in the case of varieties of general type
(in which case the proof has been translated in a purely algebraic 
form by Y.~Kawamata [Kaw99]), and by [Siu00] in general. Let us 
recall that $X$ is said to be of general type if $\kappa(K_X)=n=\dim X$.

\claim{(12.1) Theorem (Siu)} Let $X\to S$ be a proper holomorphic
map defining a family of smooth
projective of varieties of general type on an irreducible base~$S$.
Then the plurigenus $p_m(X_t)=h^0(X_t,mK_{X_t})$ of fibers is
independent of~$t$ for all $m\ge 0$.
\endclaim
%
The proof somehow involves taking ``limits'' of divisors as $m\to+\infty$,
and therefore transcendental methods are a strong contender in this circle
of ideas, because currents provide a natural compactification of the
space of divisors. Quite recently, M.~P\v{a}un
obtained a very short and elegant proof of (12.1) based merely on 
the Ohwawa-Takegoshi extension theorem, and we are going to sketch
his arguments below (see also M.~P\v{a}un [Pau07], B.~Claudon [Cla07] and
S.\ Takayama~[Taka07]). In fact, following P\u{a}un, one can prove 
more general results valid for cohomology with twisted coefficients.
Remarkably enough, no algebraic proof of these results are known 
at this point, in the case of varieties of nonnegative Kodaira dimension
which are not of general type.

Notice that by connecting any two points of $S$ by a chain of analytic 
disks, it is enough to consider the case where $S=\Delta$ is a disk.

\claim{(12.2) Theorem {\rm (further generalized version of P\u{a}un's theorem)}}
Let $\pi: \cX \to \Delta$ be a projective family over the unit disk, and 
let $(L_j,h_j)_{0\le j\le m-1}$ be $($singular$)$ hermitian 
line bundles with semi-positive curvature currents $i\Theta_{L_j,h_j}\ge 0$
on~$\cX$. Assume that
{\vskip3pt
\plainitem{\rm (i)} the restriction of $h_j$ to the central fiber $X_0$ is well
defined $($i.e.\ not identically~$+\infty)$.
\vskip3pt
\plainitem{\rm (ii)} the multiplier ideal sheaf 
$\cI(h_{j|X_0})$ is trivial for $1\le j\le m-1$.
\vskip3pt}
\noindent
Then any section $\sigma$ of 
$\cO(mK_{\cX}+\sum L_j)_{|X_0}\otimes\cI(h_{0|X_0})$ over the central 
fiber $X_0$ extends to $\cX$.\vskip0pt
\endclaim
%
The invariance of plurigenera is just the case when all line bundles
$L_j$ and their metrics $h_j$ are trivial. Since the dimension $t\mapsto 
h^0(X_t,mK_{X_t})$ is always upper semicontinuous and since (12.2)
implies the lower semicontinuity, we conclude that the dimension
is constant along analytic disks (hence along any irreducible base $S$,
by joining any two points through a chain of analytic disks).

In order to prove (12.2), we first state the technical version of the 
Ohsawa-Takegoshi $L^2$
extension theo\-rem needed for the proof, which is a special case
of the Ohsawa-Takegoshi Theorem -- the reader is
invited to check that the statement indeed follows from~(8.6).

\claim{(12.3) Lemma} Let $\pi:\cX\to\Delta$ be as before and let $(L,h)$ be
a $($singular$)$ hermitian 
line bundle with semi-positive curvature current $i\Theta_{L,h}\ge 0$ on~$\cX$.
Let $\omega$ be a global K\"ahler metric on $\cX$, and $dV_\cX$, 
$dV_{X_0}$ the respective induced volume elements on $X_0$ 
and $\cX$. Assume that $h_{X_0}$ is well defined. Then any holomorphic 
section $u$ of 
$\cO(K_\cX+L)\otimes\cI(h_{|X_0})$ extends into a section $\widetilde u$ over $\cX$
satisfying an $L^2$ estimate
$$
\int_{\cX}\Vert \widetilde u\Vert^2_{\omega\otimes h}dV_\cX\le C_0
\int_{X_0}\Vert u\Vert^2_{\omega\otimes h}dV_{X_0},
$$
where $C_0\ge 0$ is some universal constant $($independent of $\cX$, $L$, 
$\ldots\,)$.
\endclaim


\begproof{Proof of $(12.2)$.} We write $h_j=e^{-\varphi_j}$ in terms of
local plurisubharmonic weights. Fix an auxiliary line bundle $A$
(which will later be taken to be sufficiently ample), and define
inductively a sequence of line bundles $F_p$ by putting $F_0=A$ and
$$
F_p=F_{p-1}+K_\cX+L_r\qquad\hbox{if $p=mq+r$,~~$0\le r\le m-1$}.
$$
By construction we have $F_{p+m}=F_p+mK_\cX+\sum_jL_j$ and
$$
F_0=A,~F_1=A+K_\cX+L_1,\,\ldots\,,
F_p=A+pK_\cX+L_1+\ldots+L_p,~1\le p\le m-1.
$$
The game is to construct inductively families of sections, say
$(\widetilde u_j^{(p)})_{j=1\ldots N_p}$, of $F_p$ over $\cX$, together
with ad hoc $L^2$ estimates, in such a way that 
\vskip3pt
{\plainitemindent=7mm
\plainitem{\rm (a)} for $p=0,\ldots,m-1$, $F_p$ is generated by its sections
$(\widetilde u_j^{(p)})_{j=1\ldots N_p}\;$;
\vskip3pt
\plainitem{\rm (b)} we have the $m$-periodicity relations
$N_{p+m}=N_p$ and $\widetilde u_j^{(p)}$ is an extension of 
$u_j^{(p)}:=\sigma^qu_j^{(r)}$ over $\cX$ for $p=mq+r$, 
where $u_j^{(r)}:=\widetilde u^{(r)}_{j|X_0}$, $0\le r\le m-1$.
\vskip0pt}

\vskip3pt\noindent
Property (a) can certainly be achieved by taking $A$ ample enough so that
$F_0$, $\ldots\,$, $F_{m-1}$ are generated by their sections, and by choosing
the $\widetilde u_j^{(p)}$ appropriately for $p=0,\ldots,m-1$.
Now, by induction, we equip $F_{p-1}$ with the tautological metric 
$|\xi|^2/\sum|\widetilde u_j^{(p-1)}(x)|^2$, and $F_p-K_\cX=F_{p-1}+L_r$ 
with that metric multiplied by $h_r=e^{-\varphi_r}\;$; it is clear
that these metrics have semi-positive curvature currents
(the metric on $F_p$ itself if obtained by using a smooth K\"ahler metric
$\omega$ on $\cX$). In this setting,
we apply the Ohsawa-Takegoshi theorem to the line bundle
$F_{p-1}+L_r$ to extend $u_j^{(p)}$
into a section $\widetilde u_j^{(p)}$
over~$\cX$. By construction the pointwise norm of that section
in $F_{p|X_0}$ in a local trivialization of the bundles involved is the ratio
$$
{|u_j^{(p)}|^2\over\sum_\ell|u_\ell^{(p-1)}|^2}e^{-\varphi_r},
$$
up to some fixed smooth positive factor depending only on the metric induced 
by $\omega$ on~$K_\cX$. However, by the induction relations, we have
$$
{\sum_j|u_j^{(p)}|^2\over\sum_\ell|u_\ell^{(p-1)}|^2}e^{-\varphi_r}=
\plaincases{
\displaystyle
{\sum_j|u_j^{(r)}|^2\over\sum_\ell|u_\ell^{(r-1)}|^2}e^{-\varphi_r}
&\hbox{for $p=mq+r$, $0<r\le m-1$},\cr
\displaystyle
{\sum_j|u_j^{(0)}|^2\over\sum_\ell|u_\ell^{(m-1)}|^2}
|\sigma|^2e^{-\varphi_0}
&\hbox{for $p\equiv 0\mod m$}.\cr}
$$
Since the sections $(u_j^{(r)})$ generate their line bundle, the ratios
involved are positive functions without zeroes and poles, hence smooth and
bounded [possibly after shrinking the base disc $\Delta$, as is permitted].
On the other hand, assumption (ii) and the fact that $\sigma$ has coefficients
in the multiplier ideal sheaf $\cI(h_{0|X_0})$ tell us that 
$e^{-\varphi_r}$, $1\le r<m$ and $|\sigma|^2e^{-\varphi_0}$ are locally 
integrable on~$X_0$. It follows that there is a constant $C_1\ge 0$ such that
$$
\int_{X_0}
{\sum_j|u_j^{(p)}|^2\over\sum_\ell|u_\ell^{(p-1)}|^2}e^{-\varphi_r} 
dV_\omega \le C_1
$$
for all $p\ge 1$ [of course, the integral certainly involves finitely many
trivializations of the bundles involved, whereas the integrand expression is 
just local in each chart]. Inductively, the $L^2$ extension theorem produces
sections $\widetilde u_j^{(p)}$ of $F_p$ over $\cX$ such that
$$
\int_{\cX}
{\sum_j|\widetilde u_j^{(p)}|^2\over\sum_\ell|\widetilde u_\ell^{(p-1)}|^2}e^{-\varphi_r} 
dV_\omega \le C_2=C_0C_1.
$$
The next idea is to extract the limits of $p$-th roots of these sections
to get a singular hermitian metric on $mK_\cX+\sum L_j$.
As the functions $e^{-\varphi_r}$ are locally bounded below ($\varphi_r$
being psh), the H\"older inequality implies that
$$
\int_{\cX}\Big(\sum_j|\widetilde u_j^{(p)}|^2\Big)^{1/p}dV_\omega\le C_3.
$$
The mean value inequality for plurisubharmonic functions shows a fortiori
that the sequence of psh functions
${1\over p}\log\sum_j|\widetilde u_j^{(p)}|^2$ 
is locally uniformly bounded from above. These functions should be thought of 
as weights on the $\bQ$-line bundles 
$${1\over p}(A+q(mK_\cX+\sum L_j)+L_1+\ldots+L_r)~~
\hbox{converging to}~~K_\cX+{1\over m}\sum L_j~~~~\hbox{as $p\to+\infty$},
$$ 
and thus they are potentials 
of currents in a bounded subset of the K\"ahler cone. Moreover, 
the sections $\widetilde u_j^{(p)}$ extend $\sigma^qu_j^r$ on $X_0$, and so
we have in particular 
$$
\lim_{p\to+\infty}{1\over p}\log\sum_j|u_j^{(p)}|^2=
\lim_{p\to+\infty}{1\over p}
\log\big(|\sigma|^{2q}\sum_j|u_j^{(0)}|^2\big)=
{1\over m}\log|\sigma|^2\not\equiv-\infty
~~\hbox{on $X_0$.}
$$
Therefore, by well known facts of potential theory, the sequence
${1\over p}\log\sum_j|u_j^{(p)}|^2$ must have some subsequence which
converges in $L^1_\loc$ topology to the potential $\psi$
of a current in the first Chern class of
$K_\cX+{1\over m}\sum L_j$, in the form of an upper regularized limit
$$
\psi(z)=\limsup_{\zeta\to z}\lim_{\nu\to+\infty}
{1\over p_\nu}\log\sum_j|\widetilde u_j^{(p_\nu)}(\zeta)|^2,
$$
which is such that $\psi(z)\ge{1\over m}\log|\sigma|^2$ on $X_0$.
Hence $mK_\cX+\sum L_j$ possesses a hermitian metric $H=e^{-m\psi}$, and
we have by construction $\Vert\sigma\Vert_H\le 1$ and $\Theta_H\ge 0$.
In~order to conclude, we equip the bundle 
$$
G=(m-1)K_\cX+\sum L_j
$$
with the metric $\gamma=H^{1-1/m}\prod h_j^{1/m}$, and 
$mK_\cX+\sum L_j=K_\cX+G$ with the metric $\omega\otimes\gamma$.
Clearly $\gamma$ has a semi-positive curvature current on $\cX$ and in a 
local trivialization we have
$$
\Vert\sigma\Vert^2_{\omega\otimes\gamma}\le
C|\sigma|^2\exp\Big(-(m-1)\psi+{1\over m}\sum \varphi_j\Big)
\le C \Big(|\sigma|^2\prod e^{-\varphi_j}\Big)^{1/m}
$$
on $X_0$. Since $|\sigma|^2e^{-\varphi_0}$ and $e^{-\varphi_r}$, $r>0$ are
all locally integrable, we see that $\Vert\sigma\Vert^2_{\omega\otimes\gamma}$
is also locally integrable on $X_0$ by the H\"older inequality. A new (and 
final) application of the
$L^2$ extension theorem to the hermitian line bundle $(G,\gamma)$ implies 
that $\sigma$ can be 
extended to~$\cX$. Theorem (12.2) is proved.\qed
\endproof

\section{Positive cones in the (1,1) cohomology groups of 
compact K\"ahler manifolds}
\sectionrunning{Positive cones in (1,1) cohomology groups}

\subsection*{\csect.A. Nef, pseudo-effective and big cohomology classes}
We introduce again the important concepts of positivity for
cohomology classes of type~$(1,1)$ -- the only novelty is that $X$ is
an arbitrary compact K\"ahler manifold and that we do not
assume that our classes are integral or rational.

\claim{(13.1) Definition} Let $(X,\omega)$ be a compact K\"ahler manifold.
\vskip3pt
{\plainitemindent=7mm
\plainitem{\rm(i)} The K\"ahler cone is the set 
$\cK\subset H^{1,1}(X,\bR)$ of cohomology classes $\{\omega\}$
of K\"ahler forms. This is an {\rm open convex cone}.
\vskip3pt
\plainitem{\rm(ii)} The closure $\overline{\cK}$ of the K\"ahler cone consists
of classes $\{\alpha\}\in H^{1,1}(X,\bR)$ such that for every
$\varepsilon>0$ the sum $\{\alpha+\varepsilon\omega\}$
is K\"ahler, or equivalently, for every $\varepsilon>0$, there exists a 
smooth function $\varphi_\varepsilon$ on $X$ such that 
$\alpha+i\ddbar\varphi_\varepsilon\ge -\varepsilon\omega$.
We say that $\overline{\cK}$ is the cone of {\rm nef} $(1,1)$-classes.
\vskip3pt
\plainitem{\rm(iii)}The pseudo-effective cone is the set
$\cE\subset H^{1,1}(X,\bR)$ of cohomology classes $\{T\}$ of closed positive 
currents of type $(1,1)$. This is a {\rm closed convex cone}.
\vskip3pt
\plainitem{\rm(iv)}The interior $\cE^\circ$ of $\cE$ consists of classes
which still contain a closed positive current after one subtracts
$\varepsilon\{\omega\}$ for $\varepsilon>0$ small, in other words,
they are classes of closed $(1,1)$-currents $T$ such that
$T\ge \varepsilon\omega$. Such a current will be called a {\rm K\"ahler
current}, and we say that $\{T\}\in H^{1,1}(X,\bR)$ is a {\rm big 
$(1,1)$-class}.
\vskip0pt}
\endclaim
%
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\LabelTeX -19 29 $\cE$\ELTX
\LabelTeX 27 40 $\cK=\hbox{K\"ahler cone in $H^{1,1}(X,\bR)$ [open]}$\ELTX
\LabelTeX 27 30 $\overline{\cK}=\hbox{nef cone in $H^{1,1}(X,\bR)$
[closure of $\cK$]}$\ELTX
\LabelTeX 27 20 $\cE=\hbox{pseudo-effective cone in $H^{1,1}(X,\bR)$ 
[closed]}$\ELTX
\LabelTeX 27 10 $\cE^\circ=\hbox{big cone in $H^{1,1}(X,\bR)$ 
[interior of $\cE$]}$\ELTX
\EndFig
\vskip8mm

The openness of $\cK$ is clear by definition, and the closedness of $\cE$
is a consequence of the fact that bounded sets of currents are weakly
compact (as follows from the similar weak compacteness property 
for bounded sets of positive measures). It is then clear that 
$\overline\cK\subset\cE$.

In spite of the fact that cohomology groups can be defined either in terms
of forms or currents, it turns out that the cones $\overline\cK$ and
$\cE$ are in general different. To see this, it is enough to observe
that a K\"ahler class $\{\alpha\}$ satisfies $\int_Y\alpha^p>0$ for
every $p$-dimensional analytic set. On the other hand, if $X$ is the surface
obtained by blowing-up $\bP^2$ in one point, then the exceptional
divisopr $E\simeq\bP^1$ has a cohomology class $\{\alpha\}$ such that 
$\int_E\alpha= E^2=-1$, hence $\{\alpha\}\notin\overline\cK$, although
$\{\alpha\}=\{[E]\}\in\cE$.

In case $X$ is projective, it is interesting to consider also the
algebraic analogues of our ``transcendental cones'' $\cK$ and $\cE$,
which consist of suitable integral divisor classes. Since the cohomology
classes of such divisors live in $H^2(X,\bZ)$, we are led to introduce
the Neron-Severi lattice and the associated Neron-Severi space
$$
\eqalign{
\NS(X)&:= H^{1,1}(X,\bR)\cap \big(H^2(X,\bZ)/\{\hbox{torsion}\}\big),\cr
\NS_\bR(X)&:=\NS(X)\otimes_\bZ\bR.\cr
}
$$
All classes of real divisors $D=\sum c_jD_j$, $c_j\in\bR$, lie by definition 
in $\NS_\bR(X)$.
Notice that the integral lattice $H^2(X,\bZ)/\{\hbox{torsion}\}$ need not 
hit at all the subspace
$H^{1,1}(X,\bR)\subset H^2(X,\bR)$ in the Hodge decomposition,
hence in general the Picard number 
$$\rho(X)=\rank_\bZ \NS(X)=\dim_\bR \NS_\bR(X)$$
satisfies
$\rho(X)\le h^{1,1}=\dim_\bR H^{1,1}(X,\bR)$,
but the equality can be strict (actually, it is well known that a generic 
complex torus $X=\bC^n/\Lambda$ satisfies $\rho(X)=0$ and $h^{1,1}=n^2$). 
In order to deal with the case of algebraic varieties we introduce
$$\cK_{\NS} = \cK\cap \NS_\bR(X),\qquad \cE_{\NS}=\cE\cap \NS_{\bR}(X).$$ 

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\vskip8mm

A very important fact is that the ``Neron-Severi part'' of any of
the open or closed transcendental cones $\cK$, $\cE$, $\overline\cK$, $\cE^\circ$
is algebraic, i.e.\ can be characterized in simple algebraic terms.
In fact, the results of section 6.C can be reformulated as follows.

\claim{(13.2) Theorem} Let $X$ be a projective manifold. Then
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} $\cK_{\NS}$ is the open cone generated by classes of
{\rm ample} $($or very ample$)$ divisors $A$ $($Recall that a divisor
$A$ is said to be very ample if the linear system $H^0(X,\cO(A))$
provides an embedding of $X$ in projective space$)$.
\vskip3pt
\plainitem{\rm (ii)} The interior $\cE_{\NS}^\circ$ is the cone generated
by classes of {\rm big} divisors, namely divisors $D$ such that
$h^0(X,\cO(kD))\ge c\,k^{\dim X}$ for $k$ large.
\vskip3pt
\plainitem{\rm (iii)} $\cE_{\NS}$ is the closure of the cone generated by classes
of {\rm effective} divisors, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.
\vskip3pt
\plainitem{\rm (iv)} The closed cone $\overline\cK_{\NS}$ consists of the closure
of the cone generated by {\rm nef} divisors $D$ (or nef line bundles $L$), 
namely effective integral divisors $D$ such that \hbox{$D\cdot C\ge 0$} for
every curve $C$.\vskip0pt
}
\endclaim
%
Recall that (i) is just Kodaira's embedding theorem, and that the proof of
(ii) follows from the existence theorem
provided by $L^2$ estimates, since we are in a case where the curvature
is positive definite as a current. Properties (iii) and (iv) are obtained
by passing to the closure of the open cones. The terminology ``nef'', ``big'',
``pseudo-effective'' used for the full transcendental cones thus appears
to be a natural extrapolation of the algebraic situation.
\bigskip

\subsection*{\csect.B. Positive classes in intermediate $(p,p)$ bidegrees}
We describe here similar concepts for cohomology classes of type $(p,p)$,
although we will not be able to say much about these.
Recall that we have a Serre duality pairing
$$
H^{p,q}(X,\bC)\times H^{n-p,n-q}(X,\bC)\longrightarrow \bC,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta\in\bC.\leqno(13.3)
$$
In particular, if we restrict to real classes, this yields a
duality pairing
$$
H^{p,p}(X,\bR)\times H^{n-p,n-p}(X,\bR)\longrightarrow \bR,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta \in\bR.\leqno(13.4)
$$
Now, one can define $H^{p,p}_\SP(X,\bR)$ to be the closure of the
cone of classes of $d$-closed strongly positive smooth $(p,p)$-forms
(a $(p,p)$-form in $\Lambda^{p,p}T^*_X$ is by definition strongly positive 
if it is in the convex cone generated by decomposable $(p,p)$
forms $\ii u_1\wedge{\overline u}_1\wedge\ldots\wedge \ii u_p\wedge{\overline u}_p$
where the $u_j$ are $(1,0)$-forms). Clearly, 
$H^{1,1}_\SP(X,\bR)=\overline\cK$ and the cup product defines
a multilinear map
$$
\overline\cK\times \ldots\times \overline\cK\longrightarrow H^{p,p}_\SP(X,\bR)\leqno(13.5)
$$
on the $p$-fold product of the K\"ahler cone and its closure. We
also have $H^{p,p}_\SP(X,\bR)\subset H^{p,p}_{\ge 0}(X,\bR)$ where
$H^{p,p}_{\ge 0}(X,\bR)$ is the cone of classes of $d$-closed
weakly positive currents of type $(p,p)$, and the Serre duality pairing
induces a positive intersection product
$$
H^{p,p}_\SP(X,\bR)\times H^{n-p,n-p}_{\ge 0}(X,\bR)\longrightarrow \bR_+,\qquad
(\alpha,T)\longmapsto\int_X \alpha\wedge T \in\bR_+\leqno(13.6)
$$
(notice that if $\alpha$ is strongly positive and $T\ge 0$, then
$\alpha\wedge T$ is a positive measure).

If $\cC$ is a convex cone in a finite dimensional vector space $E$,
we denote by $\cC^\smallvee$ the dual cone, i.e.\ the set of linear
forms $u\in E^\star$ which take nonnegative values on all elements of $\cC$. 
By the Hahn-Banach theorem, we always have $\cC^{\smallvee\smallvee}=
\overline{\cC}$. A basic problem would
be to investigate whether $H^{p,p}_\SP(X,\bR)$ and $H^{n-p,n-p}_{\ge 0}(X,\bR)$
are always dual cones, and another even harder question, which somehow
encompasses the Hodge conjecture, would be to relate these cones to
the cones generated by cohomology classes of effective analytic cycles.
We are essentially unable to address these extremely difficult questions,
except in the special cases $p=1$ or $p=n-1$ which are much better
understood and are the main target of the next two sections.
\endsectionrunning

\section{Numerical characterization of the K\"ahler cone}

We describe here the main results obtained in [DP04]. The upshot is
that the K\"ahler cone depends only on the intersection product
of the cohomology ring, the Hodge structure and the homology classes
of analytic cycles. More precisely, we have$\,$:

\claim{(14.1) Theorem} Let $X$ be a compact K\"ahler manifold. Let $\cP$ 
be the set of real $(1,1)$ cohomology classes $\{\alpha\}$
which are {\rm numerically} {\rm positive} on analytic cycles, i.e.\ such 
that $\int_Y\alpha^p>0$ for every irreducible analytic set $Y$ in~$X$, 
$p=\dim Y$. Then the K\"ahler cone $\cK$ of $X$ is one of the 
connected components of $\cP$.
\endclaim
%
\claim{(14.2) Special case} If $X$ is projective algebraic, then $\cK=\cP$.
\endclaim
%
These results (which are new even in the projective case) can be seen 
as a generalization of the well-known Nakai-Moishezon criterion. Recall
that the Nakai-Moishezon criterion provides a necessary and sufficient 
criterion for a line bundle
to be ample: {\it a line bundle $L\rightarrow X$ on a projective
algebraic manifold $X$ is ample if and only if 
$$L^p\cdot Y = \int_Yc_1(L)^p> 0,$$ 
for every algebraic subset $Y\subset X$, $p=\dim Y$.} 

It turns out that the numerical conditions $\int_Y\alpha^p> 0$ also 
characterize arbitrary transcendental K\"ahler classes when  $X$ is 
projective$\,$: this is precisely the meaning of the special case 14.2.

\claim{(14.3) Example} {\rm The following example shows that the cone $\cP$
need not be connected (and also that the components of $\cP$ need not 
be convex, either). Let us consider for instance a complex torus 
$X=\bC^n/\Lambda$. It is well-known that a generic torus $X$ does not
possess any analytic subset except finite subsets and $X$ itself. 
In that case, the numerical positivity is expressed by the single
condition $\int_X\alpha^n>0$. However, on a torus, $(1,1)$-classes are 
in one-to-one correspondence with constant hermitian forms $\alpha$ on
$\bC^n$. Thus, for $X$ generic, $\cP$ is the set of hermitian forms on 
$\bC^n$ such that $\det(\alpha)>0$, and Theorem 14.1 just expresses the 
elementary result of linear algebra saying that the set $\cK$ of 
positive definite forms is one of the connected components of the open set
$\cP=\{\det(\alpha)>0\}$ of hermitian forms of positive determinant
(the other components, of course, are the sets of forms of signature 
$(p,q)$, $p+q=n$, $q$ even. They are not convex when $p>0$ and $q>0$).}
\endclaim
%
\medskip
{\it Sketch of proof of Theorems 14.1 and 14.2.} By definition (13.1)~(iv), a 
{\rm \it K\"ahler current} is a closed positive current $T$ of type $(1,1)$
such that $T\ge\varepsilon\omega$ for some smooth
K\"ahler metric $\omega$ and $\varepsilon>0$ small enough.
The crucial steps of the proof of Theorem~14.1 are contained in the 
following statements. 

\claim{(14.4) Proposition {\rm (P\u{a}un [Pau98a, 98b])}} Let $X$ be a compact 
complex manifold $($or more generally a compact complex space$)$. Then
\vskip3pt
\plainitem{\rm (a)} The cohomology class of a closed positive 
$(1,1)$-current $\{T\}$ is nef if and only if the restriction 
$\{T\}_{|Z}$ is nef for every irreducible component $Z$ in any of
the Lelong sublevel sets~$E_c(T)$.

\vskip3pt
\plainitem{\rm (b)} The cohomology class of a K\"ahler current $\{T\}$ is 
a K\"ahler class $($i.e.\ the class of a {\rm \rm smooth} K\"ahler form$)$ 
if and only if the restriction $\{T\}_{|Z}$ is a K\"ahler class for 
every irreducible component $Z$ in any of the Lelong sublevel sets~$E_c(T)$. 
\vskip0pt
\endclaim
%
The proof of Proposition 14.4 is not extremely hard if we take for granted the
fact that K\"ahler currents can be approximated by K\"ahler currents
with logarithmic poles, a fact which was first proved in section~9.B
(see also [Dem92]). Thus in (b), we may assume that $T=
\alpha+\ii \ddbar\varphi$ is a current with analytic singularities,
where $\varphi$ is a quasi-psh function with logarithmic poles on some
analytic set $Z$, and $\varphi$ smooth on $X\ssm Z$. Now, we proceed by 
an induction on dimension (to do this, we have to consider analytic spaces
rather than with complex manifolds, but it turns out that this makes no
difference for the proof). Hence, by the induction hypothesis, there exists
a smooth potential $\psi$ on $Z$ such that 
$\alpha_{|Z}+\ii \ddbar\psi>0$ along $Z$. It is well known that
one can then find a potential $\widetilde\psi$ on $X$ such that
$\alpha+\ii \ddbar\widetilde\psi>0$ in a neighborhood $V$ of $Z$ (but possibly
non positive elsewhere). Essentially, it is enough to take an arbitrary
extension of $\psi$ to $X$ and to add a large multiple of the square of 
the distance to $Z$, at
least near smooth points; otherwise, we stratify $Z$ by its successive
singularity loci, and proceed again by induction on the dimension of these
loci. Finally, we use a
a standard gluing procedure~: the current $T=\alpha+
\ii \max_\varepsilon(\varphi,\widetilde\psi-C)$, $C\gg 1$, will be equal
to $\alpha+\ii \ddbar\varphi>0$ on $X\ssm V$, and to a smooth
K\"ahler form on~$V$.\qed
\medskip

The next (and more substantial step) consists of the following result
which is reminiscent of the Grauert-Riemenschneider conjecture
([Siu84], [Dem85]).

\claim{(14.5) Theorem {\rm ([DP04])}} Let $X$ be a compact K\"ahler manifold
and let $\{\alpha\}$ be a nef class $($i.e.\ $\{\alpha\}\in\overline\cK)$.
Assume that $\int_X\alpha^n>0$. Then $\{\alpha\}$ contains a K\"ahler
current~$T$, in other words $\{\alpha\}\in\cE^\circ$.
\endclaim
%
\noindent
{\it Step 1}. The basic argument is to prove that for every irreducible 
analytic set $Y\subset X$ of
codimension $p$, the class $\{\alpha\}^p$ contains a closed positive
$(p,p)$-current $\Theta$ such that $\Theta\ge\delta[Y]$ for some $\delta>0$.
For this, we use in an essentail way the Calabi-Yau theorem [Yau78] on
solutions of Monge-Amp\`ere equations, which yields the following result
as a special case:

\claim{(14.6) Lemma {\rm([Yau78])}} Let $(X,\omega)$ 
be a compact K\"ahler manifold and $n=\dim X$. Then for any smooth
volume form $f>0$ such that $\int_X f = \int_X\omega^n$,
there exist a K\"ahler metric $\widetilde \omega = \omega+i\ddbar\varphi$
in the same K\"ahler class as $\omega$, such that 
$\widetilde \omega^n=f$.\qed
\endclaim
%
\noindent
We exploit this by observing that $\alpha+\varepsilon\omega$ is a
K\"ahler class. Hence we can solve the Monge-Amp\`ere equation
$$
(\alpha+\varepsilon\omega+i\ddbar\varphi_\varepsilon)^n=C_\varepsilon
\omega_\varepsilon^n
\leqno(14.6\,\hbox{a})
$$
where $(\omega_\varepsilon)$ is the family of K\"ahler metrics on $X$
produced by Lemma 3.4~(iii), such that their volume is concentrated
in an $\varepsilon$-tubular neighborhood of $Y$.
$$
C_\varepsilon = {\int_X\alpha_\varepsilon^n\over\int_X\omega_\varepsilon^n}
={\int_X(\alpha+\varepsilon\omega)^n\over\int_X\omega^n}\ge
C_0={\int_X\alpha^n\over\int_X\omega^n}>0.
$$
Let us denote by 
$$
\lambda_1(z)\le\ldots\le\lambda_n(z)
$$
the eigenvalues of $\alpha_\varepsilon(z)$ with respect to
$\omega_\varepsilon(z)$, at every point $z\in X$ (these functions
are continuous with respect to $z$, and of course depend also 
on~$\varepsilon$). The equation (14.6a) is equivalent to the fact that
$$
\lambda_1(z)\ldots\lambda_n(z)=C_\varepsilon\leqno(14.6\,\hbox{b})
$$
is constant, and the most important observation for us is that the
constant $C_\varepsilon$ is bounded away from $0$, thanks to
our assumption $\int_X\alpha^n>0$.

Fix a regular point $x_0\in Y$ and a small neighborhood $U$ (meeting only
the irreducible component of $x_0$ in $Y$). By Lemma 3.4, we have a
uniform lower bound
$$
\int_{U\cap V_\varepsilon}\omega_\varepsilon^p\wedge\omega^{n-p}
\ge\delta_p(U)>0.\leqno(14.6\,\hbox{c})
$$
Now, by looking at the $p$ smallest (resp.\ $(n-p)$ largest) eigenvalues
$\lambda_j$ of $\alpha_\varepsilon$ with respect to $\omega_\varepsilon$, 
we find
$$
\leqalignno{
\alpha_\varepsilon^p&\ge \lambda_1\ldots\lambda_p\,\omega_\varepsilon^p,
&(14.6\,\hbox{d})\cr
\alpha_\varepsilon^{n-p}\wedge\omega_\varepsilon^p
&\ge {1\over n!}\lambda_{p+1}\ldots\lambda_n\,
\omega_\varepsilon^n,&(14.6\,\hbox{e})\cr
}
$$
The last inequality $(14.6\,\hbox{e})$ implies
$$
\int_X\lambda_{p+1}\ldots\lambda_n\,\omega_\varepsilon^n\le
n!\int_X\alpha_\varepsilon^{n-p}\wedge\omega_\varepsilon^p=
n!\int_X(\alpha+\varepsilon\omega)^{n-p}\wedge\omega^p\le M
$$
for some constant $M>0$ (we assume $\varepsilon\le 1$, say). 
In particular, for every $\delta>0$, the subset 
$E_\delta\subset X$ of points $z$ such that 
$\lambda_{p+1}(z)\ldots\lambda_n(z)>M/\delta$ satisfies
$\int_{E_\delta}\omega_\varepsilon^n\le\delta$, hence
$$
\int_{E_\delta}\omega_\varepsilon^p\wedge\omega^{n-p}\le 2^{n-p}
\int_{E_\delta}\omega_\varepsilon^n\le 2^{n-p}\delta.\leqno(14.6\,\hbox{f})
$$
The combination of (14.6$\,$c) and (14.6$\,$f) yields
$$
\int_{(U\cap V_\varepsilon)\ssm E_\delta}\omega_\varepsilon^p\wedge\omega^{n-p}
\ge \delta_p(U)-2^{n-p}\delta.
$$
On the other hand (14.6$\,$b) and (14.6$\,$d) imply
$$
\alpha_\varepsilon^p\ge{C_\varepsilon\over\lambda_{p+1}\ldots\lambda_n}
\omega_\varepsilon^p
\ge{C_\varepsilon\over M/\delta}\omega_\varepsilon^p\qquad
\hbox{on $(U\cap V_\varepsilon)\ssm E_\delta$}.
$$
From this we infer
$$
\int_{U\cap V_\varepsilon}\alpha_\varepsilon^p\wedge\omega^{n-p}\ge
{C_\varepsilon\over M/\delta}
\int_{(U\cap V_\varepsilon)\ssm E_\delta}
\omega_\varepsilon^p\wedge\omega^{n-p}\ge
{C_\varepsilon\over M/\delta}(\delta_p(U)-2^{n-p}\delta)>0
\leqno(14.6\,\hbox{g})
$$
provided that $\delta$ is taken small enough, e.g.\ 
$\delta=2^{-(n-p+1)}\delta_p(U)$. The family of $(p,p)$-forms
$\alpha_\varepsilon^p$ is uniformly bounded in mass since
$$
\int_X\alpha_\varepsilon^p\wedge\omega^{n-p}
=\int_X(\alpha+\varepsilon\omega)^p\wedge\omega^{n-p}\le\hbox{Const}.
$$
Inequality (14.6$\,$g) implies that any weak limit $\Theta$ of 
$(\alpha_\varepsilon^p)$
carries a positive mass on~$U\cap Y$. By Skoda's extension theorem [Sko82], 
${\bf 1}_Y\Theta$ is a closed positive current with support in $Y$, 
hence ${\bf 1}_Y\Theta=\sum c_j[Y_j]$ is a combination of the 
various components $Y_j$ of $Y$ with coefficients $c_j>0$. 
Our construction shows that $\Theta$ belongs to the cohomology class 
$\{\alpha\}^p$. Step 1 of Theorem 14.5 is proved.
\medskip

\noindent
{\it Step 2.} The second and final step consists in using
a ``diagonal trick'': for this, we apply Step 1 to 
$$
\widetilde X=X\times X,\qquad \widetilde Y=\hbox{diagonal}\,\Delta\subset \widetilde X,\qquad
\widetilde\alpha=\pr_1^*\alpha+\pr_2^*\alpha.
$$
It is then clear that $\widetilde\alpha$ is nef on $\widetilde X$ and that
$$
\int_{\widetilde X}(\widetilde\alpha)^{2n}=
{2n\choose n}\Big(\int_X\alpha^n\Big)^2>0.
$$
It follows by Step 1 that the
class $\smash{\{\widetilde\alpha\}}^n$ contains a K\"ahler current $\Theta$ of 
bidegree $(n,n)$ such that
$\Theta\ge\delta[\Delta]$ for some $\delta>0$. Therefore 
the push-forward
$$
T:=(\pr_1)_*(\Theta\wedge\pr_2^*\omega)
$$
is a positive $(1,1)$-current such that
$$
T\ge\delta(\pr_1)_*([\Delta]\wedge\pr_2^*\omega)=\delta\omega.
$$
It follows that $T$ is a K\"ahler current. On the other hand, $T$
is numerically equivalent to $(\pr_1)_*(\widetilde\alpha^n\wedge\pr_2^*\omega)$, 
which is the form given in coordinates by
$$
x\mapsto\int_{y\in X}\big(\alpha(x)+\alpha(y)\big)^n\wedge \omega(y)=
C\alpha(x)
$$
where $C=n\int_X\alpha(y)^{n-1}\wedge\omega(y)$. Hence $T\equiv C\alpha$,
which implies that $\{\alpha\}$ contains a K\"ahler current. 
Theorem 14.5 is proved.\qed
\medskip

\noindent
{\it End of Proof of Theorems 14.1 and 14.2}. Clearly the open cone $\cK$ 
is contained in $\cP$, hence in order to show that $\cK$ is one of the
connected components of $\cP$, we need only show that $\cK$ is {\rm closed}
in $\cP$, i.e.\ that $\overline\cK\cap\cP\subset\cK$. Pick a class
$\{\alpha\}\in\overline\cK\cap\cP$. In particular $\{\alpha\}$ is
nef and satisfies $\int_X\alpha^n>0$. By Theorem~14.5 we conclude that
$\{\alpha\}$ contains a K\"ahler current $T$. However, an induction on
dimension using the assumption $\int_Y\alpha^p$ for all analytic
subsets $Y$ (we also use resolution of singularities for $Y$ at this step)
shows that the restriction $\{\alpha\}_{|Y}$ is the class of a K\"ahler
current on~$Y$. We conclude that $\{\alpha\}$ is a K\"ahler class by
14.4~(b), therefore $\{\alpha\}\in\cK$, as desired.\qed

The projective case 14.2 is a consequence of the following variant of
Theorem~14.1.

\claim{(14.7) Corollary} Let $X$ be a compact
K\"ahler manifold. A $(1,1)$ cohomology class $\{\alpha\}$ on $X$
is K\"ahler if and only if there exists a 
K\"ahler metric $\omega$ on $X$ such that $\int_Y\alpha^k\wedge\omega^{p-k}>0$
for all irreducible analytic sets $Y$ and all $k=1,2,\ldots,p=\dim Y$.
\endclaim
%
\begproof{Proof.} The assumption clearly implies that
$$
\int_Y(\alpha+t\omega)^p>0
$$
for all $t\in\bR_+$, hence the half-line $\alpha+(\bR_+)\omega$ is 
entirely contained in the cone $\cP$ of numerically positive classes.
Since $\alpha+t_0\omega$ is K\"ahler for $t_0$ large, we conclude that
the half-line in entirely contained in the connected component $\cK$,
and therefore $\alpha\in\cK$.\qed
\endproof
%
In the projective case, we can take $\omega=c_1(H)$ for a given very ample
divisor $H$, and the condition
 $\int_Y\alpha^k\wedge\omega^{p-k}>0$ is equivalent to
$$
\int_{Y\cap H_1\cap\ldots\cap H_{p-k}}\alpha^k>0
$$
for a suitable complete intersection $Y\cap H_1\cap\ldots\cap H_{p-k}$,
$H_j\in|H|$. This shows that algebraic cycles are sufficient to test
the K\"ahler property, and the special case 14.2 follows. On the other 
hand, we can pass to the limit in 14.7 by replacing $\alpha$ 
by $\alpha+\varepsilon\omega$,
and in this way we get also a characterization of nef classes.

\claim{(14.8) Corollary} Let $X$ be a compact
K\"ahler manifold. A $(1,1)$ cohomology class $\{\alpha\}$ on $X$
is nef if and only if there exists a 
K\"ahler metric $\omega$ on $X$ such that $\int_Y\alpha^k\wedge\omega^{p-k}
\ge 0$ for all irreducible analytic sets $Y$ and all $k=1,2,\ldots,p=\dim Y$.
\endclaim
%
By a formal convexity argument, one can derive from 14.7 or 14.8 the following
interesting consequence about the dual of the cone $\cK$. 

\claim{(14.9) Theorem} Let $X$ be a compact K\"ahler manifold. 
\vskip3pt
\plainitem{\rm (a)}
A $(1,1)$ cohomology class $\{\alpha\}$ on $X$ is nef if and only 
for every irreducible analytic set $Y$ in $X$, $p=\dim X$
and every K\"ahler metric $\omega$ on $X$ we have
$\int_Y\alpha\wedge\omega^{p-1}\ge 0$. $($Actually this numerical 
condition is needed only for K\"ahler classes $\{\omega\}$ which
belong to a $2$-dimensional space $\bR\{\alpha\}+\bR\{\omega_0\}$, 
where $\{\omega_0\}$ is a given K\"ahler class$)$.
\vskip3pt
\plainitem{\rm (b)} The dual of the
nef cone $\smash{\overline\cK}$ is the closed convex cone 
in $H^{n-1,n-1}(X,\bR)$ generated by cohomology 
classes of currents of the form $[Y]\wedge\omega^{p-1}$ in 
$H^{n-1,n-1}(X,\bR)$, where $Y$ runs over the 
collection of irreducible analytic subsets of $X$ and $\{\omega\}$ over
the set of K\"ahler classes of~$X$. This dual cone coincides
with $H^{n-1,n-1}_{\ge 0}(X,\bR)$.\vskip0pt
\endclaim
%
\begproof{Proof.} (a) Clearly a nef class $\{\alpha\}$ satisfies the given 
numerical condition. The proof of the converse is more tricky.
First, observe that for every 
integer $p\ge 1$, there exists a polynomial identity of the form
$$
(y-\delta x)^p - (1-\delta)^px^p
= (y-x)\int_0^1 A_p(t,\delta)\big((1-t)x+ty\big)^{p-1}\,dt
\leqno(14.10)
$$
where $A_p(t,\delta)=\sum_{0\le m\le p}a_m(t)\delta^m\in
\bQ[t,\delta]$ is a polynomial of degree${}\le p-1$ in~$t$ (moreover, 
the polynomial $A_p$ is unique under this limitation for the  degree). 
To see this, we observe that $(y-\delta x)^p - (1-\delta)^px^p$
vanishes identically for $x=y$, so it is divisible by $y-x$. By
homogeneity in $(x,y)$, we have an expansion of the form
$$
(y-\delta x)^p - (1-\delta)^px^p = (y-x)
\sum_{0\le\ell\le p-1,\,0\le m\le p} b_{\ell,m}x^\ell y^{p-1-\ell}\delta^m
$$
in the ring $\bZ[x,y,\delta]$. Formula (14.10) is then equivalent to
$$
b_{\ell,m} = \int_0^1 a_m(t){p-1\choose\ell}(1-t)^\ell t^{p-1-\ell}\,dt.
\leqno(14.10')
$$
Since $(U,V)\mapsto\int_0^1U(t)V(t)dt$ is a non degenerate linear pairing
on the space of polynomials of degree${}\le p-1$ and since
$({p-1\choose\ell}(1-t)^\ell t^{p-1-\ell})_{0\le\ell\le p-1}$ is a basis
of this space, $(14.10')$ can be achieved for a unique choice of the
polynomials $a_m(t)$. A~straightforward calculation shows that $A_p(t,0)=p$
identically. We can therefore choose $\delta_0\in[0,1[$ so small that 
$A_p(t,\delta)>0$ for all $t\in[0,1]$, $\delta\in[0,\delta_0]$ and 
$p=1,2,\ldots,n$.

Now, fix a K\"ahler metric $\omega$ such that $\omega'=\alpha+\omega$ 
yields a K\"ahler class $\{\omega'\}$ (just take a large multiple 
$\omega=k\omega_0$, $k\gg 1$, of the given  K\"ahler metric 
$\omega_0$ to initialize the process).
A substitution $x=\omega$ and $y=\omega'$ in our polynomial
identity yields
$$
(\alpha+(1-\delta)\omega)^p-(1-\delta)^p\omega^p=\int_0^1
A_p(t,\delta)\,\alpha\wedge\big((1-t)\omega+t\omega'\big)^{p-1}dt.
$$
For every irreducible analytic subset $Y\subset X$ of dimension $p$ we find
$$
\int_Y(\alpha+(1-\delta)\omega)^p-(1-\delta)^p\int_Y\omega^p
=\int_0^1A_p(t,\delta)dt\Big(
\int_Y\alpha\wedge\big((1-t)\omega+t\omega'\big)^{p-1}\Big).
$$
However, $(1-t)\omega+t\omega'$ is a K\"ahler class (contained in 
$\bR\{\alpha\}+\bR\{\omega_0\}$) and therefore
$\int_Y\alpha\wedge\big((1-t)\omega+t\omega'\big)^{p-1}\ge 0$ by the
numerical condition. This implies $\int_Y(\alpha+(1-\delta)\omega)^p>0$
for all $\delta\in[0,\delta_0]$. We have produced a segment entirely
contained in $\cP$ such that one extremity $\{\alpha+\omega\}$ is in~$\cK$,
so the other extremity $\{\alpha+(1-\delta_0)\omega\}$ is also in~$\cK$.
By repeating the argument inductively after replacing $\omega$ with 
$(1-\delta_0)\omega$, we see that 
$\{\alpha+(1-\delta_0)^\nu\omega\}\in\cK$ for every integer $\nu\ge 0$.
From this we infer that $\{\alpha\}$ is nef, as desired.
\medskip
\noindent
(b) Part (a) can be reformulated by saying that the dual cone
$\overline{\cK}^{\,\smallvee}$ is the closure of the convex cone generated by
$(n-1,n-1)$ cohomology classes of the form $[Y]\wedge\omega^{p-1}$.
Since these classes are contained in $H^{n-1,n-1}_{\ge 0}(X,\bR)$ which is
also contained in $\overline{\cK}^{\,\smallvee}$ by (13.6), we infer that
$$
\overline{\cK}^{\,\smallvee}=H^{n-1,n-1}_{\ge 0}(X,\bR)=\overline{
{\rm Cone}(\{[Y]\wedge\omega^{p-1}\})}.
\leqno\hbox{\rlap{(14.11)}\rlap{\hbox{\kern\columnwidth\llap{$\square$}}}}~
$$
\endproof
%
Our main Theorem 14.1 also has an important application to the deformation
theory of compact K\"ahler manifolds.

\claim{(14.12) Theorem} Let $\pi:\cX\to S$ be a deformation of compact K\"ahler
manifolds over an irreducible base~$S$. Then there exists a countable union 
$S'=\bigcup S_\nu$ of analytic subsets $S_\nu\subsetneq S$, such that 
the K\"ahler cones $\cK_t\subset H^{1,1}(X_t,\bC)$ of the fibers
$X_t=\pi^{-1}(t)$ are invariant over $S\ssm S'$ under 
parallel transport with respect to the $(1,1)$-projection $\nabla^{1,1}$ of
the Gauss-Manin connection $\nabla$ in the decomposition of
$$
\nabla=\plainpmatrix{
\nabla^{2,0} & * & 0\cr 
* & \nabla^{1,1} & * \cr
0 & * & \nabla^{0,2}\cr}
$$
on the Hodge bundle $H^2=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$.
\endclaim
%
We moreover conjecture that for an arbitrary deformation $\cX\to S$ of 
compact complex manifolds, the K\"ahler property is
open with respect to the countable Zariski topology on the base $S$ of
the deformation.
\vskip3pt

Let us recall the general fact that all fibers $X_t$ of a deformation over a 
connected base $S$ are diffeomorphic, since $\cX\to S$ is a locally trivial
differentiable bundle. This implies that the cohomology bundle
$$
S\ni t\mapsto H^k(X_t,\bC)
$$
is locally constant over the base $S$. The corresponding (flat) connection
of this bundle is called the Gauss-Manin connection, and will be denoted here
by $\nabla$. As is well known, the Hodge filtration
$$
F^p(H^k(X_t,\bC))=\bigoplus_{r+s=k,r\ge p}H^{r,s}(X_t,\bC)
$$
defines a {\sl holomorphic} subbundle of $H^k(X_t,\bC)$ (with respect to its
locally constant structure). On the other hand, the Dolbeault groups are 
given by
$$
H^{p,q}(X_t,\bC) = F^p(H^k(X_t,\bC))\cap\overline{F^{k-p}(H^k(X_t,\bC))},\qquad
k=p+q,
$$
and they form {\sl real analytic} subbundles of $H^k(X_t,\bC)$. We are 
interested especially in the decomposition
$$
H^2(X_t,\bC)=H^{2,0}(X_t,\bC)\oplus H^{1,1}(X_t,\bC)\oplus H^{0,2}(X_t,\bC)
$$
and the induced decomposition of the Gauss-Manin connection acting on $H^2$
$$
\nabla=\plainpmatrix{\nabla^{2,0}&*&*\cr *&\nabla^{1,1}&*\cr *&*&\nabla^{0,2}\cr}.
$$
Here the stars indicate suitable bundle morphisms -- actually with the 
lower left and upper right stars being zero by Griffiths' transversality 
property, but we do not really care here. The notation $\nabla^{p,q}$ stands 
for the induced (real analytic, not necessarily flat) connection on the
subbundle $t\mapsto H^{p,q}(X_t,\bC)$. 

\medskip
{\it Sketch of Proof of Theorem 14.12.} The result is local on the base,
hence we may assume that $S$ is contractible. Then the family is
differentiably trivial, the Hodge bundle $t\mapsto H^2(X_t,\bC)$ is
the trivial bundle and $t\mapsto H^2(X_t,\bZ)$ is a trivial lattice.
We use the existence of a relative cycle space $C^p(\cX/S)\subset C^p(\cX)$
which consists of all cycles contained in the fibres of $\pi:X\to S$.
It is equipped with a canonical holomorphic projection 
$$
\pi_p:C^p(\cX/S)\to S.
$$
We then define 
the $S_\nu$'s to be the images in $S$ of those connected components of 
$C^p(\cX/S)$ which do not project onto $S$. By the fact that the projection 
is proper on each component, we infer that $S_\nu$ is an analytic subset 
of $S$. The definition of the $S_\nu$'s imply that the cohomology classes
induced by the analytic cycles $\{[Z]\}$, $Z\subset X_t$, remain exactly 
the same for all $t\in S\ssm S'$. This result implies in its turn 
that the conditions defining the numerically positive cones $\cP_t$
remain the same, except for the fact that the spaces $H^{1,1}(X_t,\bR)\subset
H^2(X_t,\bR)$ vary along with the Hodge decomposition. At this point,
a standard calculation implies that the $\cP_t$ are invariant by parallel
transport under $\nabla^{1,1}$. This is done as follows.

Since $S$ is irreducible and $S'$ is a countable union of analytic sets, 
it follows that $S\ssm S'$ is arcwise connected by piecewise smooth analytic 
arcs. Let 
$$
\gamma:[0,1]\to S\ssm S',\qquad u\mapsto t=\gamma(u)
$$ 
be such a smooth arc,
and let $\alpha(u)\in H^{1,1}(X_{\gamma(u)},\bR)$ be a family of real 
$(1,1)$-cohomology classes which are constant by parallel transport
under $\nabla^{1,1}$. This is equivalent to assuming that 
$$
\nabla(\alpha(u))\in H^{2,0}(X_{\gamma(u)},\bC)\oplus 
H^{0,2}(X_{\gamma(u)},\bC)
$$
for all $u$. Suppose that $\alpha(0)$ is a numerically positive class
in~$X_{\gamma(0)}$. We then have
$$
\alpha(0)^p\cdot \{[Z]\}=\int_Z \alpha(0)^p>0
$$
for all $p$-dimensional analytic cycles $Z$ in $X_{\gamma(0)}$. Let us 
denote by 
$$
\zeta_Z(t)\in H^{2q}(X_t,\bZ),\qquad q=\dim X_t -p,
$$ 
the family of cohomology classes equal to $\{[Z]\}$ at $t=\gamma(0)$,
such that $\nabla\zeta_Z(t)=0$ (i.e.\ constant with respect to the Gauss-Manin
connection).  By the above discussion, $\zeta_Z(t)$ is of type
$(q,q)$ for all $t\in S$, and when $Z\subset X_{\gamma(0)}$ varies,
$\zeta_Z(t)$ generates all classes of analytic cycles in $X_t$ if
$t\in S\ssm S'$.  Since $\zeta_Z$ is $\nabla$-parallel and
$\nabla\alpha(u)$ has no component of type $(1,1)$, we find
$$
{d\over du}(\alpha(u)^p\cdot\zeta_Z(\gamma(u)) =
p \alpha(u)^{p-1}\cdot \nabla\alpha(u)\cdot \zeta_Z(\gamma(u))= 0.
$$
We infer from this that $\alpha(u)$ is a numerically positive class
for all $u\in[0,1]$. This argument shows that the set $\cP_t$ of 
numerically positive classes in $H^{1,1}(X_t,\bR)$ is invariant
by parallel transport under $\nabla^{1,1}$ over $S\ssm S'$.

By a standard result of Kodaira-Spencer [KS60] relying on elliptic PDE
theory, every K\"ahler class in $X_{t_0}$ can be deformed to a nearby 
K\"ahler class in nearby fibres $X_t$. This implies that the connected 
component of $\cP_t$ which corresponds to the K\"ahler cone $\cK_t$ must 
remain the same. The theorem is proved.\qed
\medskip

As a by-product of our techniques, especially the regularization
theorem for currents, we also get the following result
for which we refer to [DP04].

\claim{(14.13) Theorem} A compact complex manifold carries a K\"ahler
current if and only if it is bimeromorphic to a K\"ahler manifold
$($or equivalently, dominated by a K\"ahler manifold$\,)$.
\endclaim
%
This class of manifolds is called the {\it Fujiki class $\cC$}. If we 
compare this result with the solution of the Grauert-Riemenschneider
conjecture, it is tempting to make the following conjecture which would
somehow encompass both results.

\claim{(14.14) Conjecture} Let $X$ be a compact complex manifold of
dimension~$n$. Assume that $X$ possesses a nef cohomology class $\{\alpha\}$
of type $(1,1)$ such that $\int_X\alpha^n>0$. Then $X$ is in the
Fujiki class~$\cC$. $[$Also, $\{\alpha\}$ would contain a K\"ahler 
current, as it follows from Theorem 14.5 if Conjecture 14.14 is proved$\,]$.
\endclaim
%
We want to mention here that most of the above results were already known in 
the cases of complex surfaces (i.e.\ in dimension $2$),
thanks to the work of N.~Buchdahl [Buc99, 00] and A.~Lamari [Lam99a, 99b].

Shortly after the original [DP04] manuscript appeared 
in April 2001, Daniel Huybrechts [Huy01] informed us Theorem 14.1 can be 
used to calculate the K\"ahler cone of
a very general hyperk\"ahler manifold: the K\"ahler cone is then equal
to a suitable connected component of the positive cone defined by the
Beauville-Bogomolov quadratic form. In the case of an  arbitrary
hyperk\"ahler manifold, S.Boucksom [Bou02] later showed that a $(1,1)$ class
$\{\alpha\}$ is K\"ahler if and only if it lies in the positive part
of the Beauville-Bogomolov quadratic cone and moreover $\int_C\alpha>0$
for all {\rm \it rational curves} $C\subset X$ (see also [Huy99]).

\section{Structure of the pseudo-effective cone and mobile
intersection theory}
\sectionrunning{Structure of pseudo-effective cone and mobile
intersections}

\subsection*{\csect.A. Classes of mobile curves and of mobile $(n-1,n-1)$-currents}
We introduce various positive cones in $H^{n-1,n-1}(X,\bR)$, some of
which exhibit certain ``mobility'' properties, in the sense that
they can be more or less freely deformed. Ampleness is clearly such a property,
since a very ample divisor $A$ can be moved in its linear system $|A|$
so as to cover the whole ambient variety. By extension, a K\"ahler class
$\{\omega\}\in H^{1,1}(X,\bR)$ is also considered to be mobile, as
illustrated alternatively by the fact that the Monge-Amp\`ere volume form
\hbox{$(\omega+i\ddbar\varphi)^n$} of a K\"ahler metric in the same cohomology
class can be taken to be equal to an arbitrary volume form $f>0$ with 
$\int_Xf=\int_X\omega^n$ (thanks to Yau's theorem [Yau78]).
\bigskip

\claim{(15.1) Definition} Let $X$ be a smooth projective variety.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} One defines $\NE(X)$ to be the
convex cone generated by cohomology classes of all effective curves
in $H^{n-1,n-1}(X,\bR)$
\vskip3pt
\plainitem{\rm (ii)} We say that $C$ is a {\rm mobile curve} if $C=C_{t_0}$
is a member of an analytic family $(C_t)_{t\in S}$ such that 
$\bigcup_{t\in S}C_t=X$
and, as such, is a reduced irreducible $1$-cycle. We define the
mobile cone $\ME(X)$, to be the convex cone generated by all 
mobile curves.
\vskip3pt
\plainitem{\rm (iii)} If $X$ is projective, we say that an effective $1$-cycle $C$ 
is  a {\rm strongly mobile} if we have
$$
C=\mu_\star(\widetilde A_1\cap\ldots\cap \widetilde A_{n-1})
$$ 
for suitable very ample divisors $\widetilde A_j$ on $\widetilde X$, where 
$\mu:\widetilde X\to X$ is a modification. We let $\ME^s(X)$ 
be the convex cone generated by all strongly mobile effective
$1$-cycles $($notice that by taking $\widetilde A_j$ general enough these classes can 
be represented by reduced irreducible curves; also, by
Hironaka, one could just restrict oneself to compositions of blow-ups
with smooth centers$)$. 
\vskip0pt}
\endclaim
%
\noindent
Clearly, we have $\ME^s(X)\subset\ME(X)\subset\NE(X)$.
The cone $\NE(X)$ is contained in the analogue of the Neron-Severi group
for $(n-1,n-1)$-classes, namely
$$
\NS_\bR^{n-1}(X):=
(H^{n-1,n-1}(X,\bR)\cap H^{2n-2}(X,\bZ)/{\rm tors})\otimes_\bZ\bR
$$
(sometimes also denoted $N_1(X)$ in the litterature).
We wish to introduce similar concepts for cones of non necessarily integral
classes, on arbitrary compact K\"ahler manifolds. The relevant definition is
as follows.

\claim{(15.2) Definition} Let $X$ be a compact K\"ahler manifold.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} We define the set $\cN=H^{n-1,n-1}_{\ge 0}(X,\bR)$
to be the $($closed$\,)$ convex cone in
$H^{n-1,n-1}(X,\bR)$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n\,{-}\,1,n\,{-}\,1)$}, i.e., of bidimension $(1,1)$.
\vskip3pt
\plainitem{\rm (ii)} We define the cone $\cM^s\subset H^{n-1,n-1}(X,\bR)$ of
{\rm strongly mobile classes} to be the closure of the convex cone generated by
classes of currents of the form
$$\mu_\star(\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{n-1})$$
where $\mu:\widetilde X\to X$ is an arbitrary modification, and 
the $\widetilde\omega_j$ are K\"ahler forms on~$\widetilde X$.
\vskip3pt
\plainitem{\rm (iii)} We define the cone $\cM\subset H^{n-1,n-1}(X,\bR)$ of
{\rm mobile classes} to be the closure of the convex cone generated by
classes of currents of the form 
$$\mu_\star([\widetilde Y_{t_0}]\wedge\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{p-1})$$
where $\mu:\widetilde X\to X$ is an arbitrary modification, the $\widetilde\omega_j$ are 
K\"ahler forms on~$\widetilde X$ and $(\widetilde Y_t)_{t\in S}$ is an analytic family of
effective $p$-dimensional analytic cycles covering $\widetilde X$ such that 
$\widetilde Y_{t_0}$ is 
reduced and irreducible, with $p$ running over all $\{1,2,\ldots,n\}$. 
\vskip0pt
}
\endclaim
%
Clearly, we have
$$\cM^s\subset\cM\subset\cN.$$
For $X$ projective, it is also immediately clear from the definitions that
$$
\plaincases{
\kern6pt\overline{\NE(X)}\subset\cN_{\NS}:=\cN\cap\NS_\bR^{n-1}(X),\cr
\noalign{\vskip5pt}
\kern4pt\overline{\ME(X)}\subset\cM_{\NS}:=\cM\cap\NS_\bR^{n-1}(X),\cr
\noalign{\vskip4pt}
\overline{\ME^s(X)}\subset\cM^s_{\NS}:=\cM^s\cap\NS_\bR^{n-1}(X).\cr}\leqno(15.3)
$$
The upshot of these definitions lie in the following easy observation.

\claim{(15.4) Proposition} Let $X$ be a compact K\"ahler manifold. 
The Serre duality pairing
$$H^{1,1}(X,\bR)\times H^{n-1,n-1}(X,\bR)\longrightarrow \bR,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta$$
takes nonnegative values
\vskip3pt
\plainitem{\rm (a)} for all pairs $(\alpha,\beta)\in\overline\cK\times\cN$;
\vskip3pt
\plainitem{\rm (b)} for all pairs $(\alpha,\beta)\in\cE\times\cM$.
\endclaim
%
\begproof{Proof.} (a) is obvious. In order to prove (b), we may assume that
$$
\beta=\mu_\star([Y_{t_0}]\wedge\widetilde\omega_1\wedge\ldots\wedge
\widetilde\omega_{p-1})
$$ 
for some modification $\mu:\widetilde X\to X$, where $\{\alpha\}=\{T\}$ is the class
of a positive $(1,1)$-current on $X$ and $\widetilde\omega_j$ are K\"ahler
forms on~$\widetilde X$. Then for $t\in S$ generic
$$
\leqalignno{
\int_X\alpha\wedge\beta&=\int_X T\wedge
\mu_\star([\widetilde Y_t]\wedge\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{p-1})\cr
&=\int_X \mu^* T\wedge [\widetilde Y_t]\wedge
\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{p-1}\cr
&=\int_{\widetilde Y_t} (\mu^* T)_{\restriction\widetilde Y_t}\wedge
\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{p-1}\ge 0&(15.5)\cr}
$$
provided that we show that the final integral is well defined and that
the formal calulations involved in (15.5) are correct.
Here, we have used the fact that a closed positive $(1,1)$-current $T$
always has a pull-back $\mu^\star T$, which follows from the observation that
if $T=\alpha+i\ddbar\varphi$ with $\alpha$ smooth and $\varphi$ quasi-psh, 
we may always set 
$\mu^\star T=\mu^*\alpha+i\ddbar(\varphi\circ\mu)$, with
$\varphi\circ\mu$ quasi-psh and not identically $-\infty$ on~$\widetilde X$.
Similarly, we see that the restriction 
$(\mu^* T)_{\restriction\widetilde Y_t}$ is a well defined positive $(1,1)$-current
for $t$ generic, by putting
$$
(\mu^* T)_{\restriction\widetilde Y_t}=
(\mu^*\alpha)_{\restriction\widetilde Y_t}+i\ddbar\big(
(\varphi\circ\mu)_{\restriction\widetilde Y_t}\big)
$$
and choosing $t$ such that $\widetilde Y_t$ is not contained in the pluripolar 
set of $-\infty$ poles of $\varphi\circ\mu$ (this is possible 
thanks to the assumption that $\widetilde Y_t$ covers $\widetilde X$; locally
near any given point we can modify $\alpha$ so that $\alpha=0$
on a small neighborhood~$V$, and then $\varphi$ is psh on $V$).
Finally, in order to justify the formal calculations
we can use a regularization argument for $T$, writing $T=\lim T_k$
with $T_k=\alpha+i\ddbar\varphi_k$ and a decreasing sequence of smooth almost 
plurisubharmonic potentials $\varphi_k\downarrow\varphi$ such that the 
Levi forms  have a uniform lower bound $i\ddbar\varphi_k\ge -C\omega$ (such a
sequence exists by [Dem92]). Then
$(\mu^*T_k)_{\restriction\widetilde Y_t}\to (\mu^*T)_{\restriction\widetilde Y_t}$ 
in the weak topology of currents.\qed
\endproof
%
Proposition 15.4 leads to the natural question whether the 
cones $(\overline{\cK},\cN)$ and $(\cE,\cM)$ are dual under Serre duality,
The second part of the question is addressed in the next section.
The results proved in \S$\,$18 yield a complete answer to the first part 
-- even in the general K\"ahler setting.


\claim{(15.6) Theorem} Let $X$ be a compact K\"ahler manifold. Then
\vskip3pt
\plainitem{\rm(i)} $\overline{\cK}$ and $\cN$ are dual cones.
\vskip3pt
\plainitem{\rm(ii)} If $X$ is projective algebraic, then
$\overline{\cK}_{\NS}=\Nef(X)$ and $\cN_{\NS}=\overline{\NE(X)}$ and these cones are
dual.
\endclaim
%
\begproof{Proof.} (i) is a weaker version of (14.9~b). 
\medskip

\noindent
(ii) The equality $\overline{\cK}_{\NS}=\Nef(X)$ has already been discussed
and is a consequence of the Kodaira embedding theorem. Now, we know that
$$
\overline{\NE(X)}\subset\cN_{\NS}\subset
\overline{\cK}_{\NS}^\smallvee=\Nef(X)^\smallvee,
$$
where the second inclusion is a consequence of (15.4~a). However,
it is already well-known that $\overline{\NE(X)}$ and
$\overline{\NE(X)}$ are dual cones (see [Har70]), hence the inclusions are
equalities (we could also obtain a self-contained proof by reconsidering 
the arguments used for (14.9~a) when $\alpha$ and
$\omega_0$ are rational classes; one sees by the density of the rationals
that the numerical condition for $\alpha$ is needed only for
elements of the form $[Y]\wedge\omega^{p-1}$ with 
$\omega\in\bQ\{\alpha\}+\bQ\{\omega_0\}$ a rational class,
so $[Y]\wedge\omega^{p-1}$ is then a $\bQ$-effective curve).\qed
\endproof
%
\subsection*{\csect.B. Zariski decomposition and mobile intersections}
Let $X$ be compact K\"ahler and let $\alpha\in\cE^\circ$ be in the
{\it interior} of the pseudo--effective cone. In analogy with the
algebraic context such a class $\alpha$ is called ``big'', and it can
then be represented by a {\it K\"ahler current} $T$, i.e.\ a closed
positive $(1,1)$-current $T$ such that $T\ge \delta\omega$ for some
smooth hermitian metric $\omega$ and a constant $\delta\ll 1$. We first
need a variant of the regularization theorem proved in section~9.B.

\claim{(15.7) Regularization theorem for currents} 
Let $X$ be a compact complex manifold equipped with a hermitian metric
$\omega$. Let $T=\alpha+i\ddbar\varphi$ be a closed $(1,1)$-current on~$X$,
where $\alpha$ is smooth and $\varphi$ is a quasi-plurisubharmonic function. 
Assume that $T\ge\gamma$ for some real $(1,1)$-form $\gamma$ on $X$ with 
real coefficients. Then there exists 
a sequence $T_m=\alpha+i\ddbar\varphi_m$ of closed $(1,1)$-currents such that
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} $\varphi_m$ $($and thus $T_m)$ is smooth on the complement
$X\ssm Z_m$ of an analytic set $Z_m$, and the $Z_m$'s form an increasing
sequence
$$
Z_0\subset Z_1\subset\ldots\subset Z_m\subset\ldots \subset X.
$$
\vskip3pt
\plainitem{\rm (ii)} There is a uniform estimate $T_m\ge \gamma-\delta_m\omega$
with $\lim\downarrow\delta_m=0$ as $m$ tends to $+\infty$.
\vskip3pt
\plainitem{\rm (iii)} The sequence $(\varphi_m)$ is non increasing, and we 
have $\lim\downarrow\varphi_m=\varphi$. As a consequence, $T_m$ converges
weakly to $T$ as  $m$ tends to $+\infty$.
\vskip3pt
\plainitem{\rm (iv)} Near $Z_m$, the potential $\varphi_m$ has logarithmic poles,
namely, for every $x_0\in Z_m$, there is a neighborhood $U$ of $x_0$
such that $\varphi_m(z)=\lambda_m\log\sum_\ell|g_{m,\ell}|^2+O(1)$ for 
suitable holomorphic functions $(g_{m,\ell})$ on $U$ and $\lambda_m>0$.
Moreover, there is a $($global$)$ proper modification $\mu_m:\widetilde X_m\to X$ 
of $X$, obtained as a sequence of blow-ups with smooth centers, such that 
$\varphi_m\circ\mu_m$ can be written locally on $\widetilde X_m$ as
$$
\varphi_m\circ\mu_m(w)=\lambda_m\big(\sum n_\ell\log|\widetilde g_\ell|^2+f(w)\big)
$$
where $(\widetilde g_\ell=0)$ are local generators of suitable $($global$)$
divisors $D_\ell$ on $\widetilde X_m$ such that $\sum D_\ell$ has normal crossings,
$n_\ell$ are positive integers, and the $f$'s are smooth functions 
on~$\widetilde X_m$.
\vskip0pt
}
\endclaim
%
\medskip
{\it Sketch of proof.} We essentially repeat the proofs of Theorems (9.2)
and (9.12) with additional considerations. One fact that does not 
follow readily from these proofs is the
monotonicity of the sequence $\varphi_m$ (which we will not really need
anyway). For this, we can take $m=2^\nu$ and use the subadditivity
technique already explained in Step 3 of the proof  of Theorem 
(11.3$\,$b). The map $\mu_m$ is obtained by blowing-up the 
(global) ideals $\cJ_m$ defined by the holomorphic functions $(g_{j,m})$
in the local approximations $\varphi_m\sim{1\over 2m}\log\sum_j|g_{j,m}|^2$.
By Hironaka [Hir64], we can achieve that $\mu_m^*\cJ_m$ is an invertible
ideal sheaf associated with a normal crossing divisor.\qed

\claim{(15.8) Corollary} If $T$
is a K\"ahler current, then one can write $T=\lim T_m$ for a sequence of 
K\"ahler currents $T_m$ which have
logarithmic poles with coefficients in ${1\over m}\bZ$, i.e.\ there
are modifications $\mu_m:X_m\to X$ such that
$$\mu_m^\star T_m=[E_m]+\beta_m$$
where $E_m$ is an effective $\bQ$-divisor on $X_m$ with coefficients
in ${1\over m}\bZ$ $($the ``fixed part''$)$ and $\beta_m$ is a closed 
semi-positive form $($the ``mobile part''$)$.
\endclaim
%
\begproof{Proof.} We apply Theorem (15.7) with $\gamma=\varepsilon\omega$ and
$m$ so large that $\delta_m\le\varepsilon/2$. Then $T_m$ has analytic
singularities and $T_m\ge{\varepsilon\over 2}\omega$, so we get 
a composition of blow-ups $\mu_m:X_m\to X$ such 
$$\mu_m^*T_m=[E_m]+\beta_m,$$
where $E_m$ is an effective $\bQ$-divisor and 
$\beta_m\ge{\varepsilon\over 2}\mu_m^*\omega$. In particular,
$\beta_m$ is strictly positive outside the exceptional divisors, by playing 
with the multiplicities of the components of the exceptional divisors in
$E_m$, we could even achieve that $\beta_m$ is a 
K\"ahler class on $X_m$. Notice also that by construction, $\mu_m$
is obtained by blowing-up the multiplier ideal sheaves $\cI(mT)=
\cI(m\varphi)$ associated to a potential $\varphi$ of $T$.\qed
\endproof
%
The more familiar algebraic analogue would be to take \hbox{$\alpha=c_1(L)$}
with a big line bundle $L$ and to blow-up the base locus of $|mL|$,
$m\gg 1$, to get a $\bQ$-divisor decomposition
$$
\mu_m^\star L\sim E_m+D_m,\qquad E_m~~\hbox{effective},~~D_m~~\hbox{free}.
$$
Such a blow-up is usually referred to as a ``log resolution'' of the
linear system $|mL|$, and we say that $E_m+D_m$ is an approximate
Zariski decomposition of $L$. We will also use this terminology for K\"ahler
currents with logarithmic poles.
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\LabelTeX -11 17 $\widetilde\alpha$\ELTX
\LabelTeX -15 6 $[E_m]$\ELTX
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\LabelTeX 21 17 $\widetilde\alpha=\mu_m^\star\alpha=[E_m]+\beta_m$\ELTX
\EndFig
\vskip4mm
$~$

\claim{(15.9) Definition} We define the {\rm volume}, or {\rm mobile
self-intersection} of a big class $\alpha\in\cE^\circ$ to be
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\widetilde X}\beta^n>0
$$
where the supremum is taken over all K\"ahler currents $T\in \alpha$
with logarithmic poles, and $\mu^\star T=[E]+\beta$ with
respect to some modification $\mu:\widetilde X\to X$.
\endclaim
%
By Fujita [Fuj94] and Demailly-Ein-Lazarsfeld [DEL00], if $L$ 
is a big line bundle, we have
$$
\Vol(c_1(L))=\lim_{m\to+\infty}D_m^n=\lim_{m\to+\infty}{n!\over m^n}h^0(X,mL),
$$
and in these terms, we get the following statement.

\claim{(15.10) Proposition} Let $L$ be a big line bundle on the
projective manifold $X$.  Let $\epsilon > 0$. Then there exists a
modification $\mu: X_{\epsilon} \to X$ and a decomposition $\mu^*(L) =
E + \beta $ with $E$ an effective $\bQ$-divisor and $\beta$ a big and
nef $\bQ$-divisor such that
$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$
\endclaim
%
It is very useful to observe that the supremum in Definition 15.9 is actually
achieved by a collection of currents whose singularities satisfy a filtering 
property.
Namely, if $T_1=\alpha+i\ddbar\varphi_1$ and $T_2=\alpha+i\ddbar\varphi_2$ 
are two K\"ahler currents with logarithmic poles in the class of $\alpha$, 
then 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi=\max(\varphi_1,\varphi_2)\leqno(15.11)$$ 
is again a K\"ahler current with weaker 
singularities than $T_1$ and $T_2$. One could define as well 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi={1\over 2m}\log(e^{2m\varphi_1}+e^{2m\varphi_2}),\leqno(15.11')$$
where $m={\rm lcm}(m_1,m_2)$ is the lowest common multiple of the denominators
occuring in $T_1$, $T_2$. Now, take a simultaneous log-resolution
$\mu_m:X_m\to X$ for which the singularities of $T_1$ and $T_2$ are resolved 
as $\bQ$-divisors $E_1$
and $E_2$. Then clearly the associated divisor in the decomposition 
$\mu_m^\star T=[E]+\beta$ is given by $E=\min(E_1,E_2)$. By doing so, 
the volume $\int_{X_m}\beta^n$ gets increased, as we shall see in the
proof of Theorem 15.12 below.

\claim{(15.12) Theorem {\rm (Boucksom [Bou02])}} Let $X$ be a compact K\"ahler
manifold. We denote here by $H^{k,k}_{\ge 0}(X)$ the cone of cohomology classes
of type $(k,k)$ which have non-negative intersection with all closed
semi-positive smooth forms of bidegree $(n-k,n-k)$.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm(i)} For each integer $k=1,2,\ldots,n$, there exists a canonical
``mobile intersection product''
$$
\cE\times\cdots\times\cE\to H^{k,k}_{\ge 0}(X), \quad
(\alpha_1,\ldots,\alpha_k)\mapsto \langle\alpha_1\cdot\alpha_2\cdots
\alpha_{k-1}\cdot \alpha_k\rangle
$$
such that $\Vol(\alpha)=\langle\alpha^n\rangle$ whenever $\alpha$ is
a big class.
\vskip3pt
\plainitem{\rm(ii)} The product is increasing, homogeneous of 
degree $1$ and superadditive in each argument, i.e.\
$$
\langle\alpha_1\cdots(\alpha'_j+\alpha''_j)\cdots \alpha_k\rangle\ge
\langle\alpha_1\cdots\alpha'_j\cdots \alpha_k\rangle+
\langle\alpha_1\cdots\alpha''_j\cdots \alpha_k\rangle.
$$
It coincides with the ordinary intersection
product when the $\alpha_j\in\overline{\cK}$ are nef classes.
\vskip3pt
\plainitem{\rm(iii)} 
The mobile intersection product satisfies the Teissier-Hovanskii inequalities
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_n\rangle\ge
(\langle\alpha_1^n\rangle)^{1/n}\ldots(\langle\alpha_n^n\rangle)^{1/n}\qquad
\hbox{$($with $\langle\alpha_j^n\rangle=\Vol(\alpha_j)\,)$}.
$$
\vskip3pt
\plainitem{\rm(iv)} For $k=1$, the above ``product'' reduces to a
$($non linear$)$ projection operator 
$$
\cE\to\cE_1,\qquad \alpha\to\langle\alpha\rangle
$$
onto a certain convex subcone $\cE_1$ of $\cE$ such that
$\overline{\cK}\subset\cE_1\subset\cE$. Moreover, there is
a ``divisorial Zariski decomposition''
$$
\alpha=\{N(\alpha)\}+\langle\alpha\rangle
$$
where $N(\alpha)$ is a uniquely defined effective divisor which is
called the ``negative divisorial part'' of $\alpha$. The map
$\alpha\mapsto N(\alpha)$ is homogeneous and subadditive, and 
$N(\alpha)=0$ if and only if $\alpha\in\cE_1$.
\vskip3pt
\plainitem{\rm(v)} The components of $N(\alpha)$ always consist of
divisors whose cohomology classes are linearly independent,
especially $N(\alpha)$ has at most $\rho=\rank_\bZ\NS(X)$ components.
\vskip0pt}
\endclaim
%
\begproof{Proof.} We essentially repeat the arguments developped in [Bou02], with
some simplifications arising from the fact that $X$ is supposed to
be K\"ahler from the start.
\vskip3pt

\noindent
(i) First assume that all classes $\alpha_j$ are big, i.e.\ 
$\alpha_j\in\cE^\circ$. Fix a smooth closed $(n-k,n-k)$ {\it semi-positive}
form $u$ on $X$. We select K\"ahler currents $T_j\in\alpha_j$ with
logarithmic poles, and a simultaneous log-resolution
$\mu:\widetilde X\to X$ such that 
$$
\mu^\star T_j=[E_j]+\beta_j.
$$
We consider the direct image current
$\mu_\star(\beta_1\wedge\ldots\wedge\beta_k)$ (which is a closed
positive current of bidegree $(k,k)$ on $X$) and the corresponding
integrals
$$
\int_{\widetilde X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u\ge 0.
$$
If we change the representative $T_j$ with another current $T_j'$,
we may always take a simultaneous log-resolution such that
$\mu^\star T_j'=[E_j']+\beta_j'$, and by using $(15.11')$ we can always
assume that $E'_j\le E_j$. Then $D_j=E_j-E'_j$ is an effective
divisor and we find $[E_j]+\beta_j\equiv [E'_j]+\beta'_j$, hence
$\beta'_j\equiv\beta_j+[D_j]$. A substitution in the integral implies
$$
\eqalign{
\int_{\widetilde X}\beta_1'\wedge\beta_2
&{}\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&=\int_{\widetilde X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u+
\int_{\widetilde X}[D_1]\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&\ge \int_{\widetilde X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge
\mu^\star u.\cr}
$$
Similarly, we can replace successively all forms $\beta_j$ by the $\beta'_j$,
and by doing so, we find
$$
\int_{\widetilde X}\beta'_1\wedge\beta'_2\wedge\ldots\wedge\beta'_k\wedge\mu^\star u
\ge\int_{\widetilde X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u.
$$
We claim that the closed positive currents $\mu_\star(\beta_1\wedge\ldots
\wedge\beta_k)$ are uniformly bounded in mass. In fact, if $\omega$ is a 
K\"ahler metric in $X$, there exists a constant $C_j\ge 0$
such that $C_j\{\omega\}-\alpha_j$ is a K\"ahler class. Hence
$C_j\omega-T_j\equiv\gamma_j$ for some K\"ahler form $\gamma_j$ on $X$.
By pulling back with $\mu$, we find
$C_j\mu^\star \omega-([E_j]+\beta_j)\equiv\mu^\star \gamma_j$, hence 
$$\beta_j\equiv C_j\mu^\star \omega-([E_j]+\mu^\star \gamma_j).$$
By performing again a substitution in the integrals, we find
$$
\int_{\widetilde X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u
\le C_1\ldots C_k\int_{\widetilde X}\mu^\star\omega^k\wedge\mu^\star u
= C_1\ldots C_k\int_X\omega^k\wedge u
$$
and this is true especially for $u=\omega^{n-k}$. We can now arrange
that for each of the integrals associated with a countable dense family 
of forms $u$,  the supremum is achieved by a sequence of currents 
$(\mu_m)_\star(\beta_{1,m}\wedge\ldots\wedge\beta_{k,m})$ obtained
as direct images by a suitable sequence of modifications $\mu_m:
\smash{\widetilde X_m}\to X$.
By extracting a subsequence, we can achieve that this sequence is weakly 
convergent and we set
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\uparrow}\limits_{m\to+\infty}
\{(\mu_m)_\star(\beta_{1,m}\wedge\beta_{2,m}\wedge\ldots\wedge\beta_{k,m})\}
$$
(the monotonicity is not in terms of the currents themselves, but in terms
of the integrals obtained when we evaluate against a smooth closed
semi-positive form~$u$). By evaluating against a basis of positive classes
$\{u\}\in H^{n-k,n-k}(X)$, we infer by Serre duality that the class of
$\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle$ is uniquely defined
(although, in general, the representing current is not unique). 
\vskip3pt

\noindent
(ii) It is indeed clear from the definition that the mobile intersection 
product is homogeneous, increasing and superadditive in each argument, at least
when the $\alpha_j$'s are in $\cE^\circ$. However, we can extend the product 
to the closed cone $\cE$ by monotonicity, by setting
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\downarrow}\limits_{\delta\downarrow 0}
\langle(\alpha_1+\delta\omega)\cdot(\alpha_2+\delta\omega)\cdots 
(\alpha_k+\delta\omega)\rangle
$$
for arbitrary classes $\alpha_j\in\cE$ (again, monotonicity occurs
only where we evaluate against closed semi-positive forms~$u$).
By weak compactness, the mobile intersection product can always be 
represented by a closed positive current of bidegree $(k,k)$.
\vskip3pt

\noindent
(iii) The Teissier-Hovanskii inequalities are a direct consequence of the
fact that they hold true for nef classes, so we just have to apply them
to the classes $\beta_{j,m}$ on $\smash{\widetilde X_m}$ and pass to the limit.
\vskip3pt

\noindent
(iv) When $k=1$ and $\alpha\in\cE^0$, we have 
$$
\alpha=\lim_{m\to+\infty}\{(\mu_m)_\star T_m\}=
\lim_{m\to+\infty}(\mu_m)_\star[E_m]+\{(\mu_m)_\star\beta_m\}
$$
and $\langle\alpha\rangle=\lim_{m\to+\infty}\{(\mu_m)_\star\beta_m\}$ by 
definition. However, the images $F_m=(\mu_m)_\star F_m$ are effective
$\bQ$-divisors in $X$, and the filtering property implies that $F_m$ is a
decreasing sequence. It must therefore converge to a (uniquely defined)
limit $F=\lim F_m := N(\alpha)$ which is an effective $\bR$-divisor, 
and we get the asserted decomposition in the limit. 

Since $N(\alpha)=\alpha-\langle\alpha\rangle$ we easily see that
$N(\alpha)$ is subadditive and that $N(\alpha)=0$ if $\alpha$ is the
class of a smooth semi-positive form. When $\alpha$ is no longer a big 
class, we define
$$
\langle\alpha\rangle=\lim_{\delta\downarrow 0}\downarrow\langle\alpha+
\delta\omega\rangle,\qquad
N(\alpha)=\lim_{\delta\downarrow 0}\uparrow N(\alpha+\delta\omega)
$$
(the subadditivity of $N$ implies 
$N(\alpha+(\delta+\varepsilon)\omega)\le N(\alpha+\delta\omega)$).
The divisorial Zariski decomposition follows except maybe for the fact 
that $N(\alpha)$ might be a convergent countable sum of divisors. 
However, this will be ruled out when (v) is proved. As 
$N({\scriptscriptstyle\bullet})$ is subadditive
and homogeneous, the set $\cE_1=\{\alpha\in\cE\;;\;N(\alpha)=0\}$ is a 
closed convex conne, and we find that $\alpha\mapsto\langle\alpha\rangle$
is a projection of $\cE$ onto $\cE_1$ (according to [Bou02], $\cE_1$
consists of those pseudo-effective classes which are ``nef in 
codimension~$1$'').
\vskip3pt

\noindent
(v) Let $\alpha\in\cE^\circ$, and assume that $N(\alpha)$ contains
linearly dependent components $F_j$. Then already all currents
$T\in \alpha$ should be such that $\mu^\star T=[E]+\beta$ where 
$F=\mu_\star E$ contains those linearly dependent components. 
Write $F=\sum\lambda_jF_j$, $\lambda_j>0$ and assume that 
$$\sum_{j\in J}c_jF_j\equiv 0$$
for a certain non trivial linear combination. Then some of the
coefficients $c_j$ must be negative (and some other positive).
Then $E$ is numerically equivalent to
$$
E'\equiv E+t\mu^\star\Big(\sum\lambda_jF_j\Big),
$$
and by choosing $t>0$ appropriate, we obtain an effective divisor $E'$ 
which has a zero coefficient on one of the components $\mu^\star F_{j_0}$.
By replacing $E$ with $\min(E,E')$ via $(15.11')$, we eliminate the component
$\mu^\star F_{j_0}$. This is a contradiction since $N(\alpha)$
was supposed to contain $F_{j_0}$.\qed
\endproof
%
\claim{(15.13) Definition} For a class $\alpha\in H^{1,1}(X,\bR)$, 
we define the {\rm numerical dimension} $\num(\alpha)$ to be 
$\num(\alpha)=-\infty$ if $\alpha$ is not pseudo-effective, and
$$
\num(\alpha)=\max\{p\in\bN\;;\;\langle\alpha^p\rangle\ne 0\},\qquad
\num(\alpha)\in\{0,1,\ldots,n\}
$$
if $\alpha$ is pseudo-effective.
\endclaim
%
By the results of [DP04], a class is big ($\alpha\in\cE^\circ$) if and only
if $\num(\alpha)=n$. Classes of numerical dimension $0$ can be
described much more precisely, again following Boucksom [Bou02].

\claim{(15.14) Theorem} Let $X$ be a compact
K\"ahler manifold. Then the subset $\cD_0$ of irreducible divisors 
$D$ in $X$ such that $\num(D)=0$ is countable, and these divisors 
are rigid as well as their multiples. If $\alpha\in\cE$ is a 
pseudo-effective class of numerical dimension~$0$, then $\alpha$ 
is numerically equivalent to 
an effective $\bR$-divisor $D=\sum_{j\in J}\lambda_j D_j$,
for some finite subset $(D_j)_{j\in J}\subset\cD_0$ such that the 
cohomology classes $\{D_j\}$ are linearly independent and
some \hbox{$\lambda_j>0$}. If such a linear combination is of
numerical dimension~$0$, then so is any other linear combination of the
same divisors.
\endclaim
%
\begproof{Proof.} It is immediate from the definition that a pseudo-effective
class is of numerical dimension $0$ if and only if $\langle\alpha\rangle=0$,
in other words if $\alpha=N(\alpha)$. Thus $\alpha\equiv\sum\lambda_jD_j$
as described in 15.14, and since $\lambda_j\langle D_j\rangle\le 
\langle\alpha\rangle$, the divisors $D_j$ must themselves have numerical 
dimension $0$. There is at most one such divisor $D$ in any given cohomology
class in $NS(X)\cap\cE\subset H^2(X,\bZ)$, otherwise two such divisors 
$D\equiv D'$ would yield a blow-up $\mu:\widetilde X\to X$ resolving the 
intersection, and by taking
$\min(\mu^\star D,\mu^\star D')$ via $(15.11')$, we would find 
$\mu^\star D\equiv E+\beta$, $\beta\ne 0$, so that
$\{D\}$ would not be of numerical dimension $0$. This implies 
that there are at most countably many divisors of numerical 
dimension~$0$, and that these divisors are rigid as well as their
multiples.\qed
\endproof
%
\claim{(15.15) Remark} {\rm If $L$ is an arbitrary holomorphic line bundle, 
we define its numerical dimension to be $\num(L)=\num(c_1(L))$. Using 
the cananical maps $\Phi_{|mL|}$ and pulling-back the Fubini-Study metric
it is immediate to see that \hbox{$\num(L)\ge\kappa(L)$} (which generalizes
the analogue inequality already seen for nef line bundles,\break
see~(6.18)).}
\endclaim
%
The above general concept of numerical dimension leads to a very natural 
formulation of the abundance conjecture for K\"ahler varieties.

\claim{(15.16) Generalized abundance conjecture} Let $X$ be an 
arbitrary compact K\"ahler manifold~$X$.
\vskip3pt
\plainitem{\rm(a)} The Kodaira dimension of $X$ should be equal to its
numerical dimension~: $\kappa(K_X)=\num(K_X)$.
\vskip3pt
\plainitem{\rm(b)} More generally, let $\Delta$ be a $\bQ$-divisor which is
klt $($Kawamata log terminal, i.e.\ such that $c_X(\Delta)>1)$.
Then $\kappa(K_X+\Delta)=\num(K_X+\Delta)$.\vskip0pt
\endclaim
%
This appears to be a fairly strong statement. In fact, already in the case
$\Delta=0$, it is not difficult
to show that the generalized abundance conjecture would contain the
$C_{n,m}$ conjectures.

\claim{(15.17) Remark} {\rm It is obvious that abundance holds in the case
\hbox{$\num(K_X)=-\infty$} (if $L$ is not pseudo-effective, no
multiple of $L$ can have sections), or in the case \hbox{$\num(K_X)=n$}
which implies $K_X$ big (the latter property follows e.g.\ from 
the solution of the Grauert-Riemenschneider conjecture in the form proven
in [Dem85], see also [DP04]).

In the remaining cases, the most tractable situation is the case when 
$\num(K_X)=0$.
In fact Theorem 15.14 then gives $K_X\equiv\sum\lambda_j D_j$ for some
effective divisor with numerically independent components, $\num(D_j)=0$. 
It follows that the $\lambda_j$ are rational and therefore
$$
K_X\sim \sum\lambda_j D_j+F\qquad
\hbox{where $\lambda_j\in\bQ^+$, $\num(D_j)=0$ and $F\in\Pic^0(X)$.}
\leqno(*)
$$
If we assume additionally that $q(X)=h^{0,1}(X)$ is zero, then 
$mK_X$ is linearly equivalent
to an integral divisor for some multiple $m$, and it follows 
immediately that $\kappa(X)=0$. The case of a general projective
manifold with \hbox{$\num(K_X)=0$} and positive 
irregularity $q(X)>0$ has been solved by Campana-Peternell
[CP04], Corollary~3.7. It would be interesting to understand 
the K\"ahler  case as well.}
\endclaim
%
\subsection*{\csect.C. The orthogonality estimate}
The goal of this section is to show that, in an appropriate sense,
approximate Zariski decompositions are almost orthogonal.

\claim{(15.18) Theorem} Let $X$ be a projective manifold, and
let $\alpha=\{T\}\in\cE^\circ_{\NS}$ be a big class represented by
a K\"ahler current~$T$. Consider
an approximate Zariski decomposition
$$\mu_m^\star T_m = [E_m]+[D_m]$$
Then
$$
(D_m^{n-1}\cdot E_m)^2\le 20\,(C\omega)^n\big(\Vol(\alpha)-D_m^n\big)
$$
where $\omega=c_1(H)$ is a K\"ahler form and $C\ge 0$ is a constant such that
$\pm\alpha$ is dominated by $C\omega$ $($i.e., $C\omega\pm\alpha$ is nef$\,)$.
\endclaim
%
\begproof{Proof.} For every $t\in[0,1]$, we have
$$
\Vol(\alpha)=\Vol(E_m+D_m)\ge \Vol(tE_m+D_m).
$$
Now, by our choice of $C$, we can write $E_m$ as a difference of two
nef divisors
$$E_m=\mu^\star\alpha-D_m=\mu_m^\star(\alpha+C\omega)-(D_m+C\mu_m^\star
\omega).$$
\endproof
%
\claim{(15.19) Lemma} For all nef $\bR$-divisors $A$, $B$ we have
$$\Vol(A-B)\ge A^n - nA^{n-1}\cdot B$$
as soon as the right hand side is positive.
\endclaim
%
\begproof{Proof.} In case $A$ and $B$ are integral (Cartier) divisors, this is a 
consequence of the holomorphic Morse inequalities 7.4 (see [Dem01]); 
one can also argue by an elementary estimate of 
to $H^0(X, mA-B_1-\ldots-B_m)$ via the Riemann-Roch formula
(assuming $A$ and $B$ very ample, $B_1,\ldots,B_m\in|B|$ generic). If $A$ and 
$B$ are $\bQ$-Cartier, we conclude by the homogeneity of the volume. The 
general case of $\bR$-divisors follows by approximation using the
upper semi-continuity of the volume [Bou02, 3.1.26].\qed 
\endproof
%
\claim{(15.20) Remark} {\rm We hope that Lemma 15.19 also holds true on an
arbitrary K\"ahler mani\-fold for arbitrary nef (non neces\-sarily
integral) classes. This would follow from a generalization of
holomorphic Morse inequalities to non integral classes. However 
the proof of such a result seems technically much more involved than
in the case of integral classes.}
\endclaim
 
\claim{(15.21) Lemma} Let $\beta_1,\ldots,\beta_n$ and 
$\beta'_1,\ldots,\beta'_n$ be nef classes on a compact K\"ahler manifold
$\widetilde X$ such that each difference $\beta'_j-\beta_j$ is pseudo-effective. Then
the $n$-th intersection products satisfy
$$
\beta_1\cdots\beta_n \le \beta'_1\cdots\beta'_n.
$$ 
\endclaim
%
\begproof{Proof.} We can proceed step by step and replace just one $\beta_j$ by
$\beta'j\equiv\beta_j+T_j$ where $T_j$ is a closed positive
$(1,1)$-current and the other classes $\beta'_k=\beta_k$, $k\ne j$ are
limits of K\"ahler forms. The inequality is then obvious.\qed
\endproof
%
\noindent
{\it End of proof of Theorem 15.18.}
In order to exploit the lower bound of the volume, we write
$$
tE_m+D_m=A-B,\qquad
A=D_m+t\mu_m^\star(\alpha+C\omega),\quad B=t(D_m+C\mu_m^\star\omega).
$$
By our choice of the constant $C$, both $A$ and $B$ are nef. Lemma 15.19 and 
the binomial formula imply
$$\eqalign{
\Vol(tE_m+&D_m)\ge A^n - nA^{n-1}\cdot B\cr
&=D_m^n+nt\,D_m^{n-1}\cdot\mu_m^\star(\alpha+C\omega)+\sum_{k=2}^n
t^k{n\choose k}D_m^{n-k}\cdot\mu_m^\star(\alpha+C\omega)^k\cr
&\phantom{{}=D_m^n}-nt\,D_m^{n-1}\cdot(D_m+C\mu_m^\star\omega)\cr
&\phantom{{}=D_m^n}-nt^2\sum_{k=1}^{n-1}t^{k-1}{n-1\choose k}
D_m^{n-1-k}\cdot\mu_m^\star(\alpha+C\omega)^k\cdot(D_m+C\mu_m^\star\omega).
\kern-4pt\cr}
$$
Now, we use the obvious inequalities
$$
D_m\le \mu_m^\star(C\omega),\qquad
\mu_m^\star(\alpha+C\omega)\le 2\mu_m^\star(C\omega),\qquad
D_m+C\mu_m^\star\omega\le 2\mu_m^\star(C\omega)
$$
in which all members are nef (and where the inequality${}\le$ means 
that the difference of classes is pseudo-effective). We use Lemma~15.21
to bound the last summation in the estimate of the volume, and in this
way we get
$$
\Vol(tE_m+D_m)\ge D_m^n+nt D_m^{n-1}\cdot E_m-nt^2\sum_{k=1}^{n-1}2^{k+1}
t^{k-1}{n-1\choose k}(C\omega)^n.
$$
We will always take $t$ smaller than $1/10n$ so that the last summation is 
bounded by $4(n-1)(1+1/5n)^{n-2}<4ne^{1/5}<5n$. This implies
$$
\Vol(tE_m+D_m)\ge D_m^n+nt\,D_m^{n-1}\cdot E_m-5n^2t^2(C\omega)^n.
$$
Now, the choice $t={1\over 10n}(D_m^{n-1}\cdot E_m)((C\omega)^n)^{-1}$ 
gives by substituting
$$
\eqalign{
{1\over 20}{(D_m^{n-1}\cdot E_m)^2\over (C\omega)^n}
&\le \Vol(E_m+D_m)-D_m^n\le \Vol(\alpha)-D_m^n\cr}
$$
(and we have indeed $t\le {1\over 10n}$ by Lemma 15.21), whence Theorem 15.18. 
Of course, the constant 20 is certainly not optimal.\qed

\claim{(15.22) Corollary} If $\alpha\in\cE_{\NS}$, then the divisorial
Zariski decomposition $\alpha=N(\alpha)+\langle\alpha\rangle$
is such that
$$
\langle\alpha^{n-1}\rangle\cdot N(\alpha)=0.
$$
\endclaim
%
\begproof{Proof.} By replacing $\alpha$ with $\alpha+\delta c_1(H)$, one sees that
it is sufficient to consider the case where $\alpha$ is big. Then the 
orthogonality estimate implies
$$
(\mu_m)_\star(D_m^{n-1})\cdot (\mu_m)_\star E_m
= D_m^{n-1}\cdot(\mu_m)^\star (\mu_m)_\star E_m
\le D_m^{n-1}\cdot E_m \le C (\Vol(\alpha)-D_m^n)^{1/2}.
$$
Since $\langle\alpha^{n-1}\rangle = \lim (\mu_m)_\star(D_m^{n-1})$,
$N(\alpha)=\lim (\mu_m)_\star E_m$ and $\lim D_m^n=\Vol(\alpha)$,
we get the desired conclusion in the limit.\qed
\endproof
%
\subsection*{\csect.D. Dual of the pseudo-effective cone}
The following statement was first proved in [BDPP04].

\claim{(15.23) Theorem} If $X$ is projective, the cones
$\cE_{\NS}=\overline{\Eff(X)}$ and $\overline{\ME^s(X)}$ are dual.
\endclaim
%
In other words, a line bundle $L$ is pseudo-effective if (and
only if) $L\cdot C\ge 0$ for all {\it mobile curves}, i.e., $L\cdot C\ge 0$
for every very generic curve $C$ (not contained in a countable union of 
algebraic subvarieties). In fact, by definition of $\ME^s(X)$, it is enough 
to consider only those curves $C$ which are images of generic complete 
intersection of very ample divisors on some variety $\widetilde X$, under
a modification \hbox{$\mu:\widetilde X\to X$}.
By a standard blowing-up argument, it also follows that a line bundle
$L$ on a normal Moishezon variety is pseudo-effective if and only if
$L \cdot C \ge 0$ for every mobile curve~$C$. 

\begproof{Proof.} By (15.4~b) we have 
in any case
$$\cE_{\NS}\subset (\ME^s(X))^\smallvee.$$ 
If the inclusion is strict, there is an element
$\alpha\in\partial\cE_{\NS}$ on the boundary of $\cE_{\NS}$ which is in
the interior of $\ME^s(X)^\smallvee$.

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\vskip8mm

Let $\omega=c_1(H)$ be an ample class. Since $\alpha\in\partial\cE_{\NS}$, 
the class $\alpha+\delta\omega$ is big for every $\delta>0$, and since
$\alpha\in((\ME^s(X))^{\smallvee})^\circ$ we still have 
$\alpha-\varepsilon\omega\in(\ME^s(X))^{\smallvee}$ for
$\varepsilon>0$ small. Therefore
$$
\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma\leqno(15.24)
$$
for every strongly mobile curve $\Gamma$, and therefore for every
$\Gamma\in\overline{\ME^s(X)}$. We are going to contradict (15.24).
Since $\alpha+\delta\omega$ is big, we have an approximate Zariski
decomposition
$$
\mu_\delta^\star(\alpha+\delta\omega)=E_\delta+D_\delta.
$$
We pick $\Gamma=(\mu_\delta)_\star(D_\delta^{n-1})\in\overline{\ME^s(X)}$.
By the Hovanskii-Teissier concavity inequality
$$
\omega\cdot \Gamma\ge (\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}.
$$
On the other hand
$$
\eqalign{
\alpha\cdot \Gamma&=\alpha\cdot(\mu_\delta)_\star(D_\delta^{n-1})\cr
&=\mu_\delta^\star\alpha\cdot D_\delta^{n-1}\le
\mu_\delta^\star(\alpha+\delta\omega)\cdot D_\delta^{n-1}\cr
&=(E_\delta+D_\delta)\cdot D_\delta^{n-1}
=D_\delta^n+D_\delta^{n-1}\cdot E_\delta.\cr
}
$$
By the orthogonality estimate, we find
$$\eqalign{
{\alpha\cdot \Gamma\over
\omega\cdot \Gamma}
&\le {D_\delta^n + \big(20(C\omega)^n
(\Vol(\alpha+\delta\omega)-D_\delta^n)\big)^{1/2}\over
(\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}}\cr
&\le C'(D_\delta^n)^{1/n}+C''{(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}\over
(D_\delta^n)^{(n-1)/n}}.\cr}
$$
However, since $\alpha\in\partial\cE_{\NS}$, the class $\alpha$ cannot be big 
so
$$\lim_{\delta \to 0}D_\delta^n=\Vol(\alpha)=0.$$ 
We can also take $D_\delta$ to approximate $\Vol(\alpha+\delta\omega)$ 
in such a way that the term
$(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}$ tends to $0$ much faster than
$D_\delta^n$. Notice that \hbox{$D_\delta^n\ge\delta^n\omega^n$}, so in fact
it is enough to take
$$
\Vol(\alpha+\delta\omega)-D_\delta^n\le \delta^{2n},
$$
which gives $(\alpha\cdot \Gamma)/(\omega\cdot \Gamma)\le (C'+C'')\delta$. 
This contradicts (15.24) for $\delta$ small.\qed
\endproof
%
\claim{(15.25) Conjecture} {\rm The K\"ahler analogue should be~:}\newline
For an arbitrary compact K\"ahler manifold $X$, the cones $\cE$ and $\cM$ 
are dual.
\endclaim
%
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\LabelTeX 1 52 $\cK$\ELTX
\LabelTeX -5 46 $\cK_{\NS}$\ELTX
\LabelTeX 15 32 $\cE$\ELTX
\LabelTeX -19 27 $\cE_{\NS}$\ELTX
\LabelTeX -26 4 $\NS_\bR(X)$\ELTX
\LabelTeX 6.5 4 $H^{1,1}(X,\bR)$\ELTX
\LabelTeX 57 46 $\cM_{\NS}$\ELTX
\LabelTeX 63 52 $\cM$\ELTX
\LabelTeX 76.3 32 $\cN$\ELTX
\LabelTeX 43 27 $\cN_{\NS}$\ELTX
\LabelTeX 73.5 4 $\NS^{n-1}_\bR(X)$\ELTX
\LabelTeX 32 4 $H^{n-1,n-1}(X,\bR)$\ELTX
\LabelTeX 25 47 duality\ELTX
\EndFig
\vskip8mm

If holomorphic Morse inequalities were known also
in the K\"ahler case, we would infer by the same proof that
``$\alpha$ not pseudo-effective'' implies the existence of a blow-up
$\mu:\widetilde X\to X$ and a K\"ahler metric 
$\widetilde\omega$ on $\smash{\widetilde X}$ such that 
$\alpha\cdot\mu_\star(\widetilde\omega)^{n-1} < 0$. In the special case
when $\alpha=K_X$ is not pseudo-effective, we would expect the 
K\"ahler manifold $X$ to be covered by rational curves.
The main trouble is that characteristic $p$ techniques are no longer
available. On the other hand it is tempting to approach the
question via techniques of symplectic geometry~:

\claim{(15.26) Question} Let $(M,\omega)$ be a compact real symplectic manifold.
Fix an almost complex structure $J$ compatible with $\omega$, and
for this structure, assume that \hbox{$c_1(M)\cdot\omega^{n-1}>0$}. 
Does it follow that $M$ is covered by rational $J$-pseudoholomorphic curves~?
\endclaim
%
The relation between the various cones of mobile curves and currents in 
(15.1) and (15.2) is now a rather direct consequence of 
Theorem 15.23. In fact, using 
ideas hinted in [DPS96], we can say a little bit more. Given 
an irreducible curve $C \subset X$, we consider its normal ``bundle''  
$N_C = \Hom(\cI/\cI^2,\cO_C)$, where $\cI$ is the ideal sheaf of $C$. 
If $C$ is a general member of a covering family $(C_t)$, then $N_C$ 
is nef. Now [DPS96] says that the dual cone of the pseudo-effective cone
of $X$ contains the closed cone spanned by curves with nef normal 
bundle, which in turn contains the cone of mobile curves. In this
way we get~:

\claim{(15.27) Theorem} Let $X$ be a projective manifold. Then the following 
cones coincide.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} the cone $\cM_{\NS}=\cM\cap \NS_\bR^{n-1}(X)\,;$
\vskip3pt
\plainitem{\rm (ii)} the cone $\cM^s_{\NS}=\cM^s\cap \NS_\bR^{n-1}(X)\,;$
\vskip3pt
\plainitem{\rm (iii)} the closed cone $\overline{\ME^s(X)}$ of strongly mobile 
curves$\,;$
\vskip3pt
\plainitem{\rm (iv)} the closed cone $\overline{\ME(X)}$ of mobile curves$\,;$
\vskip3pt
\plainitem{\rm (v)} the closed cone $\overline{\ME_{\nef}(X)}$ of curves with nef 
normal bundle.\vskip0pt
}
\endclaim
%
\begproof{Proof.} We have already seen that 
$$\ME^s(X)\subset\ME(X)\subset\ME_{\nef}(X)\subset(\cE_{\NS})^\smallvee$$
and 
$$
\ME^s(X)\subset\cM^s_{\NS}(X)\subset\cM_{\NS}\subset(\cE_{\NS})^\smallvee
$$ 
by 15.4~(iii). Now Theorem 15.23 implies $(\cM_{\NS})^\smallvee =
\overline{\ME^s(X)}$, and 15.27 follows.\qed
\endproof
%
\claim{(15.28) Corollary} Let $X$ be a projective manifold and $L$ a 
line bundle on $X$.
\vskip3pt
\plainitem{\rm (i)} $L$ is pseudo-effective if and only if $L \cdot C \ge 0$ 
for all curves $C$ with nef
normal sheaf $N_C$.
\vskip3pt
\plainitem{\rm (ii)} If $L$ is big, then $L \cdot C > 0$ for all curves $C$ 
with nef normal sheaf $N_C$.\vskip0pt
\endclaim
%
Corollary 15.28~(i) strenghtens results from [PSS99]. It is however 
not yet clear whether 
$\cM_{\NS}$ is equal to the closed cone of curves with 
{\it ample} normal bundle (although we certainly expect this to be true).
The most important special case of Theorem 15.23 is

\claim{(15.29) Theorem} If $X$ is a projective manifold, then
$K_X$ is pseudo-effective $($i.e.\ $K_X\in\cE_{\NS})$,
if and only if $X$ is not uniruled $($i.e.\ not covered by rational curves$)$.
\endclaim
%
\begproof{Proof.} If $X$ is covered by rational curves $C_t$, then it is well-known
that the normal bundle $N_{C_t}$ is nef for a general member $C_t$, thus
$$
K_X\cdot C_t=K_{C_t}\cdot C_t-N_{C_t}\cdot C_t\le -2,
$$
and $K_X$ cannot be pseudo-effective. Conversely,
if $K_X\notin\cE_{\NS}$, Theorem 15.23 shows that
there is a mobile curve $C_t$ such that $K_X\cdot C_t<0$. The standard
``bend-and-break'' lemma of Mori theory then produces a covering
family $\Gamma_t$ of 
{\rm rational curves} with $K_X\cdot\Gamma_t<0$, so $X$ is uniruled.\qed
\endproof
%
Notice that the generalized abundance conjecture 15.16 would then yield
the stronger result~:

\claim{(15.30) Conjecture} Let $X$ be a projective manifold. 
If $X$ is not uniruled, then $K_X$ is a $\bQ$-effective
divisor and \hbox{$\kappa(X)=\num(K_X)\ge 0$}.
\endclaim
\endsectionrunning
%
\section{Super-canonical metrics and abundance}
\subsection*{\csect.A. Construction of super-canonical metrics}

Let $X$ be a compact complex manifold and $(L,h_{L,\gamma})$ a holomorphic
line bundle over $X$ equipped with a singular hermitian metric
$h_{L,\gamma}=e^{-\gamma}h_L$ with satisfies $\int e^{-\gamma}<+\infty$
locally on~$X$, where $h_L$ is a smooth metric on~$L$. In fact, we can 
more generally consider the case where
$(L,h_{L,\gamma})$ is a ``hermitian $\bR$-line bundle''; by this we mean that
we have chosen a smooth real $d$-closed $(1,1)$ form $\alpha_L$ on~$X$
(whose $dd^c$ cohomology class is equal to $c_1(L))$, and a specific 
current $T_{L,\gamma}$ representing it, namely
$T_{L,\gamma}=\alpha_L+dd^c\gamma$, such that $\gamma$ is a locally
integrable function satisfying~$\int e^{-\gamma}<+\infty$. An important
special case is obtained by considering a klt (Kawamata log terminal) 
effective divisor~$\Delta$. In this situation
$\Delta=\sum c_j\Delta_j$ with $c_j\in\bR$, and if $g_j$ is a local
generator of the ideal sheaf $\cO(-\Delta_j)$ identifying it to the trivial
invertible sheaf $g_j\cO$, we take
$\gamma=\sum c_j\log|g_j|^2$, $T_{L,\gamma}=\sum c_j[\Delta_j]$
(current of integration on $\Delta$) and $\alpha_L$ given by any smooth
representative of the same $dd^c$-cohomology class; the klt condition
precisely means that
$$
\int_V e^{-\gamma}=\int_V \prod|g_j|^{-2c_j}<+\infty
\leqno(16.1)
$$ 
on a small neighborhood~$V$ of any point in the support 
$|\Delta|=\bigcup\Delta_j$ (condition (16.1) implies $c_j<1$
for every $j$, and this in turn is sufficient to imply $\Delta$ klt
if $\Delta$ is a normal crossing divisor; the line bundle $L$ is 
then the real line bundle $\cO(\Delta)$, which makes sens as a
genuine line bundle only if $c_j\in\bZ$).
For~each klt pair $(X,\Delta)$ such that $K_X+\Delta$
is pseudo-effective, H.~Tsuji [Tsu07a, Tsu07b] has introduced a
``super-canonical metric'' which generalizes the metric introduced
by Narasimhan and Simha [NS68] for projective algebraic varieties
with ample canonical divisor. We take the opportunity to present
here a simpler, more direct and more general approach. 

We assume from now on that $K_X+L$ is {\it pseudo-effective}, i.e.\ 
that the class
$c_1(K_X)+\{\alpha_L\}$ is pseudo-effective, and under this condition, we
are going to define a ``super-canonical metric'' on $K_X+L$. Select
an arbitrary smooth hermitian metric $\omega$ on $X$. We then find 
induced hermitian
metrics $h_{K_X}$ on $K_X$ and $h_{K_X+L}=h_{K_X}h_L$ on $K_X+L$, whose 
curvature is the smooth real $(1,1)$-form
$$
\alpha=\Theta_{K_X+L,h_{K_X+L}}=\Theta_{K_X,\omega}+\alpha_L.
$$
A singular hermitian metric
on $K_X+L$ is a metric of the form $h_{K_X+L,\varphi}=e^{-\varphi}h_{K_X+L}$
where $\varphi$ is locally
integrable, and by the pseudo-effectivity assumption, we can find 
quasi-psh functions $\varphi$ such that $\alpha+dd^c\varphi\ge 0$.
The metrics on $L$ and $K_X+L$ can now be ``subtracted'' to give rise
to a metric
$$
h_{L,\gamma}h_{K_X+L,\varphi}^{-1}=e^{\varphi-\gamma}h_Lh_{K_X+L}^{-1}
=e^{\varphi-\gamma}h_{K_X}^{-1}=e^{\varphi-\gamma}dV_\omega
$$
on $K_X^{-1}=\Lambda^nT_X$, since $\smash{h_{K_X}^{-1}}=dV_\omega$ is just
the hermitian $(n,n)$ volume form on~$X$. Therefore the integral
$\int_X h_{L,\gamma}h_{K_X+L,\varphi}^{-1}$ has an intrinsic meaning, and 
it makes sense to require that
$$
\int_X h_{L,\gamma}h_{K_X+L,\varphi}^{-1}
=\int_X e^{\varphi-\gamma}dV_\omega\le 1\leqno(16.2)
$$
in view of the fact that $\varphi$ is locally bounded from above and of
the assumption $\int e^{-\gamma}<+\infty$. Observe that condition (16.2) can
always be achieved by subtracting a constant to $\varphi$. Now, 
we can generalize Tsuji's super-canonical metrics on klt pairs
(cf.\ [Tsu07b]) as follows.

\claim{(16.3) Definition} Let $X$ be a compact 
complex manifold and let $(L,h_L)$ be a hermitian $\bR$-line bundle on $X$
associated with a smooth real closed $(1,1)$ form~$\alpha_L$.
Assume that $K_X+L$ is pseudo-effective and that $L$ is
equipped with a singular hermitian metric $h_{L,\gamma}=e^{-\gamma}h_L$ 
such that \hbox{$\int e^{-\gamma}<+\infty$} locally on~$X$. Take a hermitian
metric $\omega$ on $X$ and define 
$\alpha=\Theta_{K_X+L,h_{K_X+L}}=\Theta_{K_X,\omega}+\alpha_L$.
Then we define the super-canonical metric $h_\can$ of $K_X+L$ to be 
$$
\eqalign{
&h_{K_X+L,\can}=\inf_\varphi h_{K_X+L,\varphi}\quad
\hbox{i.e.}\quad h_{K_X+L,\can}=e^{-\varphi_{\can}}h_{K_X+L},~~
\hbox{where}\cr
&\varphi_{\can}(x)=\sup_\varphi\varphi(x)~~
\hbox{for all $\varphi$ with}~~\alpha+dd^c\varphi\ge 0,
~~\int_X e^{\varphi-\gamma}dV_\omega\le 1.\cr}
$$
\endclaim
%
In particular, this gives a definition of the super-canonical metric
on $K_X+\Delta$ for every klt pair $(X,\Delta)$ such that
$K_X+\Delta$ is pseudo-effective, and as an even more special case,
a super-canonical metric on $K_X$ when $K_X$ is pseudo-effective.

In the sequel, we assume that $\gamma$ has analytic singularities, otherwise
not much can be said. The mean value inequality then immediately shows that the 
quasi-psh functions $\varphi$ involved in definition (16.3) 
are globally uniformly bounded outside of the poles of $\gamma$, and
therefore everywhere on $X$, hence the envelopes
$\varphi_\can=\sup_\varphi\varphi$ are indeed well defined and bounded
above. As a consequence, we get a ``super-canonical'' current
\hbox{$T_\can=\alpha+dd^c\varphi_\can\ge 0$} and $h_{K_X+L,\can}$ satisfies
$$
\int_Xh_{L,\gamma}h_{K_X+L,\can}^{-1}=
\int_X e^{\varphi_\can-\gamma}dV_\omega<+\infty.\leqno(16.4)
$$
It is easy to see that in Definition (16.3) the supremum is a maximum and that 
$\varphi_\can=(\varphi_\can)^*$ every\-where, so that
taking the upper semicontinuous regularization is not needed. In fact
if $x_0\in X$ is given and we write
$$
(\varphi_\can)^*(x_0)=\limsup_{x\to x_0}\varphi_\can(x)
=\lim_{\nu\to+\infty}\varphi_\can(x_\nu)
=\lim_{\nu\to+\infty}\varphi_\nu(x_\nu)
$$
with suitable sequences $x_\nu\to x_0$ and $(\varphi_\nu)$ such that
$\int_Xe^{\varphi_\nu-\gamma}dV_\omega\le 1$, the well-known 
weak compactness properties of quasi-psh functions in  $L^1$ topology 
imply the existence of
a subsequence of $(\varphi_\nu)$ converging in $L^1$ and almost everywhere to 
a quasi-psh limit $\varphi$. Since $\int_Xe^{\varphi_\nu-\gamma}dV_\omega
\le 1$ holds true for every~$\nu$, Fatou's lemma 
implies that we have $\int_Xe^{\varphi-\gamma}dV_\omega\le 1$ in
the limit. By taking a subsequence, we can 
assume that $\varphi_\nu\to\varphi$ in $L^1(X)$. Then for every $\varepsilon>0$
the mean value $-\kern-8.7pt\int_{B(x_\nu,\varepsilon)}\varphi_\nu$ 
satisfies
$$
-\kern-10.7pt\int_{B(x_0,\varepsilon)}\varphi=\lim_{\nu\to +\infty}
-\kern-10.7pt\int_{B(x_\nu,\varepsilon)}\varphi_\nu\ge \lim_{\nu\to+\infty}
\varphi_\nu(x_\nu)=(\varphi_\can)^*(x_0),
$$
hence we get $\varphi(x_0)=\lim_{\varepsilon\to 0}
-\kern-8.7pt\int_{B(x_0,\varepsilon)}\varphi\ge 
(\varphi_\can)^*(x_0)\ge \varphi_\can(x_0)$, and therefore the 
sup is a maximum and $\varphi_\can=\varphi_\can^*$.
By elaborating on this argument, one can infer certain regularity
properties of the envelope. 

\claim{(16.5) Theorem {\rm ([BmD09])}} Let $X$ be a compact complex
manifold and $(L,h_L)$ a holomorphic $\bR$-line bundle such that
$K_X+L$ is big. Assume that $L$ is equipped with a singular hermitian
metric $h_{L,\gamma}=e^{-\gamma}h_L$ with analytic singularities such
that $\int e^{-\gamma}<+\infty$ $($klt condition$)$. Denote by $Z_0$
the set of poles of a singular metric $h_0=e^ {-\psi_0}h_{K_X+L}$ with
analytic singularities on~$K_X+L$ and by $Z_\gamma$ the poles
of~$\gamma$ $($assumed analytic$)$.  Then the associated
super-canonical metric $h_\can$ is continuous on $X\ssm (Z_0\cup
Z_\gamma)$.
\endclaim
%
In fact, using the regularization techniques of [Dem94a], it is shown
in [BmD09] that $h_\can$ possesses some computable logarithmic modulus of
continuity. In order to shorten the exposition, we will only give a proof 
of the continuity in the algebraic case, using approximation by 
pluri-canonical sections.

\claim{(16.6) Algebraic version of the super-canonical metric} 
{\rm Since the klt condition is open 
and $K_X+L$ is assumed to be~big, we can always perturb $L$ a 
little bit, and after
blowing-up $X$, assume that $X$ is projective and that
$(L,h_{L,\gamma})$ is obtained as a sum of $\bQ$-divisors
$$
L=G+\Delta
$$
where $\Delta$ is klt and $G$ is equipped with a smooth
metric $h_G$ (from which $h_{L,\gamma}$ is inferred, with $\Delta$
as its poles, so that $\Theta_{L,h_{L,\gamma}}=\Theta_{G,L_G}+[\Delta]$).
Clearly this situation is ``dense'' in what we
have been considering before, just as $\bQ$ is dense in $\bR$.
In this case, it is possible to 
give a more algebraic definition of the super-canonical
metric $\varphi_\can$, following the original idea of Narasimhan-Simha
[NS68] (see also H.~Tsuji [Tsu07a])~--~the case considered by these
authors is the special situation where $G=0$, $h_G=1$ (and moreover
$\Delta=0$ and $K_X$ ample, for [NS68]). In fact, if $m$ is a 
large integer which is a multiple of the denominators involved in
$G$ and $\Delta$, we can consider sections
$$\sigma\in H^0(X,m(K_X+G+\Delta)).$$
We view them rather as sections of $m(K_X+G)$ with poles along the 
support~$|\Delta|$ of our divisor. Then 
$(\sigma\wedge \overline\sigma)^{1/m}h_G$
is a volume form with
integrable poles along~$|\Delta|$ (this is the klt condition for~$\Delta$).
Therefore one can normalize $\sigma$ by requiring that
$$
\int_X (\sigma\wedge \overline\sigma)^{1/m}h_G=1.
$$
Each of these sections defines a singular hermitian metric on 
$K_X+L=K_X+G+\Delta$, and we can take the regularized upper envelope 
$$
\varphi_\can^\alg=\bigg(\sup_{m,\sigma}{1\over m}\log|\sigma|_{h_{K_X+L}^m}^2
\bigg)^*\leqno(16.7)
$$
of the weights associated with a smooth metric~$h_{K_X+L}$. It is clear that
$\varphi_\can^\alg\le \varphi_\can$ since the supremum is
taken on the smaller set of weights $\varphi={1\over m}
\smash{\log|\sigma|^2_{h_{K_X+L}^m}}$, and the equalities
$$
e^{\varphi-\gamma}dV_\omega=|\sigma|^{2/m}_{h_{K_X+L}^m}
e^{-\gamma}dV_\omega=(\sigma\wedge\overline\sigma)^{1/m}e^{-\gamma}h_L
=(\sigma\wedge\overline\sigma)^{1/m}h_{L,\gamma}
=(\sigma\wedge\overline\sigma)^{1/m}h_G
$$
imply $\int_X e^{\varphi-\gamma}dV_\omega\le 1$. We~claim that the inequality
$\varphi_\can^\alg\le \varphi_\can$ is an equality. The proof is an 
immediate consequence of the following
statement based in turn on the Ohsawa-Takegoshi theorem and
the approximation technique of [Dem92].}
\endclaim
%
\claim{(16.8) Proposition} With $L=G+\Delta$, $\omega$, 
$\alpha=\smash{\Theta_{K_X+L,h_{K_X+L}}}$, $\gamma$ 
as above and $K_X+L$ assumed to be~big, fix a singular hermitian metric 
$e^{-\varphi}h_{K_X+L}$ of curvature $\alpha+dd^c\varphi\ge 0$, such that
$\int_X e^{\varphi-\gamma}dV_\omega\le 1$. Then
$\varphi$ is equal to a regularized limit
\vskip-5pt
$$
\varphi=\bigg(\limsup_{m\to+\infty}{1\over m}\log|\sigma_m|_{h_{K_X+L}^m}^2
\bigg)^*
$$ 
for a suitable sequence $\sigma_m\in H^0(X,m(K_X+G+\Delta))$ with
$\int_X(\sigma_m\wedge\overline\sigma_m)^{1/m}h_G\le 1$.
\endclaim
%
\begproof{Proof.} By our assumption, there exists a quasi-psh 
function $\psi_0$ with
analytic singularity set $Z_0$ such that
$$
\alpha+dd^c\psi_0\ge \varepsilon_0\omega>0
$$
and we can assume $\int_Ce^{\psi_0-\gamma}dV_\omega<1$ (the strict inequality
will be useful later). For
$m\ge p\ge 1$, this defines a singular metric 
$\exp(-(m-p)\varphi-p\psi_0)h_{K_X+L}^m$
on $m(K_X+L)$ with curvature${}\ge p\varepsilon_0\omega$, and therefore
a singular metric 
$$
h_{L'}=\exp(-(m-p)\varphi-p\psi_0)h_{K_X+L}^mh_{K_X}^{-1}
$$
on $L'=(m-1)K_X+mL$, whose curvature
$\Theta_{L',h_{L'}}\ge (p\varepsilon_0-C_0)\omega$ is arbitrary large
if $p$ is large enough. Let us fix a finite covering of $X$
by coordinate balls. Pick a point $x_0$ and one of the coordinate
balls $B$ containing~$x_0$. By the Ohsawa-Takegoshi extension theorem
applied on the ball~$B$,
we can find a section $\sigma_B$ of $K_X+L'=m(K_X+L)$
which has norm $1$ at $x_0$ with respect to the metric
$h_{K_X+L'}$ and $\int_B|\sigma_B|_{h_{K_X+L'}}^2dV_\omega\le C_1$ for some
uniform constant $C_1$ depending on the finite covering, but independent of 
$m$, $p$, $x_0$ . Now, we use a cut-off function $\theta(x)$ with
$\theta(x)=1$ near $x_0$ to truncate
$\sigma_B$ and solve a $\dbar$-equation for $(n,1)$-forms with values
in $L$ to get a global section $\sigma$ on $X$ with 
$|\sigma(x_0)|_{h_{K_X+L'}}=1$. For this we need to multiply our metric
by a truncated factor $\exp(-2n\theta(x)\log|x-x_0|)$ so as to get
solutions of $\dbar$ vanishing at~$x_0$. However, this perturbs the
curvature by bounded terms and we can absorb them again by taking
$p$ larger. In~this way we obtain
$$
\int_X|\sigma|^2_{h_{K_X+L'}}dV_\omega=
\int_X |\sigma|^2_{h_{K_X+L}^m}e^{-(m-p)\varphi-p\psi_0}dV_\omega\le C_2.
\leqno(16.9)
$$
Taking $p>1$, the H\"older inequality for congugate exponents
$m$, ${m\over m-1}$ implies
$$
\eqalign{
\int_X(\sigma\wedge\overline\sigma)^{1\over m}h_G
&=\int_X|\sigma|^{2/m}_{h_{K_X+L}^m}e^{-\gamma}dV_\omega\cr
&=\int_X\Big(|\sigma|^2_{h_{K_X+L}^m}e^{-(m-p)\varphi-p\psi_0}\Big)^{1\over m}
\Big(e^{(1-{p \over m})\varphi+{p\over m}\psi_0-\gamma}\Big)dV_\omega\cr
&\le C_2^{1\over m}\bigg(\int_X
\Big(e^{(1-{p\over m})\varphi+{p\over m}\psi_0-\gamma}
\Big)^{m\over m-1}dV_\omega\bigg)^{m-1\over m}\cr
&\le C_2^{1\over m}\bigg(\int_X
 \big(e^{\varphi-\gamma}\big)^{m-p\over m-1}
\Big(e^{{p\over p-1}(\psi_0-\gamma)}\Big)^{p-1\over m-1}dV_\omega
\bigg)^{m-1\over m}\cr
&\le C_2^{1\over m}
\bigg(\int_X e^{{p\over p-1}(\psi_0-\gamma)}dV_\omega
\bigg)^{p-1\over m}\cr}
$$
using the hypothesis $\int_X e^{\varphi-\gamma}dV_\omega\le 1$ and another
application of H\"older's inequality. Since 
klt is an open condition and $\lim_{p\to+\infty}
\smash{\int_Xe^{{p\over p-1}(\psi_0-\gamma)}dV_\omega}
=\int_Xe^{\psi_0-\gamma}dV_\omega<1$, we can take
$p$ large enough to ensure that 
$$
\int_X e^{{p\over p-1}(\psi_0-\gamma)}dV_\omega\le C_3<1.
$$
Therefore, we see that 
$$
\int_X(\sigma\wedge\overline\sigma)^{1\over m}h_G\le 
C_2^{1\over m}C_3^{p-1\over m}\le 1
$$
for $p$ large enough. On the other hand
$$
|\sigma(x_0)|^2_{h_{K_X+L'}}
=|\sigma(x_0)|^2_{h_{K_X+L}^m}e^{-(m-p)\varphi(x_0)-p\psi_0(x_0)}=1,
$$
thus
$$
{1\over m}\log |\sigma(x_0)|^2_{h_{K_X+L}^m}=
\Big(1-{p\over m}\Big)\varphi(x_0)+{p\over m}\psi_0(x_0)
\leqno(16.10)
$$
and, as a consequence
$$
{1\over m}\log |\sigma(x_0)|^2_{h_{K_X+L}^m}\longrightarrow\varphi(x_0)
$$
whenever $m\to+\infty$, ${p\over m}\to 0$, as long as
$\psi_0(x_0)>-\infty$. In the above argument, we can in fact interpolate
in finitely many points $x_1,\,x_2,\,\ldots\,,x_q$ provided that
$p\ge C_4q$. Therefore if we take a suitable dense subset
$\{x_q\}$ and a ``diagonal'' sequence associated with sections
$\sigma_m\in H^0(X,m(K_X+L))$ with 
$m\gg p=p_m\gg q=q_m\to+\infty$, we infer that
$$
\bigg(\limsup_{m\to+\infty}{1\over m}\log |\sigma_m(x)|^2_{h_{K_X+L}^m}\bigg)^*\ge
\limsup_{x_q\to x}\;\varphi(x_q)=\varphi(x)\leqno(16.11)
$$
(the latter equality occurring if $\{x_q\}$ is suitably chosen with
respect to $\varphi$). In the other direction, (16.9)
implies a mean value estimate
$$
{1\over \pi^nr^{2n}/n!}\int_{B(x,r)}
|\sigma(z)|^2_{h_{K_X+L}^m}\;dz\le {C_5\over r^{2n}}
\sup_{B(x,r)}e^{(m-p)\varphi+p\psi_0}
$$
on every coordinate ball $B(x,r)\subset X$. The function 
$|\sigma_m|^2_{h_{K_X+L}^m}$ is 
plurisubharmonic after we correct the
non necessarily positively curved smooth metric $h_{K_X+L}$ by a 
factor of the form $\exp(C_6|z-x|^2)$, hence the mean value 
inequality shows that
$$
{1\over m}\log |\sigma_m(x)|^2_{h_{K_X+L}^m}\le 
{1\over m}\log{C_5\over r^{2n}}+C_6 r^2
+\sup_{B(x,r)}\Big(1-{p_m\over m}\Big)\varphi+{p_m\over m}\psi_0.
$$
By taking in particular $r=1/m$ and letting $m\to+\infty$, $p_m/m\to 0$, 
we see that the opposite of inequality (16.9) also holds.\qed
\endproof

\claim{(16.12) Remark} \rm We can rephrase our results in slightly different
terms. In fact, let us put 
$$
\varphi^\alg_m=\sup_{\sigma}{1\over m}\log|\sigma|_{h_{K_X+L}^m}^2,
\qquad \sigma\in H^0(X,m(K_X+G+\Delta)),
$$
with normalized sections $\sigma$
such that \hbox{$\int_X (\sigma\wedge \overline\sigma)^{1/m}h_G=1$}.
Then $\varphi^\alg_m$ is quasi-psh (the supremum is taken over a
compact set in a finite dimensional vector space) and by passing 
to the regularized supremum over all $\sigma$ and all $\varphi$ 
in (16.10) we get
$$
\varphi_\can\ge\varphi^\alg_m\ge \Big(1-{p\over m}\Big)
\varphi_\can(x)+{p\over m}\psi_0(x).
$$
As $\varphi_\can$ is bounded from above, we find in particular
$$
0\le \varphi_\can-\varphi^\alg_m\le {C\over m}(|\psi_0(x)|+1).
$$
This implies that $(\varphi^\alg_m)$ converges uniformly to
$\varphi_\can$ on every compact subset of $X\subset Z_0$, and in this way
we infer again (in a purely qualitative manner) that $\varphi_\can$ 
is continuous on $X\ssm Z_0$. Moreover, we also see that in (16.7) 
the upper  semicontinuous regularization is not needed on
$X\ssm Z_0\,$; in case $K_X+L$ is ample, it is not needed at all
and we have uniform convergence of $(\varphi^\alg_m)$ towards
$\varphi_\can$ on the whole of~$X$. Obtaining such a uniform convergence
when $K_X+L$ is just big looks like a more delicate question, 
related e.g.\ to abundance
of $K_X+L$ on those subvarieties $Y$ where the restriction
$(K_X+L)_{|Y}$ would be e.g.\ nef but not big.
\endclaim
%
\claim{(16.13) Generalization} \rm In the general case
where $L$ is a $\bR$-line bundle and \hbox{$K_X+L$} is merely pseudo-effective, 
a similar algebraic approximation can be obtained. We take instead sections
$$
\sigma\in H^0(X,mK_X+\lfloor mG\rfloor+\lfloor m\Delta\rfloor +p_mA)
$$
where $(A,h_A)$ is a positive line bundle, 
$\Theta_{A,h_A}\ge \varepsilon_0\omega$,
and replace the definition of $\varphi_\can^\alg$ by
$$
\leqalignno{
&\varphi_\can^\alg=
\bigg(\limsup_{m\to+\infty}\;
\sup_\sigma\;{1\over m}\log|\sigma|_{h_{mK_X+\lfloor mG\rfloor+p_mA}}^2
\bigg)^*,&(16.14)\cr
&\int_X(\sigma\wedge\overline\sigma)^{2\over m}
h_{\lfloor mG\rfloor +p_mA}^{1\over m}\le 1,&(16.15)\cr}
$$
where $m\gg p_m\gg 1$ and $h_{\lfloor mG\rfloor}^{1/m}$ is chosen
to converge uniformly to $h_G$.

 We then find again $\varphi_\can=\varphi_\can^\alg$, with
an almost identical proof -- though we no longer have a $\sup$ in the
envelope, but just a $\limsup$. The analogue of Proposition (16.8) also
holds true in this context, with an appropriate sequence of sections
$\sigma_m\in H^0(X,mK_X+\lfloor mG\rfloor+\lfloor m\Delta\rfloor +p_mA)$.
\endclaim
%
\claim{(16.16) Remark} \rm It would be nice to have a better understanding
of the super-canonical metrics. In case $X$ is a curve, this should be easier.
In fact $X$ then has a hermitian metric $\omega$ with constant curvature,
which we normalize by requiring that $\int_X\omega=1$, and we can also
suppose $\int_Xe^{-\gamma}\omega=1$. The class
$\lambda=c_1(K_X+L)\ge 0$ is a number and we take $\alpha=\lambda\omega$. 
Our envelope is $\varphi_\can=\sup\varphi$ where 
$\lambda\omega+dd^c\varphi\ge 0$ and
$\int_Xe^{\varphi-\gamma}\omega\le 1$. If $\lambda=0$ then $\varphi$
must be constant and clearly $\varphi_\can=0$. \hbox{Otherwise}, if $G(z,a)$ 
denotes the Green function such that $\int_X G(z,a)\,\omega(z)=0$ and
$dd^cG(z,a)=\delta_a-\omega(z)$, we find
$$
\varphi_\can(z)\ge \sup_{a\in X}\bigg(\lambda
G(z,a)-\log\int_{z\in X}e^{\lambda G(z,a)-\gamma(z)}\omega(z)\bigg)
$$
by taking already the envelope over $\varphi(z)=\lambda G(z,a)-\hbox{Const}$. 
It is natural
to ask whether this is always an equality, i.e.\ whether the extremal
functions are always given by one of the Green functions, especially
when~$\gamma=0$.
\endclaim
%
\subsection*{\csect.B. Invariance of plurigenera and positivity of curvature of
super-canonical metrics}
The concept of super-canonical metric can be used to give a very interesting
result on the positivity of relative pluricanonical
divisors, which itself can be seen to imply the invariance of plurigenera.
The main idea is due to H.~Tsuji [Tsu07a], and some important details
were fixed by Berndtsson and P\u{a}un [BnP09], using techniques inspired
from their results on positivity of direct images [Bnd06], [BnP08].

\claim{(16.17) Theorem} Let $\pi:\cX\to S$ be a deformation of projective
algebraic manifolds over some irreducible complex space~$S$ $(\pi$ being
assumed locally projective over~$S)$. Let
$\cL\to\cX$ be a holomorphic line bundle equipped with a hermitian
metric $h_{\cL,\gamma}$ of weight $\gamma$ such that $\ii\smash{\Theta_{\cL,
h_{\cL,\gamma}}}\ge 0$ $($i.e.\ $\gamma$ is plurisubharmonic$)$, and
$\smash{\int_{X_t}}e^{-\gamma}<+\infty$, i.e.\ we assume the metric to
be klt over all fibers $X_t=\pi^{-1}(t)$. Then the metric defined
on $K_{\cX}+\cL$ as the fiberwise super-canonical metric has semi-positive
curvature over~$\cX$. In particular, 
$t\mapsto h^0(X_t,m(K_{X_t}+\cL_{\restriction X_t}))$ is constant for 
all~$m>0$.
\endclaim
%
Once the metric is known to have a plurisuharmonic weight on the total
space of $\cX$, the Ohsawa-Takegoshi theorem can be used exactly as
at the end of the proof of lemma (12.3). Therefore the final statement
is just an easy consequence. The cases when $\cL=\cO_\cX$ is trivial
or when $\cL_{\restriction X_t}=\cO(\Delta_t)$ for a family of klt $\bQ$-divisors
are especially interesting.

\begproof{Proof {\rm (Sketch)}.} By our assumptions, there exists (at least
locally over $S$) a rela\-tively ample line bundle $\cA$ over~$\cX$. We
have to show that the weight of the global super-canonical metric is
plurisubharmonic, and for this, it is enough to look at analytic
disks $\Delta\to S$. We may thus as well assume that $S=\Delta$ 
is the unit disk.  Consider the super-canonical metric
$h_{\can,0}$ over the fiber $X_0$.  The approximation argument seen
above (see (16.9) and remark (16.13)) show that $h_{\can,0}$ has a
weight $\varphi_{\can, 0}$ which is a regularized upper limit
$$\varphi_{\can,0}^\alg=
\bigg(\limsup_{m\to+\infty}\;{1\over m}\log|\sigma_m|^2
\bigg)^*
$$
defined by sections 
$\sigma_m\in H^0(X_0,m(K_{X_0}+\cL_{\restriction X_0})+
p_m\cA_{\restriction X_0})$ such that
$$
\int_{X_0} |\sigma|^2e^{-(m-p_m)\varphi_{\can,0}-p_m\psi_0}dV_\omega\le C_2.
$$
with the suitable weights. Now, by section 12, these sections
extend to sections $\widetilde\sigma_m$ defined on the whole family~$\cX$, 
satisfying a similar $L^2$ estimate (possibly with a slightly larger constant
$C'_2$ under control). If we set 
$$
\Phi=\bigg(\limsup_{m\to+\infty}\;{1\over m}\log|\widetilde\sigma_m|^2
\bigg)^*,
$$
then $\Phi$ is plurisubharmonic by construction, and 
$\varphi_\can\ge \Phi$ by the defining property of the super-canonical 
metric. Finally, we also have $\varphi_{\can,0}=\Phi_{\restriction X_0}$
from the approximation technique. It follows easily that $\varphi_\can$
satisfies the mean value inequality with respect to any disk centered
on the central fiber~$X_0$. Since we can consider arbitrary analytic disks
$\Delta\to S$, the plurisubharmonicity of $\varphi_\can$ follows.\qed
\endproof
%
\subsection*{\csect.C. Tsuji's strategy for studying abundance}
H.~Tsuji [Tsu07c] has recently proposed the following interesting
prospective approach of the abundance conjecture.

\claim{(16.18) Conjecture/question} Let $(X,\Delta)$ be a klt pair such that
$K_X+\Delta$ is pseudoeffective and has numerical dimension
$\num(K_X+\Delta)>0$. Then for every point $x\in X$ there exists a closed
positive current $T_x\in c_1(K_X+\Delta)$ such that the Lelong number at 
$x$ satisfies $\nu(T_x,x)>0$.
\endclaim
%
It would be quite tempting to try to produce such currents e.g.\ by a suitable
modification of the construction of super-canonical metrics, trying to
enforce singularities of the metric at any prescribed point $x\in X$. 
A~related procedure would be to enforce enough vanishing of sections
of $A+m(K_X+\Delta)$ at point $x$, where $A$ is a sufficiently ample line
bundle. The number of these sections grows as $cm^p$ where 
$p=\num(K_X+\Delta)$. Hence, by an easy linear algebra argument, one can 
prescribe a vanishing order $s\sim c'm^{p/n}$ of such a section $\sigma$, whence
a Lelong number${}\sim c'm^{{p\over n}-1}$ for the corresponding
rescaled current of integration $T={1\over m}[Z_\sigma]$ on the
zero divisor. Unfortunately, this tends to $0$ as $m\to+\infty$ whenever
$p<n$. Therefore, one should use a more clever argument which takes 
into account the fact that, most probably, all directions do not behave
in an ``isotropic way'', and vanishing should be prescribed only 
in certain directions.

Assuming that (16.17) holds true, a simple semi-continuity argument 
would imply that there exists a small number $c>0$ such
that the analytic set $Z_x=E_c(T_x)$ contains $x$, and one would expect
conjecturally that these sets can be reorganized as the generic fibers
of a reduction map $f:X\merto Y$, together with a
klt divisor $\Delta'$ on $Y$ such that (in first approximation, and 
maybe only after replacing $X$, $Y$ by suitable blow-ups), one has
$K_X+\Delta=f^*(K_Y+\Delta'+R_f)+\beta$ where $R_f$ is a suitable orbifold
divisor (in the sense of Campana [Cam04]) and $\beta$ a suitable
pseudo-effective class. The expectation is that $\dim Y=p=\num(K_X+\Delta)$
and that $(Y,\Delta')$ is of general type, i.e.\ 
$\num(K_Y+\Delta')=p$.

\section{Siu's analytic approach and P\u{a}un's non vanishing theorem}

We describe here briefly some recent developments without giving much detail
about proofs.  Recall that given a pair $(X,\Delta)$ where X is a
normal projective variety and $\Delta$ an effective $\bR$-divisor, the
transform of $(X,\Delta)$ by a birational morphism $\mu:\widetilde X\to X$ of
normal varieties is the unique pair $(\widetilde X,\widetilde\Delta)$ such that
$\smash{K_{\widetilde X}}+\widetilde\Delta=\mu^*(K_X+\Delta)+E$ where $E$ is an effective
$\mu$-exceptional divisor (we assume here that $K_X+\Delta$ and
$\smash{K_{\widetilde X}}+\widetilde\Delta$ are $\bR$-Cartier divisors).

In [BCHM06], Birkar, Cascini, Hacon and McKernan proved old-standing
conjectures concerning the existence of minimal models and finiteness
of the canonical ring for arbitrary projective varieties. The latter
result was also announced independently by Siu in [Siu06]. The main 
results can be summarized in the following
statement.

\claim{(17.1) Theorem} Let $(X,\Delta)$ be a klt pair where $\Delta$ is big.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} If $K_X+\Delta$ is pseudo-effective, $(X,\Delta)$
has a log-minimal model, i.e.\ there is a birational transformation 
$(\widetilde X,\widetilde\Delta)$ with $\widetilde X$  $\bQ$-factorial, such that 
$K_{\widetilde X}+\widetilde\Delta$ is nef and
satisfies additionally strict inequalities for the discrepancies
of $\mu$-exceptional divisors.
\vskip3pt
\plainitem{\rm (ii)} If $K_X+\Delta$ is not pseudo-effective, then $(X,\Delta)$
has a Mori fiber space, i.e.\ there exists a birational transformation
$(\widetilde X,\widetilde\Delta)$ and a morphism $\varphi:\widetilde X\to Y$ such that
$-(K_{\widetilde X}+\widetilde\Delta)$ is $\varphi$-ample.
\vskip3pt
\plainitem{\rm (iii)} If moreover $\Delta$ is a $\bQ$-divisor, the log-canonical ring
$\bigoplus_{m\ge 0} H^0(X,m(K_X+\Delta))$
is finitely generated.\vskip0pt}
\endclaim
%
The proof, for which we can only refer to [BCHM06], is an extremely
subtle induction on dimension involving finiteness of flips (a certain
class of birational transforms improving positivity of $K_X+\Delta$
step by step), and a generalization of Shokurov's non vanishing
theorem [Sho85]. The original proof of this non vanishing result was
itself based on an induction on dimension, using the existence
of minimal models in dimension $n-1$. Independently, Y.T.\ Siu
[Siu06] announced an analytic proof of the finiteness of canonical
rings $\bigoplus_{m\ge 0} H^0(X,mK_X)$, along with an analytic variant
of Shokurov's non vanishing theorem; in his approach, multiplier 
ideals and Skoda's division theorem are used in crucial ways.
Let us mention a basic statement in this direction which illustrates the 
connection with Skoda's result, and is interesting for two reasons~:
i) it does not require any strict positivity assumption, ii) it shows that
it is enough to have a sufficiently good approximation of the
minimal singularity metric $h_{\min}$ by sections of sufficiently large 
linear systems $|pK_X|$.

\claim{(17.2) Proposition} Let $X$ be a projective $n$-dimensional
manifold with $K_X$ pseudo-effective. Let $h_{\min}=e^{-\varphi_{\min}}$ be
the metric with minimal singuarity on $K_X$ $($e.g.\ 
the super-canonical metric considered in~{\rm \S$\,$16}$)$, and
let $c_0>0$ be the log canonical threshold of $\varphi_{\min}$, i.e.\
$h_{\min}^{c_0-\delta}=e^{-(c_0-\delta)\varphi_{\min}}
\in L^1$ for $\delta>0$ small. Assume that there exists an 
integer $p>0$ so that the linear system $|pK_X|$ provides a weight
$\psi_p=\smash{1\over p}\log\sum|\sigma_j|^2$ whose singularity
approximates $\varphi_{\min}$ sufficiently well, namely
$$
\psi_p\ge\Big(1+{1+c_0-\delta\over pn}\Big)\varphi_{\min}+O(1)\qquad
\hbox{for some $\delta>0$}.
$$
Then $\smash\bigoplus_{m\ge 0}H^0(X,mK_X)$ is finitely 
generated, and a set of generators is actually provided by 
a basis of sections of 
$\smash\bigoplus_{0\le m\le np+1}H^0(X,mK_X)$.
\endclaim
%
\begproof{Proof.} A simple argument based on the curve selection lemma (see
e.g.\ [Dem01], Lemma 11.16) shows that one can extract a system
$g=(g_1,\ldots g_n)$ of at most $n$
sections from $(\sigma_j)$ in such a way that
the singularities are unchanged, i.e.\ 
$C_1\log|\sigma|\le \log|g|\le C_2\log|\sigma|$. We apply
Skoda's division (8.20)
with $E=\cO_X^{\oplus n}$, $Q=\cO(pK_X)$ and $L=\cO((m-p-1)K_X)$
[so that $K_X\otimes Q\otimes L=\cO_X(mK_X)$],
and with the metric induced by $h_{\min}$ on $K_X$. By definition of a metric
with minimal singularities, every section $f$ in $H^0(X,mK_X)=
H^0(X,K_X\otimes Q\otimes L)$
is such that $|f|^2\le Ce^{m\varphi_{\min}}$. The weight of the metric on
$Q\otimes L$ is $(m-1)\varphi_{\min}$. Accordingly, we find 
$$
\eqalign{
|f|^2|g|_{h_{min}}^{-2n-2\varepsilon}e^{-(m-1)\varphi_{\min}}
&\le C \exp\Big(m\varphi_{\min}-p(n+\varepsilon)(\psi_p-\varphi_{\min})
-(m-1)\varphi_{\min}\Big)\cr
&\le C'\exp\big(-(c_0-\delta/2)\varphi_{\min}\big)\cr}
$$ 
for $\varepsilon>0$ small, thus the left hand side is in $L^1$.
Skoda's theorem implies that we can write $f=g\cdot h=\sum g_jh_j$ with
$h_j\in H^0(X,K_X\otimes L)=H^0(X,(m-p)K_X)$. The argument holds
as soon as the curvature condition $m-p-1\ge (n-1+\varepsilon)p$ is
satisfied, i.e.\ $m\ge np+2$. Therefore all multiples $m\ge np+2$ are
generated by sections of lower degree $m-p$, and the result follows.
\qed
\endproof
%
Recently, P\u{a}un [Pau08] has been able to provide a very strong
Shokurov-type analytic non vanishing statement, and in the vein of 
Siu's approach [Siu06], he gave a very detailed independent
proof which does not require any intricate induction on dimension 
(i.e.\ not involving the existence of minimal models).

\claim{(17.3) Theorem {\rm (P\u{a}un [Pau08])}}
Let $X$ be a projective manifold, and let $\alpha_L\in \NS_{\bR}(X)$ be a 
cohomology class in the real Neron-Severi space of $X$, such that~:
\vskip3pt
{\plainitemindent=7mm
\plainitem{\rm(a)} The adjoint class $c_1(K_X)+ \alpha_L$ is pseudoeffective, 
i.e.\ there exist a 
closed positive current $$ \Theta_{K_X+ L}\in c_1(K_X)+ \alpha_L ;$$
\vskip3pt
\plainitem{\rm(b)} The class $\alpha_L$ contains a K\"ahler 
current $\Theta_L$ $($so that $\alpha_L$ is big$)$, such that the 
respective potentials $\varphi_L$ of 
$\Theta_L$ and $\varphi_{K_X+ L}$ of $\Theta_{K_X+ L}$ satisfy
$$
e^{(1+\varepsilon)(\varphi_{K_X+ L}-\varphi_L)}\in L^1_\loc
$$
where $\varepsilon$ is a positive real number.\vskip3pt}
\noindent Then the adjoint class $c_1(K_X)+ \alpha_L$ contains an
effective $\bR$-divisor.
\endclaim
%
The proof is a clever application of the Kawamata-Viehweg-Nadel
vanishing theorem, combined with a perturbation trick of Shokurov
[Sho85] and with diophantine approximation to reduce the situation to
the case of $\bQ$-divisors. Shokurov's trick allows to single out
components of the divisors involved, so as to be able to take
restrictions and apply induction on~dimension.  One should notice that
the poles of $\varphi_L$ may help in achieving condition (17.3~b), so
one obtains a stronger condition by requiring (b${}'$)
$\exp((1+\varepsilon)\varphi_{K_X+ L})\in L^1_\loc$ for
$\varepsilon>0$ small, namely that $c_1(K_X)+\alpha_L$ is klt.  The
resulting weaker statement then makes sense in a pure algebraic
setting. In [BrP09], Birkar and P\u{a}un announced a relative version
of (17.3), and they have shown that this can be used to reprove a
relative version of Theorem (17.1).  A~similar purely algebraic
approach has been described by C.~Hacon in his recent Oberwolfach
notes [Hac08].

\let\fakehfill=\hfill
\section*{\fakehfill Bibliography}
{
\eightpoint

\letter KoMM87| 

\article AN54|Y.\ Akizuki {\rm and} S.\ Nakano|Note on Kodaira-Spencer's
proof of Lefschetz theorems|Proc.\ Jap.\ Acad.|30|1954|266--272|

\article AS95|U.\ Angehrn {\rm and} Y.T.\ Siu|Effective freeness and point
separation for adjoint bundles|Invent.\ Math.|122|1995|291-308|
 
\article Aub78|T.\ Aubin|Equations du type Monge-Amp\`ere sur les vari\'et\'es
k\"ahl\'e\-riennes compactes|C.R.\ Acad.\ Sci.\ Paris Ser.\ A {\bf
283}$\,$(1976), 119--121$\,$; Bull.\ Sci.\ Math|102|1978|63--95|

\article AV65|A.\ Andreotti {\rm and} E.\ Vesentini|Carleman estimates
for~the Laplace-Beltrami equation in complex manifolds|Publ.\ Math.\ 
I.H.E.S.|25|1965|81--130|

\article BBS89|M.\ Beltrametti, A.\ Biancofiore {\rm and} A.J.\
Sommese|Pro\-jective $n$-folds of log-general type, I|Trans.\ Amer.\
Math.\ Soc.|314|1989|825--849|

\miscell BeS93|M.\ Beltrametti {\rm and} A.J.\ Sommese|On $k$-jet
ampleness|Complex Analysis and Geometry, Univ.\ Series in
Math., edited by V.~Ancona and A.~Silva, Plenum Press, New-York, 1993,
355--376|

\article BFS89|M.\ Beltrametti, P.\ Francia {\rm and} A.J.\ Sommese|On
Reider's method and higher order embeddings|Duke Math.\ J.|58|1989|425--439|

\miscell BmD09|R.\ Berman {\rm and} J.-P.\ Demailly|Regularity of
plurisubharmonic upper envelopes in big cohomology
classes|arXiv:math.CV/0905.1246v1|

\article Bnd06|B.~Berndtsson|Subharmonicity properties of the Bergman
kernel and some other functions associated to pseudoconvex domains|Ann.\
Inst.\ Fourier (Grenoble)|56|2006|1633--1662|

\article BnP08|B.~Berndtsson {\rm and} M.~P\u{a}un|Bergman kernels and
the pseudoeffectivity of relative canonical bundles|Duke Math.\
J|145|2008|341--378|

\miscell BnP09|B.~Berndtsson {\rm and} M.~P\u{a}un|Canonical metrics for
projective families|Privately communicated manuscript, February 2009|

\miscell BCHM06|C.\ Birkar, P.~Cascini, C.D.~Hacon, J.~McKernan|Existence
of minimal models for varieties of log general type|arXiv:math.AG/0610203v2|   

\miscell BrP09|C.~Birkar, M.~P\u{a}un|Minimal models, flips and finite 
generation : a tribute to V.V.~Shokurov and Y.-T.~Siu|
arXiv:math.AG/0904.2936|

\article BGK05|Ch.\ P.\ Boyer, K.\ Galicki, J.\ Koll\'ar|Einstein Metrics
on Spheres|Annals of Math|162|2005|557--580|

\article Boc48|S.\ Bochner|Curvature and Betti numbers (I) and (II)|Ann.\
of Math.|49|1948|379--390$\,$; {\bf 50} (1949), 77--93|

\article Bom70|E.\ Bombieri|Algebraic values of meromorphic maps|Invent.\ 
Math.|10|1970|267--287 and {\it Addendum}, Invent.\ Math.\ {\bf 11} 
(1970), 163--166|    

\article Bom73|E.\ Bombieri|Canonical models of surfaces of general 
type|Publ.\ Math.\ IHES|42|1973|171--219|

\miscell Bon93|L.\ Bonavero|In\'egalit\'es de Morse holomorphes
singuli\`eres|Pr\'epu\-blication Universit\'e de Grenoble~I, Institut
Fourier, n${}^\circ\,259$, 1993|

\miscell Bou02|S.\ Boucksom|C\^ones positifs des vari\'et\'es
  complexes compactes|Thesis, Grenoble 2002|
  
\miscell BDPP04|S.\ Boucksom, J.-P.\ Demailly, M.\ P\u{a}un, Th.\ Peternell|The 
  pseudo-effective cone of a compact K\"ahler manifold and
  varieties of negative Kodaira dimension|manuscript May 2004,
  math.AG/0405285|

\article BSk74|J.\ Brian\c con {\rm et} H.\ Skoda|Sur la cl\^oture
int\'egrale d'un id\'eal de germes de fonctions holomorphes en un point
de $\bC^n$|C.\ R.\ Acad.\ Sc.\ Paris, s\'er.\ A|278|1974|949--951|

\article BT76|E.\ Bedford {\rm and} B.A.\ Taylor|The Dirichlet problem for the 
complex Monge-Amp\`ere equation|Invent.\ Math.|37|1976|1--44|

\article BT82|E.\ Bedford {\rm and} B.A.\ Taylor|A new capacity for 
plurisubharmonic functions|Acta Math.|149|1982|1--41|

\article Buc99|N.\ Buchdahl|On compact K\"ahler surfaces|Ann.\ 
  Inst.\ Fourier|49|1999|287--302|
  
\article Buc00|N.\ Buchdahl|A Nakai-Moishezon criterion for
  non-K\"ahler surfaces|Ann.\ Inst.\ Fourier|50|2000|1533--1538|

\article Cam92|F.\ Campana|Connexit\'e rationnelle des vari\'et\'es de
Fano|Ann.\ Sci.\ Ecole Norm.\ Sup.|25|1992|539--545|

\article Cam04|F.\ Campana|Orbifolds, special varieties and 
classification theory|Ann.\ Inst.\ Fourier (Grenoble)|54|2004|499--630|

\miscell Cat88|F.\ Catanese|Footnotes to a theorem of I.\ Reider|Proc.\
Intern.\ Conf.\ on Algebraic Geometry (L'Aquila, June 1988),
Lecture Notes in Math., Vol.~1417, Springer-Verlag, Berlin, 1990, 67--74|

\article Cla07|B.\ Claudon|Invariance of plurigenera for multiples of the 
twisted canonical bundle|Ann.\ Inst.\ Fourier|57|2007|289�-300|

\bookref CLN69|S.S.\ Chern, H.I.\ Levine {\rm and} L.\ Nirenberg|Intrinsic
norms on a complex manifold|Global Analysis (papers in honor of K.Kodaira), 
p.~119--139, Univ.\ of Tokyo Press, Tokyo|1969|

\miscell CP04|F.\ Campana, Th.\ Peternell| Geometric stability of the
cotangent bundle and the universal cover of a projective
manifold|arXiv:math.AG/0405093|

\article DeI87|P.\ Deligne {\rm and} L.\ Illusie|Rel\`evements modulo
$p^2$ et d\'e\-com\-po\-sition du complexe de De Rham|Invent.\
Math.|89|1987|247--270|

\article Dem82a|J.-P.\ Demailly|Sur les nombres de Lelong associ\'es \`a
l'image directe d'un courant positif ferm\'e|Ann.\ Inst.\ Fourier 
(Grenoble)|32|1982|37--66|

\article Dem82b|J.-P.\ Demailly|Estimations $L^2$ pour l'op\'erateur $\dbar$
d'un fibr\'e vectoriel holomorphe semi-positif au dessus d'une vari\'et\'e
k\"ahl\'e\-rienne compl\`ete|Ann.\ Sci.\ Ec.\ Norm.\ Sup.|15|1982|457--511|

\article Dem85a|J.-P.\ Demailly|Mesures de Monge-Amp\`ere et caract\'erisation 
g\'eo\-m\'e\-trique des vari\'e\-t\'es alg\'ebriques affines|M\'em.\ Soc.\ 
Math.\ France (N.S.)|19|1985|1--124|

\article Dem85b|J.-P.\ Demailly|Champs magn\'etiques et in\'egalit\'es de
Morse pour la $d''$-coho\-mo\-logie|Ann.\ Inst.\ Fourier
(Grenoble)|35|1985|189--229|

\article Dem87|J.-P.\ Demailly|Nombres de Lelong g\'en\'eralis\'es, 
th\'eor\`emes d'int\'e\-gralit\'e et 
d'analy\-ticit\'e|Acta Math.|159|1987|153--169|

\article Dem89|J.-P.\ Demailly|Transcendental proof of a generalized 
Ka\-wa\-mata-Viehweg vanishing theorem|C.\ R.\ Acad.\ Sci.\ Paris S\'er.\ 
I Math.|309|1989|123--126~~~and:\newline
Proceedings of the Conference ``Geometrical and algebraical aspects
in several complex variables" held at Cetraro, Univ.\ della Calabria,
June 1989|

\miscell Dem90|J.-P.\ Demailly|Singular hermitian metrics on positive
line bundles|Proc.\ Conf.\ Complex algebraic varieties (Bayreuth,
April~2--6, 1990), edited by K.~Hulek, T.~Peternell, M.~Schneider,
F.~Schreyer, Lecture Notes in Math., Vol.~1507, Springer-Verlag, Berlin,
1992|

\article Dem92|J.-P.\ Demailly|Regularization of closed positive currents
and Intersection Theory|J.\ Alg.\ Geom.|1|1992|361--409|

\miscell Dem93a|J.-P.\ Demailly|Monge-Amp\`ere operators, Lelong numbers
and intersection theory|Complex Analysis and Geometry, Univ.\ Series in
Math., edited by V.~Ancona and A.~Silva, Plenum Press, New-York,
1993, 115--193|

\article Dem93b|J.-P.\ Demailly|A numerical criterion for very ample line
bundles|J.\ Differential Geom.|37|1993|323--374|

\miscell Dem94a|J.-P.\ Demailly|Regularization of closed positive currents
of type $(1,1)$ by the flow
of a Chern connection|Actes du Colloque en l'honneur de P.~Dolbeault
(Juin 1992), \'edit\'e par H.~Skoda et J.-M.~Tr\'epreau,
Aspects of Mathematics, Vol.~E~26, Vieweg, 1994, 105--126|

\miscell Dem94b|J.-P.\ Demailly|$L^2$ vanishing theorems for positive
line bundles and adjunction theory|Lecture Notes of the CIME Session
``Transcendental methods in Algebraic Geometry'', Cetraro, Italy, July
1994, Ed.\ F.~Catanese, C.~Ciliberto, Lecture Notes in Math.,
Vol.~1646, 1--97|

\article Dem96|J.-P.\ Demailly|Effective bounds for very ample line
bundles|Invent.\ Math.|124|1996|243--261|

\miscell Dem98|J.-P.\ Demailly|M\'ethodes $L^2$ et r\'esultats
effectifs en g\'eom\'etrie alg\'ebrique|S\'em.\ Bourbaki, Novembre
1998, expos\'e n${}^\circ$852|

\miscell Dem00|J.-P.\ Demailly|On the Ohsawa-Takegoshi-Manivel $L^2$ extension
 theorem|Proceedings of the Conference in honour of the 85th birthday
 of Pierre Lelong, Paris, September 1997, \'ed.\ P.~Dolbeault,
 Progress in Mathematics, Birkh\"auser, Vol.~{\bf 188} (2000) 47-82|

\miscell Dem01|J.-P.\ Demailly|Multiplier ideal sheaves and 
  analytic methods in algebraic geometry|Lecture Notes, School on 
  ``Vanishing theorems and effective results in Algebraic Geometry, 
  ICTP Trieste, April 2000, Publications of ICTP, 2001|

\miscell Dem08|J.-P.\ Demailly|Appendix to I.\ Cheltsov and 
C.\ Shramov's article ``Log canonical thresholds of smooth Fano threefolds'':
On Tian's invariant and log canonical thresholds|arXiv: math.AG$\,$0806.2107|

\miscell DEL00|J.-P.\ Demailly, L.\ Ein {\rm and} R.\ Lazarsfeld|A
subadditivity property of multiplier ideals|Michigan Math.\ J., 2000,
to appear|

\article DK01]|J.-P.\ Demailly, J.\ Koll\'ar|Semicontinuity of complex 
singularity exponents and\break K\"ahler-Einstein metrics on Fano 
orbifolds|math.AG/9910118$\,$; Ann.\ Ec.\ Norm.\ 
Sup|34|2001|525--556|

\article DP04|J.-P.\ Demailly, M.\ P\v{a}un|Numerical characterization
  of the K\"ahler cone of a compact K\"ahler manifold|arXiv: 
  math.AG/0105176, Annals of Math|159|2004|1247--1274|

\article DPS94|J.-P.\ Demailly, Th.\ Peternell, M.\ Schneider|Compact
complex manifolds with numerically effective tangent bundles|J.\
Algebraic Geometry|3|1994|295--345|

\miscell DPS00|J.-P.\ Demailly, Th.\ Peternell, M.\ 
Schneider|Pseudo-effec\-tive line bundles on compact K\"ahler 
manifolds|June 2000, Algebraic Geometry e-prints math.AG/0006205|

\miscell DSk79|J.-P.\ Demailly, H.\ Skoda|Relations entre les notions de
positivit\'e de P.A.~Griffiths et de S.~Nakano|S\'eminaire P.~Lelong-H.~Skoda
(Analyse), ann\'ee 1978/79, Lecture notes in Math.\ no~{\bf 822},
Springer-Verlag, Berlin (1980) 304--309|

\article DF83|H.\ Donnelly, C.\ Fefferman|$L^2$-cohomology and
index theorem for the Bergman metric|Ann.\ Math.|118|1983|593--618|

\article DX84|H.\ Donnelly, F.\ Xavier|On the differential form spectrum 
of negatively curved Riemann manifolds|Amer.\ J.\ Math.|106|1984|169--185|

\miscell Ein94|L.\ Ein|Note on higher dimensional adjoint linear
systems|preprint 1994, personal communication to the
author|

\article EKL95|L.\ Ein, O.\ K\"uchle {\rm and} R.\ Lazarsfeld|Local
positivity of ample line bundles|J.\ Differ.\ Geom.|42|1995|193--219|

\miscell EL92|L.\ Ein and R.\ Lazarsfeld|Seshadri constants on smooth
surfaces|Journ\'ees de G\'eo\-m\'etrie Alg\'ebrique d'Orsay, July
1992, Ast\'e\-risque {\bf 282}, 1993, 177--186|

\article EL93|L.\ Ein {\rm and} R.\ Lazarsfeld|Global generation of
pluricanonical and adjoint linear series on smooth projective
threefolds|Jour.\ of Am.\ Math.\ Soc.|6|1993|875--903|

\article ELM95|L.\ Ein, R.\ Lazarsfeld {\rm and} V.\ Masek|Global 
generation of linear series on terminal threefolds|Intern.\ J.\ 
Math.|6|1995|1--18|

\miscell ELN96|L.\ Ein, R.\ Lazarsfeld {\rm and} M.\ Nakamaye|Zero 
estimates, intersection theory, and a theorem
of Demailly|in Higher Dimensional Complex
Varieties (Trento, 1994), de Gruyter (1996), 183--207|

\article EM84|H.\ El Mir|Sur le prolongement des courants positifs
ferm\'es|Acta Math.|153|1984|1--45|

\miscell Eno93|I.\ Enoki|Kawamata-Viehweg vanishing theorem for compact
K\"ahler manifolds|Einstein metrics and Yang-Mills connections
(ed.\ T.~Mabu\-chi, S.~Mukai), Marcel Dekker, 1993, 59--68|

\article EV86|H.\ Esnault {\rm and} E.\ Viehweg|Logarithmic De Rham
complexes and vanishing theorems|Invent.\ Math.|86|1986|161--194|

\miscell EV92|H.\ Esnault {\rm and} E.\ Viehweg|Lectures on vanishing
theorems|DMV Seminar, Band {\bf 20}, Birkh\"auser Verlag, 1992|

\bookref Fed69|H.\ Federer|Geometric measure theory|Springer Verlag,
Vol. 153, Berlin, Heidelberg, New-York|1969|

\miscell FdB94|G.\ Fern\'andez del Busto|UCLA Thesis|1994|

\article FdB95|G.\ Fern\'andez del Busto|Bogomolov instability and
Kawamata-Viehweg vanishing|J.\ Algebr.\ Geom.|4|1995|693--700|

\article FdB96|G.\ Fernandez del Busto|A Matsusaka-type theorem on
surfaces|J.\ Algebr.\ Geom.|5|1996|513--520|

\article Fuj83|T.\ Fujita|Semipositive line bundles|J.\ Fac.\ Sci.\ Univ.\
of Tokyo|30|1983|353--378|

\miscell Fuj87|T.\ Fujita|On polarized manifolds whose adjoint bundles
are not semi-positive|Algebraic Geometry, Sendai, 1985,
Adv.\ Stud.\ in Pure Math., Vol.~10, North Holland, T.~Oda (ed.),
1987, 167--178|

\miscell Fuj88|T.\ Fujita|Problem list|Conference held at the 
Taniguchi Foundation, Katata, Japan, August 1988|

\miscell Fuj93|T.\ Fujita|Remarks on Ein-Lazarsfeld criterion of
spannedness of adjoint bundles of polarized threefolds|Preprint
1993|

\article Fuj94|T.\ Fujita|Approximating Zariski decomposition of big line
bundles|Kodai Math.\ J.|17|1994|1--3|

\article Gir76|J.\ Girbau|Sur le th\'eor\`eme de Le Potier d'annulation de
la cohomologie|C.~R.\ Acad.\ Sci.\ Paris, s\'erie A|283|1976|355--358|

\article Gra60|H.\ Grauert|Ein Theorem der analytischen Garbentheorie und
die Modulr\"aume komplexer Strukturen|Publ.\ Math.\ I.H.E.S.|5|1960|233--292|

\bookref GR84|H.\ Grauert, R.\ Remmert|Coherent analytic sheaves|Grundlehren 
der math.\ Wissenschaften {\bf 265}, Springer-Verlag, Berlin|1984|

\miscell Gri69|P.A.\ Griffiths|Hermitian differential geometry, Chern
clas\-ses and positive vector bundles|Global Analysis, papers in honor of
K.~Kodaira, Princeton Univ.\ Press, Princeton, 1969, 181--251|

\bookref GrH78|P.A.\ Griffiths, J.\ Harris|Principles of algebraic
geometry|Wiley, New York|1978|

\miscell Hac08|C.~Hacon|Extension theorems and the existence of flips|Series
of 8 lectures given at Oberwolfach Math.\ Institute, Oct.~12-18, 2008|

\bookref HaMK07|C.~Hacon, J.~McKernan|Extension theorems and the 
existence of flips|In ``flips for 3-folds and 4-folds" (ed.\ A.~Corti), 
Oxford lecture series in mathematics and its applications {\bf 35},
Oxford University Press|2007|

\miscell HaMK08|C.~Hacon, J.~McKernan|Existence of minimal models for
varieties of general type, II|(Existence of pl-flips), 
arXiv:math.AG/0808.1929v1|

\miscell Har70|R.\ Hartshorne|Ample subvarieties of algebraic 
varieties|\rm Lecture Notes in Math., Vol.~156, Springer-Verlag,
Berlin (1970)|

\article Hel97|S.\ Helmke|On Fujita's conjecture|Duke Math.\ 
J.|88|1997|201--231|

\miscell Hel98|S.\ Helmke|Global generation of adjoint linear 
systems|Communication to the Oberwolfach Conference ``Komplexe
Analysis'', September 1998|

\article Hir64|H.\ Hironaka|Resolution of singularities of an algebraic 
variety over a field of characteristic zero|Ann.\ of Math.|79|1964|109--326|

\article H\"or65|L.\ H\"ormander|$L^2$ estimates and existence theorems for
the $\overline\partial$ operator|Acta Math.|113|1965|89--152|

\bookref H\"or66|L.\ H\"ormander|An introduction to Complex Analysis in
several variables|1966, 3rd edition, North-Holland Math.\ 
Libr., vol.7, Amsterdam, London|1990|

\article Hov79|A.G.\ Hovanski|Geometry of convex bodies and algebraic
geometry|Uspehi Mat.\ Nauk|34 {\rm(4)} |1979|160--161|

\article Huy99|D.\ Huybrechts|Compact Hyperk\"ahler Manifolds:
  Basic Results|Invent.\ Math.|135|1999|63--113|

\miscell Huy01|D.\ Huybrechts|The Projectivity Criterion for
  Hyperk\"ahler manifolds as a Consequence of the Demailly-P\u{a}un
  Theorem|personal communication, manuscript K\"oln University, May
  2001, 3~p, to appear in Invent.\ Math|
 
\article JK01|J.M.\ Johnson, J.\ Koll\'ar|K\"ahler-Einstein metrics on 
  log del Pezzo surfaces in wei\-ghted projective $3$-spaces|Ann.\ Inst.\ 
  Fourier|51|2001|69--79|

\article Kaw82|Y.\ Kawamata|A generalization of Kodaira-Ramanujam's
va\-nishing theorem|Math.\ Ann.|261|1982|43--46|

\article Kaw84|Y.\ Kawamata|The cone of curves of algebraic varieties|Ann.\
of Math.|119|1984|603--633|

\article Kaw85|Y.\ Kawamata|Pluricanonical systems of minimal
algebraic varieties|Invent.\ Math.|79|1985|567--588|

\article Kaw97|Y.\ Kawamata|On Fujita's freeness conjecture 
for $3$-folds and $4$-folds|Math.\ Ann.|308|1997|491--505|

\miscell Kaw98|Y.\ Kawamata|On the extension problem of
pluricanonical forms|Preprint Univ.\ of Tokyo at Komaba,
September 1998|

\article Kaw99|Y.\ Kawamata|Deformation of canonical 
singularities|J.~Amer.\ Math.\ Soc.|12|1999|85--92|

\article Kis78|C.O.\ Kiselman|The partial Legendre transformation for 
plu\-ri\-sub\-harmonic functions|Invent.\ Math.|49|1978|137--148|

\article Kis79|C.O.\ Kiselman|Densit\'e des fonctions
plurisousharmoniques|Bull.\ Soc.\ Math.\ France|107|1979|295--304|

\article Kis84|C.O.\ Kiselman|Sur la d\'efinition de l'op\'erateur de
Monge-Amp\`ere complexe|Analyse Complexe, Proceedings of the Jour\-n\'ees
Fermat (SMF), Toulouse, 1983, Lecture Notes in Math., Vol.\ 1094,
Springer-Verlag|\rm Berlin|1984|139--150|

\miscell KMM87|Y.\ Kawamata, K.\ Matsuda {\rm and} K.\ Matsuki|Introduction
to the minimal problem|Algebraic Geometry, Sendai, 1985,
Adv.\ Stud.\ in Pure Math., Vol.~10, T.~Oda (ed.), North Holland, Amsterdam,
1987, 283--360|

\article KobO73|S.\ Kobayashi {\rm and} T.\ Ochiai|Characterisations of
complex projective spaces and hyperquadrics|J.\ Math.\ Kyoto
Univ.|13|1973|31--47|

\article Kod53|K.\ Kodaira|On a differential geometric method in the theory
of analytic stacks|Proc.\ Nat.\ Acad.\ Sci.\ USA|39|1953|1268--1273|

\article Kod54|K.\ Kodaira|On K\"ahler varieties of restricted type|Ann.\
of Math.|60|1954|28--48|

\article Koh63|J.J.\ Kohn|Harmonic integrals on strongly pseudo-convex
manifolds~I|Ann.\ Math.|78|1963|206--213|

\article Koh64|J.J.\ Kohn|Harmonic integrals on strongly pseudo-convex
manifolds~II|Ann.\ Math.|79|1964|450--472|

\miscell Kol85|J.\ Koll\'ar|Vanishing theorems for cohomology groups|
Algebraic Geometry, Bowdoin 1985, Proc.\ Symp.\ Pure Math.,
Vol.~46, 1987, 233--243|

\article Kol92|J.\ Koll\'ar|Effective basepoint freeness|Math.\
Ann.|296|1993|595--605|

\article KoM83|J.\ Koll\'ar {\rm and} T.\ Matsusaka|Riemann-Roch type
inequalities|Amer.\ J.\ of Math.|105|1983|229--252|

\article KoMM92|J.\ Koll\'ar, Y.\ Miyaoka {\rm and} S.\ Mori|Rational
connectedness and boundedness of Fano manifolds|J.\ Differential
Geom.|36|1992|765--779|

\bookref KoM98|J.\ Koll\'ar, S.~Mori|Birational geometry of algebraic
varieties|With the collaboration of C.H.~Clemens and A.~Corti. Cambridge
Tracts in mathematics, {\bf 134}, Cambridge University press|1998|

\article Lam99a|A.\ Lamari|Courants k\"ahl\'eriens et surfaces
  compactes|Ann.\ Inst.\ Fourier|49|1999|263--285|
  
\article Lam99b|A.\ Lamari|Le cone K\"ahl\'erien d'une surface|J.\ 
  Math.\ Pures Appl.|78|1999|249--263|

\miscell Laz97|R.\ Lazarsfeld, {\rm with the assistance of} G.\ Fern\'andez
del Busto|Lec\-tu\-res on linear series|Complex Algebraic Geometry
(Park City, UT, 1993), Amer.\ Math.\ Soc., Providence, RI (1997), 
161--219|

\miscell Laz99|R.\ Lazarsfeld|Private communication|University of Ann Arbor,
1999|

\article Lel57|P.\ Lelong|Int\'egration sur un ensemble analytique 
complexe|Bull.\ Soc.\ Math.\ France|85|1957|239--262|

\miscell Lel67|P.\ Lelong|Fonctionnelles analytiques et fonctions enti\`eres
($n$ variables)|S\'em.\ de Math.\ Sup\'erieures, 6e session, \'et\'e 1967,
Presses Univ.\ Montr\'eal 1968|

\miscell Lel72|P.\ Lelong|El\'ements extr\^emaux sur le c\^one des
courants positifs ferm\'es|S\'eminaire P. Lelong (Analyse), ann\'ee
1971/72, Lecture Notes in Math., Vol.~332, Springer-Verlag, Berlin,
(1972) 112--131|

\bookref Lel69|P.\ Lelong|Plurisubharmonic functions and positive differential
forms|Gordon and Breach, New-York, and Dunod, Paris|1969|

\article LP75|J.\ Le Potier|Annulation de la cohomologie \`a valeurs dans un
fibr\'e vectoriel holomorphe de rang quelconque|Math.\ Ann.|218|1975|35--53|

\article Man93|L.\ Manivel|Un th\'eor\`eme de prolongement $L^2$ de
sections holomorphes d'un fibr\'e vectoriel|Math.\
Zeitschrift|212|1993|107--122|

\article Mat72|T.\ Matsusaka|Polarized varieties with a given Hilbert
polynomial|Amer.\ J.\ of Math.|94|1972|1027--1077|

\article Mor82|S.\ Mori|Threefolds whose canonical bundles are not
numerically effective|Ann.\ of Math.|116|1982|133--176|

\article Mou99|Ch.\ Mourougane|Th\'eor\`emes d'annulation g\'en\'eriques pour
les fibr\'es vectoriels semi-n\'egatifs|Bull.\ Soc.\ Math.\ 
Fr.|127|1999|115--133|

\article Nad89|A.M.\ Nadel|Multiplier ideal sheaves and 
K\"ahler-Einstein metrics of positive scalar curvature|Proc.\ Nat.\
Acad.\ Sci.\ U.S.A.|86|1989|7299--7300~~~and~~
Annals of Math., {\bf 132} (1990), 549--596|

\article Nak55|S.\ Nakano|On complex analytic vector bundles|J.~Math.\
Soc.\ Japan|7|1955|1--12|

\article Ohs88|T.\ Ohsawa|On the extension of $L^2$ holomorphic
functions, II|Publ.\ RIMS, Kyoto Univ.|24|1988|265--275|

\miscell Ohs94|T.\ Ohsawa|On the extension of $L^2$ holomorphic functions,
IV$\,$: A new density concept|Mabuchi, T.\ (ed.) et al.,
Geometry and analysis on complex manifolds. Festschrift for Professor
S.~Kobayashi's 60th birthday. Singapore: World Scientific,
(1994) 157--170|

\article Ohs95|T.\ Ohsawa|On the extension of $L^2$ holomorphic
functions, III$\,$: negligible weights|Math.\ Zeitschrift|219|1995|215--225|

\miscell Ohs02|T.\ Ohsawa|A precise $L^2$ division theorem|Complex 
geometry (G\"ottingen, 2000), 185--191, Springer, Berlin, 2002|

\miscell Ohs04|T.\ Ohsawa|Generalization of a precise $L^2$ division 
theorem|Complex analysis in several variables -- Memorial Conference of 
Kiyoshi Oka's Centennial Birthday, 249-�261, Adv.\ Stud.\ Pure Math., 42, 
Math.\ Soc.\ Japan, Tokyo, 2004|

\article OT87|T.\ Ohsawa {\rm and} K.\ Takegoshi|On the extension of $L^2$
holomorphic functions|Math.\ Zeitschrift|195|1987|197--204|

\article Pau98a|M.\ P\u{a}un|Sur l'effectivit\'e num\'erique des
  images inverses de fibr\'es en droites|Math.\ Ann|310|1998|411--421|
  
\miscell Pau98b|M. P\u{a}un|Fibr\'es en droites num\'eriquement
  effectifs et vari\'et\'es K\"ahl\'eriennes compactes \`a courbure de
  Ricci nef|Th\`ese, Universit\'e de Grenoble 1, (1998), 80$\,$p|
  
\miscell Pau00|M.\ P\u{a}un|Semipositive $(1,1)$-cohomology classes on
  projective manifolds|Preprint de l'I.R.M.A., Strasbourg|

\article Pau07|M.\ P\u{a}un|Siu's invariance of plurigenera: a one-tower
  proof|J.\ Differential Geom|76|2007|485�-493|

\miscell Pau08|M.~ P\u{a}un|Relative critical exponents, non-vanishing
and metrics with minimal singularities|arXiv:math.CV/\-0807.3109|

\article Rei88|I.\ Reider|Vector bundles of rank $2$ and linear systems on
algebraic surfaces|Ann.\ of Math.|127|1988|309--316|

\article Ri68|R.\ Richberg|Stetige streng pseudokonvexe Funktionen|Math.\ 
Ann.|175|1968|257--286|

\miscell Sak88|F.\ Sakai|Reider-Serrano's method on normal surfaces|Proc.\
Intern.\ Conf.\ on Algebraic Geometry (L'Aquila, June 1988),
Lecture Notes in Math., Vol.~1417, Springer-Verlag, Berlin, 1990, 301--319|

\article Sch74|M.\ Schneider|Ein einfacher Beweis des Verschwindungs\-satzes
f\"ur positive holomorphe Vektorraumb\"undel|Manuscripta Math.|11|1974|95--101|

\miscell Ser54|J.-P.\ Serre|Fonctions automorphes: quelques majorations
dans le cas o\`u $X/G$ est compact|S\'em.\ Cartan (1953-54) 2--1 \`a 2--9|

\article Ser55|J.-P.\ Serre|Un th\'eor\`eme de dualit\'e|Comment.\
Math.|29|1955|9--26|

\article Ser56|J.-P.\ Serre|G\'eom\'etrie alg\'ebrique et g\'eom\'etrie
analytique|Ann.\ Inst.\ Fourier (Grenoble)|6|1956|1--42|

\bookref ShSo85|B.\ Shiffman, A.J.\ Sommese|Vanishing theorems on
complex manifolds|Progress in Math.\ no~{\bf 56}, Birkh\"auser|1985|

\article Sho85|V.V.~Shokurov|A nonvanishing theorem|Izv.\ Akad.\ Nauk SSSR
Ser.\ Mat|49|1985|635�-651|

\article Sib85|N.\ Sibony|Quelques probl\`emes de prolongement de courants
en analyse complexe|Duke Math.\ J.|52|1985|157--197|

\article Sie55|C.L.\ Siegel|Meromorphic Funktionen auf kompakten
Mannigfaltig\-keiten~|Nach\-richten der Akademie der Wissenschaften in
G\"ottingen, Math.-Phys.\ Klasse|4|1955|71--77|

\article Siu74|Y.T.\ Siu|Analyticity of sets associated to Lelong numbers
and the extension of closed positive currents|Invent.\ Math.|27|1974|53--156|

\article Siu88|Y.T.\ Siu|The existence of K\"ahler-Einstein metrics on
  manifolds with positive anticanonical line bundle and a suitable finite
  symmetry group|Ann.\ of Math|127|1988|585--627|

\article Siu93|Y.T.\ Siu|An effective Matsusaka big theorem|Ann.\ Inst.\
Fourier|43|1993|1387--1405|

\miscell Siu95|Y.T.\ Siu|Very ampleness criterion of double
adjoint of ample line bundles|Princeton Conf.\ in honor of Robert C.~Gunning 
and Joseph J.~Kohn, Princeton University, NJ, USA, Mar.~16-20, 1992;
Ann.\ Math.\ Stud.\ {\bf 137} (1995), 291--318|

\article Siu96|Y.T.\ Siu|Effective Very Ampleness|Invent.\ 
Math.|124|1996|563--571|

\article Siu98|Y.T.\ Siu|Invariance of Plurigenera|Invent.\ 
Math.|134|1998|661--673|

\miscell Siu00|Y.T.\ Siu|Extension of twisted pluricanonical sections 
  with plurisubharmonic weight and invariance of semi-positively twisted 
  plurigenera for manifolds not necessarily of general type|Complex 
  Geometry (G\"ottingen, 2000), Springer, Berlin, 2002, 223--277|

\miscell Siu06|Y.-T.~Siu|A General Non-Vanishing Theorem and an 
Analytic Proof of the Finite Generation of the Canonical 
Ring|arXiv:math.AG/0610740|

\miscell Siu08|Y.-T.~Siu|Finite Generation of Canonical Ring by Analytic
Method|arXiv:\break math.CV/0803.2454|

\article Sko72a|H.\ Skoda|Sous-ensembles analytiques d'ordre fini ou infini
dans $\bC^n$|Bull.\ Soc.\ Math.\ France|100|1972|353--408|

\article Sko72b|H.\ Skoda|Applications des techniques $L^2$ \`a la th\'eorie
des id\'eaux d'une alg\`ebre de fonctions holomorphes avec poids|Ann.\
Scient.\ Ec.\ Norm.\ Sup.\ 4e S\'erie|5|1972|545--579|

\article Sko75|H.\ Skoda|Estimations $L^2$ pour l'op\'erateur 
$\dbar$ et applications arith\-m\'etiques|S\'eminai\-re P.\ Lelong (Analyse), 
ann\'ee 1975/76, Lecture Notes in Math., Vol.~538, 
Springer-Verlag|\rm Berlin|1977|314--323|

\article Sko78|H.\ Skoda|Morphismes surjectifs de fibr\'es vectoriels
semi-positifs|Ann.\ Sci.\ Ecole Norm.\ Sup.|11|1978|577--611|

\miscell Sko80|H.\ Skoda|Rel\`evement des sections globales dans les
fibr\'es semi-positifs|S\'em.\ P.~Le\-long-H.~Skoda (Analyse), 19e ann\'ee,
1978-1979, Lecture Notes in Math.\ 822 (1980), 259--303|

\article Sko82|H.\ Skoda|Prolongement des courants positifs ferm\'es de
masse finie|Invent.\ Math.|66|1982|361--376|

\article Som78|A.J.\ Sommese|Submanifolds of abelian varieties|Math.\
Ann.|233|1978|229--256|

\article Sug88|K-I.\ Sugiyama|A geometry of K\"ahler cones|
Math.\ Ann.|281|1988|135--144|

\article Taka06|S.\ Takayama|Pluricanonical systems on algebraic varieties 
  of general type|Invent.\ Math|165|2006|551--587|

\article Taka07|S.\ Takayama|On the invariance and lower
  semi-continuity of plurigenera of algebraic varieties|J.\ Algebraic 
  Geom|16|2007|1--18|

\article Take97|K.\ Takegoshi|On cohomology groups of nef line
bundles tensorized with multiplier ideal sheaves on compact
K\"ahler manifolds|Osaka J.\ Math.|34|1997|783--802|

\article Tei79|B.\ Teissier|Du th\'eor\`eme de l'index de Hodge aux
in\'egalit\'es isop\'eri\-m\'etriques|C.\ R.\ Acad.\ Sc.\ Paris,
s\'er.\ A|288|29\ {\rm Janvier}\ 1979|287--289|

\miscell Tei82|B.\ Teissier|Bonnesen-type inequalities in algebraic
geometry|Sem.\ on Diff.\ Geom.\ edited by S.T.\ Yau, 1982,
Princeton Univ.\ Press, 85--105|

\article Thi67|P.\ Thie|The Lelong number of a point of a complex analytic 
set|Math.\ Annalen|172|1967|269--312|

\article Tia87|G.\ Tian|On K\"ahler-Einstein metrics on certain
  K\"ahler manifolds with $c_1(M)>0$|Invent.\ Math|89|1987|225--246|
  
\article Tra95|S.\ Trapani|Numerical criteria for the positivity of the
difference of ample divisors|Math.\ Zeitschrift|219|1995|387--401|

\miscell Tsu05|H.~Tsuji|Extension of log pluricanonical forms from 
  subvarieties|arXiv:\break math.CV/0511342|
  
\miscell Tsu07a|H.\ Tsuji|Canonical singular hermitian metrics on
relative canonical bundles|prepublication Sophia University, Japan,
arXiv:math.AG/0704.0566|

\miscell Tsu07b|H.\ Tsuji|Canonical volume forms on compact K\"ahler
manifolds|prepublication Sophia University, Japan, arXiv:math.AG/0707.0111|

\miscell Tsu07c|H.\ Tsuji|Talk and personal communication|MSRI Hot-Topics
Workshop ``Minimal and canonical models in algebraic geometry'' help
at Berkeley in April 2007|

\article Vie82|E.\ Viehweg|Vanishing theorems|J.\ Reine Angew.\ 
Math.|335|1982|1--8|

\article Yau78|S.T.\ Yau|On the Ricci curvature of a complex K\"ahler
manifold and the complex Monge-Amp\`ere equation I|Comm.\ Pure and Appl.\
Math.|31|1978|339--411|

}

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