\documentclass{icmart}

%%%%%%%%%%%%%%%%
%%%%%%%%%% Use this for your addresse(s)
%%%%%%%%%% they will appear at the end of your article
%%%%%%%%%% use one \contact[]{} for each author

\contact[demailly@fourier.ujf-grenoble.fr]{Universit\'e de Grenoble I,
Institut Fourier, 100 rue des Maths, BP 74, 38402 Saint-Martin d'H\`eres}

%\contact[e-mail address]{2. Author(s)'s address}
%\contact[e-mail address]{3. Author(s)'s address}

%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Commands and environments
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\numberwithin{equation}{section}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}

\newtheorem*{coro}{Corollary} %%%% for unnumbered statements


\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{question}[theorem]{Question}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\tors}{\operatorname{torsion}}

\newcommand{\bC}{{\mathbb C}}
\newcommand{\bN}{{\mathbb N}}
\newcommand{\bP}{{\mathbb P}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bZ}{{\mathbb Z}}

\newcommand{\cC}{{\mathcal C}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cH}{{\mathcal H}}
\newcommand{\cI}{{\mathcal I}}
\newcommand{\cK}{{\mathcal K}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cX}{{\mathcal X}}

\newcommand{\dbar}{\overline\partial}
\newcommand{\ddbar}{\partial\overline\partial}
\newcommand{\ovl}{\overline}
\newcommand{\wt}{\widetilde}
\newcommand{\lra}{\longrightarrow}
\newcommand{\bul}{{\scriptscriptstyle\bullet}}

% mathematical operators
\renewcommand{\Re}{\mathop{\rm Re}\nolimits}
\renewcommand{\Im}{\mathop{\rm Im}\nolimits}
\newcommand{\Pic}{\mathop{\rm Pic}\nolimits}
\newcommand{\codim}{\mathop{\rm codim}\nolimits}
\newcommand{\Id}{\mathop{\rm Id}\nolimits}
\newcommand{\Sing}{\mathop{\rm Sing}\nolimits}
\newcommand{\Supp}{\mathop{\rm Supp}\nolimits}
\newcommand{\Vol}{\mathop{\rm Vol}\nolimits}
\newcommand{\rank}{\mathop{\rm rank}\nolimits}
\newcommand{\pr}{\mathop{\rm pr}\nolimits}

\newcommand{\NS}{\mathop{\rm NS}\nolimits}
\newcommand{\NE}{\mathop{\rm NE}\nolimits}
\newcommand{\ME}{\mathop{\rm ME}\nolimits}
\newcommand{\SME}{\mathop{\rm SME}\nolimits}
\newcommand{\alg}{{\rm alg}}
\newcommand{\nef}{{\rm nef}}
\newcommand{\num}{\nu}
\newcommand{\ssm}{\mathop{\Bbb r}}
\newcommand{\smallvee}{{\scriptscriptstyle\vee}}

% figures inserted as PostScript files
\special{header=/home/demailly/psinputs/mathdraw/grlib.ps}
\long\def\InsertFig#1 #2 #3 #4\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{"
#3}}#4$}}
\long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}}


\title[K\"ahler manifolds and transcendental techniques in algebraic geometry]
{K\"ahler manifolds and transcendental\hfil\break techniques in algebraic 
geometry}

\author[Jean-Pierre Demailly]{Jean-Pierre Demailly%\thanks{grateful to...}
}

\begin{document}

\begin{abstract}
Our goal is to survey some of the main advances which took place
recently in the study of the geometry of projective or compact
K\"ahler manifolds : very efficient new transcendental techniques, a
better understanding of the geometric structure of cones of positive
cohomology classes and of the deformation theory of K\"ahler
manifolds, new results around the invariance of plurigenera and in
the minimal model program.

\end{abstract}

\begin{classification}
Primary 14C30; Secondary 32C17, 32C30, 32L20.
\end{classification}

\begin{keywords}
Projective variety, K\"ahler manifold, Hodge theory, positive current, 
Monge-Amp\`ere equation, Lelong number,
Chern connection, curvature, Bochner-Kodaira technique, 
Kodaira embedding theorem, K\"ahler cone, ample divisor, nef divisor, 
pseudo-effective cone, Neron-Severi group, $L^2$ estimates, vanishing theorem, 
Ohsawa-Takegoshi extension theorem, pluricanonical ring, invariance
of plurigenera.\vskip0pt
\end{keywords}


\maketitle



\section{Introduction}

Modern algebraic geometry is one the most intricate crossroads between 
various branches
of mathematics : commutative algebra, complex analysis, global 
analysis on manifolds, partial differential equations, 
differential topology, symplectic geometry, number theory ... . This interplay 
has already been strongly emphasized by historical precursors, 
including Hodge, Kodaira, Hirzebruch and Grauert. Of course, there have 
been also fruitful efforts to establish purely algebraic foundations of 
the major results of algebraic geometry, and many prominent mathematicians
such as Grothendieck, Deligne and Mumford stand out among the founders of this
trend. The present contribution stands closer to the above mentioned wider 
approch ; its goal is to explain some recent applications of 
local and global complex analytic methods to the study of projective 
algebraic varieties. 

A unifying theme is the concept of positivity : ample line
bundles are characterized by the positivity of their curvature in the complex
geometric setting (Kodaira \cite{Kod54}). Projective manifolds thus appear
as a subclass of the class of compact K\"ahler manifolds, and their 
cohomological properties can be derived from the study of harmonic forms 
on K\"ahler manifolds (Hodge theory). In this vein, another central concept 
is the concept of positive current, which was 
introduced by P.~Lelong during the 50's. By carefully studying the
singularities and the intersection theory of such currents, we derive
precise structure theorems for the K\"ahler cone and for the cone
of effective divisors of arbitrary projective varieties 
(\cite{BDPP04}, \cite{DP04}).

$L^2$ estimates for solutions of $\overline\partial$ equations are another
crucial technique for pro\-ving vanishing theorems for the cohomology
of vholomorphic vector bundles or sheaves. A combination of the 
Bochner-Kodaira 
differential geometric estimate with PDE techniques of Kohn, H\"ormander and
Andreotti-Vesentini led in the 60's 
to powerful existence theorems for $\overline\partial$-equations
in hermitian vector bundles of positive curvature. A more recent and equally
decisive outcome is the $L^2$ extension theorem by Ohsawa and Takegoshi
\cite{OT87} in 1987. Among applications, we have various forms of 
approximation theorems (closed positive $(1,1)$-currents can be approximated
by algebraic divisors, and their singularities can be approximated by
algebraic singularities). In the analytic setting, this turns out to be the 
key for the study of adjunction theory (generation properties of
adjoint linear systems \hbox{$K_X+L$}, pluricanonical embeddings ...). 
As an illustration, we present a recent proof, adapted
from work by Y.T.~Siu \cite{Siu98}, \cite{Siu00}, S.~Takayama \cite{Tak06b} and
M.~P\v{a}un~\cite{Pau05}, of the 
deformation invariance of plurigenera $h^0(X_t,mK_{X_t})$, for an arbitrary
projective family $(X_t)$ of algebraic varieties.


\section{Basic concepts and results of complex geometry}

This section mostly contains only well-known definitions and results. 
However, we want to fix the notation and describe in detail our starting 
point.

\subsection{Forms, currents, K\"ahler metrics}

Let $X$ be a compact complex manifold and $n=\dim_\bC X$. In any local 
holomorphic coordinate system \hbox{$z=(z_1,\ldots,z_n)$}, a differential
form $u$ of type $(p,q)$ can be written as a sum
$u(z)=\sum_{|J|=p,|K|=q}u_{JK}(z)\,dz_J\wedge d\ovl z_K$
extended to all increasing multi-indices $J,\,K$ of length $p,\,q$, 
with the usual notation $dz_J=dz_{j_1}\wedge\ldots\wedge dz_{j_p}$.
We are especially interested in \emph{positive currents} of type 
$(p,p)$
$$
T=i^{p^2}\sum_{|J|=|K|=p}T_{JK}(z)dz_J\wedge d\ovl z_K.
$$
Recall that a current is a differential form with distribution coefficients,
and that a current is said to be positive if the distribution
$\sum\lambda_j\ovl \lambda_k T_{JK}$ is a positive real measure for all
complex numbers $\lambda_J$ (which implies
$T_{KJ}=\ovl T_{JK}$, hence $\ovl T=T$).
The coefficients $T_{JK}$ are then
complex measures -- and the diagonal ones $T_{JJ}$ are positive 
(real) measures.

A current is said to be  closed if $dT=0$ in the sense of distributions. 
Important examples of closed positive $(p,p)$-currents
are currents of integration over codimension $p$ analytic cycles
$[A]=\sum c_j[A_j]$ where the current $[A_j]$ is defined by duality as 
$$
\langle[A_j],u\rangle=\int_{A_j}u_{|A_j}
$$
for every $(n-p,n-p)$ test form $u$ on $X$. Another important
example of $(1,1)$-current is the Hessian form $T=i\ddbar\varphi$ of a 
plurisubharmonic
function on an open set $\Omega\subset X$ (plurisubharmonic functions are
upper semi-continuous functions satisfying the mean value inequality on
complex analytic disc; they are characterized by positivity of
$i\sum {\partial^2\varphi/\partial z_j\partial\ovl z_k}\;dz_j\wedge 
d\ovl z_k$). A~\emph{K\"ahler metric} on
$X$ is a positive definite hermitian $(1,1)$-form
$$
\omega(z)=i\sum_{1\le j,k\le n}\omega_{jk}(z)dz_j\wedge d\ovl z_k\qquad
\hbox{such that $d\omega=0$},
$$
with smooth coefficients. The manifold $X$ is said to be \emph{K\"ahler} 
if it possesses at least one K\"ahler metric $\omega$. 
It is clear that every complex analytic and locally closed
submanifold  $X\subset\bP_\bC^N$ is K\"ahler (the restriction of the 
Fubini-Study metric $\omega_{FS}={i\over 2\pi}
\log(|z_0|^2+|z_1|^2+\ldots+|z_N|^2)$ to $X$ is a K\"ahler metric.
Especially projective algebraic varieties are K\"ahler.

\subsection{Cohomology of compact K\"ahler manifolds} 
To every $d$-closed complex valued $k$-form or current $\alpha$ 
(resp.\ to every $\dbar$-closed complexe valued $(p,q)$-form or current 
$\alpha$)
is associated its De Rham (resp.\ Dolbeault) cohomology class
$$
\{\alpha\}\in H^{p+q}(X,\bC)\quad\hbox{(resp.\ $H^{p,q}(X,\bC))$}
$$
This definition hides a nontrivial result, namely the fact that all 
cohomology groups involved (De Rham, Dolbeault, $\ldots$) can  be 
defined either in terms of smooth forms or in terms of currents. In fact, 
if we consider the associated complexes of sheaves,
forms and currents both provide acyclic resolutions of the same sheaf
(locally constant functions, resp.\ holomorphic sections). One of the main 
results of Hodge theory, historically obtained by W.V.D.\ Hodge through the
theory of harmonic forms, is the following fundamental

\begin{theorem} Let $(X,\omega)$ be a compact K\"ahler manifold. Then there
is a canonical isomorphism
$$
H^k(X,\bC)=\bigoplus_{p+q=k}H^{p,q}(X,\bC),
$$
where each group $H^{p,q}(X,\bC)$ can ve viewed as the space of $(p,q)$-forms
$\alpha$ which are harmonic with respect to $\omega$, i.e.\ 
$\Delta_\omega\alpha=0$.
\end{theorem}

Now, observe that every analytic cycle 
$A=\sum\lambda_jA_j$ of codimension $p$ with
integral coefficients defines a cohomology class
$$
\{[A]\}\in H^{p,p}(X,\bC)\cap H^{2p}(X,\bZ)/\{\tors\}\subset
H^{p,p}(X,\bC)\cap H^{2p}(X,\bQ)
$$
where $H^{2p}(X,\bZ)/\{\tors\}\subset H^{2p}(X,\bQ)\subset H^{2p}(X,\bC)$ denotes
the image of integral classes in complex cohomology. When $X$ is a projective
algebraic manifold, this observation leads to the following statement, 
known as the
\emph{Hodge conjecture} (which was to become one of the famous seven Millenium 
problems of the Clay Mathematics Institute~$\ldots$).

\begin{conjecture} Let $X$ be a projective algebraic manifold. Then the space
of ``Hodge classes'' $H^{p,p}(X,\bC)\cap H^{2p}(X,\bQ)$ of type $(p,p)$ 
is generated by classes of algebraic cycles of codimension $p$ 
with $\bQ$-coefficients.
\end{conjecture}

At present, not much is known to support the positive direction of the Hodge 
conjecture, not even the case of \emph{abelian varieties} (i.e.\ projective 
algebraic complex tori $X=\bC/\Lambda$) -- which is the reason why several
experts believe that the conjecture could eventually lead to a counterexample.
There are however a number of cases 
where the cohomology algebra can be explicitly computed in terms of the
geometry, and which do satisfy the conjecture: flag manifolds (Schubert  
cycles generate the cohomology ring), moduli spaces of stable or
parabolic bundles over a general curve (I.~Biswas and
M.S.~Narasimhan \cite{BN97}).

In the K\"ahler case, the conjecture is trivially wrong as shown by a
general complex torus possessing a line bundle with indefinite curvature.
Moreover, by a recent result of C.~Voisin \cite{Voi02}, even a
considerably weakened form of the conjecture -- adding Chern classes
of arbitrary coherent analytic sheaves to the pool of potential generators --
is false for non projective complex tori:

\begin{theorem} {\rm (C.~Voisin \cite{Voi02})} There exists a 
$4$-dimensional complex
torus $X$ which possesses a non trivial Hodge class of degree $4$, such that
every coherent analytic sheaf $\cF$ on $X$ satisfies $c_2(\cF)=0$.
\end{theorem}

The idea is to show the existence of a $4$-dimensional complex torus 
$X=\bC^4/\Lambda$ which does not contain any analytic subset of 
positive dimension, and such
that the Hodge classes of degree $4$ are perpendicular to $\omega^{n-2}$
for a suitable choice of the K\"ahler metric $\omega$. The lattice 
$\Lambda$ is explicitly found via a number theoretic construction of Weil 
based on the number field $\bQ[i]$, also considered by S.~Zucker 
\cite{Zuc77}. The theorem of existence of Hermitian Yang-Mills connections
for stable bundles combined with L\"ubke's inequality then implies
$c_2(\cF)=0$ for every coherent sheaf $\cF$ on the torus.
\medskip

\subsection{Fundamental $L^2$ existence theorems}
Let $X$ be a complex manifold and $(E,h)$ a hermitian holomorphic vector
bundle of rank $r$ over $X$. If $E_{|U}\simeq U\times\bC^r$ is a local
holomorphic trivialization, the hermitian product can be written as
$\langle u,v\rangle = {}^tu H(z)\ovl v$ where $H(z)$ is
the hermitian matric of $h$ and $u,v\in E_z$. It is well known
that there exists a unique ``Chern connection'' $D=D^{1,0}+D^{0,1}$ such
that $D^{0,1}=\dbar$ and such that $D$ is compatible with the hermitian metric;
in the given trivialization we have 
$D^{1,0}u=\dbar u+\Gamma^{1,0}\wedge u$ where
$\Gamma^{1,0}=\ovl H^{-1}\partial\ovl H$, and its curvature operator
$\Theta_{E,h}=D^2$ is the smooth section of 
$\Lambda^{1,1}T^*_X\otimes\Hom(E,E)$ given by
$\Theta_{E,h}=\dbar(\ovl H^{-1}\partial\ovl H)$. If $E$ is of rank $r=1$,
then it is customary to write $H(z)=e^{-\varphi(z)}$, and the curvature
tensor then takes the simple expression $\Theta_{E,h}=\ddbar\varphi$. In that
case, the \emph{first Chern class} of $E$ is the cohomology class
$c_1(E)=\{{i\over2\pi}\Theta_{E,h}\}\in H^{1,1}(X,\bC)$, which is also
an integral class in $H^2(X,\bZ)$). 

In case $(X,\omega)$ is a K\"ahler manifold, the bundles $\Lambda^{p,q}T^*_X
\otimes E$ are equipped with the hermitian metric induced by
$\Lambda^{p,q}\omega \otimes h$, and we have a Hilbert space of global 
$L^2$ sections over $X$ by integrating with respect to the K\"ahler volume 
form $dV_\omega=\omega^n/n!$. If $A,\,B$ are differential operators 
acting on $L^2$ space of sections (in general, they are just closed
and densely defined operators), we denote by $A^*$ the formal adjoint
of $A$, and by $[A,B]=AB-(-1)^{\deg A\,\deg B}BA$ the usual commutator
bracket of operators. The fundamental operator $\Lambda_\omega$
of K\"ahler geometry is the adjoint of the wedge multiplication operator
$u\mapsto\omega\wedge u$.

In this context, we have the following fundamental existence theorems
for $\dbar$-equations, which is the culmination of several decades
of work by Bochner \cite{Boc49}, Kodaira \cite{Kod54}, Kohn \cite{Koh63},
Andreotti-Vesentini \cite{AV65}, H\"ormander \cite{Hor65}, Skoda 
\cite{Sko78}, Ohsawa-Takegoshi \cite{OT87} (and many others). The proofs 
always proceed through differential 
geometric inequalities relating the Laplace-Beltrami operators with the 
curvature (Bochner-Kodaira identities and inequalities).
The most basic result is the $L^2$ existence theorem for solutions
of $\dbar$-equations.

\begin{theorem} {\rm (\cite{AV65}, see also \cite{Dem82})}
\label{dbar.solutions}
Let $(X,\omega)$ be a K\"ahler manifold which is ``complete'' in the sense
that it possesses a geodesically complete K\"ahler metric $\wt\omega$. 
Let $E$ be a hermitian holomorphic vector bundle of
rank $r$ over $X$, and assume that the curvature operator
$A^{p,q}_{E,h,\omega}=[i\Theta_{E,h},\Lambda_\omega]$ is positive
definite everywhere on $\Lambda^{p,q}T^\star_X\otimes E$, $q\ge 1$.
Then for any form $g\in L^2(X,\Lambda^{p,q}T^\star_X\otimes E)$
satisfying $\dbar g=0$ and $\int_X\langle(A^{p,q}_{E,h,\omega})^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)$ such that  $\dbar f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X\langle 
(A^{p,q}_{E,h,\omega})^{-1}g,g\rangle\,dV_\omega.$$
\end{theorem}

It is thus of crucial importance to study conditions under which the operator
$A^{p,q}_{E,h,\omega}$ is positive definite. An easier case is when $E$ is
a line bundle. Then we denote by $\gamma_1(z)\leq\ldots\leq\gamma_n(z)$ the
eigenvalues of the real $(1,1)$-form $i\Theta_{E,h}(z)$ with respect to 
the metric $\omega(z)$ at each point. A straightforward calculation shows that
$$
\langle A^{p,q}_{E,h,\omega}u,u\rangle = \sum_{|J|=p,|K|=q}
\Big(\sum_{k\in K}\gamma_k-\sum_{j\in\complement J}\gamma_j\Big)|u_{JK}|^2.
$$
In particular, for $(n,q)$-forms the negative sum 
$-\sum_{j\in\complement J}\gamma_j$ disappears and we have
$$
\langle A^{n,q}_{E,h,\omega}u,u\rangle \geq(\gamma_1+\ldots+\gamma_q)|u|^2,
\quad
\langle (A^{n,q}_{E,h,\omega})^{-1}u,u\rangle \leq
(\gamma_1+\ldots+\gamma_q)^{-1}|u|^2
$$
provided the line bundle $(E,h)$ has positive definite curvature.
Therefore $\dbar$-equations can be solved for all $L^2$ $(n,q)$-forms
with $q\ge 1$, and this is the major reason why vanishing results for
$H^q$ cohomology groups are usually obtained for sections of the
``\emph{adjoint line bundle}'' $\wt E=K_X\otimes E$, where
$K_X=\Lambda^nT^*_X=\Omega^n_X$ is the ``\emph{canonical bundle}''
of~$X$, rather than for $E$ itself. Especially, if $X$ is compact (or
weakly pseudoconvex) and $i\Theta_{E,h}>0$, then $H^q(X,K_X\otimes E)=0$
for $q\ge 1$ (Kodaira), and more generally $H^{p,q}(X,E)=0$ for
$p+q\ge n+1$ (Kodaira-Nakano, take $\omega=i\Theta_{E,h}$, in
which case $\gamma_j\equiv 1$ for all $j$ and
$\sum_{k\in K}\gamma_k-\sum_{j\in\complement J}\gamma_j=p+q-n$).

As shown in \cite{Dem82}, 
Theorem~\ref{dbar.solutions} still holds true in that case when $h$
is a \emph{singular hermitian metric}, i.e.\ a metric whose weights
$\varphi$ are arbitrary locally integrable functions, provided that
the curvature is $(E,h)$ is positive in the sense of currents
(i.e., the weights $\varphi$ are strictly plurisubharmonic).
This implies the well-known Nadel vanishing theorem (\cite{Nad89},
\cite{Dem89}, \cite{Dem93}), a generalization of the Kawamata-Viehweg 
vanishing theorem \cite{Kaw82}, \cite{Vie82}.

\begin{theorem} {\rm (Nadel)} \label{nadel}
Let $(X,\omega)$ be a compact $($or weakly
pseudoconvex$)$ K\"ahler manifold, and $(L,h)$ a singular hermitian line
bundle such that $\Theta_{L,h}\geq\varepsilon\omega$ for some
$\varepsilon>0$. Then $H^q(X,K_X\otimes L\otimes\cI(h))=0$ for $q\ge 1$, 
where $\cI(h)$ is the multiplier ideal sheaf of $h$, namely the
sheaf of germs of holomorphic functions $f$ on $X$ such that
$|f|^2e^{-\varphi}$ is locally integrable with respect to the
local weights $h=e^{-\varphi}$.
\end{theorem}

It is well known that Theorems~\ref{dbar.solutions} and \ref{nadel}, 
more specifically, its 
``singular hermitian'' version, imply almost all other fundamental 
vanishing or existence theorems of algebraic geometry, as well as their 
analytic counterparts in the framework of Stein manifolds (general solution
of the Levi problem by Grauert), see e.g.\ Demailly \cite{Dem01} for 
a recent account. In particular, one gets as a consequence the
\emph{Kodaira embedding theorem} \cite{Kod54}.

\begin{theorem} Let $X$ be a compact complex $n$-dimensional manifold. 
Then the following properties are equivalent.
\begin{itemize} 
\item[\rm (i)]
$X$ can be embedded in some projective space 
$\bP^N_\bC$ as a closed analytic submanifold
$($and such a submanifold is automatically algebraic by Chow's thorem$)$.
\item[\rm (ii)]
$X$ carries a hermitian holomorphic line bundle $(L,h)$ with psoitive
definite smooth curvature form $i\Theta_{L,h}>0$.
\item[\rm (iii)]
$X$ possesses a Hodge metric, i.e., a K\"ahler metric $\omega$ such that
$\{\omega\}\in H^2(X,\bZ)$.
\end{itemize}
If property (ii) holds true, then for $m\geq m_0\gg 1$
the multiple $L^{\otimes m}$ is very ample,
namely we have an embedding given by the linear system
$V=H^0(X,L^{\otimes m})$ of sections,
$$
\Phi_{L^{\otimes m}}:X\lra P(V^*),\qquad z\mapsto H_z=\{\sigma\in V\,;\;
\sigma(z)=0\}\subset V,
$$
and $L^{\otimes m}\simeq \Phi_{L^{\otimes m}}^*\cO(1)$ is the pull-back 
of the canonical bundle on $P(V^*)$.
\end{theorem}

Another fundamental existence theorem is the $L^2$-extension result by 
Ohsawa-Take\-goshi \cite{OT87}. Many different versions and generalizations
have been given in recent years
\cite{Ohs88}, \cite{Ohs95}, \cite{Ohs94}, \cite{Ohs01}, \cite{Ohs03}. 
Here is another one, due to Manivel \cite{Man93}, which is slightly
less general but simpler to state.

\begin{theorem} {\rm (Ohsawa-Takegoshi \cite{OT87}, Manivel \cite{Man93})}
\label{Ohsawa.Takegoshi.Thm}
Let $X$ be a compact or weakly pseudoconvex $n$-dimensional complex 
manifold equipped with a K\"ahler metric $\omega$, let $L$ $($resp.\ $E)$ be
a hermitian holomorphic line bundle $($resp.\ a hermitian holomorphic 
vector bundle of rank $r$ over~$X)$, and $s$ a global holomorphic 
section of~$E$. Assume that $s$ is generically transverse to the zero
section, and let
$$
Y=\big\{x\in X\,;\;s(x)=0, \Lambda^r ds(x)\not= 0\big\},\qquad
p=\dim Y=n-r.
$$
Moreover, assume that the $(1,1)$-form 
$i\Theta(L)+r\,i\,\ddbar\log|s|^2$ is semipositive and that there is
a continuous function $\alpha\ge 1$ such that the following two inequalities
hold everywhere on $X:$
\begin{itemize}
\item[\rm (i)]
$\displaystyle i\Theta(L)+r\,i\,\ddbar\log|s|^2\ge\alpha^{-1}
{\{i\Theta(E)s,s\}\over|s|^2}\,,$
\item[\rm (ii)] $|s|\le e^{-\alpha}$.
\end{itemize}
Then for every holomorphic section $f$ over $Y$ of the adjoint line bundle 
$\wt L=K_X\otimes L$ $($restricted to~$Y)$, such that
$\int_Y|f|^2|\Lambda^r(ds)|^{-2}dV_\omega<+\infty$, 
there  exists a holomorphic extension $F$ of $f$ over $X$, with values 
in $\wt L$, such that
$$
\int_X{|F|^2\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}\,,
$$
where $C_r$ is a numerical constant depending only on~$r$.
\end{theorem}

The proof actually shows that the extension theorem holds true as well
for $\dbar$-closed $(0,q)$-forms with values in $\wt L$, of which the
stated theorem is the special case $q=0$. 

There are several other important $L^2$ existence theorems. One of them 
is Skoda's criterion
 for the surjectivity of holomorphic bundle morphisms -- more
concretely, a Bezout type division theorem for holomorphic function.
It can be derived either from Theorem \ref{dbar.solutions} on
$\dbar$-equations through sharp curvature calculations (this is
Skoda's original approch in \cite{Sko78}), or as
a consequence of the above extension theorem \ref{Ohsawa.Takegoshi.Thm}
(see Ohsawa \cite{Ohs01}).

\subsection{Positive cones}

We now introduce some further basic objects of projective or K\"ahler 
geometry, namely \emph{cones of positive cohomology classes}.

\begin{definition}
Let $X$ be a compact K\"ahler manifold and $H^{1,1}(X,\bR)$ the space of
real $(1,1)$ cohomology classes.
\begin{itemize}
\item[\rm(i)] The \emph{K\"ahler cone} is the set 
$\cK\subset H^{1,1}(X,\bR)$ of cohomology classes $\{\omega\}$
of K\"ahler forms. This is clearly an open convex cone.
\item[\rm(ii)] The \emph{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 closed convex cone $($as follows from 
the weak compactness property of bounded sets of positive measures 
or currents$)$.

\end{itemize}
\end{definition}

It is follows from this definition that $\ovl\cK\subset\cE$. In general the
inclusion is strict. 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
divisor $E\simeq\bP^1$ has a cohomology class $\{\alpha\}$ such that 
$\int_E\alpha= E^2=-1$, hence $\{\alpha\}\notin\ovl\cK$, although
$\{\alpha\}=\{[E]\}\in\cE$.
\medskip

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
\begin{align}
\NS(X)&:= H^{1,1}(X,\bR)\cap \big(H^2(X,\bZ)/\{\tors\}\big),\cr
\NS_\bR(X)&:=\NS(X)\otimes_\bZ\bR.
\nonumber
\end{align}
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\break 
$H^{1,1}(X,\bR)\subset H^2(X,\bR)$ in the Hodge decomposition,
hence in general the \emph{Picard number}, defined as
$$\rho(X)=\rank_\bZ \NS(X)=\dim_\bR \NS_\bR(X)$$
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
$$\cK_{\NS} = \cK\cap \NS_\bR(X),\qquad \cE_{\NS}=\cE\cap \NS_{\bR}(X).$$
A very important fact is that the ``Neron-Severi part'' of any of
the open or closed transcendental cones $\cK$, $\cE$, $\ovl\cK$, $\cE^\circ$
is algebraic, i.e.\ can be characterized in simple algebraic terms.

\begin{theorem} 
\label{cones.basic.results}
Let $X$ be a projective manifold. Then
\begin{itemize}
\item[\rm (i)] $\cE_{\NS}$ is the closure of the cone generated by classes
of effective divisors, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.
\item[\rm (ii)] $\cK_{\NS}$ is the open cone generated by classes of
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$)$.
\item[\rm (iii)] The interior $\cE_{\NS}^\circ$ is the cone generated
by classes of big divisors, namely divisors $D$ such that
$h^0(X,\cO(kD))\ge c\,k^{\dim X}$ for $k$ large.
\item[\rm (iv)] The closed cone $\ovl\cK_{\NS}$ consists of the closure
of the cone generated by nef divisors $D$ (or nef line bundles $L$), 
namely effective integral divisors $D$ such that $D\cdot C\ge 0$ for
every curve $C$.
\end{itemize}
\end{theorem}

By extension, we will say that $\ovl{\cK}$ is the cone of \emph{nef}
$(1,1)$-cohomology classes (even though they are not necessarily integral).
\medskip

\noindent
\emph{Sketch of proof.} (see also \cite{Dem90} for more details). 
If we denote by $\cK_\alg$ the open cone
generated by ample divisors, resp.\ by $\cE_\alg$ the closure of the cone
generated by effective divisors, we have $K_\alg\subset K_{\NS}$,
$\cE_\alg\subset\cE_{\NS}$, and clearly the interesting part lies in the 
converse inclusions. The inclusion $K_{\NS}\subset\cK_\alg$ is equivalent
to the Kodaira embedding theorem~: if a rational class $\{\alpha\}$ is
in $\cK$, then some multiple of $\{\alpha\}$ is the first Chern class
of a hermitian line bundle $L$ whose curvature form is K\"ahler. Therefore $L$
is ample and $\{\alpha\}\in \cK_\alg\,$; property (ii) follows.

Similarly, if we take a
rational class $\{\alpha\}\in\cE_{\NS}^\circ$, then we still have
$\{\alpha-\varepsilon\omega\}\in\cE_{\NS}^\circ$ by subtracting a small 
multiple $\varepsilon\omega$ of a
K\"ahler class, hence $\alpha-\varepsilon\omega\equiv T\ge 0$ for some
positive current $T$. Therefore some multiple $\{m_0\alpha\}$ is the first 
Chern class
of a hermitian line bundle $(L,h)$ with curvature current
$T$
$$
\Theta_{L,h}:=-{i\over 2\pi}i\ddbar\log h = m_0(T+\varepsilon\omega)\ge 
m_0\varepsilon\omega.
$$
Theorem \ref{dbar.solutions} on $L^2$ estimates for $\dbar$-equations 
then shows that  large multiples $L^{\otimes k}$ admit a large number 
of sections,  hence $L^{\otimes k}$ can be represented by a big 
divisor. This implies (iii) and
also that $\cE_{\NS}^\circ\subset\cE_\alg$. Therefore 
$\cE_{\NS}\subset\cE_\alg$ by passing to the closure$\,$; (i) follows.
The statement (iv) about nef divisors follows
e.g.\ from Klaiman \cite{Kle66} and Hartshorne \cite{Har70}, 
since every nef divisor is a limit 
of a sequence of ample rational divisors.\qed
\medskip

As a natural extrapolation of the algebraic situation, we say that 
$\ovl{\cK}$ is the cone of \emph{nef} $(1,1)$-cohomology classes 
(even though these classes are not necessarily integral). Property 
\ref{cones.basic.results}~(i) also explains the terminology used 
for the pseudo-effective cone.

\subsection{Approximation of currents and Zariski decomposition}

Let $X$ be compact K\"ahler manifold and let $\alpha\in\cE^\circ$ be in the
\emph{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 \emph{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 $\delta\ll 1$. Notice
that the latter definition of a K\"ahler current makes sense even if 
$X$ is an arbitrary (non necessarily K\"ahler) compact complex manifold.

\begin{theorem} {\rm (Demailly \cite{Dem92}, Boucksom 
\cite{Bou02}, 3.1.24)} 
\label{current.approx}
If $T$ is a K\"ahler current on a compact complex manifold $X$, then one 
can write $T=\lim T_m$ for a sequence of  K\"ahler currents $T_m$ in
the same cohomology class as $T$, which have
logarithmic poles and coefficients in ${1\over m}\bZ$. This means that there
are modifications \hbox{$\mu_m:\wt X_m\to X$} such that
$$\mu_m^\star T_m=[E_m]+\beta_m$$
where $E_m$ is an effective $\bQ$-divisor on $\wt X_m$ with coefficients
in ${1\over m}\bZ$ $(E_m$ is the ``fixed part'' and $\beta_m$ a closed 
semi-positive form, the ``movable part''$)$.
\end{theorem}

\proof\ We just recall the main idea and refer to \cite{Dem92} for details. 
Locally we can write $T=i\ddbar\varphi$ for some strictly 
plurisubharmonic potential $\varphi$ on $X$. The approximating potentials
$\varphi_m$ of $\varphi$ are defined as
$$
\varphi_m(z)={1\over 2m}\log\sum_\ell |g_{\ell,m}(z)|^2
$$
where $(g_{\ell,m})$ is a Hilbert basis of the space 
$\cH(\Omega,m\varphi)$ of holomorphic
functions which are $L^2$ with respect to the weight $e^{-2m\varphi}$.
The Ohsawa-Takegoshi $L^2$ extension theorem \ref{Ohsawa.Takegoshi.Thm} 
(applied to extension from a single isolated point) implies that there
are enough such holomorphic functions, and thus $\varphi_m\geq\varphi-C/m$. 
On the other hand $\varphi=\lim_{m\to+\infty}\varphi_m$ by a Bergman kernel 
trick and by the mean value inequality.

The Hilbert basis $(g_{\ell,m})$ is also a family of local generators of 
the globally defined multiplier ideal sheaf $\cI(mT)=\cI(m\varphi)$. The
modification $\mu_m:\wt X_m\to X$ is obtained by blowing-up this ideal sheaf, 
so that 
$$\mu_m^\star\cI(mT)=\cO(-mE_m).$$
for some effective $\bQ$-divisor $E_m$ with normal crossings on $\wt X_m$. 
Now, we set 
$T_m=i\ddbar\varphi_m$ and $\beta_m=\mu_m^*T_m-[E_m]$.
Then $\beta_m=i\ddbar\psi_m$ where 
$$
\psi_m={1\over 2m}\log\sum_\ell|g_{\ell,m}\circ\mu_m/h|^2
\quad\hbox{locally on $\wt X_m$}
$$
and $h$ is a generator of $\cO(-mE_m)$, and we see that $\beta_m$ is a smooth
semi-positive form on $\wt X_m$. The construction can be made global
by using a gluing technique, e.g.\ via partitions of unity.\qed
\medskip

\begin {remark} 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.
\end{remark}

\InsertFig 50 67
{
1 mm unit
-40 0 moveto -40 60 lineto 40 60 lineto stroke
0.85 0.85 1 setrgbcolor
-23 30 moveto
0 0 lineto
23 30 lineto fill
0 0 1 setrgbcolor
-23 30 moveto 23 30 lineto stroke
1 0.85 0.85 setrgbcolor
-20 50 moveto
0 0 lineto
20 50 lineto fill
1 0 0 setrgbcolor
-20 50 moveto 20 50 lineto stroke
0.35 setlinewidth
-20 50 moveto
0 0 lineto
20 50 lineto stroke
20 50 moveto
0 50 20 6 0 180 0 ellipsearc 
-15 45 lineto -5 43 lineto 5 43 lineto 15 45 lineto 20 50 lineto stroke
0 0 1 setrgbcolor
0.35 setlinewidth
-22.7 28.6 moveto
0 0 lineto
22.7 28.6 lineto stroke
23 30 moveto
0 30 23 8 0 50 0 ellipsearc stroke
23 30 moveto
0 30 23 8 0 -230 0 ellipsearc stroke
0 0 0 setrgbcolor
0.15 setlinewidth
-9 15 moveto 0.6 disk
-6 7.5 moveto 0.6 disk
-9 15 moveto -6 7.5 lineto stroke
-2.5 6.5 moveto 0.6 disk
-9 15 moveto -2.5 6.5 lineto stroke
}
\LabelTeX -3 46 $\cK_{\NS}$\ELTX
\LabelTeX -19 27 $\cE_{\NS}$\ELTX
\LabelTeX -38.5 55 $\NS_\bR(\wt X_m)$\ELTX
\LabelTeX -11 17 $\wt\alpha$\ELTX
\LabelTeX -15 6 $[E_m]$\ELTX
\LabelTeX -2.5 8.5 $\beta_m$\ELTX
\LabelTeX 21 17 $\wt\alpha=\mu_m^\star\alpha=[E_m]+\beta_m$\ELTX
\EndFig
\vskip5mm

\noindent
In the above construction, $\beta_m$ is not just semi-positive, it is even
positive definite on tangent vectors which are not mapped to $0$ by 
the differential $d\mu_m$, in parti\-cular $\beta_m$ is positive definite outside
the exceptional divisor. However, if $E$ is the exceptional divisor of the
blow-up along a smooth centre $Y\subset X$, then
$\cO(-E)$ is relatively ample with respect to the blow-up map $\pi$, hence the
negative current $-[E]$ is cohomologous to a smooth form $\theta_E$ which
is positive along the fibers of~$\pi$. As a consequence, we can slightly
perturb the decomposition of $\wt\alpha$ by increasing multiplicities in
the components of $E_m$ and adding recursively to $\beta_m$ small multiples 
$\varepsilon_E\theta_E$
in such a way that $\wt\beta_m+\sum\varepsilon_E\theta_E$ becomes
a K\"ahler metric on $\wt X_m$. This in turn
implies that $\wt X_m$ is K\"ahler and we thus get the following 
characterization of the Fujiki class $\cC$ of
compact complex manifolds which are bimeromorphic to K\"ahler manifolds:

\begin{corollary} A compact complex manifold 
is bimeromorphic to a K\"ahler mani\-fold
$($or equivalently, dominated by a K\"ahler manifold$)$
if and only if it carries a K\"ahler current $T$.
\end{corollary}

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

We describe here the main results obtained in Demailly-P\v{a}un \cite{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$\,$:

\begin{theorem} \label{char.kcone}
Let $X$ be a compact K\"ahler manifold. Let $\cP$ 
be the set of real $(1,1)$ cohomology classes $\{\alpha\}$
which are numerically 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 $\cP$.
\end{theorem}

\begin{corollary} 
\label{char.projcase}
If $X$ is projective algebraic, then $\cK=\cP$.
\end{corollary}

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: \emph{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 
Cor.~\ref{char.projcase}.

\begin{example} {\rm The following example shows that the cone $\cP$
need not be connected (and that the components of $\cP$ need not 
be convex, either). 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 \ref{char.kcone} 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 $q>0$).}
\end{example}

\noindent
\emph{Sketch of proof of Theorem~\ref{char.kcone} and 
Cor.~\ref{char.projcase}.} As is well known, the singularities of a 
closed positive current $T$ can be measured by its
\emph{Lelong numbers}
$$
\nu(T,x)=\liminf_{z\to x}{\varphi(z)\over\log|z-x|},
$$ 
where $T={i\over\pi}\ddbar\varphi$ near $x$. A fundamental theorem of Siu 
Siu \cite{Siu74} states that the Lelong sublevel sets $E_c(T):=\{x\in X\,;\,
\nu(T,x)\ge c\}$ are \emph{analytic sets} for every $c>0$
(this fact can nowadays be derived in a rather straightforward manner from 
the approximation theorem \ref{current.approx}). The crucial steps 
of the proof of Theorem~\ref{char.kcone} are contained in the 
following statements. 

\begin{proposition} {\rm (P\v{a}un \cite{Pau98a}, \cite{Pau98b})} 
\label{kahler.by.restriction}
Let $X$ be a compact 
complex manifold $($or more generally a compact complex space$)$. Then
\begin{itemize}
\item[\rm (i)] 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~$E_c(T)$.
\item[\rm (ii)] The cohomology class of a K\"ahler current $\{T\}$ is 
a K\"ahler class $($i.e.\ the class of a {\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~$E_c(T)$. 
\end{itemize}
\end{proposition}

The proof of Proposition \ref{kahler.by.restriction} 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 proved in Demailly \cite{Dem92} 
(see also  Theorem~\ref{current.approx} below).
The main point then consists in an induction on dimension and a standard 
gluing procedure~: if $T=\alpha+{i\over\pi}\ddbar\varphi$ where
$\varphi$ is smooth on $X\ssm Z$ and has $-\infty$ poles along $Z$,
then we can remove the poles of $\varphi$ by replacing $\varphi$ with
$\max(\varphi,\psi-C)$, provided $\psi$ is smooth and defined near $Z$
and $C$ is a large constant.\qed
\medskip

The next (and more substantial step) consists of the following result
which is reminiscent of the Grauert-Riemenschneider conjecture
(Siu \cite{Siu84}, Demailly \cite{Dem85}).

\begin{theorem} {\rm (Demailly-P\v{a}un \cite{DP04})}
\label{main.step} 
Let $X$ be a compact K\"ahler manifold
and let $\{\alpha\}$ be a nef class $($i.e.\ $\{\alpha\}\in\ovl\cK)$.
Assume that $\int_X\alpha^n>0$. Then $\{\alpha\}$ contains a K\"ahler
current $T$, in other words $\{\alpha\}\in\cE^\circ$.
\end{theorem}

\proof\ 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 $\delta>0$.
We check this by observing that $\alpha+\varepsilon\omega$ is a
K\"ahler class, hence by the Calabi-Yau theorem Yau \cite{Yau78} the
Monge-Amp\`ere equation
$$
(\alpha+\varepsilon\omega+i\ddbar\varphi_\varepsilon)^n=f_\varepsilon
$$
can be solved with an arbitrary right-hand side $f_\varepsilon>0$ such that 
$$
\int_Xf_\varepsilon=C_\varepsilon=\int_X(\alpha+\varepsilon\omega)^n.
$$
However, by our assumption that $\int_X\alpha^n>0$, the constant 
$C_\varepsilon$ is bounded away from~$0$. We use this fact in order 
to concentrate a fixed amount of volume of the volume form 
$f_\varepsilon$ in an $\varepsilon$-tubular neighborhood of $Y$.
We then show that the sequence of
$(p,p)$-forms $(\alpha+\varepsilon\omega+i\ddbar\varphi_\varepsilon)^p$ 
converges weakly to the desired current~$\Theta$ (this part relies
heavily on the theory of currents). The second and final part uses 
a ``diagonal trick'': apply the result just proved to 
$$
\wt X=X\times X,\qquad \wt Y=\hbox{diagonal}\subset \wt X,\qquad
\wt\alpha=\pr_1^*\alpha+\pr_2^*\alpha.
$$
It is then clear that $\wt\alpha$ is nef on $\wt X$ and that
$\int_{\wt X}(\wt\alpha)^{2n}>0$. It follows by the above that the
class $\smash{\{\wt\alpha\}}^n$ contains a K\"ahler current $\Theta$ such that
$\Theta\ge\delta[\smash{\wt Y}]$ for some $\delta>0$. Therefore 
the push-forward
$$
T:=(\pr_1)_*(\Theta\wedge\pr_2^*\omega)
$$
is numerically equivalent to a multiple of $\alpha$ and dominates
$\delta\omega$, and we see that $T$ is a K\"ahler current.\qed
\medskip

\noindent
\emph{End of Proof of Theorem \ref{char.kcone}}. 
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 closed
in $\cP$, i.e.\ that $\ovl\cK\cap\cP\subset\cK$. Pick a class
$\{\alpha\}\in\ovl\cK\cap\cP$. In particular $\{\alpha\}$ is
nef and satisfies $\int_X\alpha^n>0$. By Theorem~\ref{main.step}  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
\ref{kahler.by.restriction}~(ii), therefore $\{\alpha\}\in\cK$, as 
desired.\qed
\medskip

The projective case \ref{char.projcase} is a consequence of the
following variant of Theorem~\ref{char.kcone}.

\begin{corollary} 
\label{char.allmetrics}
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 $k=1,2,\ldots,p=\dim Y$.
\end{corollary}

\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
\medskip

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 \ref{char.projcase} follows. 
On the other 
hand, we can pass to the limit in \ref{char.allmetrics} by replacing $\alpha$ 
by $\alpha+\varepsilon\omega$,
and in this way we get also a characterization of nef classes.

\begin{corollary} 
\label{charnef.allmetrics}
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 $k=1,2,\ldots,p=\dim Y$.
\end{corollary}

By a formal convexity argument, one can derive from \ref{char.allmetrics} 
or \ref{charnef.allmetrics}  the following
interesting consequence about the dual of the cone $\cK$. 

\begin{theorem} 
\label{dual.kahlercone}
Let $X$ be a compact K\"ahler manifold. 
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
$\int_Y\alpha\wedge\omega^{p-1}\ge 0$. In other words, the dual of the
nef cone $\smash{\ovl\cK}$ is the closed convex cone 
in $H^{n-1,n-1}_\bR(X)$ 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~$X$.\qed
\end{theorem}

\section{Deformations of compact K\"ahler manifolds}

If $S$ is an analytic space, recall that a deformation of compact
complex manifolds is a proper holomorphic map $\pi:\cX\to S$ such that 
the fibers are smooth and such that $\cX$ is locally the product of the base
by a neighborhood of any point in any fiber (with $\pi$ being the first
projection of such a local decomposition). For any $t\in S$, we denote 
by $X_t=\pi^{-1}(t)$ the fiber over $t$.

Since compact K\"ahler manifolds share many common features with 
projective algebraic manifolds -- e.g.\ good Hodge theoretic
properties -- rather strong properties are expected for
their deformation theory.
Kodaira showed in the 60's that every K\"ahler surface $X$ is a limit by 
deformation of algebraic surfaces, namely there exists a deformation
$\cX\to S$ such that $X=X_{t_0}$ for some $t_0$, and $X_{t_m}$ is
projective algebraic for a sequence $t_m\to t_0$. It was therefore a natural --
and long-standing -- question whether a similar property holds in higher 
dimensions. C.~Voisin showed in a series of recent papers that the
general answer is negative, and in fact there exist rigid non projective
compact K\"ahler manifolds.

\begin{theorem} {\rm (recent results by C.~Voisin
\cite{Voi04a}, \cite{Voi04b})}
\begin{itemize}
\item[\rm (i)] In any dimension${}\ge 4$, there exist compact K\"ahler
  manifolds which do not have the homotopy type $($or even the
  homology ring$)$ of a complex projective manifold
{\rm (\cite{Voi04a})}.
\item[\rm (ii)] 
  In any dimension${}\ge 8$, there exist compact K\"ahler
  manifolds $X$ such that no compact bimeromorphic model $X'$ of $X$
  has the homotopy type of a complex projective manifold
{\rm (\cite{Voi04b})}.
\end{itemize}
\end{theorem}

The example in (i) is obtained by selecting a complex torus $T$ 
of dimension\hbox{${}\geq 2$} possessing a linear endomorphism $\varphi$ which
has non real eigenvalues (pairwise distinct and non conjugate). Then $X$
is obtained by blowing-up the finite set of pairwise intersection points 
of of the four subsets $T\times\{0\}$, $\{0\}\times T$, $\Delta={}diagonal$, 
$G_\varphi={}$ graph of $\varphi$, and then their strict transforms in
the first stage blow-up. By using rather elementary considerations of Hodge
theory, this provides an example of a rigid K\"ahler variety 
which does not have the homotopy type of a projective variety. The example
in (ii) is obtained via the Poincar\'e bundle on $T\times\hat T\,$; we refer
to \cite{Voi04a} and \cite{Voi04b} for details.\qed
\medskip

Another fundamental fact proved by Kodaira and Spencer \cite{KS60}
is the observation that the K\"ahler property is open with respect to 
deformation: if $X_{t_0}$ 
is K\"ahler for some $t_0\in S$, then the nearby fibers $X_t$ 
(for $t$ in a metric topology neighborhood of $t_0$ in $S$) is
also K\"ahler. The proof consists in showing that the desired 
K\"ahler metrics are
solutions of a suitably chosen $4$-th order elliptic differential operator
for which there is no jump of the kernel at $t_0$. However, the numerous
known examples leave hopes for a much stronger openness property.

\begin{conjecture} Let $\cX\to S$ be a deformation with irreducible base
space $S$ such that some fiber $X_{t_0}$ is K\"ahler. Then
there should exist a finite $($or possibly countable$)$
union of analytic strata $S_\nu\subset S$, $S_\nu\neq S$, such that
\begin{itemize}
\item[\rm(i)] $X_t$ is K\"ahler for $t\in S\ssm\bigcup S_\nu$.
\item[\rm(ii)] $X_t$ is bimeromorphic to a K\"ahler manifold 
for $t\in \bigcup S_\nu$.
\end{itemize}
\end{conjecture}

A crucial step in analyzing the conjecture is to describe the behaviour
of the K\"ahler cone of $X_t$ as $t$ approaches the ``bad strata''. This 
question is now fully understood thanks to the following result which is 
a direct corollary of our characterization of the K\"ahler cone 
(Theorem \ref{char.kcone}). As a consequence, a ``collapse'' of the K\"ahler
cone could only come from a degeneration of the Hodge decomposition, the
behaviour of which is complex analytic thanks to the Fr\"olicher spectral
sequence.

\begin{theorem} {\rm (Demailly-P\v{a}un \cite{DP04})}
\label{cone.deform}
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
$$
\nabla=\begin{pmatrix}
\nabla^{2,0} & * & 0\cr 
* & \nabla^{1,1} & * \cr
0 & * & \nabla^{0,2}\cr
\end{pmatrix}
$$
on the Hodge bundle $H^2=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$.
\end{theorem}

\noindent
\emph{Sketch of Proof.} 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 implies 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 $\nabla^{1,1}$. Moreover, the connected component
$\cK_t\subset\cP_t$ cannot jump from one component to the other thanks to
the already metionned results by Kodaira-Spencer \cite{KS60}. This
concludes the proof.\qed
\medskip

Theorem \ref{cone.deform} was essentially already known in 
the cases of complex surfaces (i.e.\ in dimension $2$),
thanks to the work of N.~Buchdahl \cite{Buc99}, \cite{Buc00} and 
A.~Lamari \cite{Lam99a}, \cite{Lam99b}.

Shortly after the original \cite{DP04} manuscript appeared 
in April 2001, Daniel Huybrechts \cite{Huy01} informed us 
Theorem \ref{char.kcone} 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 \emph{rational curves} $C\subset X$ (see also 
Huybrechts \cite{Huy99}).

\section{Positive cones in $H^{n-1,n-1}(X)$ and Serre duality}

\subsection{Basic definitions}

In a way which will be shown to be dual to the case of divisors and
positive $(1,1)$-currents, we consider in $H^{n-1,n-1}_\bR(X)$ the cone 
$\cN$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n-1,n-1)$} (i.e., of bidimension $(1,1)$). In the projective
case, we also consider the intersection of $\cN$ with the space 
$N_1(X)$ generated
by integral $(n-1,n-1)$-classes (by the hard Lefschetz theorem, $N_1(X)$ 
is just the dual of $NS_\bR(X)$).

\begin{definition}
Let $X$ be a compact K\"ahler manifold.
\begin{itemize}
\item[\rm (i)] We define $\cN$ to be the $($closed$)$ convex cone in 
$H^{n-1,n-1}_\bR(X)$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n-1,n-1)$} $($i.e., of bidimension $(1,1))$.

\item[\rm (ii)] We define the cone $\cM\subset H^{n-1,n-1}_\bR(X)$ of
``movable classes'' to be the closure of the convex cone generated by
classes of currents of the form
$$\mu_\star(\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1})$$
where $\mu:\wt X\to X$ is an arbitrary modification $($one could just
restrict oneself to compositions of blow-ups with smooth centers$)$, and 
the $\wt\omega_j$ are K\"ahler forms on~$\wt X$. Clearly $\cM\subset\cN$.

\item[\rm (iii)] Correspondingly, we introduce the intersections
$$ 
\cN_{\NS}=\cN\cap N_1(X),\qquad \cM_{\NS}=\cM\cap N_1(X),\qquad 
$$
in the space generated by integral bidimension $(1,1)$-classes
$$
N_1(X):=(H^{n-1,n-1}_\bR(X)\cap H^{2n-2}(X,\bZ)/\{\tors\})\otimes_\bZ\bR.
$$
\item[\rm (iv)] If $X$ is projective, we define $\NE(X)$ to be the
convex cone generated by all effective curves. Clearly  
$\ovl{\NE(X)}\subset\cN_{\NS}$.

\item[\rm (v)] If $X$ is projective, we say that $C$ is a ``strongly 
movable'' curve if 
$$
C=\mu_\star(\wt A_1\cap\ldots\cap \wt A_{n-1})
$$ 
for suitable very ample divisors $\wt A_j$ on $\wt X$, where 
$\mu:\wt X\to X$ is a modification. We let ${\rm SME}(X)$ 
be the convex cone generated by all strongly movable $($effective$)$ 
curves. Clearly  $\ovl{{\rm SME}(X)}\subset\cM_{\NS}$.

\item[\rm (vi)] We say that $C$ is a movable 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 let ${\rm ME}(X)$ 
to be the convex cone generated by all movable $($effective$)$ curves.
\end{itemize}
\end{definition}

The upshot of this definition lies in the following easy observation.

\begin{proposition} 
\label{duality.pairing}
Let $X$ be a compact K\"ahler manifold. 
Consider the Poincar\'e duality pairing
$$
H^{1,1}(X,\bR)\times H^{n-1,n-1}_\bR(X)\lra \bR,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta.
$$
Then the duality pairing takes nonnegative values
\begin{itemize}
\item[\rm (i)] for all pairs $(\alpha,\beta)\in\ovl\cK\times\cN$;
\item[\rm (ii)] for all pairs $(\alpha,\beta)\in\cE\times\cM$.
\item[\rm (iii)] for all pairs $(\alpha,\beta)$ where $\alpha\in\cE$ and 
$\beta=[C_t]\in{\rm ME}(X)$ is the class of a movable curve.
\end{itemize}
\end{proposition}

\proof\ (i) is obvious. In order to prove (ii), we may assume that
$\beta=\mu_\star(\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1})$ for
some modification $\mu:\wt X\to X$, where $\alpha=\{T\}$ is the class
of a positive $(1,1)$-current on $X$ and $\wt\omega_j$ are K\"ahler
forms on~$\wt X$. Then
$$
\int_X\alpha\wedge\beta=\int_X T\wedge
\mu_\star(\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1})=
\int_X \mu^* T\wedge
\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1}\ge 0.
$$
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 fact that
if $T=i\ddbar\varphi$ locally for some plurisubharmonic function in~$X$, 
we can set 
$\mu^\star T=i\ddbar(\varphi\circ\mu)$. For (iii), we suppose
$\alpha=\{T\}$ and $\beta=\{[C_t]\}$. Then we take an open covering
$(U_j)$ on $X$ such that $T=i\ddbar\varphi_j$ with suitable plurisubharmonic 
functions $\varphi_j$ on $U_j$. If we select a smooth partition of unity 
$\sum\theta_j=1$ subordinate to $(U_j)$, we then get
$$
\int_X\alpha\wedge\beta=\int_{C_t} T_{|C_t}=\sum_j\int_{C_t\cap U_j}
\theta_j i\ddbar\varphi_{j|C_t}\ge 0.
$$
For this to make sense, it should be noticed that $T_{|C_t}$ is a well 
defined closed positive $(1,1)$-current (i.e.\ measure) on $C_t$ for almost 
every $t\in S$, in the sense of Lebesgue measure. This is true only because
$(C_t)$ covers $X$, thus $\varphi_{j|C_t}$ is not identically
$-\infty$ for almost every $t\in S$. The equality in the last formula
is then shown by a regularization argument for $T$, writing $T=\lim T_k$
with $T_k=\alpha+i\ddbar\psi_k$ and a decreasing sequence of smooth almost 
plurisubharmonic potentials $\psi_k\downarrow\psi$ such that the Levi forms 
have a uniform lower bound $i\ddbar\psi_k\ge -C\omega$ (such a
sequence exists by Demailly \cite{Dem92}). Then, 
writing $\alpha=i\ddbar v_j$ for some
smooth potential $v_j$ on $U_j$, we have $T=i\ddbar\varphi_j$ on $U_j$
with $\varphi_j=v_j+\psi$, and this is the decreasing limit of the smooth
approximations $\varphi_{j,k}=v_j+\psi_k$ on $U_j$. Hence 
$T_{k|C_t}\to T_{|C_t}$ for the weak topology of measures on $C_t$.\qed
\medskip

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 $\cC^{\smallvee\smallvee}=
\ovl{\cC}$. Proposition \ref{duality.pairing} leads to the natural
question whether the cones $(\cK,\cN)$ and $(\cE,\cM)$ are dual under
Poincar\'e duality, according to the following schematic picture.

\InsertFig 32 67
{
1 mm unit
0.15 setlinewidth
0 0 0 setrgbcolor
-20 0 moveto -28 0 lineto -28 60 lineto 25 60 lineto stroke
0.85 0.85 1 setrgbcolor
-23.3 30 moveto
0 0 lineto
23.3 30 lineto fill
0 0 1 setrgbcolor
-23 30 moveto 23 30 lineto stroke
1 0.85 0.85 setrgbcolor
-20 50 moveto
0 0 lineto
20 50 lineto fill
1 0 0 setrgbcolor
-20 50 moveto 20 50 lineto stroke
0.35 setlinewidth
-20 50 moveto
0 0 lineto
20 50 lineto stroke
20 50 moveto
0 50 20 6 0 360 0 ellipsearc stroke
0 0 1 setrgbcolor
0.35 setlinewidth
-22.7 28.6 moveto
0 0 lineto
22.7 28.6 lineto stroke
23 30 moveto
0 30 23 8 0 50 0 ellipsearc stroke
23 30 moveto
0 30 23 8 0 -230 0 ellipsearc stroke
1 0 0 setrgbcolor
23 45 moveto 38 35 lineto stroke
25 45 moveto 23 45 lineto 23.7 43 lineto stroke
36 35 moveto 38 35 lineto 37.3 37 lineto stroke
0 0 1 setrgbcolor
23 35 moveto 38 45 lineto stroke
25 35 moveto 23 35 lineto 23.7 37 lineto stroke
36 45 moveto 38 45 lineto 37.3 43 lineto stroke
% right part
62 0 translate
0 0 0 setrgbcolor
0.15 setlinewidth
20 0 moveto 28 0 lineto 28 60 lineto -25 60 lineto stroke
1 0.85 0.85 setrgbcolor
-23.3 30 moveto
0 0 lineto
23.3 30 lineto fill
1 0 0 setrgbcolor
-23 30 moveto 23 30 lineto stroke
0.85 0.85 1 setrgbcolor
-20 50 moveto
0 0 lineto
20 50 lineto fill
0 0 1 setrgbcolor
-20 50 moveto 20 50 lineto stroke
0.35 setlinewidth
-20 50 moveto
0 0 lineto
20 50 lineto stroke
20 50 moveto
0 50 20 6 0 360 0 ellipsearc stroke
1 0 0 setrgbcolor
0.35 setlinewidth
-22.7 28.6 moveto
0 0 lineto
22.7 28.6 lineto stroke
23 30 moveto
0 30 23 8 0 50 0 ellipsearc stroke
23 30 moveto
0 30 23 8 0 -230 0 ellipsearc stroke
}
\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 8 4 $H^{1,1}(X,\bR)$\ELTX
\LabelTeX 57 46 $\cM_{\NS}$\ELTX
\LabelTeX 63 52 $\cM$\ELTX
\LabelTeX 77 32 $\cN$\ELTX
\LabelTeX 43 27 $\cN_{\NS}$\ELTX
\LabelTeX 75 4 $N_1(X)$\ELTX
\LabelTeX 35 4 $H^{n-1,n-1}_\bR(X)$\ELTX
\LabelTeX 25 47 duality\ELTX
\EndFig
\vskip6mm

\noindent
It is indeed well-known that the cone $\ovl{\cK_{\NS}}$ of nef divisors
is dual to the cone $\cN_{\NS}$ of effective curves if $X$ is projective.
The transcendental version $\cK=\cN^\vee$ also follows from our 
Theorem \ref{dual.kahlercone}.

\begin{theorem} {\rm(Demailly-P\v{a}un)} If $X$ is K\"ahler, then 
the cones $\ovl{\cK}\subset H^{1,1}(X,\bR)$ and 
$\cN\subset H^{n-1,n-1}_\bR(X)$ are dual by Poincar\'e duality, and
$\cN$ is the closed convex cone generated by classes $[Y]\wedge\omega^{p-1}$
where $Y\subset X$ ranges over $p$-dimensional analytic subsets,
$p=1,2,\ldots, n$, and $\omega$ ranges over K\"ahler forms.
\end{theorem}

\proof\ Indeed, Prop.~\ref{duality.pairing} shows that the dual 
cone $\cK^\vee$ contains
$\cN$ which itself contains the cone $\cN'$ of all classes of the form
$\{[Y]\wedge\omega^{p-1}\}$. The main result of 
Demailly-P\v{a}un \cite{DP04} conversely shows
that the dual of $(\cN')^\vee$ is equal to $\ovl\cK$, so we must
have
$$\cK^\vee = \ovl{\cN'} = \cN.\eqno\square$$
The other duality statement $\cE=\cM^\vee$ will be investigated in the 
next sections.

\subsection{Concept of volume and movable intersections}
We start with the very important concept of volume.

\begin{definition} 
\label{volume.def}
We define the \emph{volume}, or \emph{movable
self-intersection} of a big class $\alpha\in\cE^\circ$ to be
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\wt X}\beta^n>0
$$
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 $\mu:\wt X\to X$.
\end{definition}

By Fujita \cite{Fuj94} and Demailly-Ein-Lazarsfeld \cite{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.

\begin{proposition} Let $L$ be a big line bundle on the
projective manifold $X$.  Let $\varepsilon > 0$. Then there exists a
modification $\mu: X_{\varepsilon} \to X$ and a decomposition $\mu^*(L) =
E + \beta $ with $E$ an effective $\bQ$-divisor and $\beta$ a big and
nef $\bQ$-divisor such that
$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$
\end{proposition}

It is very useful to observe that the supremum in Definition \ref{volume.def} 
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 
\label{max.construction}
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi=\max(\varphi_1,\varphi_2)\leqno(\ref{max.construction})$$ 
is again a K\"ahler current with weaker 
singularities than $T_1$ and $T_2$. One could define as well
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi={1\over 2m}\log(e^{2m\varphi_1}+e^{2m\varphi_2}),
\leqno(\ref{max.construction}')$$
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
$\mu_m:\wt X_m\to X$ 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 $E=\min(E_1,E_2)$. 

\begin{theorem} {\rm (Boucksom \cite{Bou02})} 
\label{movable.inter.product}
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 $(n-k,n-k)$.
\begin{itemize}
\item[\rm(i)] For each $k=1,\ldots,n$, there exists a canonical
``movable 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.
\item[\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 $\alpha_j\in\ovl{\cK}$ are nef classes.

\item[\rm(iii)] 
The movable intersection product satisfies the Teissier-Hovanskii inequalities
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_n\rangle\ge
(\langle\alpha_1^n\rangle)^{1/n}\ldots(\langle\alpha_n^n\rangle)^{1/n}\qquad
\hbox{$($with $\langle\alpha_j^n\rangle=\Vol(\alpha_j)\,)$}.
$$

\item[\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
$\ovl{\cK}\subset\cE_1\subset\cE$. 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 $\alpha\in\cE_1$.

\item[\rm(v)] The components of $N(\alpha)$ always consist of
divisors whose cohomology classes are linearly independent,
thus $N(\alpha)$ has at most $\rho=\rank_\bZ\NS(X)$ \hbox{components}.
\end{itemize}
\end{theorem}

\proof\ We refer to S.~Boucksom's thesis \cite{Bou02} for details. Boucksom's
treatment also covers the case of compact non K\"ahler manifolds, so it is
fairly general. We only give a very rough construction of the movable
intersection product.

First assume that all classes $\alpha_j$ are big, i.e.\ 
$\alpha_j\in\cE^\circ$. We select K\"ahler currents $T_{j,m}\in\alpha_j$ with
logarithmic poles and their approximate Zariski decompositions
as in Theorem \ref{current.approx}. We can then find a simultaneous 
log-resolution
$\mu_m:\wt X_m\to X$ such that 
$$
\mu^\star T_{j,m}=[E_j{,m}]+\beta_{j,m}.
$$
We consider the direct image current
$\mu_{m\,\star}(\beta_{1,m}\wedge\ldots\wedge\beta_{k,m})$ (which is a closed
positive current of bidegree $(k,k)$ on $X$). It turns out 
by rather elementary monotonicity arguments based on the filtering
property \ref{max.construction} that one can can extract a weakly 
convergent limit
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\uparrow}\limits_{m\to+\infty}
\{(\mu_m)_\star(\beta_{1,m}\wedge\beta_{2,m}\wedge\ldots\wedge\beta_{k,m})\}
$$
and that the corresponding cohomology class in $H^{k,k}(X)$ is uniquely
defined. Now, the intersection product can be extended
to the full closed cone $\cE$ by monotonicity again, namely 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 $\alpha_j\in\cE$.\qed

\begin{definition} For a class $\alpha\in H^{1,1}(X,\bR)$, 
we define the 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.
\end{definition}

By the results of Demailly-Peternell \cite{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 \cite{Bou02}.

\begin{theorem} 
\label{decomp.numdim.zero}
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.\qed
\end{theorem}

Using the Iitaka fibration, it is immediate to see that
$\kappa(X)\leq\num(X)$ always holds true, and from the currently
known examples a natural expectation would~be

\begin{conjecture} {\rm (``generalized abundance conjecture'')} 
  For an arbitrary compact K\"ahler manifold~$X$, the Kodaira
  dimension should be equal to the numerical dimension~:
$$\kappa(X)=\num(X):=\num(c_1(K_X)).$$
\end{conjecture}

This appears to be a fairly strong statement. In fact, it is not difficult
to show that the generalized abundance conjecture contains the
$C_{n,m}$ conjectures about additivity of Kodaira dimension (since it is
not very difficult to show that the numerical dimension is additive
with respect to fibrations). A few extreme cases are known.

\begin{theorem} The generalized abundance conjecture is true at least in the
cases $\nu(X)=-\infty$, $\nu(X)=0$, $\nu(X)=n$.
\end{theorem}

\proof\ In fact $\num(X)=-\infty$ means that $K_X$ is not pseudo-effective, so
no multiple of $K_X$ can have sections and thus $\kappa(X)=-\infty$.
In case $\num(X)=n$, we have to show that $K_X$ is big ($K_X\in\cE^\circ$);
this follows from \cite{DP04} and from 
the solution of the  Grauert-Riemenschneider conjecture in the form 
proven in Demailly \cite{Dem85}. Remains the case $\num(X)=0$. Then
Theorem \ref{decomp.numdim.zero} gives $K_X\equiv\sum\lambda_j D_j$ for some
effective divisor with numerically independent components such that 
$\num(D_j)=0$. It follows that the $\lambda_j$ are rational and therefore
$$
K_X=\sum\lambda_j D_j+F\qquad
\hbox{where $\lambda_j\in\bQ^+$, $\num(D_j)=0$ and $F\in
\Pic^0(X)$.}
$$
In that case Campana and Peternell \cite{CP04} have shown that
$F$ is a torsion element of $\Pic^0(X)$, and so $\kappa(X)=0$.

\subsection{The orthogonality estimate}
The goal of this section is to show that, in an appropriate sense,
approximate Zariski decompositions are almost orthogonal.

\begin{theorem} 
\label{orthogonality.estimate}
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$\,)$.
\end{theorem}

\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
$$E_m=\mu^\star\alpha-D_m=\mu_m^\star(\alpha+C\omega)-(D_m+C\mu_m^\star
\omega).$$

\begin{lemma} 
\label{nef.inequality}
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.
\end{lemma}

\proof\ In case $A$ and $B$ are integral (Cartier) divisors, this is a 
consequence of the holomorphic Morse inequalities,
(Demailly \cite{Dem01}, 8.5). 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 
(Boucksom \cite{Bou02}, 3.1.26). In fact, we expect Lemma \ref{nef.inequality} 
to hold true also in the case of transcendental nef cohomology classes -- 
unfortunately the required generalization of Morse inequalities is still
missing at this point.\qed 
\medskip

\noindent
\emph{End of proof of Theorem \ref{orthogonality.estimate}.}
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 
\ref{nef.inequality} and the binomial formula imply
\begin{align}
\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).
\nonumber
\end{align}
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). 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
\begin{align}
{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
\nonumber
\end{align}
(and we have indeed $t\le {1\over 10n}$), whence 
Theorem \ref{orthogonality.estimate}. 
Of course, the constant 20 is certainly not optimal.\qed

\begin{corollary} If $\alpha\in\cE_{\NS}$, then the divisorial
Zariski decomposition\break 
$\alpha=N(\alpha)+\langle\alpha\rangle$ is such that
$\langle\alpha^{n-1}\rangle\cdot N(\alpha)=0$.
\end{corollary}

\proof\ By replacing $\alpha$ by $\alpha+\delta c_1(H)$, 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
\medskip

\subsection{Proof of duality between $\cE_{\NS}$ and $\cM_{\NS}$}

The main point is the following characterization of
pseudo-effective classes, proved in \cite{BDPP04} (the ``only~if'' part 
already follows from \ref{duality.pairing}~(iii)).

\begin{theorem} 
{\rm (Boucksom-Demailly-P\v{a}un-Peternell \cite{BDPP04})}
\label{char.pseudoeffective}
If $X$ is projective, then a class $\alpha\in\NS_\bR(X)$
is pseudo-effective if $($and only if$\,)$ it is in the dual cone of the cone
$\SME(X)$ of strongly movable curves.
\end{theorem}

In other words, a line bundle $L$ is pseudo-effective if (and
only if) $L\cdot C\ge 0$ for all \emph{movable 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 $\SME(X)$, it is enough 
to consider only those curves $C$ which are images of generic complete 
intersection of very ample divisors on some variety $\wt X$, under
a modification $\mu:\wt X\to X$.
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
$L \cdot C \geq 0$ for every movable curve~$C$. The K\"ahler analogue 
should be~:

\begin{conjecture} For an arbitrary compact 
K\"ahler manifold $X$, the cones $\cE$ and $\cM$ are dual.
\end{conjecture}

\InsertFig 34 67
{
1 mm unit
0.15 setlinewidth
0 0 0 setrgbcolor
-15 0 moveto -30 0 lineto -30 60 lineto 25 60 lineto stroke
0.85 0.85 1 setrgbcolor
-23.3 30 moveto
0 0 lineto
23.3 30 lineto fill
0 0 1 setrgbcolor
-23 30 moveto 23 30 lineto stroke
1 0.85 0.85 setrgbcolor
-20 50 moveto
0 0 lineto
20 50 lineto fill
1 0 0 setrgbcolor
-20 50 moveto 20 50 lineto stroke
0.35 setlinewidth
-20 50 moveto
0 0 lineto
20 50 lineto stroke
20 50 moveto
0 50 20 6 0 360 0 ellipsearc stroke
0 0 1 setrgbcolor
0.35 setlinewidth
-22.7 28.6 moveto
0 0 lineto
22.7 28.6 lineto stroke
23 30 moveto
0 30 23 8 0 50 0 ellipsearc stroke
23 30 moveto
0 30 23 8 0 -230 0 ellipsearc stroke
0.15 setlinewidth
12 20 moveto 19 17 lineto stroke
13.5 20 moveto 12 20 lineto 13 19 lineto stroke
0 0 0 setrgbcolor
0.15 setlinewidth
-10 17 moveto 0.6 disk
-10 25 moveto 0.6 disk
-10 28 moveto 0.6 disk
-10 17 moveto -10 40 lineto stroke
0.3 setlinewidth
-10 35 moveto -10 40 lineto stroke
-10.6 38 moveto -10 40 lineto -9.4 38 lineto stroke
-10.6 35 moveto -9.4 35 lineto stroke
%right part
60 0 translate
0.15 setlinewidth
20 0 moveto 28 0 lineto 28 60 lineto -25 60 lineto stroke
0.85 0.85 1 setrgbcolor
-20 50 moveto
0 0 lineto
20 50 lineto fill
0 0 1 setrgbcolor
-20 50 moveto 20 50 lineto stroke
0.35 setlinewidth
-20 50 moveto
0 0 lineto
20 50 lineto stroke
20 50 moveto
0 50 20 6 0 360 0 ellipsearc stroke
0 0 0 setrgbcolor
-3 25 moveto 0.6 disk
}
\LabelTeX 1 52 $\cE$\ELTX
\LabelTeX -5 46 $\cE_{\NS}$\ELTX
\LabelTeX 14 31 $\cM^\smallvee$\ELTX
\LabelTeX 20 15 $(\cM_{\NS})^\smallvee$\ELTX
\LabelTeX -28 4 $\NS_\bR(X)$\ELTX
\LabelTeX 15 4 $H^{1,1}(X,\bR)$\ELTX
\LabelTeX 65 38 $\cM_{\NS}$\ELTX
\LabelTeX -9 15 $\alpha-\varepsilon\omega$\ELTX
\LabelTeX -8.5 24 $\alpha$\ELTX
\LabelTeX -8.5 27.5 $\alpha+\delta\omega$\ELTX
\LabelTeX -8.5 37 $\omega$\ELTX
\LabelTeX 59 24.5 $\Gamma$\ELTX
\LabelTeX 75 4 $N_1(X)$\ELTX
\EndFig
\vskip6mm

\noindent
\emph{Proof of Theorem \ref{char.pseudoeffective}.}\ (see \cite{BDPP04}). 
We want to show that $\cE_{\NS}=\SME(X)^\vee$. By \ref{duality.pairing}~(iii) 
we have in any case
$$
\cE_{\NS}\subset (\SME(X))^\smallvee.
$$
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 $\SME(X)^\smallvee$.

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
$\alpha\in((\SME(X))^{\smallvee})^\circ$ we still have 
$\alpha-\varepsilon\omega\in(\SME(X))^{\smallvee}$ for
$\varepsilon>0$ small. Therefore
\label{cone.strict.inequality}
$$
\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma\leqno(\ref{cone.strict.inequality})
$$
for every movable curve $\Gamma$. We are going to contradict (\ref{cone.strict.inequality}).
Since $\alpha+\delta\omega$ is big, we have an approximate Zariski
decomposition
$$
\mu_\delta^\star(\alpha+\delta\omega)=E_\delta+D_\delta.
$$
We pick $\Gamma=(\mu_\delta)_\star(D_\delta^{n-1})$. By the Hovanskii-Teissier
concavity inequality
$$
\omega\cdot \Gamma\ge (\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}.
$$
On the other hand
\begin{align}
\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.
\nonumber
\end{align}
By the orthogonality estimate, we find
\begin{align}
{\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}}.
\nonumber
\end{align}
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
$(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}$ 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
$$
\Vol(\alpha+\delta\omega)-D_\delta^n\le \delta^{2n}.
$$
This is the desired contradiction by (\ref{cone.strict.inequality}).\qed
\bigskip

As a corollary, we also get a solution of the ``Hodge conjecture'' for
positive cones of $H^{n-1,n-1}(X)$, namely positive integral classes are
generated by the corresponding cones of curves. This settles in
the affirmative many of the conjectures made in \cite{DPS96}.

\begin{corollary} 
\label{neronpart.conecurves}
Let $X$ be a projective manifold. Then
\begin{itemize}
\item[\rm(i)] $\cN_{\NS}=\ovl{\NE(X)}$.
\item[\rm(ii)] $\cM_{\NS}=\ovl{{\rm SME}(X)}=\ovl{{\rm ME}(X)}$.
\end{itemize}
\end{corollary}

\proof\ (i) is indeed (mostly) a standard result of algebraic geometry, 
a restatement of the fact that the cone of effective 
curves $\NE(X)$ is dual to the cone $\ovl{\cK_{\NS}}$ of nef divisors
(see e.g.\ \cite{Har70}) : clearly $\cN_{\NS}\supset\ovl{\NE(X)}=
\cK_{\NS}^\vee$, and the other direction 
$\cN_{\NS}\subset \cK_{\NS}^\vee$ is a consequence of 
\ref{duality.pairing}~(i).
\smallskip

\noindent
(ii) It is obvious that
$\SME(X)\subset\ME(X)\subset\cM_{\NS}\subset(\cE_{\NS})^\smallvee$ 
(the latter inclusion follows from \ref{duality.pairing}~(iii)). Now 
Theorem \ref{char.pseudoeffective} 
implies $(\cE_{\NS})^\smallvee =
\ovl{\SME(X)}$, and (ii) follows.\qed

\begin{remark} {\rm 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
$\mu:\wt X\to X$ and a K\"ahler metric 
$\wt\omega$ on $\smash{\wt X}$ such that 
$\alpha\cdot\mu_\star(\wt\omega)^{n-1} < 0$. 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~: } 
\end{remark}

\begin{question} Let $(M,\omega)$ be a compact real symplectic manifold.
Fix an almost\break complex structure $J$ compatible with $\omega$, and
for this structure, assume that\break \hbox{$c_1(M)\cdot\omega^{n-1}>0$}. 
Does it follow that $M$ is covered by rational $J$-pseudoholomorphic curves~?
\end{question}

\subsection{Applications and conjectures}
The most important special case of Theorem \ref{char.pseudoeffective} is

\begin{theorem} 
\label{nonuniruled.psef}
If $X$ is a projective manifold and is not uniruled,
then $K_X$ is pseudo-effective, i.e.\ $K_X\in\cE_{\NS}$.
\end{theorem}

\proof\ If $K_X\notin\cE_{\NS}$, Prop.~\ref{duality.pairing} shows that
there is a moving curve $C_t$ such that $K_X\cdot C_t<0$. The standard
``bend-and-break'' lemma of Mori then implies that there is family 
$\Gamma_t$ of \emph{rational} curves with $K_X\cdot\Gamma_t<0$, so $X$ 
is uniruled.\qed
\medskip

Of course, if the ``abundance conjecture'' is correct, the fact that
$K_X$ is pseudo-effective would imply $\kappa(X)\ge 0$, and so every
non uniruled variety shoud satisfy $\kappa(X)\ge 0$. This still seems
beyond reach at the moment.

\section{Plurigenera and the Minimal Model Program}

In the case of algebraic surfaces, the Minimal Model Program (MMP) was
already initiated by Italian geometers at the turn of the XXth century,
and was finally completed by Zariski and Kodaira for all complex
surfaces. The case of higher dimensions (starting with dimension 3) is
a major endeavor of modern times, revitalized by Mori
\cite{Mor82}, Kawamata \cite{Kaw84}, \cite{Kaw85}, \cite{Kaw92} and 
Shokurov \cite{Sho83}, \cite{Sho96} among others
(see also \cite{KMM87} for a good survey).

The basic question is to prove that every birational class of non
uniruled algebraic varieties contains a ``minimal'' member $X$ exhibiting mild
singularities (``terminal singularities''), where ``minimal'' is taken in the 
sense of avoiding unnecessary blow-ups; minimality actually means that 
$K_X$ is nef and not just pseudo-effective (pseudo-effectivity 
follows in general from Theorem \ref{nonuniruled.psef}). This requires 
performing certain birational
transforms known as flips, and important questions are whether a) flips
are indeed possible (``existence of flips''), b) the process
terminates (``termination of flips''). Thanks to Kawamata \cite{Kaw92} and
Shokurov \cite{Sho83}, \cite{Sho96}, this has been proved in 
dimension $3$ at the end of the 80's. Very recently, C.~Hacon and J.~McKernan 
\cite{HM05b} announced that flips exist in dimension
$n$, if one assumes that a slightly stronger version of MMP 
(involving log pairs with real divisors) holds true in dimension $n-1$.
As a consequence, the existence of flips obtained by
Shokurov \cite{Sho03} in 2003 would be achieved in dimension $4$ via a
more systematic method.
Strongly related to these issues are the following fundamental questions

\begin{itemize}
\item[\rm (i)] \emph{Finiteness of the canonical ring:}\
is the \emph{canonical ring}
$R=\bigoplus H^0(X,mK_X)$
of a variety of general type always finitely generated~?\\
If true, Proj$(R)$ of this graded ring $R$ yields of course a 
``canonical model'' in the birational class of~$X$.
\item[\rm (ii)] \emph{Boundedness of pluricanonical embeddings:}
is there a bound $r_n$ depending only on dimension $\dim X=n$, such that
the pluricanonical map $\Phi_{mK_X}$ of a variety of general
type yields a birational embedding in projective space for $m\ge r_n$~?
\item[\rm (iii)] \emph{Invariance of plurigenera:} are plurigenera
$p_m=h^0(X,mK_X)$ always invariant under deformation~?
\end{itemize}

These questions involve taking ``limits'' of divisors as $m\to+\infty$,
and therefore transcendental methods are a strong contender in the arena.
Question (ii) was indeed solved in the affirmative by H.~Tsuji \cite{Tsu06},
\cite{Tsu04} under the assumption that the MMP program is solved, and
in general by S.~Takayama \cite {Tak06a}, and Ch.~Hacon-J.~McKernan 
\cite{HM05a} by pursuing further Tsuji's ideas. Question (iii) was completely
settled by Y.T.~Siu (\cite{Siu98} in the case of varieties of general type, 
and \cite{Siu00} for arbitrary varieties). Quite recently, M.~P\v{a}un
gave a very elementary proof based merely on the Ohwawa-Takegoshi
extension theorem, that we briefly sketch below. Y.T.~Siu's work also gives
strong support for the hope that (i) can be solved by a suitable combination 
of the $L^2$ existence theorems (Skoda's division theorem being one of the
main ingredients). The following is a very slight extension of results
by M.\ P\v{a}un \cite{Pau05} and B.\ Claudon \cite{Cla07}, which are
themselves based on the ideas of Y.T.~Siu~\cite{Siu00} and 
S.\ Takayama~\cite{Tak06b}.

\begin{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 semipositive curvature currents $i\Theta_{L_j,h_j}\ge 0$
on~$\cX$. Assume that
\begin{itemize}
\item[\rm (i)] the restriction of $h_j$ to the central fiber $X_0$ is well
defined $($i.e.\ not identically~$+\infty)$.
\item[\rm (ii)] additionally the multiplier ideal sheaf 
$\cI(h_{j|X_0})$ is trivial for $1\le j\le m-1$.
\end{itemize}
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 $\cX$.
\end{theorem}

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 Theorem~\ref{Ohsawa.Takegoshi.Thm} (see also Siu~\cite{Siu00}).

\begin{lemma} Let $\pi:\cX\to\Delta$ be as before and let $(L,h)$ be
a  $($singular$)$ hermitian 
line bundle with semipositive 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 $\wt u$ over $\cX$
satisfying an $L^2$ estimate
$$
\int_{\cX}\Vert \wt u\Vert^2_{\omega\otimes h}dV_\cX\le C_0
\int_{X_0}\Vert u\Vert^2_{\omega\otimes h}dV_{X_0},
$$
where $C_0\ge 0$ is some universal constant $($independent of $\cX$, $L$, 
$\ldots\,)$.
\end{lemma}

\proof\ 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
$(\wt u_j^{(p)})_{j=1\ldots N_p}$, of $F_p$ over $\cX$ in such a way that 
\begin{itemize}
\item[\rm (a)] for $p=0,\ldots,m-1$, $F_p$ is generated by its sections
$(\wt u_j^{(p)})_{j=1\ldots N_p}\;$;
\item[\rm (b)] we have the $m$-periodicity relations
$N_{p+m}=N_p$ and $\wt u_j^{(p)}$ is an extension of 
$u_j^{(p)}:=\sigma^qu_j^{(r)}$ over $\cX$ for $p=mq+r$, 
where $u_j^{(r)}:=\wt u^{(r)}_{j|X_0}$, $0\le r\le m-1$.
\end{itemize}
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 $\wt u_j^{(p)}$ appropriately for $p=0,\ldots,m-1$.
Now, by induction, we equip $F_{p-1}$ with the tautological metric 
$|\xi|^2/\sum|\wt u_j^{(p-1)}(x)|^2$, 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 semipositive 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 $\wt u_j^{(p)}$
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
$$
\frac{|u_j^{(p)}|^2}{\sum_\ell|u_\ell^{(p-1)}|^2}e^{-\varphi_r},
$$
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
$$
\frac{\sum_j|u_j^{(p)}|^2}{\sum_\ell|u_\ell^{(p-1)}|^2}e^{-\varphi_r}=
\begin{cases}
\displaystyle
\frac{\sum_j|u_j^{(r)}|^2}{\sum_\ell|u_\ell^{(r-1)}|^2}e^{-\varphi_r}
&\hbox{for $p=mq+r$, $0<r\le m-1$},\cr
\displaystyle
\frac{\sum_j|u_j^{(0)}|^2}{\sum_\ell|u_\ell^{(m-1)}|^2}
|\sigma|^2e^{-\varphi_0}
&\hbox{for $p\equiv 0\mod m$}.
\end{cases}
$$
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
$$
\int_{X_0}
\frac{\sum_j|u_j^{(p)}|^2}{\sum_\ell|u_\ell^{(p-1)}|^2}e^{-\varphi_r} 
dV_\omega \le C_1
$$
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 $\wt u_j^{(p)}$ of $F_p$ over $\cX$ such that
$$
\int_{\cX}
\frac{\sum_j|\wt u_j^{(p)}|^2}{\sum_\ell|\wt u_\ell^{(p-1)}|^2}e^{-\varphi_r} 
dV_\omega \le C_2=C_0C_1.
$$
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
$$
\int_{\cX}\Big(\sum_j|\wt u_j^{(p)}|^2\Big)^{1/p}dV_\omega\le C_3.
$$
Jensen's inequality together with well known facts of potential theory now 
show that some subsequence of
the sequence of plurisubharmonic functions\break
${1\over q}\log\sum_j|\wt u_j^{(mq)}|^2$ $\big[$which should be thought of 
as weights on the $\bQ$-line bundles 
${1\over q}(A+q(mK_\cX+\sum L_j))\,\big]$ converges almost everywhere
to the weight $\psi$ of a singular hermitian metric $H$ with semi-positive 
curvature on $mK_\cX+\sum L_j$, in the form of an upper regularized limit
$$
\psi(z)=\limsup_{\zeta\to z}\lim_{\nu\to+\infty}
{1\over q_\nu}\log\sum_j|\wt u_j^{(mq_\nu)}(\zeta)|^2.
$$
On $X_0$ we have 
$$
\lim_{q\to+\infty}{1\over q}\log\sum_j|u_j^{(mq)}|^2=
\lim_{q\to+\infty}{1\over q}
\log\big(|\sigma|^{2q}\sum_j|u_j^{(0)}|^2\big)=\log|\sigma|^2,
$$
hence $\psi(z)\ge\log|\sigma|^2$ and $\Vert\sigma\Vert_H\le 1$.
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 semipositive curvature current on $\cX$ and in a 
local trivialization we have
$$
\Vert\sigma\Vert^2_{\omega\otimes\gamma}\le
C|\sigma|^2\exp\Big(-\Big(1-{1\over m}\Big)\psi+{1\over m}\sum \varphi_j\Big)
\le C \Big(|\sigma|^2\prod e^{-\varphi_j}\Big)^{1/m}
$$
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~$\cX$. The theorem is proved.\qed
\medskip

The special case of the theorem obtained by taking all bundles $L_j$ trivial
tells us in particular that any pluricanonical section $\sigma$ of $mK_\cX$
over $X_0$ extends to $\cX$. By the upper semi-continuity of 
$t\mapsto h^0(X_t,mK_{X_t})$, this implies

\begin{corollary} {\rm (Siu \cite{Siu00})} For any projective
family $t\mapsto X_t$ of algebraic varieties, the plurigenera 
$p_m(X_t)=h^0(X_t,mK_{X_t})$ do not depend on $t$.
\end{corollary} 

At the moment, it should be observed that there are no purely algebraic 
proofs of the invariance of plurigenera, though Y.\ Kawamata \cite{Kaw99} 
has given an algebraic proof in the case of varieties of general type.

\section{References}

\renewcommand{\refname}{}    %%%% for this example
\begin{thebibliography}{20}

\bibitem{AV65} Andreotti, A., Vesentini, E., Carleman estimates for
  the Laplace-Beltrami equation in complex manifolds, \emph{Publ.\ Math.\ 
  I.H.E.S.}\ {\bf 25} (1965) 81--130.

\bibitem{BN97} Biswas, I., Narasimhan, M.S., Hodge classes of moduli 
spaces of parabolic bundles over the general curve, \emph{J.\ Algebraic 
Geom.}\ {\bf 6} (1997), 697--715.

\bibitem{Boc49} Bochner, S., Curvature and Betti numbers (I) and (II),
  \emph{Ann.\ of Math.}\ {\bf 49} (1948) 379-390$\,$; {\bf 50} (1949) 
  77--93.

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

\bibitem{Buc99} Buchdahl, N., On compact K\"ahler surfaces, \emph{Ann.\ 
  Inst.\ Fourier} {\bf 49} (1999) 287--302.
  
\bibitem{Buc00} Buchdahl, N., A Nakai-Moishezon criterion for
  non-K\"ahler surfaces, \emph{Ann.\ Inst.\ Fourier} {\bf 50} (2000)
  1533--1538.
  
\bibitem{CP04} Campana, F., Peternell, Th., Geometric stability of the
  cotangent bundle and the universal cover of a projective manifold,
  \emph{arXiv: math.AG/0405093}.

\bibitem{Cla07} Claudon, B., Invariance for multiples of the twisted 
  canonical bundle, \emph{Ann.\ Inst.\ Fourier} {\bf 57} (2007), 289--300.

\bibitem{Dem82} Demailly, J.-P., 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\'erienne
  compl\`ete, \emph{Ann.\ Sci.\ \'Ecole Norm.\ Sup.\ 4e S\'er.}\ {\bf 15}
  (1982) 457--511.
  
\bibitem{Dem85} Demailly, J.-P., Champs magn\'etiques et
  in\'egalit\'es de Morse pour la $d^{\prime \prime}$-cohomo\-logie,
  \emph{Ann.\ Inst.\ Fourier} {\bf 35} (1985), 189--229.
  
\bibitem{Dem89} Demailly, J.-P., Transcendental proof of a generalized 
  Ka\-wa\-mata-Viehweg vanishing theorem, \emph{C.\ R.\ Acad.\ Sci.\ Paris 
  S\'er.\  I Math.} {\bf 309} (1989) 123--126~~~and:~~
  \emph{Proceedings of the Conference ``Geometrical and algebraical aspects
  in several complex variables''} held at Cetraro, Univ.\ della Calabria,
  June 1989, 81--94.

\bibitem{Dem90} Demailly, J.-P., Singular hermitian metrics on
  positive line bundles, \emph{Proceedings of the Bayreuth conference
  ``Complex algebraic varieties''}, April~2-6, 1990, edited by
  K.~Hulek, T.~Peternell, M.~Schneider, F.~Schreyer, Lecture Notes in
  Math.\ ${\rm n}^\circ\,$1507, Springer-Verlag, 1992, 87--104.
  
\bibitem{Dem92} Demailly, J.-P., Regularization of closed positive
  currents and intersection theory, \emph{J.\ Algebraic Geometry} {\bf 1}
  (1992), 361--409.

\bibitem{Dem93} Demailly, J.-P., A numerical criterion for very ample line
  bundles, \emph{J.\ Differential Geom.}\ {\bf 37} (1993) 323--374.

\bibitem{Dem01} Demailly, J.-P., Multiplier ideal sheaves and 
  analytic methods in algebraic geometry, \emph{Lecture Notes, School on 
  ``Vanishing theorems and effective results in Algebraic Geometry}, 
  ICTP Trieste, April 2000, Publications of ICTP, 2001.
  
\bibitem{DEL00} Demailly, J.-P., Ein, L., Lazarsfeld, R., A
  subadditivity property of multiplier ideals, \emph{arXiv: math.AG/0002035,
  Michigan Math.\ J.}, special volume in honor of William Fulton, {\bf
  48} (2000), 137--156.

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

\bibitem{DPS96} Demailly, J.-P.; Peternell, Th.; Schneider, M.,
  Holomorphic line bundles with partially vanishing cohomology,
  \emph{Israel Math.\ Conf.\ Proc.}\ vol.~{\bf 9} (1996) 165--198.

\bibitem{DPS00} Demailly, J.-P., Peternell, Th.; Schneider, M.,
  Pseudo-effective line bundles on compact K\"ahler manifolds, 
  \emph{Intern.\ J.\ Math.}\ {\bf 6} (2001) 689-741.
  
\bibitem{Fuj94} Fujita, T., Approximating Zariski decomposition of
  big line bundles, \emph{Kodai Math.\ J.}\ {\bf 17} (1994), 1--3.
  
\bibitem{HM05a} Hacon, Ch., McKernan, J., Boundedness of pluricanonical maps
  of varieties of general type, \emph{arXiv: math.AG/0504327},
  Invent.\ Math.\ {\bf 166} (2006), 1--25

\bibitem{HM05b} Hacon, Ch., McKernan, J., On the existence of flips,
  \emph{arXiv: math.AG/0507597}.

\bibitem{Har70} Hartshorne, R., Ample subvarieties of algebraic
  varieties, \emph{Lecture Notes in Math.\ no} {\bf 156}, Springer-Verlag,
  Berlin (1970).

\bibitem{Hor65} H\"ormander, L., $L^2$ estimates and existence theorems for
  the $\dbar$ operator, \emph{Acta Math.}\ {\bf 113} (1965) 89--152.
  
\bibitem{Huy99} Huybrechts, D., Compact Hyperk\"ahler Manifolds:
  Basic Results, \emph{Invent.\ Math.}\ {\bf 135} (1999), 63--113.
  
\bibitem{Huy01} Huybrechts, D., Compact Hyperk\"ahler Manifolds:
  Basic Results, II (The Projectivity Criterion for
  Hyperk\"ahler manifolds as a Consequence of the Demailly-P\v{a}un
  Theorem), \emph{Invent.\ Math.}\ {\bf 152} (2003), 209--212.

\bibitem{Kaw82} Kawamata, Y., A generalization of Kodaira-Ramanujam's
  vanishing theorem, \emph{Math.\ Ann.}\ {\bf 261} (1982) 43--46.

\bibitem{Kaw84} Kawamata, Y., The cone of curves of algebraic
  varieties, \emph{Ann.\ of Math.} {\bf 119} (1984), 603--633.

\bibitem{Kaw85} Kawamata, Y., Minimal models and the kodaira dimension of
  algebraic fiber spaces, \emph{J.\ reine Angew.\ Math.} {\bf 363} (1985), 
  1--46.

\bibitem{Kaw92} Kawamata, Y., Termination of log flips for algebraic 
  $3$-folds, \emph{Internat.\ J.\ Math.} {\bf 3} (1992), 653--659.

\bibitem{Kaw99} Kawamata, Y., Deformation of canonical singularities,
  \emph{J.\ Amer.\ Math.\ Soc.}, {\bf 12} (1999), 85--92.

\bibitem{KMM87} Kawamata, Y., Matsuda, K., Matsuki, K., Introduction to the
  minimal model program, \emph{Adv.\ Stud.\ Pure Math.}\ {\bf 10}
  (1987) 283--360.
  
\bibitem{Kle66} Kleiman, S., Toward a numerical theory of
  ampleness, \emph{Ann.\ Math.}\ {\bf 84} (1966), 293--344.
  
\bibitem{Kod54} Kodaira, K., On K\"ahler varieties of restricted
  type, \emph{Ann.\ of Math.}\ {\bf 60} (1954) 28--48.

\bibitem{Koh63} Kohn, J.J., Harmonic integrals on strongly pseudo-convex
  manifolds~I, \emph{Ann.\ Math.}\ (2), {\bf 78} (1963) 206--213.
  
\bibitem{KS60} Kodaira, K., Spencer, D.C., On deformations of
  complex analytic structures. III. Stability theorems for complex
  structures, \emph{Ann.\ of Math.}\ {\bf 71} (1960) 43--76.
  
\bibitem{Lam99a} Lamari, A., Courants k\"ahl\'eriens et surfaces
  compactes, \emph{Ann.\ Inst.\ Fourier} {\bf 49} (1999) 263--285.
  
\bibitem{Lam99b} Lamari, A., Le cone K\"ahl\'erien d'une surface,
  \emph{J.\ Math.\ Pures Appl.}\ {\bf 78} (1999) 249--263.

\bibitem{Man93} Manivel, L., Un th\'eor\`eme de prolongement $L^2$ de
  sections holomorphes d'un fibr\'e vectoriel, \emph{Math.\ Zeitschrift}
  {\bf 212} (1993) 107--122.

\bibitem{Mor82} Mori, S., Threefolds whose canonical bundles are not
  numerically effective, \emph{Ann. of Math.} {\bf 116} (1982),
  133--176.

\bibitem{Nad89} Nadel, A.M., Multiplier ideal sheaves and 
  K\"ahler-Einstein metrics of positive scalar curvature, \emph{Proc.\ Nat.\
  Acad.\ Sci.\ U.S.A.} {\bf 86} (1989) |7299--7300~~~and~~
  \emph{Annals of Math.}\ {\bf 132} (1990), 549--596.
  
\bibitem{Ohs88} Ohsawa, T., On the extension of $L^2$ holomorphic
  functions, II, \emph{Publ.\ RIMS, Kyoto Univ.}\ {\bf 24} (1988) 265--275.

\bibitem{Ohs95} Ohsawa, T., On the extension of $L^2$ holomorphic
   functions, III$\,$: negligible weights, \emph{Math.\ Zeitschrift}
   {\bf 219} (1995) 215--225.

\bibitem{Ohs94} Ohsawa, T., On the extension of $L^2$ holomorphic functions,
  IV$\,$: A new density concept, \emph{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.

\bibitem{Ohs01} Ohsawa, T., On the extension of L2 holomorphic functions, V, 
  Effects of generalization, \emph{Nagoya Math.\ J.}\ {\bf 161} (2001), 1--21.

\bibitem{Ohs03} Ohsawa, T., On the extension of $L^2$ holomorphic
  functions, VI, A limiting case, \emph{Explorations in complex and 
  Riemannian geometry}, Contemp.\ Math., {\bf 332}, Amer.\ Math.\ Soc., 
  Providence, RI (2003) 235--239.


\bibitem{OT87} Ohsawa, T., Takegoshi, K., On the extension of $L^2$
   holomorphic functions, \emph{Math.\ Zeitschrift}, {\bf 195} (1987)
   197--204.

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

\bibitem{Pau05} P\v{a}un, M., Siu's invariance of plurigenera : a one-tower
  proof, preprint November 2005.

\bibitem{Sho83} Shokurov, V. V., Extremal contractions of three-dimensional 
  algebraic varieties, (Russian),  \emph{Yaroslavl Gos.\ Ped.\ Inst.}, 
  {\bf 102} (1983), 74--90.

\bibitem{Sho96} Shokurov, V. V., $3$-fold log models, Algebraic Geometry, 4,
  \emph{J.\ Math.\ Sci.} {\bf 81} (1996), 2667--2699.

\bibitem{Sho03} Shokurov, V. V., Prelimiting flips,
  \emph{Proc.\ Steklov Inst.\ of Math.} {\bf 240} (2003), 82--219.

\bibitem{Siu74} Siu, Y.-T., Analyticity of sets associated to Lelong
  numbers and the extension of closed positive currents, \emph{Invent.\ 
  Math.}\ {\bf 27} (1974), 53--156.
  
\bibitem{Siu84} Siu, Y.-T., Some recent results in complex manifold
  theory for the semi-positive case, \emph{survey article in the
  Proceedings of the Mathematisches Arbeitstagung}, Bonn 1984.

\bibitem{Siu98} Siu, Y.-T., Invariance of plurigenera, \emph{Invent.\ Math.}\
  {\bf 134} (1994) 631--639.

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

\bibitem{Sko78} Skoda, H., Morphismes surjectifs de fibr\'es vectoriels
  semi-positifs, \emph{Ann.\ Sci.\ Ecole Norm.\ Sup.}\ {\bf 11} (1978) 
  577--611.

\bibitem{Tak06a} Takayama, S., Pluricanonical systems on algebraic varieties 
  of general type, Invent.\ Math.\ {\bf 165} (2006) 551--587.
  
\bibitem{Tak06b} Takayama, S., On the invariance and lower
  semi-continuity of plurigenera of algebraic varieties,
  J.\ Algebraic Geom.\ {\bf 16} (2007) 1--18.

\bibitem{Tsu04} Tsuji, H., Subadjunction theorem,
  \emph{arXiv: math.AG/0111311}, Complex analysis in several
  variables -- Memorial Conference of Kiyoshi Oka's Centennial Birthday,
  313--318, Adv.\ Stud.\ Pure Math.\ {\bf 42}, Math.\ Soc.\ Japan, Tokyo, 2004.

\bibitem{Tsu06} Tsuji, H., Pluricanonical systems of projective varieties
  of general type, \emph{arXiv: math.AG/9909021},
  Osaka J.\ of Mathematics {\bf 43} (2006) 967--995.

\bibitem{Vie82} Viehweg, E., Vanishing theorems, \emph{J.~Reine Angew.\
  Math.}\ {\bf 335} (1982) 1--8.

\bibitem{Voi02} Voisin, C.,
  A counterexample to the Hodge conjecture extended to
  K\"ahler varieties, \emph{Int.\ Math.\ Res.\ Not.}\ {\bf 20} (2002) 
  1057--1075.
  
\bibitem{Voi04a} Voisin, C.,
  On the homotopy types of compact K\"ahler and complex projective manifolds
  \emph{Invent.\ Math.}\ {\bf 157} (2004), 329--343.

\bibitem{Voi04b} Voisin, C.,
  On the homotopy types of K\"ahler manifolds and the birational Kodaira
  problem, \emph{J.\ Differential Geom.} {\bf 72} (2006), 43--71.

\bibitem{Yau78} Yau, S.-T., On the Ricci curvature of a complex
  K\"ahler manifold and the complex Monge--Amp\`ere equation, \emph{Comm.\ 
  Pure Appl.\ Math.}\ {\bf 31} (1978), 339--411.

\bibitem{Zuc77} Zucker, S., The Hodge conjecture for cubic fourfolds, 
  \emph{Compositio Math.}\ {\bf 34} (1977) 199--209.

\end{thebibliography}

\end{document}

