%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Structure theorems for projective and K\"ahler varieties
%
% Lecture notes by
% Jean-Pierre Demailly (Universit\'e de Grenoble I, Institut Fourier)
%
% Park City Mathematical Institute, July 2008
% PCMI Graduate Summer School held at Park City in July 2008 :
% Analytic and Algebraic Geometry: Common Problems-Different Methods
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\tableofcontents

% The main matter.
\mainmatter

% The title of the lecture series.  Modify and uncomment.
\LectureSeries[Structure theorems for K\"ahler varieties]%
{Structure theorems for projective and K\"ahler~varieties
\author{Jean-Pierre Demailly}}

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

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

% Introductory material.	

\section*{0. Introduction}

The main purpose of these notes is to describe some basic structure
theorems for projective or compact K\"ahler varieties and their
cohomology, using recent techniques of complex analysis and potential
theory.  One central unifying concept is the concept of positivity,
which can ve viewed either in algebraic terms (positivity of divisors
and algebraic cycles), or in more analytic terms (plurisubharmonicity,
psoitive currents, hermitian connections with positive curvature). In
the course of the 20th century, following work by early precursors
such as B.~Riemann and H.~Poincar\'e, powerful $L^2$ techniques have
emerged in the hands of W.V.D.~Hodge, S.~Bochner, K.~Kodaira,
J.J.~Kohn, L.~H\"ormander and many others. We refer to Dror Varolin's
notes [Var09] in the present volume for an exposition of the
Bochner-Kodaira technique and its main consequences, namely the
fundamental $L^2$ estimates of H\"ormander [H\"or65] (resolution of
$\overline\partial$ equations), Skoda [Sko72b, 78] (surjec\-tivity theorem
for morphisms of holomorphic vector bundles) and Ohsawa-Takegoshi
[OT87] ($L^2$ extension theorem). All these theorems have an
incredible amount of geometric consequences: among those of the
Ohsawa-Takegoshi theorem, for instance, let us
mention Siu's theorem [Siu74] on the analyticity of
Lelong numbers, a more recent approximation theorem of closed positive
$(1,1)$-currents by divisors due to [Dem92], the subadditivity property
$\cI(\varphi+\psi)\subset\cI(\varphi)\cI(\psi)$ of multiplier ideals
[DEL00], the restriction formula
$\cI(\varphi_{|Y})\subset\cI(\varphi)_{|Y}$, $\ldots$~. A suitable
combination of these results can be used to reprove Fujita's result
[Fuj94] on approximate Zariski decomposition, as detailed in
section~4. Sections 5 and 6 are devoted to the study of positive cones in
K\"ahler or projective geometry. Recent ``algebro-analytic'' characterizations
of the K\"ahler cone ([DP04]) and the pseudo-effective cone of divisors
([BDPP04]) are explained in detail. This leads to a discussion
of the important concepts of volume and mobile intersections,
following S.Boucksom's PhD work [Bou02]. As a consequence, we show that
a projective algebraic manifold has a pseudo-effective canonical line bundle
if and only if it is not uniruled. The section 7 presents some important 
ideas of H.~Tsuji, later refined by Berndtsson and P\u{a}un, concerning 
the so-called ``super-canonical metrics'', and their
interpretation in terms of the invariance of plurigenera and of
the abundance conjecture. In the final section, we state
P\u{a}un's version of the Shokurov-Hacon-McKernan-Siu non vanishing
theorem and give an account of the very recent approach of the proof
of the finiteness of the canonical ring by Birkar-P\u{a}un [BiP09], based
on the ideas of Hacon-McKernan and Siu.

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

\section{Numerically effective and pseudo-effective (1,1) classes}

\subsection*{\csect.A. Pseudo-effective line bundles and metrics with 
minimal singularities}
The concept of pseudo-effectivity is quite general and makes sense on
an arbitrary compact complex manifold~$X$ (no projective or K\"ahler
assumption is needed). Let us recall that all usual concepts of
cohomology (de Rham, resp.\ Dolbeault, $\ldots$) can be calculated
with complexes of smooth forms or with complexes of currents,
alternatively. This comes from the fact that both types of complexes provide
resolutions of the same sheaves, namely germs of locally constant 
functions, resp.\ germs of holomorphic $p$-forms.

In the general context of compact complex manifolds, it is in fact more
appropriate to 
work with Bott-Chern cohomology, instead of de Rham of Dolbeault
cohomology classes: 
we define
$$H_{\BC}^{p,q}(X)=\big\{d\hbox{-closed $(p,q)$-forms}\}/
\big\{\ddbar\hbox{-exact $(p,q)$-forms}\}.\leqno(1.1)$$
By means of the Fr\"olicher spectral sequence, it is easily shown
that these cohomology groups are finite dimensional and can be computed
either with spaces of smooth forms or with currents. In both cases,
the quotient topology of $\smash{H_{\BC}^{p,q}(X)}$ induced by
the Fr\'echet topology of smooth forms or by the weak topology of
currents is Hausdorff. Clearly $H_{\BC}^\bu(X)$ is a bigraded
algebra. This algebra can be shown to be isomorphic to the usual 
De Rham cohomology
algebra $H^\bu(X,\bC)$ if $X$ is K\"ahler or more generally if $X$ is
in the Fujiki class $\cC$ of manifolds bimeromorphic to K\"ahler
manifolds.

\claim{(1.2) Definition} Let $L$ we a holomorphic line bundle on a compact
complex manifold $X$.
we say that $L$ pseudo-effective if $c_1(L)\in H_{\BC}^{1,1}(X)$ is 
the cohomology class of some
closed positive current~$T$, i.e.\ if $L$ can be equipped with a singular
hermitian metric $h$ with $T={\ii\over 2\pi}\Theta_{L,h}\ge 0$ as a current.
\endclaim
%
Recall that for any hermitian metric $h$ on a holomorphic vector 
bundle~$E$, one can define a unique compatible Chern connection $D_{E,h}$ 
such that $D_{E,h}^{0,1}=\overline\partial$. We then denote
$\Theta_{E,h}=D_{E,h}^2\in C^\infty(X,\Lambda^{1,1}T^*_X\otimes\Hom(E,E))$
the associated curvature tensor. If $E=L$ is a holomorphic line bundle
with a singular metric $h$ such that $\log h\in L^1_\loc$, the curvature 
can be computed in the sense of distributions. The locus where 
$h$ has singularities turns out to be extremely important.
The following definition was introduced in [DPS00].

\claim{(1.3) Definition} Let $L$ be a pseudo-effective line bundle
on a compact complex manifold~$X$. Consider two hermitian metrics 
$h_1$, $h_2$ on $L$ with curvature $i\Theta_{L,h_j}\ge 0$
in the sense of currents.
\vskip3pt
\plainitem{\rm(i)} We will write $h_1\preccurlyeq h_2$, and say that $h_1$ 
is less singular than $h_2$, if there exists a constant $C>0$ such that
$h_1\le C h_2$.
\vskip3pt
\plainitem{\rm(ii)} We will write $h_1\sim h_2$, and say that $h_1$, $h_2$ are
equivalent with respect to singularities, if there exists a constant
$C>0$ such that $C^{-1}h_2\le h_1\le C h_2$.
\vskip0pt
\endclaim
%
\noindent
Of course $h_1\preccurlyeq h_2$ if and only if the associated weights
in suitable trivializations locally satisfy $\varphi_2\le \varphi_1+C$.
This implies in particular $\nu(\varphi_1,x)\le\nu(\varphi_2,x)$ at each
point. The above definition is motivated by the following observation.

\claim{(1.4) Theorem} For every pseudo-effective line bundle $L$ over
a compact complex manifold $X$, there exists up to equivalence of
singularities a unique class of hermitian metrics $h$ with minimal
singularities such that $i\Theta_{L,h}\ge 0$.
\endclaim
%
\begproof{Proof.} The proof is almost trivial. We fix once for all a smooth
metric $h_\infty$ (whose curvature is of random sign and signature),
and we write singular metrics of $L$ under the form $h=h_\infty e^{-\psi}$. 
The condition $i\Theta_{L,h}\ge 0$ is equivalent to 
${i\over2\pi}\ddbar\psi\ge -u$ where $u={i\over 2\pi}\Theta_{L,h_\infty}$. This
condition implies that $\psi$ is plurisubharmonic up to the addition
of the weight $\varphi_\infty$ of $h_\infty$, and therefore locally
bounded from above.  Since we are concerned with metrics only up to
equivalence of singularities, it is always possible to adjust
$\psi$ by a constant in such a way that $\sup_X\psi=0$. We now set
$$
h_{\min}=h_\infty e^{-\psi_{\min}},\qquad \psi_{\min}(x)=\sup_\psi\psi(x)
$$
where the supremum is extended to all functions $\psi$ such that
$\sup_X\psi=0$ and ${i\over2\pi}\ddbar\psi\ge -u$. By standard
results on plurisubharmonic functions (see Lelong [Lel69]), 
$\psi_{\min}$ still satisfies ${i\over2\pi}\ddbar\psi_{\min}\ge -u$
(i.e.\ the weight $\varphi_\infty+\psi_{\min}$ of $h_{\min}$ is
plurisubharmonic), and $h_{\min}$ is obviously the metric with
minimal singularities that we were looking for. [In~principle one
should take the upper semicontinuous regularization $\psi_{\min}^*$
of $\psi_{\min}$ to really get a plurisubharmonic weight, but since 
$\psi_{\min}^*$ also participates to the upper envelope, we obtain
here $\psi_{\min}=\psi_{\min}^*$ automatically].
\qed
\endproof
%
\claim{(1.5) Remark} \rm In general, the supremum 
$\psi=\sup_{j\in I}\psi_j$ of a 
locally dominated family of plurisubharmonic functions $\psi_j$ is 
not plurisubharmonic strictly speaking, but its ``upper semi-continuous
regularization'' $\psi^*(z)=\limsup_{\zeta\to z}\psi(\zeta)$ is
plurisubharmonic and coincides almost everywhere with $\psi$, with
$\psi^*\ge\psi$. However, in the context of (1.5), $\psi^*$ still
satisfies $\psi^*\le 0$ and ${i\over2\pi}\ddbar\psi\ge -u$, hence
$\psi^*$ participates to the upper envelope. As a consequence, we have
$\psi^*\le \psi$ and thus $\psi=\psi^*$ is indeed plurisubharmonic. 
Under a strict
positivity assumption, namely if $L$ is a big line bundle (i.e.\ the
curvature can be taken to be strictly positive in the sense of
currents, see 1.12) and (1.17$\,$ii) below), then $h_{\min}$ can be shown
to possess some regularity properties. The reader may consult
[BmD09] for a rather general (but certainly non trivial) proof that 
$\psi_{\min}$ possesses locally bounded
second derivatives $\partial^2\psi_{\min}/\partial z_j\partial\overline z_k$
outside an analytic set $Z\subset X\,$; in other words, $i\Theta_{L,h_{\min}}$
has locally bounded coefficients on~$X\ssm Z$.
\endclaim
%
Following Nadel [Nad89],
one defines the {\it multiplier ideal sheaf} $\cI(\varphi)$ of a psh
function $\varphi$ to be the
sheaf of germs of holomorphic functions $f$ such that
$|f|^2e^{-2\varphi}$ is locally summable (see also [Var09] in this volume;
some authors simply write $\varphi$ where we have put $2\varphi$, but this
is just a matter of notation).
When $\varphi$ is plurisubharmonic, $\cI(\varphi)$ is a coherent ideal 
sheaf which is generated over any bounded Stein coordinate open set 
$U\compact X$ by a Hilbert basis $(g_j)$ of the $L^2$ space
$$
\cH_U(\varphi)=\big\{f\in\cO_X(U)\,;\;\int_U|f(z)|^2e^{-2\varphi(z)}dV(z)
<+\infty\big\}.
$$
Similarly, if $(L,h)$ is a singular hermitian line bundle and
$h$ is trivialized locally as $h=e^{-2\varphi}$, we simply put 
$\cI(h)=\cI(\varphi)$.
%
\claim{(1.6) Definition} Let $L$ be a pseudo-effective line bundle.
If $h$ is a singular hermitian metric such 
that $i\Theta_{L,h}\ge 0$ and
$$
H^0(X,mL\otimes\cI(h^{\otimes m}))\simeq H^0(X,mL)\qquad
\hbox{for all $m\ge 0$},
$$
we say that $h$ is an {\emph analytic Zariski decomposition} of~$L$. 
\endclaim
%
In other words, we require that $h$ has singularities so mild that
the vanishing conditions prescribed by the multiplier ideal sheaves
$\cI(h^{\otimes m})$ do not kill any sections of $L$ and its multiples.

\claim{(1.7) Exercise} {\rm A special case is when there is an 
isomorphism $pL=A+E$ where $A$ and $E$ are effective divisors such 
that $H^0(X,mpL)=H^0(X,mA)$
for all $m$ and $\cO(A)$ is generated by sections. Then $A$ possesses
a smooth hermitian metric~$h_A$, and this metric defines a singular hermitian
metric $h$ on $L$ with poles ${1\over p}E$ and curvature 
${1\over p}\Theta_{A,h_A}+{1\over p}[E]$. Show that this metric $h$
is an analytic Zariski decomposition.\newline
{\emph Note}: when $X$ projective and there is a decomposition $pL=A+E$
with $A$ nef (see (1.9) below), $E$ effective and
$H^0(X,mpL)=H^0(X,mA)$ for all $m$, one says that this is an 
{\emph algebraic Zariski decomposition} of $L$. It can be shown that Zariski
decompositions exist in dimension $2$, but in higher dimension one can see
that they do not exist.}
\endclaim
%
\claim{(1.8) Theorem} The metric $h_{\min}$ with minimal singularities
provides an analytic Zariski decomposition.
\endclaim
%
It follows that an analytic Zariski decomposition always exists (while
algebraic decompositions do not exist in general, especially in
dimension $3$ and more.)

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

\claim{(1.9) Definition} A holomorphic line bundle $L$ over a projective
manifold $X$ is said to be numerically effective, nef for short, if
$L\cdot C=\int_C c_1(L)\ge 0$ for every curve $C\subset X$.
\endclaim
%
If $L$ is nef, it can be shown that $L^p\cdot Y=\int_Y c_1(L)^p\ge 0$
for any $p$-dimensional subvariety $Y\subset X$ (see e.g.\ [Har70]).
In relation to this, let us recall the Nakai-Moishezon ampleness
criterion: a line bundle $L$ is ample if and only if $L^p\cdot Y>0$ for
every $p$-dimensional subvariety~$Y$. From this, we easily infer

\claim{(1.10) Proposition} Let $L$ be a line bundle on a projective
algebraic manifold~$X$, on which an ample line bundle $A$ and a hermitian
metric $\omega$ are given. The following properties are equivalent:
\vskip3pt
\plainitem{\rm (a)} $L$ is nef$\,;$
\vskip3pt
\plainitem{\rm (b)} for any integer $k\ge 1$, the line bundle $kL+A$ is ample$\,;$
\vskip3pt
\plainitem{\rm (c)} for every $\varepsilon>0$, there is a smooth metric
$h_\varepsilon$ on $L$ such that $\ii\Theta_{L,h_\varepsilon}\ge
-\varepsilon\omega$.
\vskip0pt
\endclaim
%
\begproof{Proof.} (a) $\Rightarrow$ (b). If $L$ is nef and $A$ is ample then
clearly $kL+A$ satisfies the Nakai-Moishezon criterion, hence
$kL+A$ is ample.
\vskip3pt
\noindent (b) $\Rightarrow$ (c). Condition (c) is independent of the choice
of the hermitian metric, so we may select a metric $h_A$ on $A$ with
positive curvature and set $\omega=\ii\Theta_{A,h_A}$. If $kL+A$ is ample,
this bundle has a metric $h_{kL+A}$ of positive curvature. Then the
metric $h_L=(h_{kL+A}\otimes h_A^{-1})^{1/k}$ has curvature
$$\ii\Theta_{L,h_L}={1\over k}\big(\ii\Theta(kL+A)-\ii\Theta_A\big)\ge
-{1\over k}\ii\Theta_{A,h_A}\,;$$
in this way the negative part can be made smaller than $\varepsilon\,\omega$
by taking $k$ large enough.
\vskip3pt
\noindent (c) $\Rightarrow$ (a). Under hypothesis (c), we get
$L\cdot C=\int_C{\ii\over2\pi}\Theta_{L,h_\varepsilon}\ge-
{\varepsilon\over2\pi}\int_C\omega$ for every curve $C$ and every
$\varepsilon>0$, hence $L\cdot C\ge 0$ and $L$ is nef.\qed
\endproof
%
Let now $X$ be an arbitrary compact complex manifold. Since there need
not exist any curve in $X$, Property 1.10~c) is simply taken as a definition
of nefness ([DPS94]):

\claim{(1.11) Definition} A line bundle $L$ on a compact complex
manifold $X$ is said to be nef if for every $\varepsilon>0$, there is a
smooth hermitian metric $h_\varepsilon$ on $L$ such that
$\ii\Theta_{L,h_\varepsilon}\ge -\varepsilon\omega$.
\endclaim
%
In general, it is not possible to extract a smooth limit $h_0$ such that
$\ii\Theta_{L,h_0}\ge 0$. The following simple example is given in
[DPS94] (Example~1.7). Let $E$ be a non trivial extension
$0\to\cO\to E\to\cO\to 0$ over an elliptic curve $C$ and let $X=\bP(E)$
(with notation as in (4.12)) be the corresponding ruled surface 
over~$C$. Then $L=\cO_{\bP(E)}(1)$ is
nef but does not admit any smooth metric of nonnegative curvature.
This example answers negatively a question raised by Fujita [Fuj83].

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

\claim{(1.12) Definition} If $L$ is a line bundle, the Kodaira-Iitaka
dimension $\kappa(L)$ is the supremum of the rank of the canonical maps
$$\Phi_m:X\ssm B_m\longrightarrow \bP(V_m),~~~~x\longmapsto H_x=\{\sigma\in
V_m\,;\,\sigma(x)=0\},~~~~m\ge 1$$
with $V_m=H^0(X,mL)$ and $B_m=\bigcap_{\sigma\in V_m}\sigma^{-1}(0)={}$base
locus of $V_m$. In case $V_m=\{0\}$ for all $m\ge 1$, we set $\kappa(L)=
-\infty$.\vskip5pt
\noindent
A line bundle is said to be {\rm big} if $\kappa(L)=\dim X$.
\endclaim
%
The following lemma is well-known (the proof is a rather elementary
consequence of the Schwarz lemma).

\claim{(1.13) Serre-Siegel lemma {\rm([Ser54], [Sie55])}} Let $L$ be any
line bundle on a compact complex manifold. Then we have
$$h^0(X,mL)\le O(m^{\kappa(L)})\qquad\hbox{for $m\ge 1$,}$$
and $\kappa(L)$ is the smallest constant for which this estimate holds.\qed
\endclaim
%

\subsection*{\csect.C. Description of the positive cones}
Let us recall that an integral cohomology class in
$H^2(X,\bZ)$ is the first Chern class of a holomorphic (or algebraic)
line bundle if and only if it lies in the
{\emph Neron-Severi} group 
$$\NS(X)=\Ker\big(H^2(X,\bZ)\to H^2(X,\cO_X)\big)\leqno(1.14)$$
(this fact is just an elementary consequence of the exponential exact
sequence\break $0\to\bZ\to\cO\to\cO^\star\to 0$). If $X$ is compact
K\"ahler, as we will suppose from now on in this section,
this is the same as saying that the class is of type $(1,1)$
with respect to Hodge decomposition.

Let us consider the real vector space $\smash{\NS_\bR(X)=\NS(X)\otimes_\bZ\bR}$,
which can be viewed as a subspace of the space $H^{1,1}(X,\bR)$ of real 
$(1,1)$ cohomology classes. Its dimension is by definition the Picard number
$$\rho(X)=\rank_\bZ \NS(X)=\dim_\bR \NS_\bR(X).\leqno(1.15)$$
We thus have $0\le\rho(X)\le h^{1,1}(X)$, and the example of complex tori shows
that all intermediate values can occur when $n=\dim X\ge 2$.

The positivity concepts for line bundles considered in sections
1.A and 1.B possess in fact natural generalizations to
$(1,1)$ classes which are not necessarily integral or rational -- and
this works at least in the category of compact K\"ahler manifolds
(in fact, by using Bott-Chern cohomology, one could even extend these
concepts to arbitrary compact complex manifolds).

\claim{(1.16) Definition} Let $(X,\omega)$ be a compact K\"ahler manifold.
\vskip3pt
{\plainitemindent=7mm
\plainitem{\rm(i)} The K\"ahler cone is the set 
$\cK\subset H^{1,1}(X,\bR)$ of cohomology classes $\{\omega\}$
of K\"ahler forms. This is an {\rm open convex cone}.
\vskip3pt
\plainitem{\rm(ii)} The closure $\overline{\cK}$ of the K\"ahler cone consists
of classes $\{\alpha\}\in H^{1,1}(X,\bR)$ such that for every
$\varepsilon>0$ the sum $\{\alpha+\varepsilon\omega\}$
is K\"ahler, or equivalently, for every $\varepsilon>0$, there exists a 
smooth function $\varphi_\varepsilon$ on $X$ such that 
$\alpha+i\ddbar\varphi_\varepsilon\ge -\varepsilon\omega$.
We say that $\overline{\cK}$ is the cone of {\rm nef} $(1,1)$-classes.
\vskip3pt
\plainitem{\rm(iii)}The pseudo-effective cone is the set
$\cE\subset H^{1,1}(X,\bR)$ of cohomology classes $\{T\}$ of closed positive 
currents of type $(1,1)$. This is a {\rm closed convex cone}.
\vskip3pt
\plainitem{\rm(iv)}The interior $\cE^\circ$ of $\cE$ consists of classes
which still contain a closed positive current after one subtracts
$\varepsilon\{\omega\}$ for $\varepsilon>0$ small, in other words,
they are classes of closed $(1,1)$-currents $T$ such that
$T\ge \varepsilon\omega$. Such a current will be called a {\rm K\"ahler
current}, and we say that $\{T\}\in H^{1,1}(X,\bR)$ is a {\rm big 
$(1,1)$-class}.
\vskip0pt}
\endclaim
%
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\LabelTeX 27 30 $\overline{\cK}=\hbox{nef cone in $H^{1,1}(X,\bR)$
[closure of $\cK$]}$\ELTX
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[closed]}$\ELTX
\LabelTeX 27 10 $\cE^\circ=\hbox{big cone in $H^{1,1}(X,\bR)$ 
[interior of $\cE$]}$\ELTX
\EndFig
\vskip8mm

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

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

In case $X$ is projective, all Chern classes $c_1(L)$ of line bundles
lie by definition in $\NS(X)$, and likewise,
all classes of real divisors $D=\sum c_jD_j$, $c_j\in\bR$, lie 
in $\NS_\bR(X)$. In order to deal with such {\emph algebraic classes}, 
we therefore introduce the intersections
$$\cK_{\NS} = \cK\cap \NS_\bR(X),\qquad \cE_{\NS}=\cE\cap \NS_{\bR}(X),$$ 
and refer to classes of $H^{1,1}(X,\bR)$ not contained in $\NS_\bR(X)$
as {\emph transcendental classes}.

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\EndFig
\vskip8mm

A very important fact is that all four cones $\cK_{\NS}$, $\cE_{\NS}$, 
$\overline\cK_{\NS}$, $\cE^\circ_{\NS}$ have simple algebraic
interpretations.

\claim{(1.17) Theorem} Let $X$ be a projective manifold. Then
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} $\cK_{\NS}$ is the open cone generated by classes of
{\rm ample} $($or very ample$)$ divisors $A$ $($Recall that a divisor
$A$ is said to be very ample if the linear system $H^0(X,\cO(A))$
provides an embedding of $X$ in projective space$)$.
\vskip3pt
\plainitem{\rm (ii)} The interior $\cE_{\NS}^\circ$ is the cone generated
by classes of {\rm big} divisors, namely divisors $D$ such that
$h^0(X,\cO(kD))\ge c\,k^{\dim X}$ for $k$ large.
\vskip3pt
\plainitem{\rm (iii)} $\cE_{\NS}$ is the closure of the cone generated by classes
of {\rm effective} divisors, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.
\vskip3pt
\plainitem{\rm (iv)} The closed cone $\overline\cK_{\NS}$ consists of the closure
of the cone generated by {\rm nef} divisors $D$ (or nef line bundles $L$), 
namely effective integral divisors $D$ such that \hbox{$D\cdot C\ge 0$} for
every curve $C$.\vskip0pt
}
\endclaim
%
In other words, the terminology ``nef'', ``big'',
``pseudo-effective'' used for classes of the full transcendental cones 
appear to be a natural extrapolation of the algebraic case.

\begproof{Proof.} (i) is just Kodaira's embedding theorem, and 
properties (iii) and (iv) are obtained easily by passing to the closure of 
the open cones. We will thus give details of the proof only for (ii) which is
possibly not as well known.

By looking at points of $\NS_{\bQ}(X)=\NS(X)\otimes_\bZ\bQ$ and multiplying
by a denominator, it is enough to check that a line bundle $L$ such that
$c_1(L)\in\cE^\circ$ is big. However, this means that $L$ possesses a
singular hermitian metric $h_L$ such that $\Theta_{L,h_L}\ge\varepsilon
\omega$ for some K\"ahler metric~$\omega$. For some integer $p_0>0$, we can then
produce a singular hermitian metric with positive curvature and with a 
given logarithmic pole
$h_L^{p_0}e^{-\theta(z)\log|z-x_0|^2}$ in a neighborhood of every point $x_0\in X$
(here $\theta$ is a smooth cut-off function supported on a neighborhood 
of~$x_0$). Then H\"ormander's $L^2$ existence theorem [H\"or65] can be 
used to produce sections of $L^k$ which generate all jets of order 
$(k/p_0)-n$ at points~$x_0$, so that $L$ is big.

Conversely, if $L$ is big and $A$ is a (smooth) very ample divisor,
the exact sequence $0\to\cO_X(kL-A)\to\cO_X(kL)\to\cO_A(kL_{\restriction A})
\to 0$ and the estimates $h^0(X,\cO_X(kL))\ge ck^n$, 
$h^0(A,\cO_A(kL_{\restriction A}))=O(k^{n-1})$ imply that
$\cO_X(kL-A)$ has a section for $k$ large, thus $kL-A\equiv D$ for some
effective divisor~$D$. This means that there exists a singular metric 
$h_L$ on $L$ such that
$${\ii\over 2\pi}\Theta_{L,h_L}={1\over k}
\Big({\ii\over2\pi}\Theta_{A,h_A}+[D]\Big)\ge{1\over k}\omega$$
where $\omega={\ii\over 2\pi}\Theta_{A,h_A}$, hence 
$c_1(L)\in\cE^\circ$.\hfill\qed
\endproof
%
Before going further, we need a lemma.

\claim{(1.18) Lemma} Let $X$ be a compact K\"ahler $n$-dimensional
manifold, let $L$ be a nef line bundle on~$X$, and let $E$ be an
arbitrary holomorphic vector bundle. Then
$h^q(X,\cO(E)\otimes\cO(kL))=o(k^n)$
as $k\to+\infty$, for every $q\ge1$. If $X$ is projective algebraic, the
following more precise bound holds:
$$h^q(X,\cO(E)\otimes\cO(kL))=O(k^{n-q}),\qquad\forall q\ge0.$$
\endclaim
%
\begproof{Proof.} The K\"ahler case will be proved in Section~12, as a
consequence of the holomorphic Morse inequalities. In the projective
algebraic case, we proceed by induction on $n=\dim X$. If $n=1$ the result
is clear, as well as if $q=0$. Now let $A$ be a nonsingular ample divisor
such that $E\otimes\cO(A-K_X)$ is Nakano positive. Then the Nakano
vanishing theorem applied to the vector bundle $F=E\otimes\cO(kL+A-K_X)$
shows that $H^q(X,\cO(E)\otimes\cO(kL+A))=0$ for all $q\ge 1$.
The exact sequence
$$0\to\cO(kL)\to\cO(kL+A)\to\cO(kL+A)_{\restriction A}\to0$$
twisted by $E$ implies
$$H^q(X,\cO(E)\otimes\cO(kL))\simeq H^{q-1}(A,\cO(E_{\restriction A}
\otimes\cO(kL+A)_{\restriction A}),$$
and we easily conclude by induction since $\dim A=n-1$. Observe that the
argument does not work any more if $X$ is not algebraic. It seems to be
unknown whether the $O(k^{n-q})$ bound still holds in that case.\qed
\endproof
%
\claim{(1.19) Corollary} If $L$ is nef, then $L$ is big $($i.e.\
$\kappa(L)=n)$ if and only if $L^n>0$. Moreover, if $L$ is nef and big,
then for every $\delta>0$, $L$ has a singular metric $h=e^{-2\varphi}$
such that $\,\max_{x\in X}\nu(\varphi,x)\le\delta$ and
$\ii\Theta_{L,h}\ge\varepsilon\,\omega$ for some $\varepsilon>0$.
The metric $h$ can be chosen to be smooth on the complement of a fixed
divisor $D$, with logarithmic poles along $D$.
\endclaim
%
\begproof{Proof.} By Lemma~1.18 and the Riemann-Roch formula, we have
$h^0(X,kL)=\chi(X,kL)+o(k^n)=k^nL^n/n!+o(k^n)$, whence the first
statement. By the proof of (1.17$\,$ii), there exists a singular metric 
$h_1$ on $L$ such that
$${\ii\over 2\pi}\Theta_{L,h_1}={1\over k}
\Big({\ii\over2\pi}\Theta_{A,h_A}+[D]\Big)\ge{1\over k}\omega,
\qquad \omega={\ii\over2\pi}\Theta_{A,h_A}.$$
Now, for every $\varepsilon>0$, there is a smooth metric $h_\varepsilon$
on $L$ such that ${\ii\over2\pi}\Theta_{L,h_\varepsilon}\ge-\varepsilon
\omega$. The convex combination
of metrics $h'_\varepsilon=h_1^{k\varepsilon}h_\varepsilon^{1-k\varepsilon}$
is a singular metric with poles along $D$ which satisfies
$${\ii\over2\pi}\Theta_{L,h'_\varepsilon}\ge \varepsilon(\omega+[D])-
(1-k\varepsilon)\varepsilon\omega\ge k\varepsilon^2\omega.$$
Its Lelong numbers are $\varepsilon\nu(D,x)$ and they can be made
smaller than $\delta$ by choosing $\varepsilon>0$ small.\qed
\endproof
%
We still need a few elementary facts about the numerical dimension of nef
line bundles.

\claim{(1.20) Definition} Let $L$ be a nef line bundle on a compact
K\"ahler manifold~$X$. One defines the numerical dimension of $L$ to be
$$\nd(L)=\max\big\{k=0,\ldots,n\,;\,c_1(L)^k\ne0~\hbox{in}~
H^{2k}(X,\bR)\big\}.$$
\endclaim
%
By Corollary 1.19, we have $\kappa(L)=n$ if and only if $\nd(L)=n$.
In general, we merely have an inequality.

\claim{(1.21) Proposition} If $L$ is a nef line bundle on a compact
K\"ahler manifold, then $\kappa(L)\le\nd(L)$.
\endclaim
%
\begproof{Proof.} By induction on $n=\dim X$. If $\nd(L)=n$ or $\kappa(L)=n$
the result is true, so we may assume $r:=\kappa(L)\le n-1$ and $k:=\nd(L)\le
n-1$. Fix $m>0$ so that $\Phi=\Phi_{|mL|}$ has generic rank $r$.
Select a nonsingular ample divisor $A$ in $X$ such that the restriction of
$\Phi_{|mL|}$ to $A$ still has rank $r$ (for this, just take $A$
passing through a point $x\notin B_{|mL|}$ at which $\rank(d\Phi_x)=r<n$,
in such a way that the tangent linear map $d\Phi_{x\restriction T_{A,x}}$
still has rank~$r$). Then $\kappa(L_{\restriction A})\ge r=\kappa(L)$
(we just have an equality because there might exist sections in
$H^0(A,mL_{\restriction A})$ which do not extend to~$X$). On the other hand,
we claim that $\nd(L_{\restriction A})=k=\nd(L)$. The inequality
$\nd(L_{\restriction A})\ge\nd(L)$ is clear. Conversely, if we set
$\omega={\ii\over2\pi}\Theta_{A,h_A}>0$, the cohomology class $c_1(L)^k$ can
be represented by a closed positive current of bidegree~$(k,k)$
$$T=\lim_{\varepsilon\to0}\Big({\ii\over 2\pi}\Theta_{L,h_\varepsilon}+
\varepsilon\omega\Big)^k$$
after passing to some subsequence (there is a uniform bound for the mass
thanks to the K\"ahler assumption, taking wedge products with
$\omega^{n-k}$). The current $T$ must be non zero since $c_1(L)^k\ne0$ by
definition of $k=\nd(L)$. Then $\{[A]\}=\{\omega\}$ as cohomology classes,
and
$$\int_A c_1(L_{\restriction A})^k\wedge\omega^{n-1-k}=
\int_X c_1(L)^k\wedge[A]\wedge\omega^{n-1-k}=
\int_X T\wedge\omega^{n-k}>0.$$
This implies $\nd(L_{\restriction A})\ge k$, as desired. The induction
hypothesis with $X$ replaced by $A$ yields
$$\kappa(L)\le\kappa(L_{\restriction A})\le
\nd(L_{\restriction A})\le\nd(L).\eqno\square$$
\endproof
%
\claim{(1.22) Remark} \rm It may happen that $\kappa(L)<\nd(L)$: take e.g.
$$L\to X=X_1\times X_2$$
equal to the total tensor product of an ample line bundle $L_1$ on a
projective manifold $X_1$ and of a unitary flat line bundle $L_2$ on an
elliptic curve $X_2$ given by a representation $\pi_1(X_2)\to U(1)$ such
that no multiple $kL_2$ with $k\ne0$ is trivial. Then
$H^0(X,kL)=H^0(X_1,kL_1)\otimes H^0(X_2,kL_2)=0$ for $k>0$, and thus
$\kappa(L)=-\infty$. However $c_1(L)=\pr_1^\star c_1(L_1)$ has
numerical dimension equal to $\dim X_1$. The same example shows
that the Kodaira dimension may increase by restriction to a
subvariety (if $Y=X_1\times\{{\rm point}\}$, then
$\kappa(L_{\restriction Y})=\dim Y$).\qed
\endclaim
%
\subsection*{\csect.D. The Kawamata-Viehweg vanishing theorem}
We derive here an algebraic version of the Nadel vanishing theorem in the
context of nef line bundles. This algebraic vanishing theorem has been
obtained independently by Kawamata [Kaw82] and Viehweg [Vie82], who both
reduced it to the Akizuki-Kodaira-Nakano vanishing theorem [AN54]
by cyclic covering constructions. Since then, a number of other proofs 
have been given, one based on connections with logarithmic
singularities [EV86], another on Hodge theory for twisted coefficient
systems [Kol85], a third one on the Bochner technique [Dem89] (see
also [EV92] for a general survey). Since the result is best expressed 
in terms of multiplier ideal sheaves (avoiding then any unnecessary
desingularization in the statement), we feel that the direct approach
via Nadel's vanishing theorem is extremely natural.

If $D=\sum\alpha_jD_j\ge 0$ is an effective $\bQ$-divisor, we define
the {\emph multiplier ideal sheaf} $\cI(D)$ to be equal to $\cI(\varphi)$
where $\varphi=\sum\alpha_j|g_j|$ is the corresponding psh function
defined by generators $g_j$ of $\cO(-D_j)$. If $D$ is a divisor with normal
crossings, it is an easy exercise to show that
$$
\cI(D)=\cO(-\lfloor D\rfloor),\qquad
\hbox{where}\quad\lfloor D\rfloor=\sum\lfloor \alpha_j\rfloor D_j
\leqno(1.23)
$$
is the integer part of $D$.
In general, the computation of $\cI(D)$ can be made algebraically thanks to
Hironaka [Hir64], by using desingularizations $\mu:\widetilde X\to X$ 
such that $\mu^\star D$ becomes a divisor with normal crossings 
on~$\widetilde X$. In fact, if $\mu:\widetilde X\to X$ is a proper
modification and $\varphi$ is a plurisubharmonic function on $X$, one 
can easily prove the direct image formula
$$
\leqalignno{
&\cO_X(K_X)\otimes\cI(\varphi)=\mu_*\big(
\cO_{\widetilde X}(K_{\widetilde X})\otimes\cI(\varphi\circ\mu)\big)
&(1.24)\cr
&\cI(\varphi)=\mu_*\big(
\cO_{\widetilde X}(K_{\widetilde X/X})\otimes\cI(\varphi\circ\mu)\big)
&(1.24')\cr}
$$
in terms of the relative canonical sheaf
$\smash{K_{\widetilde X/X}=K_{\widetilde X}\otimes\mu^*(K_X^{-1})}\,$; the proof is
immediate from the $L^2$ definition.

\claim{(1.25) Kawamata-Viehweg vanishing theorem} Let $X$ be a projective
algebraic manifold and let $F$ be a line bundle over $X$ such that some
positive multiple $mF$ can be written $mF=L+D$ where $L$ is a 
nef line bundle and $D$ an effective divisor. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(m^{-1}D)\big)=0~~~~
{\emph for}~~q>n-\nd(L).$$
\endclaim
%
\claim{(1.26) Special case} If $F$ is a nef line bundle, then
$$H^q\big(X,\cO(K_X+F)\big)=0~~~~{\emph for}~~q>n-\nd(F).$$
\endclaim
%
\begproof{Proof of Theorem 1.25.} First suppose that $\nd(L)=n$, i.e.\ that
$L$ is big. By the proof of 1.16~ii), there is a singular hermitian
metric $h_0$ on $L$ such that the corresponding weight $\varphi_0$
has algebraic singularities and
$$\ii\Theta_{L,h_0}=2\ii d'd''\varphi_0\ge\varepsilon_0\omega$$
for some $\varepsilon_0>0$. On the other hand, since $L$ is nef, there
are metrics given by weights $\varphi_\varepsilon$ such that
${\ii\over 2\pi}\Theta_{L,h_\varepsilon}\ge -\varepsilon\omega$ for
every $\varepsilon>0$, $\omega$ being a K\"ahler metric. Let
$\varphi_D=\sum\alpha_j\log|g_j|$ be the weight of the singular metric on
$\cO(D)$. We define a singular metric on $F$ by
$$\varphi_F={1\over m}
\big((1-\delta)\varphi_{L,\varepsilon}+\delta\varphi_{L,0}+\varphi_D\big)$$
with $\varepsilon\ll\delta\ll 1$, $\delta$ rational. Then $\varphi_F$
has algebraic singularities, and by taking $\delta$ small enough we
find $\cI(\varphi_F)=\cI({1\over m}\varphi_D)=\cI({1\over m}D)$.
In fact, $\cI(\varphi_F)$ can be computed by taking integer parts
of certain $\bQ$-divisors, and adding
$\delta\varphi_{L,0}$ does not change the integer part of the rational
numbers involved when $\delta$ is small. Now
$$\eqalign{dd^c\varphi_F
&={1\over m}\big((1-\delta)dd^c\varphi_{L,\varepsilon}+
\delta dd^c\varphi_{L,0}+dd^c\varphi_D\big)\cr
&\ge{1\over m}\big(-(1-\delta)\varepsilon\omega+
\delta\varepsilon_0\omega+[D]\ge{\delta\varepsilon\over m}\omega,\cr}$$
if we choose $\varepsilon\le\delta\varepsilon_0$. Nadel's theorem 
[Var09, sect.~8.4.2] thus implies the desired vanishing result
for all $q\ge 1$.

Now, if $\nd(L)<n$, we use hyperplane sections and argue by induction
on $n=\dim X$. Since the sheaf $\cO(K_X)\otimes\cI(m^{-1}D)$ behaves
functorially with respect to modifications (and since the $L^2$
cohomology complex is ``the same'' upstairs and downstairs),
we may assume after blowing-up that $D$ is a divisor with normal
crossings. Then the multiplier ideal sheaf
$\cI(m^{-1}D)=\cO(-\lfloor m^{-1}D\rfloor)$ is locally free.
By Serre duality, the expected vanishing is equivalent to
$$H^q(X,\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor))=0\qquad
\hbox{\rm for}~q<\nd(L).$$
Select a nonsingular ample divisor $A$ such that $A$ meets all
components $D_j$ transversally, and take $A$ positive enough so that
$\cO(A+F-\lfloor m^{-1}D\rfloor)$ is ample. Then
$H^q(X,\cO(-A-F)\otimes\cO(\lfloor m^{-1}D\rfloor))=0$ for $q<n$
by Kodaira vanishing, and the exact sequence
$0\to\cO_X(-A)\to\cO_X\to(i_A)_\star\cO_A\to 0$ twisted by
$\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor)$ yields an isomorphism
$$H^q(X,\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor))\simeq
H^q(A,\cO(-F_{\restriction A})\otimes\cO(\lfloor m^{-1}
D_{\restriction A}\rfloor).$$
The proof of 1.21 showed that $\nd(L_{\restriction A})=\nd(L)$, hence
the induction hypothesis implies that the cohomology group on $A$ on
the right hand side is zero for \hbox{$q<\nd(L)$}.\qed
\endproof
%
\subsection*{\csect.E. A uniform global generation property due to 
Y.T.~Siu}
Let $X$ be a projective manifold, and $(L,h)$ a pseudo-effective
line bundle.
The ``uniform global generation property'' states in some sense that the
tensor product sheaf $L\otimes\cI(h)$ has a uniform positivity property,
for any singular hermitian metric $h$ with nonnegative curvature on~$L$.

\claim{(1.27) Theorem {\rm Y.T.~Siu, ([Siu98])}} Let $X$ be a 
projective manifold. There exists an ample line bundle $G$ on $X$
such that for every pseudo-effective line bundle $(L,h)$, the sheaf
$\cO(G+L)\otimes\cI(h)$ is generated by its global sections. In fact, $G$
can be chosen as follows: pick any very ample line bundle $A$, and take
$G$ such that $G-(K_X+nA)$ is ample, e.g.\ $G=K_X+(n+1)A$.
\endclaim
%
\begproof{Proof.} Let $\varphi$ be the weight of the metric $h$ on a small
neighborhood of a point~$z_0\in X$. Assume that we have a local section 
$u$ of $\cO(G+L)\otimes\cI(h)$ on a coordinate open ball $B=B(z_0,\delta)$, 
such that
$$
\int_B|u(z)|^2e^{-2\varphi(z)}|z-z_0|^{-2(n+\varepsilon)}dV(z)<+\infty.
$$
Then Skoda's division theorem [Sko72b] (see also Corollary 8.21 below)
implies $u(z)=\sum(z_j-z_{j,0})v_j(z)$
with
$$
\int_B|v_j(z)|^2e^{-2\varphi(z)}|z-z_0|^{-2(n-1+\varepsilon)}dV(z)<+\infty,
$$
in particular $u_{z_0}\in\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$. Select
a very ample line bundle $A$ on~$X$. We take a basis
$\sigma=(\sigma_j)$ of sections of $H^0(X,G\otimes\gm_{X,z_0})$ and
multiply the metric $h$ of $G$ by the factor $|\sigma|^{-2(n+\varepsilon)}$.
The weight of the above metric has singularity $(n+\varepsilon)\log|z-z_0|^2$
at~$z_0$, and its curvature is
$$
i\Theta_G+(n+\varepsilon)i\ddbar\log|\sigma|^2\ge
i\Theta_G-(n+\varepsilon)\Theta_A.
\leqno(1.28)
$$
Now, let $f$ be a local section in $H^0(B,\cO(G+L)\otimes\cI(h))$ on 
$B=B(z_0,\delta)$, $\delta$ small. We solve the global $\dbar$ equation 
$$
\dbar u =\dbar(\theta f)\qquad\hbox{on $X$}
$$ 
with a cut-off function $\theta$ supported near $z_0$ and with the 
weight associated with our above choice of metric on $G+L$. 
Thanks to Nadel's Theorem, the solution exists if the metric of 
$G+L-K_X$ has positive curvature. As $i\Theta_{L,h}\ge 0$ in the sense 
of currents, $(1.28)$ shows that a sufficient condition is $G-K_X-nA>0$
(provided that $\varepsilon$ is small enough).
We then find a smooth solution $u$ such that $u_{z_0}\in 
\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$, hence
$$
F:=\theta f -u \in H^0(X,\cO(G+L)\otimes\cI(h))
$$ 
is a global section differing from $f$ by a germ in 
$\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$. Nakayama's lemma implies 
that $H^0(X,\cO(G+L)\otimes\cI(h))$ generates the stalks of
$\cO(G+L)\otimes\cI(h)$.
\endproof
%
\subsection*{\csect.F. Hard Lefschetz theorem with multiplier 
ideal sheaves}
We state here without proof a basic surjectivity theorem, first appeared
in [DPS00], which extends the classical ``Hard Lefschetz Theorem''.

\claim{(1.29) Theorem} Let $(L,h)$ be a pseudo-effective line bundle on a
compact K\"ahler manifold $(X,\omega)$, of dimension $n$, let 
$\Theta_{L,h}\ge 0$ be its curvature current and $\cI(h)$ the 
associated multiplier ideal sheaf. Then, the wedge multiplication 
operator $\omega^q\wedge\bu$ induces a surjective morphism
$$
\Phi^q_{\omega,h}:
H^0(X,\Omega_X^{n-q}\otimes L\otimes\cI(h))\longrightarrow
H^q(X,\Omega_X^n\otimes L\otimes\cI(h)).
$$
\endclaim
%
\noindent
The special case when $L$ is nef is due to Takegoshi [Tak97]. An even
more special case is when $L$ is semi-positive, i.e.\ possesses a
smooth metric with semi-positive curvature.
In that case the multiple ideal sheaf $\cI(h)$ coincides with $\cO_X$
and we get the following consequence already observed by Mourougane
[Mou99].

\claim{(1.30) Corollary} Let $(L,h)$ be a semi-positive line bundle
on a compact K\"ahler mani\-fold $(X,\omega)$ of dimension~$n$. Then, the wedge
multiplication operator $\omega^q\wedge\bu$ induces a surjective morphism
$$
\Phi^q_\omega:H^0(X,\Omega_X^{n-q}\otimes L)\longrightarrow
H^q(X,\Omega_X^n\otimes L).
$$
\endclaim
%
It should be observed that although all objects involved in Theorem (1.29)
are algebraic when $X$ is a projective manifold, there are no known
algebraic proof of the statement; it is not even clear how to define
algebraically $\cI(h)$ for the case when $h=h_{min}$ is a metric
with minimal singularities. However, even in the special circumstance
when $L$ is nef, the multiplier ideal sheaf is crucially needed.

In fact, let $B$ be an elliptic curve and let $V$ be the rank $2$ vector 
bundle over $B$ which is defined as the (unique) non split
extension
$$0 \to \cO_B \to V \to \cO_B \to 0.\leqno(1.31)$$
In particular, the bundle $V$ is numerically flat, i.e.\ $c_1(V)=0$,
$c_2(V)=0$. We consider the ruled surface $X=\bP(V)$. On that surface
there is a unique section $C=\bP(\cO_B) \subset X$ with $C^2=0$ and
$$\cO_X (C)= \cO_{\bP (V)} (1)$$
is thus a nef line bundle. It is easy to see that
$$h^0 (X,\cO_X(mC))=h^0 (X,\cO_{\bP (V)} (m))= h^0 (B,S^m V)=1\leqno(1.32)$$
for all $m \in \bN$ (otherwise we would have $mC=aC+M$ where
$aC$ is the fixed part of the linear system $|mC|$ and $M\ne 0$ the
moving part, thus $M^2\ge 0$ and $C\cdot M>0$, contradiction).
On the other hand, rather simple calculations show that
$$h^0 (X,\Omega_X^1 (mC))=2\leqno(1.33)$$
for all $k\ge 2$. This shows that the generalized Hard Lefschetz Theorem
cannot hold true, even for nef line bundles, when the multiplier ideal
sheaf $\cI(h)$ is omitted in the statement.

\section{Holomorphic Morse inequalities}
Let $X$ be a compact K\"ahler manifold, $E$ a holomorphic vector bundle of
rank $r$ and $L$ a line bundle over~$X$. If $L$ is equipped with a smooth 
metric $h$ of curvature form $\Theta_{L,h}$, we define the $q$-index set of $L$
to be the open subset
$$X(q,L)=\left\{x\in X~;~\ii\Theta_{L,h}(x) \hbox{~has~}
{\displaystyle q\atop\displaystyle n-q}~
{\hbox{negative eigenvalues}\atop\hbox{positive eigenvalues}}\right\}
\leqno(2.1)$$
for $0\le q\le n$. Hence $X$ admits a partition $X=\Delta\cup
\bigcup_q X(q,L)$ where $\Delta=\{\hbox{$x\in X$}\,;\,\det(\Theta_{L,h}(x))=0\}$
is the degeneracy set. We also introduce
$$X(\le q,L)=\bigcup_{0\le j\le q}X(j,L).\leqno(2.1')$$

\claim{(2.2) Morse inequalities {\rm ([Dem85])}} For any hermitian holomorphic
line bundle $L,h)$ and any holomorphic vector bundle $E$ over a compact 
complex manifold~$X$, the cohomology groups
$H^q\big(X,E\otimes\cO(kL)\big)$ satisfy the following asymptotic
inequalities as $k\to+\infty$~:
\vskip3pt
\plainitem{\rm (a)} Weak Morse inequalities
$$h^q\big(X,E\otimes\cO(kL)\big)\le
r{k^n\over n!}\int_{X(q,L)} (-1)^q\Big({\ii\over2\pi}\Theta_{L,h}\Big)^n+o(k^n).
$$
\vskip3pt
\plainitem{\rm (b)} Strong Morse inequalities
$$
\sum_{0\le j\le q}(-1)^{q-j}h^j\big(X,E\otimes\cO(kL)\big)
\le r{k^n\over n!}\int_{X(\le q,L)}(-1)^q\Big({\ii\over2\pi}\Theta_{L,h}\Big)^n
+o(k^n).$$
\endclaim
%
The proof is based on the spectral theory of the complex Laplace operator, 
using either a localization procedure or, alternatively, a heat kernel 
technique.
These inequalities are a useful complement to the Riemann-Roch formula
when information is needed about individual cohomology groups, and not
just about the Euler-Poincar\'e characteristic.

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

\claim{(2.3) Theorem} Let $L=F-G$ be a holomorphic line bundle over a
compact K\"ahler manifold~$X$, where $F$ and $G$ are numerically effective
line bundles. Then for every $q=0,1,\ldots,n=\dim X$, there is an asymptotic
strong Morse inequality
$$\sum_{0\le j\le q}(-1)^{q-j}h^j(X,kL)\le{k^n\over n!}
\sum_{0\le j\le q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j+o(k^n).$$
\endclaim
%
\begproof{Proof.} By adding $\varepsilon$ times a K\"ahler metric $\omega$ to the
curvature forms of $F$ and $G$, $\varepsilon>0$ one can write ${\ii\over
2\pi}\Theta_L=\theta_{F,\varepsilon}-\theta_{G,\varepsilon}$ where
$\theta_{F,\varepsilon}={\ii\over 2\pi}\Theta_F+\varepsilon\omega$ and
$\theta_{G,\varepsilon}={\ii\over 2\pi}\Theta_G+\varepsilon\omega$ are
positive definite. Let $\lambda_1\ge\ldots\ge\lambda_n>0$ be the
eigenvalues of $\theta_{G,\varepsilon}$ with respect to $\theta_{F,\varepsilon}$.
Then the eigenvalues of ${\ii\over 2\pi}\Theta_L$ with respect
to $\theta_{F,\varepsilon}$ are the real numbers $1-\lambda_j$ and the
set $X(\le q,L)$ is the set $\{\lambda_{q+1}<1\}$ of points $x\in X$ such
that $\lambda_{q+1}(x)<1$. The strong Morse inequalities yield 
$$\sum_{0\le j\le q}(-1)^{q-j}h^j(X,kL)\le{k^n\over n!}
\int_{\{\lambda_{q+1}<1\}}\!\!\!(-1)^q\!\!
\prod_{1\le j\le n}(1-\lambda_j)\theta_{F,\varepsilon}^n+o(k^n).$$
On the other hand we have
$${n\choose j}\theta_{F,\varepsilon}^{n-j}\wedge\theta_{G,\varepsilon}^j=
\sigma_n^j(\lambda)\,\theta_{F,\varepsilon}^n,$$
where $\sigma_n^j(\lambda)$ is the $j$-th elementary symmetric function in
$\lambda_1,\ldots,\lambda_n\,$, hence
$$\sum_{0\le j\le q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j=
\lim_{\varepsilon\to 0}\int_X\sum_{0\le j\le q}(-1)^{q-j}
\sigma_n^j(\lambda)\,\theta_{F,\varepsilon}^n.$$
Thus, to prove the Lemma, we only have to check that
$$\sum_{0\le j\le n}(-1)^{q-j}\sigma_n^j(\lambda)-
\bOne_{\{\lambda_{q+1}<1\}}(-1)^q\prod_{1\le j\le n}(1-\lambda_j)\ge 0$$
for all $\lambda_1\ge\ldots\ge\lambda_n\ge 0$, where $\bOne_
{\{\ldots\}}$ denotes the characteristic function of a set.
This is easily done by induction on~$n$ (just split apart the parameter
$\lambda_n$ and write $\sigma_n^j(\lambda)=
\sigma_{n-1}^j(\lambda)+\sigma_{n-1}^{j-1}(\lambda)\,\lambda_n$).\qed
\endproof
%
In the case $q=1$, we get an especially interesting lower bound (this
bound has been observed and used by S.~Trapani [Tra95] in a similar
context).

\claim{(2.4) Consequence}
$h^0(X,kL)-h^1(X,kL)\ge{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n)$.\newline
Therefore some multiple $kL$ has a section as soon as $F^n-nF^{n-1}\cdot
G>0$.
\endclaim
%
\claim{(2.5) Remark} \rm The weaker inequality
$$h^0(X,kL)\ge{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n)$$
is easy to prove if $X$ is projective algebraic. Indeed, by adding a
small ample $\bQ$-divisor to $F$ and $G$, we may assume that
$F$, $G$ are ample. Let $m_0G$ be very ample and let $k'$ be the smallest
integer $\ge k/m_0$. Then $h^0(X,kL)\ge h^0(X,kF-k'm_0G)$. We select
$k'$ smooth members $G_j$, $1\le j\le k'$ in the linear system $|m_0G|$
and use the exact sequence
$$0\to H^0(X,kF-\sum G_j)\to H^0(X,kF)\to\bigoplus H^0(G_j,kF_{|G_j}).$$
Kodaira's vanishing theorem yields $H^q(X,kF)=0$ and $H^q(G_j,kF_{|G_j})=0$
for $q\ge 1$ and $k\ge k_0$. By the exact sequence combined with
Riemann-Roch, we get
$$\eqalign{
h^0(X,kL)&\ge h^0(X,kF-\sum G_j)\cr
&\ge{k^n\over n!}F^n-O(k^{n-1})-
\sum\Big({k^{n-1}\over(n-1)!}F^{n-1}\cdot G_j-O(k^{n-2})\Big)\cr
&\ge{k^n\over n!}\Big(F^n-n{k'm_0\over k}F^{n-1}\cdot G\Big)-O(k^{n-1})\cr
&\ge{k^n\over n!}\Big(F^n-n\,F^{n-1}\cdot G\Big)-O(k^{n-1}).\cr}$$
(This simple proof is due to F.~Catanese.)\qed
\endclaim
%
\claim{(2.6) Corollary} Suppose that $F$ and $G$ are nef and that $F$ is
big. Some multiple of $mF-G$ has a section as soon as
$$m>n\,{F^{n-1}\cdot G\over F^n}.$$
\endclaim
%
In the last condition, the factor $n$ is sharp: this is easily seen by taking
$X=\bP_1^n$ and $F=\cO(a,\ldots,a)$ and $G=\cO(b_1,\ldots,b_n)$ over
$\bP_1^n\,$; the condition of the Corollary is then
$m>\sum b_j/a$, whereas $k(mF-G)$ has a section if and only if
$m\ge\sup b_j/a$; this shows that one cannot replace $n$ by
$n(1-\varepsilon)$.\qed

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

\subsection*{\csect.A. Local approximation theorem through Bergman kernels}
We prove here, as an application of the Ohsawa-Takegoshi extension theorem,
that every psh function on a pseudoconvex open set $\Omega\subset\bC^n$
can be approximated very accurately by functions of the form
$c\log|f|$, where $c>0$ and $f$ is a holomorphic function. The main idea
is taken from [Dem92]. For other applications to algebraic geometry,
see [Dem93] and Demailly-Koll\'ar [DK01]. Recall that 
the Lelong number of a function $\varphi\in\Psh(\Omega)$ at a point $x_0$ 
is defined to be
$$
\nu(\varphi,x_0)=\liminf_{z\to x_0}{\varphi(z)\over\log|z-x_0|}=
\lim_{r\to 0_+}{\sup_{B(x_0,r)}\varphi\over \log r}.\leqno(3.1)
$$
In particular, if $\varphi=\log|f|$ with $f\in\cO(\Omega)$, then 
$\nu(\varphi,x_0)$ is equal to
the vanishing order 
$$\ord_{x_0}(f)=\sup\{k\in\bN\,;D^{\alpha}f(x_0)=0,~
\forall|\alpha|<k\}.$$

\claim{(3.2) Theorem} Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\bC^n$. For every $m>0$, let
$\cH_\Omega(m\varphi)$ be the Hilbert space of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-2m\varphi}d\lambda<+\infty$ and
let $\varphi_m={1\over 2m}\log\sum|\sigma_\ell|^2$ where $(\sigma_\ell)$
is an orthonormal basis of~$\cH_\Omega(m\varphi)$. Then there are
constants $C_1,C_2>0$ independent of $m$ such that
{\plainitemindent=7mm
\plainitem{\rm (a)}$\displaystyle\varphi(z)-{C_1\over m}\le
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over m}
\log{C_2\over r^n}$\newline
for every $z\in\Omega$ and $r<d(z,\partial\Omega)$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$ and
\plainitem{\rm (b)}$\displaystyle\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le
\nu(\varphi,z)$~ for every $z\in\Omega$.}
\endclaim
%
\begproof{Proof.} (a) Note that $\sum|\sigma_\ell(z)|^2$ is the square of the norm
of the evaluation linear form $\ev_z:f\mapsto f(z)$ on $\cH_\Omega(m\varphi)$,
since $\sigma_\ell(z)=\ev_z(\sigma_\ell)$ is the $\ell$-th coordinate of $\ev_z$
in the orthonormal basis $(\sigma_\ell)$. In other words, we have
$$\sum|\sigma_\ell(z)|^2=\sup_{f\in B(1)}|f(z)|^2$$
where $B(1)$ is the unit ball of $\cH_\Omega(m\varphi)$ (The sum
is called the {\emph Bergman kernel} associated with $\cH_\Omega(m\varphi)$.)
As $\varphi$ is locally bounded from above, the $L^2$ topology is 
actually stronger
than the topology of uniform convergence on compact subsets of~$\Omega$.
It follows that the series $\sum|\sigma_\ell|^2$ converges uniformly on
$\Omega$ and that its sum is real analytic. Moreover, by what we just 
explained, we have
$$\varphi_m(z)=\sup_{f\in B(1)}{1\over m}\log|f(z)|.$$
For $z_0\in\Omega$ and $r<d(z_0,\partial\Omega)$, the mean value
inequality applied to the psh function $|f|^2$ implies
$$\eqalign{
|f(z_0)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|z-z-0|<r}
|f(z)|^2d\lambda(z)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|z-z_0|<r}\varphi(z)\Big)
\int_\Omega|f|^2e^{-2m\varphi}d\lambda.\cr}$$
If we take the supremum over all $f\in B(1)$ we get
$$
\varphi_m(z_0)\le\sup_{|z-z_0|<r}\varphi(z)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the second inequality in (a) is proved -- as we see, this is an easy
consequence of the mean value inequality. Conversely, the
Ohsawa-Takegoshi extension theorem (Corollary 8.9) applied
to the $0$-dimensional subvariety $\{z_0\}\subset\Omega$ shows that for any
$a\in\bC$ there is a holomorphic function $f$ on $\Omega$ such that
$f(z_0)=a$ and
$$\int_\Omega|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z_0)},$$
where $C_3$ only depends on $n$ and~${\rm diam}\,\Omega$. We fix $a$
such that the right hand side is~$1$. Then $\Vert f\Vert \le 1$ and so
we get
$$\varphi_m(z_0)\ge{1\over m}\log|f(z_0)|=
{1\over m}\log|a|=\varphi(z)-{\log C_3\over 2m}.$$
The inequalities given in (a) are thus proved. Taking $r=1/m$, we find
that $\lim_{m\to+\infty}\sup_{|\zeta-z|<1/m}\varphi(\zeta)=\varphi(z)$
by the upper semicontinuity of $\varphi$, and therefore
$\lim\varphi_m(z)=\varphi(z)$, since $\lim{1\over m}\log (C_2m^n)=0$.
\medskip

\noindent
(b) The above estimates imply
$$
\sup_{|z-z_0|<r}\varphi(z)-{C_1\over  m}\le
\sup_{|z-z_0|<r}\varphi_m(z)\le\sup_{|z-z_0|<2r}\varphi(z)+
{1\over m}\log{C_2\over r^n}.$$
After dividing by $\log r<0$ when $r\to 0$, we infer
$$
{\sup_{|z-z_0|<2r}\varphi(z)+{1\over m}\log{C_2\over r^n}\over \log r}\le
{\sup_{|z-z_0|<r}\varphi_m(z)\over \log r}\le
{\sup_{|z-z_0|<r}\varphi(z)-{C_1\over  m}\over\log r},
$$
and from this and definition (3.1), it follows immediately that 
$$
\nu(\varphi,x)-{n\over m}\le\nu(\varphi_m,z)\le\nu(\varphi,z).
\eqno\square
$$
\endproof
%
Theorem 3.2 implies in a straightforward manner the deep result of
[Siu74] on the analyticity of the Lelong number upperlevel sets.

\claim{(3.3) Corollary {\rm[Siu74])}} Let $\varphi$ be a plurisubharmonic
function on a complex manifold~$X$. Then, for every $c>0$, the Lelong number
upperlevel set
$$
E_c(\varphi)=\big\{z\in X\,;\;\nu(\varphi,z)\ge c\big\}
$$
is an analytic subset of~$X$.
\endclaim
%
\begproof{Proof.} Since analyticity is a local property, it
is enough to consider the case of a psh function $\varphi$ on a 
pseudoconvex open set $\Omega\subset\bC^n$. The
inequalities obtained in 3.2~b) imply that 
$$
E_c(\varphi)=\bigcap_{m\ge m_0}E_{c-n/m}(\varphi_m).
$$
Now, it is clear that $E_c(\varphi_m)$ is the analytic set defined by the
equations $\sigma_\ell^{(\alpha)}(z)=0$ for all multi-indices $\alpha$
such that~$|\alpha|<mc$. Thus $E_c(\varphi)$ is analytic as a (countable) 
intersection of analytic sets.\qed
\endproof
%
\claim{(3.4) Remark} {\rm It can be easily shown that the Lelong numbers
of any closed positive $(p,p)$-current coincide (at least locally) with the
Lelong numbers of a suitable plurisubharmonic potential $\varphi$
(see Skoda [Sko72a]). Hence Siu's theorem also holds true for the
Lelong number upperlevel sets $E_c(T)$ of any closed positive 
$(p,p)$-current~$T$.}
\endclaim
%
\subsection*{\csect.B. Global approximation of closed (1,1)-currents
on a compact complex manifold}
We take here $X$ to be an arbitrary compact complex manifold (no K\"ahler
assumption is needed). Now, let $T$ be a closed $(1,1)$-current
on~$X$. We assume that $T$ is {\emph almost positive}, i.e.\ that there exists
a $(1,1)$-form $\gamma$ with continuous coefficients such
that~$T\ge\gamma\;$; the case of positive currents ($\gamma=0$) is of course
the most important.

\claim{(3.5) Lemma} There exists a smooth closed $(1,1)$-form $\alpha$ 
representing the same~$\ddbar$-
cohomology class as $T$ and an {\rm almost psh} function $\varphi$
on $X$ such that \hbox{$T=\alpha+{i\over\pi}\ddbar\varphi$}. $($We say that
a function $\varphi$ is almost psh if its complex Hessian is bounded below
by a $(1,1)$-form with locally bounded coefficients, that is, if
$i\ddbar\varphi$ is almost positive$)$.
\endclaim
%
\begproof{Proof.} Select
an open covering $(U_j)$ of $X$ by coordinate balls such that 
$T=\smash{{i\over\pi}}
\ddbar\varphi_j$ over $U_j$, and construct a global function
$\varphi=\sum\theta_j \varphi_j$ by means of a partition of unity $(\theta_j)$
subordinate to~$U_j$. Now, we observe that $\varphi-\varphi_k$ is smooth 
on $U_k$  because all differences $\varphi_j-\varphi_k$ are smooth in 
the intersections
$U_j\cap U_k$ and $\varphi-\varphi_k=\sum\theta_j(\varphi_j-\varphi_k)$).
Therefore $\alpha:=T-{i\over\pi}\ddbar\varphi$ is smoth.\qed
\endproof
%
By replacing $T$ with $T-\alpha$ and $\gamma$ with
$\gamma-\alpha$, we can assume without loss of
generality that $\{T\}=0$, i.e. that $T={i\over\pi}\ddbar\varphi$
with an almost psh function~$\varphi$ on~$X$ such that
${i\over\pi}\ddbar\varphi\ge\gamma$.

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

We select a finite covering $(W_\nu)$ of $X$ with open
coordinate charts. Given $\delta>0$, we take in each $W_\nu$ a
maximal family of points with (coordinate) distance to the boundary
$\ge 3\delta$ and mutual distance $\ge\delta/2$. In this way, we get for 
$\delta>0$ small a finite covering of $X$ by open balls $U'_j$ of
radius~$\delta$ (actually every point is even at 
\hbox{distance${}\le\delta/2$} of
one of the centers, otherwise the family of points would not be maximal), 
such that the concentric ball $U_j$ of radius
$2\delta$ is relatively compact in the corresponding chart~$W_\nu$. Let
$\tau_j:U_j\longrightarrow B(a_j,2\delta)$ be the isomorphism given by the
coordinates of~$W_\nu$. Let $\varepsilon(\delta)$ be a modulus
of continuity for $\gamma$ on the sets $U_j$, such that
$\lim_{\delta\to 0}\varepsilon(\delta)=0$ and 
$\gamma_x-\gamma_{x'}\le{1\over 2}\varepsilon(\delta)\,\omega_x$ for all
$x,x'\in U_j$. We denote by $\gamma_j$ the $(1,1)$-form with constant
coefficients on $B(a_j,2\delta)$ such that $\tau_j^\star\gamma_j$
coincides with $\gamma-\varepsilon(\delta)\,\omega$ at
$\tau_j^{-1}(a_j)$. Then we have
$$0\le\gamma-\tau_j^\star\gamma_j\le 2\varepsilon(\delta)\,\omega~~~~
\hbox{\rm on}~~U'_j\leqno(3.6)$$
for $\delta>0$ small. We set $\varphi_j=\varphi\circ\tau_j^{-1}$ on
$B(a_j,2\delta)$ and let $q_j$ be the homogeneous quadratic
function in $z-a_j$ such that ${i\over\pi}\ddbar q_j=
\gamma_j$ on $B(a_j,2\delta)$. Finally, we set
$$\psi_j(z)=\varphi_j(z)-q_j(z)~~~~\hbox{\rm on}~B(a_j,2\delta).
\leqno(3.7)$$
Then $\psi_j$ is plurisubharmonic, since
$${i\over\pi}\ddbar(\psi_j\circ\tau_j)=T-\tau_j^\star\gamma_j
\ge\gamma-\tau_j^\star\gamma_j\ge 0.$$
We let $U'_j\compact U''_j\compact U_j$ be 
concentric balls of radii $\delta$, $1.5\,\delta$, $2\delta$ respectively.
On each open set $U_j$ the function
$\psi_j:=\varphi-q_j\circ\tau_j$ defined
in (3.7) is plurisubharmonic, so Theorem~(3.2) applied
with $\Omega=U_j$ produces functions
$$\psi_{j,m}={1\over 2m}\log\sum_\ell|\sigma_{j,\ell}|^2,~~~~
(\sigma_{j,\ell})=\hbox{\rm basis of}~\cH_{U_j}(m\psi_j).\leqno(3.8)$$
These functions approximate $\psi_j$ as $m$ tends to $+\infty$ and 
satisfy the inequalities
$$\psi_j(x)-{C_1\over m}\le\psi_{j,m}(x)\le\sup_{|\zeta-x|<r}
\psi_j(\zeta)+{1\over m}\log{C_2\over r^n}.\leqno(3.9)$$
The functions $\psi_{j,m}+q_j\circ\tau_j$ on $U_j$ then have
to be glued together by a partition of unity technique. For this,
we rely on the following ``discrepancy'' lemma, estimating the variation
of the approximating functions on overlapping balls.

\claim{(3.10) Lemma} There are constants $C_{j,k}$ 
independent of $m$ and $\delta$ such that the almost psh functions
$w_{j,m}=2m(\psi_{j,m}+q_j\circ\tau_j)$, i.e.\
$$w_{j,m}(x)=2m\,q_j\circ\tau_j(x)+
\log\sum_\ell\big|\sigma_{j,\ell}(x)\big|^2,~~~~
x\in U''_j,$$
satisfy
$$|w_{j,m}-w_{k,m}|\le C_{j,k}\big(\log\delta^{-1}+
m\varepsilon(\delta)\delta^2\big)
~~~~\hbox{\rm on}~~U''_j\cap U''_k.$$
\endclaim
%
\begproof{Proof.} The details will be left as an exercise to the reader.
The main idea is the following: for any holomorphic function 
$f_j\in \cH_{U_j}(m\psi_j)$,
a $\dbar$ equation $\dbar u = \dbar(\theta f_j)$ can be solved on $U_k$, 
where $\theta$ is a cut-off function with support in $U''_j\cap U''_k$,
on a ball of radius${}<\delta/4$, 
equal to $1$ on the ball of radius $\delta/8$ centered at a given 
point $x_0\in U''_j\cap U''_k$. We apply
the $L^2$ estimate with respect to the weight 
\hbox{$(n+1)\log|x-x_0|^2+2m\psi_k$}, where the first term is picked up so as 
to force the solution $u$ to vanish at $x_0$, in such a way that
$F_k=u-\theta f_j$ is holomorphic and $F_k(x_0)=f_j(x_0)$. 
The discrepancy between the weights on $U''_j$ and $U''_k$~is
$$
\psi_j(x)-\psi_k(x)=-\big(q_j\circ\tau_j(x)-q_k\circ\tau_k(x)\big)
$$
and the $\ddbar$ of this difference is $O(\varepsilon(\delta))$, so it
is easy to correct the discrepancy up to a $O(\varepsilon(\delta)\delta^2)$
term by multiplying our functions by an invertible holomorphic
function~$G_{jk}$.
In this way, we get a uniform $L^2$ control on the $L^2$ norm of the
solution $f_k=G_{jk}F_k=G_{jk}(u-\theta f_j)$ of the form
$$\int_{U_k}|f_k|^2e^{-2m\psi_k}\le C_{j,k}\delta^{-2n-4}
e^{m O(\varepsilon(\delta)\delta^2)} \int_{U_j}|f_j|^2e^{-2m\psi_j}.$$
The required estimate follows, using the fact that
$$
e^{2m\psi_{j,m}(x)}=\sum_\ell|\sigma_{j,\ell}(x)|^2=
\sup_{f\in\cH_{U_j}(m\psi_j),\,\Vert f\Vert\le 1}|f(x)|^2~~~~\hbox{on $U_j$},
$$
and the analogous equality on $U_k$.\qed
\endproof
%
Now, the actual glueing of our almost psh functions is performed
using the following elementary partition of unity argument, which
follows from a brute force calculation (see e.g.\ [Dem82]).

\claim{(3.11) Lemma} Let $U'_j\compact U''_j$ be locally finite 
open coverings
of a complex manifold $X$ by relatively compact open sets, and let
$\theta_j$ be smooth nonnegative functions with support in $U''_j$,
such that $\theta_j\le 1$ on $U''_j$ and $\theta_j=1$ on $U'_j$.
Let $A_j\ge 0$ be such that
$$i(\theta_j\ddbar\theta_j-\partial\theta_j\wedge\dbar\theta_j)\ge
-A_j\omega~~~~\hbox{\rm on}~~U''_j\ssm U'_j$$
for some positive $(1,1)$-form~$\omega$.
Finally, let $w_j$ be almost psh functions on $U_j$ with
the property that $i\partial\smash{\dbar}w_j\ge\gamma$
for some real $(1,1)$-form $\gamma$ on~$M$, and let $C_j$ be constants
such that
$$w_j(x)\le C_j+\sup_{k\ne j,\,U'_k\ni x}w_k(x)~~~~\hbox{\rm on}~~
U''_j\ssm U'_j.$$
Then the function $w=\log\big(\sum\theta_j^2 e^{w_j}\big)$
is almost psh and satisfies
$$i\ddbar w\ge\gamma-2\Big(\sum_j\bOne_{U''_j\ssm U'_j}A_je^{C_j}\Big)
\omega.$$
\endclaim
%
We apply Lemma~(3.11) to functions $\widetilde w_{j,m}$ which are just
slight modifications of the functions $w_{j,m}=2m(\psi_{j,m}+q_j\circ\tau_j)$
occurring in (3.10)~:
$$\eqalign{
\widetilde w_{j,m}(x)
&=w_{j,m}(x)+2m\Big({C_1\over m}+
C_3\varepsilon(\delta)(\delta^2/2-|\tau_j(x)|^2)\Big)\cr
&=2m\Big(\psi_{j,m}(x)+q_j\circ\tau_j(x)+{C_1\over m}+
C_3\varepsilon(\delta)(\delta^2/2-|\tau_j(x)|^2)\Big)\cr}$$
where $x\mapsto z=\tau_j(x)$ is a local coordinate identifying $U_j$
to $B(0,2\delta)$, $C_1$ is the constant occurring in (3.9) and $C_3$ is 
a sufficiently large constant. 
It is easy to see that we can take $A_j=C_4\delta^{-2}$ in Lemma~(3.11).
We have
$$
\widetilde w_{j,m}\ge w_{j,m}+2C_1+m{C_3\over 2}\varepsilon(\delta)\delta^2
~~~~\hbox{on $B(x_j,\delta/2)\subset U'_j$},
$$
since $|\tau_j(x)|\le\delta/2$ on $B(x_j,\delta/2)$, while
$$
\widetilde w_{j,m}\le w_{j,m}+2C_1-mC_3\varepsilon(\delta)\delta^2
~~~~\hbox{on $U''_j\ssm U'_j$}.
$$
For $m\ge m_0(\delta)=(\log\delta^{-1}/(\varepsilon(\delta)\delta^2)$, 
Lemma (3.10) implies $|w_{j,m}-w_{k,m}|\le C_5m\varepsilon(\delta)\delta^2$ on
$U''_j\cap U''_k$. Hence, for $C_3$ large enough, we get
$$\widetilde w_{j,m}(x)\le \sup_{k\ne j,\,B(x_k,\delta/2)\ni x}w_{k,m}(x)
\le \sup_{k\ne j,\,U'_k\ni x}w_{k,m}(x)~~~~\hbox{\rm on}~~
U''_j\ssm U'_j,$$
and we can take $C_j=0$ in the hypotheses of Lemma~(3.11). 
The associated function $w=\log\big(\sum\theta_j^2e^{\widetilde w_{j,m}}\big)$ 
is given by
$$w=\log\sum_j\theta_j^2\exp\Big(2m\big(\psi_{j,m}+q_j\circ\tau_j+
{C_1\over m}+C_3\varepsilon(\delta)(\delta^2/2-|\tau_j|^2)\big)
\Big).$$
If we define $\varphi_m={1\over 2m}w$, we get
$$
\varphi_m(x):={1\over 2m}w(x)\ge \psi_{j,m}(x)+
q_j\circ\tau_j(x)+{C_1\over m}+{C_3\over 4}\varepsilon(\delta)\delta^2
>\varphi(x)
$$
in view of (3.9), by picking an index $j$ such that $x\in B(x_j,\delta/2)$.
In the opposite
direction, the maximum number $N$ of overlapping
balls~$U_j$ does not depend on $\delta$, and we thus get
$$w\le\log N+2m\Big(
\max_j\big\{\psi_{j,m}(x)+q_j\circ\tau_j(x)\big\}+
{C_1\over m}+{C_3\over 2}\varepsilon(\delta)\delta^2\Big).$$
By definition of $\psi_j$ we have $\sup_{|\zeta-x|<r}\psi_j(\zeta)\le
\sup_{|\zeta-x|<r}\varphi(\zeta)-q_j\circ\tau_j(x)+C_5r$ thanks to
the uniform Lipschitz continuity of $q_j\circ\tau_j$, thus by 
(3.9) again we find
$$\varphi_m(x)\le
{\log N\over 2m}+\sup_{|\zeta-x|<r}\varphi(\zeta)+
{C_1\over m}+{1\over m}\log{C_2\over r^n}+
{C_3\over 2}\varepsilon(\delta)\delta^2+C_5 r$$
By taking for instance $r=1/m$ and $\delta=\delta_m\to 0$, we see that 
$\varphi_m$ converges to $\varphi$.
On the other hand $(3.6)$ implies
${i\over\pi}\ddbar q_j\circ\tau_j(x)=\tau_j^\star\gamma_j\ge
\gamma-2\varepsilon(\delta)\omega$, thus
$${i\over\pi}\ddbar\widetilde w_{j,m}\ge 
2m\big(\gamma-C_6\varepsilon(\delta)\omega\big).$$
Lemma~(3.11) then produces the lower bound
$${i\over\pi}\ddbar w
\ge 2m\big(\gamma-C_6\varepsilon(\delta)\omega\big)-C_7\delta^{-2}\omega,$$
whence
$${i\over\pi}\ddbar\varphi_m\ge\gamma-C_8\varepsilon(\delta)\omega$$
for $m\ge m_0(\delta)=(\log\delta^{-1})/(\varepsilon(\delta)\delta^2)$.
We can fix $\delta=\delta_m$ to be the smallest value of $\delta>0$ such that
$m_0(\delta)\le m$, then $\delta_m\to 0$ and we have
obtained a sequence of quasi psh functions $\varphi_m$ satisfying 
the following properties.

\claim{(3.12) Theorem} 
Let $\varphi$ be an almost psh function on a compact complex
manifold $X$ such that ${i\over\pi}\ddbar\varphi\ge\gamma$ for some
continuous $(1,1)$-form $\gamma$. Then there is a sequence of
almost psh functions $\varphi_m$ such that $\varphi_m$ has the 
same singularities
as a logarithm of a sum of squares of holomorphic functions and
a decreasing sequence $\varepsilon_m>0$ converging to $0$ such that
{\plainitemindent=7mm
\plainitem{\rm (i)}$\displaystyle\varphi(x)<\varphi_m(x)\le\sup_{|\zeta-x|<r}
\varphi(\zeta)+C\Big({|\log r|\over m}+r+\varepsilon_m\Big)$\newline
with respect to coordinate open sets covering $X$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1(X)$ and
\plainitem{\rm (ii)}$\displaystyle\nu(\varphi,x)-{n\over m}\le\nu(\varphi_m,x)\le
\nu(\varphi,x)$~ for every $x\in X\,;$
\plainitem{\rm (iii)}$\displaystyle{i\over\pi}\ddbar\varphi_m\ge
\gamma-\varepsilon_m\omega$.}
\endclaim
%
In particular, we can apply this to an arbitrary positive or 
almost positive closed
$(1,1)$-current $T=\alpha+{i\over\pi}\ddbar\varphi$.

\claim{(3.13) Corollary} 
Let $T$ be an almost positive  closed
$(1,1)$-current on a compact complex
manifold $X$ such that $T\ge\gamma$ for some
continuous $(1,1)$-form $\gamma$. Then there is a sequence of
currents $T_m$ whose local potentials have the 
same singularities as $1/m$ times a logarithm of a sum of squares 
of holomorphic functions and
a decreasing sequence $\varepsilon_m>0$ converging to $0$ such that
{\plainitemindent=7mm
\plainitem{\rm (i)} $T_m$ converges weakly to $T$,
\plainitem{\rm (ii)}$\displaystyle\nu(T,x)-{n\over m}\le\nu(T_m,x)\le
\nu(T,x)$~ for every $x\in X\,;$
\plainitem{\rm (iii)}$T_m\ge\gamma-\varepsilon_m\omega$.
\vskip0pt
}
\noindent
We say that our currents $T_m$ are approximations of $T$ possessing
logarithmic poles.
\endclaim
%
By using blow-ups of $X$, the structure of the currents $T_m$ can be
better understood.  In fact, consider the coherent ideals $\cJ_m$
generated locally by the holomorphic functions $(\sigma^{(k)}_{j,m})$
on $U_k$ in the local approximations
$$
\varphi_{k,m}={1\over 2m}\log\sum_j|\sigma^{(k)}_{j,m}|^2+O(1)
$$
of the potential $\varphi$ of~$T$ on~$U_k$. These ideals are in fact
globally defined, because the local ideals
$\cJ_m^{(k)}=(\sigma^{(k)}_{j,m})$ are integrally closed, and they
coincide on the intersections $U_k\cap U_\ell$ as they have the same
order of vanishing by the proof of Lemma~(13,10).  By Hironaka
[Hir64], we can find a composition of blow-ups with smooth centers
$\mu_m:\widetilde X_m\to X$ such that $\mu_m^*\cJ_m$ is an invertible ideal
sheaf associated with a normal crossing divisor $D_m$. Now, we can
write
$$
\mu_m^*\varphi_{k,m}=\varphi_{k,m}\circ\mu_m=
{1\over m}\log|s_{D_m}|+\widetilde\varphi_{k,m}
$$
where $s_{D_m}$ is the canonical section of $\cO(-D_m)$ and 
$\widetilde\varphi_{k,m}$ is a smooth potential. This implies
$$
\mu_m^*T_m={1\over m}[D_m]+\beta_m\leqno(3.14)
$$
where $[D_m]$ is the current of integration over $D_m$ and $\beta_m$
is a smooth closed $(1,1)$-form which satisfies the lower bound
$\beta_m\ge\mu_m^*(\gamma-\varepsilon_m\omega)$. (Recall that
the pull-back of a closed $(1,1)$-current by a holomorphic map $f$
is always well-defined, by taking a local plurisubharmonic potential 
$\varphi$ such that $T=i\ddbar\varphi$ and writing 
$f^*T=i\ddbar(\varphi\circ f)$). In the remainder of
this section, we derive from this a rather important geometric consequence,
first appeared in~[DP04]). We need two related definitions.

\claim{(3.15) Definition} A K\"ahler current on a compact
complex space $X$ is a closed positive current $T$ of bidegree $(1,1)$
which satisfies $T\ge\varepsilon\omega$ for some $\varepsilon>0$ and
some smooth positive hermitian form $\omega$ on $X$.
\endclaim
%
\claim{(3.16) Definition} A compact complex manifold is said to be
in the {\rm Fujiki class $\cC$)} if it is bimeromorphic to a K\"ahler manifold
$($or equivalently, using Hironaka's desingularization theorem, if 
it admits a proper K\"ahler modification$)$.
\endclaim
%
\claim{(3.17) Theorem} A compact complex manifold $X$ 
is bimeromorphic to a K\"ahler manifold $($i.e.\ $X\in\cC)$
if and only if it admits a K\"ahler current.
\endclaim
%
\begproof{Proof.} If $X$ is bimeromorphic to a K\"ahler manifold $Y$, 
Hironaka's desingularization theorem implies that there exists
a blow-up $\widetilde Y$ of $Y$ (obtained by a sequence of blow-ups with
smooth centers) such that the bimeromorphic map from
$Y$ to $X$ can be resolved into a modification $\mu:\widetilde Y\to X$. Then
$\widetilde Y$ is K\"ahler and the push-forward $T=\mu_*\widetilde\omega$
of a K\"ahler form $\widetilde\omega$ on $\widetilde Y$ provides a K\"ahler 
current on~$X$. In fact, if $\omega$ is a smooth hermitian form on~$X$,
there is a constant $C$ such that $\mu^*\omega\le C\widetilde\omega$ (by compactness
of $\widetilde Y$), hence
$$
T=\mu_*\widetilde\omega\ge \mu_*(C^{-1}\mu^*\omega)=C^{-1}\omega.
$$
Conversely, assume that $X$ admits a K\"ahler current $T\ge\varepsilon\omega$. 
By Theorem 3.13~(iii), there exists a K\"ahler current 
$\widetilde T=T_m\ge{\varepsilon\over 2}\omega$ (with $m\gg 1$ so large 
that $\varepsilon_m\le\varepsilon/2$) in 
the same $\ddbar$-cohomology class as~$T$, possessing logarithmic poles. 
Observation (3.14) implies the existence of a composition of blow-ups
$\mu:\widetilde X\to X$ such that
$$
\mu^*\widetilde T=[\widetilde D]+\widetilde\beta\qquad\hbox{on $\widetilde X$},
$$
where $\widetilde D$ is a $\bQ$-divisor with normal crossings and $\widetilde\beta$
a smooth closed $(1,1)$-form such that $\widetilde\beta\ge{\varepsilon\over 2}
\mu^*\omega$. In particular $\widetilde\beta$ is positive outside the exceptional
locus of~$\mu$. This is not enough yet to produce a K\"ahler
form on $\widetilde X$, but we are not very far. Suppose that $\widetilde X$ 
is obtained as a tower of blow-ups
$$
\widetilde X=X_N\to X_{N-1}\to\cdots \to X_1\to X_0=X,
$$
where $X_{j+1}$ is the blow-up of $X_j$ along a smooth center $Y_j\subset X_j$.
Denote by $E_{j+1}\subset X_{j+1}$ the exceptional divisor, and let
$\mu_j:X_{j+1}\to X_j$ be the blow-up map. Now, we use the following
simple
\endproof
%
\claim{(3.18) Lemma} For every K\"ahler current 
$T_j$ on $X_j$, there exists $\varepsilon_{j+1}>0$ and a smooth form $u_{j+1}$
in the $\ddbar$-cohomology class of $[E_{j+1}]$ such that
$$
T_{j+1}=\mu_j^\star T_j-\varepsilon_{j+1}u_{j+1}
$$ 
is a K\"ahler current on~$X_{j+1}$.
\endclaim
%
\begproof{Proof.} The line bundle
$\cO(-E_{j+1})|E_{j+1}$ is equal to $\cO_{P(N_j)}(1)$ where $N_j$
is the normal bundle to $Y_j$ in $X_j$. Pick an arbitrary smooth hermitian 
metric on $N_j$, use this metric to get an induced Fubini-Study metric on
$\cO_{P(N_j)}(1)$, and finally extend this metric as a smooth hermitian 
metric on the line bundle $\cO(-E_{j+1})$. Such a metric has positive
curvature along tangent vectors of $X_{j+1}$ which are tangent to the 
fibers of $E_{j+1}=P(N_j)\to Y_j$. Assume furthermore that 
$T_j\ge\delta_j\omega_j$ for some hermitian form $\omega_j$ on $X_j$
and a suitable $0<\delta_j\ll 1$. Then
$$
\mu_j^\star T_j-\varepsilon_{j+1}u_{j+1}\ge
\delta_j\mu_j^\star\omega_j-\varepsilon_{j+1}u_{j+1}
$$
where $\mu_j^*\omega_j$ is semi-positive on $X_{j+1}$, positive definite on 
$X_{j+1}\ssm E_{j+1}$, and also positive definite on tangent vectors of 
$T_{X_{j+1}|E_{j+1}}$ which are not tangent to the fibers of $E_{j+1}\to Y_j$.
The statement is then easily proved by taking $\varepsilon_{j+1}\ll\delta_j$ 
and by using an elementary compactness argument on the unit sphere bundle 
of~$T_{X_{j+1}}$ associated with any given hermitian metric.\qed
\endproof
%
\noindent
{\emph End of proof of Theorem 3.17.}
If $\widetilde u_j$ is the pull-back of $u_j$ to the final blow-up
$\widetilde X$, we conclude inductively that $\mu^\star
\widetilde T-\sum\varepsilon_j\widetilde u_j$ is a K\"ahler current. Therefore
the smooth form
$$
\widetilde\omega:=\widetilde\beta-\sum\varepsilon_j\widetilde u_j
=\mu^\star \widetilde T-\sum\varepsilon_j\widetilde u_j-[\widetilde D]
$$
is K\"ahler and we see that $\widetilde X$ is a K\"ahler manifold.\qed
\medskip

\claim{(3.19) Remark} {\rm A special case of Theorem (3.16) is the following
characterization of Moishezon varieties (i.e.\ manifolds which are
bimeromorphic to projective algebraic varieties or, equivalently, whose
algebraic dimension is equal to their complex dimension): 
\vskip3pt
\noindent
{\emph A compact complex manifold $X$ is Moishezon if and only if $X$ 
possesses a K\"ahler current $T$ such that the De Rham cohomology class
$\{T\}$ is rational, i.e.\ $\{T\}\in H^2(X,\bQ)$}. 
\vskip3pt
\noindent
In fact, in the above proof, we get an integral current $T$ if we take
the push forward $T=\mu_*\widetilde\omega$ of an integral ample class 
$\{\widetilde\omega\}$ on $Y$, where $\mu:Y\to X$ is a projective model 
of~$Y$. Conversely, if $\{T\}$ is rational, we can take
the $\varepsilon_j's$ to be rational in Lemma~3.5. This produces
at the end a K\"ahler metric $\widetilde\omega$ with rational De Rham cohomology
class on $\widetilde X$. Therefore $\widetilde X$ is projective by the Kodaira
embedding theorem. This result was already observed in [JS93]
(see also [Bon93, Bon98] for a more general perspective based
on a singular version of holomorphic Morse inequalities).}
\endclaim
%
\subsection*{\csect.C. Global approximation by divisors}
We now translate our previous approximation theorems into a more 
algebro-geometric setting. Namely, we assume that $T$ is a closed
positive $(1,1)$-current which belongs to the first Chern class
$c_1(L)$ of a holomorphic line bundle~$L$, and we assume here $X$
to be algebraic (i.e.\ projective or at the very least Moishezon).

Our goal is to show that $T$ can be approximated by divisors which
have roughly the same Lelong numbers as~$T$. The existence of weak
approximations by divisors has already been proved in [Lel72] for
currents defined on a pseudoconvex open set $\Omega\subset\bC^n$
with $H^2(\Omega,\bR)=0$, and in [Dem92, 93b] in the situation considered
here (cf.\ also [Dem82], although the argument given there is somewhat
incorrect). We take the opportunity to present here a slightly simpler
derivation. 

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

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

\claim{(3.20) Definition} If $K$ is a compact subset of $X$ and $\varphi$
is a quasi-psh function defined near~$K$, we define
\vskip3pt
\plainitem{\rm(a)} the
complex singularity exponent $c_K(\varphi)$ to be the supremum of all
positive numbers $c$ such that  $e^{-2c\varphi}$ is
integrable in a neighborhood of every point $z_0\in K$, with respect to 
the Lebesgue measure in holomorphic coordinates centered at~$z_0$. 
In particular $c_K(\varphi)=\inf_{z_0\in K}(\varphi)$.
\vskip3pt
\plainitem{\rm(b)} The concept is easily extended to hermitian metrics 
$h=e^{-2\varphi}$ by putting $c_K(h)=c_K(\varphi)$, to holomorphic
functions $f$ by $c_K(f)=c_K(\log|f|)$, to coherent ideals
$\cJ=(g_1,\ldots,g_N)$ by $c_K(\cJ)=c_K(\varphi)$ where
$\varphi={1\over 2}\log\sum|g_j|^2$. Also for an effective
$\bR$-divisor $D$, we put $c_K(D)=c_K(\log|\sigma_D|)$ where
$\sigma_D$ is the canonical section.\vskip0pt
\endclaim
%
\noindent
The main technical result of this section can be stated as follows, in the
case of big line bundles (cf.\ Proposition (1.17$\,$ii)).

\claim{(3.21) Theorem} Let $L$ be a line bundle on a compact complex manifold
$X$ possessing a singular hermitian metric $h$ with $\Theta_{L,h}\ge 
\varepsilon\omega$ for some $\varepsilon>0$ and some smooth positive definite
hermitian $(1,1)$-form $\omega$ on~$X$. For every real number $m>0$, consider
the space $\cH_m=H^0(X,L^{\otimes m}\otimes\cI(h^m))$ of holomorphic sections 
$\sigma$ of $L^{\otimes m}$ on $X$ such that 
$$
\int_X|\sigma|_{h^m}^2dV_\omega
=\int_X|\sigma|^2e^{-2m\varphi}dV_\omega<+\infty,
$$
where $dV_\omega={1\over m!}\omega^m$ is the hermitian volume form.
Then for $m\gg 1$, $\cH_m$ is a non zero finite dimensional Hilbert space 
and we consider the closed positive $(1,1)$-current
$$
T_m={i\over \pi}\ddbar\Big({1\over 2m}\log\sum_k|g_{m,k}|^2\Big)
={i\over \pi}\ddbar\Big({1\over 2m}\log\sum_k|g_{m,k}|_h^2\Big)
+\Theta_{L,h}
$$
where $(g_{m,k})_{1\le k\le N(m)}$ is an orthonormal basis of~$\cH_m$. 
Then$\;:$
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} For every trivialization  $L_{|U}\simeq U\times\bC$
on a cordinate open set $U$ of $X$ and every compact set $K\subset U$,
there are constants $C_1,C_2>0$ independent of $m$ and $\varphi$ such that
$$
\varphi(z)-{C_1\over m}\le
\psi_m(z):={1\over 2m}\log\sum_k|g_{m,k}(z)|^2
\le\sup_{|x-z|<r}\varphi(x)+{1\over m}\log{C_2\over r^n}
$$
for every $z\in K$ and $r\le{1\over 2}d(K,\partial U)$. In particular,
$\psi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$, hence $T_m$ converges weakly 
to $T=\Theta_{L,h}$.
\vskip3pt
\plainitem{\rm (ii)} The Lelong numbers $\nu(T,z)=\nu(\varphi,z)$ and 
$\nu(T_m,z)=\nu(\psi_m,z)$ are related by
$$
\nu(T,z)-{n\over m}\le\nu(T_m,z)\le
\nu(T,z)\quad\hbox{ for every $z\in X$.}
$$
\plainitem{\rm (iii)} For every compact set $K\subset X$, the complex singularity
exponents of the metrics given locally by $h=e^{-2\varphi}$ and 
$h_m=e^{-2\psi_m}$ satisfy
$$
c_K(h)^{-1}-{1\over m}\le c_K(h_m)^{-1}\le c_K(h)^{-1}.
$$}
\endclaim
%
\begproof{Proof.} The major part of the proof is a variation
of the arguments already explained in section 3.A.\vskip3pt

\noindent
(i) We note that $\sum|g_{m,k}(z)|^2$ is the square of the norm of 
the evaluation linear form $\sigma\mapsto \sigma(z)$ on $\cH_m$, hence
$$
\psi_m(z)=\sup_{\sigma\in B(1)}{1\over m}\log|\sigma(z)|
$$
where $B(1)$ is the unit ball of $\cH_m$. For
$r\le {1\over 2}d(K,\partial\Omega)$, the mean value inequality 
applied to the plurisubharmonic function $|\sigma|^2$ implies
$$
\eqalign{
|\sigma(z)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|x-z|<r}
|\sigma(x)|^2d\lambda(x)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|x-z|<r}\varphi(x)\Big)
\int_\Omega|\sigma|^2e^{-2m\varphi}d\lambda.\cr}
$$
If we take the supremum over all $\sigma\in B(1)$ we get
$$
\psi_m(z)\le\sup_{|x-z|<r}\varphi(x)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the right hand inequality in (i) is proved. Conversely, the
Ohsawa-Takegoshi extension theorem [OhT87], [Ohs88] applied to 
the $0$-dimensional
subvariety $\{z\}\subset U$ shows that for any $a\in\bC$ there is
a holomorphic function $f$ on $U$ such that $f(z)=a$ and
$$
\int_U|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z)},
$$
where $C_3$ only depends on $n$ and~$\diam U$. Now, provided $a$ remains in
a compact set $K\subset U$, we can use a cut-off function $\theta$ with
support in $U$ and equal to $1$ in a neighborhood of $a$, and solve the
$\dbar$-equation $\dbar g=\dbar(\theta f)$ in the $L^2$ space associated
with the
weight $2m\varphi+2(n+1)|\log|z-a|$, that is, the singular hermitian
metric $h(z)^m|z-a|^{-2(n+1)}$ on~$L^{\otimes m}$. For this, we apply
the standard Andreotti-Vesentini-H\"ormander $L^2$ estimates (see e.g.\ 
[Dem82] for the required version). This is possible for $m\ge m_0$ 
thanks to the 
hypothesis that $\Theta_{L,h}\ge \varepsilon\omega>0$, even if $X$ 
is non K\"ahler ($X$ is in any event a Moishezon variety from our
assumptions). The bound $m_0$ depends only on $\varepsilon$ and the geometry
of a finite covering of $X$ by compact sets $K_j\subset U_j$, where
$U_j$ are coordinate balls (say); it is independent of the point $a$ and 
even of the metric~$h$. It follows that $g(a)=0$ and therefore
$\sigma=\theta f-g$ is a holomorphic section of $L^{\otimes m}$ such that
$$
\int_X|\sigma|^2_{h^m}dV_\omega=
\int_X|\sigma|^2e^{-2m\varphi}dV_\omega\le C_4
\int_U|f|^2e^{-2m\varphi}dV_\omega\le C_5
|a|^2e^{-2m\varphi(z)},
$$
in particular $\sigma\in \cH_m=H^0(X,L^{\otimes m}\otimes\cI(h^m))$.
We fix $a$ such that the right hand side is~$1$. This gives
the inequality
$$
\psi_m(z)\ge{1\over m}\log|a|=\varphi(z)-{\log C_5\over 2m}
$$
which is the left hand part of statement (i).
\medskip

\noindent
(ii) The first inequality in (i) implies $\nu(\psi_m,z)\le\nu(\varphi,z)$. 
In the opposite direction, we find
$$
\sup_{|x-z|<r}\psi_m(x)\le\sup_{|x-z|<2r}\varphi(x)+
{1\over m}\log{C_2\over r^n}.
$$
Divide by $\log r<0$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a psh function over $B(x,r)$ tends to the
Lelong number at~$x$. Thus we obtain
$$
\nu(\psi_m,x)\ge\nu(\varphi,x)-{n\over m}.
$$
(iii) Again, the first inequality (in (i) immediately yields $h_m\le C_6 h$,
hence $c_K(h_m)\ge c_K(h)$. For the converse inequality, since we
have $c_{\,\cup K_j}(h)=\min_j c_{K_j}(h)$, we can assume without loss 
of generality that $K$ is contained in a trivializing open patch $U$ of~$L$.
Let us take $c<c_K(\psi_m)$. Then, by definition, if 
$V\subset X$ is a sufficiently small open neighborhood of $K$, 
the H\"older inequality
for the conjugate exponents $p=1+mc^{-1}$ and $q=1+m^{-1}c$
implies, thanks to equality ${1\over p}={c\over mq}$,
$$
\leqalignno{
\int_V & e^{-2(m/p)\varphi}dV_\omega
=\int_V\Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2e^{-2m\varphi}\Big)^{1/p}
  \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/mq}dV_\omega\cr
&\le
\left(\int_X\sum_{1\le k\le N(m)}|g_{m,k}|^2e^{-2m\varphi}dV_\omega\right)^{1/p}
\left(\int_V \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/m}dV_\omega
\right)^{1/q}\cr
&= N(m)^{1/p}
\left(\int_V \Big(\sum_{1\le k\le N(m)}|g_{m,k}|^2\Big)^{-c/m}dV_\omega
\right)^{1/q}<+\infty.
\cr}
$$
From this we infer $c_K(h)\ge m/p$, i.e., $c_K(h)^{-1}\le p/m=1/m+c^{-1}$.
As $c<c_K(\psi_m)$ was arbitrary, we get $c_K(h)^{-1}\le 1/m+c_K(h_m)^{-1}$
and the inequalities of (iii) are proved.\qed
\endproof
%
\claim{(3.22) Remark} {\rm The proof would also work, with a few
modifications, when $X$ is a Stein manifold and $L$ is an arbitrary 
holomorphic line bundle.}
\endclaim
%
\claim{(3.23) Corollary} Let $L\to X$ be a holomorphic line bundle
and $T={\ii\over 2\pi}\Theta_{L,h}$ the curvature current of some singular
hermitian metric $h$ on $L$.
\vskip3pt
{\plainitemindent=7mm
\plainitem{\rm (i)} If $L$ is big and $\Theta_{L,h}\ge \varepsilon\omega>0$, 
there exists a sequence of holomorphic sections $h_s\in H^0(X,q_sL)$ with
\hbox{$\lim q_s=+\infty$} such that the $\bQ$-divisors $D_s=\smash{1\over q_s}
\div(h_s)$ satisfy $T=\lim D_s$ in the weak topology and
\hbox{$\,\sup_{x\in X}|\nu(D_s,x)-\nu(T,x)|\to 0$} as $s\to+\infty$.
\plainitem{\rm (ii)} If $L$ is just pseudo-effective and $\Theta_{L,h}\ge 0$, 
for any ample line bundle $A$, there exists a sequence of non zero sections
$h_s\in H^0(X,p_sA+q_sL)$ with \hbox{$p_s,q_s>0$}, $\lim q_s=+\infty$
and $\lim p_s/q_s=0$, such that the divisors $D_s=\smash{1\over
q_s}\div(h_s)$ satisfy $T=\lim D_s$ in the weak topology and
$\,\sup_{x\in X}|\nu(D_s,x)-\nu(T,x)|\to 0$ as \hbox{$s\to+\infty$}.
\vskip0pt}
\endclaim
%
\begproof{Proof.} Part (ii) is a rather straightforward consequence of part (i)
applied to $mL+A$ and $T_m={1\over m}\Theta_{mL+A,h^mh_A}=T+
{1\over m}\Theta_{A,h_A}\to T$  when $m$ tends to infinity.
Therefore, it suffices to prove (i).
\vskip3pt
\noindent
(i) By Theorem (3.21), we can find sections
$g_1,\ldots,g_N\in H^0(X,mL)$ (omitting the index $m$ for simplicity of
notation), such that
$$
T_m={i\over \pi}\ddbar\Big({1\over 2m}\log\sum_{1\le j\le N}|g_j|_h^2\Big)
+\Theta_{L,h}=
{i\over \pi}\ddbar\Big({1\over 2m}\log\sum_{1\le j\le N}|g_j|^2\Big)
$$
converges weakly to $T$ and satisfies $\nu(T,x)-n/m\le\nu(T_m,x)\le\nu(T,x)$.
In fact, since the number $N$ of sections grows at most as $O(m^n)$, 
we can replace $\sum_{1\le j\le N}|g_j|^2$ by $\max_{1\le j\le N}|g_j|^2$,
as the difference of the potentials tends uniformly to $0$ with the
help of the renormalizing constant ${1\over 2m}$. Hence, we can use
instead the approximating currents
$$
\widetilde T_m={i\over \pi}\ddbar u_m,\qquad
u_m={1\over m}\log\max_{1\le j\le N}|g_j|.
$$
Now, as $L$ is big, we can write
$k_0L=A+D$ where $A$ is an ample divisor and $D$ is an effective divisor,
for some $k_0>0$. By enlarging $k_0$, we can even assume that $A$ is very
ample. Let $\sigma_D$ be the canonical section of $D$ and let
$h_1,\ldots,h_N$ be sections of $H^0(X,A)$. We get a section of
$H^0(X,(m\ell+k_0)L)$ by considering
$$
u_{\ell,m}=(g_1^\ell h_1+\ldots+g_N^\ell h_N)\sigma_D
$$
By enlarging $N$ if necessary and putting e.g.\ $g_j=g_N$ for $j>N$, 
we can assume
that the sections $h_j$ generate all $1$-jets of sections of $A$ at every
point (actually, one can always achieve this with $n+1$ sections only,
so this is not really a big demand). Then, for almost every 
$N$-tuple $(h_1,\ldots,h_N)$, Lemma~3.24 below
and the weak continuity of $\partial\dbar$ imply that
$$
\Delta_{\ell,m}={1\over \ell m}{i\over\pi}\partial\dbar\log|u_{\ell ,m}|=
{1\over \ell m}\,{\rm div}(u_{\ell ,m})
$$
converges weakly to $\widetilde T_m={i\over\pi}\ddbar u_m$ as $\ell$
tends to $+\infty$, and that
$$
\nu(T_m,x)\le\nu\Big({1\over \ell m}\Delta_{\ell ,m},x\Big)\le\nu(T,x)+
{\mu+1\over \ell m},
$$
where $\mu=\max_{x\in X}\ord_x(\sigma_D)$.
This, together with the first step, implies the proposition for some
subsequence $D_s=\Delta_{\ell(s),s}$, $\ell(s)\gg s\gg 1$. We even obtain 
the more explicit inequality
$$
\nu(T,x)-{n\over m}\le\nu\Big({1\over \ell m}\Delta_{\ell,m},x\Big)\le
\nu(T,x)+{\mu+1\over\ell m}.\qquad\qed
$$
\endproof
%
\claim{(3.24) Lemma} Let $\Omega$ be an open subset in $\bC^n$ and let
$g_1,\ldots,g_N\in H^0(\Omega,\cO_\Omega)$ be non zero functions. 
Let $S\subset H^0(\Omega,\cO_\Omega)$ be a finite dimensional subspace
whose elements generate all $1$-jets at any point of $\Omega$. 
Finally, set $u=\log\max_j|g_j|$ and
$$u_\ell =g_1^\ell h_1+\ldots+g_N^\ell h_N,~~~~h_j\in S\ssm\{0\}.$$
Then for all $(h_1,\ldots,h_N)$ in $(S\ssm\{0\})^N$ except a set of measure~$0$,
the sequence ${1\over \ell }\log|u_\ell |$ converges to $u$ in 
$L^1_{\rm loc}(\Omega)$ and
$$\nu(u,x)\le\nu\Big({1\over \ell }\log|u_\ell |\Big)\le\nu(u,x)+
{1\over \ell },~~~~\forall x\in X,~~\forall \ell \ge 1.$$
\endclaim
%
\begproof{Proof.} The argument is a fairly standard Whitney type argument based
on a count of parameters and their dimensions. The details will be omitted.\qed
\endproof
%
\claim{(3.25) Exercise} {\rm When $L$ is ample and 
$h$ is a smooth metric with $T={i\over2\pi}\Theta_{L,h}>0$, show that 
the approximating divisors can be taken smooth (and thus irreducible
if $X$ is connected).\newline
{\emph Hint.} In the above proof of Corollary (3.23), the sections $g_j$ 
have no common zeroes and one can take $\sigma_D=1$. Moreover, a smooth
divisor $\Delta$ in an ample linear system is always connected, otherwise
two disjoint parts $\Delta'$, $\Delta''$ would be big and nef and
$\Delta'\cdot\Delta''=0$ would contradict the Hovanskii-Teissier
inequality when $X$ is connected.}
\endclaim
%
\claim{(3.26) Corollary} On a projective manifold $X$, effective 
$\bQ$-divisors are dense in the weak topology in the cone
$P^{1,1}_{\NS}(X)$ of closed positive $(1,1)$-currents $T$ whose cohomology
class $\{T\}$ belongs to the Neron-Severi space $\NS_\bR(X)$.
\endclaim
%
\begproof{Proof.}We may add $\varepsilon$ times a K\"ahler metric $\omega$ so
as to get $T+\varepsilon\omega>0$, and then perturb by a small
combination $\sum\delta_j\alpha_j$ of classes ${\alpha_j}$ in a
$\bZ$-basis of $\NS(X)$ so that $\Theta=T+\varepsilon\omega+\sum
\delta_j\alpha_j\ge{\varepsilon\over 2}\omega$ and $\{\Theta\}\in H^2(X,\bQ)$.
Then $\Theta$ can be approximated by $\bQ$-divisors by Corollary (3.23),
and the conclusion follows.\qed
\endproof
%
\claim{(3.27) Comments} {\rm We can rephrase the above results by
saying that the cone of closed positive currents $P^{1,1}_{\NS}(X)$ is a
completion of the cone of effective $\bQ$-divisors. A considerable
advantage of using currents is that the cone of currents is locally
compact in the weak topology, namely the section of the cone
consisting of currents $T$ of mass $\int_XT\wedge\omega^{n-1}=1$ is
compact. This provides a very strong tool for the study of the
asymptotic behaviour of linear systems, as required for instance in
the Minimal Model Program of Kawamata-Mori-Shokurov.  One should be
aware, however, that the cone of currents is really huge and contains
objects which are very far from being algebraic in any reasonable sense.
This occurs very frequently in the realm of complex dynamics. For instance,
if $P_m(z,c)$ denotes the $m$-th iterate of the quadratic polynomial
$z\mapsto z^2+c$, then $P_m(z,c)$ defines a polynomial of degree $2^m$
on $\bC^2$, and the sequence of $\bQ$-divisors $D_m={1\over
m}{i\over\pi} \ddbar\log|P_m(z,c)|$ which have mass $1$ on
$\bC^2\subset\bP_\bC^2$ can be shown to converge to a closed positive
current $T$ of mass $1$ on~$\bP_\bC^2$. The support of this current
$T$ is extremely complicated~: its slices $c=c_0$ are the Julia sets
$J_c$ of the quadratic polynomial $z\mapsto z^2+c$, and the slice
$z=0$ is the famous Mandelbrot set $M$.  Therefore, in general, limits
of divisors in asymptotic linear systems may exhibit a fractal
behavior.}
\endclaim
%
\subsection*{\csect.D. Singularity exponents and log canonical thresholds}
The goal of this section to relate ``log canonical thresholds'' with
the $\alpha$ invariant introduced by G.~Tian [Tia87] for the study of
the existence of K\"ahler-Einstein metrics. The approximation
technique of closed positive $(1,1)$-currents introduced above can be
used to show that the $\alpha$ invariant actually coincides with the
log canonical threshold (see also [DK01], [JK01], [BGK05], [Dem08]). 

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

\claim{(3.28) Definition} Assume that $X$ is a compact manifold and that
$L$ is a pseudo-effective line bundle, i.e.\ $L$ admits a singular
hermitian metric $h_0$ with $\Theta_{L,h_0}\ge 0$. If $K$ is a compact 
subset of $X$, we put
$$
\alpha_K(L)=\inf_{\{h,\,\Theta_{L,h}\ge 0\}}c_K(h)
$$
where $h$ runs over all singular hermitian metrics on $L$ such that
$\Theta_{L,h}\ge 0$.
\endclaim
%
In algebraic geometry, it is more usual to look instead at linear systems
defined by a family of linearly independent sections 
$\sigma_0,\sigma_1,\ldots\sigma_N\in 
H^0(X,L^{\otimes m})$. We denote by $\Sigma$ the vector subspace generated by
these sections and by 
$$
|\Sigma|:=P(\Sigma)\subset|mL|:=P(H^0(X,L^{\otimes m}))
$$
the corresponding linear system. Such an $(N+1)$-tuple of sections 
$\sigma=(\sigma_j)_{0\le j\le N}$ defines a
singular hermitian metric $h$ on $L$ by putting in any trivialization
$$
\vert \xi\vert^2_h
={|\xi|^2\over\big(\sum_j|\sigma_j(z)|^2\big)^{1/m}}
={|\xi|^2\over|\sigma(z)|^{2/m}},\qquad \xi\in L_z,
$$
hence $h(z)=|\sigma(z)|^{-2/m}$ with
$\varphi(z)={1\over m}\log|\sigma(z)|={1\over 2m}\log\sum_j|\sigma_j(z)|^2$
as the associated weight function. Therefore, we are interested in the
number $c_K(|\sigma|^{-2/m})$. In the case of a single section 
$\sigma_0$ (corresponding to a one-point linear system), this is the 
same as the log canonical threshold $\lct_K(X,{1\over m}D)=c_K({1\over m}D)$ 
of the associated divisor $D$, in the notation
of~Section~1 of [CS08]. We will also use the formal notation
$c_K({1\over m}|\Sigma|)$ in the case of a higher dimensional
linear system $|\Sigma|\subset|mL|$. The main result of this section is

\claim{(3.29) Theorem} Let $L$ be a big line bundle on a compact complex
manifold $X$. Then for every compact set $K$ in $X$ we have
$$
\alpha_K(L)=\inf_{\{h,\,\Theta_{L,h}\ge 0\}} c_K(h) =\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}
c_K\Big({1\over m}D\Big).
$$
\endclaim
%
\begproof{Proof.}Observe that the inequality 
$$
\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}c_K\Big({1\over m}D\Big)
\ge \inf_{\{h,\,\Theta_{L,h}\ge 0\}} c_K(h)
$$
is trivial, since any divisor $D\in|mL|$ gives rise to a singular
hermitian metric~$h$.

The converse inequality will follow from the
approximation techniques discussed above. Given a big line bundle $L$ 
on~$X$, there exists a modification
$\mu:\widetilde X\to X$ of $X$ such that $\widetilde X$ is projective and 
$\mu^*L=\cO(A+E)$ where $A$ is an ample divisor and $E$ an effective
divisor with rational coefficients. By pushing forward by $\mu$ a smooth 
metric $h_A$ with positive curvature on $A$, we get a singular hermitian 
metric $h_1$ on $L$ such that 
$\Theta_{L,h_1}\ge\mu_*\Theta_{A,h_A}\ge \varepsilon\omega$ on~$X$. Then
for any $\delta>0$ and any singular hermitian metric $h$ on $L$ with
$\Theta_{L,h}\ge 0$, the interpolated metric 
$h_\delta=h_1^\delta h^{1-\delta}$ satisfies
$\Theta_{L,h_\delta}\ge\delta\varepsilon\omega$. Since $h_1$ is bounded
away from $0$, it follows that $c_K(h)\ge (1-\delta)c_K(h_\delta)$
by monotonicity. By theorem (3.21, iii) applied to $h_\delta$, we infer
$$
c_K(h_\delta)=\lim_{m\to+\infty} c_K(h_{\delta,m}),
$$
and we also have
$$
c_K(h_{\delta,m})\ge c_K\Big({1\over m}D_{\delta,m}\Big)
$$
for any divisor $D_{\delta,m}$ associated with a section $\sigma\in
H^0(X,L^{\otimes m}\otimes\cI(h_\delta^m))$, since the metric $h_{\delta,m}$
is given by $h_{\delta,m}=(\sum_k|g_{m,k}|^2)^{-1/m}$ for an orthornormal basis
of such sections. This clearly implies
$$
c_K(h)\ge\liminf_{\delta\to 0}\;\liminf_{m\to+\infty}\;
c_K\Big({1\over m}D_{\delta,m}\Big)\ge
\inf_{m\in\bZ_{>0}}\inf_{D\in|mL|}
c_K\Big({1\over m}D\Big).\eqno\square
$$
\endproof
%
In the applications, it is frequent to have a finite or compact group 
$G$ of automorphisms of $X$ and to look at $G$-invariant objects, namely
$G$-equivariant metrics on $G$-equivariant line bundles~$L\;$; in the case
of a reductive algebraic group $G$ we simply consider a compact real form
$G^\bR$ instead of $G$ itself.

One then gets an $\alpha$ invariant $\alpha_{K,G}(L)$ by
looking only at $G$-equivariant metrics in
Definition~(3.28). All contructions made are then
$G$-equivariant, especially $\cH_m\subset |mL|$ is a $G$-invariant
linear system. For every $G$-invariant compact set $K$ in $X$, we thus
infer
$$
\alpha_{K,G}(L)
:=\inf_{\{h\;G{\mathrm{-equiv.}},\,\Theta_{L,h}\ge 0\}}c_K(h)
=\inf_{m\in\bZ_{>0}}~~\inf_{|\Sigma|\subset|mL|,~\Sigma^G=\Sigma}~~
c_K\Big({1\over m}|\Sigma|\Big).
\leqno(3.30)
$$
When $G$ is a finite group, one can pick for $m$ large enough a
$G$-invariant divisor $D_{\delta,m}$ associated with a
$G$-invariant section $\sigma$, possibly after multiplying $m$ by
the order of $G$. One then gets the slightly simpler equality
$$
\alpha_{K,G}(L)
:=\inf_{\{h\;G{\mathrm{-equiv.}},\,\Theta_{L,h}\ge 0\}}c_K(h)
=\inf_{m\in\bZ_{>0}}~~\inf_{D\in|mL|^G}~~c_K\Big({1\over m}D\Big).
\leqno(3.31)
$$
In a similar manner, one can work on an orbifold $X$ rather than on a non
singular variety. The $L^2$ techniques work in this setting with almost
no change ($L^2$ estimates are essentially insensitive  to
singularities,  since one can just use an orbifold metric on
the open set of regular points).

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

\claim{(3.32) Theorem} Let $X$ be a Fano manifold, i.e.\ a projective
manifold with $-K_X$ ample. Assume that $X$ admits a compact group
of automorphisms $G$ such that $\alpha_{X,G}(K_X)>n/(n+1)$. Then
$X$ possesses a $G$-invariant K\"ahler-Einstein metric.
\endclaim
%
We will not give here the details  of the proof, which rely on very
delicate $C^k$-estimates (successively for $k=0,1,2,\ldots$) for 
the Monge-Amp\`ere 
operator. In fine, the required estimates can be shown to depend only on 
the boundedness of the integral
$\int_X e^{-2\gamma\varphi}$ for a suitable constant 
$\gamma\in{}]{n\over n+1},1]$, where $\varphi$ is the potential of 
the K\"ahler metric $\omega\in c_1(X)$ (also viewed as the weight 
of a hermitian metric on $K_X$). Now, one can restrict the estimate
to $G$-invariant weights $\varphi$, and this translates into the
sufficient condition (3.32). The approach explained in [DK01] 
simplifies the analysis developped in earlier works by proving first
a general semi-continuity theorem which implies the desired a priori
bound under the assumption of Theorem~3.32. The semi-continuity
theorem states as

\claim{(3.33) Theorem {\rm([DK01])}} Let $K$ be a compact set in a
complex manifold $X$. Then the map $\varphi\mapsto c_K(\varphi)^{-1}$
is upper semi-continuous with respect to the weak $({}=L^1_\loc$)
topology on the space of plurisubharmonic functions. Moreover, if
$\gamma<c_K(\varphi)$, then
$\int_K|e^{-2\gamma\psi}-e^{-2\gamma\varphi}|$ converges to $0$ when
$\psi$ converges to $\varphi$ in the weak topology.
\endclaim
%
\medskip
{\emph Sketch of proof.} We will content ourselves by explaining the main 
points. It is convenient to observe (by a quite easy integration argument
suggested to us by J.~McNeal) 
that $c_K(\varphi)$ can be calculated by estimating the Lebesgue volume
$\mu_U(\{\varphi<\log r\}$ of tubular neighborhoods as $r\to 0$~:
$$
c_K(\varphi)=\sup\big\{c\ge 0\,;\,r^{-2c}\mu_U(\{\varphi<\log r\})~~
\hbox{is bounded as $r\to 0$, for some $U\supset K$}\big\}.
\leqno(3.34)
$$
The first step is the following important monotonicity result,
which is a straightforward consequence of the $L^2$ extension theorem.

\claim{(3.35) Proposition} Let $\varphi$ be a quasi-psh function on
a complex manifold~$X$, and let $Y\subset X$ be a complex submanifold such
that $\varphi_{|Y}\not\equiv-\infty$ on every connected component of~$Y$.
Then, if $K$ is a compact subset of $Y$, we have
$$
c_K(\varphi_{|Y})\le c_K(\varphi).
$$
$($Here, of course, $c_K(\varphi)$ is computed on $X$, i.e., by means of
neighborhoods of $K$ in~$X)$.
\endclaim
%
\noindent We need only proving monotonicity for $c_{z_0}(\varphi_{|Y})$ when
$z_0$ is a point of $Y$. This is done by just extending the holomorphic
function $f(z)=1$ on $B(z_0,r)\cap Y$ with respect to the weight
$e^{-2\gamma\varphi}$ whenever $\gamma<c_{z_0}(\varphi_{|Y})$.

\claim{(3.36) Proposition} Let $X$, $Y$ be complex manifolds of 
respective dimensions $n$,~$m$, let $\cI\subset\cO_X$, $\cJ\subset\cO_Y$
be coherent ideals, and let $K\subset X$, $L\subset Y$ be compact
sets. Put $\cI\oplus\cJ:=\pr_1^\star\cI + \pr_2^\star\cJ\subset
\cO_{X\times Y}$. Then
$$
c_{K\times L}(\cI\oplus\cJ)=c_K(\cI)+c_L(\cJ).
$$
\endclaim
%
\begproof{Proof.} It is enough to show that
$c_{(x,y)}(\cI\oplus\cJ)=c_x(\cI)+c_y(\cJ)$ at every point $(x,y)\in
X\times Y$. Without loss of generality, we may assume that
\hbox{$X\subset\bC^n$}, \hbox{$Y\subset\bC^m$} are open sets and
$(x,y)=(0,0)$. Let $g=(g_1,\,\ldots\,,g_p)$, resp.\
$h=(h_1,\,\ldots\,,h_q)$, be systems of generators of $\cI$ (resp.\ $\cJ$)
on a neighborhood of~$0$. Set
$$
\varphi=\log\sum|g_j|^2,\qquad\psi=\log\sum|h_k|^2.
$$
Then $\cI\oplus\cJ$ is generated by the $p+q$-tuple of functions
$$
g\oplus h=(g_1(x),\,\ldots\,g_p(x),h_1(y),\,\ldots\,,h_q(y))
$$
and the corresponding psh function $\Phi(x,y)=\log\big(\sum|g_j(x)|^2+
\sum|h_k(y)|^2\big)$ has the same behavior along the poles as
$\Phi'(x,y)=\max(\varphi(x),\psi(y))$ (up to a term $O(1)\le\log 2$).
Now, for sufficiently small neighborhoods $U$, $V$ of $0$, we have
$$
\mu_{U\times V}\big(\big\{\max(\varphi(x),\psi(y))<\log r\big\}\big)
=\mu_U\big(\{\varphi<\log r\}\times\mu_V(\{\psi<\log r\}\big),
$$
and one can derive from this that
$$
C_1r^{2(c+c')}\le
\mu_{U\times V}\big(\big\{\max(\varphi(x),\psi(y))<\log r\big\}\big)
\le C_2r^{2(c+c')}\,|\log r|^{n-1+m-1}
$$
with $c=c_0(\varphi)=c_0(\cI)$ and $c'=c_0(\psi)=c_0(\cJ)$. We infer 
$$
c_{(0,0)}(\cI\oplus\cJ)=c+c'=c_0(\cI)+c_0(\cJ).
\eqno\square
$$
\endproof

\claim{(3.37) Proposition} Let $f$, $g$ be holomorphic on a complex 
manifold~$X$. Then, for every $x\in X$,
$$
c_x(f+g)\leq c_x(f)+c_x(g).
$$
More generally, if $\cI$ and $\cJ$ are coherent ideals, then
$$
c_x(\cI+\cJ)\leq c_x(\cI)+c_x(\cJ).
$$
\endclaim
%
\begproof{Proof.} Let $\Delta$ be the diagonal in $X\times X$. Then $\cI+\cJ$
can be seen as the restriction of $\cI\oplus\cJ$ to~$\Delta$. Hence
Prop.~3.35 and 3.36 combined imply
$$
c_x(\cI+\cJ)=c_{(x,x)}((\cI\oplus\cJ)_{|\Delta})\le
c_{(x,x)}(\cI\oplus\cJ)=c_x(\cI)+c_x(\cJ).
$$
Since $(f+g)\subset(f)+(g)$, we get
$$
c_x(f+g)\le c_x((f)+(g))\le c_x(f)+c_x(g).\eqno\square
$$
\endproof

\noindent
Now we can explain in rough terms the strategy of proof of Theorem~3.33. 
We start by approximating psh singularities with analytic singularities,
using theorem 3.21. By the argument of Corollary~3.23, we can even
reduce ourselves to the case of invertible ideals $(f)$ near $z_0=0$,
and look at what happens when we have a uniformly convergent sequence
$f_\nu\to f$. In this case, we use the Taylor expansion of $f$ at $0$ to
write $f=p_N+s_N$ where $p_N$ is a polynomial of degree $N$ and 
$s_N(z)=O(|z|^{N+1})$. Clearly $c_0(s_N)\le n/(N+1)$, and from this
we infer $|c_0(f)-c_0(P_N)|\le n/(N+1)$ by 3.37. Similarly, we get 
the uniform estimate $|c_0(f_\nu)-c_0(P_{\nu,N})|\le n/(N+1)$
for all indices~$\nu$. This means that the proof of the semi-continuity 
theorem is reduced to handling the situation of a finite dimensional space
of polynomials. This case is well-known -- one can apply 
Hironaka's desingularization theorem, in a relative version involving
the coefficients of our polynomials as parameters. The conclusion is
obtained by putting together carefully all required uniform estimates
(which involve a lot of $L^2$ estimates).\qed

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

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

\claim{(4.1) Restriction formula} Let $\varphi$ be a
plurisubharmonic function on a complex mani\-fold $X$, and let $Y\subset
X$ be a submanifold. Then
$$\cI(\varphi_{|Y})\subset\cI(\varphi)_{|Y}.$$
\endclaim
%
Thus, in some sense, the singularities of $\varphi$ can only get worse
if we restrict to a submanifold (if the restriction of $\varphi$ to some
connected component of $Y$ is identically~$-\infty$, we agree that the 
corresponding multiplier ideal sheaf is zero). The proof is straightforward
and just amounts to extending locally a germ of function $f$ on $Y$ near 
a point $y_0\in Y$ to a function $\tilde f$ on a small Stein neighborhood 
of $y_0$ in $X$, which is possible by the Ohsawa-Takegoshi extension theorem. 
As a direct consequence, we get:

\claim{(4.2) Subadditivity Theorem}
\plainitem{\rm(i)} Let $X_1$, $X_2$  be complex manifolds,
$\pi_i:X_1\times X_2\to X_i$, $i=1,2$ the projections, and
let $\varphi_i$ be a plurisubharmonic function on~$X_i$.
Then
$$
\cI(\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2) = \pi_1^\star
\cI(\varphi_1) \cdot  \pi_2^\star\cI(\varphi_2).
$$
\plainitem{\rm(ii)} Let $X$ be a complex manifold and let
$\varphi$, $\psi$ be plurisubharmonic functions on $X$.
Then
$$
\cI(\varphi + \psi) \subset \cI(\varphi) \cdot \cI(\psi)
$$
\endclaim
%
\begproof{Proof.} (i) Let us fix two relatively
compact Stein open subsets $U_1\subset X_1$, 
$U_2\subset X_2$. Then $\cH^2(U_1\times U_2,
\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2, \pi_1^\star
dV_1\otimes \pi_2^\star dV_2)$ is the Hilbert tensor
product of $\cH^2(U_1,\varphi_1,dV_1)$ and
$\cH^2(U_2,\varphi_2,dV_2)$, and admits $(f'_k\boxtimes
f''_l)$ as a Hilbert  basis, where $(f'_k)$ and
$(f''_l)$ are respective Hilbert bases. Since
$\cI(\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2)_{|U_1\times
U_2}$ is generated  as an $\cO_{U_1\times U_2}$ module
by the $(f'_k\boxtimes f''_l)$ (from the proof of the coherence
of multiplier ideals, see [Var09, section 8.4.1]), we  conclude that (i)
holds true.
\medskip

\noindent
(ii) We apply (i) to $X_1=X_2=X$ and the restriction
formula to $Y={}$diagonal of $X\times X$. Then
$$
\eqalign{
\cI(\varphi+\psi)
&=\cI\big((\varphi\circ \pi_1 + \psi\circ \pi_2)_{|Y}\big)
\subset \cI\big(\varphi\circ \pi_1 + \psi\circ \pi_2\big)_{|Y}\cr
&=\Big(\pi_1^\star\cI(\varphi)\otimes \pi_2^\star\cI(\psi)\Big)_{|Y}
=\cI(\varphi)\cdot\cI(\psi).\cr}
$$
\endproof
%
\claim{(4.3) Proposition} Let $f:X\to Y$ be an arbirary holomorphic 
map, and let $\varphi$ be a plurisubharmonic function on~$Y$. Then
$\cI(\varphi\circ f)\subset f^\star\cI(\varphi)$.
\endclaim
%
\begproof{Proof.} Let 
$$
\Gamma_f=\{(x,f(x)\,;\,x\in X\}\subset X\times Y
$$ 
be the graph of $f$, and let $\pi_X:X\times Y\to X$, $\pi_Y:X\times
Y\to Y$ be the natural projections. Then we can view $\varphi\circ f$
as the restriction of $\varphi\circ\pi_Y$ to $\Gamma_f$, as $\pi_X$ is
a biholomorphism from $\Gamma_f$ to $X$. Hence the restriction formula
implies
$$
\cI(\varphi\circ f)=\cI\big((\varphi\circ\pi_Y)_{|\Gamma_f}\big)
\subset\cI(\varphi\circ\pi_Y)_{|\Gamma_f}
=\big(\pi_Y^\star\cI(\varphi)\big)_{|\Gamma_f}=f^\star\cI(\varphi).
\eqno\square
$$
\endproof
%
As an application of subadditivity, we now reprove a result of Fujita
[Fuj94], relating the growth of sections of multiples of a line bundle
to the Chern numbers of its ``largest nef part''. Fujita's original
proof is by contradiction, using the Hodge index theorem and
intersection inequalities. 

Let $X$ be a projective $n$-dimensional algebraic variety and $L$
a line bundle over~$X$. We define the {\emph volume} of $L$ to be
$$
\Vol(L)=\limsup_{k\to+\infty}{n!\over k^n}h^0(X,kL)\in[0,+\infty[.
$$
In view of Definition 1.12 and of the Serre-Siegel Lemma 1.13, the 
line bundle is {\emph big} if and only if $\Vol(L)>0$. If $L$ is ample, 
we have $h^q(X,kL)=0$ for $q\ge 1$ and $k\gg 1$ by the Kodaira-Serre
vanishing theorem, hence 
$$
h^0(X,kL)\sim\chi(X,kL)\sim{L^n\over n!}k^n
$$ 
by the Riemann-Roch formula.  Thus $\Vol(L)=L^n\,({}=c_1(L)^n)$ if $L$ is
ample. This is still true if $L$ is nef (numerically effective), i.e.\
if $L\cdot C\ge 0$ for every effective curve~$C$. In fact, one can show 
that $h^q(X,kL)=O(k^{n-q})$ in that case. The following
well-known proposition characterizes big line bundles.

\claim{(4.4) Proposition} The line bundle $L$ is big if and only if there
is a multiple $m_0L$ such that $m_0L=E+A$, where $E$ is an effective divisor
and $A$ an ample divisor.
\endclaim
%
\begproof{Proof.} If the condition is satisfied, the decomposition 
$km_0L=kE+kA$ gives
rise to an injection $H^0(X,kA)\hookrightarrow H^0(X,km_0L)$, thus
$\Vol(L)\ge m_0^{-n}\Vol(A)>0$. Conversely, assume that $L$ is big, and take 
$A$ to be a very ample nonsingular divisor in $X$. The exact sequence
$$
0\longrightarrow \cO_X(kL-A)\longrightarrow \cO_X(kL)\longrightarrow \cO_A(kL_{|A})\longrightarrow 0
$$
gives rise to a cohomology exact sequence
$$
0\to H^0(X,kL-A) \longrightarrow H^0(X,kL) \longrightarrow H^0(A,kL_{|A}), 
$$
and $h^0(A,kL_{|A})=O(k^{n-1})$ since $\dim A=n-1$. Now, the assumption that
$L$ is big implies that $h^0(X,kL)> ck^n$ for infinitely many $k$, hence
$H^0(X,m_0L-A)\ne 0$ for some large integer $m_0$. If $E$ is the divisor 
of a section in $H^0(X,m_0L-A)$, we find $m_0L-A=E$, as required.\qed
\endproof
%
\claim{(4.5) Lemma} Let $G$ be an arbitrary line bundle. For every
$\varepsilon>0$, there exists a positive integer $m$ and a sequence
$\ell_\nu\uparrow+\infty$ such that
$$
h^0\big(X,\ell_\nu(mL-G)\big)\ge {\ell_\nu^mm^n\over n!}
\big(\Vol(L)-\varepsilon\big),
$$
in other words, $\Vol(mL-G)\ge m^n(\Vol(L)-\varepsilon)$ for $m$ large enough.
\endclaim
%
\begproof{Proof.} Clearly, $\Vol(mL-G)\ge \Vol(mL-(G+E))$ for every effective divisor
$E$. We can take $E$ so large that $G+E$ is very ample, and we are thus 
reduced to the case where $G$ is very ample by replacing $G$ with $G+E$.
By definition of $\Vol(L)$, there exists a sequence $k_\nu\uparrow+\infty$
such that
$$
h^0(X,k_\nu L)\ge {k_\nu^n\over n!}\Big(\Vol(L)-{\varepsilon\over 2}\Big).
$$
We take $m\gg 1$ (to be precisely chosen later), and $\ell_\nu=
\big[{k_\nu\over m}\big]$, so that $k_\nu=\ell_\nu m+r_\nu$,
$0\le r_\nu<m$. Then
$$
\ell_\nu(mL-G)=k_\nu L-(r_\nu L+\ell_\nu G).
$$
Fix a constant $a\in\bN$ such that $aG-L$ is an effective divisor. Then
$r_\nu L\le maG$ (with respect to the cone of effective divisors), hence
$$
h^0\big(X,\ell_\nu(mL-G)\big)\ge
h^0\big(X,k_\nu L-(\ell_\nu+am)G\big).
$$
We select a smooth divisor $D$ in the very ample linear system
$|G|$. By looking at global sections associated with the exact
sequences of sheaves
$$
0\to\cO(-(j+1)D)\otimes\cO(k_\nu L)\to\cO(-jD)\otimes\cO(k_\nu L)
\to\cO_D(k_\nu L-jD)\to 0,
$$
$0\le j<s$, we infer inductively that
$$
\eqalign{
h^0(X,k_\nu L-sD)
&\ge h^0(X,k_\nu L)-\sum_{0\le j< s}h^0\big(D,\cO_D(k_\nu L-jD)\big)\cr
&\ge h^0(X,k_\nu L)- s\,h^0\big(D,k_\nu L_{|D})\cr
&\ge {k_\nu^n\over n!}\Big(\Vol(L)-{\varepsilon\over 2}\Big) -s\,Ck_\nu^{n-1}
\cr}
$$
where $C$ depends only on $L$ and~$G$. Hence, by putting $s=\ell_\nu+am$,
we get
$$
\eqalign{
h^0\big(X,\ell_\nu(mL-G)\big) 
&\ge {k_\nu^n\over n!}\Big(\Vol(L)-{\varepsilon\over 2}\Big) 
-C(\ell_\nu+am)k_\nu^{n-1}\cr
&\ge{\ell_\nu^nm^n\over n!}\Big(\Vol(L)-{\varepsilon\over 2}\Big)
-C(\ell_\nu+am)(\ell_\nu+1)^{n-1}m^{n-1}\cr}
$$
and the desired conclusion follows by taking $\ell_\nu\gg m\gg 1$.\qed
\endproof
%
We are now ready to prove Fujita's decomposition theorem, as reproved
in [DEL00].

\claim{(4.6) Theorem (Fujita)} Let $L$ be a big line bundle. Then for 
every $\varepsilon>0$, there exists a modification $\mu:\widetilde X\to X$ and
a decomposition $\mu^\star L=E+A$, where $E$ is an effective $\bQ$-divisor
and $A$ an ample $\bQ$-divisor, such that $A^n>\Vol(L)-\varepsilon$.
\endclaim
%
\claim{(4.7) Remark} {\rm Of course, if $\mu^\star L= E+A$ with $E$ 
effective and $A$ nef, we get an injection
$$
H^0(\widetilde X,kA)\hookrightarrow H^0(\widetilde X, kE+kA)=H^0(\widetilde X, k\mu^\star L)=
H^0(X,kL)
$$
for every integer $k$ which is a multiple of the denominator of $E$, hence
$A^n\le \Vol(L)$.}
\endclaim
%
\claim{(4.8) Remark} {\rm Once Theorem 4.6 is proved, the same kind
of argument easily shows that
$$
\Vol(L)=\lim_{k\to +\infty}{n!\over k^n}h^0(X,kL),
$$
because the formula is true for every ample line bundle $A$.}
\endclaim
%
\begproof{Proof of Theorem 4.6.} It is enough to prove the theorem with $A$
being a big and nef divisor. In fact, Proposition 4.4 then shows that we can
write $A=E'+A'$ where $E'$ is an effective $\bQ$-divisor and $A'$ an
ample $\bQ$-divisor, hence
$$
E+A=E+\varepsilon E'+(1-\varepsilon)A+\varepsilon A'
$$
where $A''=(1-\varepsilon)A+\varepsilon A'$ is ample and the intersection
number $A^{\prime\prime n}$ approaches $A^n$ as closely as we want.
Let $G$ be as in Theorem (1.25) (Siu's theorem on uniform global
generation). Lemma 4.5 implies that
$\Vol(mL-G)>m^n(\Vol(L)-\varepsilon)$ for $m$ large. By Theorem~(1.8) on
the existence of analytic Zariski decomposition, there
exists a hermitian metric $h_m$ of weight $\varphi_m$ on $mL-G$ such that
$$
H^0\big(X,\ell(mL-G)\big)=H^0\big(X,\ell(mL-G)\otimes\cI(\ell\varphi_m)\big)
$$
for every $\ell\ge 0$. We take a smooth modification $\mu:\widetilde X\to X$ such
that 
$$
\mu^\star\cI(\varphi_m)=\cO_{\widetilde X}(-E)
$$
is an invertible ideal sheaf in $\cO_{\widetilde X}$. This is possible by
taking the blow-up of $X$ with respect to the ideal $\cI(\varphi_m)$
and by resolving singularities (Hironaka [Hir64]). Theorem~1.25
applied to $L'=mL-G$
implies that $\cO(mL)\otimes\cI(\varphi_m)$ is generated by its
global sections, hence its pull-back $\cO(m\,\mu^\star L-E)$ is 
also generated. This implies
$$
m\,\mu^\star L=E+A
$$ 
where $E$ is an effective divisor and $A$ is a nef (semi-ample)
divisor in~$\widetilde X$. We find
$$
\eqalign{
H^0(\widetilde X,\ell A)
  &=H^0\big(\widetilde X,\ell(m\,\mu^\star L-E)\big)\cr
  &\supset H^0\big(\widetilde X,\mu^\star\big(\cO(\ell mL)\otimes
       \cI(\varphi_m)^\ell\big)\big)\cr
  &\supset H^0\big(\widetilde X,\mu^\star\big(\cO(\ell mL)\otimes
       \cI(\ell\varphi_m)\big)\big),\cr}
$$
thanks to the subadditivity property of multiplier ideals.
Moreover, the direct image 
$\mu_\star\mu^\star\cI(\ell\varphi_m)$ coincides with the integral
closure of $\cI(\ell\varphi_m)$, hence with $\cI(\ell\varphi_m)$,
because a multiplier ideal sheaf is always integrally closed.
From this we infer
$$
\eqalign{
H^0(\widetilde X,\ell A)
  &\supset H^0\big(X,\cO(\ell mL)\otimes \cI(\ell\varphi_m)\big)\cr
  &\supset H^0\big(X,\cO(\ell(mL-G))\otimes \cI(\ell\varphi_m)\big)\cr
  &= H^0\big(X,\cO(\ell(mL-G))\big).\cr}
$$
By Lemma 4.5, we find
$$
h^0(\widetilde X,\ell A)\ge{\ell^n\over n!}m^n\big(\Vol(L)-\varepsilon\big)
$$
for infinitely many $\ell$, therefore $\Vol(A)=A^n\ge m^n(\Vol(L)-\varepsilon)$.
Theorem 4.6 is proved, up to a minor change of notation
$E\mapsto{1\over m}E$, $A\mapsto{1\over m}A$.\qed
\endproof
%
We conclude by using Fujita's theorem to establish a geometric
interpretation of the volume $\Vol(L)$. Suppose as above that $X$ is a
smooth projective variety of dimension $n$, and that $L$ is a big line
bundle on $X$. Given a large integer $k \gg 0$, denote by $B_k
\subset X$ the base-locus of the linear system $| kL |$. The
{\emph moving self-intersection number} $(kL)^{[n]}$ of
$|kL|$ is defined by choosing $n$ general divisors $D_1, \dots , D_n
\in |kL|$ and putting
$$( kL)^{[n]} = \# \Big( D_1 \cap \ldots
\cap D_n \cap ( X - B_k)  \Big).
$$
In other words, we simply count the number of intersection points
away from the base locus of $n$ general divisors in the linear system
$|kL|$. This notion arises for example in Matsusaka's proof of his
``big theorem". We show that the volume $\Vol(L)$ of $L$ 
measures the rate of growth with respect to $k$ of these moving 
self-intersection numbers:

\claim{(4.9) Proposition} One has
$$\Vol(L) = \limsup_{k \to \infty} { (kL)^{[n]} \over k^n}.$$
\endclaim
%
\begproof{Proof.} We start by interpreting 
$(kL)^{[n]}$ geometrically. Let $\mu_k: X_k \longrightarrow X$
be a modification of $|kL|$ such that $\mu_k^\star |kL| = |V_k| +  F_k $, 
where
$$ P_k := \mu_k^\star (kL) - F_k$$  
is generated by sections, and
$H^0(X,\cO_X(kL)) = V_k = H^0(X_k,\cO_{X_k}(P_k))$, so that $B_k =
\mu_k(F_k)$. Then evidently
$(kL)^{[n]}$ counts the number of intersection points
of $n$ general divisors in $P_k$, and consequently
$$(kL)^{[n]} = (P_k)^n .$$  
Since then $P_k$ is big (and nef) for $k\gg 0$, 
we have  $\Vol(P_k)=(P_k)^n$. Also, $\Vol(kL) \ge \Vol(P_k)$ 
since $P_k$ embeds in $\mu_k^\star(k L)$.  Hence
$$
\Vol(kL) \ge  ( kL)^{[n]}\qquad \forall k \gg 0.
$$
On the other hand, an easy argument in the spirit of Lemma (4.5)
shows that $\Vol(kL) = k^n \cdot \Vol(L)$
(cf.\ also [ELN96], Lemma 3.4), and so we conclude that
$$
\Vol(L) \ge { (kL)^{[n]} \over k^n}. \leqno(4.10)
$$
for every $k \gg 0$. 

For the reverse inequality we use Fujita's theorem. Fix
$\varepsilon > 0$, and consider the  decomposition $\mu^\star
L = A + E$ on $\mu : \widetilde X \longrightarrow X$ constructed in
Fujita's theorem. Let $k$ be any positive integer
such that $kA$ is integral and globally generated. By taking a
common resolution we can assume that $X_k$ dominates
$\widetilde X$, and hence we can write 
$$
\mu_k^\star  kL \sim  A_k + E_k
$$
with $A_k$ globally generated and 
$$
(A_k)^n\ge k^n \cdot ( \Vol(L) - \varepsilon ).
$$  
But then $A_k$ embeds in $P_k$ and both $\cO(A_k)$ and $\cO(P_k)$
are globally generated, consequently 
$$
(A_k)^n \le (P_k)^n =  ( kL)^{[n]}.
$$
Therefore
$$
{(kL)^{[n]}\over k^n} \ge  \Vol(L) - \varepsilon. \leqno(4.11)
$$
But (4.11) holds for any sufficiently large and divisible $k$, and in 
view of (4.10) the Proposition follows.\qed
\endproof

\endsectionrunning
%

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

\subsection*{\csect.A. Positive classes in intermediate $(p,p)$ 
bidegrees}
We first discuss some general positivity concepts for cohomology 
classes of type $(p,p)$, although we will not be able to say much about 
these. Recall that we have a Serre duality pairing
$$
H^{p,q}(X,\bC)\times H^{n-p,n-q}(X,\bC)\longrightarrow \bC,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta\in\bC.\leqno(5.1)
$$
In particular, if we restrict to real classes, this yields a
duality pairing
$$
H^{p,p}(X,\bR)\times H^{n-p,n-p}(X,\bR)\longrightarrow \bR,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta \in\bR.\leqno(5.2)
$$
Now, one can define $H^{p,p}_\SP(X,\bR)$ to be the closure of the
cone of classes of $d$-closed strongly positive smooth $(p,p)$-forms
(a $(p,p)$-form in $\Lambda^{p,p}T^*_X$ is by definition strongly positive 
if it is in the convex cone generated by decomposable $(p,p)$
forms $\ii u_1\wedge{\overline u}_1\wedge\ldots\wedge \ii u_p\wedge{\overline u}_p$
where the $u_j$ are $(1,0)$-forms). Clearly, 
$H^{1,1}_\SP(X,\bR)=\overline\cK$ and the cup product defines
a multilinear map
$$
\overline\cK\times \ldots\times \overline\cK\longrightarrow H^{p,p}_\SP(X,\bR)\leqno(5.3)
$$
on the $p$-fold product of the K\"ahler cone and its closure. We
also have $H^{p,p}_\SP(X,\bR)\subset H^{p,p}_{\ge 0}(X,\bR)$ where
$H^{p,p}_{\ge 0}(X,\bR)$ is the cone of classes of $d$-closed
weakly positive currents of type $(p,p)$, and the Serre duality pairing
induces a positive intersection product
$$
H^{p,p}_\SP(X,\bR)\times H^{n-p,n-p}_{\ge 0}(X,\bR)\longrightarrow \bR_+,\qquad
(\alpha,T)\longmapsto\int_X \alpha\wedge T \in\bR_+\leqno(5.4)
$$
(notice that if $\alpha$ is strongly positive and $T\ge 0$, then
$\alpha\wedge T$ is a positive measure).

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

\subsection*{\csect.B. Numerically positive  classes of type (1,1)}

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

\claim{(5.5) Theorem} Let $X$ be a compact K\"ahler manifold. Let $\cP$ 
be the set of real $(1,1)$ cohomology classes $\{\alpha\}$
which are {\rm numerically} {\rm positive} on analytic cycles, i.e.\ such 
that $\int_Y\alpha^p>0$ for every irreducible analytic set $Y$ in~$X$, 
$p=\dim Y$. Then the K\"ahler cone $\cK$ of $X$ is one of the 
connected components of $\cP$.
\endclaim
%
\claim{(5.6) Special case} If $X$ is projective algebraic, then $\cK=\cP$.
\endclaim
%
These results (which are new even in the projective case) can be seen 
as a generalization of the well-known Nakai-Moishezon criterion. Recall
that the Nakai-Moishezon criterion provides a necessary and sufficient 
criterion for a line bundle
to be ample: {\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 $Y\subset X$, $p=\dim Y$.}
\vskip3pt

It turns out that the numerical conditions $\int_Y\alpha^p> 0$ also 
characterize arbitrary transcendental K\"ahler classes when  $X$ is 
projective$\,$: this is precisely the meaning of the special case 5.6.

\claim{(5.7) Example} {\rm The following example shows that the cone $\cP$
need not be connected (and also that the components of $\cP$ need not 
be convex, either). Let us consider for instance a complex torus 
$X=\bC^n/\Lambda$. It is well-known that a generic torus $X$ does not
possess any analytic subset except finite subsets and $X$ itself. 
In that case, the numerical positivity is expressed by the single
condition $\int_X\alpha^n>0$. However, on a torus, $(1,1)$-classes are 
in one-to-one correspondence with constant hermitian forms $\alpha$ on
$\bC^n$. Thus, for $X$ generic, $\cP$ is the set of hermitian forms on 
$\bC^n$ such that $\det(\alpha)>0$, and Theorem 5.5 just expresses the 
elementary result of linear algebra saying that the set $\cK$ of 
positive definite forms is one of the connected components of the open set
$\cP=\{\det(\alpha)>0\}$ of hermitian forms of positive determinant
(the other components, of course, are the sets of forms of signature 
$(p,q)$, $p+q=n$, $q$ even. They are not convex when $p>0$ and $q>0$).}
\endclaim
\vskip3pt

\noindent
{\emph Sketch of proof of Theorems 5.5 and 5.6.} By definition (1.16)~(iv), a 
{\emph K\"ahler current} is a closed positive current $T$ of type $(1,1)$
such that $T\ge\varepsilon\omega$ for some smooth K\"ahler metric $\omega$ 
and $\varepsilon>0$ small enough. The crucial steps of the proof of 
Theorem~5.5 are contained in the following statements.\smallskip

\claim{(5.8) Proposition {\rm (P\u{a}un [Pau98a, 98b])}} Let $X$ be a compact 
complex manifold $($or more generally a compact complex space$)$. Then
\vskip3pt
\plainitem{\rm (a)} The cohomology class of a closed positive 
$(1,1)$-current $\{T\}$ is nef if and only if the restriction 
$\{T\}_{|Z}$ is nef for every irreducible component $Z$ in any of
the Lelong sublevel sets~$E_c(T)$.

\vskip3pt
\plainitem{\rm (b)} The cohomology class of a K\"ahler current $\{T\}$ is 
a K\"ahler class $($i.e.\ the class of a {\rm \rm smooth} K\"ahler form$)$ 
if and only if the restriction $\{T\}_{|Z}$ is a K\"ahler class for 
every irreducible component $Z$ in any of the Lelong sublevel sets~$E_c(T)$. 
\vskip0pt
\endclaim
%
The proof of Proposition 5.8 is not extremely hard if we take for granted the
fact that K\"ahler currents can be approximated by K\"ahler currents
with logarithmic poles, a fact which was first proved in section~3.B
(see also [Dem92]). Thus in (b), we may assume that $T=
\alpha+\ii \ddbar\varphi$ is a current with analytic singularities,
where $\varphi$ is a quasi-psh function with logarithmic poles on some
analytic set $Z$, and $\varphi$ smooth on $X\ssm Z$. Now, we proceed by 
an induction on dimension (to do this, we have to consider analytic spaces
rather than with complex manifolds, but it turns out that this makes no
difference for the proof). Hence, by the induction hypothesis, there exists
a smooth potential $\psi$ on $Z$ such that 
$\alpha_{|Z}+\ii \ddbar\psi>0$ along $Z$. It is well known that
one can then find a potential $\widetilde\psi$ on $X$ such that
$\alpha+\ii \ddbar\widetilde\psi>0$ in a neighborhood $V$ of $Z$ (but possibly
non positive elsewhere). Essentially, it is enough to take an arbitrary
extension of $\psi$ to $X$ and to add a large multiple of the square of 
the distance to $Z$, at
least near smooth points; otherwise, we stratify $Z$ by its successive
singularity loci, and proceed again by induction on the dimension of these
loci. Finally, we use a
a standard gluing procedure~: the current $T=\alpha+
\ii \max_\varepsilon(\varphi,\widetilde\psi-C)$, $C\gg 1$, will be equal
to $\alpha+\ii \ddbar\varphi>0$ on $X\ssm V$, and to a smooth
K\"ahler form on~$V$.\qed
\medskip

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

\claim{(5.9) Theorem {\rm ([DP04])}} Let $X$ be a compact K\"ahler manifold
and let $\{\alpha\}$ be a nef class $($i.e.\ $\{\alpha\}\in\overline\cK)$.
Assume that $\int_X\alpha^n>0$. Then $\{\alpha\}$ contains a K\"ahler
current~$T$, in other words $\{\alpha\}\in\cE^\circ$.
\endclaim
%
\noindent
{\emph Step 1}. The basic argument is to prove that for every irreducible 
analytic set $Y\subset X$ of
codimension $p$, the class $\{\alpha\}^p$ contains a closed positive
$(p,p)$-current $\Theta$ such that $\Theta\ge\delta[Y]$ for some $\delta>0$.
For this, we use in an essentail way the Calabi-Yau theorem [Yau78] on
solutions of Monge-Amp\`ere equations, which yields the following result
as a special case:

\claim{(5.10) Lemma {\rm([Yau78])}} Let $(X,\omega)$ 
be a compact K\"ahler manifold and $n=\dim X$. Then for any smooth
volume form $f>0$ such that $\int_X f = \int_X\omega^n$,
there exist a K\"ahler metric $\widetilde \omega = \omega+i\ddbar\varphi$
in the same K\"ahler class as $\omega$, such that 
$\widetilde \omega^n=f$.\qed
\endclaim
%
\noindent
We exploit this by observing that $\alpha+\varepsilon\omega$ is a
K\"ahler class. Hence we can solve the Monge-Amp\`ere equation
$$
(\alpha+\varepsilon\omega+i\ddbar\varphi_\varepsilon)^n=C_\varepsilon
\omega_\varepsilon^n
\leqno(5.10\,\hbox{a})
$$
where $(\omega_\varepsilon)$ is the family of K\"ahler metrics on $X$
produced by Lemma 3.4~(iii), such that their volume is concentrated
in an $\varepsilon$-tubular neighborhood of $Y$.
$$
C_\varepsilon = {\int_X\alpha_\varepsilon^n\over\int_X\omega_\varepsilon^n}
={\int_X(\alpha+\varepsilon\omega)^n\over\int_X\omega^n}\ge
C_0={\int_X\alpha^n\over\int_X\omega^n}>0.
$$
Let us denote by 
$$
\lambda_1(z)\le\ldots\le\lambda_n(z)
$$
the eigenvalues of $\alpha_\varepsilon(z)$ with respect to
$\omega_\varepsilon(z)$, at every point $z\in X$ (these functions
are continuous with respect to $z$, and of course depend also 
on~$\varepsilon$). The equation (5.10a) is equivalent to the fact that
$$
\lambda_1(z)\ldots\lambda_n(z)=C_\varepsilon\leqno(5.10\,\hbox{b})
$$
is constant, and the most important observation for us is that the
constant $C_\varepsilon$ is bounded away from $0$, thanks to
our assumption $\int_X\alpha^n>0$.

Fix a regular point $x_0\in Y$ and a small neighborhood $U$ (meeting only
the irreducible component of $x_0$ in $Y$). By Lemma 3.4, we have a
uniform lower bound
$$
\int_{U\cap V_\varepsilon}\omega_\varepsilon^p\wedge\omega^{n-p}
\ge\delta_p(U)>0.\leqno(5.10\,\hbox{c})
$$
Now, by looking at the $p$ smallest (resp.\ $(n-p)$ largest) eigenvalues
$\lambda_j$ of $\alpha_\varepsilon$ with respect to $\omega_\varepsilon$, 
we find
$$
\leqalignno{
\alpha_\varepsilon^p&\ge \lambda_1\ldots\lambda_p\,\omega_\varepsilon^p,
&(5.10\,\hbox{d})\cr
\alpha_\varepsilon^{n-p}\wedge\omega_\varepsilon^p
&\ge {1\over n!}\lambda_{p+1}\ldots\lambda_n\,
\omega_\varepsilon^n,&(5.10\,\hbox{e})\cr
}
$$
The last inequality $(5.10\,\hbox{e})$ implies
$$
\int_X\lambda_{p+1}\ldots\lambda_n\,\omega_\varepsilon^n\le
n!\int_X\alpha_\varepsilon^{n-p}\wedge\omega_\varepsilon^p=
n!\int_X(\alpha+\varepsilon\omega)^{n-p}\wedge\omega^p\le M
$$
for some constant $M>0$ (we assume $\varepsilon\le 1$, say). 
In particular, for every $\delta>0$, the subset 
$E_\delta\subset X$ of points $z$ such that 
$\lambda_{p+1}(z)\ldots\lambda_n(z)>M/\delta$ satisfies
$\int_{E_\delta}\omega_\varepsilon^n\le\delta$, hence
$$
\int_{E_\delta}\omega_\varepsilon^p\wedge\omega^{n-p}\le 2^{n-p}
\int_{E_\delta}\omega_\varepsilon^n\le 2^{n-p}\delta.\leqno(5.10\,\hbox{f})
$$
The combination of (5.10$\,$c) and (5.10$\,$f) yields
$$
\int_{(U\cap V_\varepsilon)\ssm E_\delta}\omega_\varepsilon^p\wedge\omega^{n-p}
\ge \delta_p(U)-2^{n-p}\delta.
$$
On the other hand (5.10$\,$b) and (5.10$\,$d) imply
$$
\alpha_\varepsilon^p\ge{C_\varepsilon\over\lambda_{p+1}\ldots\lambda_n}
\omega_\varepsilon^p
\ge{C_\varepsilon\over M/\delta}\omega_\varepsilon^p\qquad
\hbox{on $(U\cap V_\varepsilon)\ssm E_\delta$}.
$$
From this we infer
$$
\int_{U\cap V_\varepsilon}\alpha_\varepsilon^p\wedge\omega^{n-p}\ge
{C_\varepsilon\over M/\delta}
\int_{(U\cap V_\varepsilon)\ssm E_\delta}
\omega_\varepsilon^p\wedge\omega^{n-p}\ge
{C_\varepsilon\over M/\delta}(\delta_p(U)-2^{n-p}\delta)>0
\leqno(5.10\,\hbox{g})
$$
provided that $\delta$ is taken small enough, e.g.\ 
$\delta=2^{-(n-p+1)}\delta_p(U)$. The family of $(p,p)$-forms
$\alpha_\varepsilon^p$ is uniformly bounded in mass since
$$
\int_X\alpha_\varepsilon^p\wedge\omega^{n-p}
=\int_X(\alpha+\varepsilon\omega)^p\wedge\omega^{n-p}\le\hbox{Const}.
$$
Inequality (5.10$\,$g) implies that any weak limit $\Theta$ of 
$(\alpha_\varepsilon^p)$
carries a positive mass on~$U\cap Y$. By Skoda's extension theorem [Sko82], 
${\bf 1}_Y\Theta$ is a closed positive current with support in $Y$, 
hence ${\bf 1}_Y\Theta=\sum c_j[Y_j]$ is a combination of the 
various components $Y_j$ of $Y$ with coefficients $c_j>0$. 
Our construction shows that $\Theta$ belongs to the cohomology class 
$\{\alpha\}^p$. Step 1 of Theorem 5.9 is proved.
\medskip

\noindent
{\emph Step 2.} The second and final step consists in using
a ``diagonal trick'': for this, we apply Step 1 to 
$$
\widetilde X=X\times X,\qquad \widetilde Y=\hbox{diagonal}\,\Delta\subset \widetilde X,\qquad
\widetilde\alpha=\pr_1^*\alpha+\pr_2^*\alpha.
$$
It is then clear that $\widetilde\alpha$ is nef on $\widetilde X$ and that
$$
\int_{\widetilde X}(\widetilde\alpha)^{2n}=
{2n\choose n}\Big(\int_X\alpha^n\Big)^2>0.
$$
It follows by Step 1 that the
class $\smash{\{\widetilde\alpha\}}^n$ contains a K\"ahler current $\Theta$ of 
bidegree $(n,n)$ such that
$\Theta\ge\delta[\Delta]$ for some $\delta>0$. Therefore 
the push-forward
$$
T:=(\pr_1)_*(\Theta\wedge\pr_2^*\omega)
$$
is a positive $(1,1)$-current such that
$$
T\ge\delta(\pr_1)_*([\Delta]\wedge\pr_2^*\omega)=\delta\omega.
$$
It follows that $T$ is a K\"ahler current. On the other hand, $T$
is numerically equivalent to $(\pr_1)_*(\widetilde\alpha^n\wedge\pr_2^*\omega)$, 
which is the form given in coordinates by
$$
x\mapsto\int_{y\in X}\big(\alpha(x)+\alpha(y)\big)^n\wedge \omega(y)=
C\alpha(x)
$$
where $C=n\int_X\alpha(y)^{n-1}\wedge\omega(y)$. Hence $T\equiv C\alpha$,
which implies that $\{\alpha\}$ contains a K\"ahler current. 
Theorem 5.9 is proved.\qed
\medskip

\noindent
{\emph End of Proof of Theorems 5.5 and 5.6}. Clearly the open cone $\cK$ 
is contained in $\cP$, hence in order to show that $\cK$ is one of the
connected components of $\cP$, we need only show that $\cK$ is {\rm closed}
in $\cP$, i.e.\ that $\overline\cK\cap\cP\subset\cK$. Pick a class
$\{\alpha\}\in\overline\cK\cap\cP$. In particular $\{\alpha\}$ is
nef and satisfies $\int_X\alpha^n>0$. By Theorem~5.9 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
5.8~(b), therefore $\{\alpha\}\in\cK$, as desired.\qed
\vskip3pt

The projective case 5.6 is a consequence of the following variant of
Theorem~5.5.

\claim{(5.11) Corollary} Let $X$ be a compact
K\"ahler manifold. A $(1,1)$ cohomology class $\{\alpha\}$ on $X$
is K\"ahler if and only if there exists a 
K\"ahler metric $\omega$ on $X$ such that $\int_Y\alpha^k\wedge\omega^{p-k}>0$
for all irreducible analytic sets $Y$ and all $k=1,2,\ldots,p=\dim Y$.
\endclaim
%
\begproof{Proof.} The assumption clearly implies that
$$
\int_Y(\alpha+t\omega)^p>0
$$
for all $t\in\bR_+$, hence the half-line $\alpha+(\bR_+)\omega$ is 
entirely contained in the cone $\cP$ of numerically positive classes.
Since $\alpha+t_0\omega$ is K\"ahler for $t_0$ large, we conclude that
the half-line in entirely contained in the connected component $\cK$,
and therefore $\alpha\in\cK$.\qed
\endproof
%
In the projective case, we can take $\omega=c_1(H)$ for a given very ample
divisor $H$, and the condition
 $\int_Y\alpha^k\wedge\omega^{p-k}>0$ is equivalent to
$$
\int_{Y\cap H_1\cap\ldots\cap H_{p-k}}\alpha^k>0
$$
for a suitable complete intersection $Y\cap H_1\cap\ldots\cap H_{p-k}$,
$H_j\in|H|$. This shows that algebraic cycles are sufficient to test
the K\"ahler property, and the special case 5.6 follows. On the other 
hand, we can pass to the limit in 5.11 by replacing $\alpha$ 
by $\alpha+\varepsilon\omega$,
and in this way we get also a characterization of nef classes.

\claim{(5.12) Corollary} Let $X$ be a compact
K\"ahler manifold. A $(1,1)$ cohomology class $\{\alpha\}$ on $X$
is nef if and only if there exists a 
K\"ahler metric $\omega$ on $X$ such that $\int_Y\alpha^k\wedge\omega^{p-k}
\ge 0$ for all irreducible analytic sets $Y$ and all $k=1,2,\ldots,p=\dim Y$.
\endclaim
%
By a formal convexity argument, one can derive from 5.11 or 5.12 the following
interesting consequence about the dual of the cone $\cK$. 

\claim{(5.13) Theorem} Let $X$ be a compact K\"ahler manifold. 
\vskip3pt
\plainitem{\rm (a)}
A $(1,1)$ cohomology class $\{\alpha\}$ on $X$ is nef if and only 
for every irreducible analytic set $Y$ in $X$, $p=\dim X$
and every K\"ahler metric $\omega$ on $X$ we have
$\int_Y\alpha\wedge\omega^{p-1}\ge 0$. $($Actually this numerical 
condition is needed only for K\"ahler classes $\{\omega\}$ which
belong to a $2$-dimensional space $\bR\{\alpha\}+\bR\{\omega_0\}$, 
where $\{\omega_0\}$ is a given K\"ahler class$)$.
\vskip3pt
\plainitem{\rm (b)} The dual of the
nef cone $\smash{\overline\cK}$ is the closed convex cone 
in $H^{n-1,n-1}(X,\bR)$ generated by cohomology 
classes of currents of the form $[Y]\wedge\omega^{p-1}$ in 
$H^{n-1,n-1}(X,\bR)$, where $Y$ runs over the 
collection of irreducible analytic subsets of $X$ and $\{\omega\}$ over
the set of K\"ahler classes of~$X$. This dual cone coincides
with $H^{n-1,n-1}_{\ge 0}(X,\bR)$.\vskip0pt
\endclaim
%
\begproof{Proof.} (a) Clearly a nef class $\{\alpha\}$ satisfies the given 
numerical condition. The proof of the converse is more tricky.
First, observe that for every 
integer $p\ge 1$, there exists a polynomial identity of the form
$$
(y-\delta x)^p - (1-\delta)^px^p
= (y-x)\int_0^1 A_p(t,\delta)\big((1-t)x+ty\big)^{p-1}\,dt
\leqno(5.14)
$$
where $A_p(t,\delta)=\sum_{0\le m\le p}a_m(t)\delta^m\in
\bQ[t,\delta]$ is a polynomial of degree${}\le p-1$ in~$t$ (moreover, 
the polynomial $A_p$ is unique under this limitation for the  degree). 
To see this, we observe that $(y-\delta x)^p - (1-\delta)^px^p$
vanishes identically for $x=y$, so it is divisible by $y-x$. By
homogeneity in $(x,y)$, we have an expansion of the form
$$
(y-\delta x)^p - (1-\delta)^px^p = (y-x)
\sum_{0\le\ell\le p-1,\,0\le m\le p} b_{\ell,m}x^\ell y^{p-1-\ell}\delta^m
$$
in the ring $\bZ[x,y,\delta]$. Formula (5.14) is then equivalent to
$$
b_{\ell,m} = \int_0^1 a_m(t){p-1\choose\ell}(1-t)^\ell t^{p-1-\ell}\,dt.
\leqno(5.14')
$$
Since $(U,V)\mapsto\int_0^1U(t)V(t)dt$ is a non degenerate linear pairing
on the space of polynomials of degree${}\le p-1$ and since
$({p-1\choose\ell}(1-t)^\ell t^{p-1-\ell})_{0\le\ell\le p-1}$ is a basis
of this space, $(5.14')$ can be achieved for a unique choice of the
polynomials $a_m(t)$. A~straightforward calculation shows that $A_p(t,0)=p$
identically. We can therefore choose $\delta_0\in[0,1[$ so small that 
$A_p(t,\delta)>0$ for all $t\in[0,1]$, $\delta\in[0,\delta_0]$ and 
$p=1,2,\ldots,n$.

Now, fix a K\"ahler metric $\omega$ such that $\omega'=\alpha+\omega$ 
yields a K\"ahler class $\{\omega'\}$ (just take a large multiple 
$\omega=k\omega_0$, $k\gg 1$, of the given  K\"ahler metric 
$\omega_0$ to initialize the process).
A substitution $x=\omega$ and $y=\omega'$ in our polynomial
identity yields
$$
(\alpha+(1-\delta)\omega)^p-(1-\delta)^p\omega^p=\int_0^1
A_p(t,\delta)\,\alpha\wedge\big((1-t)\omega+t\omega'\big)^{p-1}dt.
$$
For every irreducible analytic subset $Y\subset X$ of dimension $p$ we find
$$
\int_Y(\alpha+(1-\delta)\omega)^p-(1-\delta)^p\int_Y\omega^p
=\int_0^1A_p(t,\delta)dt\Big(
\int_Y\alpha\wedge\big((1-t)\omega+t\omega'\big)^{p-1}\Big).
$$
However, $(1-t)\omega+t\omega'$ is a K\"ahler class (contained in 
$\bR\{\alpha\}+\bR\{\omega_0\}$) and therefore
$\int_Y\alpha\wedge\big((1-t)\omega+t\omega'\big)^{p-1}\ge 0$ by the
numerical condition. This implies $\int_Y(\alpha+(1-\delta)\omega)^p>0$
for all $\delta\in[0,\delta_0]$. We have produced a segment entirely
contained in $\cP$ such that one extremity $\{\alpha+\omega\}$ is in~$\cK$,
so the other extremity $\{\alpha+(1-\delta_0)\omega\}$ is also in~$\cK$.
By repeating the argument inductively after replacing $\omega$ with 
$(1-\delta_0)\omega$, we see that 
$\{\alpha+(1-\delta_0)^\nu\omega\}\in\cK$ for every integer $\nu\ge 0$.
From this we infer that $\{\alpha\}$ is nef, as desired.
\medskip
\noindent
(b) Part (a) can be reformulated by saying that the dual cone
$\overline{\cK}^{\,\smallvee}$ is the closure of the convex cone generated by
$(n-1,n-1)$ cohomology classes of the form $[Y]\wedge\omega^{p-1}$.
Since these classes are contained in $H^{n-1,n-1}_{\ge 0}(X,\bR)$ which is
also contained in $\overline{\cK}^{\,\smallvee}$ by (5.6), we infer that
$$
\overline{\cK}^{\,\smallvee}=H^{n-1,n-1}_{\ge 0}(X,\bR)=\overline{
{\rm Cone}(\{[Y]\wedge\omega^{p-1}\})}.
\leqno\hbox{\rlap{(5.15)}\rlap{\hbox{\kern\columnwidth\llap{$\square$}}}}~
$$
\endproof
%

\subsection*{\csect.C. Deformations of compact K\"ahler manifolds}

Our main Theorem 5.5 also has an important application to the deformation
theory of compact K\"ahler manifolds.

\claim{(5.16) Theorem} Let $\pi:\cX\to S$ be a deformation of compact K\"ahler
manifolds over an irreducible base~$S$. Then there exists a countable union 
$S'=\bigcup S_\nu$ of analytic subsets $S_\nu\subsetneq S$, such that 
the K\"ahler cones $\cK_t\subset H^{1,1}(X_t,\bC)$ of the fibers
$X_t=\pi^{-1}(t)$ are invariant over $S\ssm S'$ under 
parallel transport with respect to the $(1,1)$-projection $\nabla^{1,1}$ of
the Gauss-Manin connection $\nabla$ in the decomposition of
$$
\nabla=\plainpmatrix{
\nabla^{2,0} & * & 0\cr 
* & \nabla^{1,1} & * \cr
0 & * & \nabla^{0,2}\cr}
$$
on the Hodge bundle $H^2=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$.
\endclaim
%
We moreover conjecture that for an arbitrary deformation $\cX\to S$ of 
compact complex manifolds, the K\"ahler property is
open with respect to the countable Zariski topology on the base $S$ of
the deformation.
\vskip3pt

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

\medskip
{\emph Sketch of Proof of Theorem 5.16.} The result is local on the base,
hence we may assume that $S$ is contractible. Then the family is
differentiably trivial, the Hodge bundle $t\mapsto H^2(X_t,\bC)$ is
the trivial bundle and $t\mapsto H^2(X_t,\bZ)$ is a trivial lattice.
We use the existence of a relative cycle space $C^p(\cX/S)\subset C^p(\cX)$
which consists of all cycles contained in the fibres of $\pi:X\to S$.
It is equipped with a canonical holomorphic projection 
$$
\pi_p:C^p(\cX/S)\to S.
$$
We then define 
the $S_\nu$'s to be the images in $S$ of those connected components of 
$C^p(\cX/S)$ which do not project onto $S$. By the fact that the projection 
is proper on each component, we infer that $S_\nu$ is an analytic subset 
of $S$. The definition of the $S_\nu$'s imply that the cohomology classes
induced by the analytic cycles $\{[Z]\}$, $Z\subset X_t$, remain exactly 
the same for all $t\in S\ssm S'$. This result implies in its turn 
that the conditions defining the numerically positive cones $\cP_t$
remain the same, except for the fact that the spaces $H^{1,1}(X_t,\bR)\subset
H^2(X_t,\bR)$ vary along with the Hodge decomposition. At this point,
a standard calculation implies that the $\cP_t$ are invariant by parallel
transport under $\nabla^{1,1}$. This is done as follows.

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

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

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

\claim{(5.17) Theorem} A compact complex manifold carries a K\"ahler
current if and only if it is bimeromorphic to a K\"ahler manifold
$($or equivalently, dominated by a K\"ahler manifold$\,)$.
\endclaim
%
This class of manifolds is called the {\emph Fujiki class $\cC$}. If we 
compare this result with the solution of the Grauert-Riemenschneider
conjecture, it is tempting to make the following conjecture which would
somehow encompass both results.

\claim{(5.18) Conjecture} Let $X$ be a compact complex manifold of
dimension~$n$. Assume that $X$ possesses a nef cohomology class $\{\alpha\}$
of type $(1,1)$ such that $\int_X\alpha^n>0$. Then $X$ is in the
Fujiki class~$\cC$. $[$Also, $\{\alpha\}$ would contain a K\"ahler 
current, as it follows from Theorem 5.9 if Conjecture 5.18 is proved$\,]$.
\endclaim
%
We want to mention here that most of the above results were already known in 
the cases of complex surfaces (i.e.\ in dimension $2$),
thanks to the work of N.~Buchdahl [Buc99, 00] and A.~Lamari [Lam99a, 99b].

Shortly after the original [DP04] manuscript appeared 
in April 2001, Daniel Huybrechts [Huy01] informed us that Theorem 5.5 can be 
used to calculate the K\"ahler cone of
a very general hyperk\"ahler manifold: the K\"ahler cone is then equal
to a suitable connected component of the positive cone defined by the
Beauville-Bogomolov quadratic form. In the case of an  arbitrary
hyperk\"ahler manifold, S.Boucksom [Bou02] later showed that a $(1,1)$ class
$\{\alpha\}$ is K\"ahler if and only if it lies in the positive part
of the Beauville-Bogomolov quadratic cone and moreover $\int_C\alpha>0$
for all {\rm \emph rational curves} $C\subset X$ (see also [Huy99]).

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

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

\claim{(6.1) Definition} Let $X$ be a smooth projective variety.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} One defines $\NE(X)$ to be the
convex cone generated by cohomology classes of all effective curves
in $H^{n-1,n-1}(X,\bR)$
\vskip3pt
\plainitem{\rm (ii)} We say that $C$ is a {\rm mobile curve} if $C=C_{t_0}$
is a member of an analytic family $(C_t)_{t\in S}$ such that 
$\bigcup_{t\in S}C_t=X$
and, as such, is a reduced irreducible $1$-cycle. We define the
mobile cone $\ME(X)$, to be the convex cone generated by all 
mobile curves.
\vskip3pt
\plainitem{\rm (iii)} If $X$ is projective, we say that an effective $1$-cycle $C$ 
is  a {\rm strongly mobile} if we have
$$
C=\mu_\star(\widetilde A_1\cap\ldots\cap \widetilde A_{n-1})
$$ 
for suitable very ample divisors $\widetilde A_j$ on $\widetilde X$, where 
$\mu:\widetilde X\to X$ is a modification. We let $\ME^s(X)$ 
be the convex cone generated by all strongly mobile effective
$1$-cycles $($notice that by taking $\widetilde A_j$ general enough these classes can 
be represented by reduced irreducible curves; also, by
Hironaka, one could just restrict oneself to compositions of blow-ups
with smooth centers$)$. 
\vskip0pt}
\endclaim
%
\noindent
Clearly, we have $\ME^s(X)\subset\ME(X)\subset\NE(X)$.
The cone $\NE(X)$ is contained in the analogue of the Neron-Severi group
for $(n-1,n-1)$-classes, namely
$$
\NS_\bR^{n-1}(X):=
(H^{n-1,n-1}(X,\bR)\cap H^{2n-2}(X,\bZ)/{\rm tors})\otimes_\bZ\bR
$$
(sometimes also denoted $N_1(X)$ in the litterature).
We wish to introduce similar concepts for cones of non necessarily integral
classes, on arbitrary compact K\"ahler manifolds. The relevant definition is
as follows.

\claim{(6.2) Definition} Let $X$ be a compact K\"ahler manifold.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} We define the set $\cN=H^{n-1,n-1}_{\ge 0}(X,\bR)$
to be the $($closed$\,)$ convex cone in
$H^{n-1,n-1}(X,\bR)$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n\,{-}\,1,n\,{-}\,1)$}, i.e., of bidimension $(1,1)$.
\vskip3pt
\plainitem{\rm (ii)} We define the cone $\cM^s\subset H^{n-1,n-1}(X,\bR)$ of
{\rm strongly mobile classes} to be the closure of the convex cone generated by
classes of currents of the form
$$\mu_\star(\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{n-1})$$
where $\mu:\widetilde X\to X$ is an arbitrary modification, and 
the $\widetilde\omega_j$ are K\"ahler forms on~$\widetilde X$.
\vskip3pt
\plainitem{\rm (iii)} We define the cone $\cM\subset H^{n-1,n-1}(X,\bR)$ of
{\rm mobile classes} to be the closure of the convex cone generated by
classes of currents of the form 
$$\mu_\star([\widetilde Y_{t_0}]\wedge\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{p-1})$$
where $\mu:\widetilde X\to X$ is an arbitrary modification, the $\widetilde\omega_j$ are 
K\"ahler forms on~$\widetilde X$ and $(\widetilde Y_t)_{t\in S}$ is an analytic family of
effective $p$-dimensional analytic cycles covering $\widetilde X$ such that 
$\widetilde Y_{t_0}$ is 
reduced and irreducible, with $p$ running over all $\{1,2,\ldots,n\}$. 
\vskip0pt
}
\endclaim
%
Clearly, we have
$$\cM^s\subset\cM\subset\cN.$$
For $X$ projective, it is also immediately clear from the definitions that
$$
\plaincases{
\kern6pt\overline{\NE(X)}\subset\cN_{\NS}:=\cN\cap\NS_\bR^{n-1}(X),\cr
\noalign{\vskip5pt}
\kern4pt\overline{\ME(X)}\subset\cM_{\NS}:=\cM\cap\NS_\bR^{n-1}(X),\cr
\noalign{\vskip4pt}
\overline{\ME^s(X)}\subset\cM^s_{\NS}:=\cM^s\cap\NS_\bR^{n-1}(X).\cr}\leqno(6.3)
$$
The upshot of these definitions lie in the following easy observation.

\claim{(6.4) Proposition} Let $X$ be a compact K\"ahler manifold. 
The Serre duality pairing
$$H^{1,1}(X,\bR)\times H^{n-1,n-1}(X,\bR)\longrightarrow \bR,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta$$
takes nonnegative values
\vskip3pt
\plainitem{\rm (a)} for all pairs $(\alpha,\beta)\in\overline\cK\times\cN$;
\vskip3pt
\plainitem{\rm (b)} for all pairs $(\alpha,\beta)\in\cE\times\cM$.
\endclaim
%
\begproof{Proof.} (a) is obvious. In order to prove (b), we may assume that
$$
\beta=\mu_\star([Y_{t_0}]\wedge\widetilde\omega_1\wedge\ldots\wedge
\widetilde\omega_{p-1})
$$ 
for some modification $\mu:\widetilde X\to X$, where $\{\alpha\}=\{T\}$ is the class
of a positive $(1,1)$-current on $X$ and $\widetilde\omega_j$ are K\"ahler
forms on~$\widetilde X$. Then for $t\in S$ generic
$$
\leqalignno{
\int_X\alpha\wedge\beta&=\int_X T\wedge
\mu_\star([\widetilde Y_t]\wedge\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{p-1})\cr
&=\int_X \mu^* T\wedge [\widetilde Y_t]\wedge
\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{p-1}\cr
&=\int_{\widetilde Y_t} (\mu^* T)_{\restriction\widetilde Y_t}\wedge
\widetilde\omega_1\wedge\ldots\wedge\widetilde\omega_{p-1}\ge 0&(6.5)\cr}
$$
provided that we show that the final integral is well defined and that
the formal calulations involved in (6.5) are correct.
Here, we have used the fact that a closed positive $(1,1)$-current $T$
always has a pull-back $\mu^\star T$, which follows from the observation that
if $T=\alpha+i\ddbar\varphi$ with $\alpha$ smooth and $\varphi$ quasi-psh, 
we may always set 
$\mu^\star T=\mu^*\alpha+i\ddbar(\varphi\circ\mu)$, with
$\varphi\circ\mu$ quasi-psh and not identically $-\infty$ on~$\widetilde X$.
Similarly, we see that the restriction 
$(\mu^* T)_{\restriction\widetilde Y_t}$ is a well defined positive $(1,1)$-current
for $t$ generic, by putting
$$
(\mu^* T)_{\restriction\widetilde Y_t}=
(\mu^*\alpha)_{\restriction\widetilde Y_t}+i\ddbar\big(
(\varphi\circ\mu)_{\restriction\widetilde Y_t}\big)
$$
and choosing $t$ such that $\widetilde Y_t$ is not contained in the pluripolar 
set of $-\infty$ poles of $\varphi\circ\mu$ (this is possible 
thanks to the assumption that $\widetilde Y_t$ covers $\widetilde X$; locally
near any given point we can modify $\alpha$ so that $\alpha=0$
on a small neighborhood~$V$, and then $\varphi$ is psh on $V$).
Finally, in order to justify the formal calculations
we can use a regularization argument for $T$, writing $T=\lim T_k$
with $T_k=\alpha+i\ddbar\varphi_k$ and a decreasing sequence of smooth almost 
plurisubharmonic potentials $\varphi_k\downarrow\varphi$ such that the 
Levi forms  have a uniform lower bound $i\ddbar\varphi_k\ge -C\omega$ (such a
sequence exists by [Dem92]). Then
$(\mu^*T_k)_{\restriction\widetilde Y_t}\to (\mu^*T)_{\restriction\widetilde Y_t}$ 
in the weak topology of currents.\qed
\endproof
%
Proposition 6.4 leads to the natural question whether the 
cones $(\overline{\cK},\cN)$ and $(\cE,\cM)$ are dual under Serre duality,
The second part of the question is addressed in the next section.
The results proved in \S$\,$5 yield a complete answer to the first part 
-- even in the general K\"ahler setting.


\claim{(6.6) Theorem} Let $X$ be a compact K\"ahler manifold. Then
\vskip3pt
\plainitem{\rm(i)} $\overline{\cK}$ and $\cN$ are dual cones.
\vskip3pt
\plainitem{\rm(ii)} If $X$ is projective algebraic, then
$\overline{\cK}_{\NS}=\Nef(X)$ and $\cN_{\NS}=\overline{\NE(X)}$ and these cones are
dual.
\endclaim
%
\begproof{Proof.} (i) is a weaker version of (5.13~b). 
\medskip

\noindent
(ii) The equality $\overline{\cK}_{\NS}=\Nef(X)$ has already been discussed
and is a consequence of the Kodaira embedding theorem. Now, we know that
$$
\overline{\NE(X)}\subset\cN_{\NS}\subset
\overline{\cK}_{\NS}^\smallvee=\Nef(X)^\smallvee,
$$
where the second inclusion is a consequence of (5.8~a). However,
it is already well-known that $\overline{\NE(X)}$ and
$\overline{\NE(X)}$ are dual cones (see [Har70]), hence the inclusions are
equalities (we could also obtain a self-contained proof by reconsidering 
the arguments used for (5.13~a) when $\alpha$ and
$\omega_0$ are rational classes; one sees by the density of the rationals
that the numerical condition for $\alpha$ is needed only for
elements of the form $[Y]\wedge\omega^{p-1}$ with 
$\omega\in\bQ\{\alpha\}+\bQ\{\omega_0\}$ a rational class,
so $[Y]\wedge\omega^{p-1}$ is then a $\bQ$-effective curve).\qed
\endproof
%
\subsection*{\csect.B. Zariski decomposition and mobile intersections}
Let $X$ be compact K\"ahler and let $\alpha\in\cE^\circ$ be in the
{\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 $\delta\ll 1$. We first
need a variant of the regularization theorem proved in section~3.B.

\claim{(6.7) Regularization theorem for currents} 
Let $X$ be a compact complex manifold equipped with a hermitian metric
$\omega$. Let $T=\alpha+i\ddbar\varphi$ be a closed $(1,1)$-current on~$X$,
where $\alpha$ is smooth and $\varphi$ is a quasi-plurisubharmonic function. 
Assume that $T\ge\gamma$ for some real $(1,1)$-form $\gamma$ on $X$ with 
real coefficients. Then there exists 
a sequence $T_m=\alpha+i\ddbar\varphi_m$ of closed $(1,1)$-currents such that
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} $\varphi_m$ $($and thus $T_m)$ is smooth on the complement
$X\ssm Z_m$ of an analytic set $Z_m$, and the $Z_m$'s form an increasing
sequence
$$
Z_0\subset Z_1\subset\ldots\subset Z_m\subset\ldots \subset X.
$$
\vskip3pt
\plainitem{\rm (ii)} There is a uniform estimate $T_m\ge \gamma-\delta_m\omega$
with $\lim\downarrow\delta_m=0$ as $m$ tends to $+\infty$.
\vskip3pt
\plainitem{\rm (iii)} The sequence $(\varphi_m)$ is non increasing, and we 
have $\lim\downarrow\varphi_m=\varphi$. As a consequence, $T_m$ converges
weakly to $T$ as  $m$ tends to $+\infty$.
\vskip3pt
\plainitem{\rm (iv)} Near $Z_m$, the potential $\varphi_m$ has logarithmic poles,
namely, for every $x_0\in Z_m$, there is a neighborhood $U$ of $x_0$
such that $\varphi_m(z)=\lambda_m\log\sum_\ell|g_{m,\ell}|^2+O(1)$ for 
suitable holomorphic functions $(g_{m,\ell})$ on $U$ and $\lambda_m>0$.
Moreover, there is a $($global$)$ proper modification $\mu_m:\widetilde X_m\to X$ 
of $X$, obtained as a sequence of blow-ups with smooth centers, such that 
$\varphi_m\circ\mu_m$ can be written locally on $\widetilde X_m$ as
$$
\varphi_m\circ\mu_m(w)=\lambda_m\big(\sum n_\ell\log|\widetilde g_\ell|^2+f(w)\big)
$$
where $(\widetilde g_\ell=0)$ are local generators of suitable $($global$)$
divisors $D_\ell$ on $\widetilde X_m$ such that $\sum D_\ell$ has normal crossings,
$n_\ell$ are positive integers, and the $f$'s are smooth functions 
on~$\widetilde X_m$.
\vskip0pt
}
\endclaim
%
\medskip
{\emph Sketch of proof.} We essentially repeat the proofs of Theorems (3.2)
and (3.12) with additional considerations. One fact that does not 
follow readily from these proofs is the
monotonicity of the sequence $\varphi_m$ (which we will not really need
anyway). For this, we can take $m=2^\nu$ and use the subadditivity
technique already explained in Step 3 of the proof  of Theorem 
(11.3$\,$b). The map $\mu_m$ is obtained by blowing-up the 
(global) ideals $\cJ_m$ defined by the holomorphic functions $(g_{j,m})$
in the local approximations $\varphi_m\sim{1\over 2m}\log\sum_j|g_{j,m}|^2$.
By Hironaka [Hir64], we can achieve that $\mu_m^*\cJ_m$ is an invertible
ideal sheaf associated with a normal crossing divisor.\qed

\claim{(6.8) Corollary} If $T$
is a K\"ahler current, then one can write $T=\lim T_m$ for a sequence of 
K\"ahler currents $T_m$ which have
logarithmic poles with coefficients in ${1\over m}\bZ$, i.e.\ there
are modifications $\mu_m:X_m\to X$ such that
$$\mu_m^\star T_m=[E_m]+\beta_m$$
where $E_m$ is an effective $\bQ$-divisor on $X_m$ with coefficients
in ${1\over m}\bZ$ $($the ``fixed part''$)$ and $\beta_m$ is a closed 
semi-positive form $($the ``mobile part''$)$.
\endclaim
%
\begproof{Proof.} We apply Theorem (6.7) with $\gamma=\varepsilon\omega$ and
$m$ so large that $\delta_m\le\varepsilon/2$. Then $T_m$ has analytic
singularities and $T_m\ge{\varepsilon\over 2}\omega$, so we get 
a composition of blow-ups $\mu_m:X_m\to X$ such 
$$\mu_m^*T_m=[E_m]+\beta_m,$$
where $E_m$ is an effective $\bQ$-divisor and 
$\beta_m\ge{\varepsilon\over 2}\mu_m^*\omega$. In particular,
$\beta_m$ is strictly positive outside the exceptional divisors, by playing 
with the multiplicities of the components of the exceptional divisors in
$E_m$, we could even achieve that $\beta_m$ is a 
K\"ahler class on $X_m$. Notice also that by construction, $\mu_m$
is obtained by blowing-up the multiplier ideal sheaves $\cI(mT)=
\cI(m\varphi)$ associated to a potential $\varphi$ of $T$.\qed
\endproof
%
The more familiar algebraic analogue would be to take \hbox{$\alpha=c_1(L)$}
with a big line bundle $L$ and to blow-up the base locus of $|mL|$,
$m\gg 1$, to get a $\bQ$-divisor decomposition
$$
\mu_m^\star L\sim E_m+D_m,\qquad E_m~~\hbox{effective},~~D_m~~\hbox{free}.
$$
Such a blow-up is usually referred to as a ``log resolution'' of the
linear system $|mL|$, and we say that $E_m+D_m$ is an approximate
Zariski decomposition of $L$. We will also use this terminology for K\"ahler
currents with logarithmic poles.
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\EndFig
\vskip4mm
$~$

\claim{(6.9) Definition} We define the {\rm volume}, or {\rm mobile
self-intersection} of a big class $\alpha\in\cE^\circ$ to be
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\widetilde X}\beta^n>0
$$
where the supremum is taken over all K\"ahler currents $T\in \alpha$
with logarithmic poles, and $\mu^\star T=[E]+\beta$ with
respect to some modification $\mu:\widetilde X\to X$.
\endclaim
%
By Fujita [Fuj94] and Demailly-Ein-Lazarsfeld [DEL00], if $L$ 
is a big line bundle, we have
$$
\Vol(c_1(L))=\lim_{m\to+\infty}D_m^n=\lim_{m\to+\infty}{n!\over m^n}h^0(X,mL),
$$
and in these terms, we get the following statement.

\claim{(6.10) Proposition} Let $L$ be a big line bundle on the
projective manifold $X$.  Let $\epsilon > 0$. Then there exists a
modification $\mu: X_{\epsilon} \to X$ and a decomposition $\mu^*(L) =
E + \beta $ with $E$ an effective $\bQ$-divisor and $\beta$ a big and
nef $\bQ$-divisor such that
$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$
\endclaim
%
It is very useful to observe that the supremum in Definition 6.9 is actually
achieved by a collection of currents whose singularities satisfy a filtering 
property.
Namely, if $T_1=\alpha+i\ddbar\varphi_1$ and $T_2=\alpha+i\ddbar\varphi_2$ 
are two K\"ahler currents with logarithmic poles in the class of $\alpha$, 
then 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi=\max(\varphi_1,\varphi_2)\leqno(6.11)$$ 
is again a K\"ahler current with weaker 
singularities than $T_1$ and $T_2$. One could define as well 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi={1\over 2m}\log(e^{2m\varphi_1}+e^{2m\varphi_2}),\leqno(6.11')$$
where $m={\rm lcm}(m_1,m_2)$ is the lowest common multiple of the denominators
occuring in $T_1$, $T_2$. Now, take a simultaneous log-resolution
$\mu_m:X_m\to X$ for which the singularities of $T_1$ and $T_2$ are resolved 
as $\bQ$-divisors $E_1$
and $E_2$. Then clearly the associated divisor in the decomposition 
$\mu_m^\star T=[E]+\beta$ is given by $E=\min(E_1,E_2)$. By doing so, 
the volume $\int_{X_m}\beta^n$ gets increased, as we shall see in the
proof of Theorem 6.12 below.

\claim{(6.12) Theorem {\rm (Boucksom [Bou02])}} Let $X$ be a compact K\"ahler
manifold. We denote here by $H^{k,k}_{\ge 0}(X)$ the cone of cohomology classes
of type $(k,k)$ which have non-negative intersection with all closed
semi-positive smooth forms of bidegree $(n-k,n-k)$.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm(i)} For each integer $k=1,2,\ldots,n$, there exists a canonical
``mobile intersection product''
$$
\cE\times\cdots\times\cE\to H^{k,k}_{\ge 0}(X), \quad
(\alpha_1,\ldots,\alpha_k)\mapsto \langle\alpha_1\cdot\alpha_2\cdots
\alpha_{k-1}\cdot \alpha_k\rangle
$$
such that $\Vol(\alpha)=\langle\alpha^n\rangle$ whenever $\alpha$ is
a big class.
\vskip3pt
\plainitem{\rm(ii)} The product is increasing, homogeneous of 
degree $1$ and superadditive in each argument, i.e.\
$$
\langle\alpha_1\cdots(\alpha'_j+\alpha''_j)\cdots \alpha_k\rangle\ge
\langle\alpha_1\cdots\alpha'_j\cdots \alpha_k\rangle+
\langle\alpha_1\cdots\alpha''_j\cdots \alpha_k\rangle.
$$
It coincides with the ordinary intersection
product when the $\alpha_j\in\overline{\cK}$ are nef classes.
\vskip3pt
\plainitem{\rm(iii)} 
The mobile intersection product satisfies the Teissier-Hovanskii inequalities
([Hov79], [Tei79, 82])
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_n\rangle\ge
(\langle\alpha_1^n\rangle)^{1/n}\ldots(\langle\alpha_n^n\rangle)^{1/n}\qquad
\hbox{$($with $\langle\alpha_j^n\rangle=\Vol(\alpha_j)\,)$}.
$$
\vskip3pt
\plainitem{\rm(iv)} For $k=1$, the above ``product'' reduces to a
$($non linear$)$ projection operator 
$$
\cE\to\cE_1,\qquad \alpha\to\langle\alpha\rangle
$$
onto a certain convex subcone $\cE_1$ of $\cE$ such that
$\overline{\cK}\subset\cE_1\subset\cE$. Moreover, there is
a ``divisorial Zariski decomposition''
$$
\alpha=\{N(\alpha)\}+\langle\alpha\rangle
$$
where $N(\alpha)$ is a uniquely defined effective divisor which is
called the ``negative divisorial part'' of $\alpha$. The map
$\alpha\mapsto N(\alpha)$ is homogeneous and subadditive, and 
$N(\alpha)=0$ if and only if $\alpha\in\cE_1$.
\vskip3pt
\plainitem{\rm(v)} The components of $N(\alpha)$ always consist of
divisors whose cohomology classes are linearly independent,
especially $N(\alpha)$ has at most $\rho=\rank_\bZ\NS(X)$ components.
\vskip0pt}
\endclaim
%
\begproof{Proof.} We essentially repeat the arguments developped in [Bou02], with
some simplifications arising from the fact that $X$ is supposed to
be K\"ahler from the start.
\vskip3pt

\noindent
(i) First assume that all classes $\alpha_j$ are big, i.e.\ 
$\alpha_j\in\cE^\circ$. Fix a smooth closed $(n-k,n-k)$ {\emph semi-positive}
form $u$ on $X$. We select K\"ahler currents $T_j\in\alpha_j$ with
logarithmic poles, and a simultaneous log-resolution
$\mu:\widetilde X\to X$ such that 
$$
\mu^\star T_j=[E_j]+\beta_j.
$$
We consider the direct image current
$\mu_\star(\beta_1\wedge\ldots\wedge\beta_k)$ (which is a closed
positive current of bidegree $(k,k)$ on $X$) and the corresponding
integrals
$$
\int_{\widetilde X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u\ge 0.
$$
If we change the representative $T_j$ with another current $T_j'$,
we may always take a simultaneous log-resolution such that
$\mu^\star T_j'=[E_j']+\beta_j'$, and by using $(6.11')$ we can always
assume that $E'_j\le E_j$. Then $D_j=E_j-E'_j$ is an effective
divisor and we find $[E_j]+\beta_j\equiv [E'_j]+\beta'_j$, hence
$\beta'_j\equiv\beta_j+[D_j]$. A substitution in the integral implies
$$
\eqalign{
\int_{\widetilde X}\beta_1'\wedge\beta_2
&{}\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&=\int_{\widetilde X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u+
\int_{\widetilde X}[D_1]\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&\ge \int_{\widetilde X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge
\mu^\star u.\cr}
$$
Similarly, we can replace successively all forms $\beta_j$ by the $\beta'_j$,
and by doing so, we find
$$
\int_{\widetilde X}\beta'_1\wedge\beta'_2\wedge\ldots\wedge\beta'_k\wedge\mu^\star u
\ge\int_{\widetilde X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u.
$$
We claim that the closed positive currents $\mu_\star(\beta_1\wedge\ldots
\wedge\beta_k)$ are uniformly bounded in mass. In fact, if $\omega$ is a 
K\"ahler metric in $X$, there exists a constant $C_j\ge 0$
such that $C_j\{\omega\}-\alpha_j$ is a K\"ahler class. Hence
$C_j\omega-T_j\equiv\gamma_j$ for some K\"ahler form $\gamma_j$ on $X$.
By pulling back with $\mu$, we find
$C_j\mu^\star \omega-([E_j]+\beta_j)\equiv\mu^\star \gamma_j$, hence 
$$\beta_j\equiv C_j\mu^\star \omega-([E_j]+\mu^\star \gamma_j).$$
By performing again a substitution in the integrals, we find
$$
\int_{\widetilde X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u
\le C_1\ldots C_k\int_{\widetilde X}\mu^\star\omega^k\wedge\mu^\star u
= C_1\ldots C_k\int_X\omega^k\wedge u
$$
and this is true especially for $u=\omega^{n-k}$. We can now arrange
that for each of the integrals associated with a countable dense family 
of forms $u$,  the supremum is achieved by a sequence of currents 
$(\mu_m)_\star(\beta_{1,m}\wedge\ldots\wedge\beta_{k,m})$ obtained
as direct images by a suitable sequence of modifications $\mu_m:
\smash{\widetilde X_m}\to X$.
By extracting a subsequence, we can achieve that this sequence is weakly 
convergent and we set
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\uparrow}\limits_{m\to+\infty}
\{(\mu_m)_\star(\beta_{1,m}\wedge\beta_{2,m}\wedge\ldots\wedge\beta_{k,m})\}
$$
(the monotonicity is not in terms of the currents themselves, but in terms
of the integrals obtained when we evaluate against a smooth closed
semi-positive form~$u$). By evaluating against a basis of positive classes
$\{u\}\in H^{n-k,n-k}(X)$, we infer by Serre duality that the class of
$\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle$ is uniquely defined
(although, in general, the representing current is not unique). 
\vskip3pt

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

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

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

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

\noindent
(v) Let $\alpha\in\cE^\circ$, and assume that $N(\alpha)$ contains
linearly dependent components $F_j$. Then already all currents
$T\in \alpha$ should be such that $\mu^\star T=[E]+\beta$ where 
$F=\mu_\star E$ contains those linearly dependent components. 
Write $F=\sum\lambda_jF_j$, $\lambda_j>0$ and assume that 
$$\sum_{j\in J}c_jF_j\equiv 0$$
for a certain non trivial linear combination. Then some of the
coefficients $c_j$ must be negative (and some other positive).
Then $E$ is numerically equivalent to
$$
E'\equiv E+t\mu^\star\Big(\sum\lambda_jF_j\Big),
$$
and by choosing $t>0$ appropriate, we obtain an effective divisor $E'$ 
which has a zero coefficient on one of the components $\mu^\star F_{j_0}$.
By replacing $E$ with $\min(E,E')$ via $(6.11')$, we eliminate the component
$\mu^\star F_{j_0}$. This is a contradiction since $N(\alpha)$
was supposed to contain $F_{j_0}$.\qed
\endproof
%
\claim{(6.13) Definition} For a class $\alpha\in H^{1,1}(X,\bR)$, 
we define the {\rm numerical dimension} $\nd(\alpha)$ to be 
$\nd(\alpha)=-\infty$ if $\alpha$ is not pseudo-effective, and
$$
\nd(\alpha)=\max\{p\in\bN\;;\;\langle\alpha^p\rangle\ne 0\},\qquad
\nd(\alpha)\in\{0,1,\ldots,n\}
$$
if $\alpha$ is pseudo-effective.
\endclaim
%
By the results of [DP04], a class is big ($\alpha\in\cE^\circ$) if and only
if $\nd(\alpha)=n$. Classes of numerical dimension $0$ can be
described much more precisely, again following Boucksom [Bou02].

\claim{(6.14) Theorem} Let $X$ be a compact
K\"ahler manifold. Then the subset $\cD_0$ of irreducible divisors 
$D$ in $X$ such that $\nd(D)=0$ is countable, and these divisors 
are rigid as well as their multiples. If $\alpha\in\cE$ is a 
pseudo-effective class of numerical dimension~$0$, then $\alpha$ 
is numerically equivalent to 
an effective $\bR$-divisor $D=\sum_{j\in J}\lambda_j D_j$,
for some finite subset $(D_j)_{j\in J}\subset\cD_0$ such that the 
cohomology classes $\{D_j\}$ are linearly independent and
some \hbox{$\lambda_j>0$}. If such a linear combination is of
numerical dimension~$0$, then so is any other linear combination of the
same divisors.
\endclaim
%
\begproof{Proof.} It is immediate from the definition that a pseudo-effective
class is of numerical dimension $0$ if and only if $\langle\alpha\rangle=0$,
in other words if $\alpha=N(\alpha)$. Thus $\alpha\equiv\sum\lambda_jD_j$
as described in 6.14, and since $\lambda_j\langle D_j\rangle\le 
\langle\alpha\rangle$, the divisors $D_j$ must themselves have numerical 
dimension $0$. There is at most one such divisor $D$ in any given cohomology
class in $NS(X)\cap\cE\subset H^2(X,\bZ)$, otherwise two such divisors 
$D\equiv D'$ would yield a blow-up $\mu:\widetilde X\to X$ resolving the 
intersection, and by taking
$\min(\mu^\star D,\mu^\star D')$ via $(6.11')$, we would find 
$\mu^\star D\equiv E+\beta$, $\beta\ne 0$, so that
$\{D\}$ would not be of numerical dimension $0$. This implies 
that there are at most countably many divisors of numerical 
dimension~$0$, and that these divisors are rigid as well as their
multiples.\qed
\endproof
%
\claim{(6.15) Remark} {\rm If $L$ is an arbitrary holomorphic line bundle, 
we define its numerical dimension to be $\nd(L)=\nd(c_1(L))$. Using 
the cananical maps $\Phi_{|mL|}$ and pulling-back the Fubini-Study metric
it is immediate to see that \hbox{$\nd(L)\ge\kappa(L)$} (which generalizes
the analogue inequality already seen for nef line bundles,\break
see~(1.21)).}
\endclaim
%
The above general concept of numerical dimension leads to a very natural 
formulation of the abundance conjecture for K\"ahler varieties.

\claim{(6.16) Generalized abundance conjecture} Let $X$ be an 
arbitrary compact K\"ahler manifold~$X$.
\vskip3pt
\plainitem{\rm(a)} The Kodaira dimension of $X$ should be equal to its
numerical dimension~: $\kappa(K_X)=\nd(K_X)$.
\vskip3pt
\plainitem{\rm(b)} More generally, let $\Delta$ be a $\bQ$-divisor which is
klt $($Kawamata log terminal, i.e.\ such that $c_X(\Delta)>1)$.
Then $\kappa(K_X+\Delta)=\nd(K_X+\Delta)$.\vskip0pt
\endclaim
%
This appears to be a fairly strong statement. In fact, already in the case
$\Delta=0$, it is not difficult
to show that the generalized abundance conjecture would contain the
$C_{n,m}$ conjectures.

\claim{(6.17) Remark} {\rm It is obvious that abundance holds in the case
\hbox{$\nd(K_X)=-\infty$} (if $L$ is not pseudo-effective, no
multiple of $L$ can have sections), or in the case \hbox{$\nd(K_X)=n$}
which implies $K_X$ big (the latter property follows e.g.\ from 
the solution of the Grauert-Riemenschneider conjecture in the form proven
in [Dem85], see also [DP04]).

In the remaining cases, the most tractable situation is the case when 
$\nd(K_X)=0$.
In fact Theorem 6.14 then gives $K_X\equiv\sum\lambda_j D_j$ for some
effective divisor with numerically independent components, $\nd(D_j)=0$. 
It follows that the $\lambda_j$ are rational and therefore
$$
K_X\sim \sum\lambda_j D_j+F\qquad
\hbox{where $\lambda_j\in\bQ^+$, $\nd(D_j)=0$ and $F\in\Pic^0(X)$.}
\leqno(*)
$$
If we assume additionally that $q(X)=h^{0,1}(X)$ is zero, then 
$mK_X$ is linearly equivalent
to an integral divisor for some multiple $m$, and it follows 
immediately that $\kappa(X)=0$. The case of a general projective
manifold with \hbox{$\nd(K_X)=0$} and positive 
irregularity $q(X)>0$ has been solved by Campana-Peternell
[CP04], Corollary~3.7. It would be interesting to understand 
the K\"ahler  case as well.}
\endclaim
%
\subsection*{\csect.C. The orthogonality estimate}
The goal of this section is to show that, in an appropriate sense,
approximate Zariski decompositions are almost orthogonal.

\claim{(6.18) Theorem} Let $X$ be a projective manifold, and
let $\alpha=\{T\}\in\cE^\circ_{\NS}$ be a big class represented by
a K\"ahler current~$T$. Consider
an approximate Zariski decomposition
$$\mu_m^\star T_m = [E_m]+[D_m]$$
Then
$$
(D_m^{n-1}\cdot E_m)^2\le 20\,(C\omega)^n\big(\Vol(\alpha)-D_m^n\big)
$$
where $\omega=c_1(H)$ is a K\"ahler form and $C\ge 0$ is a constant such that
$\pm\alpha$ is dominated by $C\omega$ $($i.e., $C\omega\pm\alpha$ is nef$\,)$.
\endclaim
%
\begproof{Proof.} For every $t\in[0,1]$, we have
$$
\Vol(\alpha)=\Vol(E_m+D_m)\ge \Vol(tE_m+D_m).
$$
Now, by our choice of $C$, we can write $E_m$ as a difference of two
nef divisors
$$E_m=\mu^\star\alpha-D_m=\mu_m^\star(\alpha+C\omega)-(D_m+C\mu_m^\star
\omega).$$
\endproof
%
\claim{(6.19) Lemma} For all nef $\bR$-divisors $A$, $B$ we have
$$\Vol(A-B)\ge A^n - nA^{n-1}\cdot B$$
as soon as the right hand side is positive.
\endclaim
%
\begproof{Proof.} In case $A$ and $B$ are integral (Cartier) divisors, this is a 
consequence of the holomorphic Morse inequalities 2.4 (see [Dem01]); 
one can also argue by an elementary estimate of 
to $H^0(X, mA-B_1-\ldots-B_m)$ via the Riemann-Roch formula
(assuming $A$ and $B$ very ample, $B_1,\ldots,B_m\in|B|$ generic). If $A$ and 
$B$ are $\bQ$-Cartier, we conclude by the homogeneity of the volume. The 
general case of $\bR$-divisors follows by approximation using the
upper semi-continuity of the volume [Bou02, 3.1.26].\qed 
\endproof
%
\claim{(6.20) Remark} {\rm We hope that Lemma 6.19 also holds true on an
arbitrary K\"ahler mani\-fold for arbitrary nef (non neces\-sarily
integral) classes. This would follow from a generalization of
holomorphic Morse inequalities to non integral classes. However 
the proof of such a result seems technically much more involved than
in the case of integral classes.}
\endclaim
 
\claim{(6.21) Lemma} Let $\beta_1,\ldots,\beta_n$ and 
$\beta'_1,\ldots,\beta'_n$ be nef classes on a compact K\"ahler manifold
$\widetilde X$ such that each difference $\beta'_j-\beta_j$ is pseudo-effective. Then
the $n$-th intersection products satisfy
$$
\beta_1\cdots\beta_n \le \beta'_1\cdots\beta'_n.
$$ 
\endclaim
%
\begproof{Proof.} We can proceed step by step and replace just one $\beta_j$ by
$\beta'j\equiv\beta_j+T_j$ where $T_j$ is a closed positive
$(1,1)$-current and the other classes $\beta'_k=\beta_k$, $k\ne j$ are
limits of K\"ahler forms. The inequality is then obvious.\qed
\endproof
%
\noindent
{\emph End of proof of Theorem 6.18.}
In order to exploit the lower bound of the volume, we write
$$
tE_m+D_m=A-B,\qquad
A=D_m+t\mu_m^\star(\alpha+C\omega),\quad B=t(D_m+C\mu_m^\star\omega).
$$
By our choice of the constant $C$, both $A$ and $B$ are nef. Lemma 6.19 and 
the binomial formula imply
$$\eqalign{
\Vol(tE_m+&D_m)\ge A^n - nA^{n-1}\cdot B\cr
&=D_m^n+nt\,D_m^{n-1}\cdot\mu_m^\star(\alpha+C\omega)+\sum_{k=2}^n
t^k{n\choose k}D_m^{n-k}\cdot\mu_m^\star(\alpha+C\omega)^k\cr
&\phantom{{}=D_m^n}-nt\,D_m^{n-1}\cdot(D_m+C\mu_m^\star\omega)\cr
&\phantom{{}=D_m^n}-nt^2\sum_{k=1}^{n-1}t^{k-1}{n-1\choose k}
D_m^{n-1-k}\cdot\mu_m^\star(\alpha+C\omega)^k\cdot(D_m+C\mu_m^\star\omega).
\kern-4pt\cr}
$$
Now, we use the obvious inequalities
$$
D_m\le \mu_m^\star(C\omega),\qquad
\mu_m^\star(\alpha+C\omega)\le 2\mu_m^\star(C\omega),\qquad
D_m+C\mu_m^\star\omega\le 2\mu_m^\star(C\omega)
$$
in which all members are nef (and where the inequality${}\le$ means 
that the difference of classes is pseudo-effective). We use Lemma~6.21
to bound the last summation in the estimate of the volume, and in this
way we get
$$
\Vol(tE_m+D_m)\ge D_m^n+nt D_m^{n-1}\cdot E_m-nt^2\sum_{k=1}^{n-1}2^{k+1}
t^{k-1}{n-1\choose k}(C\omega)^n.
$$
We will always take $t$ smaller than $1/10n$ so that the last summation is 
bounded by $4(n-1)(1+1/5n)^{n-2}<4ne^{1/5}<5n$. This implies
$$
\Vol(tE_m+D_m)\ge D_m^n+nt\,D_m^{n-1}\cdot E_m-5n^2t^2(C\omega)^n.
$$
Now, the choice $t={1\over 10n}(D_m^{n-1}\cdot E_m)((C\omega)^n)^{-1}$ 
gives by substituting
$$
\eqalign{
{1\over 20}{(D_m^{n-1}\cdot E_m)^2\over (C\omega)^n}
&\le \Vol(E_m+D_m)-D_m^n\le \Vol(\alpha)-D_m^n\cr}
$$
(and we have indeed $t\le {1\over 10n}$ by Lemma 6.21), whence Theorem 6.18. 
Of course, the constant 20 is certainly not optimal.\qed

\claim{(6.22) Corollary} If $\alpha\in\cE_{\NS}$, then the divisorial
Zariski decomposition $\alpha=N(\alpha)+\langle\alpha\rangle$
is such that
$$
\langle\alpha^{n-1}\rangle\cdot N(\alpha)=0.
$$
\endclaim
%
\begproof{Proof.} By replacing $\alpha$ with $\alpha+\delta c_1(H)$, one sees that
it is sufficient to consider the case where $\alpha$ is big. Then the 
orthogonality estimate implies
$$
(\mu_m)_\star(D_m^{n-1})\cdot (\mu_m)_\star E_m
= D_m^{n-1}\cdot(\mu_m)^\star (\mu_m)_\star E_m
\le D_m^{n-1}\cdot E_m \le C (\Vol(\alpha)-D_m^n)^{1/2}.
$$
Since $\langle\alpha^{n-1}\rangle = \lim (\mu_m)_\star(D_m^{n-1})$,
$N(\alpha)=\lim (\mu_m)_\star E_m$ and $\lim D_m^n=\Vol(\alpha)$,
we get the desired conclusion in the limit.\qed
\endproof
%
\subsection*{\csect.D. Dual of the pseudo-effective cone}
The following statement was first proved in [BDPP04].

\claim{(6.23) Theorem} If $X$ is projective, the cones
$\cE_{\NS}=\overline{\Eff(X)}$ and $\overline{\ME^s(X)}$ are dual.
\endclaim
%
In other words, a line bundle $L$ is pseudo-effective if (and
only if) $L\cdot C\ge 0$ for all {\emph mobile curves}, i.e., $L\cdot C\ge 0$
for every very generic curve $C$ (not contained in a countable union of 
algebraic subvarieties). In fact, by definition of $\ME^s(X)$, it is enough 
to consider only those curves $C$ which are images of generic complete 
intersection of very ample divisors on some variety $\widetilde X$, under
a modification \hbox{$\mu:\widetilde X\to X$}.
By a standard blowing-up argument, it also follows that a line bundle
$L$ on a normal Moishezon variety is pseudo-effective if and only if
$L \cdot C \ge 0$ for every mobile curve~$C$. 

\begproof{Proof.} By (6.4~b) we have 
in any case
$$\cE_{\NS}\subset (\ME^s(X))^\smallvee.$$ 
If the inclusion is strict, there is an element
$\alpha\in\partial\cE_{\NS}$ on the boundary of $\cE_{\NS}$ which is in
the interior of $\ME^s(X)^\smallvee$.

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\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
\vskip8mm

Let $\omega=c_1(H)$ be an ample class. Since $\alpha\in\partial\cE_{\NS}$, 
the class $\alpha+\delta\omega$ is big for every $\delta>0$, and since
$\alpha\in((\ME^s(X))^{\smallvee})^\circ$ we still have 
$\alpha-\varepsilon\omega\in(\ME^s(X))^{\smallvee}$ for
$\varepsilon>0$ small. Therefore
$$
\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma\leqno(6.24)
$$
for every strongly mobile curve $\Gamma$, and therefore for every
$\Gamma\in\overline{\ME^s(X)}$. We are going to contradict (6.24).
Since $\alpha+\delta\omega$ is big, we have an approximate Zariski
decomposition
$$
\mu_\delta^\star(\alpha+\delta\omega)=E_\delta+D_\delta.
$$
We pick $\Gamma=(\mu_\delta)_\star(D_\delta^{n-1})\in\overline{\ME^s(X)}$.
By the Hovanskii-Teissier concavity inequality
$$
\omega\cdot \Gamma\ge (\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}.
$$
On the other hand
$$
\eqalign{
\alpha\cdot \Gamma&=\alpha\cdot(\mu_\delta)_\star(D_\delta^{n-1})\cr
&=\mu_\delta^\star\alpha\cdot D_\delta^{n-1}\le
\mu_\delta^\star(\alpha+\delta\omega)\cdot D_\delta^{n-1}\cr
&=(E_\delta+D_\delta)\cdot D_\delta^{n-1}
=D_\delta^n+D_\delta^{n-1}\cdot E_\delta.\cr
}
$$
By the orthogonality estimate, we find
$$\eqalign{
{\alpha\cdot \Gamma\over
\omega\cdot \Gamma}
&\le {D_\delta^n + \big(20(C\omega)^n
(\Vol(\alpha+\delta\omega)-D_\delta^n)\big)^{1/2}\over
(\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}}\cr
&\le C'(D_\delta^n)^{1/n}+C''{(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}\over
(D_\delta^n)^{(n-1)/n}}.\cr}
$$
However, since $\alpha\in\partial\cE_{\NS}$, the class $\alpha$ cannot be big 
so
$$\lim_{\delta \to 0}D_\delta^n=\Vol(\alpha)=0.$$ 
We can also take $D_\delta$ to approximate $\Vol(\alpha+\delta\omega)$ 
in such a way that the term
$(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}$ tends to $0$ much faster than
$D_\delta^n$. Notice that \hbox{$D_\delta^n\ge\delta^n\omega^n$}, so in fact
it is enough to take
$$
\Vol(\alpha+\delta\omega)-D_\delta^n\le \delta^{2n},
$$
which gives $(\alpha\cdot \Gamma)/(\omega\cdot \Gamma)\le (C'+C'')\delta$. 
This contradicts (6.24) for $\delta$ small.\qed
\endproof
%
\claim{(6.25) Conjecture} {\rm The K\"ahler analogue should be~:}\newline
For an arbitrary compact K\"ahler manifold $X$, the cones $\cE$ and $\cM$ 
are dual.
\endclaim
%
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\LabelTeX 6.5 4 $H^{1,1}(X,\bR)$\ELTX
\LabelTeX 57 46 $\cM_{\NS}$\ELTX
\LabelTeX 63 52 $\cM$\ELTX
\LabelTeX 76.3 32 $\cN$\ELTX
\LabelTeX 43 27 $\cN_{\NS}$\ELTX
\LabelTeX 73.5 4 $\NS^{n-1}_\bR(X)$\ELTX
\LabelTeX 32 4 $H^{n-1,n-1}(X,\bR)$\ELTX
\LabelTeX 25 47 duality\ELTX
\EndFig
\vskip8mm

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

\claim{(6.26) Question} Let $(M,\omega)$ be a compact real symplectic manifold.
Fix an almost complex structure $J$ compatible with $\omega$, and
for this structure, assume that \hbox{$c_1(M)\cdot\omega^{n-1}>0$}. 
Does it follow that $M$ is covered by rational $J$-pseudoholomorphic curves~?
\endclaim
%
The relation between the various cones of mobile curves and currents in 
(6.1) and (6.2) is now a rather direct consequence of 
Theorem 6.23. In fact, using 
ideas hinted in [DPS96], we can say a little bit more. Given 
an irreducible curve $C \subset X$, we consider its normal ``bundle''  
$N_C = \Hom(\cI/\cI^2,\cO_C)$, where $\cI$ is the ideal sheaf of $C$. 
If $C$ is a general member of a covering family $(C_t)$, then $N_C$ 
is nef. Now [DPS96] says that the dual cone of the pseudo-effective cone
of $X$ contains the closed cone spanned by curves with nef normal 
bundle, which in turn contains the cone of mobile curves. In this
way we get~:

\claim{(6.27) Theorem} Let $X$ be a projective manifold. Then the following 
cones coincide.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} the cone $\cM_{\NS}=\cM\cap \NS_\bR^{n-1}(X)\,;$
\vskip3pt
\plainitem{\rm (ii)} the cone $\cM^s_{\NS}=\cM^s\cap \NS_\bR^{n-1}(X)\,;$
\vskip3pt
\plainitem{\rm (iii)} the closed cone $\overline{\ME^s(X)}$ of strongly mobile 
curves$\,;$
\vskip3pt
\plainitem{\rm (iv)} the closed cone $\overline{\ME(X)}$ of mobile curves$\,;$
\vskip3pt
\plainitem{\rm (v)} the closed cone $\overline{\ME_{\nef}(X)}$ of curves with nef 
normal bundle.\vskip0pt
}
\endclaim
%
\begproof{Proof.} We have already seen that 
$$\ME^s(X)\subset\ME(X)\subset\ME_{\nef}(X)\subset(\cE_{\NS})^\smallvee$$
and 
$$
\ME^s(X)\subset\cM^s_{\NS}(X)\subset\cM_{\NS}\subset(\cE_{\NS})^\smallvee
$$ 
by 6.4~(iii). Now Theorem 6.23 implies $(\cM_{\NS})^\smallvee =
\overline{\ME^s(X)}$, and 6.27 follows.\qed
\endproof
%
\claim{(6.28) Corollary} Let $X$ be a projective manifold and $L$ a 
line bundle on $X$.
\vskip3pt
\plainitem{\rm (i)} $L$ is pseudo-effective if and only if $L \cdot C \ge 0$ 
for all curves $C$ with nef
normal sheaf $N_C$.
\vskip3pt
\plainitem{\rm (ii)} If $L$ is big, then $L \cdot C > 0$ for all curves $C$ 
with nef normal sheaf $N_C$.\vskip0pt
\endclaim
%
Corollary 6.28~(i) strenghtens results from [PSS99]. It is however 
not yet clear whether 
$\cM_{\NS}$ is equal to the closed cone of curves with 
{\emph ample} normal bundle (although we certainly expect this to be true).
The most important special case of Theorem 6.23 is

\claim{(6.29) Theorem} If $X$ is a projective manifold, then
$K_X$ is pseudo-effective $($i.e.\ $K_X\in\cE_{\NS})$,
if and only if $X$ is not uniruled $($i.e.\ not covered by rational curves$)$.
\endclaim
%
\begproof{Proof.} If $X$ is covered by rational curves $C_t$, then it is well-known
that the normal bundle $N_{C_t}$ is nef for a general member $C_t$, thus
$$
K_X\cdot C_t=K_{C_t}\cdot C_t-N_{C_t}\cdot C_t\le -2,
$$
and $K_X$ cannot be pseudo-effective. Conversely,
if $K_X\notin\cE_{\NS}$, Theorem 6.23 shows that
there is a mobile curve $C_t$ such that $K_X\cdot C_t<0$. The standard
``bend-and-break'' lemma of Mori theory then produces a covering
family $\Gamma_t$ of 
{\rm rational curves} with $K_X\cdot\Gamma_t<0$, so $X$ is uniruled.\qed
\endproof
%
Notice that the generalized abundance conjecture 6.16 would then yield
the stronger result~:

\claim{(6.30) Conjecture} Let $X$ be a projective manifold. 
If $X$ is not uniruled, then $K_X$ is a $\bQ$-effective
divisor and \hbox{$\kappa(X)=\nd(K_X)\ge 0$}.
\endclaim
\endsectionrunning
%
\section{Super-canonical metrics and abundance}
\subsection*{\csect.A. Construction of super-canonical metrics}

Let $X$ be a compact complex manifold and $(L,h_{L,\gamma})$ a holomorphic
line bundle over $X$ equipped with a singular hermitian metric
$h_{L,\gamma}=e^{-\gamma}h_L$ with satisfies $\int e^{-\gamma}<+\infty$
locally on~$X$, where $h_L$ is a smooth metric on~$L$. In fact, we can 
more generally consider the case where
$(L,h_{L,\gamma})$ is a ``hermitian $\bR$-line bundle''; by this we mean that
we have chosen a smooth real $d$-closed $(1,1)$ form $\alpha_L$ on~$X$
(whose $dd^c$ cohomology class is equal to $c_1(L))$, and a specific 
current $T_{L,\gamma}$ representing it, namely
$T_{L,\gamma}=\alpha_L+dd^c\gamma$, such that $\gamma$ is a locally
integrable function satisfying~$\int e^{-\gamma}<+\infty$. An important
special case is obtained by considering a klt (Kawamata log terminal) 
effective divisor~$\Delta$. In this situation
$\Delta=\sum c_j\Delta_j$ with $c_j\in\bR$, and if $g_j$ is a local
generator of the ideal sheaf $\cO(-\Delta_j)$ identifying it to the trivial
invertible sheaf $g_j\cO$, we take
$\gamma=\sum c_j\log|g_j|^2$, $T_{L,\gamma}=\sum c_j[\Delta_j]$
(current of integration on $\Delta$) and $\alpha_L$ given by any smooth
representative of the same $dd^c$-cohomology class; the klt condition
precisely means that
$$
\int_V e^{-\gamma}=\int_V \prod|g_j|^{-2c_j}<+\infty
\leqno(7.1)
$$ 
on a small neighborhood~$V$ of any point in the support 
$|\Delta|=\bigcup\Delta_j$ (condition (7.1) implies $c_j<1$
for every $j$, and this in turn is sufficient to imply $\Delta$ klt
if $\Delta$ is a normal crossing divisor; the line bundle $L$ is 
then the real line bundle $\cO(\Delta)$, which makes sens as a
genuine line bundle only if $c_j\in\bZ$).
For~each klt pair $(X,\Delta)$ such that $K_X+\Delta$
is pseudo-effective, H.~Tsuji [Tsu07a, Tsu07b] has introduced a
``super-canonical metric'' which generalizes the metric introduced
by Narasimhan and Simha [NS68] for projective algebraic varieties
with ample canonical divisor. We take the opportunity to present
here a simpler, more direct and more general approach. 

We assume from now on that $K_X+L$ is {\emph pseudo-effective}, i.e.\ 
that the class
$c_1(K_X)+\{\alpha_L\}$ is pseudo-effective, and under this condition, we
are going to define a ``super-canonical metric'' on $K_X+L$. Select
an arbitrary smooth hermitian metric $\omega$ on $X$. We then find 
induced hermitian
metrics $h_{K_X}$ on $K_X$ and $h_{K_X+L}=h_{K_X}h_L$ on $K_X+L$, whose 
curvature is the smooth real $(1,1)$-form
$$
\alpha=\Theta_{K_X+L,h_{K_X+L}}=\Theta_{K_X,\omega}+\alpha_L.
$$
A singular hermitian metric
on $K_X+L$ is a metric of the form $h_{K_X+L,\varphi}=e^{-\varphi}h_{K_X+L}$
where $\varphi$ is locally
integrable, and by the pseudo-effectivity assumption, we can find 
quasi-psh functions $\varphi$ such that $\alpha+dd^c\varphi\ge 0$.
The metrics on $L$ and $K_X+L$ can now be ``subtracted'' to give rise
to a metric
$$
h_{L,\gamma}h_{K_X+L,\varphi}^{-1}=e^{\varphi-\gamma}h_Lh_{K_X+L}^{-1}
=e^{\varphi-\gamma}h_{K_X}^{-1}=e^{\varphi-\gamma}dV_\omega
$$
on $K_X^{-1}=\Lambda^nT_X$, since $\smash{h_{K_X}^{-1}}=dV_\omega$ is just
the hermitian $(n,n)$ volume form on~$X$. Therefore the integral
$\int_X h_{L,\gamma}h_{K_X+L,\varphi}^{-1}$ has an intrinsic meaning, and 
it makes sense to require that
$$
\int_X h_{L,\gamma}h_{K_X+L,\varphi}^{-1}
=\int_X e^{\varphi-\gamma}dV_\omega\le 1\leqno(7.2)
$$
in view of the fact that $\varphi$ is locally bounded from above and of
the assumption $\int e^{-\gamma}<+\infty$. Observe that condition (7.2) can
always be achieved by subtracting a constant to $\varphi$. Now, 
we can generalize Tsuji's super-canonical metrics on klt pairs
(cf.\ [Tsu07b]) as follows.

\claim{(7.3) Definition} Let $X$ be a compact 
complex manifold and let $(L,h_L)$ be a hermitian $\bR$-line bundle on $X$
associated with a smooth real closed $(1,1)$ form~$\alpha_L$.
Assume that $K_X+L$ is pseudo-effective and that $L$ is
equipped with a singular hermitian metric $h_{L,\gamma}=e^{-\gamma}h_L$ 
such that \hbox{$\int e^{-\gamma}<+\infty$} locally on~$X$. Take a hermitian
metric $\omega$ on $X$ and define 
$\alpha=\Theta_{K_X+L,h_{K_X+L}}=\Theta_{K_X,\omega}+\alpha_L$.
Then we define the super-canonical metric $h_\can$ of $K_X+L$ to be 
$$
\eqalign{
&h_{K_X+L,\can}=\inf_\varphi h_{K_X+L,\varphi}\quad
\hbox{i.e.}\quad h_{K_X+L,\can}=e^{-\varphi_{\can}}h_{K_X+L},~~
\hbox{where}\cr
&\varphi_{\can}(x)=\sup_\varphi\varphi(x)~~
\hbox{for all $\varphi$ with}~~\alpha+dd^c\varphi\ge 0,
~~\int_X e^{\varphi-\gamma}dV_\omega\le 1.\cr}
$$
\endclaim
%
In particular, this gives a definition of the super-canonical metric
on $K_X+\Delta$ for every klt pair $(X,\Delta)$ such that
$K_X+\Delta$ is pseudo-effective, and as an even more special case,
a super-canonical metric on $K_X$ when $K_X$ is pseudo-effective.

In the sequel, we assume that $\gamma$ has analytic singularities, otherwise
not much can be said. The mean value inequality then immediately shows that the 
quasi-psh functions $\varphi$ involved in definition (7.3) 
are globally uniformly bounded outside of the poles of $\gamma$, and
therefore everywhere on $X$, hence the envelopes
$\varphi_\can=\sup_\varphi\varphi$ are indeed well defined and bounded
above. As a consequence, we get a ``super-canonical'' current
\hbox{$T_\can=\alpha+dd^c\varphi_\can\ge 0$} and $h_{K_X+L,\can}$ satisfies
$$
\int_Xh_{L,\gamma}h_{K_X+L,\can}^{-1}=
\int_X e^{\varphi_\can-\gamma}dV_\omega<+\infty.\leqno(7.4)
$$
It is easy to see that in Definition (7.3) the supremum is a maximum and that 
$\varphi_\can=(\varphi_\can)^*$ every\-where, so that
taking the upper semicontinuous regularization is not needed. In fact
if $x_0\in X$ is given and we write
$$
(\varphi_\can)^*(x_0)=\limsup_{x\to x_0}\varphi_\can(x)
=\lim_{\nu\to+\infty}\varphi_\can(x_\nu)
=\lim_{\nu\to+\infty}\varphi_\nu(x_\nu)
$$
with suitable sequences $x_\nu\to x_0$ and $(\varphi_\nu)$ such that
$\int_Xe^{\varphi_\nu-\gamma}dV_\omega\le 1$, the well-known 
weak compactness properties of quasi-psh functions in  $L^1$ topology 
imply the existence of
a subsequence of $(\varphi_\nu)$ converging in $L^1$ and almost everywhere to 
a quasi-psh limit $\varphi$. Since $\int_Xe^{\varphi_\nu-\gamma}dV_\omega
\le 1$ holds true for every~$\nu$, Fatou's lemma 
implies that we have $\int_Xe^{\varphi-\gamma}dV_\omega\le 1$ in
the limit. By taking a subsequence, we can 
assume that $\varphi_\nu\to\varphi$ in $L^1(X)$. Then for every $\varepsilon>0$
the mean value $-\kern-8.7pt\int_{B(x_\nu,\varepsilon)}\varphi_\nu$ 
satisfies
$$
-\kern-10.7pt\int_{B(x_0,\varepsilon)}\varphi=\lim_{\nu\to +\infty}
-\kern-10.7pt\int_{B(x_\nu,\varepsilon)}\varphi_\nu\ge \lim_{\nu\to+\infty}
\varphi_\nu(x_\nu)=(\varphi_\can)^*(x_0),
$$
hence we get $\varphi(x_0)=\lim_{\varepsilon\to 0}
-\kern-8.7pt\int_{B(x_0,\varepsilon)}\varphi\ge 
(\varphi_\can)^*(x_0)\ge \varphi_\can(x_0)$, and therefore the 
sup is a maximum and $\varphi_\can=\varphi_\can^*$.
By elaborating on this argument, one can infer certain regularity
properties of the envelope. 

\claim{(7.5) Theorem {\rm ([BmD09])}} Let $X$ be a compact complex
manifold and $(L,h_L)$ a holomorphic $\bR$-line bundle such that
$K_X+L$ is big. Assume that $L$ is equipped with a singular hermitian
metric $h_{L,\gamma}=e^{-\gamma}h_L$ with analytic singularities such
that $\int e^{-\gamma}<+\infty$ $($klt condition$)$. Denote by $Z_0$
the set of poles of a singular metric $h_0=e^ {-\psi_0}h_{K_X+L}$ with
analytic singularities on~$K_X+L$ and by $Z_\gamma$ the poles
of~$\gamma$ $($assumed analytic$)$.  Then the associated
super-canonical metric $h_\can$ is continuous on $X\ssm (Z_0\cup
Z_\gamma)$.
\endclaim
%
In fact, using the regularization techniques of [Dem94], it is shown
in [BmD09] that $h_\can$ possesses some computable logarithmic modulus of
continuity. In order to shorten the exposition, we will only give a proof 
of the continuity in the algebraic case, using approximation by 
pluri-canonical sections.

\claim{(7.6) Algebraic version of the super-canonical metric} 
{\rm Since the klt condition is open 
and $K_X+L$ is assumed to be~big, we can always perturb $L$ a 
little bit, and after
blowing-up $X$, assume that $X$ is projective and that
$(L,h_{L,\gamma})$ is obtained as a sum of $\bQ$-divisors
$$
L=G+\Delta
$$
where $\Delta$ is klt and $G$ is equipped with a smooth
metric $h_G$ (from which $h_{L,\gamma}$ is inferred, with $\Delta$
as its poles, so that $\Theta_{L,h_{L,\gamma}}=\Theta_{G,L_G}+[\Delta]$).
Clearly this situation is ``dense'' in what we
have been considering before, just as $\bQ$ is dense in $\bR$.
In this case, it is possible to 
give a more algebraic definition of the super-canonical
metric $\varphi_\can$, following the original idea of Narasimhan-Simha
[NS68] (see also H.~Tsuji [Tsu07a])~--~the case considered by these
authors is the special situation where $G=0$, $h_G=1$ (and moreover
$\Delta=0$ and $K_X$ ample, for [NS68]). In fact, if $m$ is a 
large integer which is a multiple of the denominators involved in
$G$ and $\Delta$, we can consider sections
$$\sigma\in H^0(X,m(K_X+G+\Delta)).$$
We view them rather as sections of $m(K_X+G)$ with poles along the 
support~$|\Delta|$ of our divisor. Then 
$(\sigma\wedge \overline\sigma)^{1/m}h_G$
is a volume form with
integrable poles along~$|\Delta|$ (this is the klt condition for~$\Delta$).
Therefore one can normalize $\sigma$ by requiring that
$$
\int_X (\sigma\wedge \overline\sigma)^{1/m}h_G=1.
$$
Each of these sections defines a singular hermitian metric on 
$K_X+L=K_X+G+\Delta$, and we can take the regularized upper envelope 
$$
\varphi_\can^\alg=\bigg(\sup_{m,\sigma}{1\over m}\log|\sigma|_{h_{K_X+L}^m}^2
\bigg)^*\leqno(7.7)
$$
of the weights associated with a smooth metric~$h_{K_X+L}$. It is clear that
$\varphi_\can^\alg\le \varphi_\can$ since the supremum is
taken on the smaller set of weights $\varphi={1\over m}
\smash{\log|\sigma|^2_{h_{K_X+L}^m}}$, and the equalities
$$
e^{\varphi-\gamma}dV_\omega=|\sigma|^{2/m}_{h_{K_X+L}^m}
e^{-\gamma}dV_\omega=(\sigma\wedge\overline\sigma)^{1/m}e^{-\gamma}h_L
=(\sigma\wedge\overline\sigma)^{1/m}h_{L,\gamma}
=(\sigma\wedge\overline\sigma)^{1/m}h_G
$$
imply $\int_X e^{\varphi-\gamma}dV_\omega\le 1$. We~claim that the inequality
$\varphi_\can^\alg\le \varphi_\can$ is an equality. The proof is an 
immediate consequence of the following
statement based in turn on the Ohsawa-Takegoshi theorem and
the approximation technique of [Dem92].}
\endclaim
%
\claim{(7.8) Proposition} With $L=G+\Delta$, $\omega$, 
$\alpha=\smash{\Theta_{K_X+L,h_{K_X+L}}}$, $\gamma$ 
as above and $K_X+L$ assumed to be~big, fix a singular hermitian metric 
$e^{-\varphi}h_{K_X+L}$ of curvature $\alpha+dd^c\varphi\ge 0$, such that
$\int_X e^{\varphi-\gamma}dV_\omega\le 1$. Then
$\varphi$ is equal to a regularized limit
\vskip-5pt
$$
\varphi=\bigg(\limsup_{m\to+\infty}{1\over m}\log|\sigma_m|_{h_{K_X+L}^m}^2
\bigg)^*
$$ 
for a suitable sequence $\sigma_m\in H^0(X,m(K_X+G+\Delta))$ with
$\int_X(\sigma_m\wedge\overline\sigma_m)^{1/m}h_G\le 1$.
\endclaim
%
\begproof{Proof.} By our assumption, there exists a quasi-psh 
function $\psi_0$ with
analytic singularity set $Z_0$ such that
$$
\alpha+dd^c\psi_0\ge \varepsilon_0\omega>0
$$
and we can assume $\int_Ce^{\psi_0-\gamma}dV_\omega<1$ (the strict inequality
will be useful later). For
$m\ge p\ge 1$, this defines a singular metric 
$\exp(-(m-p)\varphi-p\psi_0)h_{K_X+L}^m$
on $m(K_X+L)$ with curvature${}\ge p\varepsilon_0\omega$, and therefore
a singular metric 
$$
h_{L'}=\exp(-(m-p)\varphi-p\psi_0)h_{K_X+L}^mh_{K_X}^{-1}
$$
on $L'=(m-1)K_X+mL$, whose curvature
$\Theta_{L',h_{L'}}\ge (p\varepsilon_0-C_0)\omega$ is arbitrary large
if $p$ is large enough. Let us fix a finite covering of $X$
by coordinate balls. Pick a point $x_0$ and one of the coordinate
balls $B$ containing~$x_0$. By the Ohsawa-Takegoshi extension theorem
applied on the ball~$B$,
we can find a section $\sigma_B$ of $K_X+L'=m(K_X+L)$
which has norm $1$ at $x_0$ with respect to the metric
$h_{K_X+L'}$ and $\int_B|\sigma_B|_{h_{K_X+L'}}^2dV_\omega\le C_1$ for some
uniform constant $C_1$ depending on the finite covering, but independent of 
$m$, $p$, $x_0$ . Now, we use a cut-off function $\theta(x)$ with
$\theta(x)=1$ near $x_0$ to truncate
$\sigma_B$ and solve a $\dbar$-equation for $(n,1)$-forms with values
in $L$ to get a global section $\sigma$ on $X$ with 
$|\sigma(x_0)|_{h_{K_X+L'}}=1$. For this we need to multiply our metric
by a truncated factor $\exp(-2n\theta(x)\log|x-x_0|)$ so as to get
solutions of $\dbar$ vanishing at~$x_0$. However, this perturbs the
curvature by bounded terms and we can absorb them again by taking
$p$ larger. In~this way we obtain
$$
\int_X|\sigma|^2_{h_{K_X+L'}}dV_\omega=
\int_X |\sigma|^2_{h_{K_X+L}^m}e^{-(m-p)\varphi-p\psi_0}dV_\omega\le C_2.
\leqno(7.9)
$$
Taking $p>1$, the H\"older inequality for congugate exponents
$m$, ${m\over m-1}$ implies
$$
\eqalign{
\int_X(\sigma\wedge\overline\sigma)^{1\over m}h_G
&=\int_X|\sigma|^{2/m}_{h_{K_X+L}^m}e^{-\gamma}dV_\omega\cr
&=\int_X\Big(|\sigma|^2_{h_{K_X+L}^m}e^{-(m-p)\varphi-p\psi_0}\Big)^{1\over m}
\Big(e^{(1-{p \over m})\varphi+{p\over m}\psi_0-\gamma}\Big)dV_\omega\cr
&\le C_2^{1\over m}\bigg(\int_X
\Big(e^{(1-{p\over m})\varphi+{p\over m}\psi_0-\gamma}
\Big)^{m\over m-1}dV_\omega\bigg)^{m-1\over m}\cr
&\le C_2^{1\over m}\bigg(\int_X
 \big(e^{\varphi-\gamma}\big)^{m-p\over m-1}
\Big(e^{{p\over p-1}(\psi_0-\gamma)}\Big)^{p-1\over m-1}dV_\omega
\bigg)^{m-1\over m}\cr
&\le C_2^{1\over m}
\bigg(\int_X e^{{p\over p-1}(\psi_0-\gamma)}dV_\omega
\bigg)^{p-1\over m}\cr}
$$
using the hypothesis $\int_X e^{\varphi-\gamma}dV_\omega\le 1$ and another
application of H\"older's inequality. Since 
klt is an open condition and $\lim_{p\to+\infty}
\smash{\int_Xe^{{p\over p-1}(\psi_0-\gamma)}dV_\omega}
=\int_Xe^{\psi_0-\gamma}dV_\omega<1$, we can take
$p$ large enough to ensure that 
$$
\int_X e^{{p\over p-1}(\psi_0-\gamma)}dV_\omega\le C_3<1.
$$
Therefore, we see that 
$$
\int_X(\sigma\wedge\overline\sigma)^{1\over m}h_G\le 
C_2^{1\over m}C_3^{p-1\over m}\le 1
$$
for $p$ large enough. On the other hand
$$
|\sigma(x_0)|^2_{h_{K_X+L'}}
=|\sigma(x_0)|^2_{h_{K_X+L}^m}e^{-(m-p)\varphi(x_0)-p\psi_0(x_0)}=1,
$$
thus
$$
{1\over m}\log |\sigma(x_0)|^2_{h_{K_X+L}^m}=
\Big(1-{p\over m}\Big)\varphi(x_0)+{p\over m}\psi_0(x_0)
\leqno(7.10)
$$
and, as a consequence
$$
{1\over m}\log |\sigma(x_0)|^2_{h_{K_X+L}^m}\longrightarrow\varphi(x_0)
$$
whenever $m\to+\infty$, ${p\over m}\to 0$, as long as
$\psi_0(x_0)>-\infty$. In the above argument, we can in fact interpolate
in finitely many points $x_1,\,x_2,\,\ldots\,,x_q$ provided that
$p\ge C_4q$. Therefore if we take a suitable dense subset
$\{x_q\}$ and a ``diagonal'' sequence associated with sections
$\sigma_m\in H^0(X,m(K_X+L))$ with 
$m\gg p=p_m\gg q=q_m\to+\infty$, we infer that
$$
\bigg(\limsup_{m\to+\infty}{1\over m}\log |\sigma_m(x)|^2_{h_{K_X+L}^m}\bigg)^*\ge
\limsup_{x_q\to x}\;\varphi(x_q)=\varphi(x)\leqno(7.11)
$$
(the latter equality occurring if $\{x_q\}$ is suitably chosen with
respect to $\varphi$). In the other direction, (7.9)
implies a mean value estimate
$$
{1\over \pi^nr^{2n}/n!}\int_{B(x,r)}
|\sigma(z)|^2_{h_{K_X+L}^m}\;dz\le {C_5\over r^{2n}}
\sup_{B(x,r)}e^{(m-p)\varphi+p\psi_0}
$$
on every coordinate ball $B(x,r)\subset X$. The function 
$|\sigma_m|^2_{h_{K_X+L}^m}$ is 
plurisubharmonic after we correct the
non necessarily positively curved smooth metric $h_{K_X+L}$ by a 
factor of the form $\exp(C_6|z-x|^2)$, hence the mean value 
inequality shows that
$$
{1\over m}\log |\sigma_m(x)|^2_{h_{K_X+L}^m}\le 
{1\over m}\log{C_5\over r^{2n}}+C_6 r^2
+\sup_{B(x,r)}\Big(1-{p_m\over m}\Big)\varphi+{p_m\over m}\psi_0.
$$
By taking in particular $r=1/m$ and letting $m\to+\infty$, $p_m/m\to 0$, 
we see that the opposite of inequality (7.9) also holds.\qed
\endproof

\claim{(7.12) Remark} \rm We can rephrase our results in slightly different
terms. In fact, let us put 
$$
\varphi^\alg_m=\sup_{\sigma}{1\over m}\log|\sigma|_{h_{K_X+L}^m}^2,
\qquad \sigma\in H^0(X,m(K_X+G+\Delta)),
$$
with normalized sections $\sigma$
such that \hbox{$\int_X (\sigma\wedge \overline\sigma)^{1/m}h_G=1$}.
Then $\varphi^\alg_m$ is quasi-psh (the supremum is taken over a
compact set in a finite dimensional vector space) and by passing 
to the regularized supremum over all $\sigma$ and all $\varphi$ 
in (7.10) we get
$$
\varphi_\can\ge\varphi^\alg_m\ge \Big(1-{p\over m}\Big)
\varphi_\can(x)+{p\over m}\psi_0(x).
$$
As $\varphi_\can$ is bounded from above, we find in particular
$$
0\le \varphi_\can-\varphi^\alg_m\le {C\over m}(|\psi_0(x)|+1).
$$
This implies that $(\varphi^\alg_m)$ converges uniformly to
$\varphi_\can$ on every compact subset of $X\subset Z_0$, and in this way
we infer again (in a purely qualitative manner) that $\varphi_\can$ 
is continuous on $X\ssm Z_0$. Moreover, we also see that in (7.7) 
the upper  semicontinuous regularization is not needed on
$X\ssm Z_0\,$; in case $K_X+L$ is ample, it is not needed at all
and we have uniform convergence of $(\varphi^\alg_m)$ towards
$\varphi_\can$ on the whole of~$X$. Obtaining such a uniform convergence
when $K_X+L$ is just big looks like a more delicate question, 
related e.g.\ to abundance
of $K_X+L$ on those subvarieties $Y$ where the restriction
$(K_X+L)_{|Y}$ would be e.g.\ nef but not big.
\endclaim
%
\claim{(7.13) Generalization} \rm In the general case
where $L$ is a $\bR$-line bundle and \hbox{$K_X+L$} is merely pseudo-effective, 
a similar algebraic approximation can be obtained. We take instead sections
$$
\sigma\in H^0(X,mK_X+\lfloor mG\rfloor+\lfloor m\Delta\rfloor +p_mA)
$$
where $(A,h_A)$ is a positive line bundle, 
$\Theta_{A,h_A}\ge \varepsilon_0\omega$,
and replace the definition of $\varphi_\can^\alg$ by
$$
\leqalignno{
&\varphi_\can^\alg=
\bigg(\limsup_{m\to+\infty}\;
\sup_\sigma\;{1\over m}\log|\sigma|_{h_{mK_X+\lfloor mG\rfloor+p_mA}}^2
\bigg)^*,&(7.14)\cr
&\int_X(\sigma\wedge\overline\sigma)^{2\over m}
h_{\lfloor mG\rfloor +p_mA}^{1\over m}\le 1,&(7.15)\cr}
$$
where $m\gg p_m\gg 1$ and $h_{\lfloor mG\rfloor}^{1/m}$ is chosen
to converge uniformly to $h_G$.

 We then find again $\varphi_\can=\varphi_\can^\alg$, with
an almost identical proof -- though we no longer have a $\sup$ in the
envelope, but just a $\limsup$. The analogue of Proposition (7.8) also
holds true in this context, with an appropriate sequence of sections
$\sigma_m\in H^0(X,mK_X+\lfloor mG\rfloor+\lfloor m\Delta\rfloor +p_mA)$.
\endclaim
%
\claim{(7.16) Remark} \rm It would be nice to have a better understanding
of the super-canonical metrics. In case $X$ is a curve, this should be easier.
In fact $X$ then has a hermitian metric $\omega$ with constant curvature,
which we normalize by requiring that $\int_X\omega=1$, and we can also
suppose $\int_Xe^{-\gamma}\omega=1$. The class
$\lambda=c_1(K_X+L)\ge 0$ is a number and we take $\alpha=\lambda\omega$. 
Our envelope is $\varphi_\can=\sup\varphi$ where 
$\lambda\omega+dd^c\varphi\ge 0$ and
$\int_Xe^{\varphi-\gamma}\omega\le 1$. If $\lambda=0$ then $\varphi$
must be constant and clearly $\varphi_\can=0$. \hbox{Otherwise}, if $G(z,a)$ 
denotes the Green function such that $\int_X G(z,a)\,\omega(z)=0$ and
$dd^cG(z,a)=\delta_a-\omega(z)$, we find
$$
\varphi_\can(z)\ge \sup_{a\in X}\bigg(\lambda
G(z,a)-\log\int_{z\in X}e^{\lambda G(z,a)-\gamma(z)}\omega(z)\bigg)
$$
by taking already the envelope over $\varphi(z)=\lambda G(z,a)-\hbox{Const}$. 
It is natural
to ask whether this is always an equality, i.e.\ whether the extremal
functions are always given by one of the Green functions, especially
when~$\gamma=0$.
\endclaim
%
\subsection*{\csect.B. Invariance of plurigenera and positivity of curvature of
super-cano\-nical metrics}
The concept of super-canonical metric can be used to give a very interesting
result on the positivity of relative pluricanonical
divisors, which itself can be seen to imply the invariance of plurigenera.
The main idea is due to H.~Tsuji [Tsu07a], and some important details
were fixed by Berndtsson and P\u{a}un [BnP09], using techniques inspired
from their results on positivity of direct images [Bnd06], [BnP08].

\claim{(7.17) Theorem} Let $\pi:\cX\to S$ be a deformation of projective
algebraic manifolds over some irreducible complex space~$S$ $(\pi$ being
assumed locally projective over~$S)$. Let
$\cL\to\cX$ be a holomorphic line bundle equipped with a hermitian
metric $h_{\cL,\gamma}$ of weight $\gamma$ such that $\ii\smash{\Theta_{\cL,
h_{\cL,\gamma}}}\ge 0$ $($i.e.\ $\gamma$ is plurisubharmonic$)$, and
$\smash{\int_{X_t}}e^{-\gamma}<+\infty$, i.e.\ we assume the metric to
be klt over all fibers $X_t=\pi^{-1}(t)$. Then the metric defined
on $K_{\cX}+\cL$ as the fiberwise super-canonical metric has semi-positive
curvature over~$\cX$. In particular, 
$t\mapsto h^0(X_t,m(K_{X_t}+\cL_{\restriction X_t}))$ is constant for 
all~\hbox{$m>0$}.
\endclaim
%
Once the metric is known to have a plurisuharmonic weight on the total
space of $\cX$, the Ohsawa-Takegoshi theorem can be used exactly as
at the end of the proof of lemma (12.3). Therefore the final statement
is just an easy consequence. The cases when $\cL=\cO_\cX$ is trivial
or when $\cL_{\restriction X_t}=\cO(\Delta_t)$ for a family of klt $\bQ$-divisors
are especially interesting.

\begproof{Proof {\rm (Sketch)}.} By our assumptions, there exists (at least
locally over $S$) a rela\-tively ample line bundle $\cA$ over~$\cX$. We
have to show that the weight of the global super-canonical metric is
plurisubharmonic, and for this, it is enough to look at analytic
disks $\Delta\to S$. We may thus as well assume that $S=\Delta$ 
is the unit disk.  Consider the super-canonical metric
$h_{\can,0}$ over the fiber $X_0$.  The approximation argument seen
above (see (7.9) and remark (7.13)) show that $h_{\can,0}$ has a
weight $\varphi_{\can, 0}$ which is a regularized upper limit
$$\varphi_{\can,0}^\alg=
\bigg(\limsup_{m\to+\infty}\;{1\over m}\log|\sigma_m|^2
\bigg)^*
$$
defined by sections 
$\sigma_m\in H^0(X_0,m(K_{X_0}+\cL_{\restriction X_0})+
p_m\cA_{\restriction X_0})$ such that
$$
\int_{X_0} |\sigma|^2e^{-(m-p_m)\varphi_{\can,0}-p_m\psi_0}dV_\omega\le C_2.
$$
with the suitable weights. Now, by section 12, these sections
extend to sections $\widetilde\sigma_m$ defined on the whole family~$\cX$, 
satisfying a similar $L^2$ estimate (possibly with a slightly larger constant
$C'_2$ under control). If we set 
$$
\Phi=\bigg(\limsup_{m\to+\infty}\;{1\over m}\log|\widetilde\sigma_m|^2
\bigg)^*,
$$
then $\Phi$ is plurisubharmonic by construction, and 
$\varphi_\can\ge \Phi$ by the defining property of the super-canonical 
metric. Finally, we also have $\varphi_{\can,0}=\Phi_{\restriction X_0}$
from the approximation technique. It follows easily that $\varphi_\can$
satisfies the mean value inequality with respect to any disk centered
on the central fiber~$X_0$. Since we can consider arbitrary analytic disks
$\Delta\to S$, the plurisubharmonicity of $\varphi_\can$ follows.\qed
\endproof
%
\subsection*{\csect.C. Tsuji's strategy for studying abundance}
H.~Tsuji [Tsu07c] has recently proposed the following interesting
prospective approach of the abundance conjecture.

\claim{(7.18) Conjecture/question} Let $(X,\Delta)$ be a klt pair such that
$K_X+\Delta$ is pseudoeffective and has numerical dimension
$\nd(K_X+\Delta)>0$. Then for every point $x\in X$ there exists a closed
positive current $T_x\in c_1(K_X+\Delta)$ such that the Lelong number at 
$x$ satisfies $\nu(T_x,x)>0$.
\endclaim
%
It would be quite tempting to try to produce such currents e.g.\ by a suitable
modification of the construction of super-canonical metrics, trying to
enforce singularities of the metric at any prescribed point $x\in X$. 
A~related procedure would be to enforce enough vanishing of sections
of $A+m(K_X+\Delta)$ at point $x$, where $A$ is a sufficiently ample line
bundle. The number of these sections grows as $cm^p$ where 
$p=\nd(K_X+\Delta)$. Hence, by an easy linear algebra argument, one can 
prescribe a vanishing order $s\sim c'm^{p/n}$ of such a section $\sigma$, whence
a Lelong number${}\sim c'm^{{p\over n}-1}$ for the corresponding
rescaled current of integration $T={1\over m}[Z_\sigma]$ on the
zero divisor. Unfortunately, this tends to $0$ as $m\to+\infty$ whenever
$p<n$. Therefore, one should use a more clever argument which takes 
into account the fact that, most probably, all directions do not behave
in an ``isotropic way'', and vanishing should be prescribed only 
in certain directions.

Assuming that (7.18) holds true, a simple semi-continuity argument 
would imply that there exists a small number $c>0$ such
that the analytic set $Z_x=E_c(T_x)$ contains $x$, and one would expect
conjecturally that these sets can be reorganized as the generic fibers
of a reduction map $f:X\merto Y$, together with a
klt divisor $\Delta'$ on $Y$ such that (in first approximation, and 
maybe only after replacing $X$, $Y$ by suitable blow-ups), one has
$K_X+\Delta=f^*(K_Y+\Delta'+R_f)+\beta$ where $R_f$ is a suitable orbifold
divisor (in the sense of Campana [Cam04]) and $\beta$ a suitable
pseudo-effective class. The expectation is that $\dim Y=p=\nd(K_X+\Delta)$
and that $(Y,\Delta')$ is of general type, i.e.\ 
$\nd(K_Y+\Delta')=p$.

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

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

In [BCHM06], Birkar, Cascini, Hacon and McKernan proved several
fundamental conjectures which had been expected for more than two
decades, concerning the existence of minimal models and the finiteness of
the canonical ring for arbitrary projective varieties. The latter
result was also announced independently by Siu in [Siu06]. The main
results can be summarized in the following statement.

\claim{(8.1) Theorem} Let $(X,\Delta)$ be a klt pair where $\Delta$ is big.
{\plainitemindent=7mm
\vskip3pt
\plainitem{\rm (i)} If $K_X+\Delta$ is pseudo-effective, $(X,\Delta)$
has a log-minimal model, i.e.\ there is a birational transformation 
$(\widetilde X,\widetilde\Delta)$ with $\widetilde X$  $\bQ$-factorial,
such that $K_{\widetilde X}+\widetilde\Delta$ is nef and
satisfies additionally strict inequalities for the discrepancies
of $\mu$-exceptional divisors.
\vskip3pt
\plainitem{\rm (ii)} If $K_X+\Delta$ is not pseudo-effective, then $(X,\Delta)$
has a Mori fiber space, i.e.\ there exists a birational transformation
$(\widetilde X,\widetilde\Delta)$ and a morphism $\varphi:\widetilde X\to Y$ such that
$-(K_{\widetilde X}+\widetilde\Delta)$ is $\varphi$-ample.
\vskip3pt
\plainitem{\rm (iii)} If moreover $\Delta$ is a $\bQ$-divisor, the log-canonical ring
$\bigoplus_{m\ge 0} H^0(X,m(K_X+\Delta))$
is finitely generated.\vskip0pt}
\endclaim
%
The proof, for which we can only refer to [BCHM06], is an extremely
subtle induction on dimension involving finiteness of flips (a certain
class of birational transforms improving positivity of $K_X+\Delta$
step by step), and a generalization of Shokurov's non vanishing
theorem [Sho85]. The original proof of this non vanishing result was
itself based on an induction on dimension, using the existence
of minimal models in dimension $n-1$. Independently, Y.T.\ Siu
[Siu06] announced an analytic proof of the finiteness of canonical
rings $\bigoplus_{m\ge 0} H^0(X,mK_X)$, along with an analytic variant
of Shokurov's non vanishing theorem; in his approach, multiplier 
ideals and Skoda's division theorem are used in crucial ways.
Let us mention a basic statement in this direction which illustrates the 
connection with Skoda's result, and is interesting for two reasons~:
i) it does not require any strict positivity assumption, ii) it shows that
it is enough to have a sufficiently good approximation of the
minimal singularity metric $h_{\min}$ by sections of sufficiently large 
linear systems $|pK_X|$.

\claim{(8.2) Proposition} Let $X$ be a projective $n$-dimensional
manifold with $K_X$ pseudo-effective. Let $h_{\min}=e^{-\varphi_{\min}}$ be
a metric with minimal singularity on $K_X$ $($e.g.\ 
the super-canonical metric), and
let $c_0>0$ be the log canonical threshold of $\varphi_{\min}$, i.e.\
$h_{\min}^{c_0-\delta}=e^{-(c_0-\delta)\varphi_{\min}}
\in L^1$ for $\delta>0$ small. Assume that there exists an 
integer $p>0$ so that the linear system $|pK_X|$ provides a weight
$\psi_p=\smash{1\over p}\log\sum|\sigma_j|^2$ whose singularity
approximates $\varphi_{\min}$ sufficiently well, namely
$$
\psi_p\ge\Big(1+{1+c_0-\delta\over pn}\Big)\varphi_{\min}+O(1)\qquad
\hbox{for some $\delta>0$}.
$$
Then $\smash\bigoplus_{m\ge 0}H^0(X,mK_X)$ is finitely 
generated, and a set of generators is actually provided by 
a basis of sections of 
$\smash\bigoplus_{0\le m\le np+1}H^0(X,mK_X)$.
\endclaim
%
\begproof{Proof.} A simple argument based on the curve selection lemma (see
e.g.\ [Dem01], Lemma 11.16) shows that one can extract a system
$g=(g_1,\ldots g_n)$ of at most $n$
sections from $(\sigma_j)$ in such a way that
the singularities are unchanged, i.e.\ 
$C_1\log|\sigma|\le \log|g|\le C_2\log|\sigma|$. We apply
Skoda's division (7.20)
with $E=\cO_X^{\oplus n}$, $Q=\cO(pK_X)$ and $L=\cO((m-p-1)K_X)$
[so that $K_X\otimes Q\otimes L=\cO_X(mK_X)$],
and with the metric induced by $h_{\min}$ on $K_X$. By definition of a metric
with minimal singularities, every section $f$ in $H^0(X,mK_X)=
H^0(X,K_X\otimes Q\otimes L)$
is such that $|f|^2\le Ce^{m\varphi_{\min}}$. The weight of the metric on
$Q\otimes L$ is $(m-1)\varphi_{\min}$. Accordingly, we find 
$$
\eqalign{
|f|^2|g|_{h_{min}}^{-2n-2\varepsilon}e^{-(m-1)\varphi_{\min}}
&\le C \exp\Big(m\varphi_{\min}-p(n+\varepsilon)(\psi_p-\varphi_{\min})
-(m-1)\varphi_{\min}\Big)\cr
&\le C'\exp\big(-(c_0-\delta/2)\varphi_{\min}\big)\cr}
$$ 
for $\varepsilon>0$ small, thus the left hand side is in $L^1$.
Skoda's theorem implies that we can write $f=g\cdot h=\sum g_jh_j$ with
$h_j\in H^0(X,K_X\otimes L)=H^0(X,(m-p)K_X)$. The argument holds
as soon as the curvature condition $m-p-1\ge (n-1+\varepsilon)p$ is
satisfied, i.e.\ $m\ge np+2$. Therefore all multiples $m\ge np+2$ are
generated by sections of lower degree $m-p$, and the result follows.
\qed
\endproof
%
Recently, P\u{a}un [Pau08] has been able to provide a very strong
Shokurov-type analytic non vanishing statement, and in the vein of 
Siu's approach [Siu06], he gave a very detailed independent
proof which does not require any intricate induction on dimension 
(i.e.\ not involving the existence of minimal models).

\claim{(8.3) Theorem {\rm (P\u{a}un [Pau08])}}
Let $X$ be a projective manifold, and let $\alpha_L\in \NS_{\bR}(X)$ be a 
cohomology class in the real Neron-Severi space of $X$, such that~:
\vskip3pt
{\plainitemindent=7mm
\plainitem{\rm(a)} The adjoint class $c_1(K_X)+ \alpha_L$ is pseudoeffective, 
i.e.\ there exist a 
closed positive current $$ \Theta_{K_X+ L}\in c_1(K_X)+ \alpha_L ;$$
\vskip3pt
\plainitem{\rm(b)} The class $\alpha_L$ contains a K\"ahler 
current $\Theta_L$ $($so that $\alpha_L$ is big$)$, such that the 
respective potentials $\varphi_L$ of 
$\Theta_L$ and $\varphi_{K_X+ L}$ of $\Theta_{K_X+ L}$ satisfy
$$
e^{(1+\varepsilon)(\varphi_{K_X+ L}-\varphi_L)}\in L^1_\loc
$$
where $\varepsilon$ is a positive real number.\vskip3pt}
\noindent Then the adjoint class $c_1(K_X)+ \alpha_L$ contains an
effective $\bR$-divisor.
\endclaim
%
The proof is a clever application of the Kawamata-Viehweg-Nadel
vanishing theorem, combined with a perturbation trick of Shokurov
[Sho85] and with diophantine approximation to reduce the situation to
the case of $\bQ$-divisors. Shokurov's trick allows to single out
components of the divisors involved, so as to be able to take
restrictions and apply induction on~dimension.  One should notice that
the poles of $\varphi_L$ may help in achieving condition (8.3~b), so
one obtains a stronger condition by requiring (b${}'$)
$\exp((1+\varepsilon)\varphi_{K_X+ L})\in L^1_\loc$ for
$\varepsilon>0$ small, namely that $c_1(K_X)+\alpha_L$ is klt. The
resulting weaker statement then makes sense in a pure algebraic
setting. In [BrP09], Birkar and P\u{a}un announced a relative version
of (8.3), and they showed that this can be used to reprove a
relative version of Theorem (8.1). The notes from
Mihai P\u{a}un [Pau09] in the present volume give a fairly precise 
account of these techniques, and incorporate as well some ideas of 
Ein-Lazarsfeld-Musta\c{t}\u{a}-Nakamaye-Popa [E-P06] 
and of A.~Corti and V.~Lazi\'c (see [Laz09]). A~similar purely algebraic
approach has been described by C.~Hacon last year in his Oberwolfach
lectures [Hac08]. The reader will find it thoroughly explained in 
the present volume.
\let\fakehfill=\hfill
\section*{\fakehfill Bibliography}
{
\eightpoint

\letter KoMM87| 

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\miscell BmD09|R.\ Berman {\rm and} J.-P.\ Demailly|Regularity of
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}

\vskip15pt
\noindent
(version of November 25, 2009, printed on \today)
\vskip5pt
\noindent
Jean-Pierre Demailly, Universit\'e de Grenoble I

\end{document}

