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%%On the geometry of positive cones of projective and K\"ahler varieties
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% Jean-Pierre Demailly,
% Universit\'e de Grenoble I, BP 74
% UMR 5582 du CNRS, Institut Fourier
% 38402 Saint-Martin d'H\`eres, France
%
% Thomas Peternell
% Universit\"at Bayreuth
% Mathematisches Institut
% D-95440 Bayreuth, Deutschland
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{On the geometry of positive cones}
\title{of projective and K\"ahler varieties}
\titlerunning{On the geometry of positive cones of projective and K\"ahler 
varieties}
\medskip
\centerline{Jean-Pierre Demailly}
\centerline{\it Institut Fourier, Universit\'e de Grenoble I}
\centerline{\it 38402 Saint-Martin d'H\`eres, France}
\vskip20pt
\line{\it \hfill To the memory of Guido Fano}
\vskip20pt
\noindent
{\bf Abstract.} The goal of these notes is to give a short
introduction to several works by S\'ebastien Boucksom, Mihai Paun,
Thomas Peternell and myself on the geometry of positive cones of
projective or K\"ahler manifolds.  Mori theory has shown that the
structure of projective algebraic manifolds is -- up to a large extent
-- governed by the geometry of its cones of divisors or curves. In the
case of divisors, two cones are of primary importance: the cone of
ample divisors and the cone of effective divisors (and the closure
of these cones as well). We introduce here the analogous transcendental 
cones for
arbitrary compact K\"ahler manifolds, and show that these cones depend
only on analytic cycles and on the Hodge structure of the base
manifold.  Also, we obtain new very precise duality statements 
connecting the cones of curves and divisors via Serre duality. 
As a consequence, we are able to prove one of the basic conjectures 
in the classification of projective algebraic
varieties -- a subject which Guido Fano contributed to in many ways~:
a projective algebraic manifold $X$ is uniruled (i.e.\ covered by
rational curves) if and only if its canonical class $c_1(K_X)$ does
not lie in the closure of the cone spanned by effective divisors.

\section{\S1. Nef and pseudo-effective cones}

Let $X$ be a compact complex manifold and $n=\dim_\bC X$. We are especially
interested in {\it closed positive currents} of type $(1,1)$
$$
T=i\sum_{1\le j,k\le n}T_{jk}(z)dz_j\wedge d\ovl z_k,\qquad
dT=0.
$$
Recall that a current is a differential form with distribution coefficients,
and that a current is said to be positive if the distribution
$\sum\lambda_j\overline \lambda_k T_{jk}$ is a positive measure for all
complex numbers $\lambda_j$. The coefficients $T_{jk}$ are then
complex measures. Important examples of closed positive $(1,1)$-currents
are currents of integration over divisors~:
$$
D=\sum c_jD_j,\qquad [D]=\sum c_j[D_j]
$$
where the current $[D_j]$ is defined by duality as 
$$
\langle[D_j],u\rangle=\int_{D_j}u_{|D_j}
$$
for every $(n-1,n-1)$ test form $u$ on $X$. Another important
example is the Hessian form $T=i\ddbar\varphi$ of a plurisubharmonic
function on an open set $\Omega\subset X$. A~{\it K\"ahler metric} on
$X$ is a positive definite hermitian $(1,1)$-form
$$
\omega(z)=i\sum_{1\le j,k\le n}\omega_{jk}(z)dz_j\wedge d\ovl z_k\qquad
\hbox{such that $d\omega=0$},
$$
with smooth coefficients. To every closed real $(1,1)$-form (or current) 
$\alpha$ is associated its cohomology class
$$
\{\alpha\}\in H^{1,1}_\bR(X)\subset H^2(X,\bR).
$$
This assertion hides a nontrivial fact, namely the fact that all 
cohomology groups involved (De Rham, Dolbeault, $\ldots$) can  be 
defined either in terms of smooth forms or in terms of currents. In fact, 
if we consider the associated complexes of sheaves,
forms and currents both provide acyclic resolutions of the same sheaf
(locally constant functions, resp.\ holomorphic sections).

The manifold $X$ is said to be {\it K\"ahler} if it possesses at least 
one K\"ahler metric $\omega$. It is well
known that every projective manifold $X\subset\bP_\bC^N$ is K\"ahler
(the restriction of the Fubini-Study metric $\omega_{FS}$ to $X$ is
a K\"ahler metric with integral cohomology class $\{\omega_{FS}\}\in
H^2(X,\bZ))$. Conversely, the Kodaira embedding theorem [Kod54] states
that every compact K\"ahler manifold $X$ possessing a K\"ahler metric
$\omega$ with an integral cohomology class $\{\omega\}\in H^2(X,\bZ)$ 
can be embedded in projective space as a projective algebraic 
subvariety.

\claim 1.1. Definition|Let $X$ be a compact K\"ahler manifold.
\smallskip
\item{\rm(i)} The K\"ahler cone is the set 
$\cK\subset H^{1,1}_\bR(X)$ of cohomology classes $\{\omega\}$
of K\"ahler forms. This is an \ul{open convex cone}.
\smallskip
\item{\rm(ii)}The pseudo-effective cone is the set
$\cE\subset H^{1,1}_\bR(X)$ of cohomology classes $\{T\}$ of closed positive 
currents of type $(1,1)$. This is a \ul{closed convex cone}.\vskip0pt
\endclaim

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

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

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

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\LabelTeX -3 46 $\cK_{\NS}$\ELTX
\LabelTeX -19 27 $\cE_{\NS}$\ELTX
\LabelTeX -38.5 55 $\NS_\bR(X)$\ELTX
\EndFig
\vskip8mm

A very important fact is that the ``Neron-Severi part'' of any of
the open or closed transcendental cones $\cK$, \cE$, \ovl\cK$, $\cE^\circ$
is algebraic, i.e.\ can be characterized in simple algebraic terms.

\claim 1.2. Theorem|Let $X$ be a projective manifold. Then
\smallskip
\item{\rm (i)} $\cE_{\NS}$ is the closure of the cone generated by classes
of \ul{effective} divisors, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.
\smallskip
\item{\rm (ii)} $\cK_{\NS}$ is the open cone generated by classes of
\ul{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$)$.
\smallskip
\item{\rm (iii)} The interior $\cE_{\NS}^\circ$ is the cone generated
by classes of \ul{big} divisors, namely divisors $D$ such that
$h^0(X,\cO(kD))\ge c\,k^{\dim X}$ for $k$ large.
\smallskip
\item{\rm (iv)} The closed cone $\ovl\cK_{\NS}$ consists of the closure
of the cone generated by \ul{nef} divisors $D$ (or nef line bundles $L$), 
namely effective integral divisors $D$ such that $D\cdot C\ge 0$ for
every curve $C$.\vskip0pt
\endclaim

\noindent
{\it Sketch of proof.} These results were already observed (maybe in a slightly
different terminology) in [Dem90]. If we denote by $\cK_\alg$ the open cone
generated by ample divisors, resp.\ by $\cE_\alg$ the closure of the cone
generated by effective divisors, it is obvious that
$$
K_\alg\subset K_{\NS},\qquad \cE_\alg\subset\cE_{\NS}.
$$
As was to be expected, the interesting part lies in the converse
inclusions. The inclusion $K_{\NS}\subset\cK_\alg$ is more or 
less equivalent
to the Kodaira embedding theorem~: if a rational class $\{\alpha\}$ is
in $\cK$, then some multiple of $\{\alpha\}$ is the first Chern class
of a hermitian line bundle $L$ whose curvature form is K\"ahler. Therefore $L$
is ample and $\{\alpha\}\in \cK_\alg\,$; property (ii) follows.

Similarly, if we take a
rational class $\{\alpha\}\in\cE_{\NS}^\circ$, then it is still in
$\cE$ by subtracting a small multiple $\varepsilon\omega$ of a
K\"ahler class, hence some multiple of $\{\alpha\}$ is the first Chern class
of a hermitian line bundle $(L,h)$ with curvature form
$$
T=\Theta_h(L):=-{i\over 2\pi}i\ddbar\log h \ge \varepsilon\omega.
$$
The standard theory of $L^2$ estimates for the $\dbar$-operator ([AV65], 
[Dem82]) then shows that large multiples $kL$ admit a large number of 
sections, 
hence $kL$ can be represented by a big divisor. This implies (iii) and
also that $\cE_{\NS}^\circ\subset\cE_\alg$. Therefore 
$\cE_{\NS}\subset\cE_\alg$ by passing to the closure$\,$; (i) follows.
The statement (iv) about nef divisors follows
e.g.\ from [Kle66], [Har70], since every nef divisor is a limit 
of a sequence of ample rational divisors.
\qed

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

\section{\S2. Numerical characterization of the K\"ahler cone}

We describe here the main results obtained in [DP03]. 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 2.1. Theorem|Let $X$ be a compact K\"ahler manifold. Let $\cP$ 
be the set of real $(1,1)$ cohomology classes $\{\alpha\}$
which are \ul{numerically} \ul{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 2.2. Special case|If $X$ is projective algebraic, then $\cK=\cP$.
\endclaim

These results (which are new even in the projective case) can be seen 
as a generalization of the well-known Nakai-Moishezon criterion. Recall
that the Nakai-Moishezon criterion provides a necessary and sufficient 
criterion for a line bundle
to be ample: {\it a line bundle $L\rightarrow X$ on a projective
algebraic manifold $X$ is ample if and only if 
$$L^p\cdot Y = \int_Yc_1(L)^p> 0,$$ 
for every algebraic subset $Y\subset X$, $p=\dim Y$.} 

It turns out that the numerical conditions $\int_Y\alpha^p> 0$ also 
characterize arbitrary transcendental K\"ahler classes when  $X$ is 
projective$\,$: this is precisely the meaning of the special case 2.2.

\claim 2.3. Example|{\rm The following example shows that the cone $\cP$
need not be connected (and also that the components of $\cP$ need not 
be convex, either). Let us consider for instance a complex torus 
$X=\bC^n/\Lambda$. It is well-known that a generic torus $X$ does not
possess any analytic subset except finite subsets and $X$ itself. 
In that case, the numerical positivity is expressed by the single
condition $\int_X\alpha^n>0$. However, on a torus, $(1,1)$-classes are 
in one-to-one correspondence with constant hermitian forms $\alpha$ on
$\bC^n$. Thus, for $X$ generic, $\cP$ is the set of hermitian forms on 
$\bC^n$ such that $\det(\alpha)>0$, and Theorem 2.1 just expresses the 
elementary result of linear algebra saying that the set $\cK$ of 
positive definite forms is one of the connected components of the open set
$\cP=\{\det(\alpha)>0\}$ of hermitian forms of positive determinant
(the other components, of course, are the sets of forms of signature 
$(p,q)$, $p+q=n$, $q$ even. They are not convex when $p>0$ and $q>0$).}
\endclaim

\noindent
{\it Sketch of proof of Theorems 2.1 and 2.2.} By definition a 
\ul{\it K\"ahler current} is a closed positive current $T$ of type $(1,1)$
such that $T\ge\varepsilon\omega$ for some smooth
K\"ahler metric $\omega$ and $\varepsilon>0$ small enough.
The singularities of a closed positive current $T$ are measured by its
\ul{\it Lelong numbers}
$$
\nu(T,x)=\liminf_{z\to x}{\varphi(z)\over\log|z-x|},
$$ 
where $T={i\over\pi}\ddbar\varphi$ near $x$. A fundamental theorem of Siu 
[Siu74] states that the Lelong sublevel sets $E_c(T):=\{x\in X\,;\,
\nu(T,x)\ge c\}$ are {\it analytic sets} for every $c>0$. The crucial steps 
of the proof of Theorem~2.1 are contained in the following statements. 

\claim 2.4. Proposition {\rm (Paun [Pau98a, 98b])}|Let $X$ be a compact 
complex manifold $($or more generally a compact complex space$)$. Then
\smallskip
\item{\rm (i)} The cohomology class of a closed positive 
$(1,1)$-current $\{T\}$ is nef if and only if the restriction 
$\{T\}_{|Z}$ is nef for every irreducible component $Z$ in any of
the Lelong sublevel sets~$E_c(T)$.

\smallskip
\item{\rm (ii)} The cohomology class of a K\"ahler current $\{T\}$ is 
a K\"ahler class $($i.e.\ the class of a \ul{\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 2.4 is not extremely hard if we take for granted the
fact that K\"ahler currents can be approximated by K\"ahler currents
with logarithmic poles, a fact which was proved in [Dem92] (see also 
Theorem~5.1 below).
The main point then consists in an induction on dimension and a standard 
gluing procedure~: if $T=\alpha+{i\over\pi}\ddbar\varphi$ where
$\varphi$ is smooth on $X\ssm Z$ and has $-\infty$ poles along $Z$,
then we can remove the poles of $\varphi$ by replacing $\varphi$ with
$\max(\varphi,\psi-C)$, provided $\psi$ is smooth and defined near $Z$
and $C$ is a large constant.\qed

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

\claim 2.5. Theorem {\rm ([DP03])}|Let $X$ be a compact K\"ahler manifold
and let $\{\alpha\}$ be a nef class $($i.e.\ $\{\alpha\}\in\ovl\cK)$.
Assume that $\int_X\alpha^n>0$. Then $\{\alpha\}$ contains a K\"ahler
current $T$, in other words $\{\alpha\}\in\cE^\circ$.
\endclaim

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

\noindent
{\it End of Proof of Theorems 2.1 and 2.2}. Clearly the open cone $\cK$ 
is contained in $\cP$, hence in order to show that $\cK$ is one of the
connected components of $\cP$, we need only show that $\cK$ is \ul{closed}
in $\cP$, i.e.\ that $\ovl\cK\cap\cP\subset\cK$. Pick a class
$\{\alpha\}\in\ovl\cK\cap\cP$. In particular $\{\alpha\}$ is
nef and satisfies $\int_X\alpha^n>0$. By Theorem~2.5 we conclude that
$\{\alpha\}$ contains a K\"ahler current $T$. However, an induction on
dimension using the assumption $\int_Y\alpha^p$ for all analytic
subsets $Y$ (we also use resolution of singularities for $Y$ at this step)
shows that the restriction $\{\alpha\}_{|Y}$ is the class of a K\"ahler
current on~$Y$. We conclude that $\{\alpha\}$ is a K\"ahler class by
2.4~(ii), therefore $\{\alpha\}\in\cK$, as desired.\qed

The projective case 2.2 is a consequence of the following variant of
Theorem 2.1.

\claim 2.6. 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

\proof. The assumption clearly implies that
$$
\int_Y(\alpha+t\omega)^p>0
$$
for all $t\in\bR_+$, hence the half-line $\alpha+(\bR_+)\omega$ is 
entirely contained in the cone $\cP$ of numerically positive classes.
Since $\alpha+t_0\omega$ is K\"ahler for $t_0$ large, we conclude that
the half-line in entirely contained in the connected component $\cK$,
and therefore $\alpha\in\cK$.\qed
\medskip

In the projective case, we can take $\omega=c_1(H)$ for a given very ample
divisor $H$, and the condition
 $\int_Y\alpha^k\wedge\omega^{p-k}>0$ is equivalent to
$$
\int_{Y\cap H_1\cap\ldots\cap H_{p-k}}\alpha^k>0
$$
for a suitable complete intersection $Y\cap H_1\cap\ldots\cap H_{p-k}$,
$H_j\in|H|$. This shows that algebraic cycles are sufficient to test
the K\"ahler property, and the special case 2.2 follows. On the other 
hand, we can pass to the limit in 2.6 by replacing $\alpha$ 
by $\alpha+\varepsilon\omega$,
and in this way we get also a characterization of nef classes.

\claim 2.7. 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 2.6 or 2.7 the following
interesting consequence about the dual of the cone $\cK$. We will not 
give the proof here, because it is just a simple tricky argument which 
does not require any new analysis.

\claim 2.8. Theorem|Let $X$ be a compact K\"ahler manifold. 
A $(1,1)$ cohomology class $\{\alpha\}$ on $X$ is nef if and only 
for every irreducible analytic set $Y$ in $X$, $p=\dim X$
and every K\"ahler metric $\omega$ on $X$ we have
$\int_Y\alpha\wedge\omega^{p-1}\ge 0$. In other words, the dual of the
nef cone $\smash{\ovl\cK}$ is the closed convex cone 
in $H^{n-1,n-1}_\bR(X)$ generated by cohomology 
classes of currents of the form $[Y]\wedge\omega^{p-1}$ in 
$H^{n-1,n-1}(X,\bR)$, where $Y$ runs over the 
collection of irreducible analytic subsets of $X$ and $\{\omega\}$ over
the set of K\"ahler classes of~$X$.
\endclaim

Our main Theorem 2.1 has an important application to the deformation theory of
compact K\"ahler manifolds.

\claim 2.9. 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
$$
\nabla=\pmatrix{
\nabla^{2,0} & * & 0\cr 
* & \nabla^{1,1} & * \cr
0 & * & \nabla^{0,2}\cr}
$$
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.
\medskip

\noindent
{\it Sketch of Proof of Theorem 2.9.} 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 $H^{1,1}_\bR(X_t)\subset
H^2(X_t,\bR)$ vary along with the Hodge decomposition. At this point,
a standard calculation implies that the $\cP_t$ are invariant by parallel
transport under $\nabla^{1,1}$. Moreover, the connected component
$\cK_t\subset\cP_t$ cannot jump from one component to the other thanks to
Kodaira-Spencer theory [KS60]~: {\it every K\"ahler class in $X_{t_0}$ can be 
deformed to a nearby K\"ahler class in nearby fibres $X_t$}
(a result which relies on perturbations of elliptic PDE's).\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 [DP03].

\claim 2.10. Theorem|A compact complex manifold carries a K\"ahler
current if and only if it is bimeromorphic to a K\"ahler manifold
$($or equivalently, dominated by a K\"ahler manifold$)$.
\endclaim

This class of manifolds is called the {\it Fujiki class $\cC$}. If we 
compare this result with the solution of the Grauert-Riemenschneider
conjecture, it is tempting to make the following conjecture which would
somehow encompass both results.

\claim 2.11 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 2.5 if Conjecture 2.11 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 [DP03] manuscript appeared 
in April 2001, Daniel Huybrechts [Huy01] informed us Theorem 2.1 can be 
used to calculate the K\"ahler cone of
a very general hyperk\"ahler manifold: the K\"ahler cone is then equal
to a suitable connected component of the positive cone defined by the
Beauville-Bogomolov quadratic form. In the case of an  arbitrary
hyperk\"ahler manifold, S.Boucksom [Bou02] later showed that a $(1,1)$ class
$\{\alpha\}$ is K\"ahler if and only if it lies in the positive part
of the Beauville-Bogomolov quadratic cone and moreover $\int_C\alpha>0$
for all \ul{\it rational curves} $C\subset X$ (see also [Huy99]).

\section{\S3. Cones of curves}

In a dual way, we consider in $H^{n-1,n-1}_\bR(X)$ the cone 
$\cN$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n-1,n-1)$} (i.e., of bidimension $(1,1)$). In the projective
case, we also consider the intersection

By extension, we will say that $\overline{\cK}$ is the cone of {\it nef}
$(1,1)$-cohomology classes (even though they are not necessarily integral).
We now turn ourselves to cones in cohomo\-logy of bidegree $(n-1,n-1)$.
\bigskip

\claim 3.1. Definition|
Let $X$ be a compact K\"ahler manifold.
\smallskip
\item {\rm (i)} We define $\cN$ to be the $($closed$)$ convex cone in 
$H^{n-1,n-1}_\bR(X)$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n-1,n-1)$} $($i.e., of bidimension $(1,1))$.

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

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

\item{\rm (v)} If $X$ is projective, we say that $C$ is a \ul{strongly 
movable curve} if 
$$
C=\mu_\star(\wt A_1\cap\ldots\cap \wt A_{n-1})
$$ 
for suitable very ample divisors $\wt A_j$ on $\wt X$, where 
$\mu:\wt X\to X$ is a modification. We let ${\rm SME}(X)$ 
to be the convex cone generated by all strongly movable $($effective$)$ 
curves. Clearly  $\overline{{\rm SME}(X)}\subset\cM_{\NS}$.
\smallskip
\item{\rm (vi)} We say that $C$ is a \ul{movable curve} if $C=C_{t_0}$
is a member of an analytic family $(C_t)_{t\in S}$ such that 
$\bigcup_{t\in S}C_t=X$
and, as such, is a reduced irreducible $1$-cycle. We let ${\rm ME}(X)$ 
to be the convex cone generated by all movable $($effective$)$ curves.
\vskip0pt
\endclaim

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

\claim 3.2. Proposition|Let $X$ be a compact K\"ahler manifold. 
Consider the Poincar\'e duality pairing
$$
H^{1,1}_\bR(X)\times H^{n-1,n-1}_\bR(X)\lra \bR,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta.
$$
Then the duality pairing takes nonnegative values
\smallskip
\item{\rm (i)} for all pairs $(\alpha,\beta)\in\ol\cK\times\cN$;
\item{\rm (ii)} for all pairs $(\alpha,\beta)\in\cE\times\cM$.
\item{\rm (iii)} for all pairs $(\alpha,\beta)$ where $\alpha\in\cE$ and 
$\beta=[C_t]\in{\rm ME}(X)$ is the class of a movable curve.
\vskip0pt
\endclaim

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

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 $\cC^{\smallvee\smallvee}=
\overline{\cC}$. 

Proposition 3.2 leads to the natural question whether the 
cones $(cK,\cN)$ and $(\cE,\cM)$ are dual under Poincar\'e duality. 
This question is addressed in the next section. Before doing so,
we observe that the algebraic and transcendental cones
of $(n-1,n-1)$ cohomology classes are related by the following 
equalities (similar to what we already noticed for $(1,1)$-classes, 
see Theorem~1.2).

\claim 3.3. Theorem|Let $X$ be a projective manifold. Then
\smallskip
\item{\rm(i)} $\overline{\NE(X)} =\cN_{\NS}$.
\smallskip
\item{\rm(ii)} $\overline{{\rm SME}(X)}=\overline{{\rm ME}(X)} =\cM_{\NS}$.
\endclaim

\proof. (i) It is a standard result of algebraic geometry (see e.g.\
[Har70]), that the cone of effective cone $\NE(X)$ is dual to the cone
$\overline{\cK_{\NS}}$ of nef divisors, hence 
$$\cN_{\NS}\supset\overline{\NE(X)}=\cK^\vee.$$ 
On the other hand, (3.3)~(i) implies that $\cN_{\NS}\subset\cK^\vee$, so we must
have equality and (i) follows.

Similarly, (ii) requires a duality statement which will be established only
in the next sections, so we postpone the proof.\qed

\section{\S4. Main results and conjectures}

It is very well-known that the cone $\overline{\cK_{\NS}}$ of nef divisors
is dual to the cone $\cN_{\NS}$ of effective curves if $X$ is projective.
The transcendental case is Theorem 2.8 which we can restate as follows.

\claim 4.1. Theorem {\rm(Demailly-Paun, 2001)}|If $X$ is K\"ahler, then 
the cones $\overline{\cK}\subset H^{1,1}_\bR(X)$ and 
$\cN\subset H^{n-1,n-1}_\bR(X)$ are dual by Poincar\'e duality, and
$\cN$ is the closed convex cone generated by classes $[Y]\wedge\omega^{p-1}$
where $Y\subset X$ ranges over $p$-dimensional analytic subsets,
$p=1,2,\ldots, n$, and $\omega$ ranges over K\"ahler forms.
\endclaim

\proof. Indeed, Prop.~3.4 shows that the dual cone $\cK^\vee$ contains
$\cN$ which itself contains the cone $\cN'$ of all classes of the form
$\{[Y]\wedge\omega^{p-1}\}$. The main result of [DP03] conversely shows
that the dual of $(\cN')^\vee$ is equal to $\overline\cK$, so we must
have
$$\cK^\vee = \overline{\cN'} = \cN.\eqno\square$$

The other important duality result is the following characterization of
pseudo-effective classes, proved in [BDPP03] (the ``only if'' part 
already follows from 3.4~(iii)).

\claim 4.2. Theorem|If $X$ is projective, then a class $\alpha\in\NS_\bR(X)$
is pseudo-effective if $($and only if$\,)$ it is in the dual cone of the cone
$\SME(X)$ of strongly movable curves.
\endclaim

In other words, a line bundle $L$ is pseudo-effective if (and
only if) $L\cdot C\ge 0$ for all {\it movable curves}, i.e., $L\cdot C\ge 0$
for every very generic curve $C$ (not contained in a countable union of 
algebraic subvarieties). In fact, by definition of $\SME(X)$, it is enough 
to consider only those curves $C$ which are images of generic complete 
intersection of very ample divisors on some variety $\wt X$, under
a modification $\mu:\wt X\to X$.

By a standard blowing-up argument, it also follows that a line bundle
$L$ on a normal Moishezon variety is pseudo-effective if and only if
$L \cdot C \geq 0$ for every movable curve~$C$. 
\smallskip

The K\"ahler analogue should be~:

\claim 4.3. Conjecture| For an arbitrary compact 
K\"ahler manifold $X$, the cones $\cE$ and $\cM$ are dual.
\endclaim


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\vskip8mm

The relation between the various cones of movable curves and currents in 
(3.5) is now a rather direct consequence of Theorem 4.2. 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 movable curves. In this
way we get~:

\claim 4.4. Theorem|Let $X$ be a projective manifold. Then the following 
cones coincide.
\smallskip
\item {\rm (i)} the cone $\cM_{\NS}=\cM\cap N_1(X)\,;$
\smallskip
\item {\rm (ii)} the closed cone $\overline{\SME(X)}$ of strongly movable 
curves$\,;$
\smallskip
\item {\rm (iii)} the closed cone $\overline{\ME(X)}$ of movable curves$\,;$
\smallskip
\item {\rm (iv)} the closed cone $\overline{\ME_{\nef}(X)}$ of curves with nef 
normal bundle.\vskip0pt
\endclaim

\proof. We have already seen that 
$$\SME(X)\subset\ME(X)\subset\ME_{\nef}(X)\subset(\cE_{\NS})^\smallvee$$
and 
$$
\SME(X)\subset\ME(X)\subset\cM_{\NS}\subset(\cE_{\NS})^\smallvee
$$ 
by 3.4~(iii). Now Theorem 4.2 implies $(\cM_{\NS})^\smallvee =
\overline{\SME(X)}$, and 4.4 follows.\qed

\claim 4.5. Corollary| Let $X$ be a projective manifold and $L$ a line bundle 
on $X$.
\smallskip
\item {\rm (i)} $L$ is pseudo-effective if and only if $L \cdot C \geq 0$ 
for all curves $C$ with nef
normal sheaf $N_C$.
\smallskip
\item {\rm (ii)} If $L$ is big, then $L \cdot C > 0$ for all curves $C$ 
with nef normal sheaf $N_C$.\vskip0pt
\endclaim

4.5~(i) strenghtens results from [PSS99]. It is however not yet clear whether 
$\cM_{\NS}=\cM\cap N_1(X)$ is equal to the closed cone of curves with 
{\it ample} normal bundle (although we certainly expect this to be true).

The most important special case of Theorem 4.2 is

\claim 4.6. Theorem| If $X$ is a projective manifold and is not uniruled,
then $K_X$ is pseudo-effective, i.e.\ $K_X\in\cE_{\NS}$.
\endclaim

\proof. If $K_X\notin\cE_{\NS}$, Theorem 3.2 shows that
there is a moving curve $C_t$ such that $K_X\cdot C_t<0$. The
``bend-and-break'' lemma then implies that there is family $\Gamma_t$ of 
\ul{rational curves} with $K_X\cdot\Gamma_t<0$, so $X$ is uniruled.\qed
\medskip

A stronger result is expected to be true, namely~:

\claim 4.7. Conjecture {\rm (special case of the ``abundance conjecture'')}|
If $K_X$ is pseudo-effective, then $\kappa(X)\ge 0$.
\endclaim

\section{\S5 Zariski decomposition and movable intersections}

Let $X$ be compact K\"ahler and let $\alpha\in\cE^\circ$ be in the
{\it interior} of the pseudo--effective cone. In analogy with the
algebraic context such a class $\alpha$ is called ``big'', and it can
then be represented by a {\it K\"ahler current} $T$, i.e.\ a closed
positive $(1,1)$-current $T$ such that $T\ge \delta\omega$ for some
smooth hermitian metric $\omega$ and a constant $\delta\ll 1$.

\claim 5.1. Theorem {\rm (Demailly [Dem92], [Bou02, 3.1.24]}|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 ``movable part''$)$.
\endclaim

\proof. Since this result has already been studied extensively, we just recall
the main idea. Locally we can write $T=i\ddbar\varphi$ for some strictly 
plurisubharmonic potential $\varphi$. By a Bergman kernel trick and the
Ohsawa-Takegoshi $L^2$ extension theorem, we get local approximations
$$
\varphi=\lim\varphi_m,\qquad \varphi_m(z)={1\over 2m}\log\sum_\ell
|g_{\ell,m}(z)|^2
$$
where $(g_{\ell,m})$ is a Hilbert basis of the set of holomorphic
functions which are $L^2$ with respect to the weight $e^{-2m\varphi}$.
This Hilbert basis is also a family of local generators of the globally defined
multiplier ideal sheaf $\cI(mT)=\cI(m\varphi)$. Then $\mu_m:X_m\to X$
is obtained by blowing-up this ideal sheaf, so that 
$$\mu_m^\star\cI(mT)=\cO(-mE_m).$$
We should notice that by approximating $T-{1\over m}\omega$ instead of $T$,
we can replace $\beta_m$ by $\beta_m+{1\over m}\mu^\star\omega$ which is
a big class on $X_m\,$; by playing with the multiplicities of the components
of the exceptional divisor, we could even achieve that $\beta_m$ is a 
K\"ahler class on $X_m$, but this will not be needed here.\qed
\medskip

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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\vskip4mm

\claim 5.2. Definition|We define the {\bf volume}, or {\bf movable
self-intersection} of a big class $\alpha\in\cE^\circ$ to be
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\wt X}\beta^n>0
$$
where the supremum is taken over all K\"ahler currents $T\in \alpha$
with logarithmic poles, and $\mu^\star T=[E]+\beta$ with
respect to some modification $\mu:\wt X\to X$.
\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 5.3. 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 5.2 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 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi=\max(\varphi_1,\varphi_2)\leqno(5.4)$$ 
is again a K\"ahler current with weaker 
singularities than $T_1$ and $T_2$. One could define as well 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi={1\over 2m}\log(e^{2m\varphi_1}+e^{2m\varphi_2}),\leqno(5.4')$$
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 5.5 below.

\claim 5.5. 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 $(n-k,n-k)$.
\smallskip
\item{\rm(i)} For each integer $k=1,2,\ldots,n$, there exists a canonical
``movable intersection product''
$$
\cE\times\cdots\times\cE\to H^{k,k}_{\ge 0}(X), \quad
(\alpha_1,\ldots,\alpha_k)\mapsto \langle\alpha_1\cdot\alpha_2\cdots
\alpha_{k-1}\cdot \alpha_k\rangle
$$
such that $\Vol(\alpha)=\langle\alpha^n\rangle$ whenever $\alpha$ is
a big class.
\smallskip
\item{\rm(ii)} The product is increasing, homogeneous of 
degree $1$ and superadditive in each argument, i.e.\
$$
\langle\alpha_1\cdots(\alpha'_j+\alpha''_j)\cdots \alpha_k\rangle\ge
\langle\alpha_1\cdots\alpha'_j\cdots \alpha_k\rangle+
\langle\alpha_1\cdots\alpha''_j\cdots \alpha_k\rangle.
$$
It coincides with the ordinary intersection
product when the $\alpha_j\in\overline{\cK}$ are nef classes.
\smallskip
\item{\rm(iii)} 
The movable intersection product satisfies the Teissier-Hovanskii inequalities
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_n\rangle\ge
(\langle\alpha_1^n\rangle)^{1/n}\ldots(\langle\alpha_n^n\rangle)^{1/n}\qquad
\hbox{$($with $\langle\alpha_j^n\rangle=\Vol(\alpha_j)\,)$}.
$$
\smallskip
\item{\rm(iv)} For $k=1$, the above ``product'' reduces to a
$($non linear$)$ projection operator 
$$
\cE\to\cE_1,\qquad \alpha\to\langle\alpha\rangle
$$
onto a certain convex subcone $\cE_1$ of $\cE$ such that
$\overline{\cK}\subset\cE_1\subset\cE$. 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 $\alpha\in\cE_1$.
\smallskip
\item{\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

\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.
\smallskip

\noindent
(i) First assume that all classes $\alpha_j$ are big, i.e.\ 
$\alpha_j\in\cE^\circ$. Fix a smooth closed $(n-k,n-k)$ {\it semi-positive}
form $u$ on $X$. We select K\"ahler currents $T_j\in\alpha_j$ with
logarithmic poles, and a simultaneous log-resolution
$\mu:\wt X\to X$ 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
$$
\int_{\wt X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u\ge 0.
$$
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 $(5.4')$ 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
$$
\eqalign{
\int_{\wt X}\beta_1'\wedge\beta_2
&{}\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&=\int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u+
\int_{\wt X}[D_1]\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&\ge \int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge
\mu^\star u.\cr}
$$
Similarly, we can replace successively all forms $\beta_j$ by the $\beta'_j$,
and by doing so, we find
$$
\int_{\wt X}\beta'_1\wedge\beta'_2\wedge\ldots\wedge\beta'_k\wedge\mu^\star u
\ge\int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u.
$$
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
$$
\int_{\wt X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u
\le C_1\ldots C_k\int_{\wt X}\mu^\star\omega^k\wedge\mu^\star u
= C_1\ldots C_k\int_X\omega^k\wedge u
$$
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 $\mu_m:
\smash{\wt X_m}\to X$.
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 Poincar\'e 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). 
\smallskip

\noindent
(ii) It is indeed clear from the definition that the movable 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 movable intersection product can always be 
represented by a closed positive current of bidegree $(k,k)$.
\smallskip

\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 $\smash{\wt X_m}$ and pass to the limit.
\smallskip

\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 $1$'').
\smallskip

\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 $(5.4')$, we eliminate the component
$\mu^\star F_{j_0}$. This is a contradiction since $N(\alpha)$
was supposed to contain $F_{j_0}$.\qed
\medskip

\claim 5.6. Definition|For a class $\alpha\in H^{1,1}_\bR(X)$, 
we define the \ul{numerical dimension} $\num(\alpha)$ to be 
$\num(\alpha)=-\infty$ if $\alpha$ is not pseudo-effective, and
$$
\num(\alpha)=\max\{p\in\bN\;;\;\langle\alpha^p\rangle\ne 0\},\qquad
\num(\alpha)\in\{0,1,\ldots,n\}
$$
if $\alpha$ is pseudo-effective.
\endclaim

By the results of [DP03], a class is big ($\alpha\in\cE^\circ$) if and only
if $\num(\alpha)=n$. Classes of numerical dimension $0$ can be
described much more precisely, again following Boucksom [Bou02].

\claim 5.7. Theorem|Let $X$ be a compact
K\"ahler manifold. Then the subset $\cD_0$ of irreducible divisors 
$D$ in $X$ such that $\num(D)=0$ is countable, and these divisors 
are rigid as well as their multiples. If $\alpha\in\cE$ is a 
pseudo-effective class of numerical dimension~$0$, then $\alpha$ 
is numerically equivalent to 
an effective $\bR$-divisor $D=\sum_{j\in J}\lambda_j D_j$,
for some finite subset $(D_j)_{j\in J}\subset\cD_0$ such that the 
cohomology classes $\{D_j\}$ are linearly independent and
some \hbox{$\lambda_j>0$}. If such a linear combination is of
numerical dimension~$0$, then so is any other linear combination of the
same divisors.
\endclaim

\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 5.7, 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 $\mu:\wt X\to X$ resolving the 
intersection, and by taking
$\min(\mu^\star D,\mu^\star D')$ via $(5.4')$, 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
\medskip

The above general concept of numerical dimension leads to a very natural 
formulation of the abundance conjecture for non-minimal (K\"ahler) varieties.

\claim 5.8. Generalized abundance conjecture|For an arbitrary compact 
K\"ahler manifold~$X$, the Kodaira dimension should be equal to the
numerical dimension~:
$$\kappa(X)=\num(X):=\num(c_1(K_X)).$$
\endclaim

This appears to be a fairly strong statement. In fact, it is not difficult
to show that the generalized abundance conjecture would contain the
$C_{n,m}$ conjectures.

\claim 5.9. Remark|{\rm Using the Iitaka fibration, it is immediate to 
see that $\kappa(X)\leq\num(X)$.}
\endclaim

\claim 5.10. Remark|{\rm It is known that abundance holds in case
\hbox{$\num(X)=-\infty$} (if $K_X$ is not pseudo-effective, no
multiple of $K_X$ can have sections), or in case $\num(X)=n$. The
latter follows from the solution of the Grauert-Riemenschneider
conjecture in the form proven in [Dem85] (see also [DP03]).

In the remaining cases, the most tractable situation is probably the
case when $\num(X)=0$.
In fact Theorem 5.7 then gives $K_X\equiv\sum\lambda_j D_j$ for some
effective divisor with numerically independent components, $\num(D_j)=0$. 
It follows that the $\lambda_j$ are rational and therefore
$$
K_X\sim \sum\lambda_j D_j+F\qquad
\hbox{where $\lambda_j\in\bQ^+$, $\num(D_j)=0$ and $F\in\Pic^0(X)$.}
\leqno(*)
$$
Especially, 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
(or compact K\"ahler) manifold with \hbox{$\num(X)=0$} and positive 
irregularity $q(X)>0$ would be interesting to understand.}
\endclaim

\section{\S6 The orthogonality estimate}

The goal of this section is to show that, in an appropriate sense,
approximate Zariski decompositions are almost orthogonal.

\claim 6.1. 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

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

\claim 6.2. 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

\proof. In case $A$ and $B$ are integral (Cartier) divisors, this is a 
consequence of the holomorphic Morse inequalities, [Dem01, 8.4]. 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 

\claim 6.3. Remark|{\rm We hope that Lemma 6.2 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 6.4. Lemma|Let $\beta_1,\ldots,\beta_n$ and 
$\beta'_1,\ldots,\beta'_n$ be nef classes on a compact K\"ahler manifold
$\wt X$ 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

\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

\noindent
{\it End of proof of Theorem 6.1.}
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 6.2 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~6.4
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 6.4), whence Theorem 6.1. 
Of course, the constant 20 is certainly not optimal.\qed

\claim 6.5. 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

\proof. By replacing $\alpha$ by $\alpha+\delta c_1(H)$, one sees that
it is sufficient to consider the case where $\alpha$ is big. Then the 
orthogonality estimate implies
$$
(\mu_m)_\star(D_m^{n-1})\cdot (\mu_m)_\star E_m
= D_m^{n-1}\cdot(\mu_m)^\star (\mu_m)_\star E_m
\le D_m^{n-1}\cdot E_m \le C (\Vol(\alpha)-D_m^n)^{1/2}.
$$
Since $\langle\alpha^{n-1}\rangle = \lim (\mu_m)_\star(D_m^{n-1})$,
$N(\alpha)=\lim (\mu_m)_\star E_m$ and $\lim D_m^n=\Vol(\alpha)$,
we get the desired conclusion in the limit.\qed

\section{\S7 Proof of the duality theorem}

The proof is reproduced from [BDPP03]. We want to show that 
$\cE_{\NS}$ and $\SME(X)$ are dual (Theorem 4.2). By 3.4~(iii) we have 
in any case
$$\cE_{\NS}\subset (\SME(X))^\smallvee.$$ 
If the inclusion is strict, there is an element
$\alpha\in\partial\cE_{\NS}$ on the boundary of $\cE_{\NS}$ which is in
the interior of $\SME(X)^\smallvee$.


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\vskip8mm

Let $\omega=c_1(H)$ be an ample class. Since $\alpha\in\partial\cE_{\NS}$, 
the class $\alpha+\delta\omega$ is big for every $\delta>0$, and since
$\alpha\in((\SME(X))^{\smallvee})^\circ$ we still have 
$\alpha-\varepsilon\omega\in(\SME(X))^{\smallvee}$ for
$\varepsilon>0$ small. Therefore
$$
\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma\leqno(7.1)
$$
for every movable curve $\Gamma$. We are going to contradict (7.1).
Since $\alpha+\delta\omega$ is big, we have an approximate Zariski
decomposition
$$
\mu_\delta^\star(\alpha+\delta\omega)=E_\delta+D_\delta.
$$
We pick $\Gamma=(\mu_\delta)_\star(D_\delta^{n-1})$. By the Hovanskii-Teissier
concavity inequality
$$
\omega\cdot \Gamma\ge (\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}.
$$
On the other hand
$$
\eqalign{
\alpha\cdot \Gamma&=\alpha\cdot(\mu_\delta)_\star(D_\delta^{n-1})\cr
&=\mu_\delta^\star\alpha\cdot D_\delta^{n-1}\le
\mu_\delta^\star(\alpha+\delta\omega)\cdot D_\delta^{n-1}\cr
&=(E_\delta+D_\delta)\cdot D_\delta^{n-1}
=D_\delta^n+D_\delta^{n-1}\cdot E_\delta.\cr
}
$$
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
$(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}$ tends to $0$ much faster than
$D_\delta^n$. Notice that \hbox{$D_\delta^n\ge\delta^n\omega^n$}, so in fact
it is enough to take
$$
\Vol(\alpha+\delta\omega)-D_\delta^n\le \delta^{2n}.
$$
This is the desired contradiction by (7.1).\qed
\bigskip

\claim 7.2. Remark|{\rm If holomorphic Morse inequalities were known also
in the K\"ahler case, we would infer by the same proof that
``$\alpha$ not pseudo-effective'' implies the existence of a blow-up
$\mu:\wt X\to X$ and a K\"ahler metric 
$\wt\omega$ on $\smash{\wt X}$ such that 
$\alpha\cdot\mu_\star(\wt\omega)^{n-1} < 0$. In the special case
when $\alpha=K_X$ is not pseudo-effective, we would expect the 
K\"ahler manifold $X$ to be covered by rational curves.
The main trouble is that characteristic $p$ techniques are no longer
available. On the other hand it is tempting to approach the
question via techniques of symplectic geometry~: } \endclaim

\claim 7.3. Question|Let $(M,\omega)$ be a compact real symplectic manifold.
Fix an almost\break complex structure $J$ compatible with $\omega$, and
for this structure, assume that\break \hbox{$c_1(M)\cdot\omega^{n-1}>0$}. 
Does it follow that $M$ is covered by rational $J$-pseudoholomorphic curves~?
\endclaim
\vskip20pt

\noindent {\twelvebf References}
\medskip

{\eightpoint
\itemindent = 1.5cm

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

\bibitem [Bou02]&Boucksom, S.:& C\^ones positifs des vari\'et\'es
  complexes compactes.& Thesis, Grenoble 2002&
  
\bibitem [BDPP03]&Boucksom, S., Demailly, J.P., Paun, M., Peternell,
  Th.& The pseudo-effective cone of a compact K\"ahler manifold and
  varieties of negative Kodaira dimension;& manuscript, July 2003&
  
\bibitem [Buc99]&Buchdahl, N.;& On compact K\"ahler surfaces;& Ann.\ 
  Inst.\ Fourier {\bf 49} (1999) 287--302&
  
\bibitem [Buc00]&Buchdahl, N.;& A Nakai-Moishezon criterion for
  non-K\"ahler surfaces;& Ann.\ Inst.\ Fourier {\bf 50} (2000)
  1533--1538&
  
\bibitem [Dem82]&Demailly, J.-P.;& Estimations $L^2$ pour
  l'op\'erateur $\overline\partial$ d'un fibr\'e vectoriel holomorphe
  semi-positif au-dessus d'une vari\'et\'e k\"ahl\'erienne
  compl\`ete;& Ann.\ Sci.\ \'Ecole Norm.\ Sup.\ 4e S\'er.\ {\bf 15}
  (1982) 457--511&
  
\bibitem [Dem85]&Demailly, J.-P.;& Champs magn\'etiques et
  in\'egalit\'es de Morse pour la $d^{\prime \prime}$-cohomo\-logie;&
  Ann.\ Inst.\ Fourier\ {\bf 35} (1985)&
  
\bibitem [Dem90]&Demailly, J.-P.;& Singular hermitian metrics on
  positive line bundles;& Proceedings of the Bayreuth conference
  ``Complex algebraic varieties'', April~2-6, 1990, edited by
  K.~Hulek, T.~Peternell, M.~Schneider, F.~Schreyer, Lecture Notes in
  Math.\ ${\rm n}^\circ\,$1507, Springer-Verlag, 1992&
  
\bibitem [Dem92]&Demailly, J.-P.;& Regularization of closed positive
  currents and intersection theory;& J.\ Algebraic Geometry, {\bf 1}
  (1992), 361--409&

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

\bibitem [DP03]&Demailly, J.-P., Paun, M.;& Numerical characterization
  of the K\"ahler cone of a compact K\"ahler manifold;& manuscript
  Universit\'e Joseph fourier, Grenoble et Universit\'e Louis Pasteur,
  Strasbourg, septembre 2001, math.AG/0105176, accepted for
  publication in Annals of Math&

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

\bibitem [DPS00]&Demailly, J.-P., Peternell, Th.; Schneider, M.;& 
  Pseudo-effective line bundles on compact K\"ahler manifolds;& 
  Intern.\ J.\ Math. {\bf 6} (2001) 689-741&
  
\bibitem [Fuj94]&Fujita, T.;& Approximating Zariski decomposition of
  big line bundles;& Kodai Math.\ J.\ {\bf 17} (1994), 1--3&
  
\bibitem [Har70]&Hartshorne, R.;& Ample subvarieties of algebraic
  varieties;& Lecture Notes in Math.\ no {\bf 156}, Springer-Verlag,
  Berlin (1970)&
  
\bibitem [Huy99]&Huybrechts, D.;& Compact Hyperk\"ahler Manifolds:
  Basic Results;& Invent.\ math.\ {\bf 135} (1999), 63--113&
  
\bibitem [Huy01]&Huybrechts, D.;& The Projectivity Criterion for
  Hyperk\"ahler manifolds as a Consequence of the Demailly-Paun
  Theorem;& personal communication, manuscript K\"oln University, May
  2001, 3~p, to appear in Invent.\ Math&
  
\bibitem [Kle66]&Kleiman, S.;& Toward a numerical theory of
  ampleness;& Ann.\ Math.\ {\bf 84} (1966), 293--344&
  
\bibitem [Kod54]&Kodaira, K.;& On K\"ahler varieties of restricted
  type;& Ann.\ of Math.\ {\bf 60} (1954) 28--48&
  
\bibitem [KS60]&Kodaira, K., Spencer, D.C.;& On deformations of
  complex analytic structures. III. Stability theorems for complex
  structures;& Ann.\ of Math.\ {\bf 71} (1960) 43--76&
  
\bibitem [Lam99a]&Lamari, A.;& Courants k\"ahl\'eriens et surfaces
  compactes;& Ann.\ Inst.\ Fourier {\bf 49} (1999) 263--285&
  
\bibitem [Lam99b]&Lamari, A.;& Le cone K\"ahl\'erien d'une surface;&
  J.\ Math.\ Pures Appl.\ {\bf 78} (1999) 249--263&
  
\bibitem [Pau98a]&Paun, M.;& Sur l'effectivit\'e num\'erique des
  images inverses de fibr\'es en droites;& Math.\ Ann.\ {\bf 310}
  (1998), 411--421&
  
\bibitem [Pau98b]&Paun, M.;& Fibr\'e en droites num\'eriquement
  effectifs et vari\'et\'es K\"ahl\'eriennes compactes \`a courbure de
  Ricci nef;& Th\`ese, Universit\'e de Grenoble 1 (1998), 80$\,$p&
  
\bibitem [Pau00]&Paun, M.;& Semipositive (1,1)--cohomology classes on
  projective manifolds;& Preprint de l'I.R.M.A&
  
\bibitem [Siu74]&Siu, Y.-T.;& Analyticity of sets associated to Lelong
  numbers and the extension of closed positive currents;& Invent.\ 
  Math.\ {\bf 27} (1974)&
  
\bibitem [Siu84]&Siu, Y.-T.;& Some recent results in complex manifold
  theory for the semi-positive case;& survey article in the
  Proceedings of the international congress held in Bonn, 1984&
  
\bibitem [Yau78]&Yau, S.-T.;& On the Ricci curvature of a complex
  K\"ahler manifold and the complex Monge--Amp\`ere equation;& Comm.\ 
  Pure Appl.\ Math.\ {\bf 31} (1978)&

\vskip10pt
\noindent
Jean-Pierre Demailly\\
Universit\'e de Grenoble I, Institut Fourier, UMR 5582 du CNRS\\
BP74, 100 rue des Maths, 38402 Saint-Martin d'H\`eres Cedex, France\\
{\sl E-mail:}\/ demailly@fourier.ujf-grenoble.fr
\medskip

\noindent
(version of July 26, 2003; printed on \today)

}

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