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\null
\vskip 3cm
\centerline{\fourt A Nakai--Moishezon criterion}
\bigskip
\centerline{\fourt for the (1, 1)--cohomology classes on}
\bigskip
\centerline{\fourt compact K\"ahler manifolds}

\null
\vskip 5pt
\centerline{\rm Jean--Pierre DEMAILLY and Mihai PAUN}
\null
\vskip 30pt
\noindent{\twel Introduction.} 
\vskip 5pt
\noindent Let $X$ be a compact complex manifold, and $\{\alpha \} \in
H^2(X, \bR)$ be a
cohomology class of $(1, 1)$--type. Assume --for the moment-- that 
$\{\alpha \} \in H^2(X, \bZ)$, then we have the following theorem,
due to Y.-T. Siu (which gives a positive answer to a conjecture 
of Grauert--Riemenschneider, see [15])

\noindent {\sl {\tx Theorem {\rm (Y.-T. Siu).}} Let $X$ be a 
$n$--dimensional compact
complex manifold, and let $L$ be a Hermitian holomorphic 
line bundle on $X$, whose curvature form $\alpha $ is everywhere 
semipositive and strictly positive at some point. Then 
$h^0(X, L^m)\geq Cm^n$, for some positive constant $C$ (in particular,
$X$ is Moishezon).
}

\noindent  A first generalization of 
Siu's theorem was given by J.-P. Demailly (see [5]), by weakening the 
positivity assumptions on the differential form $\alpha$
(the pointwise hypothesis are replaced by integral ones). Other 
results were obtained by T. Bouche  
and L. Bonavero, (see [1], [2]), by replacing the compactness of $X$
with convexity properties,
respectively by authorizing differential forms with analytic
singularities.

\noindent If we have a Hermitian line bundle with the metric
properties stated in the theorem above,
its curvature form contains a strongly positive current (not
necessarily smooth). Then as a consequence of the theorem of
Siu, we see that if $\{\alpha \} \in H^2(X, \bZ)$ is of $(1, 1)$--type
and has the properties that $\alpha \geq
0$ at each point of $X$ and $\alpha ^n\not\equiv 0$, there 
exist a function $\varphi\in L^1(X)$ and a positive constant 
$\lambda$ such that 
$\alpha + i\partial\overline\partial\varphi\geq \lambda g,$
as currents on $X$ ($g$ is a fixed hermitian metric).
\vskip 5pt
\noindent In this perspective, one of the goals of 
this paper is to obtain some results when
the integrality of the class $\{\alpha\}$ is not assumed. 
Actually, one has
the following conjecture, formulated by ??:

\vskip 5pt
\noindent {\sl {\tx Conjecture.} Let $(X, g)$ be a compact complex 
manifold, and $\alpha$ be a closed differential form of $(1,
1)$--type, such that $\alpha\geq 0$ on $X$ and $\alpha> 0$
at some point of the manifold $X$. Then there exist a function
 $\varphi\in L^1(X)$ and some positive constant $\lambda $
such that we have:
$\alpha + i\partial\overline\partial\varphi\geq \lambda g,$
as currents on $X$.}

\vskip 5pt
\noindent Before stating a result which go in this 
direction, let us recall the following notion of 
weak positivity for a cohomology class (see [7]):

\vskip 5pt
\noindent {\sl {\tx Definition 1.} Let $(X, g)$ be a compact complex 
manifold, and $\alpha$ be a closed differential form of $(1,1)$--type; 
we say that the cohomology class $\{\alpha\}$ is nef in the metric sense
if there exist 
$\displaystyle (f_\varepsilon)_{\varepsilon}\subset 
{\cal C}^\infty (X)$ such that
$$\alpha + i\partial\overline\partial f_\varepsilon\geq -\varepsilon g
\hbox { on } X$$

\noindent for each $\ep>0$.}

\vskip 5pt
\noindent Of course, the property of being nef in the metric sense
does not depend on the choice of a 
(smooth!) representative $\alpha$ in the respective 
cohomology class; unless specified,
when we will speak in what follows about ``nef cohomology class'', we always 
understand this notion as defined above, even if the class in question 
is entire.  

\vskip 5pt
\noindent The first result which will be proved here is.

\vskip 5pt
\noindent {\sl {\tx Theorem 1.} Let $(X, \omega)$ be a compact K\"ahler
manifold, and $\alpha$ be a d--closed differential form of $(1,
1)$--type, such that $\{\alpha\}$ is nef and $\displaystyle \int_X\alpha^n> 0$.
Then there exist a function
 $\varphi\in L^1(X)$ and some positive constant $C>0$
such that we have:
$\alpha + i\partial\overline\partial\varphi\geq C^{-1}\omega,$
as currents on $X$.}
\vskip 10pt

\noindent The proof of the theorem 1 relies heavily on the Aubin--Calabi--Yau
theorem, and in few words it goes as follows: we consider the product 
manifold $X\times X$ and the pull--back differential form 
$\widetilde\alpha:= \pi_1^*\alpha+
\pi_2^*\alpha$, where $\pi_1, \pi_2$ are the two projections. The cohomology 
class of $\widetilde\alpha$ is still nef, and 
$\displaystyle \int_{X\times X}\!\!\widetilde\alpha^{2n}> 0$, 
as it will be (easily) verified. The main step in the proof is to show that 
the cohomology class of $\widetilde\alpha^n$ contains a positive current 
$\widetilde T$, which has strictly positive generic Lelong numbers on the diagonal
$\Delta\subset X\times X$; this is shown to be so by using a sequence of
Monge--Amp\`ere equations, in the following way: the class $\{\widetilde \alpha+
\varepsilon \widetilde\omega\}$ contains a K\"ahler metric, say 
$\widetilde\omega_\varepsilon$, whose determinant converge as 
$\varepsilon\mapsto 0$
to the current $[\Delta]\wedge\widetilde\omega^n$. Then it is reasonable to 
expect that the limit of the sequence 
$\widetilde\omega^n_\varepsilon$ has positive 
Lelong numbers on the diagonal, and we will show that it is actually the case. 
Once we have this, then we can take 
$T:= \pi_{1 *}\bigl(\widetilde T\wedge \pi_2^*(\omega)\bigr)$, and it is quite 
easy to see that $T$ is strongly positive and it belongs to the cohomology
class of $\alpha$.
  
\par\noindent 
The idea of using Aubin--Calabi--Yau 
theorem in order to force the occurrence
of positive Lelong numbers in the limit is due to
J.-P.~Demailly (see [8]). In contrast with 
the arguments presented in [8], (where one of the 
goals was to obtain positive Lelong numbers at a
finite numbers of points in $X$), if 
we have a analytic set of dimension greater then 0, say 
$Y\subset B(1)$ 
(where $B(1)$ is the unit ball in $\bC^n$),
then we cannot find a quasi--plurisubharmonic function
$\phi$ with analytic poles on $Y$ and $\phi= 0$ on the 
boundary of the ball of radius 2; the existence of 
such a function is clear if $Y$ is a point.
\vskip 5pt 
\noindent The theorem 1 combined with a result of J.--P. 
Demailly concerning the regularization of closed positive currents
gives a numerical criterion for the K\"ahler cone of compact K\"ahler
manifolds stated in the theorem 2.
\vskip 5pt
\noindent {\sl {\tx Theorem 2.} Let $(X,\omega)$ be a compact 
K\"ahler manifold, and let $\{\alpha\}$ be a $(1, 1)$--cohomology 
class on $X$. Then $\{\alpha \}$ is a K\"ahler class iff
$\displaystyle \int_Z\alpha^k\wedge\omega^{p- k}> 0$
for all $p$--dimensional analytic subset $Z\subset X$
and all $k= 1,...,p$.}

 
\noindent This result can be seen as a generalization of the
Nakai--Moishezon criterion in algebraic geometry.
Actually, let us recall the following theorem.
\vskip 5pt
\noindent {\sl {\tx Theorem {\rm (Nakai--Moishezon Criterion.)}}
Let $X$ be a projective manifold, and $L\rightarrow X$ a line bundle.
Then $L$ is ample iff $\displaystyle \int_Zc_1(L)^k> 0$, for all
$Z\subset X$ algebraic subset.}

\noindent Now even if $X$ is projective it may happen 
that the ample cone of $X$ is strictly
contained in the K\"ahler cone, and one can ask if the K\"ahler cone 
can be characterised in a similar manner; 
as a corollary of the theorem 2, we 
give a positive answer to this question.

\vskip 5pt
\noindent {\sl {\tx Theorem 2$^\prime$}. 
Let $X$ be a projective manifold, and
let $\{\alpha\}$ be a $(1, 1)$--cohomology class on $X$. 
Then $\{\alpha\}$ is a K\"ahler class iff 
$\displaystyle \int_Y\alpha^k> 0$
for all $k$--dimensional algebraic subset $Y\subset X$.}

\vskip 5pt
\noindent In the case of compact complex surfaces
the theorem 2 was known, see [3].
A similar result was obtained by Campana--Peternell for
projective manifolds of arbitrary dimension, under the assumption that 
the class $\{\alpha \}$ is in the closure of
the Neron--Severi cone, see [4]; another partial 
result was obtained by Eyssidieux, see [9].

\noindent As for the proof of the theorem 2, it relies on
the following simple observation (and on the theorem 1): 
consider the ``reference'' K\"ahler metric 
$\omega$ given in the hypothesis and 
consider also the set 
$${\cal J}:= \bigl\{t\geq 0/\hbox{ the class } \{\alpha + t\omega\}
\hbox{ is K\"ahler }\bigr\}\subset \bR$$

\noindent Then it is clear that ${\cal J}$ is non--empty and open;
thanks to the theorem 1 we show that it is also closed, 
and this implies in particular that $0\in {\cal J}$, so 
the cohomology class of $\alpha $ is K\"ahler
(it may very well happen that a limit of K\"ahler cohomology classes 
is not K\"ahler, but anyway it inherits some weak positiveness
properties; those properties mixed up with the hypothesis on 
$\alpha$ are enough to get the conclusion). 


\vskip 40pt 
\noindent {\twel Proof of the theorem 1.}
\vskip 5pt
\noindent Let us start by fixing some notations.
Consider the product manifold $X\times X$ with the 2 
projections, denoted by $\pi_1$ and respectively $\pi_2$;
we also fix a K\"ahler metric $\omega $ on $X$, and take the 
product metric $\widetilde\omega:= \pi_1^*\omega+ \pi_2^*\omega$
on $X\times X$. We denote by 
$\widetilde\alpha:= \pi_1^*\alpha+ \pi_2^*\alpha$; 
the main fact in what will 
follow is that the hypothesis which we've made about 
the cohomology class of 
$\alpha$ provide $\{\widetilde\alpha\}$ with enough 
positivity in order to 
get a positive current $\widetilde T\in \{\widetilde\alpha^n\}$ 
whose generic
Lelong number on the diagonal $\Delta\subset X\times X$ is positive.

\noindent A first observation is that the cohomology class 
$\{\widetilde\alpha\}$ is still nef on $X\times X$; 
indeed, the nefness of 
$\{\alpha\}$ gives a sequence $(f_\varepsilon)_{\varepsilon }\subset 
{\cal C}^\infty (X)$ such that 
$$\alpha + i\partial\overline\partial 
f_\varepsilon\geq -\varepsilon/2 \omega$$

\noindent for all $\varepsilon > 0$, so if we take the pull--back on 
$X\times X$ with respect to the two projections, we get
$$\widetilde\alpha + i\partial\overline\partial 
(f_\varepsilon\circ \pi_1+ 
f_\varepsilon\circ \pi_2)\geq -\varepsilon\widetilde\omega$$

\noindent on the product manifold, and this proves our claim. 
On the other hand, we have $\displaystyle \int _X\widetilde \alpha^{2n}=
{\pmatrix 
{2n\cr n\cr}}\Bigl(\int _X\alpha^n\Bigr)^2> 0$. 

\noindent Next we will manufacture
an appropriate sequence of volume elements for the 
Monge--Amp\`ere equation, 
in the sense that we want their limit 
to degenerate on the diagonal.
In order to illustrate the construction which will be given, suppose that 
we have a holomorphic line bundle $L\mapsto X$, and 
$\sigma_1, ..., \sigma_k$
are holomorphic sections of $L$; denote by $Y:= \cap_j (\sigma_j=0)$.
Then there is a natural way of constructing a sequence of functions 
$(f_\varepsilon)_\ep\subset {\cal C}^\infty(X)$, which converge to a quasi-psh
function with logarithmic poles on $Y$, namely 
$\displaystyle f_\varepsilon(z)= 
\log(\varepsilon+ \sum_j\vert\sigma_j(z)\vert_h^2)$,
where $h$ is a hermitian metric on $L$. 

\noindent One can find a substitute for the sequence 
$(f_\varepsilon)_\ep$ 
even if the analytic set $Y$ is not given by the zero set of a section
of some vector bundle; such a construction is explained in [6], 
but as we shall need a very slight modification of it, we try to recall
in the next few lines the main facts.

\vskip 5pt
\noindent {\sl {\tx Lemma 1.{\rm (J.-P. Demailly.)}} Let $Y\subset Z$
a sub-manifold of a compact complex manifold $Z$ and let 
$\omega$ be a hermitian metric on $Z$. Then there exist a positive 
constant $C> 0$ and a sequence of functions 
$(f_\varepsilon)_{\varepsilon}$ such that:
\smallskip
\item {i)} For all $\varepsilon> 0$ we have $i\partial
\overline\partial f_\varepsilon \geq -C^{-1}\omega $.
\smallskip 
\item {ii)} For almost all $p\in Y$, there exist a neighbourhood
$V= V(p)$ and local coordinates $(z_j)$ in $V$ 
such that 
$\displaystyle f_\varepsilon (z)= 
\log\bigl( \varepsilon+ \sum_{j=1}^p\vert z_j\vert^2\bigr)+ 
{\cal O}(1)$, where $(z_j=0)$ for $j=1,..., p$ are the equations of
$Y\cap V$ and the ${\cal O}(1)$ quantity 
represent a ${\cal C}^\infty$ function which
does not depend on $\varepsilon$.}

\noindent The proof goes along the following lines 
(see [6] for more details): consider a covering of $Z$
with balls $B(p_j, 1)_{j\geq 1}$ such that the centers 
$p_1, ..., p_N$ belong to $Y$, the balls of radius 
1/2 are disjoint and $\cup _{j=1}^NB(p_j, 1)$ is a neighbourhood of 
$Y$ in $Z$. For each $j= 1,...,N$, we take a coordinate system
$(z^{(j)}_k)_{k=1,...,n}$ on $B(p_j, 1)$, such that 
$Y\cap B(p_j, 1)$ is given by $z^{(j)}_k= 0$ for $k=1,...,p$.
Then the local form of the functions we are looking for is 
$$f_{\varepsilon, j}(z)= 
\log\bigl( \varepsilon+ \sum_{k=1}^p\vert z^{(j)}_k\vert^2\bigr)
- {{1}\over {1- \vert z^{(j)}\vert^2}} \hbox{ on } B(p_j, 1).$$

\noindent Now if one needs only a ${\cal C}^0$ 
sequence of functions, then just take the maximum of
$f_{\varepsilon, j}$ because the quantity 
$$\displaystyle {{\varepsilon+ \sum_{k=1}^p\vert z^{(j)}_k\vert^2}\over
{\varepsilon+ \sum_{k=1}^p\vert z^{(l)}_k\vert^2}}$$

\noindent  is bounded
away from zero on $B(p_j, 1)\cap B(p_l, 1)$, so when we are close to the 
boundary of $B(p_j, 1)$, we have 
$f_{\varepsilon, j}(z)< f_{\varepsilon, l}(z)$.
As we need a sequence of ${\cal C}^\infty$ functions,
we use a regularized maximum in order to glue the different 
local expressions above, as explained in [6]--; the ``almost all`` in $ii)$ 
is due to the regularized maximum function.
\vskip 5pt
\noindent We apply the lemma in the case of the 
diagonal $\Delta\subset X \times X$, and so we get our sequence of
${\cal C}^\infty$ functions $(f_\varepsilon)_{\varepsilon}$ on $X\times X$,
such that locally at almost all the points of $\Delta$ we have
$f_\varepsilon(z, \xi)= \log(\varepsilon+ \vert z- \xi\vert^2)$ modulo
some uniform terms (with respect to $\varepsilon$), and we can assume
by rescaling the metric that 
$\widetilde\omega+ i\partial\overline\partial f_\varepsilon> 0$.
\vskip 15pt
\noindent Let us recall the following deep 
result concerning the Monge--Amp\`ere
equations (see [16]):

\noindent {\sl {\tx Theorem {\rm (Yau.)}} Let $(X, \omega)$ 
be a compact K\"ahler
manifold. Then for any smooth
volume form $\exp(f)\omega ^n$ such that $\int_X\omega^n=
\int_X\exp(f)\omega ^n$, there exist a K\"ahler metric $\widetilde \omega$
in the same K\"ahler class as $\omega$, such that $\widetilde \omega^n= 
\exp(f)\omega ^n.$}

\noindent In order to apply it in our situation, remark that
the nefness of the $(1, 1)$--class $\{\widetilde\alpha\}$ on 
$X\times X$ implies the existence of the sequence 
$(\psi_\varepsilon)_{\varepsilon}\subset {\cal C}^\infty(X\times X$
such that $\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\psi_\varepsilon> 0$. Then by Yau's result, 
for each $\varepsilon> 0$ there exist a ${\cal C}^\infty $
function $\phi_\varepsilon$, unique up to some normalization,
which is solution of the following Monge--Amp\`ere equation

$$\bigl(\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial(\psi_\varepsilon+ \phi_\varepsilon)\bigr)^{2n}=
\lambda_\varepsilon(
\widetilde\omega+ i\partial\overline\partial f_\varepsilon )^{2n}\leqno (1)$$

\noindent We denote by $\varphi_\varepsilon:=
\psi_\varepsilon+ \phi_\varepsilon$, normalized such that 
$\displaystyle \int_{X\times X}\!\!
\varphi_\varepsilon dV_{\widetilde\omega}= 0$; in conclusion, 
the previous result gives us the positive $(1, 1)$ form
$\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon$ on $X\times X$ such that

$$(\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon)^{2n}=
\lambda_\varepsilon(
\widetilde\omega+ i\partial\overline\partial f_\varepsilon )^{2n}\leqno (2)$$

\noindent provided that we take 
$\displaystyle \lambda_\varepsilon:= 
{{\int_{X\times X}\!(\widetilde\alpha+ \varepsilon \widetilde\omega)^{2n}}
\over
{\int_{X\times X}\!\widetilde\omega^{2n}}}$. 
The sequence of normalization
constants $(\lambda_\varepsilon )_\ep$ is uniformly 
bounded from above by a strictly 
positive constant, thanks to the fact that 
$\int_{X\times X}\!\!\widetilde\alpha^{2n}> 0$.
The convention in what follows is that we call all the 
constants by the same letter ``$C$'', as long as they don't depends on 
$\ep, r$, and so on. 

\noindent Next let us take for each $\varepsilon > 0$ 
the following differential form of $(n, n)$--type
 $\widetilde T_\varepsilon:= 
(\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon)^{n}$; it is 
$d$--closed, positive and 
$$\int_{X\times X}\widetilde T_\varepsilon\wedge \widetilde\omega^n=
\int_{X\times X}(\widetilde\alpha+ \varepsilon \widetilde\omega)^n
\wedge \widetilde\omega^n\leq C$$

\noindent for some constant $C> 0$. This shows that if we
consider $(\widetilde T_\varepsilon)_{\varepsilon}$
as a sequence of $(n, n)$--currents on $X\times X$, we have a 
uniformly bounded mass, and standard results in the theory 
of closed positive currents show that we can extract a
weak limit of $(\widetilde T_\varepsilon)_{\varepsilon}$,
as $\varepsilon\mapsto 0$; we denote by $\widetilde T$ 
the corresponding limit. Now $\widetilde T$ is also a closed
positive current of $(n, n)$--type, and thanks to the 
closeness of $\widetilde\alpha$ and $\widetilde\omega$,
the cohomology class of $\widetilde T$ is $\{\widetilde\alpha^n\}$.

\noindent Our next task is to show that the current $\widetilde T$
dominates a strictly positive multiple of the current of
integration on the diagonal $\Delta$; by a result of Y.-T. Siu
which we shall recall later, it is enough to show that the generic
Lelong number of $\widetilde T$ on $\Delta$ is positive.
This will be proved by analysing the sequence of equations which 
defines the potential $\varphi_\varepsilon$--we need to show that 
$(\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon)^{n}\wedge
\widetilde\omega^n$ becomes singular on the diagonal, as
$\ep\mapsto 0$, and we show that it is the case 
by using the fact that we have the exact value of the determinant
$(\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon)^{2n}$
and on the other hand we will morally get an upper bound for 
the eigenvalues of $\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon$ in the tangent directions of
$\Delta$--.

\noindent We start by finding a lower bound for the
``singular'' part of the 
left-hand side term in the equality $(2)$.
As we only need a statement about the generic Lelong number,
we shall restrict ourself in a small neighbourhood 
of a point $(x_0, x_0)\in \Delta $
given by the lemma 1. If $(z, \xi)$ are local coordinates at this point,
we recall that the expression of the functions $f_\varepsilon $ 
obtained above is $f_\varepsilon= \log (\varepsilon + \vert z- \xi\vert^2)+
{\cal O}(1)$; as one can easily compute, we have
$$i\partial\overline\partial\log (\varepsilon + \vert z- \xi\vert^2)\geq
{{\varepsilon}\over{(\varepsilon+ \vert z- \xi\vert^2)^2}}
i\sum_{j=1}^nd(z_j- \xi_j)\wedge d(\overline {z_j- \xi_j})
$$
\noindent and we get a similar lower bound for the quantity
$\widetilde\omega+ i\partial\overline\partial f_\varepsilon$
(the Hessian of the ${\cal O }(1)$ term is absorbed by the metric).
This shows that in a neighbourhood $U$ of the fixed
point $(x_0, x_0)$ we have
$$(\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon)^{2n}\geq
C^{-1}
{{\varepsilon^n}\over{(\varepsilon+ \vert z- \xi\vert^2)^{2n}}}
d\lambda(z, \xi)\leqno (3)
$$
\noindent for some positive constant $C$--the inequality (3)
is an easy consequence of the equation $(2)$, and the fact that 
the normalisation constants verify $\lambda_\varepsilon\geq C^{-1}$,
as pointed out before--.

\noindent We show next that at least in ``many points''
of the open set $U$, the eigenvalues of the $(1, 1)$--form
$\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon$ corresponding 
to the tangent directions of the diagonal are bounded --just 
as it is the case for the potential 
$\log (\varepsilon + \vert z- \xi\vert^2)$--. For this, let us compute the
integral
$$\int_{X\times X}
(\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon)\wedge
(\widetilde\omega+ i\partial\overline\partial f_\varepsilon)^n
\wedge\widetilde\omega^{n- 1}= 
\int_{X\times X}(\widetilde\alpha+ \varepsilon \widetilde\omega)
\wedge\widetilde\omega^{2n- 1}\leq C$$

\noindent uniformly with respect to $\varepsilon> 0$. As all
the differential forms involved in the 
previous integral are positive, we have
$$\int_{U}
(\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon)\wedge
(\widetilde\omega+ i\partial\overline\partial f_\varepsilon)^n
\wedge\widetilde\omega^{n- 1}\leq C.\leqno (4)$$

\noindent In order to exploit this inequality, we define 
new local coordinates on $U$ as follows: $Z_j:= z_j- \xi_j$
and $W_j:= z_j+ \xi_j$, for $j= 1,...,n$; in terms of $(Z, W),$
we can write

$$\eqalign{
\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon= &  
i\sum_{j, k}T_{j, k}^{1, \varepsilon}(Z, W) dZ_j\wedge d\overline Z_k+
+ i\sum_{j, k}T_{j, k}^{2, \varepsilon}(Z, W) dW_j\wedge d\overline W_k+ \cr 
+ &  2\Re\bigl(i\sum_{j, k}T_{j, k}^{3, \varepsilon}(Z, W) 
dZ_j\wedge d\overline W_k\bigr). \cr
}$$

\noindent As we clearly have $\displaystyle 
\widetilde\omega^{n- 1}\geq C^{-1}\sum_{j\geq 1}
\prod _{k\neq j} idW_k\wedge d\overline W_k$, 
the inequality $4$ shows that

$$\int_{U}\bigl(\sum_{j= 1}^nT_{j, j}^{2, \varepsilon}\bigr)
{{\varepsilon^n}\over{(\varepsilon+ \vert Z\vert^2)^{2n}}}
d\lambda(Z, W)\leq C
\leqno (5)$$
\noindent (because as we saw before,  
$\displaystyle (\widetilde\omega+ i\partial\overline\partial f_\ep)^n\geq 
C^{-1}{{\varepsilon^n}\over{(\varepsilon+ \vert Z\vert^2)^{2n}}}
\prod _jidZ_j\wedge d\overline Z_j$ on $U$). 
In conclusion, we obtain a $L^1$--bound
of the eigenvalues of the K\"ahler metric 
$\widetilde\alpha+ \varepsilon \widetilde\omega+ 
i\partial\overline\partial\varphi_\varepsilon$ 
in the $W$--directions, if the $L^1$ norm in question is taken
with respect to the measure
$$d\lambda_\varepsilon(Z, W):= 
{{\varepsilon^n}\over{(\varepsilon+ \vert Z\vert^2)^{2n}}}
d\lambda(Z, W)$$

\noindent We fix now some notations for the real and imaginary part of
$Z$ and $W$, namely: $Z_j= x_j+ iy_j, W_j= u_j+ iv_j$, and consider 
the following neighbourhood of $\Delta\cap U$
$\displaystyle U_\varepsilon:= \lbrace(Z, W)\in U/
\prod_{j= 1}^n(\vert x_j\vert^2<\varepsilon)
\times(\vert y_j\vert^2<\varepsilon)\hbox{ and } \vert W\vert< 1\rbrace.
$

\noindent Then the previous inequality $(5)$ gives
$$\int_{U_\ep}\bigl(\sum_{j= 1}^nT_{j, j}^{2, \varepsilon}\bigr)
d\lambda_\varepsilon(Z, W)\leq C\leqno (6)$$

\noindent and again a quick computation shows that 
$\displaystyle \Vol_{\lambda_\varepsilon}(U_\varepsilon)= C^{-1}> 0$
for all $\varepsilon > 0$. We have the obvious lemma

\noindent {\sl {\tx Lemma.} Let $(X, \mu)$ be a topological 
space endowed with a measure $\mu$ such that $\int_Xd\mu= 1$
and $f: X\rightarrow \bR_+$ a continuous function such that
$\int_Xfd\mu\leq C$. Then there exist a open set $W\subset X$
$\int_Wd\mu\geq  1/2$ such that $f(w)\leq 2C$ for all 
$w\in W$.}

\noindent By applying this result in our situation, we get
a open subset $\Lambda_\varepsilon\subset U_\varepsilon$,
such that
\smallskip 
\item {i)} For all $\varepsilon > 0$, we have 
$\Vol_{\lambda_\varepsilon}(\Lambda_\varepsilon)\geq 
1/2\Vol_{\lambda_\varepsilon}(U_\varepsilon)$
\smallskip 
\item {ii)} The eigenvalues corresponding to the tangent directions
verify $\displaystyle T_{j, j}^{2, \varepsilon}(Z, W)\leq
C$ for all $j=1,...,n$ and $(Z, W)\in \Lambda_\varepsilon$.
\vskip 10pt
\noindent For a small enough real $r> 0$ and a point $p\in U$ 
at a distance at least $r$ from the boundary of $U$, 
we denote by
$\displaystyle {\cal C}_{4n}(p, r):= 
\lbrace (Z, W)\in U/ \prod_{j=1}^n(\vert x_j- x_j(p)\vert\leq r)
\times(\vert y_j- y_j(p)\vert\leq r)
\times(\vert u_j- u_j(p)\vert\leq r)
\times(\vert v_j- v_j(p)\vert\leq r)\rbrace$ the $4n$--dimensional real cube 
centred at $p$. If $p\in \Delta$, we denote by 
$\displaystyle {\cal C}_{2n}(p, r):= {\cal C}_{4n}(p, r)\cap \Delta$, 
i.e. the trace of the cube on the diagonal, which is a $2n$--dimensional 
cube. Let
$$\Lambda_\varepsilon(r):= \lbrace
p\in \Delta/\Vol_{\lambda_\varepsilon}
({\cal C}_{4n}(p, r)\cap \Lambda_\varepsilon)\geq
2^{-4n- 2}\Vol_{\lambda_\varepsilon}
({\cal C}_{4n}(p, r)\cap U_\varepsilon)\rbrace .$$

\noindent Now if $r$ is fixed, we show in what follows that
$\displaystyle \Vol_{\lambda}\big(\Lambda_\varepsilon(r)\big)\geq C^{-1}$,
uniformly with respect to $\varepsilon$ and $r$, provided that
$\varepsilon< \varepsilon_0(r)$ (here $\lambda$ denotes the Lebesgue 
measure on $\Delta$). 

\noindent For this, we take a subdivision of 
${\cal C}_{2n}(x_0, 1)\subset \Delta$ into cubes of side 
$r/2$ (we can do this for instance if 
$r= 1/m$ with $m\in \bZ_+$; this is not at all a 
serious restriction for what we shall need);
denote by $(p_j)_j$ the corresponding 
centers of the cubes. 
Then consider $\cup_j{\cal C}_{4n}(p_j, r/2)$
; it covers $U_\ep$ if $r/4> \sqrt \ep$
(so we take $\ep_0(r)= r^2/16$). 
\noindent The relevant set of centers is 
$$J:= \{j\in \bN/ 
\Vol_{\lambda_\ep}\big({\cal C}_{4n}(p_j, r/2)\cap \Lambda_\ep\big)\geq 1/4
\Vol_{\lambda_\ep}\big({\cal C}_{4n}(p_j, r/2)\cap U_\ep\big)\big\}$$

\noindent From the inequality i) we have
$$\Vol_{\lambda_\ep}\big
(\cup_{j\in J}{\cal C}_{4n}(p_j, r/2)\cap \Lambda_\ep\big)+
\Vol_{\lambda\ep}\big
(\cup_{j\in {\cal C}_J}{\cal C}_{4n}(p_j, r/2)\cap \Lambda_\ep\big)
\geq 1/2\Vol_{\lambda_\ep}(U_\ep) $$

\noindent But the construction of the covering 
of $U_\ep$ shows that
$\Vol_{\lambda_\ep}(U_\ep)\geq (r/2)^{2n}\big(\vert J\vert+ 
\vert {\cal C}_J\vert\big)$,
(where $\vert J\vert$ denotes the cardinal of the set $J$),
and on the other hand we have
(by the definition of $J$), 
$$\Vol_{\lambda_\ep}\big
(\cup_{j\in {\cal C}_J}{\cal C}_{4n}(p_j, r/2)\cap \Lambda_\ep\big)=
\sum_{j\in {\cal C}_J}\Vol_{\lambda_\ep}
\big({\cal C}_{4n}(p_j, r/2)\cap \Lambda_\ep\big)
\leq 1/4(r/2)^{2n}\vert {\cal C}_J\vert $$

\noindent as well as
$$\Vol_{\lambda_\ep}\big
(\cup_{j\in J}{\cal C}_{4n}(p_j, r/2)\cap \Lambda_\ep\big)
\leq (r/2)^{2n}\vert J\vert .$$

\noindent By combining the previous inequalities,
we immediately get
$$\vert J\vert + 1/4\vert {\cal C}_J\vert\geq
1/2(\vert J\vert + \vert {\cal C}_J\vert )$$

\noindent and this relation implies $\vert J\vert \geq 1/3
(\vert J\vert + \vert {\cal C}_J\vert )$. In this way we obtain
$$\Vol_{\lambda} \big(\cup_{j\in J}{\cal C}_{4n}(p_j, r/2)\cap \Delta\big)=
(r/2)^{2n}\vert J\vert\geq 1/3\Vol_{\lambda}
\big({\cal C}_{2n}(x_0, 1)\big).$$


\noindent  Now if
$p\in \cup_{j\in J}{\cal C}_{4n}(p_j, r/2)\cap \Delta$, 
then $p\in {\cal C}_{4n}(p_j, r/2)\cap \Delta$ for some $j\in J$
and we have
$$\eqalign{
\Vol_{\lambda_\ep}\big({\cal C}_{4n}(p, r)\cap \Lambda_\ep\big)\geq &
\Vol_{\lambda_\ep}\big({\cal C}_{4n}(p_j, r/2)\cap \Lambda_\ep\big)\geq\cr
\geq & 1/4\Vol_{\lambda_\ep }\big({\cal C}_{4n}(p_j, r/2)\cap U_\ep\big)= \cr
= & 1/2^{4n+ 2}\Vol_{\lambda_\ep}\big({\cal C}_{4n}(p, r)\cap U_\ep\big)\cr
}$$

\noindent and this proves the assertion concerning the 
volume of $\Lambda_\ep(r)$.
\vskip 15pt
\noindent Recall now the equation which gives the function
$\varphi_\ep$; we already saw that it implies
$$(\widetilde\alpha+ \ep\widetilde\omega+
i\partial\overline\partial\varphi_\ep)^{2n}
\geq 
C^{-1}
{{\varepsilon^n}\over{(\varepsilon+ \vert Z\vert^2)^{2n}}}
d\lambda(Z, W)\leqno (7)$$

\noindent for all the points $(Z, W)\in U_\ep$. In terms of 
the local functions $T_{j, k}^{p, \ep}$ defining $
\widetilde\alpha+ \ep\widetilde\omega+
i\partial\overline\partial\varphi_\ep$ locally on $U_\ep$,
the inequality $(7)$ gives
$$\det\Big(T_{j, k}^{1, \ep}(Z, W)\Big)
\prod_jT_{j, j}^{2, \ep}(Z, W)\geq 
C^{-1}
{{\varepsilon^n}\over{(\varepsilon+ \vert Z\vert^2)^{2n}}}
$$

\noindent (this follows by the 
lemma 2 which will be stated and proved below).
\vskip 15pt
For some $r>0$ fixed, and $\ep$ small enough, let us integrate
the previous inequality over ${\cal C}_{4n}(p,\ r)\cap\Lambda_\ep$;
here we take $p\in\Lambda_\ep(r)$. We obtain 
$$\eqalign{
\int_{{\cal C}_{4n}(p, r)\cap\Lambda_\ep}  
 & \det\bigl(T^{1,\ep}_{j,k}(Z,W)\bigr)
\prod_{j=1}^n T^{2,\ep}_{j,j}(Z,W)d\lambda(Z,W) \geq \cr  
\geq  & C^{-1}
\int_{{\cal C}_{4n}(p, r)\cap\Lambda_\ep}
{{\ep^n}\over {(\ep+ \vert Z\vert^2)^{2n}}}d\lambda(Z, W)=\cr 
= & C^{-1}\Vol_{\lambda_\ep}({\cal C}_{4n}(p, r)\cap\Lambda_\ep)\geq \cr
\geq & C^{-1}\Vol_{\lambda_\ep}({\cal C}_{4n}(p, r)\cap U_\ep)= 
C^{-1}r^{2n} \cr
}$$


\noindent (we have used the fact that $p\in\Lambda_\ep(r)$).
If $(Z, W)\in \Lambda_\ep$, then we have a uniform upper bound
for the eigenvalues $\displaystyle T^{2,\ep}_{j,j}(Z,W)$, so from
the inequality just proved it follows

$$\int_{{\cal C}_{4n}(p, r)}
\det\bigl(T^{1,\ep}_{j,k}(Z,W)\bigr)d\lambda(Z,W)\geq
C^{-1}r^{2n}.
$$

\noindent The current $\displaystyle \widetilde T_\ep
= (\widetilde\alpha+ \ep\widetilde\omega+
i\partial\overline\partial\varphi_\ep)^{n}$ verify 
the pointwise inequality
$$\widetilde T_\ep\wedge \widetilde\omega^n\geq C^{-1}
\widetilde T_\ep\wedge\prod_{j=1}^nidW_j\wedge d\overline W_j= C^{-1}
\det\big(T^{1,\ep}_{j,k}(Z,W)\bigr)d\lambda(Z,W)$$

\noindent so we get
$$\int_{{\cal C}_{4n}(p, r)}\!\! \widetilde T_\ep
\wedge \widetilde\omega^n\geq C^{-1}r^{2n}\leqno (8)$$

\noindent if $p\in\Lambda_\ep(r)\subset \Delta$ and $\ep< \ep_0(r)$.
\vskip10pt
\noindent Consider now $\displaystyle 
\Lambda_{\infty}(r):= \lim\sup_{\ep< \ep_0(r)}\Lambda_\ep(r)$; by Fatou's
lemma we observe that $\Vol_{\lambda}\big(\Lambda_{\infty}(r)\big)
\geq C^{- 1}\Vol(\Delta)$, and here we 
recall that the constant $C$ does not depends on $r$.
On the other hand, if $p\in \Lambda_{\infty}(r)$, then there exists
a sequence $\displaystyle p_{m_k}\in \Lambda_{\ep_k}(r)$ such that
$\displaystyle p_{m_k}= p$ and $\ep_k\mapsto 0$ as $k\mapsto \infty$.
Then for all $k\geq 0$, we get
$$\int _{{\cal C}_{4n}(p, r)}\widetilde T_{\ep_k}
\wedge 
\big(i\partial\overline\partial(\vert Z\vert^2+ \vert W\vert ^2\big)^{n}
\geq C^{- 1}r^{2n}$$

\noindent for all $k\geq 0$, so if we take the limit 
as $k\mapsto \infty$, we obtain
$$\int _{{\cal C}_{4n}(p, r)}
\widetilde T\wedge \widetilde\omega^{n}\geq C^{- 1}r^{2n}$$

\noindent for all $p\in \Lambda_{\infty}(r)$ 
(the metric ``$\omega$'' in the previous relation
and in what follows is the flat one). 
\par In conclusion,
the current $\widetilde T$ has the following property: 
there exists a positive
constant $C$ such that for all $0< r\ll 1$ there exists a 
set $\Lambda_{\infty}(r)\subset \Delta$, with 
$\Vol_{\lambda}\big(\Lambda_{\infty}(r)\big)\geq
C^{- 1}\Vol(\Delta)$ 
and if $p\in \Lambda_{\infty}(r)$, we have
$${{1}\over {r^{2n}}}
\int _{B(p, r)}\widetilde T\wedge \widetilde\omega^{n}
\geq C^{-1}.\leqno (9)$$

\noindent We shall see next that this property of our current
$\widetilde T$ implies that its 
generic Lelong number on $\Delta$ is strictly
positive. Assume the contrary; then in the complementary of a set of 
measure zero in $\Delta$ (the set in question is a countable union
of analytic sets) we have $\nu(\widetilde T, z)= 0$. There exists a
compact set $K\subset \Delta$ such that 
$\Vol(K)\geq (1- C^{- 1}/2)\Vol(\Delta)$
and $\nu(T, z)= 0$ for all $z\in K$ (here and in what follows the
constant ``$C$'' is really the one in the inequality (9)). 
Take a point $z_0\in K$; then
$$\lim_{r\mapsto 0}{{1}\over {r^{2n}}}
\int _{B(z_0, r)}\widetilde T\wedge \widetilde\omega^{n}= 0$$

\noindent so there exists a $r(z_0)$ such that
if $r\leq r(z_0)$, we have
$${{1}\over {r^{2n}}}
\int _{B(z_0, r)}\widetilde T\wedge \widetilde\omega^{n}
\leq C^{- 1}/2^{2n+ 1}.$$

\noindent If $z\in B(z_0, r(z_0)/2)$, then $B(z, r(z_0)/2)\subset
B(z_0, r(z_0))$ so we obtain by the previous inequality
$${{1}\over {{(r(z_0)/2)}^{2n}}}
\int _{B(z, r(z_0)/2)}\widetilde T\wedge \widetilde\omega^{n}
\leq C^{- 1}/2$$

\noindent and as the function $\displaystyle 
r\mapsto {{1}\over {r^{2n- 2}}}
\int _{B(z, r)}\widetilde T\wedge \widetilde\omega^{n- 1}$ 
is non-decreasing 
for each $z$, if $r\leq r(z_0)/2$ we have
$${{1}\over {r^{2n}}}
\int _{B(z, r)}\widetilde T\wedge \widetilde\omega^{n}
\leq C^{- 1}/2.$$

\noindent This proves that we can choose a $r_0$ uniform
for all $z\in K$ such that if $r\leq r_0$, we have
$${{1}\over {r^{2n}}}
\int _{B(z, r)}\widetilde T\wedge \widetilde\omega^{n}
\leq C^{- 1}/2.\leqno (10)$$

\noindent But $\Vol(K)\geq (1- C^{- 1}/2)\Vol(\Delta)$ 
and for all $r$ small enough, $\Vol\big(\Lambda_\infty(r)\big)\geq
C^{- 1}\Vol(\Delta)$, so $K\cap \Lambda_\infty(r)\neq \emptyset$,
and the inequality (9) contradicts (10).

\vskip 15pt
\noindent Let us recall now
some results in the theory of currents, which are
due to Siu (see [14])--as they have been (or will be) 
used in the proof--. 
Let $\widetilde T$ be a closed positive
current of $(n, n)$--type; then we have the notion
of generic Lelong number of $\widetilde T$ on the diagonal $\Delta$; 
this is formulated in the following lemma.

\noindent {\sl {\tx Lemma\hskip 2pt {\rm (Y.-T. Siu).}} If $\widetilde T$
is a closed positive current of bidegree $(n, n)$ and
$\Delta$ is the diagonal of $X\times X$, we set
$$m_\Delta:= \inf\{z\in \Delta;\nu(\widetilde T, z)\}$$

\noindent Then $\nu\widetilde (T, x)= m_\Delta$ 
for $x$ in the complementary
of a countable union of analytic sets in $H$.}

\noindent In view of our preceeding considerations,
the next result completes the main part in the proof of the
theorem 1: 

\noindent {\sl {\tx Proposition \hskip 2pt {\rm (Y.-T. Siu).}} 
Suppose $\widetilde T$ is a closed 
$(n, n)$--current and $H$ is a $n$--di\-men\-sional 
irreducible analytic subset of 
$X\times X$. If we note by $m_H$ the generic 
Lelong number of $T$ on $H$, then the current 
$\widetilde T- m_H[H]$ is positive.} 

\noindent So by the previous lemma, 
and the fact that in our case the generic Lelong number
is greater than $C^{-1}$, we obtain that
$\widetilde T\geq C^{-1}[\Delta]$, and moreover
we have $\widetilde T\in\{\widetilde \alpha\}^n$.
By taking the wedge product of $\widetilde T$ with the pull--back
of the K\"ahler metric $\omega$ we get
$$\widetilde T\wedge \pi_2^*\omega\geq 1/C[\Delta]\wedge \pi_2^*\omega.$$

\noindent To finish with, we denote 
$T:= C(n,\alpha)\pi_{1, *}(\widetilde T\wedge \pi_2^*\omega )$,
where the positive constant 
$\displaystyle C(n, \alpha):= 
{{n^{-1}}\over {\int_X\alpha^{n- 1}\wedge\omega}}$; 
we show that it is a 
closed and strictly positive current in the cohomology class
$\{\alpha \}$. Indeed, let $\gamma$ be a positive differential
form on $X$ of $(n- 1, n- 1)$--type; then
$$\eqalign{
\langle T, \gamma \rangle = &  C(n, \alpha)\int _{X\times X}\!\! 
\widetilde T\wedge \pi_2^*\omega\wedge \pi_1^*\gamma\geq\cr
\geq & C^{-1}\int _{X\times X}\!\![\Delta]\wedge \pi_2^*\omega
\wedge \pi_1^*\gamma
= C^{-1}\int _{\Delta}\pi_2^*\omega
\wedge \pi_1^*\gamma= C^{-1} \int_ X\omega\wedge \gamma\cr
}$$ 

\noindent so we get $T\geq C^{-1}\omega$.
To find the cohomology class of $T$,
consider $\beta$ a $d$--closed form of $(n- 1, n- 1)$--type;
we have
$$\eqalign{
\langle T, \beta \rangle = & C(\alpha, n)\int _{X\times X}\!\! 
\widetilde T\wedge \pi_2^*\omega\wedge \pi_1^*\gamma= 
C(\alpha, n)\int_{X\times X}\!\!(\pi_1^*\alpha+ \pi_2^*\alpha)^n\wedge
\pi_2^*\omega\wedge \pi_1^*\beta\cr
= & nC(\alpha, n)\int _{X\times X}\!\!\pi_1^*\alpha \wedge\pi_1^*\beta
\wedge(\pi_2^*\alpha)^{n- 1}\wedge\pi_2^*\omega=\cr
= & \int_X\!\!\alpha\wedge \beta\cr
}$$

\noindent and the theorem 1 is proved, modulo the following
lemma about positive differential forms. 
\vskip 10pt
\noindent {\sl {\tx Lemma 2.} Consider a positive 
$(1, 1)$--form with constants coefficients on $\bC^{2n}$ 
$\displaystyle  
T= i\sum_{j, k= 1}^nT_{j, k}^{1}dZ_j\wedge d\overline Z_k
+ i\sum_{j, k= 1}^nT_{j, k}^{2}dW_j\wedge d\overline W_k+ 
2\Re\bigl(i\sum_{j, k= 1}^nT_{j, k}^{3} 
dZ_j\wedge d\overline W_k\bigr). 
$ 
 Then $\det (T)\leq C(n)\det (T_{j, k}^{1}) 
\det (T_{j, k}^{2}),$ where $C(n)$ is a constant which depends 
only on $n$.}



\noindent {\tx Proof.} We can take some linear changes of coordinates 
in the $Z$ and respectively $W$--variables separately, such that
in the new coordinates --still denoted $(Z, W)$ we have
$$T= i\sum_{j= 1}^nT_{j}^{1}dZ_j\wedge d\overline Z_k+
+ i\sum_{j= 1}^nT_{j}^{2}dW_j\wedge d\overline W_k+ 
2\Re\bigl(i\sum_{j, k= 1}^nT_{j, k}^{3} 
dZ_j\wedge d\overline W_k\bigr);$$

\noindent moreover, we can do this such that the 
quantity $\det (T)$ remaines 
unchanged, and in addition
$\displaystyle \det (T_{j, k}^{1})=\prod_jT_{j}^{1}$, and 
$\displaystyle \det (T_{j, k}^{2})= \prod_jT_{j}^{2}$. 
In conclusion, we have
to prove the following assertion: let $\displaystyle T= 
i\sum_{j, k=1}^nT_{j, k}dz_j\wedge dz_k$ be a real positive 
form on $\bC^n$; then $\displaystyle 
\det (T_{j,k})\leq C(n)\prod_j T_{j,j}$ for some constant which
depends only on $n$. But this is quite clear, as the positiveness
of T shows that for each pairs of indices $(j, l)$ we have 
$\vert T_{j, l}\vert^2\leq T_{j, j} T_{l,l}$, 
and the lemma is proved simply by writing down
the expression of the determinant
(the constant $C(n)$ can be taken $n!$ for instance).
\vskip 5pt
\noindent {\tx Remark.} If $X$ is a projective manifold, then
one can prove by the same method that in a 
nef cohomology
class $\{\alpha\}$ of (1,1)--type such that $\int_X\alpha^n> 0$,
there exist a positive current whose
Lelong numbers on some ample divisor are positive (see [13]). 

\vskip 15pt
\noindent {\twel Proof of the Theorem 2.}
\vskip 5pt
\noindent One implication is clear, namely if the class
$\{\alpha\}$ is K\"ahler, then we obviously have 
$\int_Z\alpha^k\wedge\omega^{p- k}> 0$ 
for all $p$--dimensional analytic subset $Z\subset X$
and all $k= 1,...,p$. 
For the other implication, we use the induction on $n$, the complex 
dimension of $X$. If $n= 1$ there is nothing to prove, because
on a curve $C$ all the $(1,1)$--classes are proportional with 
a fixed K\"ahler metric, so the hypothesis we have on the
class $\{\alpha \}$ imply that the multiple in question is a 
positive real. Thus we can assume the result proved for all
the manifolds of dimension less than $n$, and we start by considering 
the set

$${\cal J}:= \bigl\{t\geq 0/\hbox{ the class } \{\alpha + t\omega\}
\hbox{ is K\"ahler }\bigr\}.$$

\noindent The compactness of $X$ show that ${\cal J}\neq \emptyset$ 
(as the differential form $\omega$ is positive); it is also clear that 
${\cal J}$ is open, because if $t_0\in {\cal J}$, then
$\{\alpha + t_0\omega\}$ is a K\"ahler class, so for all 
$t\in \bR_+$ close enough to $t_0$, the class  
$\{\alpha + t\omega\}$ is still K\"ahler. 

\noindent We shall prove now that ${\cal J}$ is closed; this will be a 
consequence of the theorem 1. Let 
$\displaystyle t_0= \lim_\nu t_\nu$ with $t_\nu\in {\cal J}$. Then for each
$\nu\geq 1$ the class $\{\alpha + t_\nu\omega\}$ is K\"ahler, and 
a first observation is that the limit class 
$\{\alpha + t_0\omega\}$ is nef. Actually, let $\ep> 0$; for each
$\nu\geq 1$, we have a function $\varphi_\nu\in {\cal C}^\infty (X)$
such that 
$\alpha + t_\nu\omega+ i\partial\overline\partial\varphi_\nu> 0$ 
on $X$, and then we get 
$$\alpha + t_0\omega+ i\partial\overline\partial\varphi_\nu
\geq (t_0- t_\nu)\omega\geq -\ep\omega$$

\noindent if $\nu\geq \nu_0$; so as claimed, our limit class
$\{\alpha + t_0\omega\}$ is nef (actually, there is a general
result due to J.-P. Demailly, which says that on a K\"ahler manifold, 
the closure of the K\"ahler cone is the nef cone, see [7]).
On the other hand, we have 
$$\eqalign{
\int_X\big(\alpha + t_0\omega\big)^n= &  \int_X\alpha^n+
\sum_{k= 1}^n{\pmatrix 
{n\cr k\cr}}t_0^k\int_X\omega^k\wedge\alpha^{n- k}>\cr
> & \hskip 1pt 0\cr
}$$

\noindent for all $t_0\geq 0$ by the hypothesis. 
The theorem 1 shows that the cohomology class
$\{\alpha + t_0\omega\}$ contains a representative, say
$T_0:= \alpha + t_0\omega+ i\partial\overline\partial \phi_0$, such that
$T_0\geq C^{-1}\omega$, for some positive constant $C$; the function $\phi_0$
is in $L^1(X)$. Now the singularities of  $\phi_0$ can be quite terrible, but
as $T_0$ is strongly positive, we can obtain another current, say $T_1$ 
in the same cohomology class, which is
still strongly positive and such that its potential has logarithmic singularities.
This follows from a regularization theorem of J.-P. Demailly, which we state next.

\vskip 15pt
\noindent {\sl {\tx Theorem {\rm (J.--P. Demailly).}} Let $\psi$ be an almost
psh function on a compact complex manifold $X$, such that 
$i\partial\overline\partial \psi\geq \rho$ for some continuous $(1,1)$--form
$\rho$. Then there is a sequence of almost psh functions $\psi_m$ with logarithmic
singularities, such that $i\partial\overline\partial \psi\geq \rho- \ep_m\omega$
where $\omega$ is a fixed hermitian metric on $X$ and we have
$\ep_m\mapsto 0$.}
\vskip 5pt
\noindent Thus by a mild regularization of our current $T_0$, we get
$T_1= \alpha + t_0\omega+ i\partial\overline\partial \phi_1$ such that
$T_1\geq C^{-1}\omega$; 
the singularities of $\phi_1$ are now concentrated 
on a finite union of irreducible algebraic sets 
(by the Siu's theorem quoted above), namely 
$Y_1\cup ...\cup Y_N$ (so in a neighbourhood of 
each point of the previous union,
the shape of the singular part of $\phi_1$ is $\log(\vert f_1\vert^2+...+  
\vert f_m\vert^2)$ for some holomorphic functions $(f_j)$ whose common 
zero set describe locally $Y_1\cup ...\cup Y_N$). To finish the proof, 
let us point out how the induction hypothesis 
can be used in order to
remove the poles of $\phi_1$. This is a slightly technical matter, 
--but in some sense, quite standard--
because of the possible singularities of $Y_j$, and will be exposed in the 
next lemma (see [7] and also [12]).

\vskip 5pt
\noindent {\sl {\tx Lemma 3.} There exist a constant $C> 0$, a neighbourhood
${\cal V}$ of $Y_1\cup ...\cup Y_N$ and a function 
$\phi_2\in {\cal C}^\infty ({\cal V})$ such that

$$\alpha + t_0\omega+ i\partial\overline\partial \phi_2\geq 
C^{-1}\omega
\hbox{ on }{\cal V.}$$}

\smallskip
\noindent {\tx Proof of the lemma}. We will proceed by induction on 
$k= \max(\dim Y_j)$.
The case of curves is left to the reader, 
so we can assume the statement proved 
for all algebraic sets of dimension less than $k$. We start with a single
irreducible $k$--dimensional algebraic set $Y_1$. According to 
Hironaka's theorem, there exist a modification 
$\pi:\widetilde X\mapsto X$ such that
the proper transform, say $\widetilde Y$ of $Y_1$ is smooth. 
As we have already seen, the cohomology 
class $\{\alpha+ t_0\omega\}$ is nef, so its inverse image on 
$\widetilde X$ 
is equally nef (see for example [7]), and moreover the restriction 
$\displaystyle \pi^*\{\alpha+ t_0\omega\}_{\vert \widetilde Y}$ 
has the same property. 
On the other hand, we have $\displaystyle \int_{\widetilde Y}
\pi^*\big(\alpha+ t_0\omega\big)^k= \int_{Y_1}
\big(\alpha+ t_0\omega\big)^k
> 0$ by the hypothesis on $\alpha$. 
The theorem 1 show that there exist 
a strongly positive current in the cohomology class 
$\displaystyle \pi^*\{\alpha+ t_0\omega\}_{\vert \widetilde Y}$, 
so in particular we have
$$T_1:= \pi^*\big(\alpha+ t_0\omega\big)+
i\partial\overline\partial{\widetilde\psi }
\geq C^{-1}\pi^*\big(\omega\big)$$
\noindent on the manifold $\widetilde Y$; we can assume that the poles of 
$\widetilde \psi $ are concentrated on a union of algebraic subsets of 
$\widetilde Y$, by another application of Demailly's theorem.   
So if we take the direct image of $T_1$ by $\pi$, 
we get a strongly positive current $T:=\alpha+ t_0\omega+
i\partial\overline\partial\psi_1$ on $Y_1$, which is moreover smooth on 
the complement of some proper algebraic subset of 
$Y_1$, say $Z_1\cup ...\cup Z_M$.
By the induction hypothesis, there exist a neighbourhood $V$ of 
$Z_1\cup ...\cup Z_M$ in $X$ and a function $\psi_2\in {\cal C}^\infty (V)$
such that 
$\alpha+ t_0\omega+
i\partial\overline\partial\psi_2\geq C^{-1}\omega$ on $V$. 
Now we glue the functions $\psi_1$ and $\psi_2$ 
together, in the following way:
let $\tau$ be a smooth function on $Y\setminus (Z_1\cup ...\cup Z_M)$, such that
$\tau$ has logarithmic poles on $Z_1\cup ...\cup Z_M$. Then we take
$\psi= \max_{\reg}(\psi_1+ \ep\tau, \psi_2- C)$, where $C$ is a positive 
enough real such that $\psi_1+ \ep\tau> \psi_2- C$ at the boundary 
of $V\cap Y_1$
and $\max_{\reg}$ is a regularized maximum function, see [6]. So we have 
proved that the restriction of the class $\{\alpha+ t_0\omega\}$
on $Y_1$ has a smooth and strictly positive representative, if 
$\ep$ is small enough; by another result 
of J.--P. Demailly (see [6]) 
we can modify the function $\psi$ 
in order to get positiveness also in the normal directions of $Y_1$, 
so the case of an irreducible algebraic set of dimension $k$ is done. 

\noindent To finish the proof, we construct recursively
a neighbourhood $V_m$ of $Y_1\cup...\cup Y_m$ and a function 
$\psi_m\in {\cal C}^\infty (V_m)$ for all $m=1,...,N$ 
with the properties required in the lemma. The case $m=1$ has been 
already treated, let us point out how to add a new set $Y_{m}$. We consider
$Y_{m}\cap (Y_1\cup...\cup Y_{m- 1})= W_1\cup...\cup W_l$. There exist a 
neighbourhood $V^\prime$ of $Y_m$ in $X$ and a function 
$\psi^\prime_m\in {\cal C}^\infty (V^\prime)$ such that 
$\alpha+ t_0\omega+
i\partial\overline\partial{\psi^\prime_m}\geq C^{-1}\omega$ 
on $V^\prime$. By the same procedure as before, we glue now the functions
$\psi_{m- 1}$ and $\psi^\prime_m$; so take $\tau$ a function which is
smooth on $Y_m\setminus (W_1\cup...\cup W_l)$, with logarithmic poles on
$W_1\cup...\cup W_l$, and consider 
$\psi_m:= \max_{\reg}(\psi_{m- 1}+ \ep\tau, \psi^\prime_m- C)$ on the open set
$V_{m- 1}$. 
If the constant $C$ is big enough and $\ep$ small enough, we still have 
$\alpha+ t_0\omega+
i\partial\overline\partial{\psi_m}\geq C_1^{-1}\omega$ on $V_{m- 1}$,
for some positive $C_1$, and in addition, we have $\psi_m= \psi^\prime_m- C$
in a neighbourhood of $W_1\cup...\cup W_l$. Then we can define $\psi_m$ in a 
neighbourhood of $Y_m$ as $\psi_m:= \psi^\prime_m- C$ and this ends 
the proof of the lemma. 
\vskip 15pt
\noindent Now it is clear how to get a K\"ahler metric in the cohomology class
$\{\alpha+ t_0\omega\}$: we remove the poles of $\phi_1$ using the 
function $\phi_2$ given by the lemma 3 and the procedure described (several times)
in the lines above.

\vskip 10pt  
\noindent Finally, the set ${\cal J}$ is also closed, 
and in view of our preceeding considerations, this means that
${\cal J}= (t\geq 0)$, so the cohomology class $\{\alpha \}$ is K\"ahler.
The theorem 2 is proved; as for the theorem $2^\prime$, 
it is an obvious corollary of the theorem 2, because we can consider 
$A\mapsto X$ a very ample line bundle endowed with a metric
$h$ of strictly positive curvature; then take $\omega:= \Theta_h(A)$
and we can easily see that if $\displaystyle \int_Z\alpha^p> 0$
for all $p$--dimensional algebraic subset $Z\subset X$, then 
$\displaystyle \int_Z\alpha^k\wedge\omega^{p- k}> 0$.

\vskip 5pt 
\noindent {\tx Remark.} An obvious application of the theorem 2
gives the following
result: let $(X, \omega)$ be a compact K\"ahler manifold, and 
let $\{\alpha \}$ be a $(1, 1)$--cohomology class on $X$. Then 
$\{\alpha \}$ is nef iff $\int_Z\alpha^k\wedge\omega^{p- k}\geq 0$
for all $p$--dimensional analytic subset $Z\subset X$ and all 
$k= 1,...,p$ (and an analogue statement in projective context).

\vskip 5pt
\noindent Finally, we point out a ``linear'' characterization for
the nefness of a cohomology class, as follows 

\vskip 5pt
\noindent {\sl {\tx Theorem 3.} Let $X$ be a compact K\"ahler
manifold, and let $\alpha$ be a $d$--closed form of
$(1, 1)$--type; then the class $\{\alpha\}$ is nef iff
$\displaystyle \int_Z\alpha\wedge\omega_{1}\wedge...
\wedge\omega_{k- 1}\geq 0$,
for all $(Z, \omega_j)$, where $Z$ is a k--dimensional 
analytic subset of $X$, and $(\omega_j)_j$ are K\"ahler metrics on $Z$.}

\noindent The proof of the theorem 3 is very easy
and left to the reader
(one can use Kleiman's arguments in the case of line bundles, and the
theorem 2). 









\vskip30pt

\noindent {\twel R{e}f{e}rences:}
\vskip5pt
{\parindent=0.7cm


\item{\bf [1]} Bonavero, L.\ ---{\sl In\'egalit\'es de Morse
holomorphes singuli\`eres}, C.\ R.\ Acad.\ Sci.\ {\bf
317} (1993).

\item{\bf [2]} Bouche, T.\ ---{\sl In\'egalit\'es de Morse 
pour la $d^{\prime \prime}$--cohomologie sur une vari\'et\'e
non--compacte}, Ann.\ Sci.\ Ecole Norm.\ Sup.\  {\bf 22} 
(1989).

\item{\bf [3]} Buchdal, N.\ ---{\sl On compact K\"ahler surfaces}, 
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\item{\bf [4]} Campana, F., Peternell, T.\ ---{\sl Algebraicity of the 
ample cone of
projective varieties} J.\ Reine Angew.\ Math.\ {\bf 407} (1990) 

\item{\bf [5]} Demailly, J.-P.\ --- {\sl Champs magn\'etiques et
in\'egalit\'es de Morse pour la $d^{\prime \prime}$--cohomologie},
Ann.\ Inst.\ Fourier\ {\bf 35} (1985). 

\item{\bf [6]} Demailly, J.P.\ --- {\sl Cohomology of q--convex 
Spaces in Top Degrees}, Math.\ Z.\ 
{\bf 204} (1990), 283--295

\item{\bf [7]} Demailly, J.P.\ --- {\sl Regularization of closed
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{\bf 1} (1992), 361--409

\item{\bf [8]} Demailly, J.-P.\ --- {\sl A numerical criterion for
very ample line bundles}, J.\ Diff.\ Geom.\ {\bf 37} (1993). 

\item{\bf [9]} Eyssidieux, P.\ ---{\sl Th\'eor\`emes de 
Nakai--Moishezon pour certaines classes de type (1,1)
et applications}, preprint.

\item{\bf [10]} Kleiman, S.\ --- {\sl Toward a numerical theory of
ampleness}, Ann.\ Math.\ {\bf 84} (1966). 

\item{\bf [11]} Paun, M.\ --- {\sl Sur l'effectivit\'e num\'erique 
des images inverses de fibr\'es en droites}, 
Math.\ Ann.\ {\bf 310} (1998). 

\item{\bf [12]} Paun, M.\ --- {\sl 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). 


\item{\bf [13]} Paun, M.\ --- {\sl Semipositive (1,1)--cohomology 
classes on projective manifolds}, Preprint de l'I.R.M.A.

\item{\bf [14]} Siu, Y.-T.\ ---{\sl Analyticity of sets associated to
Lelong numbers and the extension of closed positive currents},  
Invent.\ Math.\ {\bf 27} (1974).

\item{\bf [15]} Siu, Y.-T.\ ---{\sl 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. 

\item{\bf [16]} Yau, S.-T.\ ---{\sl On the Ricci curvature of a 
complex K\"ahler manifold and the complex Monge--Amp\`ere equation},
Comm.\ Pure Appl.\ Math.\ {\bf 31} (1978). 


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