%  Article J.-P. Demailly
%``Regularization of closed positive currents and intersection theory"

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\centerline{\bftwelve Regularization of closed positive currents}
\centerline{\bftwelve and Intersection Theory}

\vskip10pt
{\noindent\hangindent4cm\hangafter-1
{\bf Jean-Pierre Demailly

}}

{\noindent\hangindent4cm\hangafter-4{\sl
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
URA 188 du C.N.R.S.,\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\vskip15pt\noindent {\bf Abstract}\pointir{\sl Let $X$ be a compact
complex manifold and let $T$ be a closed positive current of bidegree
$(1,1)$ on~$X$. It is shown that $T$ is the weak limit of a sequence
$(T_k)$ of smooth closed real $(1,1)$-currents with small negative
part. The negative part of the $T_k\,$'s can be bounded in terms of
the Lelong numbers of $T$, once a lower bound for the curvature of the
tangent bundle $TX$ is known. Moreover, Kiselman's procedure for killing
Lelong numbers of a plurisubharmonic function is extended to manifolds
by an alternative method based on H\"ormander's $L^2$ estimates
for~$\op$. These results are then applied to derive various results
concerning divisors or intersection theory in the context of analytic
geometry. Especially, we obtain a relation between effective and
numerically effective divisors on arbitrary compact manifolds, and
we show that every manifold $X$ in the Fujiki class ${\cal C}$ with
nef tangent bundle is K\"ahler. If $D$ is an effective divisor in a
K\"ahler manifold, we also obtain a general self-intersection
inequality giving a bound of the degrees of the constant multiplicity
strata of $D$, in terms of a polynomial in the cohomology class $\{D\}\in
H^2(X,\rb)$.} \vskip15pt

\noindent{\bf Contents}\vskip1pt
{\eightpoint\noindent
1. Introduction and main results\dotfill p. 2\break
2. Kiselman's singularity attenuation technique\dotfill p. 6\break
3. Approximation of psh functions by logarithms of holomorphic
   functions\dotfill p. 13\break
4. Construction of hermitian metrics by $L^2$ techniques\dotfill p. 17\break
5. An alternative singularity attenuation technique\dotfill p. 23\break
6. Application to numerically effective divisors\dotfill p. 31\break
7. Application to self-intersection inequalities\dotfill p. 35\break
A. Appendix: basic results on $L^2$ estimates\dotfill p. 38\break
References\dotfill p. 40
}\vskip15pt
\noindent
{\bf Key words:} Closed positive current, plurisubharmonic function,
Lelong number, $L^2$ estimates for $\op$, effective and numerically
effective divisors, Fujiki class ${\cal C}$, intersection theory,
self-intersection inequality.

\vskip10pt
\noindent
{\bf A.M.S. Classification:} 32J25, 32C30, 32C40
\vfill\eject

\titre 1. Introduction and main results|

Let $X$ be a compact complex manifold of dimension~$n$. According to
[Le~57], a current $T$ of bidimension $(p,p)$ $\big($that is, of
bidegree $(n-p,n-p)\big)$ over $X$ is said to be {\sl positive} if
$T\wedge i\alpha_1\wedge\ov\alpha_1\wedge\ldots\alpha_p\wedge\ov\alpha_p$
is a positive measure for all smooth $(1,0)$-forms $\alpha_1\ld\alpha_p$
on~$X$. We say that $T$ is {\sl almost positive} if there exists a
smooth form $v$ of bidegree $(n-p,n-p)$ such that $T+v\ge 0$.
Similarly, a function $\varphi$ on $X$ is said to be {\sl almost psh}
if $\varphi$ is locally equal to the sum of a psh (plurisubharmonic)
function and of a smooth function; then the $(1,1)$-current
$i\partial\op\varphi$ is almost positive; conversely, if a locally
integrable function $\varphi$ is such that $i\partial\op\varphi$ is
almost positive, then $\varphi$ is equal a.e. to an almost psh function.
If $T$ is closed and almost positive, the Lelong numbers (or density
numbers) $\nu(T,x)$ are well defined, since the negative part always
contributes for zero. For every $c>0$, we introduce the Lelong number
upperlevel sets
$$E_c(T)=\big\{x\in X\,;\,\nu(T,x)\ge c\big\}.$$
A well-known theorem of Siu [Siu~74] asserts that $E_c(T)$ is an analytic
subset of~$X$. Our main result is the following approximation-regularization
theorem:

\tha Main Theorem 1.1|Let $T$ be a closed almost positive $(1,1)$-current
and let $\alpha$ be a smooth real $(1,1)$-form in the same
$\partial\op$-cohomology class as $T$, i.e. $T=\alpha+{i\over\pi}\partial\op
\psi$ where $\psi$ is an almost psh function.
Let $\gamma$ be a continuous real $(1,1)$-form such that $T\ge\gamma$.
Suppose that $\oc_{TX}(1)$ is equipped with a smooth hermitian
metric such that the Chern curvature form satisfies
$$c\big(\oc_{TX}(1)\big)+\pi^\star_X u\ge 0$$
with $\pi_X:P(T^\star X)\to X$ and with some nonnegative smooth $(1,1)$-form
$u$ on~$X$. Fix a hermitian metric $\omega$ on~$X$. Then for every
$c>0$, there is a sequence of closed almost positive $(1,1)$-currents
$T_{c,k}=\alpha+{i\over\pi}\partial\op\psi_{c,k}$ such that $\psi_{c,k}$
is smooth on $X\ssm E_c(T)$ and decreases to $\psi$ as $k$ tends to~$+\infty$
$($in particular, the current $T_{c,k}$ is smooth on $X\ssm E_c(T)$ and
converges weakly to $T$ on~$X)$, and such that
\item{\rm(i)}$T_{c,k}\ge\gamma-\min\{\lambda_k,c\}u-\varepsilon_k\omega$~~
where:
\item{\rm(ii)}$\lambda_k(x)$ is a decreasing sequence of continuous functions
on $X$ such that $\lim_{k\to+\infty}\lambda_k(x)=\nu(T,x)$ at every point,
\item{\rm(iii)}$\varepsilon_k$ is positive decreasing and
$\lim_{k\to+\infty}\varepsilon_k=0$,
\item{\rm(iv)}$\nu(T_{c,k},x)=\big(\nu(T,x)-c\big)_+$
at every point $x\in X$.\fintha

Here $\oc_{TX}(1)$ is the tautological line bundle associated to the
tangent bundle $TX$ over the projectivized hyperplane bundle~$P(T^\star X)$.
Observe that the theorem gives in particular approximants $T_{c,k}$
which are smooth everywhere on $X$ if we take $c>\max_{x\in
X}\nu(T,x)$. Such a regularization result was already proved in [De~82]
under somewhat stronger curvature hypotheses (namely, $X$ has a K\"ahler
metric for which the induced metric on $\oc_{TX}(1)$ satisfies the above
curvature assumption). Properties (i) and (ii) show that the presence of
positive Lelong numbers is an obstruction to smoothing when no loss of
positivity is admitted; the old regularization theorem of Richberg
[Ri~68] precisely solved the special case when the potential $\psi$ is
finite and continuous (the Lelong numbers are then always zero). For a
general choice of the constant~$c$, the inequalities (iv) tell us that
the procedure kills all Lelong numbers that are $\le c$ and shifts all
others by~$-c$. Thus we obtain an analogue over arbitrary
manifolds of Kiselman's procedure [Ki~78] for killing Lelong numbers
(here we assumed $X$ compact only for the sake of simplicity; our
method can be easily extended to noncompact manifolds, but uniform
estimates only hold on relatively compact subsets). Unfortunately,
Kiselman's method, which depends on the Legendre transformation for psh
functions, only works with the stronger curvature assumption
$c(TX)+u\otimes\id_{TX}\ge 0$ in the sense of Griffiths. Since
this method is simpler and more natural, we first explain it in
Section~2. With the $\oc_{TX}(1)$ lower bound assumption (which is
much better from the point of view of algebraic geometry), we
have to use more complicated arguments based on a combination of
different types of $L^2$ estimates for~$\op$, due chronologically to
H\"ormander~[H\"o~66], Skoda~[Sk~72b] and Ohsawa-Takegoshi
[OT~87,~Oh~88]. The proof occupies Sections 3,4,5.
H\"ormander's estimates are used in Section~4 to
construct hermitian metrics on high symmetric powers $S^mTX$ and
$S^mT^\star X$, when starting from a metric on~$\oc_{TX}(1)$. The
Ohsawa-Takegoshi estimates are needed to approximate psh functions
locally by logarithms of holomorphic functions, while Skoda's $L^2$
estimates are used in the gluing process, to compare the size of high
order jets of functions produced in different coordinate patches.

The Main Theorem can be applied in many ways to get interesting results
of intersection theory. Since we consider non necessarily algebraic
manifolds, it is better to work with $\partial\op$-cohomology classes
instead of De Rham classes: we define
$$H_{\partial\op}^{p,q}(X)=\big\{d\hbox{-closed $(p,q)$-forms}\}/
\big\{\partial\op\hbox{-exact $(p,q)$-forms}\}.\leqno(1.2)$$
By means of the Fr\"olicher spectral sequence, it is easily shown
that these cohomology groups are finite dimensional and can be computed
either with spaces of smooth forms or with currents. In both cases,
the quotient topology of $\smash{H_{\partial\op}^{p,q}(X)}$ induced by
the Fr\'echet topology of smooth forms or by the weak topology of
currents is Hausdorff. Clearly $H_{\partial\op}^\bu(X)$ is a bigraded
algebra. This algebra is isomorphic to the usual De Rham cohomology
algebra $H^\bu(X,\cb)$ if $X$ is K\"ahler or more generally if $X$ is
in the Fujiki class ${\cal C}$ of manifolds bimeromorphic to K\"ahler
manifolds. Our first application concerns the relation between
pseudo-effective and nef cohomology classes (nef${}={}$numerically effective).

\tha Definition 1.3|A cohomology class $\{\alpha\}\in
\smash{H_{\partial\op}^{1,1}(X)}$ is said to be pseudo-effective if it
can be represented by a closed positive $(1,1)$-current, and
nef if for some fixed hermitian metric $\omega$ on~$X$ and for every
$\varepsilon>0$ there is a smooth form $\alpha_\varepsilon\in\{\alpha\}$
such that $\alpha_\varepsilon\ge -\varepsilon\omega$. We denote
respectively by $\smash{H^{1,1}_{\rm psef}(X)}$ and $\smash{H^{1,1}_{\rm nef}
(X)}$ the cones of pseudo-effective and nef cohomology classes.\fintha

It is not difficult to see (cf. section~6) that $H^{1,1}_{\rm psef}(X)$ and
$H^{1,1}_{\rm nef}(X)$ are closed convex cones in $H_{\partial\op}^{1,1}(X)$
and that $H^{1,1}_{\rm nef}(X)\subset H^{1,1}_{\rm psef}(X)$. When $X$ is
K\"ahler, $H^{1,1}_{\rm nef}(X)$ coincides with the closure of the
K\"ahler cone, that is, the convex cone generated by K\"ahler classes.
Recall that the real Neron-Severi space $NS_\rb(X)$ is the real
subspace of $H^{1,1}_{\partial\op}(X)$ generated by integral classes,
i.e. by Chern classes of holomorphic line bundles. The algebraic
situation is then described by the following properties (these properties
are often taken as a definition by algebraists; compare with [De~90a]).

\tha Proposition 1.4|\kern-4pt When $X$ is projective algebraic,
$H^{1,1}_{\rm psef}(X)\cap NS_\rb(X)$ is the closed convex cone generated
by cohomology classes of effective divisors, and $H^{1,1}_{\rm nef}(X)\cap
NS_\rb(X)$ is the closed convex cone generated by ample divisors. This
last cone is characterized by the following property: a class
$\{\alpha\}\in NS_\rb(X)$ is nef if and only if
$\{\alpha\}\cdot C=\int_C\alpha\ge 0$ for every curve $C\subset X$.\fintha

It is of course always true that $\{\alpha\}\in H^{1,1}_{\rm nef}(X)$ implies
$\int_C\alpha\ge 0$ for every curve, but even for $\{\alpha\}\in
NS_\rb(X)$ the converse is not always true when $X$ is not projective,
because $X$ might not have enough curves; similarly, a K\"ahler manifold
$X$ may have closed positive $(1,1)$-currents but no divisors; this is
why we felt preferable to choose 1.3 as the general definition of
pseudo-effectivity and nefness. It follows from Prop.~1.4 that the
cone $H^{1,1}_{\rm nef}(X)$ is in general strictly smaller
than  $H^{1,1}_{\rm psef}(X)$, even on projective manifolds: for example,
when $X$ is the blow-up of a manifold $X'$ at a point and
$E\simeq\pb^{n-1}$ is the exceptional divisor, then
$\{E\}=c_1(\oc(E))$ and $\oc(E)_{\restriction E}\simeq\oc(-1)$, so
$\{E\}\cdot C=-{\rm deg}\,C<0$ for every curve $C\subset E\,$;
therefore $\{E\}\in H^{1,1}_{\rm psef}(X)\ssm H^{1,1}_{\rm nef}(X)$.
The Main Theorem gives the following interesting relation between
pseudo-effective and nef classes.

\tha Corollary 1.5|Suppose that there is a nef cohomology class
$\{u\}$ in $H_{\partial\op}^{1,1}(X)$ such that $c_1\big(\oc_{TX}(1)\big)+
\pi^\star_X\{u\}$ is nef over $P(T^\star X)$.
Then for any closed positive $(1,1)$-current $T$ on $X$
and $b_n=\inf\{c>0\,;\,\codim E_c(T)\ge n\}$, the cohomology class
$\{T\}+b_n\{u\}$ is nef. In particular, if $TX$ is nef, we have
$H^{1,1}_{\rm nef}(X)=H^{1,1}_{\rm psef}(X)$.\fintha

By definition, we say that a vector bundle $E$ is nef if
$c_1\big(\oc_E(1)\big)$ is nef over the projectivized bundle $P(E^\star)$
of hyperplanes of~$E$. We refer to [DPS~91] for the basic properties of nef
vector bundles over general compact complex manifolds. The following related
result is an easy consequence of the Main Theorem.

\tha Corollary 1.6|Let $X$ be a compact complex manifold with $TX$ nef.
Then $X$ is K\"ahler if and only if $X$ is in the Fujiki class ${\cal C}$,
and $X$ is projective if and only if $X$ is Moishezon.\fintha

 Finally, it is possible to extend the self-intersection inequality
proved in [De~90b] to an arbitrary closed positive $(1,1)$-current $T$
on a K\"ahler manifold~$X$ (in the original version, we had to suppose
$X$ projective algebraic and $\{T\}$ in $H^2(X,\zb)$). To the current
$T$ we associate the sequence $0=b_1\le\ldots\le b_n\le b_{n+1}$ of
``{\sl jumping values}" $b_p$ such that the dimension of $E_c(T)$ drops
by one unit when $c$ gets larger than~$b_p$, namely $\codim E_c(T)=p$
when $c\in{}]b_p,b_{p+1}]$ (by this, we mean that all components have
codimension $\ge p$, with equality for at least one of them). Let
$(Z_{p,k})_{k\ge 1}$ be the collection of $p$-codimensional components
of all sets $E_c(T)$, $c\in{}]b_p,b_{p+1}]$, and let
$\nu_{p,k}=\min_{x\in Z_{p,k}}\nu(T,x)\in{}]b_p,b_{p+1}]$ be the
generic Lelong number of $T$ along~$Z_{p,k}$. We are interested in the
following question:\break is it possible to derive a bound for the degrees of
the Lelong number strata $Z_{p,k}$ in terms of the cohomology class
$\{T\}\in H^2_{DR}(X,\rb)\,$? The answer is affirmative:

\tha Theorem 1.7|Suppose that $X$ is K\"ahler and that $\oc_{TX}(1)$ has
a hermitian metric such that $c\big(\oc_{TX}(1)\big)+\pi^\star_X u\ge 0$,
where $u$ is a smooth closed nonnegative $(1,1)$-form. For each $p=1\ld n$,
the De Rham cohomology class
$(\{T\}+b_1\{u\})\cdots(\{T\}+b_p\{u\})$ can be represented by a closed 
positive current $\Theta_p$ of bidegree $(p,p)$ such that
$$\Theta_p\ge\sum_{k\ge 1}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)\,[Z_{p,k}]
+(T_{\rm abc}+b_1u)\wedge\ldots\wedge(T_{\rm abc}+b_pu)$$
where $T_{\rm abc}\ge 0$ is the absolutely continuous part in the
Lebesgue decomposition of the coefficients of $T$ into absolutely continuous
and singular measures. As a consequence, if $\omega$ is a
K\"ahler metric on $X$ and if $\{u\}$ is a nef cohomology class such
that $c_1\big(\oc_{TX}(1)\big)+\pi^\star_X\{u\}$ is nef, the degrees of
the components $Z_{p,k}$ with respect to $\omega$ satisfy the estimate
$$\eqalign{
\sum_{k=1}^{+\infty}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)
&\int_X[Z_{p,k}]\wedge\omega^{n-p}\cr
&\le\big(\{T\}+b_1\{u\}\big)\cdots\big(\{T\}+b_p\{u\}\big)
\cdot\{\omega\}^{n-p}.\cr}$$\fintha

To give the flavour of inequality~1.7 in a simple case, suppose that $X$
is a K\"ahler surface and that $T=[C]$ is the current of integration on
an irreducible curve $C\subset X$. Then $b_1=0$, $b_2=1$ and $b_3$ is the
maximum of the multiplicities $\nu_k\ge 2$ of the singular points of~$C$.
For $p=2$, the inequality gives $\sum\nu_k(\nu_k-1)\le\{C\}\cdot(\{C\}
+\{u\})$ (in that case, the inequality is well-known and is related to
the genus formula for curves).
The general proof follows essentially the same lines as the algebraic
case considered in [De~90b] (this case was an essential ingredient in
our proof of numerical criteria for very ample line bundles). The main
idea is to kill the Lelong numbers of $T$ up to the level~$b_j\,$; then
the singularities of the resulting current $T_j$ occur only in
codimension~$j$ and the wedge product $T_1\wedge\ldots\wedge T_p$ is
well defined by means of Bedford-Taylor's definition of the
Monge-Amp\`ere operator and suitable integration by parts
(see~[De~91]).


The above results, especially corollaries 1.5 and 1.6, have been strongly
moti\-vated by our common work with Thomas Peternell and Michael Schneider
on the classification of compact K\"ahler varieties with nef tangent
bundle ([DPS~91], to appear). The author wishes to thank both of them for
very fruitful discussions.

\titre 2. Kiselman's singularity attenuation technique|

Let us first recall some basic facts concerning Chern connection and
curvature. If $E$ is a hermitian holomorphic vector bundle on
a complex $n$-dimensional manifold~$M$, we
denote by $D_E$ (or simply $D$ when no confusion can occur) the Chern
connection on~$E$. We denote by $c(E)={i\over 2\pi}D_E^2$ the
associated Chern curvature tensor, which is a hermitian $(1,1)$-form
with values in $\hom(E,E)$. With respect to a local orthonormal frame
$(e_\lambda)_{1\le\lambda\le r}$ of $E$ and to local coordinates
$(z_j)_{1\le j\le n}$ on $M$, such a form can be written 
$$c(E)=i\sum c_{jk\lambda\mu}\,dz_j\wedge d\ov z_k\otimes e_\lambda^\star\otimes
e_\mu$$
with $c_{kj\mu\lambda}=\ov c_{jk\lambda\mu}$ for $1\le j,k\le n$ and
$1\le\lambda,\mu\le r$. As usual, we identify $c(E)$ with the hermitian
form on $TM\otimes E$ defined by
$$\langle c(E)\bu,\bu\rangle=
\sum c_{jk\lambda\mu}\,(dz_j\otimes e_\lambda^\star)\otimes(\ov{
dz_k\otimes e_\mu^\star}),\leqno(2.1)$$
and we say that $c(E)$ is {\sl semipositive (resp. positive) in the sense of
Griffiths} if $c(E)(t\otimes v\,,\,t\otimes v)\ge 0$ (resp. $>0$) for
all non zero vectors $t\in TM$, $v\in E$.

Now, let $X$ be a compact $n$-dimensional complex manifold and let
$T$ be a closed almost positive current of bidegree $(1,1)$ on~$X$. 
We select a smooth closed $(1,1)$-form $\alpha$ representing the same
$\partial\op$-cohomology class as $T$ and an almost psh function $\psi$
on $X$ such that $T=\alpha+{i\over\pi}\partial\op\psi$. Such a
decomposition exists even when $X$ is non-K\"ahler, since we can always
find an open covering $(U_j)$ of $X$ such that $T=\smash{{i\over\pi}}
\partial\op\psi_j$ over $U_j$, and construct a global function
$\psi=\sum\theta_j \psi_j$ by means of a partition of unity $(\theta_j)$
subordinate to~$U_j$ (observe that $\psi-\psi_k$ is smooth on $U_k$ 
because all differences $\psi_j-\psi_k$ are smooth in the intersections
$U_j\cap U_k$). By replacing $T$ with $T-\alpha$ and $\gamma$ with
$\gamma-\alpha$ in the Main Theorem, we can suppose without loss of
generality that $\{T\}=0$, i.e. that $T={i\over\pi}\partial\op\psi$
with an almost psh function~$\psi$ on~$X$. Thus we have to devise
a singularity attenuation technique for almost psh functions.
We first describe Kiselman's method [Ki~78]. An alternative method
working under slightly more general hypotheses will be explained in
section~5.

{\petcap Local procedure}\pointir
To begin with, let $\varphi$ be a psh function on an open set
$U\subset\cb^n$. Suppose that the trivial vector bundle
$E=U\times\cb^n$ is equipped with a smooth hermitian metric
$||\xi||_z$ for $\xi\in\{z\}\times\cb^n$ (later on, the hermitian
structure of $U\times\cb^n$ will come from a trivialization
of $TX_{\restriction U}$). For $w\in\cb$, we set
$$\Phi_\infty(z,w)=\sup_{||\xi||_z\le 1}\varphi(z+e^w\xi).\leqno(2.2)$$
Clearly $\Phi_\infty(z,w)$ depends only on the real part $\re w$ and
is defined on the open set $\Omega$ of points $(z,w)\in U\times\cb$
such that $\re w<\log d_z(z,\partial U)$, where $d_z$ denotes
euclidean distance with respect to $||~~||_z$. Moreover $\Phi_\infty(z,w)$
is a convex increasing function of~$\re w$ by the well known
properties of plurisubharmonic functions. Next, for $(z,w)\in\Omega$,
we introduce the {\sl Legendre transform}
$$\Phi_c(z,w)=\inf_{t\le 0}~\Phi_\infty(z,w+t)-ct,~~~~c>0.\leqno(2.3)$$
It is easy to see that the family $(\Phi_c)$ is increasing in $c$
and that
$$\lim_{c\to 0}\Phi_c(z,w)=\varphi(z),~~~~\lim_{c\to +\infty}\Phi_c=\Phi_\infty.
\leqno(2.4)$$
When $||\xi||_z$ is taken to be a constant metric, we know by
Kiselman [Ki~78] that $\Phi_\infty$ and $\Phi_c$ are plurisubharmonic functions
of the pair~$(z,w)$, and that the Lelong numbers of $\Phi_c(\bu,w)$ are
given by
$$\nu\big(\Phi_c(\bu,w),z\big)=\big(\nu(\varphi,z)-c\big)_+,~~~~
\forall(z,w)\in\Omega.\leqno(2.5)$$
In general, $\Phi_\infty$ is continuous on $\Omega$ and
its right derivative $\partial\Phi_\infty(z,w)/\partial\re w_+$ is upper
semicontinuous; indeed, this partial derivative is the decreasing limit of
$\big(\Phi_\infty(z,w+t)-\Phi_\infty(z,w)\big)/t$ as~$t\downarrow 0_+$.
It follows that $\Phi_c$ is continuous on $\Omega\ssm(E_c(\varphi)\times\cb)$:
in fact, we have $\nu(\varphi,z)=\lim_{t\to-\infty}\partial\Phi_\infty(z,t)/
\partial t_+<c$ on every compact set $K\subset\Omega\ssm(E_c(\varphi)
\times\cb)$, so by the upper semicontinuity there is a constant $t_0$ such
that $\partial\Phi_\infty(z,w+t)/\partial t_+<c$ for $(z,w)\in K$ and~$t<t_0$.
Therefore
$$\Phi_c(z,w)=\inf_{t_0\le t\le 0}\Phi_\infty(z,w+t)-ct~~~~\hbox{\rm on}~K,$$
and this infimum with compact range is continuous.
Our first goal is to investigate the plurisubharmonicity of $\Phi_c$ when
$||\xi||_z$ is a variable hermitian metric.

\tha Proposition 2.6|Suppose that the Chern curvature of the hermitian metric
on $E=U\times\cb^n$ satisfies $c(E)+u\otimes\id_E\ge 0$ in the sense
of Griffiths, for some nonnegative $(1,1)$-form $u$ on~$X$. Then 
\item{\rm(i)}For all $v\ge 0$, we have
$$\Phi_c(z,w-v)\ge\Phi_c(z,w)-\min\big\{{\partial\Phi_\infty(z,w)\over
\partial\re w_-}\,,\,c\Big\}v\,;$$
\item{\rm(ii)}For $(\zeta,\eta)\in TU\times\cb$ and $c\in{}]0,+\infty]$,
the Hessian of $\Phi_c$ satisfies
$${i\over\pi}\partial\op(\Phi_c)_{(z,w)}(\zeta,\eta)
\ge -\min\Big\{{\partial\Phi_\infty(z,w)\over\partial\re w_+}\,,\,
c\Big\}\,u_z(\zeta).$$\fintha

\remarque Proof|(i) For $v\ge 0$ and $t\le 0$, we have
$$\Phi_\infty(z,w+t-v)\ge\Phi_\infty(z,w+t)-v{\partial\Phi_\infty(z,w+t)
\over\partial\re w_-}$$
by convexity of $\Phi_\infty(z,w)$ in $\re w$. As $\partial\Phi_\infty(z,w)/
\partial\re w_-$ is increasing in $\re w$, the infimum of both sides minus
$ct$ gives
$$\Phi_c(z,w-v)\ge\Phi_c(z,w)-v{\partial\Phi_\infty(z,w)\over\partial\re w_-}.$$
On the other hand, the change of variables $t=t'+v$ yields
$$\Phi_c(z,w-v)\ge\inf_{t'\le-v}\Phi_\infty(z,w+t')-c(t'+v)\ge
\Phi_c(z,w)-cv$$
and (i) follows. The proof of (ii) requires the following simple lemma.

\tha Lemma 2.7|Let $z_0\in U$ be fixed and let $v$ be the hermitian
quadratic function with constant Hessian ${i\over\pi}\partial\op
v=u_{z_0}$. Then there is a holomorphic $n\times n$ matrix $g(z)$ near
$z_0$ such that $g(z_0)=\id$ and
$$||g(z)\cdot\xi||_z\le||\xi||_{z_0}e^{v(z-z_0)+O(|z-z_0|^3)}.$$\fintha

\remarque Proof|Without loss of generality, we may suppose that $z_0=0$ and
that $||\xi||_{z_0}$ is given by the unit hermitian matrix (otherwise take
any orthonormal basis $(e_\lambda)$ as the new basis of $\cb^n$). We have a
Taylor expansion
$$||\xi||_z^2=\sum_{1\le\lambda\le n}|\xi_\lambda|^2
+\sum_{1\le j,\lambda,\mu\le n}(a_{j\lambda\mu}z_j+\ov a_{j\mu\lambda}\ov z_j)
\xi_\lambda\ov\xi_\mu+O(|z|^2|\xi|^2)$$
at first order. We can get rid of the first order terms by setting
$$g_1(z)\cdot\xi=\xi-
\sum_{j,\lambda,\mu}a_{j\lambda\mu}z_j\xi_\lambda\,e_\mu.$$
A simple calculation shows indeed that $||g_1(z)\cdot\xi||_z^2=\sum
|\xi_\lambda|^2+O(|z|^2|\xi|^2)$. We then look at the second order
terms in the Taylor expansion
$$\eqalign{
||g_1(z)\cdot\xi||_z^2=\sum_\lambda|\xi_\lambda|^2&{}+
\sum_{j,k,\lambda,\mu}(a_{jk\lambda\mu}z_jz_k+\ov a_{jk\mu\lambda}
\ov z_j\ov z_k)\xi_\lambda\ov\xi_\mu\cr
&{}+\sum_{j,k,\lambda,\mu}
b_{jk\lambda\mu}z_j\ov z_k\xi_\lambda\ov\xi_\mu+O(|z|^3|\xi|^2)\cr}$$
and kill the coefficients $a_{jk\lambda\mu}$ by setting
$$g(z)\cdot\xi=g_1(z)\cdot\xi-\sum_{j,k,\lambda,\mu}
a_{jk\lambda\mu}z_jz_k\xi_\lambda e_\mu.$$
The remaining coefficients $(b_{jk\lambda\mu})$ cannot be killed and
are related to the curvature tensor $c(E)$ by $c_{jk\lambda\mu}=
-{1\over\pi}b_{jk\lambda\mu}$. If we write $v_{jk}=\partial^2v/
\partial z_j\partial\ov z_k(0)$, we have $u(0)={i\over\pi}\sum v_{jk}
dz_j\wedge d\ov z_k$ and the curvature assumption
$c(E)+u\otimes\id_E\ge 0$ at the origin implies
$$\sum_{j,k,\lambda,\mu}-b_{jk\lambda\mu}z_j\ov z_k\xi_\lambda\ov\xi_\mu
+\sum_{j,k}v_{jk}z_j\ov z_k\sum_\lambda|\xi_\lambda|^2\ge 0.$$
Therefore we get
$$\eqalign{
||g(z)\cdot\xi||^2_z&=\sum_\lambda|\xi_\lambda|^2+
\sum_{j,k,\lambda,\mu}b_{jk\lambda\mu}z_j\ov z_k\xi_\lambda\ov\xi_\mu
+O(|z|^3|\xi|^2)\cr
&\le ||\xi||_0^2\big(1+v(z)+O(|z|^3)\big).\cr}$$
The asserted inequality 2.7 is equivalent to this.\finpr

\remarque Proof of {\rm 2.6 (ii)}|Fix $(z_0,w_0)\in\Omega$ and an arbitrary
quadratic function $v$ such that ${i\over\pi}\partial\op v>u_{z_0}$.
Then Lemma~2.7 yields an invertible matrix $g(z)$ such that
$||g(z)\cdot\xi||_z\le||\xi||_{z_0}e^{v(z-z_0)}$ near~$z_0$. A change
of variable $\xi\mapsto g(z)\cdot\xi$ in (2.2) implies
$$\eqalign{
\Phi_\infty(z,w)&=\sup_{||g(z)\cdot\xi||_z\le 1}\varphi(z+e^wg(z)\cdot\xi)\cr
&\ge\sup_{||\xi||_{z_0}\le e^{-v(z-z_0)}}\varphi(z+e^wg(z)\cdot\xi)
~~~~\hbox{\rm near}~(z_0,w_0),\cr}$$
with the inequality being an equality for $z=z_0$. The function
$$h(z,w)=\sup_{||\xi||_{z_0}\le 1}\varphi(z+e^wg(z)\cdot\xi)$$
is plurisubharmonic and, if $h_c$ is its Legendre transform, we obtain
successively
$$\eqalign{
\Phi_\infty(z,w)&{}\ge h\big(z,w-v(z-z_0)\big),\cr
\Phi_c(z,w)&{}\ge h_c\big(z,w-v(z-z_0)\big)\cr}$$
near $(z_0,w_0)$, with equality for $z=z_0$. For small $(\zeta,\eta)\in
TU\times\cb$, we combine this with the mean value inequality to get
$$\eqalign{\int_0^{2\pi}\Phi_c(z_0+e^{i\theta}\zeta,w_0+e^{i\theta}\eta)\,
{d\theta\over 2\pi}
&\ge\int_0^{2\pi}h_c\big(z_0+e^{i\theta}\zeta,
w_0+e^{i\theta}\eta-v(\zeta)\big)\,{d\theta\over 2\pi}\cr
&\ge h_c\big(z_0,w_0-v(\zeta)\big)=\Phi_c\big(z_0,w_0-v(\zeta)\big).\cr}$$
This inequality is still valid at points $(z,w)$ near $(z_0,w_0)$, provided
that $u_z<{i\over\pi}\partial\op v$ and that $|\zeta|^2+|\eta|^2<r^2_{z,w}$
is small enough. By 2.6~(i), we get the inequality
$$\int_0^{2\pi}\Phi_c(z+e^{i\theta}\zeta,w+e^{i\theta}\eta)\,
{d\theta\over 2\pi}\ge\Phi_c(z,w)-
\min\Big\{{\partial\Phi_\infty(z,w)\over\partial\re w_+}\,,\,c\Big\}v(\zeta)$$
near $(z_0,w_0)$. For $A>\partial\Phi_\infty(z_0,w_0)/\partial\re w_+$,
we still have $A>\partial\Phi_\infty(z,w)/\partial\re w_+\kern-0.14pt$
in a neighborhood by the upper semicontinuity, and we conclude that the
function $\Phi_c(z,w)+\min\{A,c\}v(z)$ satisfies the mean value
inequality near~$(z_0,w_0)$. Hence $\Phi_c(z,w)+\min\{A,c\}v(z)$
is psh near $(z_0,w_0)$ and (ii) is proved.\finpr

{\petcap Gluing process}\pointir
The next step is to describe a gluing process for the construction of
global regularizations of almost psh functions. We suppose that $X$ is
equipped with a hermitian metric $\omega$ such that its Chern curvature
satisfies $c(TX)+u\otimes\id_{TX}\ge 0$ in the sense of Griffiths,
where $u$ is a smooth semipositive $(1,1)$-form on~$X$. Let
$T={i\over\pi}\partial\op\psi$ where $\psi$ is an almost psh function
on~$X$, and let $\gamma$ be a real continuous $(1,1)$-form such that
$T\ge\gamma$. We select a finite covering $(W_\nu)$ of $X$ with open
coordinate charts. Given $\delta>0$, we take in each $W_\nu$ a
maximal family of points with (coordinate) distance to the boundary
$\ge 3\delta$ and mutual distance $\ge\delta$. In this way, we get for 
$\delta>0$ small a finite covering of $X$ by open balls $U'_j$ of
radius~$\delta$, such that the concentric ball $U_j$ of radius
$2\delta$ is relatively compact in the corresponding chart~$W_\nu$. Let
$\tau_j:U_j\la B(a_j,2\delta)$ be the isomorphism given by the
coordinates of~$W_\nu$. Let $\varepsilon(\delta)$ be a modulus
of continuity for $\gamma$ on the sets $U_j$, such that
$\lim_{\delta\to 0}\varepsilon(\delta)=0$ and 
$\gamma_x-\gamma_{x'}\le{1\over 2}\varepsilon(\delta)\,\omega_x$ for all
$x,x'\in U_j$. We denote by $\gamma_j$ the $(1,1)$-form with constant
coefficients on $B(a_j,2\delta)$ such that $\tau_j^\star\gamma_j$
coincides with $\gamma-\varepsilon(\delta)\,\omega$ at
$\tau_j^{-1}(a_j)$. Then we have
$$0\le\gamma-\tau_j^\star\gamma_j\le 2\varepsilon(\delta)\,\omega~~~~
\hbox{\rm on}~~U'_j\leqno(2.8)$$
for $\delta>0$ small. We set $\psi_j=\psi\circ\tau_j^{-1}$ on
$B(a_j,2\delta)$ and let $\wt\gamma_j$ be the homogeneous quadratic
function in $z-a_j$ such that ${i\over\pi}\partial\op\wt\gamma_j=
\gamma_j$ on $B(a_j,2\delta)$. Finally, we set
$$\varphi_j(z)=\psi_j(z)-\wt\gamma_j(z)~~~~\hbox{\rm on}~B(a_j,2\delta).
\leqno(2.9\,{\rm a})$$
Then $\varphi_j$ is plurisubharmonic, since
$${i\over\pi}\partial\op(\varphi_j\circ\tau_j)=T-\tau_j^\star\gamma_j
\ge\gamma-\tau_j^\star\gamma_j\ge 0.$$
Let $||\xi||_{j,z}$ be the metric induced by $\omega$ at a point $z\in
B(a_j,2\delta)$ in the trivialization $d\tau_j:TX_{\restriction U_j}\to
B(a_j,2\delta)\times\cb^n$. We combine (2.2) and (2.3) to define functions
$$\leqalignno{
\Phi_{j,c}(z,w)&=\inf_{t\le 0}\Big(\sup_{||\xi||_{j,z}\le 1}
\varphi_j(z+e^{w+t}\xi)-ct\Big),~~~~z\in B(a_j,\sqrt{2}\,\delta),
&(2.9\,{\rm b})\cr
\Psi_{j,c}(z,w)&=\Phi_{j,c}(z,w)+\wt\gamma_j(z)-\delta|z-a_j|^2,
\phantom{\Big)}~~~~z\in B(a_j,\sqrt{2}\,\delta),
&(2.9\,{\rm c})\cr
\Psi_c(x,w)&=\sup_{U''_j\ni x}\Psi_{j,c}(\tau_j(x),w),~~~~x\in X,
\phantom{\Big)}&(2.9\,{\rm d})\cr}$$
where $U''_j=\tau_j^{-1}(B(a_j,\sqrt{2}\,\delta))$. Note that
$U'_j\compact U''_j\compact U_j$ by construction and that there exists
$w_0=\log\delta-C_1<0$ such that $z+e^{w+t}\xi\in B(a_j,2\delta)$
when $z\in B(a_j,\sqrt{2}\,\delta)$ and $\re w<w_0$. Then $\Psi_c(x,w)$
is well defined on $X\times\{\re w<w_0\}$. We have to check that the
sup used in the definition of $\Psi_c$ does not create discontinuities when
$x$ passes through a boundary~$\partial U''_j$. For this, we must compare
$\Psi_{k,c}(z,w)$ and $\Psi_{j,c}(\tau_{jk}(z),w)$ for $z\in\tau_k(U''_j
\cap U''_k)$, where $\tau_{jk}=\tau_j\circ\tau_k^{-1}$. By (2.9$\,$a,b)
there is a uniform estimate
$$\Phi_{j,c}(z,w)=\inf_{t\le 0}\Big(\sup_{||\xi||_{j,z}\le 1}
\psi_j(z+e^{w+t}\xi)-ct\Big)-\wt\gamma_j(z)+O(|e^w|),$$
therefore (2.9$\,$c) gives
$$\Psi_{j,c}(z,w)=\inf_{t\le 0}\Big(\sup_{||\xi||_{j,z}\le 1}
\psi_j(z+e^{w+t}\xi)-ct\Big)-\delta|z-a_j|^2+O(|e^w|).\leqno(2.10)$$
The error terms $O(|e^w|)$ come from the variation of the quadratic function
$\wt\gamma_j(z)$ when $z$ is replaced by $z+e^{w+t}\xi$. This variation is
bounded by $C_2|e^w|$ with a constant $C_2$ independent of~$\delta$.

When we make the substitution $z\mapsto\tau_{jk}(z)$, the trivialization
of the tangent bundle is submitted to a corresponding transformation
$\xi\mapsto d\tau_{jk}(z)\cdot\xi$, in particular
$||\xi||_{k,z}=||d\tau_{jk}(z)\cdot\xi||_{j,\tau_{jk}(z)}$.
As $\psi_k=\psi_j\circ\tau_{jk}$, we get
$$\sup_{||\xi||_{k,z}\le 1}\psi_k(z+e^{w+t}\xi)=
\sup_{||\xi||_{k,z}\le 1}\psi_j\circ\tau_{jk}(z+e^{w+t}\xi),\leqno(2.11_k)$$
to be compared with
$$\sup_{||\xi||_{j,\tau_{jk}(z)}\le 1}\psi_j(\tau_{jk}(z)+e^{w+t}\xi)=
\sup_{||\xi||_{k,z}\le 1}\psi_j\big(\tau_{jk}(z)+e^{w+t}
d\tau_{jk}(z)\cdot\xi\big).\leqno(2.11_j)$$
However $\tau_{jk}(z+\xi)-[\tau_{jk}(z)+d\tau_{jk}(z)\cdot\xi]=O(|\xi|^2)$,
thus if $B_j(z,r)$ denotes the euclidean ball of center $z$ and radius $r$
with respect to $||~~||_{j,z}$, there is a constant $C_3>0$ such that
$$B_j(\tau_{jk}(z),r-C_3r^2)\subset\tau_{jk}(B_k(z,r))\subset
B_j(\tau_{jk}(z),r+C_3r^2)$$
for all $j,k$ and $r<r_0$. By taking $r=|e^{w+t}|$, these inclusions show that
$$\eqalign{
|(2.11_j)-(2.11_k)|
&\le\Psi_{j,\infty}\big(\tau_{jk}(z),\log(r+C_3r^2)\big)-
    \Psi_{j,\infty}\big(\tau_{jk}(z),\log(r-C_3r^2)\big)\cr
&\le C_4\big(\log(r+C_3r^2)-\log(r-C_3r^2)\big)\le C_5|e^{w+t}|,\cr}$$
because $\partial\Psi_{j,\infty}(z,w)/\partial w_+$ is uniformly bounded on
$U_j\times{}]-\infty,w_0[$ (the bound is independent of $\delta$).
Therefore (2.10) implies
$$\Psi_{j,c}(\tau_{jk}(z),w)-\Psi_{k,c}(z,w)=
\delta(|z-a_k|^2-|\tau_{jk}(z)-a_j|^2)+O(|e^w|).$$
Now, if $x\in X$, there exists by construction an index $k$ such that
$x\in U'_k$, i.e. $z=\tau_k(x)\in B(a_k,\delta)$. If $x\in U''_j$ tends to a
point in the boundary $\partial U''_j$, then $|\tau_{jk}(z)-a_j|$ tends to
$\sqrt{2}\,\delta$ and
$$\Psi_{j,c}(\tau_{jk}(z),w)-\Psi_{k,c}(z,w)\la
\delta(|z-a_k|^2-2\delta^2)+O(|e^w|)\le -\delta^3+O(|e^w|).$$
This implies that formula (2.9$\,$d) does not involve values of
$\Psi_{j,c}$ near $\partial U''_j$ when $w\le w_1=3\log\delta-C_6$.
In particular $\Psi_c$ is continuous on $X\ssm E_c(T)$, since each function
$\Psi_{j,c}$ has this property. Also, by Kiselman's result~2.5, we get
$$\nu\big(\Psi_c(\bu,w),x\big)=\nu\big(\Phi_{j,c}(\bu,w),x\big)
=\big(\nu(\varphi,x)-c\big)_+=\big(\nu(\psi,x)-c\big)_+\leqno(2.12)$$
when $x\in U'_j$. Clearly $\Psi_c(x,w)$ is convex and increasing in~$w$,
for each function $\Phi_{j,c}$ has this property. Finally, let
$x\in X$ with $x\in U''_k$, let $z=\tau_k(x)$ and let $v_k$ be a positive
quadratic function on $B(a_k,2\delta)$ with ${i\over\pi}\partial\op v_k>
(\tau_k^{-1})^\star u_x$. As $\tau_{jk}(z+e^{i\theta}\zeta)=\tau_{jk}(z)
+e^{i\theta}d\tau_{jk}(z)\cdot\zeta+O(|\zeta|^2)$, the definition
of $\Phi_{j,c}$, $\Psi_{j,c}$ in (2.9$\,$b,c) and the mean value
argument used in the proof of~2.6 imply
$$\leqalignno{
\int_0^{2\pi}\Phi_{j,c}&\big(\tau_{jk}(z+e^{i\theta}\zeta),
w+e^{i\theta}\eta\big)\,{d\theta\over 2\pi}\ge
\Phi_{j,c}\big(\tau_{jk}(z),w-v_k(\zeta)\big),\cr
\int_0^{2\pi}\Psi_{j,c}&\big(\tau_{jk}(z+e^{i\theta}\zeta),
w+e^{i\theta}\eta\big)\,{d\theta\over 2\pi}
\ge\Phi_{j,c}\big(\tau_{jk}(z),w-v_k(\zeta)\big)\cr
&{}+\wt\gamma_j(\tau_{jk}(z))+i\partial\op\wt\gamma_j(d\tau_{jk}(z)\cdot\zeta)
-O(|\zeta|^3)-\delta|\tau_{jk}(z)-a_j|^2-O(\delta)|\zeta|^2&(2.13)\cr}$$
for every $j$ such that $U''_j\ni x\,$; here we use the fact that
for a suitable quadratic function $v_j$ on $B(a_j,x)$ approximating
closely $(\tau_j^{-1})^\star u_x$ (more accurately than $v_k$ did), we have
$$v_j\big(e^{i\theta}d\tau_{jk}(z)\cdot\zeta+O(|\zeta|^2)\big)
\le v_k(\zeta)$$
for $\zeta$ small. Since ${i\over\pi}\partial\op\wt\gamma_j\circ\tau_j=
\tau_j^\star\gamma_j\ge\gamma-2\varepsilon(\delta)\omega$ by construction,
we find
$${i\over\pi}\partial\op\wt\gamma_j(d\tau_{jk}(z)\cdot\zeta)\ge
(\gamma_x-2\varepsilon(\delta)\omega_x)\big(d\tau_k^{-1}(z)\cdot\zeta\big).$$
When we take the supremum over~$j$ in (2.13) and use definition (2.9$\,$d),
we get
$$\int_0^{2\pi}\Psi_c\big(\tau_k^{-1}(z+e^{i\theta}\zeta),w+e^{i\theta}
\eta\big)\,{d\theta\over 2\pi}
\ge\Psi_c(x,w-v_k(\zeta)\big)+\pi(\gamma_x-\wt\varepsilon(\delta)
\omega_x)\big(d\tau_k^{-1}(z)\cdot\zeta\big)$$
with $\wt\varepsilon(\delta)=2\varepsilon(\delta)+O(\delta)$. By the
arguments already explained in the proof of 2.6~(ii), we infer
$${i\over\pi}\partial\op(\Psi_c)_{(x,w)}(\xi,\eta)
\ge-\min\Big\{{\partial\Psi_\infty(x,w)\over\partial\re w_+}\,
,\,c\Big\}\,u_x(\xi)+\gamma_x(\xi)-\wt\varepsilon(\delta)\omega_x(\xi)
\leqno(2.14)$$
when $w<w_1(\delta)=3\log\delta-C_6$. Finally, we obtain the sequence
$\psi_{c,k}$ needed in the Main Theorem by setting
$$\psi_{c,k}(x)=\Psi_c(x,w_k)+{1\over k}~~~~\hbox{\rm with}~
\delta=\delta_k:=2^{-k}$$
and with a rapidly decreasing sequence $w_k\to-\infty$. The covering
$(U_j)$ changes with the parameter $\delta$ when $\delta$ tends to~$0$,
so it is not a priori clear that $\psi_{c,k}$ is decreasing; this is
the reason why we introduced the additional term $1/k\,$; the change of
covering between $\psi_{c,k}$ and $\psi_{c,k+1}$ can only modify our
functions by terms of order of magnitude $\delta_k^2=2^{-2k}$ (coming
from the change of quadratic functions) and $|e^{w_k}|$ (coming from
the change of chart in the definition of $\Phi_{j,c}$). These terms are
negligible with respect to $1/k-1/(k+1)$ if we take for instance
$w_k\le-3k\le 4\log\delta_k$. Property (2.12) shows that
$\nu(\psi_{c,k},x)=(\nu(\psi,x)-c)_+$ at every point. Moreover, (2.14) implies
$${i\over\pi}\partial\op\psi_{c,k}(x)\ge-\min\Big\{
{\partial\Psi_\infty(x,w_k)\over\partial\re w_+}\,
,\,c\Big\}\,u_x+\gamma_x-\wt\varepsilon(\delta_k)\omega_x.$$
Since $\partial\Psi_\infty(x,w)/\partial\re
w_+$ is upper semi-continuous and converges to
$\nu({i\over\pi}\partial\op\psi,x)$ for each $\delta$~fixed, the following
claim is easy to show: given any strictly decreasing sequence of
continuous functions $\lambda_k\ge 0$ on $X$ such that
$\lim\lambda_k(x)= \nu({i\over\pi}\partial\op\psi,x)$, there is a choice
of $w_k$ such that $\partial\Psi_\infty(x,w_k)/\partial\re w_+\le\lambda_k(x)$
for $\delta=\delta_k$. Therefore the currents $T_{c,k}={i\over\pi}
\partial\op\psi_{c,k}$ satisfy all properties 1.1~(i)-(iv).

Our construction only showed a priori that $T_{c,k}$ is continuous
on~$X\ssm E_c(T)$. In fact $T_{c,k}$ can be made smooth on $X\ssm E_c(T)$
by Richberg's approximation theorem [Ri~68]: by Lemma 2.15 below, there
are smooth approximants $\psi'_{c,k}$ of $\psi_{c,k}$ on $X\ssm E_c(T)$
with $|\psi'_{c,k}-\psi_{c,k}|<2^{-k}$ and with an arbitrary small loss
of positivity on the continuous lower bound of the Hessian, say a
loss~$\le 2^{-k}\omega$. It is then clear that $\psi'_{c,k}$ extends to an
almost psh function on $X$ and that $T'_{c,k}=\alpha+{i\over\pi}\partial\op
\psi'_{c,k}$ satisfies essentially the same estimates as~$T_{c,k}$.
Theorem~1.1 is thus proved under the more restrictive condition that
$c(TX)+u\otimes\id_{TX}\ge 0$ in the sense of Griffiths.

\tha Lemma 2.15 {\rm(Richberg [Ri~68], Greene-Wu [GW~75])}|
Let $\psi$ be a quasi psh function on a complex manifold $M$ such that
$i\partial\op\psi\ge \gamma$ where $\gamma$ is a continuous
$(1,1)$-form. For any hermitian metric $\omega$ and any
continuous function $\delta>0$ on~$M$, there is a smooth function
$\psi'$ such that $\psi<\psi'<\psi+\delta$ and
$i\partial\op\psi\ge\gamma-\delta\,\omega$ on~$M$.\fintha

\remarque Sketch of proof|Take a locally finite covering $(B_j)$ of $M$
by open balls and take a convolution $\psi\star\rho_{\varepsilon_j}$ by a
regularizing kernel on a neighborhood of each $\smash{\ov B}_j$. Let $z^j$
be a local coordinate identifying $B_j$ to the ball $B(0,r_j)$.
The smoothing process consists in gluing the functions
$\psi\star\rho_{\varepsilon_j}(z)+\eta_j(r_j^2-|z^j|^2)$ by taking the
maximum with respect to all overlapping balls at the given point~$z$. 
A~smooth function can be obtained by means of regularized max functions. For a
suitable choice of $\varepsilon_j\ll\inf_{B_k\cap B_j\ne\emptyset}\eta_k$,
the boundary values of the functions do not contribute to the maximum.
The estimates are then clearly satisfied if the $\varepsilon_j$'s and
$\eta_j$'s are small enough.\finpr

\longtitre 3. Approximation of plurisubharmonic functions by|
logarithms of holomorphic functions|

Let $T$ be a closed almost positive current of bidegree $(1,1)$ on a
compact complex manifold~$X$. We write $T=\alpha+{i\over\pi}\partial\op
\psi$ with an almost psh function $\psi$ on~$X$. The first step in the
proof of Th.~1.1 consists in showing that $\psi$ can be replaced
locally by the logarithm of a sum of squares of holomorphic functions,
in such a way that the Lelong numbers and the almost positive lower
bound of $T$ are preserved. For this, we use basically the same $L^2$
technique as in Section~9 of [De~90b] (namely the Ohsawa-Takegoshi
$L^2$ extension theorem [OT~87]), but we work with functions on
pseudoconvex open sets rather than with global sections of ample line
bundles. We begin by a description of this local procedure.

\tha Proposition 3.1|Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\cb^n$. For every $m>0$, let
$\hc_\Omega(m\varphi)$ be the Hilbert space of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-2m\varphi}d\lambda<+\infty$ and
let $\varphi_m={1\over 2m}\log\sum|\sigma_\ell|^2$ where $(\sigma_\ell)$
is an orthonormal basis of~$\hc_\Omega(m\varphi)$. Then there are
constants $C_1,C_2>0$ independent of $m$ such that
\item{\rm (i)}$\displaystyle\varphi(z)-{C_1\over m}\le
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over m}
\log{C_2\over r^n}$\hfill\break
for every $z\in\Omega$ and $r<d(z,\partial\Omega)$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$ and
\item{\rm (ii)}$\displaystyle\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le
\nu(\varphi,z)$~ for every $z\in\Omega$.\fintha

\remarque Remark 3.2|Proposition~3.1 can be used to derive a very simple
proof of Siu's result [Siu~74] that $E_c(T)$ is an analytic set, at
least in the case of a bidegree $(1,1)$-current $T$ (the case of an
arbitrary bidegree can be actually reduced to the $(1,1)$ case by a
standard argument due to P.~Lelong). To see this, it is enough to
consider the case of a current $T={i\over\pi}\partial\op \varphi$ on a
pseudoconvex open set $\Omega\subset\cb^n$. The inequalities in~(ii)
imply that $E_c(T)=\bigcap_{m\ge m_0}E_{c-n/m}(T_m)$, where
$T_m={i\over\pi}\partial\op\varphi_m$. Now, it is clear that $E_c(T_m)$ is
the analytic set defined by the equations $\sigma_\ell^{(\alpha)}(z)=0$ for all
multi-indices $\alpha$ such that~$|\alpha|<mc$. Thus $E_c(T)$ is
analytic.\finpr

\remarque Proof|Note that $\sum|\sigma_\ell(z)|^2$ is the square of the norm
of the evaluation linear form $f\mapsto f(z)$ on $\hc_\Omega(m\varphi)$. As
$\varphi$ is locally bounded above, the $L^2$ topology is actually stronger
than the topology of uniform convergence on compact subsets of~$\Omega$.
It follows that the series $\sum|\sigma_\ell|^2$ converges uniformly on
$\Omega$ and that its sum is real analytic. Moreover we have
$$\varphi_m(z)=\sup_{f\in B(1)}{1\over m}\log|f(z)|$$
where $B(1)$ is the unit ball of $\hc_\Omega(m\varphi)$. For
$r<d(z,\partial\Omega)$, the mean value inequality applied to the psh
function $|f|^2$ implies
$$\eqalign{
|f(z)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|\zeta-z|<r}
|f(\zeta)|^2d\lambda(\zeta)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|\zeta-z|<r}\varphi(\zeta)\Big)
\int_\Omega|f|^2e^{-2m\varphi}d\lambda.\cr}$$
If we take the supremum over all $f\in B(1)$ we get
$$\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}$$
and the second inequality in (i) is proved. Conversely, the
Ohsawa-Takegoshi extension theorem [OT~87,~Oh~88] (Appendix, A.3) applied
to the $0$-dimensional subvariety $\{z\}\subset\Omega$ shows that for any
$a\in\cb$ there is a holomorphic function $f$ on $\Omega$ such that
$f(z)=a$ and
$$\int_\Omega|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z)},$$
where $C_3$ only depends on $n$ and~${\rm diam}\,\Omega$. We fix $a$
such that the right hand side is~$1$. This gives the other
inequality
$$\varphi_m(z)\ge{1\over m}\log|a|=\varphi(z)-{\log C_3\over 2m}.$$
The above inequality implies $\nu(\varphi_m,z)\le\nu(\varphi,z)$. In the
opposite direction, we find
$$\sup_{|x-z|<r}\varphi_m(x)\le\sup_{|\zeta-z|<2r}\varphi(\zeta)+
{1\over m}\log{C_2\over r^n}.$$
Divide by $\log r$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a psh function over $B(x,r)$ tends to the
Lelong number at~$x$. Thus we obtain
$$\nu(\varphi_m,x)\ge\nu(\varphi,x)-{n\over m}.\finpr$$

{\petcap Gluing process}\pointir When $T={i\over\pi}\partial\op\psi$
is a global current on~$X$, we use a covering of $X$ by coordinate open balls
$U_j$ of radius $2\delta$ constructed as in section~2, and we let
$U'_j\compact U''_j\compact U_j$, $\gamma_j$, $\wt\gamma_j$ be defined in
the same way. On each open set $U_j$ the function
$\varphi_j:=\psi-\wt\gamma_j\circ\tau_j$ defined
in (2.9$\,$a) is plurisubharmonic, so Proposition~3.1 gives functions
$$\varphi_{j,m}={1\over 2m}\log\sum_\ell|\sigma_{j,\ell}|^2,~~~~
(\sigma_{j,\ell})=\hbox{\rm basis of}~\hc_{U_j}(m\varphi_j),\leqno(3.3)$$
approximating $\varphi_j$ as $m$ tends to $+\infty$ and satisfying
the inequalities
$$\varphi_j(x)-{C_1\over m}\le\varphi_{j,m}(x)\le\sup_{|\zeta-x|<r}
\varphi_j(\zeta)+{1\over m}\log{C_2\over r^n}.\leqno(3.4)$$
The functions $\varphi_{j,m}+\wt\gamma_j\circ\tau_j$ on $U_j$ are then
glued together by a partition of unity technique developed in
the following lemma.

\tha Lemma 3.5|Let $U'_j\compact U''_j$ be locally finite open coverings
of a complex manifold $X$ by relatively compact open sets, and let
$\theta_j$ be smooth nonnegative functions with support in $U''_j$,
such that $\theta_j\le 1$ on $U''_j$ and $\theta_j=1$ on $U'_j$.
Let $A_j\ge 0$ be such that
$$i(\theta_j\partial\op\theta_j-\partial\theta_j\wedge\op\theta_j)\ge
-A_j\omega~~~~\hbox{\rm on}~~U''_j\ssm U'_j.$$
Finally, let $w_j$ be almost psh functions on $U_j$ with
the property that $i\partial\smash{\op}w_j\ge\gamma$
for some real $(1,1)$-form $\gamma$ on~$M$, and let $C_j$ be constants
such that
$$w_j(x)\le C_j+\sup_{k\ne j,\,U'_k\ni x}w_k(x)~~~~\hbox{\rm on}~~
U''_j\ssm U'_j.$$
Then the function $w=\log\big(\sum\theta_j^2 e^{w_j}\big)$
is almost psh and satisfies
$$i\partial\op w\ge\gamma-2\Big(\sum_j\oneb_{U''_j\ssm U'_j}A_je^{C_j}\Big)
\omega.$$\fintha

\remarque Proof|If we set $\alpha_j=\theta_j\partial w_j+2\partial\theta_j$,
a straightforward computation shows that
$$\eqalign{
\partial w&={\sum(\theta_j^2\partial w_j+2\theta_j\partial\theta_j)e^{w_j}
\over\sum\theta_j^2e^{w_j}}={\sum\theta_je^{w_j}\alpha_j
\over\sum\theta_j^2e^{w_j}},\cr
\partial\op w&={\sum\!\big(\alpha_j\wedge\ov\alpha_j{+}\theta_j^2\partial\op w_j
{+}2\theta_j\partial\op\theta_j{-}2\partial\theta_j{\wedge}\op\theta_j\big)
e^{w_j}\over\sum\theta_j^2e^{w_j}}
-{\sum_{j,k}\theta_je^{w_j}\theta_ke^{w_k}\alpha_j{\wedge}
\ov\alpha_k\over\big(\sum\theta_j^2e^{w_j}\big)^2}\cr
&={\sum_{j<k}\!\big|\theta_j\alpha_k{-}\theta_k\alpha_j\big|^2e^{w_j}e^{w_k}
\over\big(\sum\theta_j^2e^{w_j}\big)^2}{+}{\sum\theta_j^2e^{w_j}\partial\op w_j
\over\sum\theta_j^2e^{w_j}}{+}{\sum\!\big(2\theta_j\partial\op\theta_j{-}
2\partial\theta_j{\wedge}\op\theta_j\big)e^{w_j}\over\sum\theta_j^2e^{w_j}}\cr}$$
by using the Legendre identity. The first term in the last line is nonnegative
and the second one is $\ge\gamma$. In the third term, if $x$ is in the support
of $\theta_j\partial\op\theta_j-\partial\theta_j\wedge\op\theta_j$, then
$x\in U''_j\ssm U'_j$ and so $w_j(x)\le C_j+w_k(x)$ for some $k\ne j$
with $U'_k\ni x$ and $\theta_k(x)=1$. This gives
$$i{\sum\big(
2\theta_j\partial\op\theta_j-2\partial\theta_j\wedge\op\theta_j\big)e^{w_j}
\over\sum\theta_j^2e^{w_j}}\ge
-2\sum_j\oneb_{U''_j\ssm U'_j}e^{C_j}A_j\omega.$$
The expected lower bound follows.\finpr

We now apply Lemma~3.5 to the functions
$$w_j(x)=2m\wt\gamma_j(z^j)+\sqrt{m}(\delta^2-|z^j|^2)+\log\sum_\ell
|\sigma_{j,\ell}(x)|^2$$
where $z^j=\tau_j(x)$ is a local coordinate identifying $U_j$
to $B(0,2\delta)$. The associated function $w$ is defined by
$$w(x)=\log\Big(\sum_j\theta_j(x)^2\exp\big(2m\wt\gamma_j(z^j)+
\sqrt{m}(\delta^2-|z^j|^2)\big)\sum_\ell|\sigma_{j,\ell}(x)|^2\Big)$$
and we set $\psi_m={1\over 2m}w$. On $U'_j$ we have $\theta_j=1$ and
$|z^j|<\delta$. Hence on every compact subset of $U'_j$ there is a small
constant $c(\delta)>0$ such that
$$\eqalign{
w(x)&\ge 2m\wt\gamma_j(z^j)+2c(\delta)\sqrt{m}
     +\log\sum_\ell|\sigma_{j,\ell}(x)|^2,\cr
\psi_m(x)&\ge\wt\gamma_j(z^j)+\varphi_{j,m}(x)+{c(\delta)\over\sqrt{m}}
\ge\wt\gamma_j(z^j)+\varphi_j(x)-{C_1\over m}+{c(\delta)\over\sqrt{m}}>\psi(x)
\kern-1.1pt\cr}$$
for $m$ large, in view of (3.4) and the definition of~$\varphi_j$.
Therefore $\psi_m>\psi$ for $m\ge m_0(\delta)$ large. In the opposite
direction, if $N(\delta)$ is the maximum number of overlapping
balls~$U_j$, we have
$$w(x)\le\log N(\delta)+\delta^2\sqrt{m}+\max_j\Big\{
2m\wt\gamma_j(z^j)+\log\sum_\ell|\sigma_{j,\ell}(x)|^2\Big\}.$$
By definition of $\varphi_j$ we have $\sup_{|\zeta-x|<r}\varphi_j(\zeta)\le
\sup_{|\zeta-x|<r}\psi(\zeta)-\wt\gamma_j(z^j)+C_4r$, thus by (3.4) again
we find
$$\eqalign{
\psi_m(x)&\le\max_j\big\{\wt\gamma_j(z^j)+\varphi_{j,m}(x)\big\}+{1\over 2m}
\log N(\delta)+{\delta^2\over\sqrt{m}}\cr
&\le\sup_{|\zeta-x|<r}\psi(\zeta)+C_4r+{\delta^2\over\sqrt{m}}+
{1\over m}\log{C_2N(\delta)^{1/2}\over r^n}.
\cr}$$
By taking $r=1/m$ for instance, we see that $\psi_m$ converges to $\psi$.
On the other hand $(2.8)$ implies
${i\over\pi}\partial\op\wt\gamma_j(z^j)=\tau_j^\star\gamma_j\ge
\gamma-2\varepsilon(\delta)\omega$, thus for $m$ large
$${i\over\pi}\partial\op w_j\ge 2m\tau_j^\star\gamma_j-O(\sqrt{m})
\ge 2m(\gamma-3\varepsilon(\delta)\omega).$$
Lemma~3.6 below will be used to show that $w_j(x)\le
C_j+\sup_{k\ne j,\,U'_k\ni x}w_k(x)$ for $C_j=0$ and $m$ large enough.
Hence Lemma~3.5 gives an additional bounded negative term in the
Hessian of $w$ and we finally get
$${i\over\pi}\partial\op\psi_m\ge\gamma-4\varepsilon(\delta)\omega$$
for $m\ge m'_0(\delta)$ large.

\tha Lemma 3.6|There are constants $C_{j,k}=C_{j,k}(\delta)$ independent of
$m$ such that the almost psh functions
$$\wt w_j(x)=2m\,\wt\gamma_j(z^j)+
\log\sum_\ell\big|\sigma_{j,\ell}(x)\big|^2,~~~~
x\in U''_j$$
satisfy
$$|\wt w_j-\wt w_k|\le(2n+2)\log m+C_{j,k}~~~~\hbox{\rm on}~~
U''_j\cap U''_k.$$\fintha

\remarque Proof|The proof will be given in a more general context in the next
sec\-tion: we apply 4.6~(i) when $E$ is the trivial bundle $\oc_X$ equipped
with the weight $e^{-\psi}$. In that case we simply have $P(E^\star)=X$,
$p=\id_X$, $\oc_E(1)=\oc_X$ and $v=\gamma$. Note that the
smoothness of the metric of $E$ is not used in the proof of 4.6~(i).
The main idea is as follows: for any holomorphic function $h$ on~$U_j$,
a $\op$ equation can be solved on $U_k$, after some truncation, to get
a holomorphic function assuming the same value as $h$ at any prescribed
point of $U''_j\cap U''_k$; this can be done with a uniform control on
$L^2$ norms.\finpr

By Lemma 3.6, we see that $w_j(x)=\wt w_j(x)+\sqrt{m}(\delta^2-|z^j|^2)$
satisfies $w_j(x)\le w_k(x)$ for $x\in(\ov U''_j\ssm U'_j)\cap U'_k$
and $m$ large enough (note that $\delta^2-|z^j|^2\le 0$
and $\delta^2-|z^k|^2>0$ at $x$). Since $X$ is covered by a collection
of compact sets contained in $(U'_k)$, we see that the assumption of
Lemma~3.5 on $w_j$ is actually satisfied with $C_j=0$ for $m\ge m''_0(\delta)$
large enough. If we take $\delta=\delta_m$ decreasing to $0$ very slowly
with $m$, we obtain a sequence $\psi_m$ satisfying the following properties.

\tha Proposition 3.7|Let $\psi$ be an almost psh function on a compact complex
manifold $X$ such that ${i\over\pi}\partial\op\psi\ge\gamma$ for some
continuous $(1,1)$-form $\gamma$. Then there is a sequence of
almost psh functions $\psi_m$ such that $\psi_m$ has the same singularities
as a logarithm of a sum of squares of holomorphic functions and
\item{\rm (i)}$\displaystyle\psi(x)<\psi_m(x)\le\sup_{|\zeta-x|<r}
\psi(\zeta)+C\Big({|\log r|\over m}+r+m^{-1/2}\Big)$\hfill\break
with respect to coordinate open sets covering $X$. In particular,
$\psi_m$ converges to $\psi$ pointwise and in $L^1(X)$ and
\item{\rm (ii)}$\displaystyle\nu(\psi,x)-{n\over m}\le\nu(\psi_m,x)\le
\nu(\psi,x)$~ for every $x\in X\,;$
\item{\rm (iii)}$\displaystyle{i\over\pi}\partial\op\psi_m\ge
\gamma-\varepsilon_m\omega$~ with $\varepsilon_m>0$ decreasing to $0$.
\fintha

\titre 4. Construction of hermitian metrics by $L^2$ techniques|

Let $E$ be a holomorphic vector bundle of rank $r$ over a complex
manifold~$M$, and let $\oc_E(1)$ be the associated canonical line
bundle over the projectivized bundle $P(E^\star)$ of hyperplanes
of~$E$. By definition there is an exact sequence
$$0\la H\la \pi^\star E\la \oc_E(1)\la 0$$
where $\pi:P(E^\star)\to M$ is the projection and $H\subset\pi^\star E$
the tautological hyperplane subbundle. Suppose that $\oc_E(1)$ is equipped
with a hermitian metric. As the total space of $\oc_E(-1)$ is the blow-up
of the total space of $E^\star$ along the zero section, a hermitian metric
on $\oc_E(-1)$ corresponds to a {\sl Finsler metric}
on~$E^\star$, that is, to an arbitrary positive homogeneous function of
degree~1 along the fibers of~$E^\star$. Our goal is to construct hermitian
metrics on all high symmetric powers $S^mE$ which induce approximations
of the given metric on $\oc_E(1)$ with a good control of the curvature.
For this, we use the fact that $\pi_\star\oc_E(m)\simeq S^mE$
and we apply H\"ormander's $L^2$ estimates for $\op$ to construct
holomorphic sections of~$\oc_E(m)$ locally over $M$ (of course, no
global holomorphic section need exist in general).

\tha Theorem 4.1|Suppose that $\oc_E(1)$ is equipped with a
smooth hermitian metric such that $c\big(\oc_E(1)\big)\ge\pi^\star v$
for some continuous real $(1,1)$-form $v$ on~$M$. Then, for each positive
integer~$m$, there is a smooth hermitian metric on $S^mE$ such that
\item{\rm(i)}the sequence of metrics on $\oc_E(1)$ induced by the surjective
morphisms $\pi^\star S^mE\to\oc_E(m)$ converges locally uniformly to the
original metric;
\item{\rm(ii)}for every relatively compact subset $M'\compact M$ and every
$\varepsilon>0$, there exists an integer $m_0$ such that $m\ge m_0$ implies
$$c(S^mE)\ge m(v-\varepsilon\omega)\otimes\id_{S^mE}~~~~\hbox{\rm over}~~M'$$
\vskip-5pt\item{}in the sense of Griffiths, where $\omega$ is a fixed
hermitian metric on~$M$.\fintha

\remarque Proof|For every $M'\compact M$ and $\varepsilon>0$ fixed, we are
going to construct a sequence of metrics on $S^mE$ satisfying (i) and (ii)
with this choice of~$\varepsilon$ and~$M'$. This will be enough to prove
the result, by taking some suitable diagonal subsequence with $\varepsilon_k$
decreasing to $0$ and $M'_k$ increasing to~$M$. 
Let $(U_j)$ be a locally finite covering of $M$ by open balls such that
each of these balls is relatively compact in a coordinate open chart. Let
$z^j=(z_1\ld z_n)$ be coordinates near $\smash{\ov U_j}$ and let
$v_j$ be the $(1,1)$-form with constant coefficients in $U_j$ which
coincides with $v-\varepsilon\,\omega$ at the center of~$U_j$. Finally,
let $\wt v_j(z^j)=\pi\sum z_k\ov z_\ell\big(\partial^2v/
\partial z_k\partial\ov z_\ell(0)-\varepsilon\omega_{k\ell}(0)\big)$
be the associated quadratic function such that $v_j={i\over\pi}\partial\op
\wt v_j$. By choosing the radius of $U_j$ small enough, we can suppose that
$0\le v-v_j\le 2\varepsilon\,\omega$ on $U_j$ for every~$j$.
Now, we consider the Hilbert space
$\hc_j(m)$ of holomorphic sections $f\in H^0\big(\pi^{-1}(U_j),\oc_E(m)\big)$
with the $L^2$~norm
$$||f||_j^2=\int_{\pi^{-1}(U_j)}|f|^2 e^{2m\,\wt v_j(z^j)}dV,$$
where $dV$ is a volume element on $P(E^\star)$ (fixed once for all)
and $|f|^2$ is the pointwise norm on $\oc_E(m)$ induced by the given hermitian
metric on $\oc_E(1)$. This means that we have multiplied the original metric
of $\oc_E(1)$ by the weight $\exp(2\wt v_j)$. The corresponding curvature
form is
$$c\big(\oc_E(1)\big)-{i\over\pi}\partial\op\wt v_j(z^j)
=c\big(\oc_E(1)\big)-v_j\ge\pi^\star(v-v_j)\ge 0\leqno(4.2)$$
thanks to the curvature hypothesis. Finally, let $U'_j\compact U''_j
\compact U_j$ be concentric balls such that $(U'_j)$ still cover $M$ and
let $\theta_j$ be smooth functions with support in $U''_j$, such that
$0\le\theta_j\le 1$ on $U''_j$ and $\theta_j=1$ on~$\smash{\ov U'_j}$.
We define a hermitian metric on $S^mE^\star$ as follows: for all
$x\in M$ and $\xi\in S^mE^\star_x$ we set
$$||\xi||_{(m)}^2=\sum_j\theta_j^2(x)\exp\big(2m\wt v_j(z^j)+\sqrt{m}
(r_j^{\prime 2}-|z^j|^2)\big)
\sum_\ell\big|\sigma_{j,\ell}(x)\cdot\xi\big|^2,\leqno(4.3)$$
where $r_j'$ is the radius of $U'_j$ and $(\sigma_{j,\ell})_{\ell\ge 1}$
is an orthonormal basis of $\hc_j(m)$. Here, we consider
$\sigma_{j,\ell}$ as a section
in $H^0(U_j,S^mE)$ and we use the obvious pairing between $S^mE$ and
$S^mE^\star$ to compute $\sigma_{j,\ell}(x)\cdot\xi$. It is easy to
see that the sum $\sum_\ell|\sigma_{j,\ell}(x)\cdot\xi|^2$ does not depend
on the choice of the orthonormal basis and converges locally uniformly
above every compact subset of $U_j\,$: in fact the evaluation linear form
$f\mapsto f(x)\cdot \xi$ is continuous on $\hc_j(m)$ and the above sum
is the square of its norm; the local uniform convergence comes from the
fact that the linear form varies continuously with $(x,\xi)$ when $x\in U_j$.
The metric on $S^mE^\star$ defined by (4.3) is thus smooth. It induces
by duality a metric on $S^mE$ with the desired properties. To check this,
we use similar techniques to those of section~3. The following
elementary lemma is needed.

\tha Lemma 4.4|Given a hermitian holomorphic vector bundle $F$ and a
real $(1,1)$-form $\gamma$ on $M$, we have the equivalent properties
$$c(F)\ge\gamma\otimes\id_F\Longleftrightarrow
-c(F^\star)\ge\gamma\otimes\id_{F^\star}\Longleftrightarrow
{i\over 2\pi}\partial\op\log||\xi||^2\ge p^\star\gamma,~~
\xi\in F^\star$$
where $\log||\xi||^2$ is seen as a function on $F^\star$ and where
$p:F^\star\to M$ is the projection.\fintha

\remarque Proof|The first equivalence is obvious, for $-c(F^\star)$ is the
transposition of~$c(F)$. The second equivalence follows from the fact that
for any local holomorphic section $\sigma$ of $F^\star$, we have
$${i\over 2\pi}\partial\op\log||\sigma||^2\ge
-{\langle c(F^\star)\sigma,\sigma\rangle\over||\sigma||^2},$$
with equality at any point $x\in M$ where $D\sigma(x)=0$ and $\sigma(x)
\ne 0$.\finpr

In order to prove the curvature estimate 4.1~(ii), we apply Lemma 3.5 with
$w(x,\xi)=\log||\xi||_{(m)}^2$ and
$$w_j(x,\xi)=2m\,\wt v_j(z^j)+\sqrt{m}(r^{\prime 2}_j-|z^j|^2)+
\log\sum_\ell\big|\sigma_{j,\ell}(x)\cdot\xi\big|^2,\leqno(4.5)$$
on the total space $X=S^mE^\star$ covered by $p^{-1}(U''_j)$.
For this, we need to estimate the differences $|w_j(\xi)-w_k(\xi)|$ over
$p^{-1}(U''_j\cap U''_k)$. This is the crucial point where $L^2$ estimates
for $\op$ are needed.

\tha Lemma 4.6|There are constants $C_{j,k}$ and $C'_j$ independent of $m$
such that the almost psh functions
$$\wt w_j(x,\xi)=2m\,\wt v_j(z^j)+
\log\sum_\ell\big|\sigma_{j,\ell}(x)\cdot\xi\big|^2,~~~~
(x,\xi)\in p^{-1}(U''_j)\subset S^mE^\star$$
satisfy
$$\leqalignno{
&~~~~|\wt w_j-\wt w_k|\le(2n+2)\log m+C_{j,k}~~~~\hbox{\rm on}~~
p^{-1}(U''_j\cap U''_k)\,;&\hbox{\rm(i)}\cr
&~~~~\big|\wt w_j(x,\eta^m)-2m\log|\eta|\,\big|\le(2n+2r-2)\log m+C'_j~~~~
\hbox{\rm for}~~x\in U''_j,~~\eta\in E^\star_x,&\hbox{\rm(ii)}\cr}$$
where $|\eta|$ denotes the original Finsler metric on $E^\star$.\fintha

\remarque Proof|As $E_{\restriction\ov U_j}\simeq\ov U_j\times\cb^r$ is
trivial near $\ov U_j$, we can define a hermitian metric on
$E_{\restriction\ov U_j}$ with positive definite curvature by taking
$|\xi|^2=\sum|\xi_\lambda|^2e^{-|z^j|^2}$. The associated curvature
form on $\oc_E(1)$ is positive and defines a K\"ahler metric $\omega_j$
on $\pi^{-1}(U_j)
\simeq U_j\times\pb^{r-1}$, equal to the direct sum of the euclidean
metric of $U_j$ and of the Fubini-Study metric of $\pb^{r-1}$.
In particular, the Ricci curvature of $\omega_j$ is nonnegative
(it is even positive along each fiber $\pb^{r-1}$). Let $e^{-\rho_j}$
be the ratio between the new metric on $\oc_E(1)_{\restriction U_j}$ and
the original one, which will be kept for the calculation of the
curvature form $c(\oc_E(1))$. By definition of $\rho_j$ we have
$$c\big(\oc_E(1)\big)+{i\over\pi}\partial\op\rho_j=\omega_j>0.
\leqno(4.7)$$
Let $\tau$ be an arbitrary almost psh function on a neighborhood of
$\pi^{-1}(\ov U_j)$ and let $\mu\ge 0$ be a large integer such that
$${i\over\pi}\partial\op\tau+\mu\omega_j\ge\omega_j~~~~\hbox{\rm on}~~
\pi^{-1}(U_j).\leqno(4.8)$$
For each $q=0,1\ld n$, we consider the
Hilbert space $\fc^{0,q}_j(m)$ of $(0,q)$-forms $f$ on $\pi^{-1}(U_j)$ with
values in $\oc_E(m)$, equipped with the $L^2$ norm
$$||f||^2_{j,q}=\int_{\pi^{-1}(U_j)}|f|^2_je^{2(m-\mu)\wt v_j(z^j)-2\mu\rho_j
-2\tau}dV_j\,,$$
where $dV_j=\omega_j^{n+r-1}/(n+r-1)!$ and where the pointwise norm $|f|_j$
is computed by means of $\omega_j$ and of the original metric on $\oc_E(m)$.
Thanks to (4.2), (4.7), (4.8), the weight involved in the norm
$||~~~||_{j,q}$ corresponds to a curvature form on $\oc_E(m)$ given by
$$\eqalign{
m\,c\big(\oc_E(1)\big)&{}-(m-\mu){i\over\pi}\partial\op\wt v_j+\mu{i\over\pi}
\partial\op\rho_j+{i\over\pi}\partial\op\tau\cr
&{}={i\over\pi}\partial\op\tau+\mu\omega_j+(m-\mu)\big(c\big(\oc_E(1)\big)
-v_j\big)\ge\omega_j.\cr}$$
Hence, by H\"ormander's $L^2$ estimates (Appendix, Theorem~A.1), for every
$(0,q)$-form $g\in \fc^{0,q}_j(m)$ with $q\ge 1$ and $\op g=0$, there is a
$(0,q-1)$-form $f\in\fc^{0,q-1}_j(m)$ such that
$$\op f=g,~~~~||f||_{j,q-1}\le q^{-1/2}||g||_{j,q}.\leqno(4.9)$$
Observe that this estimate is valid for $(0,q)$-forms because
Ricci$(\omega_j)\ge 0$. Moreover, if $\tau$ is bounded, the norm
$||~~~||_{j,0}$ is equivalent to the norm $||f||_j$ of $\hc_j(m)$,
up to constants independent of $m$.

(i) Fix a point $x\in U''_j\cap U''_k$ and $\xi\in S^mE^\star_x$. There exists
$h\in\hc_k(m)$ with $||h||_k=1$ such that
$$|h(x)\cdot\xi|^2=\sum_\ell|\sigma_{k,\ell}(x)\cdot\xi|^2\,;$$
in fact, $h$ satisfies this relation if and only if $h$ is orthogonal to
the hyperplane of sections $\sigma\in\hc_k(m)$ such that $\sigma(x)\cdot\xi=0$.
Let $\chi$ be a cut-off function with support in the ball $B(x,1/m)$, equal
to $1$ on $B(x,1/2m)$ and with $|\op\chi|\le C_1 m$. For~$m\ge m_0$
large enough (independent of $x$) we have $B(x,1/m)\subset U_j\cap U_k$,
and (4.9) allows us to solve the equation
$$\op f=h\,\op(\chi\circ\pi)~~~\hbox{\rm on}~~\pi^{-1}(U_j),~~~~
||f||_{j,0}\le||h\,\op(\chi\circ\pi)||_{j,1}.$$
We then obtain a holomorphic section
$$h'=h(\chi\circ\pi)-f\in H^0\big(\pi^{-1}(U_j),\oc_E(m)\big).$$
We want $h'$ to coincide with $h$ along the fiber $\pi^{-1}(x)$, so we take
$\tau(z)=n\log|z^j-z^j(x)|$ depending only on the base variables with a
logarithmic pole at~$x\,$: the $L^2$ condition on $f$ then implies that
$f$ vanishes identically along $\pi^{-1}(x)$. As $\tau$ is psh, we can take
in fact $\mu=1$ here. By the bound on $|\op\chi|$, we have
$$\eqalign{
||h\,\op(\chi\circ\pi)||_{j,1}^2
&\le C_2m^2\int_{\pi^{-1}\big(B(x,1/m)\ssm B(x,1/2m)\big)}{|h|^2
e^{2(m-1)\wt v_j(z^j)-2\rho_j}\over|z^j-z^j(x)|^{2n}}dV_j\cr
&\le C_3m^{2n+2}\int_{\pi^{-1}B(x,1/m)}|h|^2e^{2m\wt v_j(z^j)}dV_j\cr
&\le C_4m^{2n+2}e^{2m\big(\wt v_j(z^j(x))-\wt v_k(z^k(x))\big)}||h||^2_k\cr}$$
with constants $C_2$, $C_3$, $C_4$ independent of $m$ and $x$. In the last
inequality, we used the fact that the oscillation of $\wt v_j$ and $\wt v_k$
on $B(x,1/m)$ is $O(1/m)$. Hence we get the estimate
$$||f||_j^2\le C_5||f||_{j,0}^2\le C_4C_5m^{2n+2}
e^{2m\big(\wt v_j(z^j(x))-\wt v_k(z^k(x))\big)}||h||^2_k$$
where $||h||_k=1$ (the constant $C_5$ exists because $\tau$ is bounded above).
The norm $||h(\chi\circ\pi)||_j$ satisfies a similar estimate. This implies
$$\eqalign{||h'||_j^2
&\le C_6m^{2n+2}e^{2m\big(\wt v_j(z^j(x))-\wt v_k(z^k(x))\big)},\cr
\sum_\ell\big|\sigma_{j,\ell}(x)\cdot\xi\big|^2
&\ge C_6^{-1}m^{-(2n+2)}e^{-2m\big(\wt v_j(z^j(x))-\wt v_k(z^k(x))\big)}
|h'(x)\cdot\xi|^2\cr
&\ge C_6^{-1}m^{-(2n+2)}e^{-2m\big(\wt v_j(z^j(x))-\wt v_k(z^k(x))\big)}
\sum_\ell\big|\sigma_{k,\ell}(x)\cdot\xi\big|^2,\cr}$$
since by construction $h'(x)=h(x)$ and $|h(x)\cdot\xi|^2=
\sum\big|\sigma_{k,\ell}(x)\cdot\xi\big|^2$. By taking logarithms and
reversing the roles of $j$ and~$k$, we obtain the desired inequality~(i).

(ii) The proof is similar, except that we take $h$ to be a local holomorphic
section of $\oc_E(m)$ near $[\eta]\in P(E^\star_x)$ such that
$|h([\eta])|=1\,$; for this, we select a vector $a\in E^\star$ such that
$\langle a,\eta\rangle= |\eta|$ and $\sum|a_\lambda|^2\le C_7$, and we
take $h$ to be the image of the constant section $a^m\in H^0(U_j,S^mE)$
in~$\oc_E(m)$. Denote by $B([\eta],r)$ the geodesic ball of center $[\eta]$
and radius $r$ in $P(E^\star)$, and let $\chi$ be a cut-off function
with support in $B([\eta],1/m)$ equal to $1$ on $B([\eta],1/2m)$. If
$x\in U''_j$ and $m$ is large, then $B([\eta],1/m)\subset\pi^{-1}(U_j)$.
By (4.9), we solve the equation
$$\op f=h\,\op\chi~~~\hbox{\rm on}~~\pi^{-1}(U_j),~~~~
||f||_{j,0}\le||h\,\op\chi||_{j,1},$$
and we choose a weight $\tau$ with an isolated logarithmic pole of order
$n+r-1=\dim P(E^\star)$ at~$[\eta]\,$; the associated constant $\mu$ can be
chosen independent of $x$, $[\eta]$ and $m$. Then the $L^2$ condition
gives $f([\eta])=0$, thus
$$h'=h\chi-f\in H^0\big(\pi^{-1}(U_j),\oc_E(m)\big)$$
satisfies $h'([\eta])=h([\eta])$. Moreover we have
$$\eqalign{
||h\,\op\chi||_{j,1}^2
&\le C_8m^{2(n+r-1)+2}\int_{B([\eta],1/m)}|h|^2e^{2m\wt v_j(z^j)}dV_j
\le C_9m^2e^{2m\wt v_j(z^j(x))},\cr
||h'||_j^2&\le C_{10}m^2e^{2m\wt v_j(z^j(x))}.\cr}$$
This implies
$$\leqalignno{
\sum_\ell\big|\sigma_{j,\ell}(x)\cdot\eta^m\big|^2
&\ge C_{10}^{-1}m^{-2}e^{-2m\wt v_j(z^j(x))}|h'(x)\cdot\eta^m|^2\cr
&=C_{10}^{-1}m^{-2}e^{-2m\wt v_j(z^j(x))}|\eta|^{2m}&(4.10)\cr}$$
because $|h'(x)\cdot\eta^m|=|h([\eta])\cdot\eta^m|=|\eta|^m$ by construction.
Conversely, if $h\in\hc_j(m)$ is a section of norm $||h||_j=1$ such that
$$\sum_\ell\big|\sigma_{j,\ell}(x)\cdot\eta^m\big|^2=
|h(x)\cdot\eta^m|^2=|h([\eta])\cdot\eta^m|^2,$$
the mean value inequality on the ball $B([\eta],1/m)\subset P(E^\star)$
shows that
$$\eqalign{\big|h([\eta])\big|^2
&\le C_{11}m^{2(n+r-1)}e^{-2\wt v_j(z^j(x))}\int_{B([\eta],1/m)}|h|^2
e^{2m\wt v_j(z^j)}dV_j\cr
&\le C_{11}m^{2(n+r-1)}e^{-2\wt v_j(z^j(x))},\cr}$$
because the oscillation of the weight of $\oc_E(m)$ is bounded by a constant
on a ball of radius $1/m$. Hence
$$\sum_\ell\big|\sigma_{j,\ell}(x)\cdot\eta^m\big|^2
\le C_{11}m^{2(n+r-1)}e^{-2\wt v_j(z^j(x))}|\eta|^{2m}\leqno(4.11)$$
and inequality (ii) follows from the combination of (4.10) and (4.11).\finpr

\remarque End of proof of Theorem 4.1|Thanks to Lemma 4.6~(i), the
functions $w_j(x,\xi)$ defined in (4.5) satisfy
$w_j(x,\xi)\le w_k(x,\xi)$ for any $x\in(\ov U''_j\ssm U'_j)\cap U'_k$
and $m$ large enough (note that $r^{\prime 2}_j-|z^j|^2\le 0$
and $r^{\prime 2}_k-|z^k|^2>0$ at $x$). By an obvious compactness
argument, we see that the assumption of Lemma~3.5 on $w_j$ is satisfied
with $C_j=0$ for $m\ge m'_0$ large enough:
$$w_j(x,\xi)\le\sup_{k\ne j,\,U'_k\ni x}w_k(x,\xi)~~~~\hbox{\rm for}~~
x\in U''_j\ssm U'_j.$$
Moreover, as $v_j\ge v-2\varepsilon\omega$ and $\sqrt{m}\ll m\varepsilon$
for $m$ large, we have
$${i\over 2\pi}\partial\op w_j\ge m\,p^\star v_j-\sqrt{m}
{i\over 2\pi}\partial\op|z^j|^2\ge m\,p^\star(v-3\varepsilon\,\omega)$$
for $m\ge m''_0$. Lemma~3.5 then shows that $w(x,\xi)=\log||\xi||_{(m)}^2$
satisfies
$${i\over 2\pi}\partial\op w\ge m\,p^\star(v-3\varepsilon\,\omega)
-p^\star\Big(2\sum_j\oneb_{U''_j\ssm U'_j}A_j\omega\Big)
\ge m\,p^\star(v-4\varepsilon\,\omega)$$
for $m\ge m'''_0$ large. By Lemma~4.4 applied to $F=S^mE$, we infer
$$c(S^mE)\ge m(v-4\varepsilon\,\omega)\otimes\hbox\id_{S^mE}~~~~
\hbox{\rm for}~~m\ge m_0=\max(m'_0,m''_0,m'''_0),$$
hence 4.1~(ii) is proved with $4\varepsilon$ instead of~$\varepsilon$.
Now, Lemma~4.6~(ii) shows that ${1\over m}w_j(x,\eta^m)$ converges
uniformly to $\log|\eta|$ on $E^\star_{\restriction U''_j}$. Therefore
the global weight
\hbox{${1\over m}w(x,\eta^m)={1\over m}\log||\eta^m||^2_{(m)}$}
converges locally uniformly to $\log|\eta|$ on $E^\star$, i.e., the inclusions
$\oc_E(-m)\subset\pi^\star S^mE^\star$ induce metrics on $\oc_E(-1)$
which converge to the original metric. This statement is equivalent to
4.1~(i) by duality.\finpr

\titre 5. An alternative singularity attenuation technique|

As already remarked in section~2, it is sufficient to prove the Main
Theorem when $T=\smash{{i\over\pi}}\partial\op\psi$ has zero cohomology class.
Let $U'_j\compact U''_j\compact U_j$ be concentric balls of respective
radius $\delta$, $\smash{\sqrt{2}}\delta$, $2\delta$ such that the
family $(U'_j)$ covers $X$, and let $\gamma_j$, $\wt\gamma_j$ be as in
sections~2,3. We set $\varphi_j:=\psi-\wt\gamma_j\circ\tau_j$ on~$U_j$.
By (3.3), (3.4) and Lemma~3.5 we obtain local approximations
$\varphi_{j,m}$ satisfying the following properties:
$$\leqalignno{
&\qquad\varphi_{j,m}={1\over 2m}\log\sum_\ell|\sigma_{j,\ell}|^2~~~~\hbox{\rm
with $(\sigma_{j,\ell})_{\ell\ge 0}=$ basis of $\hc_{U_j}(m\varphi_j)$},
&(5.1)\cr
&\qquad\Big(\nu(\psi,x)-{n\over m}\Big)_+\le\nu(\varphi_{j,m}\,,x)\le\nu(\psi,x)
~~~~\hbox{\rm on}~~U_j,&(5.2)\cr
&\qquad\big|(\varphi_{j,m}+\wt\gamma_j\circ\tau_j)-
 (\varphi_{k,m}+\wt\gamma_k\circ\tau_k)\big|\le(n+1){\log m+C(\delta)\over m}
~~~~\hbox{\rm on}~~U''_j\cap U''_k.&(5.3)\cr}$$
The inequalities (5.2) show that $\varphi_{j,m}$ roughly has the same
singularities as~$\psi$. We want to ``attenuate" these singularities,
i.e. to shift the Lelong numbers by $-c$ for some positive number~$c$.
For this, the idea is to replace the functions $\sigma_{j,\ell}$ in
$\varphi_{j,m}$ by their derivatives of order $q\le [mc]$. Since we
must do this in such a way that the resulting functions defined on
various sets $U_j$ can be patched together, it is preferable to deal
with jet sections $J^q\sigma_{j,\ell} \in H^0(U_j,J^q\oc_X)$. Here
$J^q\oc_X$ denotes the global vector bundle of $q$-jets of holomorphic
functions on~$X$.

\tha Lemma 5.4|Suppose that $\oc_{TX}(1)$ is equipped with a smooth hermitian
metric such that $c\big(\oc_{TX}(1)\big)+\pi^\star_X u\ge 0$
for some nonnegative smooth $(1,1)$-form $u$ on~$X$. Let $\omega$ be a fixed
hermitian metric on~$X$. Then there are smooth hermitian metrics on $J^q\oc_X$
for all $q\ge 0$, such that
$$c(J^q\oc_X)\le q(u+\varepsilon_q\omega)\otimes\id_{J^q\oc_X}~~~~
\hbox{\rm with}~~\lim_{q\to+\infty}\varepsilon_q=0.$$
\fintha

\remarque Proof|By Theorem~4.1 applied to $E=TX$ and $v=-u$, there are
hermitian metrics on $S^qTX$ such that
$$c(S^qTX)\ge -q(u+\delta_q\omega)\otimes\id_{S^qTX}~~~~
\hbox{\rm with}~~\lim_{q\to+\infty}\delta_q=0.$$
We argue by induction on $q$ and use the canonical exact sequence
$$0\la S^qT^\star X\buildo j\over\la J^q\oc_X\buildo\pi\over\la J^{q-1}\oc_X
\la 0.$$
When $q=0$, we equip $J^q\oc_X=\oc_X$ with the trivial flat metric.
Assume that $q\ge 1$ and that a metric has already been constructed
on~$J^{q-1}\oc_X$. Fix a $C^\infty$-splitting $j':J^q\oc_X\to S^qT^\star X$
and the corresponding right inverse $\pi':J^{q-1}\oc_X\to J^q\oc_X$ onto the
kernel of~$j'$. For every $\eta>0$, we define a hermitian metric
on $J^q\oc_X$ by
$$|\xi|_\eta^2=\eta^2|j'\xi|^2+|\pi\xi|^2,~~~~~\forall\xi\in J^q\oc_X.$$
With respect to the given metrics on $S^qT^\star X$, $J^{q-1}\oc_X$ and
to $|~~|_\eta$ on $J^q\oc_X$, the adjoints $j^\star_\eta$ and
$\pi^\star_\eta$ of $j,\pi$ are given by $j^\star_\eta=\eta
j'$ and $\pi^\star_\eta=\pi'$. It follows that the second
fundamental form with values in ${\rm Hom}(S^qT^\star X,J^{q-1}\oc_X)$
is equal to $\eta\beta$ for some fixed matrix valued
$(1,0)$-form~$\beta$ (apply the standard formula
$\beta^\star=-j^\star\op\pi^\star$). In the orthogonal
decomposition $(j^\star_\eta,\pi):J^q\oc_X\la S^{q-1}T^\star
X\oplus J^{q-1}\oc_X$, the curvature of $J^q\oc_X$ is then given by
$$c(J^q\oc_X)_\eta=\pmatrix{
  c(S^qT^\star X)-\eta^2{i\over 2\pi}\beta^\star\wedge\beta
                                  &-{i\over 2\pi}\eta\op\beta^\star\cr
  {i\over 2\pi}\eta\partial\beta
       &c(J^{q-1}\oc_X)-\eta^2{i\over 2\pi}\beta\wedge\beta^\star\cr}.$$
It follows that
$$\eqalign{c(J^q\oc_X)_\eta
&\le\pmatrix{
  q(u+\delta_q\omega)\otimes\id_{S^qT^\star X} &\kern-46pt\qquad\qquad 0\cr
  0\qquad\qquad&\kern-46pt (q-1)(u+\varepsilon_{q-1}\omega)\otimes
  \id_{J^{q-1}\oc_X}\cr}+O(\eta)\,\omega\otimes\id_{J^q\oc_X}\cr
&\le q(u+\varepsilon_q\omega)\otimes\id_{J^q\oc_X}\cr}$$
where $\varepsilon_q$ is given by the induction formula
$$\varepsilon_q=\max\Big\{\delta_q,\Big(1-{1\over q}\Big)\varepsilon_{q-1}
\Big\}+O(\eta).$$
We take $\eta=\eta_q$ so small that $O(\eta)\le 2^{-q}$. Since $\prod(1-1/q)=0$,
it is then easy to check that $\lim\varepsilon_q=0$.

\remarque Remark 5.5|This construction can be reformulated as follows:
take a $C^\infty$-splitting $J^q\oc_X=\bigoplus_{0\le k\le q}S^kT^\star X$,
and rescale the metric on each summand $S^kT^\star X$ by a positive
factor~$\eta_k$. In general, the sequence $(\eta_k)$ will have to
decay very quickly to~$0$ (in a way that we do not try to control). In
order to emphasize this dependence on the sequence $(\eta_k)$, we
denote by $|~~|_{(\eta)}$ the resulting metric on~$J^q\oc_X$.\finpr

Now, we would like to compare the jet norms
$\sum_\ell|J^q\sigma_{j,\ell}|^2_{(\eta)}$ for various open sets~$U_j$.
Inequality (5.3) does this for~$q=0$. Unfortunately, it seems very
difficult to derive precise estimates for $q$-jets from estimates on
the functions (the Cauchy inequalities are not precise enough because
they involve high powers of the radius). We shall therefore modify our
functions $\sigma_{j,\ell}$ in such a way that the jets can be
estimated more easily. Our method relies heavily on Skoda's $L^2$
estimates for ideals of holomorphic functions~[Sk~72b]. We set
$\sigma_j=(\sigma_{j,\ell})_{\ell\ge 0}$ and $|\sigma_j|^2=
\sum_\ell|\sigma_{j,\ell}|^2$. For all integers $m,p\ge 1$, we consider
the Hilbert space $\hc_{U''_j}(mp\varphi_{j,m})$ of holomorphic functions
$f\in H^0(U''_j,\oc_X)$ equipped with the $L^2$-norm
$$||f||^2_{j,m,p}=\int_{U''_j}|f|^2e^{-2mp\varphi_{j,m}}d\lambda
=\int_{U''_j}|f|^2|\sigma_j|^{-2p}d\lambda,$$
and we introduce the functions
$$\varphi''_{j,m,p}={1\over 2mp}\log\sum_\ell|\sigma''_{j,\ell}|^2~~~~
\hbox{\rm on}~~U''_j,$$
where $(\sigma''_{j,\ell})$ is an orthonormal basis of
$\hc_{U''_j}(mp\varphi_{j,m})$. By 3.1~(ii) applied to
$\varphi=\log|\sigma_j|=m\,\varphi_{j,m}$ and $m$ replaced by $p$, we get
$$\nu(\varphi_{j,m}\,,x)-{n\over mp}\le\nu(\varphi''_{j,m,p}\,,x)
\le\nu(\varphi_{j,m}\,,x)~~~~\hbox{\rm on}~~U''_j.\leqno(5.6)$$
Thus, for $p$ large, the Lelong numbers have not been substantially modified.
Our goal is to compare the norms $\sum_\ell|J^q\sigma''_{j,\ell}|^2_{(\eta)}$
for different indices~$j$. By $(2.8)$ the Hessian of
$\wt\gamma_j\circ\tau_j-\wt\gamma_k\circ\tau_k$ is bounded by
$4\varepsilon(\delta)\omega$ on $U_j\cap U_k$. Since $\wt\gamma_j$ is
quadratic, all its translates differ by a real affine function. Hence
$$\big[\wt\gamma_j(\tau_j(x))-\wt\gamma_j(\tau_j(x)-b_j)\big]-
\big[\wt\gamma_k(\tau_k(x))-\wt\gamma_k(\tau_k(x)-b_k)\big]$$
is a pluriharmonic function, equal to the real part of a holomorphic
function $h_{jk}$ on~$U_j\cap U_k$. We choose $b_j$ and $b_k$ such that
the critical points of $\wt\gamma_j(\tau_j(x)-b_j)$ and
$\wt\gamma_k(\tau_k(x)-b_k)$ coincide at some point of~$U_j\cap U_k$. Then
$$|\wt\gamma_j\circ\tau_j-\wt\gamma_k\circ\tau_k-\re h_{jk}|=
|\wt\gamma_j(\tau_j-b_j)-\wt\gamma_k(\tau_k-b_k)|
\le C_1\varepsilon(\delta)\delta^2~~~~\hbox{\rm on}~~U_j\cap U_k,\leqno(5.7)$$
because the Hessian of this function is $O(\varepsilon(\delta))$ and
diam$(U_j\cap U_k)=O(\delta)$. The required estimates for jets will be
obtained by the following lemmas when $p\gg m$.

\tha Lemma 5.8|For every $f\in\hc_{U''_j}(mp\varphi_{j,m})$ with~$p>n$ and
every $L=(\ell_1\ld\ell_{p-n-1})\in\nb^{p-n-1}$, there are holomorphic
functions $g_L$ on $U''_j\cap U''_k$ such that
$$f=e^{-mph_{jk}}\sum_{L\in\nb^{p-n-1}}g_L\sigma_k^L,~~~~~
\sigma_k^L:=\prod_{1\le s\le p-n-1}\sigma_{k,\ell_s},$$
satisfying the $L^2$ estimates
$$\eqalign{
\int_{U''_j\cap U''_k}&\sum_L|g_L|^2|\sigma_k|^{-2(n+1)}d\lambda
\le C(\delta,m,p)\int_{U''_j}|f|^2e^{-2mp\varphi_{j,m}}d\lambda,\cr
C(\delta,m,p)&=(p-n)\exp\Big(2mp\Big(C_1\varepsilon(\delta)\delta^2+
C_2(\delta){\log m\over m}\Big)\Big).\cr}$$\fintha

\remarque Proof|By definition of $\varphi_{k,m}$ we have
$$\int_{U''_j\cap U''_k}|fe^{mph_{jk}}|^2|\sigma_k|^{-2p}d\lambda
=\int_{U''_j\cap U''_k}|f|^2e^{2mp(\re h_{jk}-\varphi_{k,m})}d\lambda.$$
On the other hand, inequalities (5.7) and (5.3) imply
$$\eqalign{
\re h_{jk}-\varphi_{k,m}&\le(\wt\gamma_j\circ\tau_j-\wt\gamma_k\circ\tau_k)
-\varphi_{k,m}+C_1\varepsilon(\delta)\delta^2\cr
&\le-\varphi_{j,m}+C_1\varepsilon(\delta)\delta^2
 +C_2(\delta){\log m\over m}~~~~\hbox{\rm on $U''_j\cap U''_k$}.\cr}$$
Therefore
$$\int_{U''_j\cap U''_k}|fe^{mph_{jk}}|^2|\sigma_k|^{-2p}d\lambda\le
{1\over p-n}C(\delta,m,p)\int_{U''_j}|f|^2e^{-2mp\varphi_{j,m}}d\lambda$$
and the desired functions $g_L$ follow from Skoda's $L^2$ existence
theorem [Sk~72b] (see Appendix, corollary A.5).\finpr

\tha Corollary 5.9|Let $U^{(3)}_j$ be a ball such that
$U'_j\compact U^{(3)}_j\compact U''_j$. For every
$f\in\hc_{U''_j}(mp\varphi_{j,m})$ with~$p>n$, there are holomorphic
functions $b_\ell$ on $U^{(3)}_j\cap U^{(3)}_k$ such that
$$\eqalign{
&f=e^{-mph_{jk}}\sum_{\ell\in\nb}b_\ell\sigma''_{k,\ell}~~~~
\hbox{\rm on~ $U^{(3)}_j\cap U^{(3)}_k$},\cr
\sum_{\ell\in\nb}~&\sup_{U^{(3)}_j\cap U^{(3)}_k}|b_\ell|^2
\le C'(\delta,m,p)\int_{U''_j}|f|^2e^{-2mp\varphi_{j,m}}d\lambda\cr}$$
with
$$C'(\delta,m,p)=(p-n)\exp\Big(2mp\Big(C_1\varepsilon(\delta)\delta^2+
C_2(\delta){\log m\over m}\Big)+C_3(\delta,m)\Big).$$\fintha

\remarque Proof|By Nadel's coherence theorem [Na~89] (see also
[De~90b], Lemma 4.4), the sheaf of germs of holomorphic functions $h$
such that $|h|^2|\sigma_k|^{ -2(n+1)}\in L^2_{\rm loc}$ is coherent
on~$U_k$. Let $v_1\ld v_N$ be generators of this sheaf on a
neighborhood of~$\smash{\ov{U}}\strut''_k$, chosen in such a way that
$\int_{U''_k}|v_s|^2|\sigma_k|^{-2(n+1)}d\lambda\le 1$ (note that the
covering depends on $\delta$ and that $\sigma_k$ depends on $m$, so
$N=N(\delta,m)$ is independent of~$p$). Since the functions $g_L$
constructed in Lemma~5.8 are sections of this sheaf on~$U''_j\cap
U''_k$, Cartan's theorem~A shows that there are holomorphic functions
$w_{L,1}\ld w_{L,N}$ with $g_L=\sum_s v_s w_{L,s}$ on $U''_j\cap
U''_k$, such that the family $(w_{L,s})_{1\le s\le N}$ depends
continuously on $g_L$ in the topology of uniform convergence on compact
subsets. In particular, there is a constant $C_4(\delta,m)$ independent
of $p$ such that
$$\sum_{1\le s\le N}~\sup_{U^{(3)}_j\cap U^{(3)}_k}|w_{L,s}|^2\le
C_4(\delta,m)\int_{U''_j\cap U''_k}|g_L|^2|\sigma_k|^{-2(n+1)}d\lambda.$$
From Lemma~5.8, we infer
$$\eqalign{
&f=e^{-mph_{jk}}\sum_{L,s}v_s w_{L,s}\sigma_k^L~~~~\hbox{\rm where}\cr
\sum_{|L|=p-n-1}&\int_{U''_k}\big|v_s\sigma_k^L\big|^2|\sigma_k|^{-2p}d\lambda
 =\int_{U''_k}|v_s|^2|\sigma_k|^{-2(n+1)}d\lambda\le 1.\cr}$$
Therefore, there are scalars $a_{L,s,\ell}\in\cb$ with
$\sum_{L,\ell}|a_{L,s,\ell}|^2\le 1$, such that
$$v_s\sigma_k^L=\sum_\ell a_{L,s,\ell}\sigma''_{k,\ell}$$
in terms of the basis $(\sigma''_{k,\ell})$. On $U^{(3)}_j\cap U^{(3)}_k$,
we finally get
$$f=e^{-mph_{jk}}\sum_{L,s,\ell}a_{L,s,\ell}w_{L,s}\sigma''_{k,\ell} 
   =e^{-mph_{jk}}\sum_\ell b_\ell\sigma''_{k,\ell}$$
where $b_\ell=\sum_{L,s}a_{L,s,\ell}w_{L,s}$ satisfies the estimate
$$\eqalign{
\sum_{\ell\in\nb}~\sup_{U^{(3)}_j\cap U^{(3)}_k}|b_\ell|^2
&\le \sum_{L,s,\ell}|a_{L,s,\ell}|^2
\sum_{L,s}~\sup_{U^{(3)}_j\cap U^{(3)}_k}|w_{L,s}|^2\cr
&\le N\,C_4(\delta,m)\sum_L
\int_{U''_j\cap U''_k}|g_L|^2|\sigma_k|^{-2(n+1)}d\lambda\cr
&\le N\,C_4(\delta,m)C(\delta,m,p)\int_{U''_j}
|f|^2e^{-2mp\varphi_{j,m}}d\lambda.\cr}$$
The last inequality is a consequence of Lemma~5.8. Corollary~5.9
follows.\finpr

\tha Corollary 5.10|Let $U^{(4)}_j$ be balls such that
$U'_j\compact U^{(4)}_j\compact U^{(3)}_j$. For every $\delta,m,p,q$
there is a sequence $\eta=(\eta_k)$ such that for all $j,k$
$$e^{2mp\wt\gamma_j\circ\tau_j}\sum_\ell|J^q\sigma''_{j,\ell}|_{(\eta)}^2\le
C''(\delta,m,p,q)e^{2mp\wt\gamma_k\circ\tau_k}\sum_\ell
|J^q\sigma''_{k,\ell}|_{(\eta)}^2~~~~\hbox{\rm on}~U^{(4)}_j\cap U^{(4)}_k,$$
with
$$C''(\delta,m,p,q)=2(p-n){q+n\choose n}\exp\Big(2mp\Big(2C_1
\varepsilon(\delta)\delta^2+C_2(\delta){\log m\over m}\Big)+
C_3(\delta,m)\Big).$$\fintha

\remarque Proof|Let $f\in\hc_{U''_j}(mp\varphi_{j,m})$. We compute the
$q$-jet $J^qf$ in local coordinates by means of the identity obtained
in Corollary~5.9. Leibnitz' rule implies
$$J^qf=e^{-mph_{jk}}\sum_{\ell\in\nb}b_\ell J^q\sigma''_{k,\ell}+
\sum_{\ell\in\nb}\sum_{0<|\alpha|<q}{1\over\alpha!}\partial^\alpha
(e^{-mph_{jk}}b_\ell)\,A^\alpha(J^q\sigma''_{k,\ell})$$
where $A^\alpha:J^q\oc_X\to J^q\oc_X$ denotes the (local) bundle morphism
which maps a jet $z\mapsto\sum a_\beta(z-x)^\beta$ to
$z\mapsto\sum a_\beta(z-x)^{\alpha+\beta}$ mod $O(|z-x|^{q+1})$.
In the natural decreasing filtration
$F^tJ^q\oc_X=\ker(J^q\oc_X\to J^{t-1}\oc_X)$
whose graded module is $\bigoplus S^tT^\star X$, we have
$A^\alpha(F^tJ^q\oc_X)\subset F^{t+|\alpha|}J^q\oc_X$. It follows from
this and the definition of $|~~|_{(\eta)}$ that the operators $A^\alpha$
have a very small $|~~|_{(\eta)}$-norm when the quotients $\eta_{t+1}/\eta_t$
are taken small enough. Moreover, for every holomorphic function $b$ on
$U^{(3)}_j\cap U^{(3)}_k$, the Cauchy inequalities yield constants
$C_5(\delta,\alpha,m,p,q)$ such that
$$\sup_{U^{(4)}_j\cap U^{(4)}_k}|\partial^\alpha(be^{-mph_{jk}})|\le
C_5(\delta,\alpha,m,p,q)\sup_{U^{(3)}_j\cap U^{(3)}_k}|b|.$$
Hence by taking the quotients $\eta_{t+1}/\eta_t$ small enough, we get
at each point of $U^{(4)}_j\cap U^{(4)}_k$ the estimate
$$\eqalign{|J^qf|_{(\eta)}^2
&\le 2e^{-2mp\re h_{jk}}\Big(\sum_{\ell\in\nb}~\sup_{U^{(3)}_j\cap U^{(3)}_k}
|b_\ell|^2\Big)\sum_{\ell\in\nb}|J^q\sigma''_{k,\ell}|_{(\eta)}^2\cr
&\le 2e^{-2mp\re h_{jk}}C'(\delta,m,p)||f||_{j,m,p}^2\sum_{\ell\in\nb}
|J^q\sigma''_{k,\ell}|_{(\eta)}^2,\cr}$$
thanks to corollary~5.9. We take the supremum over all elements $f$
in the unit ball of $\hc_{U''_j}(mp\varphi_{j,m})$ and combine with (5.7)
to obtain
$$e^{2mp\wt\gamma_j\circ\tau_j}\sup_{f\in B(1)}|J^qf|_{(\eta)}^2\le
2e^{2mpC_1\varepsilon(\delta)\delta^2}C'(\delta,m,p)e^{2mp\wt\gamma_k
\circ\tau_k}\sum_{\ell\in\nb}|J^q\sigma''_{k,\ell}|_{(\eta)}^2.$$
Now, for every point $x\in X$, the quadratic form $f\mapsto
|J^qf(x)|_{(\eta)}^2$ has rank equal to the rank ${q+n\choose n}$ of the
bundle $J^q\oc_X$. By the elementary Lemma 5.11 below, we thus get
$$\sum_{\ell\in\nb}|J^q\sigma''_{j,\ell}(x)|_{(\eta)}^2\le{q+n\choose n}
\sup_{f\in B(1)}|J^qf(x)|_{(\eta)}^2$$
and corollary 5.10 follows.\finpr

\tha Lemma 5.11|Let $Q$ be a semi-positive quadratic form of finite rank $r$
on a Hilbert space~$\hc$. Then, for any orthonormal basis $(e_\ell)$ of
$\hc$ we have
$$\sum_\ell Q(e_\ell)\le r\sup_{x\in B(1)}Q(x).$$\fintha

\remarque Proof|If $Q=|f|^2$ is the square of a linear form ($r=1$), both
sides coincide with the square of the norm of~$f$. If $Q=\sum_{1\le j\le r}
|f_j|^2$ has rank~$r$, then
$$\sum_\ell Q(e_\ell)\le\sum_{1\le j\le r}~\sup_{x\in B(1)}|f_j(x)|^2
\le r\sup_{x\in B(1)}Q(x).\eqno\finpr$$

\tha Lemma 5.12|Suppose that the concentric balls $U^{(5)}_k\compact U'_k$ of
radius $\delta/2$ still cover~$X$. Fix a nonnegative real number $c$ and take
$q=\lceil mpc\rceil$ equal to the smallest integer $\ge mpc$. Then there are
constants $C_6,C_7\ge 0$ such that the functions
$$w_j(x)=2mp\Big(\wt\gamma_j(\tau_j(x))+\sqrt{\varepsilon(\delta)}
\big(\delta^2-|\tau_j(x)|^2\big)+{1\over\sqrt{m}}\Big)+
\log\sum_\ell|J^q\sigma''_{j,\ell}|_{(\eta)}^2~~~~\hbox{\rm on}~U_j^{(4)}$$
enjoy the following properties for $p\gg m\gg 1/\delta$ and
$\eta_{t+1}/\eta_t\ll 1/p\,$:
\item{\rm(i)}$\displaystyle{i\over\pi}\partial\op w_j\ge
2mp\big(\gamma-C_6\sqrt{\varepsilon(\delta)}\,\omega\big)-2q(u+\varepsilon_q
\omega)$~ on $U^{(4)}_j$,
\item{\rm(ii)}$\displaystyle\psi(x)\le{1\over 2mp}w_j(x)\le
\sup_{|z-x|<r}\psi(z)+C_7\Big({|\log r|\over m}+r+m^{-1/2}+\delta^2\Big)$~
on $U'_j\,$,
\item{\rm(iii)}$w_j(x)\le\sup_{k\ne j,\,U'_k\ni x}w_k(x)~~~~\hbox{\rm on}~~
U^{(4)}_j\ssm U'_j\,$,
\item{\rm(iv)}$\displaystyle\Big(\nu(\psi,x)-c-{2n+1\over m}\Big)_+
\le{1\over 2mp}\nu(w_j,x)\le\big(\nu(\psi,x)-c\big)_+$~ on $U^{(4)}_j$.\fintha

\remarque Proof|(i) We have ${i\over\pi}\partial\op(\wt\gamma_j\circ\tau_j)
\ge\gamma-2\varepsilon(\delta)\omega$ by (2.8). On the other hand, Lemma~5.4
implies
$${i\over 2\pi}\partial\op\log|J^q\sigma''_{j,\ell}|^2_{(\eta)}\ge
-{\langle c(J^q\oc_X)\,J^q\sigma''_{j,\ell}\,,J^q\sigma''_{j,\ell}\rangle
\over|J^q\sigma''_{j,\ell}|_{(\eta)}^2}\ge -q(u+\varepsilon_q\omega).$$
These inequalities imply (i).

(ii) By definition of $|~~|_{(\eta)}$, we have $|J^q\sigma''_{j,\ell}|_{(\eta)}
\ge |\sigma''_{j,\ell}|$. Proposition~3.1~(i) applied first to
$\varphi_j=\psi-\wt\gamma_j\circ\tau_j$ and then to
$\log|\sigma_j|=m\varphi_{j,m}$ implies the existence of $C_8,C_9\ge 0$
such that
$$\eqalign{
\varphi_j(x)-{C_8\over m}&\le{1\over m}\log|\sigma_j(x)|\le
\sup_{|z-x|<r}\varphi_j(z)+{1\over m}\log{C_8\over r^n},\cr
\log|\sigma_j(x)|-{C_9\over p}&\le{1\over p}\log|\sigma''_j(x)|\le
\sup_{|z-x|<r}\log|\sigma_j(z)|+{1\over p}\log{C_9\over r^n}.\cr}$$
Thus on $U'_j$ we have
$$\eqalign{
w_j&\ge 2mp\,\wt\gamma_j\circ\tau_j+2\log|\sigma''_j|\cr
&\ge 2mp\Big(\wt\gamma_j\circ\tau_j+\varphi_j+{1\over\sqrt{m}}-{C_8\over m}
 -{C_9\over mp}\Big)\ge 2mp\,\psi\cr}$$
for $p\gg m\gg 1$. In the opposite direction, we find
$${1\over mp}\log|\sigma''_j(x)|\le
\sup_{|z-x|<2r}\varphi_j(z)+C_{10}{|\log r|\over m}$$
and therefore
$${1\over 2mp}\log\sum_\ell|J^q\sigma''_{j,\ell}(x)|^2_{(\eta)}\le
\sup_{|z-x|<2r}\varphi_j(z)+C_{10}{|\log r|+1\over m}$$
for small $\eta$ (the left hand side decreases to ${1\over mp}\log
|\sigma''_j(x)|$ when all $\eta_t$ decrease to~$0$, and we can apply
Dini's lemma since the right hand side is continuous).
The right hand inequality in (ii) follows by adding $\wt\gamma_j\circ\tau_j+
\sqrt{\varepsilon(\delta)}\delta^2+m^{-1/2}$ to this.

(iii) As ${q+n\choose n}\le (q+1)^n\le(mpc+2)^n$, corollary~(5.10) gives
$$\eqalign{
\Big|\Big(\wt\gamma_j\circ\tau_j
&{}+{1\over 2mp}\log\sum_\ell|J^q\sigma''_{j,\ell}|^2_{(\eta)}\Big)-
\Big(\wt\gamma_k\circ\tau_k+{1\over 2mp}
\log\sum_\ell|J^q\sigma''_{k,\ell}|^2_{(\eta)}\Big)\Big|\cr
&\le 2C_1\varepsilon(\delta)\delta^2+C_2(\delta){\log m\over m}+
{1\over 2mp}\big(C_3(\delta,m)+\log 2p(mpc+2)^n\big).\cr}$$
When we take $p\gg m\gg 1/\delta$, this upper bound is less than
$3C_1\varepsilon(\delta) \delta^2$. On the other hand, the additional terms
$\sqrt{\varepsilon(\delta)}(\delta^2-|\tau_j(x)|^2)$ create a
discrepancy at least equal to ${3\over 4}\sqrt{\varepsilon(\delta)}\delta^2$
when $x\in (U^{(4)}_j\ssm U'_j)\cap U^{(5)}_k$. This is sufficient to
compensate $3C_1\varepsilon(\delta)\delta^2$ and (iii) follows.

(iv) A combination of inequalities (5.2) and (5.6) gives
$$\nu(\psi,x)-{2n\over m}\le\nu(\varphi_{j,m,p}\,,x)\le\nu(\psi,x)
~~~~\hbox{\rm on}~~U''_j,$$
where $2mp\varphi_{j,m,p}=\log\sum_\ell|\sigma''_{j,\ell}|^2$. The Lelong
numbers of $\log\sum_\ell|J^q\sigma''_{j,\ell}|^2_{(\eta)}$ are shifted
down by $2q$ with $mpc\le q<mpc+1$. Hence we get
$$\nu(\psi,x)-{mpc+1\over mp}-{2n\over m}
\le{1\over 2mp}\nu(w_j,x)\le\big(\nu(\psi,x)-c\big)_+$$
and (iv) is proved.\finpr

\remarque Proof of Theorem 1.1|We use Lemma 3.5 to glue the functions
$w_j$ of Lemma~5.12 into a single function $w=\log\sum\theta_j^2e^{w_j}$,
by means of cut-off functions $\theta_j$ such that $\theta_j=1$ on $U'_j$
and ${\rm Supp}\,\theta_j\subset U^{(4)}_j$. We then have a loss of
positivity $-C_{11}(\delta)\omega$ in the Hessian of $w$ with respect
to~5.12~(i). Since the differences $|w_j-w_k|$ are bounded, we get
$\nu(w,x)=\nu(w_j,x)$ on~$U^{(4)}_j$. Also, the obvious inequalities
$$w_j(x)\le w(x)\le\sup_{U^{(4)}_k\ni x}w_k(x)+\log N(\delta)~~~~
\hbox{\rm on}~~U'_j$$
(with $N(\delta)=$ maximal number of overlapping sets $U^{(4)}_k$)
show that 5.12~(ii) is essentially preserved for~$p\gg\log N(\delta)$.
Therefore $\Psi_{c,\delta,m,p}:={1\over 2mp}w$ satisfies on $X$ the
global estimates
$$\leqalignno{
&{i\over\pi}\partial\op\Psi_{c,\delta,m,p}\ge\gamma-c\,u
-2C_6\sqrt{\varepsilon(\delta)}\,\omega,&(5.13~\hbox{\rm i)}\cr
&\psi(x)\le\Psi_{c,\delta,m,p}(x)\le\sup_{|z-x|<r}\psi(z)+
C_7\Big({|\log r|\over m}+r+\delta\Big),&(5.13~\hbox{\rm ii)}\cr
&\Big(\nu(\psi,x)-c-{2n+1\over m}\Big)_+
\le\nu(\Psi_{c,\delta,m,p}\,,x)\le\big(\nu(\psi,x)-c\big)_+
&(5.13~\hbox{\rm iv)}\cr}$$
when $p\gg m\gg 1/\delta$. Moreover $\Psi_{c,\delta,m,p}$
has the same singularities as the logarithm of a sum of squares of
holomorphic functions, thus $\Psi_{c,\delta,m,p}$ is smooth on~$X\ssm
E_c(\psi)$.

In order to complete the proof of Theorem~1.1, we still have to organize
the functions $\Psi_{c,\delta,m,p}$ into a decreasing sequence
$(\psi_{c,k})_{k\ge 1}$ converging to $\psi$ and satisfying properties
1.1~(i,ii) (which are stronger than inequality (5.13~i)). The
arguments are almost purely formal. We first note
that for given $\delta$, $m$, $p$ there is a choice of $(\eta)$ which
works for all $c'\in[0,c]$ and $q'=\lceil mpc'\rceil\le q$. Then by
construction, we see that $w_j$, $w$ and $\Psi_{c',\delta,m,p}$ are
increasing in~$c'$. Fix a strictly decreasing sequence of continuous
functions $\lambda_k>0$ on $X$ such that $\lim_{k\to+\infty}\lambda_k(x)
=\nu(\psi,x)$ at every point. We select inductively a sequence
$\delta_\ell\downarrow 0$ and strictly increasing sequences $(m_\ell)$,
$(p_\ell)$ such that the above estimates (5.13) are satisfied and such that
for all $k=1\ld\ell$
$$\Psi_{(1-k/(\ell+1))c,\delta_{\ell+1},m_{\ell+1},p_{\ell+1}}+
{1\over\ell+1}<\Psi_{(1-k/\ell)c,\delta_\ell,m_\ell,p_\ell}+{1\over\ell}~~~~
\hbox{\rm on}~~A_{k,\ell},\leqno(5.14)$$
where $A_{k,\ell}:=\lambda_\ell^{-1}\big([0,(1-k/\ell)c]\big)$. This is
actually possible by Dini's lemma, because $A_{k,\ell}$ is a compact set
contained in $X\ssm E_{(1-k/\ell)c}(\psi)$: thus, the right hand side of
$(5.14)$ is smooth and $\ge\psi+1/\ell$ on $A_{k\ell}$, whereas the left
hand side converges to the smaller limit $\psi+1/(\ell+1)$
as $\delta_{\ell+1}\to 0$ and $m_{\ell+1},p_{\ell+1}\to+\infty$. For a
reason that will be apparent later, we also take $m_\ell\ge(2n+1)\ell/c$.
We claim that the (upper semi-continuous) regularized upper envelope
$$\psi_{c,k}=\Big(\sup_{\ell\ge k}\Psi_{(1-k/\ell)c,\delta_\ell,m_\ell,p_\ell}
+{1\over\ell}\Big)^\star$$
has all required properties. In fact, it is clear that the sequence
$(\psi_{c,k})_{k\ge 1}$ is\break decreasing and (5.13~ii) shows that it
converges to~$\psi$. Now, we have by definition
$\psi_{c,k}\ge\Psi_{(1-k/\ell)c,\delta_\ell,m_\ell,p_\ell}$ and therefore
$$\nu(\psi_{c,k},x)\le\nu(\Psi_{(1-k/\ell)c,\delta_\ell,m_\ell,p_\ell},x)
\le\big(\nu(\psi,x)-(1-k/\ell)c\big)_+$$
for all~$\ell\ge k$. This implies $\nu(\psi_{c,k},x)\le(\nu(\psi,x)-c)_+$
when $\ell$ tends to~$+\infty$. On the other hand, all terms in the supremum
have Lelong numbers at least equal to
$$\nu(\psi,x)-\Big(1-{k\over\ell}\Big)c-{2n+1\over m_\ell}\ge
\nu(\psi,x)-c$$
thanks to our choice $(2n+1)/m_\ell\le c/\ell$. By standard arguments,
we get therefore the converse inequality $\nu(\psi_{c,k},x)\ge\nu(\psi,x)-c$.
The monotonicity condition (5.14) implies
$$\psi_{c,k}=\sup_{k\le\ell\le s}\Psi_{(1-k/\ell)c,\delta_\ell,m_\ell,p_\ell}
+{1\over\ell}~~~~\hbox{\rm on}~~A_{k,s}$$
(note that $A_{k,\ell}$ is increasing in $\ell$ and that
$\bigcup_\ell A_{k,\ell}=X\ssm E_c(\psi)\,)$. Therefore the Hessian of
$\psi_{c,k}$ satisfies the weakest bound valid for each term in this
finite envelope, namely
$${i\over\pi}\partial\op\psi_{c,k}\ge
\gamma-(1-k/s)c\,u-2C_6\sqrt{\varepsilon(\delta_k)}\omega~~~~\hbox{\rm on}~~
\{\lambda_s<(1-k/s)c\}\subset A_{k,s}^\circ$$
thanks to (5.13~i). If $x\in X$ with $\lambda_k(x)<c$ is given, we select the
smallest $s>k$ such that $\lambda_s(x)<(1-k/s)c$. Then $(1-k/(s-1))c\le
\lambda_{s-1}(x)$ and we see that
$$(1-k/s)c\le\lambda_{s-1}(x)+kc/s(s-1)<\lambda_k(x)+c/k.$$
We infer from this
$${i\over\pi}\partial\op\psi_{c,k}\ge
\gamma-(\lambda_k+c/k)u-2C_6\sqrt{\varepsilon(\delta_k)}\omega
~~~~\hbox{\rm on}~~\{\lambda_k<c\}.$$
However, all terms in $\psi_{c,k}$ have a Hessian bounded below by
$\gamma-c\,u-2C_6\sqrt{\varepsilon(\delta_k)}\omega$, so we get in all cases
$${i\over\pi}\partial\op\psi_{c,k}\ge
\gamma-\min\{\lambda_k,c\}u
-\big(2C_6\sqrt{\varepsilon(\delta_k)}+O(1/k)\big)\omega
~~~~\hbox{\rm on}~~X.$$
Finally, as $\psi_{c,k}$ is a finite supremum on each set $A_{k,\ell}$ in
$X\ssm E_c(\psi)=\bigcup_\ell A_{k,\ell}$, we conclude that $\psi_{c,k}$
is continuous on $X\ssm E_c(\psi)$. Richberg's theorem~2.15 can be applied
to make $\psi_{c,k}$ smooth on~$X\ssm E_c(\psi)$. Theorem~1.1 is thus
completely proved.\finpr

\remarque Remark 5.15|By a further application of Proposition 3.7 to each
function $\psi_{c,k}$, one could achieve that $\psi_{c,k}$ has
the same singularities along $E_c(\psi)$ as a logarithm of a sum of
squares of holomorphic functions, divided by some large integer~$2m$.
Of course, the Lelong numbers are then rational numbers with
denominator~$m$, so one cannot expect in that case that condition 1.1~(iv)
holds exactly. Only a weaker condition of the type
$$(\nu(\psi,x)-c-1/k\big)_+\le\nu(\psi_{c,k},x)\le\big(\nu(\psi,x)-c\big)_+$$
will be obtained.

\titre 6. Application to numerically effective divisors|

We first prove the list of easy facts mentioned in the introduction about
pseudoeffective and nef cohomology classes.

\tha Proposition 6.1|Let $X$ be a compact complex manifold. Then
\item{\rm(i)} $H^{1,1}_{\rm psef}(X)$ and $H^{1,1}_{\rm nef}(X)$ are closed
convex cones in $H^{1,1}_{\partial\op}(X)$.
\item{\rm(ii)} $H^{1,1}_{\rm nef}(X)\subset H^{1,1}_{\rm psef}(X)$.
\item{\rm(iii)} If $X$ is K\"ahler, then $H^{1,1}_{\rm nef}(X)$ is the
closure of the K\"akler cone.
\item{\rm(iv)} For any $\{\alpha\}\in H^{1,1}_{\rm nef}(X)$ and any curve
$C$, then $\{\alpha\}\cdot C=\int_C\alpha\ge 0$.
\item{\rm(v)} If $X$ is projective and if $\{\alpha\}\in NS_\rb(X)$ is such
that $\{\alpha\}\cdot C\ge 0$ for every curve $C$, then $\{\alpha\}\in
H^{1,1}_{\rm nef}(X)$.
\item{\rm(vi)} If $X$ is projective, $H^{1,1}_{\rm psef}(X)\cap NS_\rb(X)$
is the closure of the cone of effective divisors in $NS_\rb(X)$.
\item{\rm(vii)} If $X$ is projective, $H^{1,1}_{\rm nef}(X)\cap NS_\rb(X)$
is the closure of the cone of ample divisors in $NS_\rb(X)$.
\smallskip\fintha

\remarque Proof|Unless otherwise specified, we take $\omega$ to be a
special hermitian metric such that $\partial\op(\omega^{n-1})=0$
(where $n=\dim X$). Such a metric always exists by Gauduchon [Ga~77].
Note the following useful fact: if $T$ is a closed positive
$(1,1)$-current, the total mass $\int_X T\wedge\omega^{n-1}$
only depends on the $\partial\op$-cohomology class of~$T$.

(i) Let $T_k$ be closed positive currents such that the classes
$\{T_k\}$ converge to a limit $\{\Theta\}$. Then $\int_X
T_k\wedge\omega^{n-1}$ converges to $\int_X\Theta\wedge\omega^{n-1}$.
In particular the sequence $T_k$ is bounded in mass, and therefore
weakly compact. If $T_{k_\nu}\to T$ is a weakly convergent subsequence,
the limit $T$ is a closed positive current, so $\{T\}=\{\Theta\}$ is
pseudoeffective. Therefore $H^{1,1}_{\rm psef}(X)$ is closed.
Similarly, if $\{\alpha_k\}$ converges to $\{\alpha\}$, we can select
smooth representatives $\beta_k\in\{\alpha\}-\{\alpha_k\}$ converging
to $0$ in the $C^\infty$ topology. Fix $\varepsilon>0$.
If $\{\alpha_k\}$ is nef, there is a smooth representative
$\alpha_{k,\varepsilon}$ such that $\alpha_{k,\varepsilon}\ge-\varepsilon
\omega$. For $k$ large, we have $\beta_k\ge-\varepsilon\omega$, thus
$\{\alpha\}=\{\alpha_k\}+\{\beta_k\}$ contains
$\alpha_{k,\varepsilon}+\beta_k\ge-2\varepsilon\omega$. Hence
$\{\alpha\}$ is nef and $H^{1,1}_{\rm nef}(X)$ is closed.

(ii) Let $\alpha_\varepsilon\in\{\alpha\}$ be a smooth form such that
$\alpha_\varepsilon \ge-\varepsilon\omega$. Then
$\alpha_\varepsilon+\varepsilon\omega\ge 0$ and the mass
$\int_X(\alpha_\varepsilon+\varepsilon\omega)\wedge\omega^{n-1}=
\int_X\alpha\wedge\omega^{n-1}+\varepsilon\int_X\omega^n$ is bounded.
Therefore the family $\alpha_\varepsilon+\varepsilon\omega$ admits a
weakly convergent subsequence in the sense of currents. If $T$ is the
limit, then $T$ is closed positive and $\{\alpha\}=\{T\}$, so
$\{\alpha\}$ is pseudoeffective.

(iii) Obviously, the K\"ahler cone is contained in $H^{1,1}_{\rm
nef}(X)$. Conversely, if $\omega$ is a K\"ahler metric and if
$\alpha_\varepsilon\in\{\alpha\}$ is a smooth representative with
$\alpha_\varepsilon\ge-\varepsilon\omega$, then
$\alpha_\varepsilon+2\varepsilon\omega$ is a K\"ahler form whose
cohomology class converges to~$\{\alpha\}$.

(iv) We have $\int_C\alpha=\int_C\alpha_\varepsilon\ge-\varepsilon\int_C
\omega$ for every $\varepsilon>0$.

(v) If $X$ is projective, there is a basis of $NS_\rb(X)$ whose members
are ample divisor classes $\gamma_1\ld\gamma_r$ ($r=$ Picard number of~$X$).
Then for any $\{\alpha\}\in NS_\rb(X)$, there are arbitrary small numbers
$\varepsilon_j>0$ such that $\alpha+\sum\varepsilon_j\gamma_j$ is a
rational class. Let $p$ be an integer such that $p(\alpha+\sum\varepsilon_j
\gamma_j)$ is integral. Then $p(\alpha+\sum\varepsilon_j\gamma_j)$ is the
first Chern class of a line bundle~$L$. Suppose that $\{\alpha\}\cdot C\ge 0$
for every curve~$C$. We have
$$c_1(L)\cdot C\ge p\sum\varepsilon_j\{\gamma_j\}\cdot C\ge
\hbox{\rm const}\,\int_C\omega$$
and Kleiman's criterion shows that $L$ is ample. Hence $L$ is positive
and there is a representative $\alpha'\in\{\alpha\}$ such that
$p(\alpha'+\sum\varepsilon_j\gamma_j)>0$. In particular
$\alpha'\ge-\sum\varepsilon_j\gamma_j\ge-\varepsilon\omega$ for
$\varepsilon_j$ small enough, and we see that $\{\alpha\}$ is nef.

(vii) The proof of (v) actually shows that every nef class $\{\alpha\}\in
NS_\rb(X)$ can be approximated by ample rational classes $\{\alpha'+
\sum\varepsilon_j\gamma_j\}$.

(vi) By definition $H^{1,1}_{\rm psef}(X)\cap NS_\rb(X)$ contains all
classes of effective divisors. Conversely, if $T$ is a closed positive
current such that $\{T\}\in NS_\rb(X)$, the arguments of (v) show that
there are arbitrary small numbers $\varepsilon_j$, an integer $p$ and a
line bundle $L$ such that
$c_1(L)=p\big\{T+\sum\varepsilon_j\gamma_j\big\}$. This means that we
can equip $L$ with a singular metric (locally of the form
$e^{-\varphi}$ for some psh function~$\varphi$), such that the Chern
curvature form of $L$ is precisely
$$c(L)=p\Big(T+\sum\varepsilon_j\gamma_j\Big),$$
in particular $c(L)\ge\varepsilon\omega$ for some K\"ahler metric~$\omega$.
We can then apply H\"ormander's estimates (cf. corollary A.5) to construct
sections of $K_X\otimes L^m$ for $m\ge m_0$ large. Hence $c_1(K_X\otimes L^m)$
contains the class $\{D\}$ of an effective divisor. We infer
$$\{T\}={1\over p}c_1(L)-\sum\varepsilon_j\{\gamma_j\}=
{1\over mp}\big(\{D\}-c_1(K_X)\big)-{1\over p}\sum\varepsilon_j\{\gamma_j\}.$$
By taking $m$ large and $\varepsilon_j$ small, we see that
$\{T\}-{1\over mp}\{D\}$ can be made arbitrarily small.\finpr

We now use our Main Theorem to give a proof of corollaries 1.5 and~1.6.
The following elementary lemmas are needed.

\tha Lemma 6.2|Let $\{u\}$ be a nef cohomology class on $X$ such that
$c_1(\oc_{TX}(1))+\pi_X^\star\{u\}$ is nef over~$P(T^\star X)$. Let
$\omega$ be a hermitian metric on~$X$. Then for every $\varepsilon>0$
there exists a smooth form $u_\varepsilon\in\{u\}$ with $u_\varepsilon
\ge-\varepsilon\omega$ and a smooth hermitian metric on $\oc_{TX}(1)$
such that
$$c\big(\oc_{TX}(1)\big)+\pi_X^\star(u_\varepsilon+\varepsilon\omega)\ge 0.$$
\fintha

\remarque Proof|Let $h_\omega$ be the metric on $\oc_{TX}(1)$
associated to the metric of $TX$ defined by~$\omega$, and
let $c(\oc_{TX}(1))_\omega$ be the curvature of~$h_\omega$.
Since $c(\oc_{TX}(1))_\omega$ is positive along the fibers of
$\pi_X:P(T^\star X)\to X$, there is a constant $C\ge 1$ such that
$$\beta:=c\big(\oc_{TX}(1)\big)_\omega+\pi_X^\star(u+C\omega)>0~~~~
\hbox{\rm on}~~P(T^\star X).$$
Now $(1+\varepsilon)u_\varepsilon-\varepsilon u\in\{u\}$. Thus, by definition
of nefness, there is for each $\varepsilon>0$ a hermitian metric
$h_\varepsilon$ on $\oc_{TX}(1)$ such that the associated curvature form
satisfies
$$c\big(\oc_{TX}(1)\big)_\varepsilon+\pi_X^\star\big((1+\varepsilon)
u_\varepsilon-\varepsilon u\big)\ge-\varepsilon\beta.$$
Hence we get
$$c\big(\oc_{TX}(1)\big)_\varepsilon+\varepsilon\,
c\big(\oc_{TX}(1)\big)_\omega+\pi_X^\star\big((1+\varepsilon)u_\varepsilon
+\varepsilon C\omega\big)\ge 0.$$
This implies that the curvature of the barycenter metric
$(h_\varepsilon h_\omega^\varepsilon)^{1/(1+\varepsilon)}$ satisfies
$$c\big(\oc_{TX}(1)\big)+\pi_X\Big(u_\varepsilon+{\varepsilon C\over
1+\varepsilon}\omega\Big)\ge 0.$$
The lemma follows by replacing $\varepsilon$ with $\varepsilon/C$.\finpr

\tha Lemma 6.3|Let $T$ be a closed positive $(1,1)$-current which is smooth
on $X\ssm\Sigma$ for some finite subset~$\Sigma\subset X$. Then
$\{T\}$ is nef.\fintha

\remarque Proof|For all $x_j\in\Sigma$, fix disjoint small coordinate
neighborhoods $V_j$ of $x_j$ and write $T={i\over\pi}\partial\op\varphi_j$
with psh functions $\varphi_j\in C^\infty(\ov V_j\ssm\{x_j\})$.
Let $\theta_j$ be a cut-off function equal to $1$ near $x_j$ with support
in~$V_j$. Then $\theta_j(z)\log|z-x_j|$ has a Hessian which is bounded
below by $-C\omega$ for some positive constant~$C$. We denote by
$\max_\varepsilon=\max\star\rho_\varepsilon$ a regularized max function
and set
$$\varphi'_j(z)={\scriptstyle\max_\varepsilon}
\big(\varphi_j(z)+\varepsilon\theta_j(z)
\log|z-x_j|~,~-A\big)~~~~\hbox{\rm on}~~V_j,$$
with a large constant $A>0$. Then $\varphi'_j$ is smooth on~$\ov V_j$,
coincides with $\varphi_j$ near $\partial V_j$ and satisfies
${i\over\pi}\partial\op\varphi'_j\ge -C\varepsilon\,\omega$. Therefore
the current $T'$ equal to $T$ on $X\ssm\bigcup V_j$ and to
${i\over\pi}\partial\op\varphi'_j$ on $V_j$ is smooth. We have
$T'\in\{T\}$ and $T'\ge-C\varepsilon\,\omega$, thus $\{T\}$ is nef.\finpr

\remarque Proof of corollary 1.5|Let $T$ be a closed positive $(1,1)$-current
and let $b_n=\inf\{c>0\,;\,\codim E_c(T)\ge n\}$. Fix $u_\varepsilon$ as
in Lemma 6.2 and $c>b_n$. Then Theorem~1.1 applied with $\gamma=0$ and
$u=u_\varepsilon+\varepsilon\omega$ shows that there is $T_{c,k}\in\{T\}$
such that $T_{c,k}$ is smooth on $X\ssm E_c(T)$ and
$$T_{c,k}\ge-c(u_\varepsilon+\varepsilon\omega)-\varepsilon\omega,$$
thus $T_{c,k}+c\,u_\varepsilon\ge-(c+1)\varepsilon\omega$. Now,
$E_c(T)$ is a finite set. The arguments given in Lemma~6.3 show that
$\{T_{c,k}+c\,u_\varepsilon\}=\{T\}+c\{u\}$ is nef. Since this is true
for all $c>b_n$, we conclude that $\{T\}+b_n\{u\}$ is nef and
corollary~1.5 is proved. When $b_n=0$ we get in particular
the following much stronger version of Lemma~6.3 (the existence of
a nef class $\{u\}$ is not needed in that statement).\finpr

\tha Corollary~6.4|Let $T$ be a closed positive current which has zero
Lelong numbers except perhaps on a countable subset of~$X$. Then
$\{T\}$ is nef.\fintha

\remarque Proof of corollary 1.6|Let $X$ be a manifold in the Fujiki
class ${\cal C}$ such that $TX$ is nef. We know by Fujiki [Fu~78] that
$X$ has a smooth K\"ahler modification $\mu:\wt X\to X$. Let $\alpha$
be a K\"ahler metric on $\wt X$ and let $T=\mu_\star\alpha$ be the direct
image current. If $\omega$ is a hermitian metric on $X$, we have
$\alpha\ge\delta\,\mu^\star\omega$ on $\wt X$ for some small constant
$\delta>0$, thus by taking the direct image we get
$T=\mu_\star\alpha\ge\delta\omega$ on~$X$. Now Lemma 6.2 shows the existence
of smooth hermitian metrics on $\oc_{TX}(1)$ such that
$c(\oc_{TX}(1))+\pi_X^\star(\varepsilon\omega)\ge 0$. Fix $c>\max_X
\nu(T,x)$. The Main Theorem produces a smooth approximation $T_{c,k}$
of $T$ such that $T_{c,k}\ge\delta\omega-c\varepsilon\,
\omega-\varepsilon_k\omega$. Therefore $T_{c,k}\ge{\delta\over 2}\omega$
if $\varepsilon$ is small enough and $k$ large. Thus $T_{c,k}$ is
a K\"ahler metric and $X$ is K\"ahler. In particular, if $X$ is Moishezon
with $TX$ nef, we can take $\alpha$ to be the curvature form of an
ample line bundle $\oc(D)$ over~$\wt X$. Then $T$ and $T_{c,k}$ are
cohomologous to $\mu_\star([D])=[D']$ where $D'=\mu(D)$ is the image
of $D$ in~$X$. This implies that $\oc(D')$ is ample and therefore
$X$ is projective algebraic (we could also have applied the well-known
result of Moishezon [Mo~67] that every K\"ahler Moishezon manifold is
projective algebraic). The converse implications in corollary~1.6 are
trivial.\finpr

\titre 7. Application to self-intersection inequalities|

An interesting application of our regularization technique is the
estimation of a global intersection number $\int_X\{T_1\}\wedge
\ldots\wedge\{T_n\}$ of closed positive currents of type~$(1,1)$, when
the pointwise product $T_1\wedge\ldots\wedge T_n$ is not well defined
as a current. In fact, explicit examples show that $T_1\wedge\ldots
\wedge T_n$ cannot be defined in a reasonable way without additional
hypotheses (see e.g. [Ki~84] and [De~91]); in the case of a
self-intersection with $T_1=\ldots=T_n$ equal to the current of
integration over an exceptional divisor~$E$, it may even happen that
$\int_X\{E\}^n<0$. In order to overcome this difficulty, the idea is to
replace the currents $T_j$ by smooth approximations and to compute the
limit. Such a method has been already used in [De~90a] to get
asymptotic bounds for the dimensions of cohomology groups with values
in line bundles $\oc(kD)$, when $D$ is an effective divisor.

The self-intersection inequality~1.7 presented here is another direct
application which uses the full force of Theorem~1.1 (see also [De~90b]
for an independent proof in the special case where $X$ is projective
algebraic and $T\in H^2(X,\zb)$). In fact, inequality~1.7 can be
somewhat improved if the cohomology class $\{T\}$ is supposed to be
nef. In that case, it becomes possible to kill the singularities
``locally" on the manifold. More precisely, suppose we are given an
arbitrary subset $\Xi\subset X$. We define the sequence of {\sl
jumping values of $T$ with respect to $\Xi$} by
$$b_p=b_p(T,\Xi)=\inf\big\{c>0\,;\,
\codim\big(E_c(T),x\big)\ge p,~\forall x\in \Xi\big\},\leqno(7.1)$$
with the convention that a germ has codimension $>n$ if and only if it
is empty. Then $0=b_1\le b_2\le\ldots\le b_n\le b_{n+1}$ with
$b_{n+1}=\max_{x\in \Xi}\nu(T,x)<+\infty$, and for $c\in{}]b_p,b_{p+1}]$,
the set $E_c(T)$ has codimension $\ge p$ at every point of $\Xi$ and has
at least one component of codimension~$p$ exactly intersecting~$\Xi$.

\tha Theorem 7.2|If $\{T\}$ is nef, the inequalities of Theorem~{\rm 1.7}
hold with \hbox{$b_p=b_p(T,\Xi)$} and $Z_{p,k}$ associated to an arbitrary
subset $\Xi$ of~$X$.\eject\fintha

Before entering into the proof, let us recall a few basic facts
of current theory (see [Siu~74] and [De~91] for details).

\tha Lemma 7.3 {\rm (Siu's decomposition formula [Siu~74])}|Let
$\Theta$ be a closed positive current of bidimension $(p,p)$ and let
$Z_k$ be the collection of all $p$-dimensional components appearing
in the upperlevel sets $E_c(T)$. Then
$$\Theta=\sum\lambda_k[Z_k]+R$$
where $\lambda_k=\inf_{x\in Z_k}\nu(\Theta,x)$ is the generic Lelong number
of $\Theta$ along $Z_k$ and $R$ is a closed positive current such that
$\dim E_c(R)<p$ for every $c>0$.\fintha

We also need a few results about intersection theory of closed positive
currents. Let $T_1\ld T_q$ be closed positive currents of bidegree $(1,1)$ on
a complex manifold~$M$. We can write $T_j={i\over\pi}\partial\op
\varphi_j$ locally on every contractible Stein open subset $U\subset M$,
for some plurisubharmonic function~$\varphi_j$. If $\varphi$ is psh or
almost psh, we define the {\sl unbounded locus} $L(\varphi)$ to be the
set of points $x$ such that $\varphi$ is not bounded below near~$x$.

\tha Lemma 7.4 {\rm ([De~91])}|Let $\Theta$ be a closed positive
current of bidimension $(p,p)$. The wedge product
$T_1\wedge\ldots\wedge T_q\wedge\Theta$ is well defined as soon as
$$L(\varphi_{j_1})\cap\ldots\cap L(\varphi_{j_m})\cap{\rm Supp}\,\Theta$$
has zero Hausdorff measure ${\cal H}_{2p-2m+1}$ for every choice of indices
$j_1<\ldots<j_m$ in $\{1\ld q\}$. In particular, this is the case if
$L(\varphi_{j_1})\cap\ldots\cap L(\varphi_{j_m})$ is contained in
an analytic subset of dimension $\le p-m$. Moreover, the
product is weakly continuous with respect to monotone decreasing
sequences of plurisubharmonic functions $\varphi_j^k\to\varphi_j$
when $k$ tends to~$+\infty$.\fintha

The proof of this result is obtained essentially by using Bedford and
Taylor's definition of Monge-Amp\`ere operators (see [De~91]): we set
$i\partial\op\varphi\wedge \Theta:=i\partial\op(\varphi \Theta)$ when
$\Theta$ is a closed positive current and $\varphi$ is a locally bounded
psh function. Suitable integration by parts then give inequalities
of Chern-Levine-Nirenberg type and show that the product
$i\partial\op\varphi_1\wedge\ldots\wedge i\partial\op\varphi_q\wedge\Theta$
can be extended through the singularities of the potentials~$\varphi_j$.
Finally, when the product $i\partial\op\varphi\wedge\Theta$ is well
defined, we have at every point $x$
$$\nu(i\partial\op\varphi\wedge\Theta,x)\ge
\nu(i\partial\op\varphi,x)\,\nu(\Theta,x).\leqno(7.5)$$

\remarque Proof of Theorem 1.7|We argue by induction on~$p$.
For $p=1$, Siu's decomposition formula shows that
$$T=\sum\nu_{1,k}[Z_{1,k}]+R,$$
where $R$ is a closed positive current such that $\codim E_c(R)\ge 2$
for every~$c>0$. We have $R\ge T_{\rm abc}$ since the other part has
singular measures as coefficients. The result is thus true with~$\Theta_1=T$.

Now, suppose that $\Theta_{p-1}$ has been constructed. For $c>b_p$, the
current $T_{c,k}=\alpha+{i\over\pi}\partial\op\psi_{c,k}$ produced by
the Main Theorem is such that $T_{c,k}+c\,u+\varepsilon_k\omega\ge 0$
for some sequence $\varepsilon_k>0$ converging to~$0$, thanks to
(1.1$\,$i,$\,$ii). Since $L(\psi_{c,k})\subset E_c(T)$ and
$\codim E_c(T)\ge p$, Lemma~7.4 shows that
$$\Theta_{p,c,k}=\Theta_{p-1}\wedge(T_{c,k}+c\,u+\varepsilon_k\omega)$$
is a well defined closed positive current. As $\{T_{c,k}\}=\{T\}$, the
cohomology class $\{\Theta_{p,c,k}\}$
converges to $\{\Theta_{p-1}\}\cdot(\{T\}+c\{u\})$. In particular, the mass
$\int_X\Theta_{p,c,k}\wedge\omega^{n-p}$ remains uniformly bounded and the
family $(\Theta_{p,c,k})_{c\in{}]b_p,b_p+1],k\ge 1}$ is relatively
compact in the weak topology (observe that the K\"ahler assumption
has been used here). We define
$$\Theta_p=\lim_{c\to b_p+0}\lim_{k\to+\infty}\Theta_{p,c,k},$$
possibly after extracting some weakly convergent subsequence. Then
$\{\Theta_p\}=\{\Theta_{p-1}\}\cdot(\{T\}+b_p\{u\})$, and so
$\{\Theta_p\}=(\{T\}+b_1\{u\})\cdots(\{T\}+b_p\{u\})$. Moreover,
we have
$$\eqalign{
\nu(\Theta_p,x)&\ge\limsup_{c\to b_p+0}~\limsup_{k\to+\infty}~
\nu\big(\Theta_{p-1}\wedge(T_{c,k}+c\,u+\varepsilon_k\omega),x\big)\cr
&\ge\nu(\Theta_{p-1},x)\times
\limsup_{c\to b_p+0}~\limsup_{k\to+\infty}~\nu(T_{c,k},x)\cr
&\ge\nu(\Theta_{p-1},x)\big(\nu(T,x)-b_p\big)_+\cr}$$
by application of (1.1$\,$iv) and (7.5) (and by the fact that the Lelong
number of a weak limit is always at least equal to the limsup of the
Lelong numbers). By induction we thus get
$$\nu(\Theta_p,x)\ge
\big(\nu(T,x)-b_1\big)_+\ldots\big(\nu(T,x)-b_p\big)_+,$$
in particular, the generic Lelong number of $\Theta_p$ along $Z_{p,k}$
is at least equal to $(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)$. Siu's
decomposition formula implies
$$\Theta_p\ge\sum_{k\ge 1}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)\,[Z_{p,k}].$$
Since the right hand side is Lebesgue singular, the desired inequality
will be proved if we show in addition that
$$\Theta_{p,{\rm abc}}\ge(T_{\rm abc}+b_1u)\wedge\ldots
\wedge(T_{\rm abc}+b_pu),$$
or inductively, that $\Theta_{p,{\rm abc}}\ge\Theta_{p-1,{\rm abc}}
\wedge (T_{\rm abc}+b_pu)$. In order to do this, we simply have to
make sure that $\lim_{k\to+\infty}T_{c,k,{\rm abc}}=T_{\rm abc}$ almost
everywhere and to use induction again. But our arguments are not affected
if we replace $\psi_{c,k}$ by $\psi'_{c,k}=\max\{\psi,\psi_{c,k}-A_k\}$ with
$A_k$ converging quickly to~$+\infty$. It is then easy to show that a
suitable choice of $A_k$ gives $\smash{\lim(i\partial\op\psi'_{c,k})_{\rm abc}
=(i\partial\op\psi)_{\rm abc}}$ almost everywhere
(see Lemma~10.12 in [De~90b]).\finpr

\remarque Proof of Theorem 7.2|We apply a similar induction on~$p$.
When $\Xi\ne X$ and $c>b_p=b_p(T,\Xi)$, the current $T_{c,k}$  may have
singularities along some components of $E_c(T)$ of codimension $<p$
which do not intersect $\Xi$. In view of Lemma~7.3, this might create
trouble in the definition of $\Theta_{p,c,k}$. When $\{T\}$ is supposed
to be nef, we can modify the inductive construction of $\Theta_p$ as
follows. Let $\alpha_k \in\{T\}$ be a smooth representative such that
$\alpha_k\ge-\varepsilon_k\omega$. Then we have
$\alpha_k=\alpha+{i\over\pi}\partial\op\varphi_k$
for some smooth function $\varphi_k$. We set
$$\psi_{c,k,A}=\max\{\psi_{c,k}\,,\,\varphi_k-A\}$$
with a large constant~$A>0$, and replace $T_{c,k}$, $\Theta_{p,c,k}$ by
$$T_{c,k,A}=\alpha+{i\over\pi}\partial\op\psi_{c,k,A},~~~~~~
\Theta_{p,c,k,A}=\Theta_{p-1}\wedge(T_{c,k,A}+c\,u+\varepsilon_k\omega).$$
We have ${i\over\pi}\partial\op\psi_{c,k}\ge-\alpha-c\,u-
\varepsilon_k\omega$ by (1.1$\,$i,$\,$ii) and ${i\over\pi}\partial\op\varphi_k=
\alpha_k-\alpha\ge -\alpha-\varepsilon_k\omega$.\break Therefore
${i\over\pi}\partial\op\psi_{c,k,A}$ satisfies the same lower bound
as $\psi_{c,k}$ and we get the inequality
$T_{c,k,A}+c\,u+\varepsilon_k\omega\ge 0$
as before. Since the potential $\psi_{c,k,A}$ is bounded everywhere,
the product $\Theta_{p,c,k,A}$ is well defined on~$X$. We then define
$\Theta_{p,c,k}$ by extracting a weak limit
$$\Theta_{p,c,k}=\lim_{A\to+\infty}\Theta_{p,c,k,A}~~~~\hbox{\rm on}~X.$$
By definition of $b_p=b_p(T,\Xi)$, there is a neighborhood $V$ of $\Xi$
such that $E_c(T)\cap V$ only contains components of codimension $\ge p$
when~$c>b_p$. As $\lim_{A\to+\infty}\psi_{c,k,A}=\psi_{c,k}$, the
continuity result of Lemma~7.4 implies
$$\Theta_{p,c,k}=\Theta_{p-1}\wedge(T_{c,k}+c\,u+\varepsilon_k\omega)~~~~
\hbox{\rm on}~V.$$
The rest of the proof is identical to that of Th.~1.7.\finpr

In the case $b_n=0$, we get the following interesting complement to
Corol\-lary~6.4 by applying inequality 1.7 with $p=n$.

\tha Corollary 7.6|Let $T$ be a closed positive $(1,1)$-current which
has zero Lelong numbers except perhaps on a countable subset of~$X$.
Then we have
$$\int_X\{T\}^n\ge\sum_{x\in X}\nu(T,x)^n+\int_X T_{\rm abc}^n.$$\fintha

\titre A. Appendix: basic results on $L^2$ estimates|

We state here the basic $L^2$ existence theorems used in the above
sections, concerning $\op$ equations or holomorphic functions. The
first of these is the intrinsic manifold version of H\"ormander's $L^2$
estimates [H\"o~65,66], based on the Bochner-Kodaira-Nakano technique
(see also Andreotti-Vesentini [AV~65]).

\tha Theorem A.1|Let $L$ be a holomorphic line bundle on a weakly
pseudoconvex $n$-dimensional manifold~$X$ equipped with a K\"ahler
metric~$\omega$. Suppose that $L$ has a smooth hermitian metric
whose curvature form satisfies
$$2\pi\,c(L)+i\partial\op\varphi\ge A\,\omega$$
where $\varphi$ is an almost psh function and $A$ a positive continuous
function on~$X$. Then for every form $v$ of type $(n,q)$, $q\ge 1$, with
values in $L$, such that $\op v=0$ and
$$\int_X {1\over A}|v|^2e^{-\varphi}dV_\omega<+\infty,$$
there exists a form $u$ of type $(n,q-1)$ with values in $L$
such that $\op u=v$ and
$$\int_X|u|^2e^{-\varphi}dV_\omega\le{1\over q}
\int_X {1\over A}|v|^2e^{-\varphi}dV_\omega.$$\fintha

A {\sl weakly pseudoconvex manifold} is by definition a complex
manifold possessing a smooth weakly pseudoconvex exhaustion function
(examples: Stein manifolds, compact manifolds, the total space of a
Griffiths weakly negative vector bundle, $\ldots$). Suppose that
$\varphi$ has Lelong number $\nu(\varphi,x)=0$ at a given point~$x$.
Then for every $m$ the weight $e^{-m\varphi}$ is integrable in a small
neighborhood $V$ of~$x$ (see [Sk~72a]). Let $\theta$ be a cut-off
function equal to $1$ near $x$, with support in~$V$. Let $z$ be
coordinates and let $e$ be a local frame of $L$ on~$V$. For $\varepsilon$
small enough, the curvature form
$$2\pi\,c(L)+i\partial\op\big(\varphi(z)+2\varepsilon\theta(z)\log|z-x|\big)$$
is still positive definite. We apply A.1 to the bundle $L^m$
equipped with the corresponding weight $m(\varphi(z)+2\varepsilon
\theta(z)\log|z-x|)$, and solve the equation $\op u=v$ for the $(n,1)$-form
$v=\op(\theta(z)P(z)dz_1\wedge\ldots\wedge dz_n\otimes e^m)$ associated to
an arbitrary polynomial~$P$. The $L^2$ estimate shows that the solution $u$
has to vanish at order $\ge q+1$ at~$x$ where $q=[m\varepsilon]-n$,
hence
$$\theta(z)P(z)dz_1\wedge\ldots\wedge dz_n\otimes e^m-u(z)$$
is a holomorphic section of $K_X\otimes L^m$ with prescribed jet of
order $q$ at~$x$.

\tha Corollary A.2|Suppose that $2\pi\,c(L)+i\partial\op\varphi\ge
\delta\,\omega$ for some $\delta>0$. Let $x\in X$ be such that
$\nu(\varphi,x)=0$. Then there exists $\varepsilon>0$ such that the
sections in $H^0(X,K_X\otimes L^m)$ generate all jets of order
$\le m\varepsilon$ at $x$ for $m$ large.\finpr\fintha

The Ohsawa-Takegoshi estimates were needed in section~3 in the
following very special case ([Oh~88], Cor.$\,$2 p.$\,$266).

\tha Proposition A.3|Let $Y$ be a pure dimensional closed complex
submanifold of~$\cb^n$, let $\Omega$ be a bounded pseudoconvex open set
and let $\varphi$ be a plurisubharmonic function on~$\Omega$. Then for
any holomorphic function $f$ on $Y\cap\Omega$ with
$$\int_{Y\cap\Omega}|f|^2e^{-\varphi}dV_Y<+\infty,$$
there exists a holomorphic extension F to $\Omega$ such that
$$\int_{\Omega}|F|^2e^{-\varphi}dV\le
A\int_{Y\cap\Omega}|f|^2e^{-\varphi}dV_Y<+\infty.$$
Here $A$ depends only on $Y$ and on the diameter of~$\Omega$.\fintha

Finally, a crucial application of Skoda's $L^2$ estimates [Sk~72b] for
ideals of holomorphic functions was made in section~5:

\tha Theorem A.4|Let $\varphi$ be a plurisubharmonic function
on a pseudoconvex open set~$\Omega\subset\cb^n$ and let
$\sigma_1\ld \sigma_N$ be holomorphic functions on~$\Omega$
$($the sequence $\sigma_j$ can be infinite$)$.
Set $q=\min\{N-1,n\}$ and $|\sigma|^2=\sum|\sigma_j|^2$.
Then, for every holomorphic function $f$ on $\Omega$ such that
$$\int_\Omega|f|^2|\sigma|^{-2(q+1+\alpha)}e^{-\varphi}dV<+\infty,~~~~
\alpha>0,$$
there exist holomorphic functions $g_1\ld g_N$ on $\Omega$ such that
$f=\sum_{1\le j\le N}g_j\sigma_j$ and
$$\int_\Omega|g|^2|\sigma|^{-2(q+\alpha)}e^{-\varphi}dV\le
{\alpha+1\over\alpha}
\int_\Omega|f|^2|\sigma|^{-2(q+1+\alpha)}e^{-\varphi}dV<+\infty.$$
\fintha

\tha Corollary A.5|With the same notations, suppose that
$$\int_\Omega|f|^2|\sigma|^{-2(q+m+\alpha)}e^{-\varphi}dV<+\infty$$ for
some $\alpha>0$ and some integer $m\ge 1$. Then there exist holomorphic
functions $g_L$ for all $L=(\ell_1\ld\ell_m)\in\{1\ld N\}^m$ such that
$$\eqalign{
f=\sum_L g_L\sigma^L~~~~&\hbox{\rm with}~~\sigma^L=
\sigma_{\ell_1}\sigma_{\ell_2}\ldots\sigma_{\ell_m},\cr
\int_\Omega\sum_L|g_L|^2|\sigma|^{-2(q+\alpha)}e^{-\varphi}dV&\le
{\alpha+m\over\alpha}
\int_\Omega|f|^2|\sigma|^{-2(q+m+\alpha)}e^{-\varphi}dV<+\infty.\cr}$$
\fintha

\remarque Proof|Use induction on $m$: if the result is true for
$(m-1,\alpha+1)$ then $f=\sum_\Lambda g_\Lambda\sigma^\Lambda$ with
$\Lambda$ of length $m-1$, and each function $g_\Lambda$ can be written
$g_\Lambda=\sum_{\ell_m}g_L\sigma_{\ell_m}$ with $L=(\Lambda,\ell_m)$ and
$$\eqalignno{
&\int_\Omega\sum_{\ell_m}|g_L|^2|\sigma|^{-2(q+\alpha)}e^{-\varphi}dV\le
{\alpha+1\over\alpha}
\int_\Omega|G_\Lambda|^2|\sigma|^{-2(q+1+\alpha)}e^{-\varphi}dV<+\infty,\cr
&\int_\Omega\sum_\Lambda|g_\Lambda|^2|\sigma|^{-2(q+1+\alpha)}e^{-\varphi}dV
\le{\alpha+m\over\alpha+1}
\int_\Omega|f|^2|\sigma|^{-2(q+m+\alpha)}e^{-\varphi}dV<+\infty.&\finpr\cr}$$
\vskip20pt

\centerline{\bf References}
\vskip15pt
\eightpoint
\lettre KMM~87| 

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\vbox{\article
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\vskip30pt
\tenpoint\noindent(July 9, 1991)

\end

