\input demail.mac
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\def\er{{\cal E}}
\def\hc{{\cal H}}
\def\ic{{\cal I}}
\def\mr{{\cal M}}
\def\oc{{\cal O}}

\def\div{{\rm div}}
\def\Div{{\rm Div}}
\def\psh{{\rm Psh}}
\def\reg{{\rm reg}}
\def\sing{{\rm sing}}
\def\supp{{\rm Supp}}
\def\loc{{\rm loc}}
\def\gge{{\scriptscriptstyle\ge}}

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\titlea{~ Monge-Amp\`ere Operators, Lelong Numbers and
Intersection Theory}
\titlearunning{~ Lelong Numbers and Intersection Theory}
\vskip-55pt
{\noindent\hangindent3.2cm\hangafter-1
{\bf Jean-Pierre Demailly

}}

{\noindent\hangindent3.2cm\hangafter-4{\sl
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
URA 188 associ\'ee au C.N.R.S.\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\vskip15pt\noindent{\bf Abstract}
\vskip5pt\noindent\eightpoint
 This article is a survey on the theory of Monge-Amp\`ere
operators and Lelong numbers. The definition of complex Monge-Amp\`ere
operators is extended in such a way that wedge products of closed positive
currents become admissible in a large variety of situations; the only
basic requirement is that the polar set singularities have mutual
intersections of the correct codimension. This makes possible to develope
the intersection theory of analytic cycles by means of current theory
and Lelong numbers. The advantage of this point of view, in addition to
its wider generality, is to produce simpler proofs of previously known
results, as well as to relate some of these results to other questions
in analytic geometry or number theory. For instance, the generalized
Lelong-Jensen formula provides a useful tool for studying the location
and multiplicities of zeros of entire functions on $\bbbc^n$ or on a
manifold, in relation with the growth at infinity (Schwarz lemma type
estimates). Finally, we obtain a general self-intersection inequality
for divisors and positive $(1,1)$-currents on compact K\"ahler manifolds,
based on a singularity attenuation technique for quasi-plurisubharmonic
functions.
\vskip15pt\tenpoint
\noindent{\bf Contents}\vskip5pt
{\eightpoint\noindent
0. Introduction\dotfill p. 2\break
1. Definition of Monge-Amp\`ere Operators\dotfill p. 5\break
2. Case of Unbounded Plurisubharmonic Functions\dotfill p. 12\break
3. Generalized Lelong Numbers\dotfill p. 20\break
4. The Lelong-Jensen Formula\dotfill p. 26\break
5. Comparison Theorems for Lelong Numbers\dotfill p. 31\break
6. Siu's Semicontinuity Theorem\dotfill p. 40\break
7. Transformation of Lelong Numbers by Direct Images\dotfill p. 51\break
8. A Schwarz Lemma. Application to Number Theory\dotfill p. 58\break
9. Global Intersection Class and Self-intersection\dotfill p. 63\break
References\dotfill p. 68
}\vfill\eject

\tenpoint

\titleb{0. Introduction}
This contribution is a survey article on the theory of Lelong numbers,
viewed as a tool for studying intersection theory by complex differential
geometry. We have not attempted to make an exhaustive compilation
of the existing literature on the subject, nor to present a complete
account of the state-of-the-art. Instead, we have tried
to present a coherent unifying frame for the most basic results of
the theory, based in part on our earlier works [De1,2,3,4] and on Siu's
fundamental work~[Siu]. To a large extent, the asserted results are
given with complete proofs, many of them substantially shorter and simpler
than their original counterparts. We only assume that the reader has some
familiarity with differential calculus on complex manifolds and with the
elementary facts concerning analytic sets and plurisubharmonic functions.
The reader can consult Lelong's books [Le2,3] for an introduction to
the subject. Most of our results still work on arbitrary complex
analytic spaces, provided that suitable definitions are given for
currents, plurisubharmonic functions, etc, in this more general situation.
We have refrained ourselves from doing so for simplicity of exposition;
we refer the reader to [De3] for the technical definitions required in the
context of analytic spaces.

Let us first recall a few basic definitions. A {\sl current} of degree
$q$ on an oriented differentiable manifold $M$ is simply a differential
$q$-form $\Theta$ with distribution coefficients. Alternatively, a
current of degree $q$ is an element $\Theta$ in the dual space
$\dr'_p(M)$ of the space $\dr_p(M)$ of smooth differential forms of
degree $p=\dim M-q$ with compact support; the duality pairing is given by
$$\langle\Theta,\alpha\rangle=\int_M\Theta\wedge\alpha,~~~~\alpha\in\dr_p(M).
\leqno(0.1)$$
A basic example is the {\sl current of integration} $[S]$ over a compact
oriented submanifold $S$ of $M\,$:
$$\langle[S],\alpha\rangle=\int_S\alpha,~~~~{\rm deg}\alpha=p=\dim_\bbbr S.
\leqno(0.2)$$
Then $[S]$ is a current with measure coefficients, and
Stokes' formula shows that $d[S]=(-1)^{q-1}[\partial S]$, in particular
$d[S]=0$ if $S$ has no boundary. Because of this example, the integer $p$
is said to be the dimension of~$\Theta$ when $\Theta\in\dr_p'(M)$. The
current $\Theta$ is said to be {\sl closed} if $d\Theta=0$.

On a complex manifold $X$, we have similar notions of
bidegree and bidimension. According to Lelong [Le1], a current $T$
of bidimension $(p,p)$ is said to be {\sl (weakly) positive} if
for every choice of smooth $(1,0)$-forms $\alpha_1\ld\alpha_p$ on~$X$
the distribution
$$T\wedge\ii\alpha_1\wedge\ovl\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\ovl\alpha_p~~~~\hbox{\rm is a positive measure.}\leqno(0.3)$$
Then, the coefficients $T_{I,J}$ of $T$ are complex measures, and up
to constants, they are dominated by the trace measure $\sum T_{I,I}$
which is positive. With every closed analytic set $A\subset X$ of pure
dimension $p$ is associated a current of integration
$$\langle [A],\alpha\rangle=\int_{A_\reg}\alpha,~~~~\alpha\in\dr_{p,p}(X),
\leqno(0.4)$$
obtained by integrating over the regular points of~$A$. It is easy to see
that $[A]$ is positive. Lelong~[Le1] has shown that $[A]$ has locally finite
mass near $A_\sing$ and that $[A]$ is closed in~$X$. This last result can be
seen today as a\break
consequence of the Skoda-El Mir extension theorem ([EM], [Sk3];
also~[Sib]).

\theorem{0.5}{Let $E$ be a closed complete pluripolar set in $X$, and
let $\Theta$ be a closed positive current on $X\ssm E$ such that the
coefficients $\Theta_{I,J}$ of $\Theta$ are measures with locally finite
mass near~$E$. Then the trivial extension $\wt\Theta$ obtained by extending
the measures $\Theta_{I,J}$ by $0$ on $E$ is still closed.}

A {\sl complete pluripolar} set is by definition a set $E$ such that there is
an open covering $(\Omega_j)$ of $X$ and plurisubharmonic functions $u_j$ on
$\Omega_j$ with $E\cap\Omega_j=u_j^{-1}(-\infty)$. Any (closed) analytic
set is of course complete pluripolar. Lelong's result $d[A]=0$
is obtained by applying the El Mir-Skoda theorem to $\Theta=[A_\reg]$
on $X\ssm A_\sing$. Another interesting consequence is

\corollary{0.6}{Let $T$ be a closed positive current on $X$ and let
$E$ be a complete pluripolar set. Then $\bbbone_E T$ and
$\bbbone_{X\ssm E}T$ are closed positive currents. In fact
$\bbbone_E T=T\ssm \wt T$, where $\wt T$ is the trivial extension
of $T_{\restriction X\ssm E}$ to~$X$.}

The other main tool used in this paper is the theory of plurisubharmonic
functions. If $u$ is a plurisubharmonic function on $X$, we can associate with
$u$ a closed positive current $T=\ii\partial\ovl\partial u$ of bidegree~$(1,1)$.
Conversely, every closed positive current of bidegree $(1,1)$ can be
written under this form if $H^2_{DR}(X,\bbbr)=H^1(X,\oc)=0$. In the special
case $u=\log|F|$ with a non zero holomorphic function $F\in\oc(X)$,
we have the important {\sl Lelong-Poincar\'e} equation
$${\ii\over\pi}\partial\ovl\partial\log|F|=[Z_F],\leqno(0.7)$$
where $Z_F=\sum m_jZ_j$, $m_j\in\bbbn$, is the zero divisor of~$F$ and
$[Z_F]=\sum m_j[Z_j]$ is the associated current of integration.

Our goal is to develope the intersection theory of analytic cycles from
this point of view. In particular, we would like to define the wedge
product $T\wedge\ii\partial\ovl\partial u_1\wedge\ldots\wedge
\ii\partial\ovl\partial u_q$ of a closed positive current $T$ by
``generalized" divisors $\ii\partial\ovl\partial u_j$. In general
this is not possible, because measures cannot be multiplied. However,
we will show in sections 1,2 that Monge-Amp\`ere operators of this type
are well defined as soon as the set of poles of the $u_j$'s have
intersections of sufficiently low dimension. The proof rests on a
procedure due to Bedford and Taylor~[B-T1], [B-T2] and consists mostly in
rather simple integration by parts. In spite of its simple nature, this
result seems to be new.

Then, following [De2,4], we introduce the generalized Lelong numbers of
a closed positive current $T\in\dr'_{p,p}(X)$ with respect to a
plurisubharmonic weight~$\varphi$. Under suitable exhaustivity conditions
for~$\varphi$, we define $\nu(T,\varphi)$ as the residue
$$\nu(T,\varphi)=\int_{\varphi^{-1}(\infty)}T\wedge\Big({\ii\over 2\pi}
\partial\ovl\partial\varphi\Big)^p.\leqno(0.8)$$
The standard Lelong number $\nu(T,x)$ corresponds to the ``isotropic" weight
$\varphi(z)=\log|z-x|\,$; it can also be seen as the euclidean density of
$T$ at~$x$, when $T$ is compared to the current of integration over a
$p$-dimensional vector subspace in~$\bbbc^n$. However the generalized
definition is more flexible and allows us to give very simple proofs of
several basic properties: in particular, the Lelong number $\nu(T,x)$
does not depend on the choice of coordinates, and coincides with
the algebraic multiplicity in the case of a current of integration
$T=[A]$ (Thie's theorem [Th]). These facts are obtained as a consequence
of a comparison theorem for the Lelong numbers $\nu(T,\varphi)$ and
$\nu(T,\psi)$ associated with different weights.

Next, we prove Siu's semicontinuity theorem in this general setting: if
$\varphi_y$ is a family of plurisubharmonic functions on $X$ depending
on a parameter $y\in Y$, such that $\varphi(x,y):=\varphi_y(x)$ is
plurisubharmonic and satisfies some natural exhaustivity and continuity
conditions, then $y\mapsto\nu(T,\varphi_y)$ is upper semicontinuous
with respect to the (analytic) Zariski topology. Explicitly, the
upperlevel sets
$$E_c(T)=\{y\in Y\,;\,\nu(T,\varphi_y)\ge c\},~~~~c>0\leqno(0.9)$$
are analytic in~$Y$. The result of [Siu] concerning ordinary
Lelong numbers is obtained for $\varphi(x,y)=\log|x-y|$, but the above
result allows much more general variations of the weight. The proof
uses ideas of Kiselman [Ki$\,$1,2] and rests heavily on $L^2$ estimates
for $\ovl\partial$, specifically on the H\"ormander-Bombieri-Skoda
theorem ([H\"o], [Bo], [Sk2]).

Next, following [De2], we investigate the behaviour of Lelong numbers
under direct images by proper holomorphic maps. Let $T$ be a closed
positive current of bidimension $(p,p)$ on~$X$, let $F:X\to Y$ be a
proper and finite analytic map and let $F_\star T$ be the direct image
of $T$ by~$F$. We prove inequalities of the type
$$\leqalignno{
\nu(F_\star T,y)&\ge\sum_{x\in\supp\,T\cap F^{-1}(y)}\mu_p(F,x)\,\nu(T,x),
&(0.10)\cr
\nu(F_\star T,y)&\le\sum_{x\in\supp\,T\cap F^{-1}(y)}\ovl\mu_p(F,x)\,\nu(T,x),
&(0.10')\cr}$$
where $\mu_p(F,x)$ and $\ovl\mu_p(F,x)$ are suitable multiplicities
attached to $F$ at each point. In case $p=\dim X$, the multiplicity
$\mu_p(F,x)$ coincides with the one introduced by Stoll~[St]
(see also Draper~[Dr]).

As an application of these inequalities, we prove a general Schwarz
lemma for entire functions in $\bbbc^n$, relating the growth at
infinity of such a function and the location of its zeros ([De1]). The
proof is essentially based on the Lelong-Jensen formula. Finally, we
use the Schwarz lemma to derive a simple proof of Bombieri's theorem
[Bo] on algebraic values of meromorphic maps satisfying algebraic
differential equations.

The above techniques are also useful for relating local intersection
invariants to global ones, e.g. intersection numbers of analytic cycles
in compact K\"ahler manifolds. In that case, it is very important to
consider self-intersections or situations with excess dimension of
intersection. Such situations can also be handled by the Monge-Amp\`ere
techniques, thanks to a general regularization process for closed
positive~$(1,1)$-currents on compact manifolds~[De7]. The smooth
approximations of a positive current are no longer positive, but they
have a small negative part depending on the curvature of the tangent
bundle of the ambient manifold. As an application, we obtain a general
self-intersection inequality giving a bound for the degree of the strata
of constant multiplicity in an effective divisor~$D$, in terms of a
polynomial in the cohomology class $\{D\}\in H^2_{DR}(X,\bbbr)$.
This inequality can be seen as a generalization of the usual bound
$d(d-1)/2$ for the number of multiple points of a plane curve of
degree~$d$. Other applications are presented in [De5,6]. 

This paper is an expanded version of a course made in Nice in July~1989,
at a summer school on Complex Analysis organized by the ICPAM. Since
then, the author has benefited from many valuable remarks made by
several mathematicians, in particular M.S.~Narasimhan at the Tata
Institute, Th.~Peternell and M.~Schneider at Bayreuth University,
and Z.~B\l ocki, S.~Ko\l odziej, J.~Siciak and T.~Winiarski
at the Jagellonian University in Cracow. The author expresses his warm
thanks to these institutions for their hospitality.

\titleb{1. Definition of Monge-Amp\`ere Operators}
Let $X$ be a $n$-dimensional complex manifold. We denote by $d=d'+d''$
the usual decomposition of the exterior derivative in terms of its $(1,0)$
and $(0,1)$ parts, and we set
$$d^c={1\over2\ii\pi}(d'-d'').$$
It follows in particular that $d^c$ is a real operator, i.e. $\ovl{d^cu}=
d^c\ovl u$, and that $dd^c={\ii\over\pi}d'd''$. Although not quite standard,
the $1/2\ii\pi$ normalization is very convenient for many purposes, since
we may then forget $2\pi$ almost everywhere (e.g. in the Lelong-Poincar\'e
equation (0.7)). In this context, we have the following integration by part
formula.

\statement{(1.1) Formula}{Let $\Omega\compact X$ be a smoothly bounded open
set in $X$ and let $f,g$ be forms of class $C^2$ on $\smash{\ovl\Omega}$ of
pure bidegrees $(p,p)$ and $(q,q)$ with~\hbox{$p+q=n-1$}. Then
$$\int_\Omega f\wedge dd^c g-dd^c f\wedge g=
\int_{\partial\Omega} f\wedge d^c g-d^c f\wedge g.$$}

\proof{By Stokes' theorem the right hand side is the integral over $\Omega$ of
$$d(f\wedge d^c g-d^c f\wedge g)= f\wedge dd^c g-dd^c f\wedge g+
(df\wedge d^c g+d^c f\wedge dg).$$
As all forms of total degree $2n$ and bidegree $\ne(n,n)$ are zero, we get
$$df\wedge d^cg={1\over 2\ii\pi}(d''f\wedge d'g-d'f\wedge d''g)=-d^cf\wedge
dg.\eqno{\sq}$$}

Let $u$ be a plurisubharmonic function on $X$ and let $T$ be a closed
positive current of bidimension $(p,p)$, i.e. of bidegree $(n-p,n-p)$. 
Our desire is to define the wedge product $dd^cu\wedge T$ even when
neither $u$ nor $T$ are smooth.  A priori, this product does not make
sense because $dd^cu$ and $T$ have measure coefficients and measures
cannot be multiplied.  The discussion made in section~9 shows that
there is no way of defining $dd^cu\wedge T$ as a closed positive current
without further hypotheses (see also [Ki3] for interesting
counterexamples). Assume however that $u$ is a {\sl locally
bounded} plurisubharmonic function. Then the current $uT$ is well
defined since $u$ is a locally bounded Borel function and $T$ has
measure coefficients.  According to Bedford-Taylor [B-T2] we define
    $$dd^cu\wedge T=dd^c(uT)$$
where $dd^c(~~)$ is taken in the sense of distribution (or current) theory.

\statement{(1.2) Proposition}{The wedge product $dd^c u\wedge T$ is 
again a closed positive current.}

\proof{The result is local. In an open set $\Omega\subset\bbbc^n$,
we can use convolution with a family of regularizing kernels to
find a decreasing sequence of smooth plurisubharmonic
functions $u_k=u\star\rho_{1/k}$ converging pointwise to $u$.
Then $u\le u_k\le u_1$ and Lebesgue's dominated 
convergence theorem shows that $u_k T$ converges weakly to $uT\,$;
thus $dd^c(u_k T)$ converges weakly to $dd^c(u T)$ by the weak
continuity of differentiations. However, since $u_k$ is smooth,
$dd^c(u_k T)$ coincides with the product $dd^c u_k\wedge T$ in its usual
sense. As $T\ge0$ and as $dd^cu_k$ is a positive $(1,1)$-form, we have 
$dd^c u_k\wedge T\ge 0$, hence the weak limit $dd^c u\wedge T$ is 
$\ge 0$ (and obviously closed).\qed}

Given locally bounded plurisubharmonic functions $u_1\ld u_q$, we define
inductively
$$dd^c u_1\wedge dd^cu_2\wedge\ldots\wedge dd^c u_q\wedge T=
dd^c(u_1dd^c u_2\ldots\wedge dd^c u_q\wedge T).$$
By (1.2) the product is a closed positive current. In particular, when $u$ is
a locally bounded plurisubharmonic function, the bidegree $(n,n)$ current
$(dd^cu)^n$ is well defined and is a positive measure. If $u$ is of class $C^2$,
a computation in local coordinates gives
$$(dd^cu)^n=\det\big({\partial^2u\over\partial z_j\partial\ovl z_k}\Big)
\cdot{n!\over\pi^n}\,\ii dz_1\wedge d\ovl z_1\wedge\ldots\wedge
\ii dz_n\wedge d\ovl z_n.$$
The expression ``Monge-Amp\`ere operator" classically refers to the
non-linear partial differential operator
$u\longmapsto\det(\partial^2u/\partial z_j\partial\ovl z_k)$. By extension,
all operators $(dd^c)^q$ defined above are also called Monge-Amp\`ere
operators.


Now, let $\Theta$ be a current of order $0$. When $K\compact X$ is 
an arbitrary compact subset, we define a {\sl mass} semi-norm
$$||\Theta||_K=\sum_j\int_{K_j}~~\sum_{I,J}|\Theta_{I,J}|$$
by taking a partition $K=\bigcup K_j$ where each $\smash{\ovl K_j}$ 
is contained in a coordinate patch and where $\Theta_{I,J}$ are the
corresponding measure coefficients. Up to constants, the semi-norm 
$||\Theta||_K$ does not depend on the choice of the coordinate systems 
involved. When $K$ itself is contained in a coordinate patch, we set
$\beta=dd^c|z|^2$ over $K\,$; then, if $\Theta\ge 0$, there are constants
$C_1,C_2>0$ such that
$$C_1||\Theta||_K\le\int_K\Theta\wedge\beta^p\le C_2||\Theta||_K.$$
We denote by $L^1(K)$, resp. by $L^\infty(K)$, the space of integrable
(resp. bounded measurable) functions on $K$ with respect to any smooth
positive density on~$X$.

\statement{(1.3) Chern-Levine-Nirenberg inequalities {\rm([C-L-N])}}{For 
all compact subsets
$K,L$ of $X$ with $L\subset K^\circ$, there exists a constant $C_{K,L}\ge0$ 
such that
$$||dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T||_L\le C_{K,L}~
||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}\,||T||_K.$$}

\proof{By induction, it is sufficient to prove the result for $q=1$ and
$u_1=u$. There is a covering of $L$ by a family of balls
$B'_j\compact B_j\subset K$ contained in coordinate patches of $X$.
Let $\chi\in\dr(B_j)$ be equal to $1$ on $\smash{\ovl B'_j}$. Then
$$||dd^cu\wedge T||_{L\cap\ovl B'_j}\le C\int_{\ovl B'_j}dd^c u\wedge T
\wedge\beta^{p-1}\le C\int_{B_j}\chi\,dd^c u\wedge T\wedge\beta^{p-1}.$$
As $T$ and $\beta$ are closed, an integration by parts yields
$$||dd^cu\wedge T||_{L\cap\ovl B'_j}\le C\int_{B_j}u\,T\wedge dd^c\chi
\wedge\beta^{p-1}\le C'||u||_{L^\infty(K)}||T||_K$$
where $C'$ is equal to $C$ multiplied by a bound for the coefficients
of the smooth form $dd^c\chi\wedge\beta^{p-1}$.\qed}

\remark{1.4}{With the same notations as above, any plurisubharmonic
function $V$ on $X$ satisfies inequalities of the type
\medskip
\item{\rm(a)} $\quad||dd^cV||_L\le C_{K,L}\,||V||_{L^1(K)}$.
\medskip
\item{\rm(b)} $\quad{\displaystyle\sup_L}\,V_+\le C_{K,L}\,||V||_{L^1(K)}$.
\medskip
\noindent In fact the inequality
$$\int_{L\cap\ovl B'_j}dd^cV\wedge\beta^{n-1}\le\int_{B_j}\chi dd^cV\wedge
\beta^{n-1}=\int_{B_j}Vdd^c\chi\wedge\beta^{n-1}$$
implies (a), and (b) follows from the mean value inequality.}

\remark{1.5}{Products of the form $\Theta=\gamma_1\wedge\ldots\wedge
\gamma_q\wedge T$ with mixed $(1,1)$-forms $\gamma_j=dd^cu_j$ or 
$\gamma_j=dv_j\wedge d^cw_j+dw_j\wedge d^cv_j$ are also well defined 
whenever $u_j$, $v_j$, $w_j$ are locally bounded plurisubharmonic functions.
Moreover, for $L\subset K^\circ$, we have
$$||\Theta||_L\le C_{K,L}||T||_K\prod||u_j||_{L^\infty(K)}
\prod||v_j||_{L^\infty(K)}\prod||w_j||_{L^\infty(K)}.$$
To check this, we may suppose $v_j,w_j\ge 0$ and $||v_j||=||w_j||=1$ 
in~$L^\infty(K)$. Then the inequality 
follows from (1.3) by the polarization identity
$$2(dv_j\wedge d^cw_j+dw_j\wedge d^cv_j)=dd^c(v_j+w_j)^2
-dd^cv_j^2-dd^cw_j^2-v_jdd^cw_j-w_jdd^cv_j$$
in which all $dd^c$ operators act on plurisubharmonic functions.}

\corollary{1.6}{Let $u_1\ld u_q$ be continuous $($finite$)$ plurisubharmonic
functions and let $u_1^k\ld u_q^k$ be sequences of plurisubharmonic functions
converging locally uniformly to $u_1\ld u_q$. If $T_k$ is a sequence of closed
positive currents converging weakly to $T$, then
\smallskip
\item{\rm(a)} $u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k
\wedge T_k\lra u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\sl weakly}$.
\smallskip
\item{\rm(b)} $dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T_k\lra
dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\sl weakly}$.}

\proof{We observe that (b) is an immediate consequence of (a) by the weak
continuity of $dd^c$. By using induction on $q$, it is enough to prove result
(a) when~$q=1$. If $(u^k)$ converges locally uniformly to a finite
continuous plurisubharmonic function $u$, we introduce local regularizations
$u_\varepsilon=u\star\rho_\varepsilon$ and write
$$u^kT_k-uT=(u^k-u)T_k+(u-u_\varepsilon)T_k+u_\varepsilon(T_k-T).$$
As the sequence $T_k$ is weakly convergent, it is locally uniformly bounded
in mass, thus $||(u^k-u)T_k||_K\le ||u^k-u||_{L^\infty(K)}||T_k||_K$ converges
to $0$ on every compact set~$K$. The same argument shows that
$||(u-u_\varepsilon)T_k||_K$ can be made arbitrarily small by choosing
$\varepsilon$ small enough. Finally $u_\varepsilon$ is smooth, so
$u_\varepsilon(T_k-T)$ converges weakly to~$0$.\qed}

Now, we prove a deeper monotone continuity theorem due to
Bedford-Taylor [B-T2], according to which the
continuity and uniform convergence assumptions can be dropped if
each sequence $(u_j^k)$ is decreasing and $T_k$ is a constant sequence.

\theorem{1.7}{Let $u_1\ld u_q$ be locally bounded plurisubharmonic functions
and let $u_1^k\ld u_q^k$ be decreasing sequences of plurisubharmonic functions
converging pointwise to $u_1\ld u_q$. Then 
\medskip
\item{\rm(a)} $u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k
\wedge T\lra u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T~~~{\sl weakly}$.
\medskip
\item{\rm(b)} $dd^cu_1^k\wedge\ldots\wedge dd^cu_q^k\wedge T\lra
dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T~~~{\sl weakly}$.}

\proof{Again by induction, observing that (a)~$\Longrightarrow$~(b) and that
(a) is obvious for $q=1$ thanks to Lebesgue's bounded convergence theorem.
To proceed with the induction step, we first have to make some
slight modifications of our functions $u_j$ and $u_j^k$.

As the sequence $\smash{(u_j^k)}$ is decreasing and as $u_j$ is
locally bounded, the family $\smash{(u_j^k)_{k\in\bbbn}}$ is locally
uniformly bounded. The results are local, so we can work on a Stein open
set $\Omega\compact X$ with strongly pseudoconvex boundary. We use the
following notations:
\medskip
\item{$(1.8)$} ~~Let $\psi$ be a strongly plurisubharmonic function of class
$\ci$ near $\smash{\ovl\Omega}$\break
${}$ ~~with $\psi<0$ on $\Omega$ and $\psi=0$,
$d\psi\ne 0$ on~$\partial\Omega$.
\smallskip
\item{$(1.8')$} ~~We set $\Omega_\delta=\{z\in\Omega\,;\,\psi(z)<-\delta\}$
for all $\delta>0$.
\medskip
\noindent After addition of a constant we can
assume that $-M\le\smash{u_j^k}\le-1$ near~$\smash{\ovl\Omega}$. Let us denote
by $\smash{(u_j^{k,\varepsilon})}$, $\varepsilon\in{}]0,\varepsilon_0]$, an
increasing family of regularizations converging to $\smash{u_j^k}$ as 
$\varepsilon\to 0$ and such that $-M\le\smash{u^{k,\varepsilon}_j}\le-1$
on $\smash{\ovl\Omega}$. Set $A=M/\delta$ with $\delta>0$ small and replace
$\smash{u_j^k}$ by $\smash{v_j^k=\max\{A\psi,u_j^k\}}$ and
$\smash{u_j^{k,\varepsilon}}$ by $\smash{v_j^{k,\varepsilon}=
\max_\varepsilon\{A\psi,u_j^{k,\varepsilon}\}}$
where $\max_\varepsilon=\max~\star~\rho_\varepsilon$ is a regularized 
max function (the construction of $v_j^k$ is described by Fig.~1).

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\begfig -1 cm
\figure{1} {Construction of $v_j^k$}
\endfig
\pritem{}Then $\smash{v_j^k}$ coincides with $\smash{u_j^k}$ on 
$\Omega_\delta$ since $A\psi<-A\delta=-M$ on
$\Omega_\delta$, and $\smash{v_j^k}$ is equal to 
$A\psi$ on the corona $\Omega\ssm\Omega_{\delta/M}$.  Without loss 
of generality, we can therefore assume that all $\smash{u_j^k}$ 
(and similarly all $\smash{u_j^{k,\varepsilon}})$ coincide with $A\psi$ 
on a fixed neighborhood of $\partial\Omega$. We need a lemma.}

\lemma{1.9}{Let $f_k$ be a decreasing sequence of upper
semi-continuous functions converging to $f$ on some separable locally
compact space $X$ and $\mu_k$ a sequence of positive measures
converging weakly to $\mu$ on $X$. Then every weak limit $\nu$ of $f_k\mu_k$
satisfies $\nu\le f\mu$.}

Indeed if $(g_p)$ is a decreasing sequence of continuous functions
converging to $f_{k_0}$ for some $k_0$, then $f_k\mu_k\le f_{k_0}\mu_k
\le g_p\mu_k$ for $k\ge k_0$, thus $\nu\le g_p\mu$ as $k\to+\infty$.
The monotone convergence theorem then gives $\nu\le f_{k_0}\mu$ as
$p\to+\infty$ and $\nu\le f\mu$ as $k_0\to+\infty$.\qed

\pproof{End of proof of Theorem 1.7}{Assume that (a) has been proved
for $q-1$.  Then 
$$S^k=dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k\wedge T\lra 
S=dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T.$$ 
By 1.3 the sequence $(u_1^k S^k)$ has locally bounded mass, hence is 
relatively compact for the weak topology.  In order to prove (a), we only
have to show that every weak limit $\Theta$ of $u_1^k S^k$ is equal to 
$u_1 S$.  Let $(m,m)$ be the bidimension of $S$ and let $\gamma$ be an
arbitrary smooth and strongly positive~form of bidegree $(m,m)$.  Then
the positive measures $S^k\wedge\gamma$ converge weakly to
$S\wedge\gamma$ and Lemma~1.9 shows that $\Theta\wedge\gamma\le
u_1S\wedge\gamma$, hence $\Theta\le u_1S$.  To get~the equality, we set
$\beta=dd^c\psi>0$ and show that
$\int_\Omega u_1 S\wedge\beta^m\le\int_\Omega\Theta\wedge\beta^m$, i.e. 
$$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\le
\liminf_{k\to+\infty}\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge dd^cu_q^k
\wedge T\wedge\beta^m.$$ 
As $u_1\le u_1^k\le u_1^{k,\varepsilon_1}$ for every $\varepsilon_1>0$, we get
$$\eqalign{\int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge
T\wedge\beta^m\cr
&\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2
\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr 
&=\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge u_2dd^cu_3
\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr}$$ 
after an integration by parts (there is no boundary term because
$\smash{u_1^{k,\varepsilon_1}}$ and $u_2$ both vanish on $\partial\Omega$).  
Repeating this argument with $u_2\ld u_q$, we obtain
$$\eqalign{
\int_\Omega u_1&dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\beta^m\cr 
&\le\int_\Omega dd^c u_1^{k,\varepsilon_1}\wedge\ldots\wedge dd^c
u_{q-1}^{k,\varepsilon_{q-1}}\wedge u_q T\wedge\beta^m\cr
&\le\int_\Omega u_1^{k,\varepsilon_1}dd^cu_2^{k,\varepsilon_2}
\wedge\ldots\wedge dd^c u_q^{k,\varepsilon_q}\wedge T\wedge\beta^m.\cr}$$ 
Now let $\varepsilon_q\to 0\ld\varepsilon_1\to 0$ in this order.  We
have weak convergence at each step and $\smash{u_1^{k,\varepsilon_1}}=0$
on the boundary; therefore the integral in the last line converges and we
get the desired inequality 
$$\int_\Omega u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge
T\wedge\beta^m\le\int_\Omega u_1^k dd^cu_2^k\wedge\ldots\wedge
dd^cu_q^k\wedge T\wedge\beta^m.\eqno{\sq}$$}

\corollary{1.10}{The product $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$
is symmetric with respect to $u_1\ld u_q$.}

\proof{Observe that the definition was unsymmetric. The
result is true when $u_1\ld u_q$ are smooth and follows in general from
Th.~1.7 applied to the sequences $u_j^k=u_j\star\rho_{1/k}$,
$1\le j\le q$.\qed}

\vbox{%
\statement{(1.11) Proposition}{Let $K,L$ be compact subsets of $X$ such that
$L\subset K^\circ$. For any plurisubharmonic functions $V,u_1\ld u_q$ on $X$ 
such that $u_1\ld u_q$ are locally bounded, there is an inequality
$$||Vdd^cu_1\wedge\ldots\wedge dd^cu_q||_L\le 
C_{K,L}\,||V||_{L^1(K)}||u_1||_{L^\infty(K)}\ldots||u_q||_{L^\infty(K)}.$$}}


\proof{We may assume that $L$ is contained in a strongly pseudoconvex open
set $\Omega=\{\psi<0\}\subset K$ (otherwise we cover $L$ by
small balls contained in $K$).  A suitable normalization gives
$-2\le u_j\le -1$ on $K\,$;  then we can modify $u_j$ on 
$\Omega\ssm L$ so that $u_j=A\psi$ on
$\Omega\ssm\Omega_\delta$ with a fixed constant $A$ and
$\delta>0$ such that $L\subset\Omega_\delta$.  Let $\chi\ge 0$
be a smooth function equal to $-\psi$ on $\Omega_\delta$ with compact
support in $\Omega$.  If we take $||V||_{L^1(K)}=1$, we see that $V_+$
is uniformly bounded on $\Omega_\delta$ by 1.4 (b); after subtraction
of a fixed constant we can assume $V\le 0$ on $\Omega_\delta$.  First
suppose $q\le n-1$. As $u_j=A\psi$ on $\Omega\ssm\Omega_\delta$,
we find
$$\eqalign{
&\int_{\Omega_\delta}-V\,dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q}\cr
&~\,{}=\int_\Omega V\,dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q-1}\wedge dd^c\chi-A^q\int_{\Omega\ssm
\Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi\cr
&~\,{}=\int_\Omega\chi\,dd^cV\wedge dd^cu_1\wedge\ldots\wedge
dd^cu_q\wedge\beta^{n-q-1}-A^q\int_{\Omega\ssm
\Omega_\delta} V\,\beta^{n-1}\wedge dd^c\chi.\cr}$$
The first integral of the last line is uniformly bounded thanks to
1.3 and 1.4~(a), and the second one is bounded by $||V||_{L^1(\Omega)}\le$
constant. Inequality 1.11 follows for $q\le n-1$. If $q=n$,
we can work instead on $X\times\bbbc$ and consider $V,u_1\ld u_q$ as
functions on $X\times\bbbc$ independent of the extra variable
in~$\bbbc$.\qed}\vskip-10pt

\titleb{2. Case of Unbounded Plurisubharmonic Functions}
We would like to define $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$
also in some cases when $u_1\ld u_q$ are not bounded below everywhere,
especially when the $u_j$ have logarithmic poles.
Consider first the case $q=1$ and let $u$ be a plurisubharmonic function
on $X$.  The {\sl pole set} of $u$ is by definition $P(u)=u^{-1}(-\infty)$.
We define the {\sl unbounded locus} $L(u)$ to be the set of points $x\in X$
such that $u$ is unbounded in every neighborhood of~$x$. Clearly $L(u)$ is
closed and we have $L(u)\supset\smash{\ovl{P(u)}}$ but in general
these sets are different: in fact, $u(z)=\sum k^{-2}\log(|z-1/k|+e^{-k^3})$
is everywhere finite in $\bbbc$ but $L(u)=\{0\}$.

\statement{(2.1) Proposition}{We make two additional assumptions:
\smallskip
\item{\rm(a)} $T$ has non zero bidimension $(p,p)$ (i.e. degree of $T<2n$).
\smallskip
\item{\rm(b)} $X$ is covered by a family of Stein open
sets $\Omega\compact X$ whose boundaries $\partial\Omega$ do not
intersect $L(u)\cap\supp\,T$.
\smallskip
\noindent Then the current $uT$ has locally finite mass in X.}


For any current $T$, hypothesis 2.1~(b) is clearly satisfied 
when $u$ has a discrete unbounded locus $L(u)$; an interesting example is
$u=\log|F|$ where $F=(F_1\ld F_N)$ are holomorphic functions having
a discrete set of common zeros.
Observe that the current $uT$ need not have locally finite
mass when $T$ has degree $2n$ (i.e. $T$ is a measure); example:
$T=\delta_0$ and $u(z)=\log|z|$\break in $\bbbc^n$. The result also fails
when the sets $\Omega$ are not assumed to be Stein; example: $X={}$
blow-up of $\bbbc^n$ at $0$, $T=[E]={}$ current of integration on the
exceptional divisor and $u(z)=\log|z|$.

\proof{By shrinking $\Omega$ slightly, we may assume that $\Omega$ has a
smooth strongly pseudoconvex boundary. Let $\psi$ be a defining
function of $\Omega$ as in (1.8). By subtracting a constant to $u$, we may
assume $u\le-\varepsilon$ on $\ovl\Omega$. We fix $\delta$ so small that
$\ovl\Omega\ssm\Omega_\delta$ does not intersect $L(u)\cap\supp\,T$ and
we select a neighborhood $\omega$ of $(\ovl\Omega\ssm\Omega_\delta)\cap
\supp\,T$ such that $\ovl\omega\cap L(u)=\emptyset$. Then we define
$$u_s(z)=\cases{
\max\{u(z),A\psi(z)\}&on $\omega$,\cr
\max\{u(z),s\}&on $\Omega_\delta=\{\psi<-\delta\}$.\cr}$$
By construction $u\ge -M$ on $\omega$ for some constant $M>0$. We fix
$A\ge M/\delta$ and take $s\le -M$, so
$$\max\{u(z),A\psi(z)\}=\max\{u(z),s\}=u(z)~~~~\hbox{\rm on}~~
\omega\cap\Omega_\delta$$
and our definition of $u_s$ is coherent. Observe that $u_s$ is defined
on $\omega\cup\Omega_\delta$, which is a neighborhood of
$\ovl\Omega\cap\supp\,T$. Now, $u_s=A\psi$ on
$\omega\cap(\Omega\ssm\Omega_{\varepsilon/A})$, hence Stokes' theorem implies
$$\eqalign{
\int_\Omega dd^cu_s\wedge T\wedge(dd^c\psi)^{p-1}&{}-
\int_\Omega Add^c\psi\wedge T\wedge(dd^c\psi)^{p-1}\cr
&=\int_\Omega dd^c\big[(u_s-A\psi)T\wedge(dd^c\psi)^{p-1}\big]=0\cr}$$
because the current $[\ldots]$ has a compact support contained in
$\ovl\Omega_{\varepsilon/A}$. Since $u_s$ and $\psi$ both vanish on
$\partial\Omega$, an integration by parts gives
$$\eqalign{
\int_\Omega u_sT\wedge(dd^c\psi)^p
&=\int_\Omega\psi dd^cu_s\wedge T\wedge(dd^c\psi)^{p-1}\cr
&\ge-||\psi||_{L^\infty(\Omega)}\int_\Omega T\wedge dd^cu_s\wedge
(dd^c\psi)^{p-1}\cr
&=-||\psi||_{L^\infty(\Omega)}A\int_\Omega T\wedge(dd^c\psi)^p.\cr}$$
Finally, take $A=M/\delta$, let $s$ tend to $-\infty$ and use the inequality
$u\ge-M$ on~$\omega$. We obtain
$$\eqalign{
\int_\Omega u\,T\wedge(dd^c\psi)^p
&\ge -M\int_\omega T\wedge(dd^c\psi)^p+\lim_{s\to-\infty}
\int_{\Omega_\delta}u_sT\wedge(dd^c\psi)^p\cr
&\ge -\big(M+||\psi||_{L^\infty(\Omega)}M/\delta\big)
\int_\Omega T\wedge(dd^c\psi)^p.\cr}$$
The last integral is finite. This concludes the proof.\qed}

\remark{2.2}{If $\Omega$ is smooth and strongly pseudoconvex, the above
proof shows in fact that
$$||uT||_{\ovl\Omega}\le{C\over\delta}||u||_{L^\infty(
(\ovl\Omega\ssm\Omega_\delta)\cap\supp\,T)}||T||_{\ovl\Omega}$$
when $L(u)\cap\supp\,T\subset\Omega_\delta$. In fact, if $u$ is continuous
and if $\omega$ is chosen sufficiently small, the constant $M$ can be taken
arbitrarily close to $||u||_{L^\infty(
(\ovl\Omega\ssm\Omega_\delta)\cap\supp\,T)}$. Moreover, the maximum
principle implies 
$$||u_+||_{L^\infty(\ovl\Omega\cap\supp\,T)}=
||u_+||_{L^\infty(\partial\Omega\cap\supp\,T)},$$
so we can achieve $u<-\varepsilon$ on a neighborhood of $\ovl\Omega
\cap\supp\,T$ by subtracting $||u||_{L^\infty((\ovl\Omega\ssm\Omega_\delta)
\cap\supp\,T)}+2\varepsilon$ $[$Proof of maximum principle: if $u(z_0)>0$ at
$z_0\in\Omega\cap\supp\,T$ and $u\le 0$ near $\partial\Omega\cap\supp\,T$, 
then
$$\int_\Omega u_+T\wedge(dd^c\psi)^p=\int_\Omega\psi dd^cu_+\wedge T
\wedge(dd^c\psi)^{p-1}\le 0,$$
a contradiction$]$.\qed}

\corollary{2.3}{Let $u_1\ld u_q$ be plurisubharmonic functions on $X$
such that $X$ is covered by Stein open sets $\Omega$ with
$\partial\Omega\cap L(u_j)\cap\supp\,T=\emptyset$.
We use again induction to define
$$dd^c u_1\wedge dd^cu_2\wedge\ldots\wedge dd^c u_q\wedge T=
dd^c(u_1dd^c u_2\ldots\wedge dd^c u_q\wedge T).$$
Then, if $u_1^k\ld u_q^k$ are decreasing sequences of plurisubharmonic
functions converging pointwise to $u_1\ld u_q$, $q\le p$,
properties~$(1.7\,{\rm a,b})$ hold.}

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\begfig -0.8 cm
\figure{2} {Modified construction of $v_j^k$}
\endfig
\vskip7mm

\proof{Same proof as for Th.~1.7, with the following minor
modification: the max procedure $v_j^k:=\max\{u_j^k,A\psi\}$ is applied
only on a neighborhood $\omega$ of $\supp\,T\cap(\ovl\Omega\ssm\Omega_\delta)$
with $\delta>0$ small, and $u_j^k$ is left
unchanged near \hbox{$\supp\,T\cap\ovl\Omega_\delta$} (see Fig.~2).
Observe that the integration by part process requires the
functions $u_j^k$ and $\smash{u_j^{k,\varepsilon}}$ to be defined only 
near $\ovl\Omega\cap\supp\,T$.\qed}

\statement{(2.4) Proposition}{Let $\Omega\compact X$ be a Stein open subset.
If $V$ is a plurisubharmonic function on $X$ and  $u_1\ld u_q$,
$1\le q\le n-1$, are plurisubharmonic functions such that
$\partial\Omega\cap L(u_j)=\emptyset$,
then $Vdd^cu_1\wedge\ldots \wedge dd^cu_q$ has locally finite mass
in $\Omega$.}

\proof{Same proof as for 1.11, when $\delta>0$ is taken so small that
$\Omega_\delta\supset L(u_j)$ for all $1\le j\le q$.\qed}

Finally, we show that Monge-Amp\`ere operators can also be defined
in the case of plurisubharmonic functions with non compact pole sets,
provided that the mutual intersections of the pole sets are of
sufficiently small Hausdorff dimension with respect to the dimension
$p$ of~$T$.

\theorem{2.5}{Let $u_1\ld u_q$ be plurisubharmonic functions on~$X$.
The currents $u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T$ and
$dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ are well defined and have
locally finite mass in~$X$ as soon as $q\le p$ and
$$\hc_{2p-2m+1}\big(L(u_{j_1})\cap\ldots\cap L(u_{j_m})\cap\supp\,T\big)=0$$
for all choices of indices $j_1<\ldots<j_m$ in $\{1\ld q\}$.}

The proof is an easy induction on $q$, thanks to the following improved
version of the Chern-Levine-Nirenberg inequalities.

\statement{(2.6) Proposition}{Let $A_1\ld A_q\subset X$ be closed sets
such that 
$$\hc_{2p-2m+1}\big(A_{j_1}\cap\ldots\cap A_{j_m}\cap\supp\,T\big)=0$$
for all choices of $j_1<\ldots<j_m$ in $\{1\ld q\}$. Then for all compact
sets $K$, $L$ of $X$ with $L\subset K^\circ$, there exist neighborhoods
$V_j$ of $K\cap A_j$ and a constant $C=C(K,L,A_j)$ such that the
conditions $u_j\le 0$ on $K$ and $L(u_j)\subset A_j$ imply
\medskip
\item{\rm(a)} $||u_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T||_L\le
C||u_1||_{L^\infty(K\ssm V_1)}\ldots||u_q||_{L^\infty(K\ssm V_q)}||T||_K$
\medskip
\item{\rm(b)} $||dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T||_L\le
C||u_1||_{L^\infty(K\ssm V_1)}\ldots||u_q||_{L^\infty(K\ssm V_q)}||T||_K$.}

\proof{We need only show that every point $x_0\in K^\circ$ has a neighborhood
$L$ such that (a), (b) hold. Hence it is enough to work in a coordinate open
set. We may thus assume that $X\subset\bbbc^n$ is open, and after a
regularization process $u_j=\lim u_j\star\rho_\varepsilon$ for $j=q$,
$q-1\ld 1$ in this order, that $u_1\ld u_q$ are smooth.
We proceed by induction on $q$ in two steps:
\smallskip
\noindent{\sl Step 1}.~~$({\rm b}_{q-1})\Longrightarrow({\rm b}_q),$\hfill\break
\noindent{\sl Step 2}.~~$({\rm a}_{q-1})~{\rm and}~({\rm b}_q)
\Longrightarrow({\rm a}_q),$
\smallskip
\noindent where $({\rm b}_0)$ is the trivial statement $||T||_L\le||T||_K$
and $({\rm a}_0)$ is void. Observe that we have
$({\rm a}_q)\Longrightarrow({\rm a}_\ell)$
and $({\rm b}_q)\Longrightarrow({\rm b}_\ell)$ for $\ell\le q\le p$ by taking
$u_{\ell+1}(z)=\ldots=u_q(z)=|z|^2$. We need the following elementary fact.}

\lemma{2.7}{Let $F\subset\bbbc^n$ be a closed set such that $\hc_{2s+1}(F)=0$
for some integer $0\le s<n$. Then for almost all choices of unitary
coordinates $(z_1\ld z_n)=(z',z'')$ with $z'=(z_1\ld z_s)$,
$z''=(z_{s+1}\ld z_n)$ and almost all radii of balls
$B''=B(0,r'')\subset\bbbc^{n-s}$, the set $\{0\}\times\partial B''$
does not intersect~$F$.}

\proof{The unitary group $U(n)$ has real dimension $n^2$. There is a proper
submersion
$$\Phi:U(n)\times\big(\bbbc^{n-s}\ssm\{0\}\big)\lra\bbbc^n\ssm\{0\},~~~~
(g,z'')\longmapsto g(0,z''),$$
whose fibers have real dimension $N=n^2-2s$. It follows that the inverse
image $\Phi^{-1}(F)$ has zero Hausdorff measure $\hc_{N+2s+1}=\hc_{n^2+1}$.
The set of pairs $(g,r'')\in U(n)\times\bbbr^\star_+$ such that
$g(\{0\}\times\partial B'')$ intersects $F$ is precisely the image of
$\Phi^{-1}(F)$ in $U(n)\times\bbbr^\star_+$ by the Lipschitz map
$(g,z'')\mapsto (g,|z''|)$. Hence this set has zero
$\hc_{n^2+1}$-measure.\qed}

\pproof{Proof of step 1}{Take $x_0=0\in K^\circ$. Suppose first
$0\in A_1\cap\ldots\cap A_q$ and set $F=A_1\cap\ldots\cap A_q\cap\supp\,T$.
Since $\hc_{2p-2q+1}(F)=0$, Lemma~2.7 implies that there are
coordinates $z'=(z_1\ld z_s)$, $z''=(z_{s+1}\ld z_n)$ with $s=p-q$ and
a ball $\ovl B''$ such that $F\cap\big(\{0\}\times\partial B''\big)=\emptyset$
and $\{0\}\times\ovl B''\subset K^\circ$. By compactness of $K$, we can
find neighborhoods $W_j$ of $K\cap A_j$ and a ball $B'=B(0,r')\subset\bbbc^s$
such that $\ovl B'\times\ovl B''\subset K^\circ$ and
$$\ovl W_1\cap\ldots\cap\ovl W_q\cap\supp\,T\cap\Big(\ovl B'\times\big(
\ovl B''\ssm(1-\delta)B''\big)\Big)=\emptyset\leqno(2.8)$$
for $\delta>0$ small. If $0\notin A_j$ for some~$j$, we choose instead
$W_j$ to be a small neighborhood of~$0$ such that $\ovl W_j\subset
(\ovl B'\times(1-\delta)B'')\ssm A_j\,$; property (2.8) is then automatically
satisfied. Let $\chi_j\ge 0$ be a function with compact support in $W_j$,
equal to $1$ near $K\cap A_j$ if $A_j\ni 0$ (resp. equal to $1$ near $0$
if $A_j\not\ni 0$) and let $\chi(z')\ge 0$ be a function equal to $1$
on $1/2\,B'$ with compact support in~$B'$. Then
$$\int_{B'\times B''}dd^c(\chi_1u_1)\wedge\ldots\wedge dd^c(\chi_qu_q)
\wedge T\wedge\chi(z')\,(dd^c|z'|^2)^s=0$$
because the integrand is $dd^c$ exact and has compact support 
in $B'\times B''$ thanks to~(2.8). If we expand all factors
$dd^c(\chi_ju_j)$, we find a term
$$\chi_1\ldots\chi_q\chi(z')dd^cu_1\wedge\ldots\wedge dd^c u_q\wedge T\ge 0$$
which coincides with $dd^cu_1\wedge\ldots\wedge dd^c u_q\wedge T$ on a
small neighborhood of $0$ where $\chi_j=\chi=1$. The other terms involve
$$d\chi_j\wedge d^cu_j+du_j\wedge d^c\chi_j+u_jdd^c\chi_j$$
for at least one index $j$. However $d\chi_j$ and $dd^c\chi_j$ vanish on
some neighborhood $V'_j$ of~$K\cap A_j$ and therefore $u_j$ is bounded on
$\smash{\ovl B'\times\ovl B''}\ssm V'_j$. We then apply the induction
hypothesis $({\rm b}_{q-1})$ to the current
$$\Theta=dd^cu_1\wedge\ldots\wedge\wh{dd^cu_j}\wedge
\ldots\wedge dd^cu_q\wedge T$$
and the usual Chern-Levine-Nirenberg inequality to the product of $\Theta$
with the mixed term $d\chi_j\wedge d^cu_j+du_j\wedge d^c\chi_j$.
Remark 1.5 can be applied because $\chi_j$ is smooth and is therefore a
difference $\chi^{(1)}_j-\chi^{(2)}_j$ of locally bounded plurisubharmonic
functions in $\bbbc^n$. Let $K'$ be a compact neighborhood of $\smash{
\ovl B'\times\ovl B''}$ with $K'\subset K^\circ$, and let $V_j$ be a 
neighborhood of $K\cap A_j$ with $\ovl V_j\subset V'_j$. Then with
$L':=(\smash{\ovl B'\times\ovl B''})\ssm V'_j\subset(K'\ssm V_j)^\circ$
we obtain
$$\eqalign{
||(d\chi_j{\wedge}d^cu_j+du_j{\wedge}d^c&\chi_j)\wedge
\Theta||_{\ovl B'\times \ovl B''}
=||(d\chi_j{\wedge}d^cu_j+du_j{\wedge}d^c\chi_j)\wedge\Theta||_{L'}\cr
&\le C_1||u_j||_{L^\infty(K'\ssm V_j)}||\Theta||_{K'\ssm V_j},\cr
||\Theta||_{K'\ssm V_j}\le||\Theta||_{K'}
&\le C_2||u_1||_{L^\infty(K\ssm V_1)}\ldots\wh{||u_j||}\ldots
||u_q||_{L^\infty(K\ssm V_q)}||T||_K.\cr}$$
Now, we may slightly move the unitary basis in $\bbbc^n$ and get coordinate
systems $z^m=(z^m_1\ld z^m_n)$ with the same properties as above, such that
the forms
$$(dd^c|z^{m\prime}|^2)^s={s!\over\pi^s}
\ii\,dz^m_1\wedge d\ovl z^m_1\wedge\ldots\wedge\ii\,dz^m_s
\wedge d\ovl z^m_s,~~~~1\le m\le N$$
define a basis of $\bigwedge^{s,s}(\bbbc^n)^\star$. It follows that all
measures
$$dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T\wedge
\ii\,dz^m_1\wedge d\ovl z^m_1\wedge\ldots\wedge\ii\,dz^m_s\wedge d\ovl z^m_s$$
satisfy estimate $({\rm b}_q)$ on a small neighborhood $L$ of~$0$.}

\pproof{Proof of Step 2}{We argue in a similar way with the integrals
$$\eqalign{
\int_{B'\times B''}&\chi_1u_1dd^c(\chi_2u_2)\wedge\ldots dd^c(\chi_qu_q)
\wedge T\wedge\chi(z')(dd^c|z'|^2)^s\wedge dd^c|z_{s+1}|^2\cr
&=\int_{B'\times B''}|z_{s+1}|^2dd^c(\chi_1u_1)\wedge
\ldots dd^c(\chi_qu_q)\wedge T\wedge\chi(z')(dd^c|z'|^2)^s.\cr}$$
We already know by $({\rm b}_q)$ and Remark~1.5 that all terms in the right
hand integral admit the desired bound. For $q=1$, this shows that
$({\rm b}_1)\Longrightarrow({\rm a}_1)$. Except for
$\chi_1\ldots\chi_q\chi(z')\,u_1dd^cu_2\wedge\ldots\wedge
dd^cu_q\wedge T$, all terms in the left hand integral involve derivatives
of~$\chi_j$. By construction, the support of these derivatives is disjoint
from~$A_j$, thus we only have to obtain a bound for
$$\int_Lu_1dd^cu_2\wedge\ldots\wedge dd^cu_q\wedge T\wedge\alpha$$
when $L=\ovl B(x_0,r)$ is disjoint from $A_j$ for some $j\ge 2$,
say $L\cap A_2=\emptyset$, and $\alpha$ is a constant positive form of
type~$(p-q,p-q)$. Then $\ovl B(x_0,r+\varepsilon)\subset K^\circ\ssm\ovl V_2$
for some $\varepsilon>0$ and some neighborhood $V_2$ of~$K\cap A_2$. By the
max construction used e.g. in Prop.~2.1, we can replace $u_2$ by a
plurisubharmonic function $\wt u_2$ equal to $u_2$ in $L$ and
to $A(|z-x_0|^2-r^2)-M$ in $\ovl B(x_0,r+\varepsilon)\ssm B(x_0,r+
\varepsilon/2)$, with $M=||u_2||_{L^\infty(K\ssm V_2)}$ and
$A=M/\varepsilon r$. Let $\chi\ge 0$ be a smooth function equal to $1$
on $B(x_0,r+\varepsilon/2)$ with support in $B(x_0,r)$. Then
$$\eqalign{
\int_{B(x_0,r+\varepsilon)}u_1&dd^c(\chi\wt u_2)\wedge dd^cu_3\wedge
\ldots\wedge dd^cu_q\wedge T\wedge\alpha\cr
&=\int_{B(x_0,r+\varepsilon)}\chi\wt u_2dd^cu_1\wedge dd^cu_3\wedge\ldots
\wedge dd^cu_q\wedge T\wedge\alpha\cr
&\le O(1)~||u_1||_{L^\infty(K\ssm V_1)}\ldots||u_q||_{L^\infty(K\ssm V_q)}
||T||_K\cr}$$
where the last estimate is obtained by the induction hypothesis
$({\rm b}_{q-1})$ applied to
$dd^cu_1\wedge dd^cu_3\wedge\ldots\wedge dd^cu_q\wedge T$. By construction
$$dd^c(\chi\wt u_2)=\chi\,dd^c\wt u_2+(\hbox{\rm smooth terms
involving $d\chi$})$$
coincides with $dd^cu_2$ in $L$, and $({\rm a}_{q-1})$ implies the
required estimate for the other terms in the left hand integral.\qed}

\statement{Proposition 2.9}{With the assumptions of Th.~$2.5$, the
analogue of the monotone convergence Theorem {\rm 1.7~(a,b)} holds.}

\proof{By the arguments already used in the proof of Th.~1.7 (e.g. by
Lemma~1.9), it is enough to show that
$$\eqalign{
\int_{B'\times B''}&\chi_1\ldots\chi_q\,u_1\wedge dd^cu_2\wedge
\ldots\wedge dd^cu_q\wedge T\wedge\alpha\cr
&\le\liminf_{k\to+\infty}\int_{B'\times B''}\chi_1\ldots\chi_q\,
u^k_1dd^cu^k_2\wedge\ldots\wedge dd^cu^k_q\wedge T\wedge\alpha\cr}$$
where $\alpha=\chi(z')(dd^c|z'|^2)^s$ is closed.
Here the functions $\chi_j$, $\chi$ are chosen as in the proof of Step~1
in~2.7, especially their product has compact support in $B'\times B''$
and $\chi_j=\chi=1$ in a neighborhood of the given point~$x_0$. We argue
by induction on $q$ and also on the number $m$ of functions $(u_j)_{j\ge 1}$
which are unbounded near~$x_0$. If $u_j$ is bounded near $x_0$, we take $W_j''
\compact W'_j\compact W_j$ to be small balls of center $x_0$ on which $u_j$
is bounded and we modify the sequence $u_j^k$ on the corona $W_j\ssm W''_j$
so as to make it constant and equal to a smooth function $A|z-x_0|^2+B$ on
the smaller corona~$W_j\ssm W'_j$. In that case, we take $\chi_j$
equal to $1$ near~$\smash{\ovl W'_j}$ and $\supp\,\chi_j\subset W_j$.
For every $\ell=1\ld q$, we are going to check that
$$\eqalign{
\liminf_{k\to+\infty}\int_{B'\times B''}
\chi_1u^k_1&dd^c(\chi_2u^k_2)\wedge\ldots\cr
dd^c(\chi_{\ell-1}u^k_{\ell-1})\wedge{}&dd^c(\chi_\ell u_\ell)\wedge
dd^c(\chi_{\ell+1}u_{\ell+1})\ldots dd^c(\chi_qu_q)\wedge T\wedge\alpha\cr
{}\le\liminf_{k\to+\infty}\int_{B'\times B''}
\chi_1u^k_1&dd^c(\chi_2u^k_2)\wedge\ldots\cr
dd^c(\chi_{\ell-1}u^k_{\ell-1})\wedge{}&dd^c(\chi_\ell u^k_\ell)\wedge
dd^c(\chi_{\ell+1}u_{\ell+1})\ldots dd^c(\chi_qu_q)\wedge T\wedge\alpha.\cr}$$
In order to do this, we integrate by parts $\chi_1u^k_1dd^c(\chi_\ell
u_\ell)$ into $\chi_\ell u_\ell dd^c(\chi_1u^k_1)$ for $\ell\ge 2$,
and we use the inequality $u_\ell\le u^k_\ell$. Of course, the derivatives
$d\chi_j$, $d^c\chi_j$, $dd^c\chi_j$ produce terms which are no longer
positive and we have to take care of these. However, $\supp\,d\chi_j$ is
disjoint from the unbounded locus of $u_j$ when $u_j$ is
unbounded, and contained in $W_j\ssm\smash{\ovl W'_j}$ when $u_j$ is bounded.
The number $m$ of unbounded functions is therefore replaced by $m-1$ in the
first case, whereas in the second case $u^k_j=u_j$ is constant and smooth
on $\supp\,d\chi_j$, so $q$ can be replaced by $q-1$. By induction on $q+m$
(and thanks to the polarization technique~1.5), the limit of the terms
involving derivatives of $\chi_j$ is equal on both sides to the
corresponding terms obtained by suppressing all indices~$k$. Hence
these terms do not give any contribution in the inequalities.\qed}

We finally quote the following simple consequences of Th.~2.5 when 
$T$ is arbitrary and $q=1$, resp. when $T=1$ has bidegree $(0,0)$ and
$q$ is arbitrary.

\corollary{2.10}{Let $T$ be a closed positive current of bidimension $(p,p)$
and let $u$ be a plurisubharmonic function on $X$ such that $L(u)\cap\supp\,T$
is contained in an analytic set of dimension at most $p-1$. Then
$uT$ and $dd^cu\wedge T$ are well defined and have locally finite mass
in~$X$.\qed}

\corollary{2.11}{Let $u_1\ld u_q$ be plurisubharmonic functions on $X$
such that $L(u_j)$ is contained in an analytic set $A_j\subset X$ for 
every~$j$. Then $dd^cu_1\wedge\ldots\wedge dd^cu_q$ is well defined as
soon as $A_{j_1}\cap\ldots\cap A_{j_m}$ has codimension at least $m$ for
all choices of indices $j_1<\ldots<j_m$ in $\{1\ld q\}$.\qed}

In the particular case when $u_j=\log|f_j|$ for some non zero
holomorphic function $f_j$ on $X$, we see that the intersection
product of the associated zero divisors $[Z_j]=dd^cu_j$ is well
defined as soon as the supports $|Z_j|$ satisfy
$\codim|Z_{j_1}|\cap\ldots\cap |Z_{j_m}|=m$ for every~$m$. Similarly,
when $T=[A]$ is an analytic $p$-cycle, Cor.~2.10 shows that
$[Z]\wedge[A]$ is well defined for every divisor $Z$ such that
$\dim|Z|\cap|A|=p-1$. These observations easily imply the following

\statement{(2.12) Proposition}{Suppose that the divisors $Z_j$ satisfy the
above codimension condition and let $(C_k)_{k\ge 1}$ be the irreducible
components of the point set intersection $|Z_1|\cap\ldots\cap|Z_q|$. Then
there exist integers $m_k>0$ such that
$$[Z_1]\wedge\ldots\wedge[Z_q]=\sum m_k[C_k].$$
The number $m_k$ is called the multiplicity of intersection of $Z_1\ld Z_q$
along~$C_k$.}

\proof{The wedge product has bidegree $(q,q)$ and support in
$C=\bigcup C_k$ where $\codim C=q$, so
it must be a sum as above with $m_k\in\bbbr_+$. We check by induction
on $q$ that $m_k$ is a positive integer. If we denote by $A$ some
irreducible component of $|Z_1|\cap\ldots\cap |Z_{q-1}|$, we need
only check that $[A]\wedge[Z_q]$ is an integral analytic cycle of
codimension $q$ with positive coefficients on each component $C_k$ of
the intersection. However $[A]\wedge[Z_q]=dd^c(\log|f_q|~[A])$. First
suppose that no component of $A\cap f_q^{-1}(0)$ is contained in the
singular part $A_\sing$. Then the Lelong-Poincar\'e equation applied
on $A_\reg$ shows that $dd^c(\log|f_q|~[A])=\sum m_k[C_k]$ on $X\ssm
A_\sing$, where $m_k$ is the vanishing order of $f_q$ along $C_k$ in
$A_\reg$. Since $C\cap A_\sing$ has codimension $q+1$ at least, the
equality must hold on $X$. In general, we replace $f_q$ by
$f_q-\varepsilon$ so that the divisor of $f_q-\varepsilon$ has no
component contained in $A_\sing$. Then
$dd^c(\log|f_q-\varepsilon|~[A])$ is an integral codimension $q$ cycle
with positive multiplicities on each component of $A\cap
f_q^{-1}(\varepsilon)$ and we conclude by letting $\varepsilon$ tend
to zero.\qed}

\titleb{3. Generalized Lelong Numbers}
The concepts we are going to study mostly concern the behaviour of
currents or plurisubharmonic functions in a neighborhood of a point
at which we have for instance a logarithmic pole. Since the interesting
applications are local, we assume from now on (unless otherwise stated)
that $X$ is a Stein manifold, i.e. that $X$ has a strictly plurisubharmonic
exhaustion function. Let $\varphi:X\longrightarrow[-\infty,+\infty[$ 
be a continuous plurisubharmonic function (in general $\varphi$ may have
$-\infty$ poles, our continuity assumption means that $e^\varphi$ is
continuous). The sets
$$\leqalignno{
S(r)&=\{x\in X\,;\,\varphi(x)=r\},&(3.1)\cr
B(r)&=\{x\in X\,;\,\varphi(x)<r\},&(3.1')\cr
\ovl B(r)&=\{x\in X\,;\,\varphi(x)\le r\}&(3.1'')\cr}$$
will be called {\sl pseudo-spheres} and {\sl pseudo-balls}
associated with $\varphi$. Note that $\ovl B(r)$ is not necessarily equal
to the closure of $B(r)$, but this is often true in concrete situations.
The most simple example we have in mind is the case of the
function $\varphi(z)=\log|z-a|$ on
an open subset $X\subset\bbbc^n\,$; in this case $B(r)$ is the euclidean
ball of center $a$ and radius $e^r\,$; moreover, the forms
$${1\over 2}dd^ce^{2\varphi}={\ii\over 2\pi}d'd''|z|^2,~~~~
dd^c\varphi={\ii\over\pi}d'd''\log|z-a|\leqno(3.2)$$
can be interpreted respectively as the flat hermitian metric on $\bbbc^n$
and as the pull-back over $\bbbc^n$ of the Fubini-Study
metric of $\bbbp^{n-1}$, translated by $a$.

\definition{3.3}{We say that $\varphi$ is semi-exhaustive
if there exists a real number $R$ such that $B(R)\compact X$. 
Similarly, $\varphi$ is said to be
semi-exhaustive on a closed subset $A\subset X$ if there exists
$R$ such that $A\cap B(R)\compact X$.}

We are interested especially in the set of poles $S(-\infty)=
\{\varphi=-\infty\}$ and in the behaviour of $\varphi$ near $S(-\infty)$.
Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$.
Assume that $\varphi$ is semi-exhaustive on $\supp\,T$ and that
$B(R)\cap\supp\,T\compact X$.
Then $P=S(-\infty)\cap{\rm Supp T}$ is compact and the results
of \S 2 show that the measure $T\wedge(dd^c\varphi)^p$ 
is well defined. Following [De2], [De4], we introduce:

\definition{3.4}{If $\varphi$ is semi-exhaustive on $\supp\,T$ and if
$R$ is such that $B(R)\cap\supp\,T\compact X$, we set for all
$r\in{}]-\infty,R[$
$$\eqalign{
\nu(T,\varphi,r)&=\int_{B(r)}T\wedge(dd^c\varphi)^p,\cr
\nu(T,\varphi)&=\int_{S(-\infty)}T\wedge(dd^c\varphi)^p=\lim_{r\to-\infty}
\nu(T,\varphi,r).\cr}$$
The number $\nu(T,\varphi)$ will be called the (generalized) Lelong
number of $T$ with respect to the weight $\varphi$.}

If we had not required $T\wedge(dd^c\varphi)^p$ to be
defined pointwise on $\varphi^{-1}(-\infty)$, the assumption
that $X$ is Stein could have been dropped: in fact, the integral over
$B(r)$ always makes sense if we define
$$\nu(T,\varphi,r)=\int_{B(r)}T\wedge\big(dd^c\max\{\varphi,s\}\big)^p~~~~
\hbox{\rm with}~~s<r.$$
Stokes' formula shows that the right hand integral is
actually independent of~$s$. The example given after (2.1) shows however
that $T\wedge(dd^c\varphi)^p$ need not exist on $\varphi^{-1}(-\infty)$
if $\varphi^{-1}(-\infty)$ contains an exceptional compact analytic
subset. We leave the reader consider by himself this more general situation
and extend our statements by the $\max\{\varphi,s\}$ technique.
Observe that $r\longmapsto\nu(T,\varphi,r)$ is always an increasing
function of $r$. Before giving examples, we need a formula.

\statement{(3.5) Formula}{For any convex increasing function 
$\chi:\bbbr\lra\bbbr$ we have
$$\int_{B(r)}T\wedge(dd^c\chi\circ\varphi)^p=
\chi'(r-0)^p\,\nu(T,\varphi,r)$$
where $\chi'(r-0)$ denotes the left derivative of $\chi$ at $r$.}

\proof{Let $\chi_\varepsilon$ be the convex function equal to
$\chi$ on $[r-\varepsilon,+\infty[$ and to
a linear function of slope $\chi'(r-\varepsilon-0)$
on $]-\infty,r-\varepsilon]$. We get
$dd^c(\chi_\varepsilon\circ\varphi)=\chi'(r-\varepsilon-0)dd^c\varphi$
on $B(r-\varepsilon)$ and Stokes' theorem implies
$$\eqalign{
\int_{B(r)}T\wedge(dd^c\chi\circ\varphi)^p
&=\int_{B(r)}T\wedge(dd^c\chi_\varepsilon\circ\varphi)^p\cr
&\ge\int_{B(r-\varepsilon)}T\wedge(dd^c\chi_\varepsilon\circ\varphi)^p\cr
&=\chi'(r-\varepsilon-0)^p\nu(T,\varphi,r-\varepsilon).\cr}$$
Similarly, taking $\wt\chi_\varepsilon$ equal to $\chi$ on 
$]-\infty,r-\varepsilon]$ and linear on $[r-\varepsilon,r]$, we obtain
$$\int_{B(r-\varepsilon)}T\wedge(dd^c\chi\circ\varphi)^p
\le\int_{B(r)}T\wedge(dd^c\wt\chi_\varepsilon\circ\varphi)^p
=\chi'(r-\varepsilon-0)^p\nu(T,\varphi,r).$$
The expected formula follows when $\varepsilon$ tends to $0$.\qed}

We get in particular
$\int_{B(r)}T\wedge(dd^ce^{2\varphi})^p=(2e^{2r})^p\nu(T,\varphi,r)$,
whence the formula
$$\nu(T,\varphi,r)=e^{-2pr}\int_{B(r)}T\wedge
\Big({1\over 2}dd^ce^{2\varphi}\Big)^p.\leqno(3.6)$$

Now, assume that $X$ is an open subset of $\bbbc^n$ and that 
$\varphi(z)=\log|z-a|$ for some $a\in X$. Formula (3.6) gives
$$\nu(T,\varphi,\log r)=r^{-2p}\int_{|z-a|<r}T\wedge\Big({\ii\over 2\pi}
d'd''|z|^2\Big)^p.$$
The positive measure $\sigma_T={1\over p!}T\wedge({\ii\over 2}d'd''|z|^2)^p
=2^{-p}\sum T_{I,I}\,.\,\ii^n dz_1\wedge\ldots\wedge d\ovl z_n$ 
is called the {\sl trace measure} of $T$. We get
$$\nu(T,\varphi,\log r)={\sigma_T\big(B(a,r)\big)\over
\pi^p r^{2p}/p!}\leqno(3.7)$$
and $\nu(T,\varphi)$ is the limit of this ratio as $r\to 0$. This
limit is called the ({\sl ordinary}) {\sl Lelong number}
of $T$ at point~$a$ and is denoted~$\nu(T,a)$.
This was precisely the original definition of Lelong 
(cf. [Le3]). Let us mention a simple but important consequence.

\statement{(3.8) Consequence}{The ratio $\sigma_T\big(B(a,r)\big)/r^{2p}$
is an increasing function of~$r$. Moreover, for every compact
subset $K\subset X$ and every $r_0<d(K,\partial X)$ we have
$$\sigma_T\big(B(a,r)\big)\le Cr^{2p}~~~~\hbox{\rm for}~~a\in K~
\hbox{\rm and}~r\le r_0,$$
where $C=\sigma_T\big(K+\ovl B(0,r_0)\big)/r_0^{2p}$.}

All these results are particularly interesting when $T=[A]$ is the 
current of integration
over an analytic subset $A\subset X$ of pure dimension $p$. Then
$\sigma_T\big(B(a,r)\big)$ is the euclidean area of $A\cap B(a,r)$,
while $\pi^pr^{2p}/p!$ is the area of a ball of radius $r$ in a
$p$-dimensional subspace of $\bbbc^n$. Thus $\nu(T,\varphi,\log r)$ is
the ratio of these areas and the Lelong number $\nu(T,a)$
is the limit ratio.

\remark{3.9}{It is immediate to check that 
$$\nu([A],x)=\cases{0&for~ $x\notin A$,\cr
                    1&when $x\in A$ is a regular point.\cr}$$
We will see later that $\nu([A],x)$ is always an integer (Thie's theorem~5.8).}

\remark{3.10}{When $X=\bbbc^n$, $\varphi(z)=\log|z-a|$ and 
$A=X$ (i.e. $T=1$), we obtain in particular $\int_{B(a,r)}
(dd^c\log|z-a|)^n=1$ for all $r$. This implies 
$$(dd^c\log|z-a|)^n=\delta_a.$$
This fundamental formula can be viewed as a higher dimensional analogue
of the usual formula $\Delta\log|z-a|=2\pi\delta_a$ in $\bbbc$.\qed}

We next prove a result which shows in particular
that the Lelong numbers of a closed positive current are zero except
on a very small set.

\statement{(3.11) Proposition}{If $T$ is a closed positive current of
bidimension $(p,p)$, then for each $c>0$ the set
$E_c=\{x\in X\,;\,\nu(T,x)\ge c$ is a closed set of locally finite
$\hc_{2p}$ Hausdorff measure in $X$.}

\proof{By (3.7), we infer
$\nu(T,a)=\lim_{r\to 0}\sigma_T\big(\ovl B(a,r)\big)p!/\pi^pr^{2p}$.
The function $a\mapsto\sigma_T\big(\ovl B(a,r)\big)$ is clearly
upper semicontinuous. Hence the decreasing limit $\nu(T,a)$ as
$r$ decreases to $0$ is also upper semicontinuous in~$a$. This implies
that $E_c$ is closed. Now, let $K$ be a compact subset in $X$ and
let $\{a_j\}_{1\le j\le N}$, $N=N(\varepsilon)$, be a maximal collection
of points in $E_c\cap K$ such that $|a_j-a_k|\ge 2\varepsilon$ for $j\ne k$.
The balls $B(a_j,2\varepsilon)$ cover $E_c\cap K$,
whereas the balls $B(a_j,\varepsilon)$ are disjoint.
If $K_{c,\varepsilon}$ is the set of points which are at distance
$\le\varepsilon$ of $E_c\cap K$, we get
$$\sigma_T(K_{c,\varepsilon})\ge\sum\sigma_T\big(B(a_j,\varepsilon)\big)
\ge N(\varepsilon)\,c\pi^p\varepsilon^{2p}/p!,$$
since $\nu(T,a_j)\ge c$. By the definition of Hausdorff measure, we infer
$$\eqalignno{
\hc_{2p}(E_c\cap K)&\le\liminf_{\varepsilon\to 0}
\sum\big(\hbox{\rm diam}\,B(a_j,2\varepsilon)\big)^{2p}\cr
&\le\liminf_{\varepsilon\to 0}N(\varepsilon)(4\varepsilon)^{2p}
\le{p!4^{2p}\over c\pi^p}\sigma_T(E_c\cap K).&\sq\cr}$$}

Finally, we conclude this section by proving two simple
semi-continuity results for Lelong numbers.

\statement{(3.12) Proposition}{Let $T_k$ be a sequence of closed positive
currents of bidimension $(p,p)$ converging weakly to a limit~$T$.
Suppose that there is a closed set $A$ such that $\supp\,T_k\subset A$
for all~$k$ and such that $\varphi$ is semi-exhaustive on $A$
with $A\cap B(R)\compact X$. Then for all $r<R$ we have
$$\eqalign{
\int_{B(r)}T\wedge(dd^c\varphi)^p
&\le\liminf_{k\to+\infty}\int_{B(r)}T_k\wedge(dd^c\varphi)^p\cr
&\le\limsup_{k\to+\infty}\int_{\ovl B(r)}T_k\wedge(dd^c\varphi)^p
\le\int_{\ovl B(r)}T\wedge(dd^c\varphi)^p.\cr}$$
When $r$ tends to $-\infty$, we find in particular
$$\limsup_{k\to+\infty}\nu(T_k,\varphi)\le\nu(T,\varphi).$$}

\proof{Let us prove for instance the third inequality. Let $\varphi_\ell$
be a sequence of smooth plurisubharmonic approximations of $\varphi$ with
$\varphi\le\varphi_\ell<\varphi+1/\ell$ on $\{r-\varepsilon\le\varphi\le
r+\varepsilon\}$. We set
$$\psi_\ell=\cases{
\varphi&on $\ovl B(r)$,\cr
\max\{\varphi,(1+\varepsilon)(\varphi_\ell-1/\ell)-r\varepsilon\}
&on $X\ssm B(r)$.\cr}$$
This definition is coherent since $\psi_\ell=\varphi$ near $S(r)$, and we have
$$\psi_\ell=(1+\varepsilon)(\varphi_\ell-1/\ell)-r\varepsilon~~~~
\hbox{\rm near}~~S(r+\varepsilon/2)$$
as soon as $\ell$ is large enough, i.e. 
$(1+\varepsilon)/\ell\le\varepsilon^2/2$. Let $\chi_\varepsilon$
be a cut-off function equal to $1$ in $B(r+\varepsilon/2)$ with support
in $B(r+\varepsilon)$. Then
$$\eqalign{
\int_{\ovl B(r)}T_k\wedge(dd^c\varphi)^p
&\le\int_{B(r+\varepsilon/2)}T_k\wedge(dd^c\psi_\ell)^p\cr
&=(1+\varepsilon)^p\int_{B(r+\varepsilon/2)}T_k\wedge(dd^c\varphi_\ell)^p\cr
&\le(1+\varepsilon)^p\int_{B(r+\varepsilon)}\chi_\varepsilon
T_k\wedge(dd^c\varphi_\ell)^p.\cr}$$
As $\chi_\varepsilon(dd^c\varphi_\ell)^p$ is smooth with compact support and
as $T_k$ converges weakly to~$T$, we infer
$$\limsup_{k\to+\infty}\int_{\ovl B(r)}T_k\wedge(dd^c\varphi)^p
\le(1+\varepsilon)^p\int_{B(r+\varepsilon)}\chi_\varepsilon
T\wedge(dd^c\varphi_\ell)^p.$$
We then let $\ell$ tend to $+\infty$ and $\varepsilon$ tend to $0$ to get
the desired inequality. The first inequality is obtained in a similar way,
we define $\psi_\ell$ so that $\psi_\ell=\varphi$ on $X\ssm B(r)$ and
$\psi_\ell=\max\{(1-\varepsilon)(\varphi_\ell-1/\ell)+r\varepsilon\}$ on
$\ovl B(r)$, and we take $\chi_\varepsilon=1$ on $B(r-\varepsilon)$
with $\supp\,\chi_\varepsilon\subset B(r-\varepsilon/2)$. Then
for $\ell$ large
$$\eqalignno{
\int_{B(r)}T_k\wedge(dd^c\varphi)^p
&\ge\int_{B(r-\varepsilon/2)}T_k\wedge(dd^c\psi_\ell)^p\cr
&\ge(1-\varepsilon)^p\int_{B(r-\varepsilon/2)}\chi_\varepsilon
T_k\wedge(dd^c\varphi_\ell)^p.&\sq\cr}$$}

\statement{(3.13) Proposition}{Let $\varphi_k$ be a $($non necessarily
monotone$)$ sequence of continuous plurisubharmonic functions such that
$e^{\varphi_k}$ converges uniformly to $e^\varphi$ on every compact
subset of $X$. Suppose that $\{\varphi<R\}\cap\supp\,T\compact X$.
Then for $r<R$ we have
$$\limsup_{k\to+\infty}\int_{\{\varphi_k\le r\}\cap\{\varphi<R\}}
T\wedge(dd^c\varphi_k)^p\le
\int_{\{\varphi\le r\}}T\wedge(dd^c\varphi)^p.$$
In particular $\limsup_{k\to+\infty}\nu(T,\varphi_k)\le\nu(T,\varphi)$.}

When we take $\varphi_k(z)=\log|z-a_k|$ with $a_k\to a$,
Prop.~3.13 implies the upper semicontinuity of $a\mapsto\nu(T,a)$ which
was already noticed in the proof of Prop.~3.11.

\proof{Our assumption is equivalent to saying that $\max\{\varphi_k,t\}$
converges locally uniformly to $\max\{\varphi,t\}$ for every~$t$.
Then Cor.~1.6 shows that $T\wedge(dd^c\max\{\varphi_k,t\})^p$
converges weakly to $T\wedge(dd^c\max\{\varphi,t\})^p$. If $\chi_\varepsilon$
is a cut-off function equal to $1$ on $\{\varphi\le r+\varepsilon/2\}$
with support in $\{\varphi<r+\varepsilon\}$, we get
$$\lim_{k\to+\infty}\int_X\chi_\varepsilon T\wedge(dd^c\max\{\varphi_k,t\})^p=
\int_X\chi_\varepsilon T\wedge(dd^c\max\{\varphi,t\})^p.$$
For $k$ large, we have $\{\varphi_k\le r\}\cap\{\varphi<R\}\subset
\{\varphi<r+\varepsilon/2\}$, thus when $\varepsilon$ tends to $0$ we infer
$$\limsup_{k\to+\infty}\int_{\{\varphi_k\le r\}\cap\{\varphi<R\}}
T\wedge(dd^c\max\{\varphi_k,t\})^p\le
\int_{\{\varphi\le r\}}T\wedge(dd^c\max\{\varphi,t\})^p.$$
When we choose $t<r$, this is equivalent to the first inequality in
statement (3.13).\qed}

\titleb{4. The Lelong-Jensen Formula}
We assume in this section that $X$ is Stein, that $\varphi$ is
{\sl semi-exhaustive} on $X$ and that $B(R)\compact X$. We set
for simplicity $\varphi_{\gge r}=\max\{\varphi,r\}$.
For every $r\in{}]-\infty,R[$, the measures
$dd^c(\varphi_{\gge r})^n$ are well defined.  By Cor.~1.6, the map 
$r\longmapsto(dd^c\varphi_{\gge r})^n$ is continuous on $]-\infty,R[$
with respect to the weak topology. As
$(dd^c\varphi_{\gge r})^n=(dd^c\varphi)^n$ on $X\ssm\ovl B(r)$
and as $\varphi_{\gge r}\equiv r$, $(dd^c\varphi_{\gge r})^n=0$
on~$B(r)$, the left continuity implies 
$(dd^c\varphi_{\gge r})^n\ge\bbbone_{X\ssm B(r)}(dd^c\varphi)^n$. 
Here $\bbbone_A$ denotes the characteristic function of any subset
$A\subset X$. According to the definition introduced in [De3], the
collection of {\sl Monge-Amp\`ere measures} associated with $\varphi$ is 
the family of positive measures $\mu_r$ such that
$$\mu_r=(dd^c\varphi_{\gge r})^n-\bbbone_{X\ssm B(r)}(dd^c\varphi)^n,
~~~~r\in{}]-\infty,R[.\leqno(4.1)$$ 
The measure $\mu_r$ is supported on $S(r)$ and $r\longmapsto\mu_r$ is 
weakly continuous on the left by the bounded convergence theorem. 
Stokes' formula shows
that $\int_{B(s)}(dd^c\varphi_{\gge r})^n-(dd^c\varphi)^n=0$ for $s>r$,
hence the total mass $\mu_r(S(r))=\mu_r(B(s))$ is equal
to the difference between the masses of $(dd^c\varphi)^n$ and 
$\bbbone_{X\ssm B(r)}(dd^c\varphi)^n$ over $B(s)$, i.e.
$$\mu_r\big(S(r)\big)=\int_{B(r)}(dd^c\varphi)^n.\leqno(4.2)$$

\example{4.3}{When $(dd^c\varphi)^n=0$ on
$X\ssm\varphi^{-1}(-\infty)$, formula (4.1) can be simplified into
$\mu_r=(dd^c\varphi_{\gge r})^n$. This is so for $\varphi(z)=\log|z|$.
In this case, the invariance of $\varphi$
under unitary transformations implies that $\mu_r$ is also invariant.
As the total mass of $\mu_r$ is equal to $1$ by 3.10 and (4.2), we see
that $\mu_r$ is the invariant measure of mass $1$ on the euclidean sphere
of radius $e^r$.} 

\statement{Proposition 4.4}{Assume that $\varphi$ is smooth near $S(r)$ and
that $d\varphi\ne 0$ on $S(r)$, i.e. $r$ is a non critical value. Then
$S(r)=\partial B(r)$ is a smooth oriented real hypersurface and
the measure $\mu_r$ is given by the $(2n-1)$-volume form
$(dd^c\varphi)^{n-1}\wedge d^c\varphi_{\restriction S(r)}$.}

\proof{Write $\max\{t,r\}=\lim_{k\to+\infty}\chi_k(t)$ where $\chi$ is
a decreasing sequence of smooth convex functions with
$\chi_k(t)=r$ for $t\le r-1/k$, $\chi_k(t)=t$\break for $t\ge r+1/k$.
Corollary~1.6 shows that $(dd^c\chi_k\circ\varphi)^n$ converges
weakly to~$(dd^c\varphi_{\gge r})^n$.
Let $h$ be a smooth function with compact support near~$S(r)$.
We first compute the limit, noting that $\lim\chi'_k\circ\varphi=
\bbbone_{X\ssm B(r)}$ a.e., and then apply Stokes' theorem 
on $X\ssm B(r)$ (the boundary is $S(r)$ with opposite orientation):
$$\leqalignno{
\int_X h(dd^c\varphi_{\gge r})^n&=\lim_{k\to+\infty}
\int_X h(dd^c\chi_k\circ\varphi)^n\cr
&=\lim_{k\to+\infty}\int_X -dh\wedge(dd^c\chi_k\circ\varphi)^{n-1}
\wedge d^c(\chi_k\circ\varphi)\cr
&=\lim_{k\to+\infty}\int_X -(\chi'_k\circ\varphi)^n\,dh\wedge(dd^c\varphi)^{n-1}
\wedge d^c\varphi\cr
&=\int_{X\ssm B(r)}-dh\wedge(dd^c\varphi)^{n-1}\wedge d^c\varphi\cr
&=\int_{S(r)}h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi+
\int_{X\ssm B(r)}h\,(dd^c\varphi)^{n-1}\wedge d^c\varphi.\cr}$$
Near $S(r)$ we thus have an equality of measures
$$(dd^c\varphi_{\gge r})^n=(dd^c\varphi)^{n-1}\wedge d^c
\varphi_{\restriction S(r)}+\bbbone_{X\ssm B(r)}(dd^c\varphi)^n.
\eqno{\sq}$$}

\statement{(4.5) Lelong-Jensen formula}{Let $V$ be any plurisubharmonic function 
on~$X$. Then $V$ is $\mu_r$-integrable for every $r\in{}]-\infty,R[$ and
$$\mu_r(V)-\int_{B(r)}V(dd^c\varphi)^n=
\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt.$$}

\proof{Proposition 1.11 shows that $V$ is integrable with respect to
$(dd^c\varphi_{\gge r})^n$, hence $V$ is $\mu_r$-integrable.
By definition 
$$\nu(dd^cV,\varphi,t)=\int_{\varphi(z)<t}dd^cV\wedge(dd^c\varphi)^{n-1}$$
and the Fubini theorem gives
$$\leqalignno{
\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt
&=\int\!\!\!\int_{\varphi(z)<t<r}dd^cV(z)\wedge(dd^c\varphi(z))^{n-1}\,dt\cr
&=\int_{B(r)}(r-\varphi)dd^cV\wedge(dd^c\varphi)^{n-1}.&(4.6)\cr}$$
We first show that formula 4.5 is true when $\varphi$ and $V$ are
smooth. As both members of the formula are left continuous with
respect to $r$ and as almost all values of $\varphi$ are non critical
by Sard's theorem, we may assume $r$ non critical. Formula 1.1 applied
with $f=(r-\varphi)(dd^c\varphi)^{n-1}$ and $g=V$ shows 
that integral $(4.6)$ is equal to
$$\int_{S(r)}V(dd^c\varphi)^{n-1}\wedge d^c\varphi-\int_{B(r)}
V\,(dd^c\varphi)^n=\mu_r(V)-\int_{B(r)}V\,(dd^c\varphi)^n.$$
Formula 4.5 is thus proved when $\varphi$ and $V$ are smooth.
If $V$ is smooth and $\varphi$ merely continuous and finite,
one can write $\varphi=\lim\varphi_k$ where $\varphi_k$ is a 
decreasing sequence of smooth plurisubharmonic functions 
(because $X$ is Stein). Then $dd^cV\wedge(dd^c\varphi_k)^{n-1}$
converges weakly to $dd^cV\wedge(dd^c\varphi)^{n-1}$ and (4.6)
converges, since $\bbbone_{B(r)}(r-\varphi)$ is continuous with compact
support on~$X$. The left hand side of formula 4.5 also
converges because the definition of $\mu_r$ implies
$$\mu_{k,r}(V)-\int_{\varphi_k<r}V(dd^c\varphi_k)^n=
\int_X V\big((dd^c\varphi_{k,\gge r})^n-(dd^c\varphi_k)^n\big)$$
and we can apply again weak convergence on a neighborhood of
$\ovl B(r)$. If $\varphi$ assumes $-\infty$ as a value, we replace
$\varphi$ by $\varphi_{\gge-k}$ where $k\to+\infty$. Then $\mu_r(V)$
is unchanged, $\int_{B(r)}V(dd^c\varphi_{\gge -k})^n$ converges
to $\int_{B(r)}V(dd^c\varphi)^n$ and the right hand side of formula 4.5
is replaced by $\int_{-k}^r\nu(dd^cV,\varphi,t)\,dt$.
Finally, for $V$ arbitrary, write $V=\lim\downarrow V_l$
with a sequence of smooth functions $V_l$.
Then $dd^cV_l\wedge(dd^c\varphi)^{n-1}$ converges weakly to 
$dd^cV\wedge(dd^c\varphi)^{n-1}$ by Prop.~2.4, so
$$\liminf_{k\to+\infty}\int_{B(r)}(r-\varphi)dd^cV_l\wedge(dd^c\varphi)^{n-1}
\ge\int_{B(r)}(r-\varphi)dd^cV\wedge(dd^c\varphi)^{n-1},$$
with equality if $\varphi$ is finite. As $\mu_r(V_l)$ and
$\int_{B(r)}V_l(dd^c\varphi)^n$ converge to the expected
limits by the monotone convergence theorem, we get
$$\mu_r(V)-\int_{B(r)}V(dd^c\varphi)^n\ge
\int_{B(r)}(r-\varphi)dd^cV\wedge(dd^c\varphi)^{n-1},$$
with equality if $\varphi$ does not assume $-\infty$ as a value. In
particular, replacing $\varphi$ by $\varphi_{\gge-k}$, we find
$$\mu_r(V)-\int_{B(r)}V(dd^c\varphi_{\gge-k})^n=\int_k^r\nu(dd^cV,\varphi,t)
\,dt,$$
and from the equality $-V=\sup -V_l$ we easily get
$$\int_{B(r)}-V(dd^c\varphi)^n\le\liminf_{k\to+\infty}\int_{B(r)}
-V(dd^c\varphi_{\gge-k})^n.$$
A limit as $k\to+\infty$ yields the expected converse inequality
$$\mu_r(V)-\int_{B(r)}V(dd^c\varphi)^n\le\int_{-\infty}^r\nu(dd^cV,\varphi,t)
\,dt.\eqno{\sq}$$}

For $r<r_0<R$, the Lelong-Jensen formula implies
$$\mu_r(V)-\mu_{r_0}(V)+\int_{B(r_0)\ssm B(r)}V(dd^c\varphi)^n=
\int_{r_0}^r\nu(dd^cV,\varphi,t)\,dt.\leqno(4.7)$$

\corollary{4.8}{Assume that $(dd^c\varphi)^n=0$ on $X\ssm
S(-\infty)$. Then $r\longmapsto\mu_r(V)$ is a convex increasing
function of $r$ and the lelong number $\nu(dd^cV,\varphi)$ is given by
$$\nu(dd^cV,\varphi)=\lim_{r\to-\infty}{\mu_r(V)\over r}.$$}

\proof{By (4.7) we have
$$\mu_r(V)=\mu_{r_0}(V)+\int_{r_0}^r\nu(dd^cV,\varphi,t)\,dt.$$
As $\nu(dd^cV,\varphi,t)$ is increasing and nonnegative, it follows
that $r\longmapsto\mu_r(V)$ is convex and increasing. The formula
for $\nu(dd^cV,\varphi)=\lim_{t\to-\infty}\nu(dd^cV,\varphi,t)$ is then
obvious.\qed}

\example{4.9}{Let $X$ be an open subset of $\bbbc^n$ equipped with the
semi-exhaustive function $\varphi(z)=\log|z-a|$, $a\in X$. Then 
$(dd^c\varphi)^n=\delta_a$ and the Lelong-Jensen formula becomes
$$\mu_r(V)=V(a)+\int_{-\infty}^r\nu(dd^cV,\varphi,t)\,dt.$$
As $\mu_r$ is the mean value measure on the sphere $S(a,e^r)$, we
make the change of variables $r\mapsto\log r$, $t\mapsto\log t$ and
obtain the more familiar formula
$$\mu(V,S(a,r))=V(a)+\int_0^r\nu(dd^cV,a,t)\,{dt\over t}\leqno
(4.9\,{\rm a)}$$
where $\nu(dd^cV,a,t)=\nu(dd^cV,\varphi,\log t)$ is given by (3.7):
$$\nu(dd^cV,a,t)=
{1\over\pi^{n-1}t^{2n-2}/(n-1)!}\int_{B(a,t)}{1\over2\pi}\Delta V.
\leqno(4.9\,{\rm b)}$$
In this setting, Cor.~4.8 implies
$$\nu(dd^cV,a)=\lim_{r\to 0}{\mu\big(V,S(a,r)\big)\over\log r}=
\lim_{r\to 0}{\sup_{S(a,r)}V\over\log r}.\leqno(4.9\,{\rm c)}$$
To prove the last equality, we may assume $V\le 0$ after subtraction of
a constant. Inequality $\ge$ follows from the
obvious estimate $\mu(V,S(a,r))\le\sup_{S(a,r)}V$, while inequality $\le$
follows from the standard Harnack estimate
$$\sup_{S(a,\varepsilon r)}V\le{1-\varepsilon\over(1+\varepsilon)^{2n-1}}\,
\mu\big(V,S(a,r)\big)\leqno(4.9\,{\rm d)}$$
when $\varepsilon$ is small. As $\sup_{S(a,r)}V=\sup_{B(a,r)}V$,
Formula (4.9$\,$c) can also be rewritten $\nu(dd^cV,a)=\liminf_{z\to a}
V(z)/\log|z-a|$. Since $\sup_{S(a,r)}V$ is a convex (increasing) function
of $\log r$, we infer that
$$V(z)\le\gamma\log|z-a|+{\rm O}(1)\leqno(4.9\,{\rm e)}$$
with $\gamma=\nu(dd^cV,a)$, and $\nu(dd^cV,a)$ is the largest constant
$\gamma$ which satisfies this inequality. Thus $\nu(dd^cV,a)=\gamma$
is equivalent to $V$ having a logarithmic pole of coefficient $\gamma$.}

\pproof{\bf (4.10) Special case}{Take in particular $V=\log|f|$ where
$f$ is a holomorphic function on~$X$.
The Lelong-Poincar\'e formula shows that $dd^c\log|f|$ is
equal to the zero divisor $[Z_f]=\sum m_j[H_j]$, where $H_j$ are the
irreducible components of $f^{-1}(0)$ and $m_j$ is the multiplicity of
$f$ on $H_j$. The trace ${1\over 2\pi}\Delta f$ is then the euclidean
area measure of $Z_f$ (with corresponding multiplicities $m_j$).
By Formula (4.9$\,$c), we see that the Lelong number $\nu([Z_f],a)$
is equal to the vanishing order ${\rm ord}_a(f)$, that is, the smallest
integer $m$ such that $D^\alpha f(a)\ne 0$ for some multiindex $\alpha$
with~$|\alpha|=m$. In
dimension $n=1$, we have ${1\over2\pi}\Delta f=\sum m_j\delta_{a_j}$.
Then (4.9$\,a$) is the usual Jensen formula
$$\mu\big(\log|f|,S(0,r)\big)-\log|f(0)|=\int_0^r\nu(t){dt\over t}=
\sum m_j\log{r\over|a_j|}$$
where $\nu(t)$ is the number of zeros $a_j$ in the disk $D(0,t)$,
counted with multi\-plicities $m_j$.}

\example{4.11}{Take $\varphi(z)=\log\max|z_j|^{\lambda_j}$
where $\lambda_j>0$. Then $B(r)$ is the polydisk of
radii $(e^{r/\lambda_1}\ld e^{r/\lambda_n})$. If some coordinate $z_j$
is non zero, say $z_1$, we can write $\varphi(z)$ as
$\lambda_1\log|z_1|$ plus some function depending only on
the $(n-1)$ variables $z_j/z_1^{\lambda_1/\lambda_j}$. Hence
$(dd^c\varphi)^n=0$ on $\bbbc^n\ssm\{0\}$. It will be shown
later that 
$$(dd^c\varphi)^n=\lambda_1\ldots\lambda_n\,\delta_0.\leqno(4.11\,{\rm a})$$
We now determine the measures $\mu_r$.
At any point $z$ where not all terms $|z_j|^{\lambda_j}$
are equal, the smallest one can be omitted without changing $\varphi$
in a neighborhood of $z$. Thus $\varphi$ depends only on
$(n-1)$-variables and $(dd^c\varphi_{\ge r})^n=0$, $\mu_r=0$
near $z$. It follows that $\mu_r$ is supported by the
distinguished boundary $|z_j|=e^{r/\lambda_j}$ of the polydisk $B(r)$.
As $\varphi$ is invariant by all rotations $z_j\longmapsto
e^{\ii\theta_j}z_j$, the measure $\mu_r$ is also invariant and we see that
$\mu_r$ is a constant multiple of $d\theta_1\ldots d\theta_n$.
By formula (4.2) and (4.11$\,$a) we get
$$\mu_r=\lambda_1\ldots\lambda_n\,(2\pi)^{-n}d\theta_1\ldots d\theta_n.
\leqno(4.11\,{\rm b})$$
In particular, the Lelong number $\nu(dd^cV,\varphi)$ is given by
$$\nu(dd^cV,\varphi)=\lim_{r\to-\infty}{\lambda_1\ldots\lambda_n\over r}
\int_{\theta_j\in[0,2\pi]}\kern-6pt V(e^{r/\lambda_1+\ii\theta_1}
\ld e^{r/\lambda_n+\ii\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}.$$
These numbers have been introduced and studied by Kiselman [Ki4].
We call them {\sl directional Lelong numbers} with coefficients
$(\lambda_1\ld\lambda_n)$. For an arbitrary current $T$, we define
$$\nu(T,x,\lambda)=\nu\big(T,\log\max|z_j-x_j|^{\lambda_j}\big).
\leqno(4.11\,{\rm c})$$
The above formula for $\nu(dd^cV,\varphi)$ combined with the analogue of
Harnack's inequality (4.9$\,$d) for polydisks gives
$$\leqalignno{
\nu(dd^cV,x,\lambda)&=\lim_{r\to 0}{\lambda_1\ldots\lambda_n\over\log r}
\int V(r^{1/\lambda_1}e^{\ii\theta_1}\ld r^{1/\lambda_n}
e^{\ii\theta_n})\,{d\theta_1\ldots d\theta_n\over(2\pi)^n}\cr
&=\lim_{r\to 0}{\lambda_1\ldots\lambda_n\over\log r}
\sup_{\theta_1\ld\theta_n}V(r^{1/\lambda_1}e^{\ii\theta_1}\ld r^{1/\lambda_n}
e^{\ii\theta_n}).&(4.11\,{\rm d})\cr}$$}

\titleb{5. Comparison Theorems for Lelong Numbers}
Let $T$ be a closed positive current of bidimension $(p,p)$ on a Stein
manifold $X$ equipped with a semi-exhaustive plurisubharmonic
weight~$\varphi$. We first show that the Lelong numbers $\nu(T,\varphi)$ only
depend on the asymptotic behaviour of $\varphi$ near the polar set
$S(-\infty)$. In a precise way:

\statement{(5.1) First comparison theorem}{Let
$\varphi,\psi:X\longrightarrow[-\infty,+\infty[$
be continuous plurisubharmonic functions. We assume that $\varphi,\psi$ are 
semi-exhaustive on $\supp\,T$ and that
$$\ell:=\limsup{{\psi(x)}\over{\varphi(x)}}<+\infty~~~~{\sl as}~~
x\in\supp\,T~~{\sl and}~~\varphi(x)\to -\infty.$$
Then $\nu(T,\psi)\le \ell^p\nu(T,\varphi)$, and the equality holds if
$\ell=\lim\psi/\varphi$.}

\proof{Definition 3.4 shows immediately that
$\nu(T,\lambda\varphi)=\lambda^p\nu(T,\varphi)$
for every scalar $\lambda>0$. It is thus sufficient to verify the
inequality $\nu(T,\psi)\le\nu(T,\varphi)$ under the hypothesis 
$\limsup\psi/\varphi<1$. For all $c>0$, consider the plurisubharmonic
function 
$$u_c=\max(\psi-c,\varphi).$$
Let $R_\varphi$ and $R_\psi$
be such that $B_\varphi(R_\varphi)\cap\supp\,T$ and
$B_\psi(R_\psi)\cap\supp\,T$ be relatively compact in $X$.
Let $r<R_\varphi$ and $a<r$ be fixed. For $c>0$ large enough, we have 
$u_c=\varphi$ on $\varphi^{-1}([a,r])$ and Stokes' formula gives
$$\nu(T,\varphi,r)=\nu(T,u_c,r)\ge\nu(T,u_c).$$
The hypothesis $\limsup\psi/\varphi<1$ implies on the other hand that
there exists $t_0<0$ such that $u_c=\psi-c$ on $\{u_c<t_0\}\cap
\supp\,T$. We infer 
$$\nu(T,u_c)=\nu(T,\psi-c)=\nu(T,\psi),$$
hence $\nu(T,\psi)\le\nu(T,\varphi)$. The equality case is obtained by
reversing the roles of $\varphi$ and $\psi$ and observing that
$\lim\varphi/\psi=1/l$.\qed}

  Assume in particular that $z^k=(z^k_1\ld z^k_n)$, $k=1,2$, are
coordinate systems centered at a point $x\in X$ and let
   $$\varphi_k(z)=\log|z^k|=\log\bigl(|z^k_1|^2+\ldots+|z^k_n|^2\bigr)^{1/2}.$$
We have $\lim_{z\to x}{\varphi_2(z)/\varphi_1(z)}=1$, hence 
$\nu(T,\varphi_1)=\nu(T,\varphi_2)$ by Th.~5.1. 

\corollary{5.2}{The usual Lelong numbers $\nu(T,x)$ are independent
of the choice of local coordinates.\qed}

This result had been originally proved by [Siu] with a much more
delicate proof. Another interesting consequence is:

\corollary{5.3}{On an open subset of $\bbbc^n$, the Lelong numbers
and Kiselman numbers are related by
     $$\nu(T,x)=\nu\big(T,x,(1\ld 1)\big).$$}

\proof{By definition, the Lelong number $\nu(T,x)$ is associated with the 
weight $\varphi(z)=\log|z-x|$ and the Kiselman number 
$\nu\big(T,x,(1\ld 1)\big)$ to the weight
$\psi(z)=\log\max|z_j-x_j|$. It is clear that
$\lim_{z\to x}\psi(z)/\varphi(z)=1$, whence the conclusion.\qed}

Another consequence of Th.~5.1 is that $\nu(T,x,\lambda)$ is an
increasing function of each variable~$\lambda_j$. Moreover, if
$\lambda_1\le\ldots\le\lambda_n$, we get the inequalities
$$\lambda_1^p\nu(T,x)\le\nu(T,x,\lambda)\le\lambda_n^p\nu(T,x).$$
These inequalities will be improved in section~7
(see Cor.~7.14). For the moment, we just prove the following
special case.

\corollary{5.4}{For all $\lambda_1\ld\lambda_n>0$ we have
$$\big(dd^c\log\max_{1\le j\le n}|z_j|^{\lambda_j}\big)^n=
\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\lambda_j}\Big)^n=
\lambda_1\ldots\lambda_n\,\delta_0.$$}

\proof{In fact, our measures vanish on $\bbbc^n\ssm\{0\}$ by the arguments
explained in example~4.11. Hence they are equal to $c\,\delta_0$
for some constant $c\ge 0$ which is simply the Lelong number of the
bidimension $(n,n)$-current $T=[X]=1$ with the corresponding weight.
The comparison theorem shows that the first equality holds and that
$$\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\lambda_j}\Big)^n=
\ell^{-n}\Big(dd^c\log\sum_{1\le j\le n}|z_j|^{\ell\lambda_j}\Big)^n$$
for all $\ell>0$. By taking $\ell$ large and approximating $\ell\lambda_j$
with $2[\ell\lambda_j/2]$, we may assume that $\lambda_j=2s_j$ is
an even integer. Then formula (3.6) gives
$$\eqalign{
&\int_{\sum|z_j|^{2s_j}<r^2}\Big(dd^c\log\sum|z_j|^{2s_j}\Big)^n=
r^{-2n}\int_{\sum|z_j|^{2s_j}<r^2}\Big(dd^c\sum|z_j|^{2s_j}\Big)^n\cr
&\qquad{}=s_1\ldots s_n\,r^{-2n}\int_{\sum|w_j|^2<r^2}2^n\Big({\ii\over 2\pi}
d'd''|w|^2\Big)^n=\lambda_1\ldots\lambda_n\cr}$$
by using the $s_1\ldots s_n$-sheeted change of variables
$w_j=z_j^{s_j}$.\qed}


Now, we assume that $T=[A]$ is the current of integration over an
analytic set $A\subset X$ of pure dimension $p$ (cf. P. Lelong[Le1]).
The above comparison theorem will enable us to give a simple
proof of P.~Thie's main result~[Th]: the Lelong number $\nu([A],x)$ 
can be interpreted as the multiplicity of the analytic set $A$ at point
$x$.

   Let $x\in A$ be a given point and $\ic_{A,x}$ the ideal of
germs of holomorphic functions at $x$ vanishing on $A$.
Then, one can find local coordinates $z=(z_1,\ldots,z_n)$ on $X$
centered at $x$ such that there exist distinguished Weierstrass
polynomials $P_j\in\ic_{A,x}$ in the variable $z_j$, $p<j\le n$,
of the type
$$P_j(z)=z_j^{d_j}+\sum_{k=1}^{d_j}a_{j,k}(z_1,\ldots,z_{j-1})z_j^{d_j-k}\ ,\ 
                a_{j,k}\in{\cal M}^k_{\bbbc^{j-1},0}\leqno(5.5)$$
where ${\cal M}_{X,x}$ is the maximal ideal of $X$ at $x$.

Indeed, we will prove this property by induction on codim$\,X=n-p$.
We~fix a coordinate system $(w_1\ld w_n)$ by which we identify
the germ $(X,x)$ to $(\bbbc^n,0)$.

   If $n-p\ge 1$, there exists a non zero element $f\in\ic_{A,x}$.
Let $d$ be the smallest integer such that $f\in{\cal M}_{\bbbc^n,0}^d$ and
let $e_n\in\bbbc^n$ be a non zero vector such that 
$\lim_{t\to 0}{f(te_n)/t^d\ne 0}$. Complete $e_n$ into a basis 
$(\tilde e_1,\ldots,\tilde e_{n-1},e_n)$ of $\bbbc^n$ and denote by
$(\tilde z_1,\ldots,\tilde z_{n-1},z_n)$ the corresponding coordinates.
The Weierstrass preparation theorem gives a factorization 
$f=gP$ where $P$ is a distinguished polynomial of type (5.5) in the
variable $z_n$ and where $g$ is an invertible holomorphic function 
at point $x$. If $n-p=1$, the polynomial $P_n=P$ satisfies the
requirements. Observe that a generic choice of $e_n$ actually works,
since $e_n$ only has to avoid the algebraic hypersurface $f_d(z)=0$
where $f_d$ is the polynomial of lowest degree in the Taylor expansion
of $f$ at~$0$.

   If $n-p\ge 2$, $\oc_{A,x}=\oc_{X,x}/\ic_{A,x}$ is a
$\oc_{\bbbc^{n-1},0}=\bbbc\{\tilde z_1,\ldots,\tilde z_{n-1}\}$-module
of finite type, i.e. the projection ${\rm pr}:(X,x)\approx
(\bbbc^n,0)\longrightarrow(\bbbc^{n-1},0)$ is a finite morphism 
of $(A,x)$ onto a germ $(Z,0)\subset(\bbbc^{n-1},0)$ of dimension~$p$.
The induction hypothesis applied to
$\ic_{Z,0}=\oc_{\bbbc^{n-1},0}\cap\ic_{A,x}$ implies the
existence of a new basis $(e_1,\ldots,e_{n-1})$ of $\bbbc^{n-1}$ and
of Weierstrass polynomials\break $P_{p+1},\ldots,P_{n-1}\in\ic_{Z,0}$,
of the type (5.5) with respect to the coordinates $(z_1,\ldots,z_{n-1})$
associated with $(e_1,\ldots,e_{n-1})$. The polynomials
$P_{p+1}\ld P_n$ show that the expected property also holds in
codimension~$n-p$.

  For any polynomial $Q(w)=w^d+a_1w^{d-1}+\ldots+a_d\in\bbbc[w]$,
the roots $w$ of $Q$ satisfy
         $$|w|\le 2\max_{1\le k\le d}|a_k|^{1/k},\leqno(5.6)$$
otherwise $Q(w)w^{-d}=1+a_1w^{-1}+\ldots+a_dw^{-d}$ would have a
modulus larger than $1-(2^{-1}+\ldots+2^{-d})=2^{-d}$, a contradiction.
Let us denote $z=(z',z'')$ with $z'=(z_1,\ldots,z_p)$ and
$z''=(z_{p+1},\ldots,z_n)$. As $a_{j,k}\in{\cal M}^k_{\bbbc^{j-1},0}$,
we get
   $$|a_{j,k}(z_1,\ldots,z_{j-1})|=
    {\rm O}\big((|z_1|+\ldots+|z_{j-1}|)^k\big)~~~{\rm if}~~j>p,$$
and we deduce from (5.5), (5.6) that $|z_j|=O(|z_1|+\ldots+|z_{j-1}|)$
on $(A,x)$. Therefore, we get:

\lemma{5.7}{For a generic choice of coordinates $z'=(z_1\ld z_p)$ and
$z''=(z_{p+1}\ld z_n)$ on $(X,x)$, the germ $(A,x)$ is contained in a
cone\break $|z''|\le C|z'|$.\qed}

   We use this property to compute the Lelong number
of $[A]$ at point~$x$. When $z\in A$ tends to $x$, the functions
$$\varphi(z)=\log|z|=\log(|z'|^2+|z''|^2)^{1/2},~~~~\psi(z)=\log|z'|.$$
are equivalent. As $\varphi,\psi$ are semi-exhaustive on $A$, 
Th.~5.1 implies
        $$\nu([A],x)=\nu([A],\varphi)=\nu([A],\psi).$$
Let $B'\subset\bbbc^p$ the ball of center $0$ and radius $r'$, 
$B''\subset\bbbc^{n-p}$ the ball of center $0$ and radius $r''=Cr'$. 
The inclusion of germ $(A,x)$ in the cone $|z''|\le C|z'|$ shows that
for $r'$ small enough the projection
      $${\rm pr}:A\cap(B'\times B'')\longrightarrow B'$$
is proper. The fibers are finite by (5.5). Hence this projection
is a ramified covering with finite sheet number~$m$ (see Fig.~3).

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\figure{3} {Ramified covering $\pi$ and ramification locus $S$}
\endfig
\vskip3mm


\pritem{}Let us apply formula (3.6) to $\psi\,$: for every $t<r'$ we get
$$\eqalign{\nu([A],\psi,\log t)
&=t^{-2p}\int_{\{\psi<\log t\}}[A]\wedge\Big({1\over 2}dd^ce^{2\psi}\Big)^p\cr
&=t^{-2p}\int_{A\cap\{|z'|<t\}}\Big({1\over 2}{\rm pr}^\star dd^c|z'|^2\Big)^p\cr
&=m\,t^{-2p}\int_{\bbbc^p\cap\{|z'|<t\}}\Big({1\over 2}dd^c|z'|^2\Big)^p=m,\cr}$$
hence $\nu(T,\psi)=m$.  Here, we used the fact that pr is actually a
covering with $m$ sheets over the complement of the ramification locus
$S\subset B'$, which is of zero Lebesgue measure.  We thus obtain a new
proof of Thie's result~[Th] that $\nu([A],x)$ is equal to the multiplicity
of $A$ at~$x\,$:

\theorem{5.8}{Let $A$ be an analytic set of dimension $p$ in a complex
manifold of dimension~$n$. For every point $x\in A$, there exist local
coordinates
    $$z=(z',z''),~~~z'=(z_1,\ldots,z_p),~~~z''=(z_{p+1},\ldots,z_n)$$
centered at $x$ and balls $B'\subset\bbbc^p$, $B''\subset\bbbc^{n-p}$
of radii $r',r''$ in these coordi\-nates, such that
$A\cap(B'\times B'')$ is contained in the cone \hbox{$|z''|\le(r''/r')|z'|$.}
The multiplicity of $A$ at $x$ is defined as the number $m$ of sheets of any
such ramified covering map $A\cap(B'\times B'')\longrightarrow B'$.
Then $\nu([A],x)=m$.}

There is another interesting version of the comparison theorem which
compares the Lelong numbers of two currents obtained as
intersection products (in that case, we take the same weight for both).

\statement{(5.9) Second comparison theorem}{Let $u_1\ld u_q$ and $v_1\ld v_q$
be plurisubharmonic functions such that each $q$-tuple satisfies the
hypotheses of Th.~2.5 with respect to~$T$. Suppose moreover that
$u_j=-\infty$ on $\supp\,T\cap\varphi^{-1}(-\infty)$ and that
$$\ell_j:=\limsup{v_j(z)\over u_j(z)}<+\infty~~~~\hbox{\rm when}~~
z\in\supp\,T\ssm u_j^{-1}(-\infty),~~\varphi(z)\to -\infty.$$
Then
$$\nu(dd^cv_1\wedge\ldots\wedge dd^cv_q\wedge T,\varphi)\le
\ell_1\ldots\ell_q\,\nu(dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T,\varphi).$$}

\proof{By homogeneity in each factor $v_j$, it is enough to prove the
inequality with constants $\ell_j=1$ under the hypothesis $\limsup v_j/u_j<1$.
We set
$$w_{j,c}=\max\{v_j-c,u_j\}.$$
Our assumption implies that $w_{j,c}$ coincides with $v_j-c$ on a
neighborhood $\supp\,T\cap\{\varphi<r_0\}$ of
$\supp\,T\cap\{\varphi<-\infty\}$, thus
$$\nu(dd^cv_1\wedge\ldots\wedge dd^cv_q\wedge T,\varphi)=
\nu(dd^cw_{1,c}\wedge\ldots\wedge dd^cw_{q,c}\wedge T,\varphi)$$
for every $c$. Now, fix $r<R_\varphi$. Proposition~2.9 shows that the
current\break $dd^cw_{1,c}\wedge\ldots\wedge dd^cw_{q,c}\wedge T$ converges
weakly to $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ when $c$ tends
to~$+\infty$. By Prop.~3.12 we get
$$\limsup_{c\to+\infty}~\nu(dd^cw_{1,c}\wedge\ldots\wedge dd^cw_{q,c}\wedge T,
\varphi)\le\nu(dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T,\varphi).\eqno\sq$$}

\corollary{5.10}{If $dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T$ is
well defined, then at every point $x\in X$ we have
$$\nu\big(dd^cu_1\wedge\ldots\wedge dd^cu_q\wedge T,x\big)
  \ge\nu(dd^cu_1,x)\ldots\nu(dd^cu_q,x)\,\nu(T,x).$$}

\proof{Apply (5.9) with $\varphi(z)=v_1(z)=\ldots=v_q(z)=\log|z-x|$ and
observe that $\ell_j:=\limsup v_j/u_j=1/\nu(dd^cu_j,x)$ (there is nothing
to prove if $\nu(dd^cu_j,x)=0$).\qed}

Finally, we present an interesting stability property of Lelong
numbers due to~[Siu]: almost all slices of a closed positive current $T$
along linear subspaces passing through a given point have the same
Lelong number as~$T$. Before giving a proof of this, we need a useful
formula known as {\sl Crofton's formula}.

\lemma{5.11}{Let $\alpha$ be a closed positive $(p,p)$-form on
$\bbbc^n\ssm\{0\}$ which is invariant under the unitary group~$U(n)$.
Then $\alpha$ has the form
$$\alpha=\big(dd^c\chi(\log|z|)\big)^p$$
where $\chi$ is a convex increasing function. Moreover $\alpha$ is
invariant by homotheties if and only if $\chi$ is an affine function, i.e.
$\alpha=\lambda\,(dd^c\log|z|)^p$.}

\proof{A radial convolution $\alpha_\varepsilon(z)=\int_\bbbr
\rho(t/\varepsilon)\,\alpha(e^tz)\,dt$ produces a smooth form with the
same properties as~$\alpha$ and $\lim_{\varepsilon\to 0}\alpha_\varepsilon=
\alpha$. Hence we can suppose that $\alpha$ is smooth on $\bbbc^n\ssm\{0\}$.
At a point $z=(0\ld 0,z_n)$, the $(p,p)$-form $\alpha(z)\in\bigwedge^{p,p}
(\bbbc^n)^\star$ must be invariant by $U(n-1)$ acting on the first
$(n-1)$ coordinates. We claim that the subspace of $U(n-1)$-invariants
in $\bigwedge^{p,p}(\bbbc^n)^\star$ is generated by
$(dd^c|z|^2)^p$ and $(dd^c|z|^2)^{p-1}\wedge\ii dz_n\wedge d\ovl z_n$.
In fact, a form $\beta=\sum\beta_{I,J}dz_I\wedge d\ovl z_J$ is invariant
by $U(1)^{n-1}\subset U(n-1)$ if and only if $\beta_{I,J}=0$ for
$I\ne J$, and invariant by the permutation group
${\cal S}_{n-1}\subset U(n-1)$ if and only if all coefficients
$\beta_{I,I}$ (resp. $\beta_{Jn,Jn})$ with
$I,J\subset\{1\ld n-1\}$ are equal. Hence
$$\beta=\lambda\sum_{|I|=p}dz_I\wedge d\ovl z_I+\mu
\Big(\sum_{|J|=p-1}dz_J\wedge d\ovl z_J\Big)\wedge dz_n\wedge d\ovl z_n.$$
This proves our claim. As $d|z|^2\wedge d^c|z|^2={\ii\over\pi}|z_n|^2
dz_n\wedge d\ovl z_n$ at $(0\ld 0,z_n)$, we conclude that
$$\alpha(z)=f(z)(dd^c|z|^2)^p+g(z)(dd^c|z|^2)^{p-1}\wedge
d|z|^2\wedge d^c|z|^2$$
for some smooth functions $f,g$ on $\bbbc^n\ssm\{0\}$.
The $U(n)$-invariance of $\alpha$ shows that $f$ and $g$ are radial functions.
We may rewrite the last formula as
$$\alpha(z)=u(\log|z|)(dd^c\log|z|)^p+v(\log|z|)(dd^c\log|z|)^{p-1}\wedge
d\log|z|\wedge d^c\log|z|.$$
Here $(dd^c\log|z|)^p$ is a positive $(p,p)$-form coming from $\bbbp^{n-1}$,
hence it has zero contraction in the radial direction, while the
contraction of the form $(dd^c\log|z|)^{p-1}\wedge d\log|z|\wedge d^c\log|z|$
by the radial vector field is $(dd^c\log|z|)^{p-1}$. This shows easily that
$\alpha(z)\ge 0$ if and only if $u,v\ge 0$. Next, the closedness
condition $d\alpha=0$ gives $u'-v=0$. Thus $u$ is increasing and we define
a convex increasing function $\chi$ by $\chi'=u^{1/p}$. Then
$v=u'=p\chi^{\prime p-1}\chi''$ and
$$\alpha(z)=\big(dd^c\chi(\log|z|)\big)^p.$$
If $\alpha$ is invariant by homotheties, the functions
$u$ and $v$ must be constant, thus $v=0$ and $\alpha=\lambda
(dd^c\log|z|)^p$.\qed}

\statement{(5.12) Corollary {\rm(Crofton's formula)}}{Let $dv$ be the
unique $U(n)$-invariant measure of mass $1$ on the Grassmannian
$G(p,n)$ of $p$-dimensional subspaces in $\bbbc^n$. Then
$$\int_{S\in G(p,n)}[S]\,dv(S)=(dd^c\log|z|)^{n-p}.$$}

\proof{The left hand integral is a closed positive bidegree $(n-p,n-p)$
current which is invariant by $U(n)$ and by homotheties.
By lemma~5.11, this current must coincide with the form
$\lambda(dd^c\log|z|)^{n-p}$ for some $\lambda\ge 0$. The coefficient
$\lambda$ is the Lelong number at~$0$. As $\nu([S],0)=1$ for every~$S$,
we get $\lambda=\int_{G(p,n)}dv(S)=1$.\qed}

We now recall the basic facts of slicing theory (see Federer~[Fe]
and Harvey~[Ha]). Let $\sigma:M\to M'$ be a submersion of smooth differentiable
manifolds and let $\Theta$ be a {\sl locally flat} current on~$M$,
that is a current which can be written locally as $\Theta=U+dV$
where $U$, $V$ have locally integrable coefficients. It can be shown
that every current $\Theta$ such that both $\Theta$ and $d\Theta$ have
measure coefficients is locally flat; in particular,
closed positive currents are locally flats. Then, for almost every~$x'\in M'$,
there is a well defined slice~$\Theta_{x'}$,
which is the current on the fiber $\sigma^{-1}(x')$ defined by
$$\Theta_{x'}=U_{\restriction\sigma^{-1}(x')}+
  dV_{\restriction\sigma^{-1}(x')}.$$
The restrictions of $U$, $V$ to the fibers exist for almost all $x'$ by
the Fubini theorem. It is easy to show by a regularization
$\Theta_\varepsilon=\Theta\star\rho_\varepsilon$ that
the slices of a closed positive current are again closed and positive:
in fact $U_{\varepsilon,x'}$ and $V_{\varepsilon,x'}$ converge to $U_{x'}$
and $V_{x'}$ in $L^1_{\rm loc}$, thus
$\Theta_{\varepsilon,x'}$ converges weakly to $\Theta_{x'}$
for almost every~$x'$. This kind of slicing can be referred to as
{\sl parallel slicing} (if~we think of $\sigma$ as being a projection map).
The kind of slicing we need (where~the slices are taken over linear subspaces
passing through a given point) is of a slightly different nature and is
called {\sl concurrent slicing}.

The possibility of concurrent slicing is proved as follows. Let $T$ be a closed
positive current of bidimension $(p,p)$ in the ball $B(0,R)\subset\bbbc^n$.
Let
$$Y=\big\{(x,S)\in\bbbc^n\times G(q,n)\,;\,x\in S\big\}$$
be the total space of the tautological rank $q$ vector bundle over the
Grassmannian $G(q,n)$, equipped with the obvious projections
$$\sigma:Y\lra G(q,n),~~~~\pi:Y\lra\bbbc^n.$$
We set $Y_R=\pi^{-1}(B(0,R))$ and $Y_R^\star=\pi^{-1}
(B(0,R)\ssm\{0\})$. The restriction $\pi_0$ of $\pi$ to $Y_R^\star$ is a
submersion onto $B(0,R)\ssm\{0\}$, so we have a well defined pull-back
$\pi_0^\star T$ over $Y_R^\star$. We would like to extend it as
a pull-back $\pi^\star T$ over~$Y_R$, so as to define slices
$T_{\restriction S}=(\pi^\star T)_{\restriction\sigma^{-1}(S)}\,$;
of course, these slices can be non zero only if the dimension of $S$
is at least equal to the degree of~$T$, i.e. if $q\ge n-p$.
We first claim that $\pi_0^\star T$ has
locally finite mass near the zero section $\pi^{-1}(0)$ of~$\sigma$.
In fact let $\omega_G$ be a unitary invariant K\"ahler metric over
$G(q,n)$ and let $\beta=dd^c|z|^2$ in~$\bbbc^n$. Then we get a
K\"ahler metric on $Y$ defined by $\omega_Y=\sigma^\star\omega_G+
\pi^\star\beta$. If $N=(q-1)(n-q)$ is the dimension of the fibers
of~$\pi$, the projection formula $\pi_\star(u\wedge\pi^\star v)=
(\pi_\star u)\wedge v$ gives
$$\pi_\star\omega_Y^{N+p}=\sum_{1\le k\le p}{N+p\choose k}
\beta^k\wedge\pi_\star(\sigma^\star\omega_G^{N+p-k}).$$
Here $\pi_\star(\sigma^\star\omega_G^{N+p-k})$ is a unitary and homothety
invariant $(p-k,p-k)$ closed positive form on $\bbbc^n\ssm\{0\}$,
so $\pi_\star(\sigma^\star\omega_G^{N+p-k})$ is proportional to
$(dd^c\log|z|)^{n-k}$. With some constants $\lambda_k>0$, we thus get
$$\eqalign{
\int_{Y_r^\star}\pi_0^\star T\wedge\omega_Y^{N+p}
&=\sum_{0\le k\le p}\lambda_k\int_{B(0,r)\ssm\{0\}}T\wedge\beta^k\wedge
(dd^c\log|z|)^{k-p}\cr
&=\sum_{0\le k\le p}\lambda_k2^{-(p-k)}r^{-2(p-k)}
\int_{B(0,r)\ssm\{0\}}T\wedge\beta^p<+\infty.\cr}$$
The Skoda-El Mir theorem 0.5 shows that the trivial extension
$\wt\pi_0^\star T$ of $\pi_0^\star T$ is a closed positive current on~$Y_R$.
Of course, the zero section $\pi^{-1}(0)$ might also carry some extra
mass of the desired current~$\pi^\star T$. Since $\pi^{-1}(0)$ has
codimension~$q$, this extra mass cannot exist when
$q>n-p=\codim\pi^\star T$ and we simply set $\pi^\star T=\wt\pi_0^\star T$.
On the other hand, if $q=n-p$, we set
$$\pi^\star T:=\wt\pi_0^\star T+\nu(T,0)\,[\pi^{-1}(0)].\leqno(5.13)$$
We can now apply parallel slicing with respect to $\sigma:Y_R\to G(q,n)$,
which is a submersion: for almost all $S\in G(q,n)$, there is a
well defined slice $T_{\restriction S}=
(\pi^\star T)_{\restriction\sigma^{-1}(S)}$. These slices
coincide with the usual restrictions of $T$ to $S$ if $T$ is smooth. 

\statement{(5.14) Theorem {\rm ([Siu])}}{For almost all $S\in G(q,n)$
with $q\ge n-p$, the slice $T_{\restriction S}$ satisfies~
$\nu(T_{\restriction S},0)=\nu(T,0)$.}

\proof{If $q=n-p$, the slice $T_{\restriction S}$ consists of some positive
measure with support in $S\ssm\{0\}$ plus a Dirac
measure $\nu(T,0)\,\delta_0$ coming from the slice of
$\nu(T,0)\,[\pi^{-1}(0)]$. The equality $\nu(T_{\restriction S},0)=
\nu(T,0)$ thus follows directly from~(5.13). 

In the general case $q>n-p$, it is clearly sufficient to prove the
following two properties:
\smallskip
\noindent(a)~~ $\nu(T,0,r)=\displaystyle\int_{S\in G(q,n)}
\nu(T_{\restriction S},0,r)\,dv(S)$~~ for all $r\in{}]0,R[\,$;
\smallskip
\noindent(b)~~ $\nu(T_{\restriction S},0)\ge\nu(T,0)$~~ for almost all $S$.
\smallskip
\noindent In fact, (a) implies that $\nu(T,0)$ is the average
of all Lelong numbers $\nu(T_{\restriction S},0)$ and the conjunction
with (b) implies that these numbers must be equal to $\nu(T,0)$ for
almost all $S$. In order to prove (a) and (b), we can suppose without loss
of generality that $T$ is smooth on $B(0,R)\ssm\{0\}$. Otherwise,
we perform a small convolution with respect to the action of
${\rm Gl}_n(\bbbc)$ on $\bbbc^n$:
$$T_\varepsilon=\int_{g\in{\rm Gl}_n(\bbbc)}\rho_\varepsilon(g)\,
g^\star T\,dv(g)$$
where $(\rho_\varepsilon)$ is a regularizing family with support in
an $\varepsilon$-neighborhood of the unit element of ${\rm Gl}_n(\bbbc)$.
Then $T_\varepsilon$ is smooth in $B(0,(1-\varepsilon)R)\ssm\{0\}$ and
converges weakly to $T$. Moreover, we have
$\nu(T_\varepsilon,0)=\nu(T,0)$ by (5.2) and
$\nu(T_{\restriction S},0)\ge\limsup_{\varepsilon\to 0}\nu(T_{\varepsilon,
\restriction S},0)$ by (3.12), thus (a), (b) are preserved in the limit.
If $T$ is smooth on $B(0,R)\ssm\{0\}$, the slice $T_{\restriction S}$
is defined for all $S$ and is simply the restriction of $T$ to
$S\ssm\{0\}$ (carrying no mass at the origin).
\smallskip
\noindent(a) Here we may even assume that $T$ is smooth at $0$ by
performing an ordinary convolution. As $T_{\restriction S}$ has
bidegree $(n-p,n-p)$, we have
$$\nu(T_{\restriction S},0,r)=\int_{S\cap B(0,r)}T\wedge\alpha_S^{q-(n-p)}
=\int_{B(0,r)}T\wedge [S]\wedge\alpha_S^{p+q-n}$$
where $\alpha_S=dd^c\log|w|$ and $w=(w_1\ld w_q)$ are orthonormal
coordinates on~$S$. We simply have to check that
$$\int_{S\in G(q,n)}[S]\wedge\alpha_S^{p+q-n}\,dv(S)=(dd^c\log|z|)^p.$$
However, both sides are unitary and homothety invariant $(p,p)$-forms
with Lelong number $1$ at the origin, so they must coincide by Lemma~5.11.
\smallskip
\noindent(b) We prove the inequality when $S=\bbbc^q\times\{0\}$.
By the comparison theorem~5.1, for every $r>0$ and
$\varepsilon>0$ we have
$$\leqalignno{
&\int_{B(0,r)}T\wedge\gamma_\varepsilon^p\ge\nu(T,0)~~~~\hbox{\rm where}
&(5.15)\cr
&\gamma_\varepsilon={1\over 2}dd^c\log(\varepsilon|z_1|^2+\ldots+
\varepsilon|z_q|^2+|z_{q+1}|^2+\ldots+|z_n|^2).\cr}$$
We claim that the current $\gamma_\varepsilon^p$ converges weakly to
$$[S]\wedge\alpha_S^{p+q-n}=
[S]\wedge\Big({1\over2}dd^c\log(|z_1|^2+\ldots+|z_q|^2)\Big)^{p+q-n}$$
as $\varepsilon$ tends to $0$. In fact, the Lelong number of
$\gamma_\varepsilon^p$ at $0$ is $1$, hence by homogeneity
$$\int_{B(0,r)}\gamma_\varepsilon^p\wedge(dd^c|z|^2)^{n-p}=(2r^2)^p$$
for all $\varepsilon,r>0$. Therefore the family $(\gamma_\varepsilon^p)$
is relatively compact in the weak topology. Since
$\gamma_0=\lim\gamma_\varepsilon$ is smooth on $\bbbc^n\ssm S$ and
depends only on $n-q$ variables ($n-q\le p$), we have
$\lim\gamma_\varepsilon^p=\gamma_0^p=0$ on $\bbbc^n\ssm S$. This
shows that every weak limit of $(\gamma_\varepsilon^p)$ has
support in $S$. Each of these is the direct image by inclusion
of a unitary and homothety invariant $(p+q-n,p+q-n)$-form
on $S$ with Lelong number equal to $1$ at $0$. Therefore we must have
$$\lim_{\varepsilon\to 0}\gamma_\varepsilon^p=(i_S)_\star
(\alpha_S^{p+q-n})=[S]\wedge\alpha_S^{p+q-n},$$
and our claim is proved (of course, this can also be checked by direct
elementary calculations). By taking the limsup in (5.15) we obtain
$$\nu(T_{\restriction S},0,r+0)=\int_{\ovl B(0,r)}T\wedge[S]\wedge
\alpha_S^{p+q-n}\ge\nu(T,0)$$
(the singularity of $T$ at $0$ does not create any difficulty because
we can modify $T$ by a $dd^c$-exact form near $0$ to make it smooth
everywhere). Property (b) follows when $r$ tends to $0$.\qed}

\titleb{6. Siu's Semicontinuity Theorem}
  Let $X$, $Y$ be complex manifolds of dimension $n$, $m$ such that $X$ is
Stein. Let $\varphi:X\times Y\longrightarrow[-\infty,+\infty[$ be a continuous 
plurisubharmonic function. We assume that $\varphi$ is {\sl semi-exhaustive}
with respect to $\supp\,T$ , i.e. that for every compact subset 
$L\subset Y$ there exists $R=R(L)<0$ such that
 $$\{(x,y)\in\supp\,T\times L\,;
            \,\varphi(x,y)\le R\}\compact X\times Y.\leqno(6.1)$$
Let $T$ be a closed positive current of bidimension $(p,p)$ on $X$. 
For every point $y\in Y$, the function $\varphi_y(x):=\varphi(x,y)$ 
is semi-exhaustive on $\supp\,T\,$; one can therefore associate with $y$ 
a generalized Lelong number $\nu(T,\varphi_y)$. Proposition 3.13 implies
that the map $y\mapsto\nu(T,\varphi_y)$ is upper semi-continuous,
hence the upperlevel sets
$$E_c=E_c(T,\varphi)=\{y\in Y ;\nu(T,\varphi_y)\ge c\}\ ,\ c>0\leqno(6.2)$$
are closed. Under mild additional hypotheses, we are going to show
(following [De4]) that the sets $E_c$ are in fact analytic subsets of $Y$. 

\definition{6.3}{We say that a function $f(x,y)$ is locally
H\"older continuous with respect to $y$ on $X\times Y$
if every point of $X\times Y$ has a neighborhood $\Omega$ on which
$$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$
for all $(x,y_1)\in\Omega$, $(x,y_2)\in\Omega$,
with some constants $M>0$, $\gamma\in{}]0,1]$, and suitable
coordinates on $Y$.}

\statement{(6.4) Theorem {\rm([De4])}}{Let $T$ be a closed positive current
on $X$ and let
       $$\varphi:X\times Y\longrightarrow[-\infty,+\infty[$$
be a continuous plurisubharmonic function. Assume that $\varphi$ is
semi-exhaustive on $\supp\,T$ and that $e^{\varphi(x,y)}$ is locally
H\"older continuous with respect to $y$ on~$X\times Y$. Then the upperlevel
sets
          $$E_c(T,\varphi)=\{y\in Y;\nu(T,\varphi_y)\ge c\}$$
are analytic subsets of $Y$.}

This theorem can be rephrased by saying that $y\longmapsto\nu(T,\varphi_y)$
is upper semi-continuous with respect to the analytic Zariski topology.
As a special case, we get the following important result of~[Siu]:

\corollary{6.5}{If $T$ is a closed positive current of bidimension
$(p,p)$ on a complex manifold~$X$, the upperlevel sets
$E_c(T)=\{x\in X\,;\,\nu(T,x)\ge c\}$ of the usual Lelong numbers are
analytic subsets of dimension $\le p$.}

\proof{The result is local, so we may assume that $X\subset\bbbc^n$ is an open
subset. Theorem 6.4 with $Y=X$ and
$\varphi(x,y)=\log|x-y|$ shows that $E_c(T)$ is analytic. Moreover,
Prop.~3.11 implies $\dim E_c(T)\le p$.\qed}

\noindent{\bf(6.6) Generalization.} Theorem 6.4 can be applied more
generally to weight functions of the type
  $$\varphi(x,y)=\max_j~\log\Big(\sum_k|F_{j,k}(x,y)|^{\lambda_{j,k}}\Big)$$
where $F_{j,k}$ are holomorphic functions on $X\times Y$ and where
$\gamma_{j,k}$ are positive real constants; in this case $e^\varphi$ is
H\"older continuous of exponent $\gamma=\min\{\lambda_{j,k},1\}$ and
$\varphi$ is semi-exhaustive with respect to the whole space $X$ as soon
as the projection
${\rm pr}_2:\bigcap F_{j,k}^{-1}(0)\lra Y$ is proper and finite.

For example, when $\varphi(x,y)=\log\max|x_j-y_j|^\lambda_j$ on
an open subset $X$ of $\bbbc^n$ , we see that the upperlevel sets for
Kiselman's numbers $\nu(T,x,\lambda)$ are analytic in $X$ (a result
first proved in [Ki4]). More generally,
set $\psi_\lambda(z)=\log\max|z_j|^{\lambda_j}$
and $\varphi(x,y,g)=\psi_\lambda\big(g(x-y)\big)$ where $x,y\in\bbbc^n$
and $g\in{\rm Gl}(\bbbc^n)$. Then $\nu(T,\varphi_{y,g})$ 
is the Kiselman number of $T$ at $y$ when the coordinates have been rotated
by $g$. It is clear that $\varphi$ is plurisubharmonic in $(x,y,g)$ and
semi-exhaustive with respect to $x$, and that $e^\varphi$ is locally H\"older
continuous with respect to $(y,g)$. Thus the upperlevel sets
$$E_c=\{(y,g)\in X\times{\rm Gl}(\bbbc^n)\,;\,\nu(T,\varphi_{y,g})\ge c\}$$
are analytic in $X\times{\rm Gl}(\bbbc^n)$. 
However this result is not meaningful on a manifold,
because it is not invariant under coordinate changes. One can obtain
an invariant version as follows. Let $X$ be a manifold and let $J^k\oc_X$
be the bundle of $k$-jets of holomorphic functions on~$X$. We consider
the bundle $S_k$ over $X$ whose fiber $S_{k,y}$ is the set of $n$-tuples
of $k$-jets $u=(u_1\ld u_n)\in(J^k\oc_{X,y})^n$ such that $u_j(y)=0$ and
$du_1\wedge\ldots\wedge du_n(y)\ne 0$. Let $(z_j)$ be local coordinates on
an open set $\Omega\subset X$. Modulo $O(|z-y|^{k+1})$, we can write 
$$u_j(z)=\sum_{1\le|\alpha|\le k}a_{j,\alpha}(z-y)^\alpha$$
with ${\rm det}(a_{j,(0\ld 1_k\ld 0)})\ne 0$. The numbers
$((y_j),(a_{j,\alpha}))$ define a coordinate system on the total space of
$S_{k\,\restriction\Omega}$. For $(x,(y,u))\in X\times S_k$, we introduce
the function
$$\varphi(x,y,u)=\log\max|u_j(x)|^{\lambda_j}=
\log\max\Big|\sum_{1\le|\alpha|\le k}a_{j,\alpha}(x-y)^\alpha
\Big|^{\lambda_j}$$
which has all properties required by Th.~6.4 on a neighborhood of the diagonal
$x=y$, i.e. a neighborhood of $X\times_X S_k$ in $X\times S_k$. For
$k$ large, we claim that Kiselman's directional Lelong numbers
$$\nu(T,y,u,\lambda):=\nu(T,\varphi_{y,u})$$
with respect to the coordinate system $(u_j)$ at $y$
do not depend on the selection of the jet representives and are therefore
canonically defined on~$S_k$. In fact, a change of $u_j$ by $O(|z-y|^{k+1})$
adds $O(|z-y|^{(k+1)\lambda_j})$ to $e^\varphi$, and we have $e^\varphi\ge
O(|z-y|^{\max\lambda_j})$. Hence by (5.1) it is enough to take
$k+1\ge\max\lambda_j/\min\lambda_j$. Theorem~6.4 then shows that the
upperlevel sets $E_c(T,\varphi)$ are analytic in~$S_k$.\qed
\medskip

Now we give the detailed proof of Th.~6.4. As the
result is local on $Y$, we may assume without loss of
generality that $Y$ is a ball in $\bbbc^m$. After addition of 
a constant to $\varphi$, we may also assume that
there exists a compact subset $K\subset X$ such that
$$\{(x,y)\in X\times Y ;\varphi(x,y)\le 0\}\subset K\times Y.$$
By Th.~5.1, the Lelong numbers depend only on the
asymptotic behaviour of $\varphi$ near the (compact) polar set 
$\varphi^{-1}(-\infty)\cap({\rm Supp T}\times Y)$. We can add 
a smooth strictly plurisubharmonic function on $X\times Y$ to make
$\varphi$ strictly plurisuharmonic. Then Richberg's approximation
theorem for continuous plurisubharmonic functions shows that there exists
a smooth plurisubharmonic function $\wt\varphi$ such that
$\varphi\le\wt\varphi\le\varphi+1$. We may therefore assume that
$\varphi$ is smooth on $(X\times Y)\ssm\varphi^{-1}(-\infty)$.
\medskip
\pritem{$\bullet$} {\bf First step}: 
{\sl construction of a local plurisubharmonic potential.}

   Our goal is to generalize the usual construction of plurisubharmonic
potentials associated with a closed positive current (cf. P.~Lelong[Le2]
and H.~Skoda[Sk1]). We replace here the usual kernel
$|z-\zeta|^{-2p}$ arising from the hermitian metric of $\bbbc^n$
by a kernel depending on the weight $\varphi$.
Let $\chi\in C^\infty(\bbbr,\bbbr)$ be an increasing function such that
$\chi(t)=t$ for $t\le-1$ and $\chi(t)=0$ for $t\ge 0$. We consider
the half-plane $H=\{z\in\bbbc\,;\,{\rm Re}z<-1\}$ and associate with
$T$ the potential function $V$ on $Y\times H$ defined by
$$V(y,z)=-\int^0_{{\rm Re}z}\nu(T,\varphi_y,t)\chi'(t)\,dt.\leqno(6.7)$$
For every $t>\re z$, Stokes' formula gives
$$\nu(T,\varphi_y,t)=
       \int_{\varphi(x,y)<t}T(x)\wedge(dd^c_x\wt\varphi(x,y,z))^p$$
with $\wt\varphi(x,y,z):=\max\{\varphi(x,y),{\rm Re}z\}$. The Fubini
theorem applied to (6.7) gives
$$\eqalign{V(y,z)
&=-\int_{{\scriptstyle x\in X,\varphi(x,y)<t}\atop{\scriptstyle\re z<t<0}}
 T(x)\wedge(dd^c_x\wt\varphi(x,y,z))^p~\chi'(t)dt\cr
&=\int_{x\in X}T(x)\wedge\chi(\wt\varphi(x,y,z))\,(dd^c_x\wt\varphi(x,y,z))^p.
\cr}$$
For all $(n-1,n-1)$-form $h$ of class $C^{\infty}$ with compact support 
in $Y\times H$, we get
$$\eqalign{
\langle dd^cV,h\rangle&=\langle V,dd^ch\rangle\cr
&=\int_{X\times Y\times H}T(x)\wedge\chi(\wt\varphi(x,y,z))
(dd^c\wt\varphi(x,y,z))^p\wedge dd^ch(y,z).\cr}$$
Observe that the replacement of $dd^c_x$ by the total differentiation
$dd^c=dd^c_{x,y,z}$ does not modify the integrand, because the
terms in $dx$, $d\ovl x$ must have total bidegree $(n,n)$.
The current $T(x)\wedge\chi(\wt\varphi(x,y,z))h(y,z)$ has compact support in
$X\times Y\times H$. An integration by parts can thus be performed
to obtain
$$\langle dd^cV,h\rangle=\int_{X\times Y\times H}T(x)\wedge
  dd^c(\chi\circ\wt\varphi(x,y,z))\wedge(dd^c\wt\varphi(x,y,z))^p.h(y,z).$$
On the corona $\{-1\le\varphi(x,y)\le0\}$ we have $\wt\varphi(x,y,z)
=\varphi(x,y)$, whereas for $\varphi(x,y)<-1$ we get
$\wt\varphi<-1$ and $\chi\circ\wt\varphi=\wt\varphi$. As $\wt\varphi$
is plurisubharmonic, we see that $dd^cV(y,z)$ is the sum of the positive $(1,1)$-form
   $$(y,z)\longmapsto\int_{\{x\in X;\varphi(x,y)<-1\}}
                      T(x)\wedge(dd^c_{x,y,z}\wt\varphi(x,y,z))^{p+1}$$
and of the $(1,1)$-form independent of $z$
   $$y\longmapsto\int_{\{x\in X;-1\le\varphi(x,y)\le0\}}T\wedge 
                 dd^c_{x,y}(\chi\circ\varphi)\wedge(dd^c_{x,y}\varphi)^p.$$
As $\varphi$ is smooth outside $\varphi^{-1}(-\infty)$, this last
form has locally bounded coefficients. Hence $dd^cV(y,z)$ is $\ge 0$ except
perhaps for locally bounded terms. In addition, $V$ is
continuous on $Y\times H$ because $T\wedge(dd^c\wt\varphi_{y,z})^p$
is weakly continuous in the variables $(y,z)$ by Cor.~1.6. 
We therefore obtain the following result.

\statement{(6.8) Proposition}{There exists a positive plurisubharmonic
function $\rho\in C^\infty(Y)$ such that $\rho(y)+V(y,z)$ is plurisubharmonic
on $Y\times H$.}

If we let $\re z$ tend to $-\infty$, we see that the function
    $$U_0(y)=\rho(y)+V(y,-\infty)
            =\rho(y)-\int_{-\infty}^0\nu(T,\varphi_y,t)\chi'(t)dt$$
is locally plurisubharmonic or identically $-\infty$ on $Y$. Moreover, it
is clear that $U_0(y)=-\infty$ at every point $y$ such that
$\nu(T,\varphi_y)>0$. If $Y$ is connected and $U_0\not\equiv-\infty$,
we already conclude that the density set $\bigcup_{c>0}E_c$
is pluripolar in~$Y$.
\medskip
\pritem{$\bullet$} {\bf Second step}: 
{\sl application of Kiselman's minimum principle.}

We refer to Kiselman [Ki1] for a proof (in a somewhat more general
situation) of the following basic minimum principle for plurisubharmonic
functions.

\statement{(6.9) Kiselman's minimum principle}{Let $M$ be a 
complex manifold, let $\omega\subset\bbbr^n$ be a convex open subset and
$\Omega$ be the ``tube domain" $\Omega=\omega+\ii\bbbr^n$. For
every plurisubharmonic function $v(\zeta,z)$ on $M\times\Omega$ that
does not depend on $\im z$, the function
$$u(\zeta)=\inf_{z\in\Omega}~v(\zeta,z)$$
is plurisubharmonic or locally $\equiv-\infty$ on $M$.\qed}

   Let $a\ge 0$ be arbitrary. The function
     $$Y\times H\ni(y,z)\longmapsto\rho(y)+V(y,z)-a{\rm Re}z$$
is plurisubharmonic and independent of ${\rm Im}\,z$. By 6.9, the Legendre
transform
     $$U_a(y)=\inf_{r<-1}\big\{\rho(y)+V(y,r)-ar\big\}$$
is locally plurisubharmonic or $\equiv-\infty$ on $Y$.

\lemma{6.10}{Let $y_0\in Y$ be a given point.
\medskip
\item{\rm(a)} If $a>\nu(T,\varphi_{y_0})$, then $U_a$ is bounded
below on a neighborhood of $y_0$.
\medskip
\item{\rm(b)} If $a<\nu(T,\varphi_{y_0})$, then $U_a(y_0)=-\infty$.}

\proof{By definition of $V$ (cf. $(6.7)$) we have
$$V(y,r)\le-\nu(T,\varphi_y,r)\int_r^0\chi'(t)dt=r\nu(T,\varphi_y,r)
                 \le r\nu(T,\varphi_y).\leqno(6.11)$$
Then clearly $U_a(y_0)=-\infty$ if $a<\nu(T,\varphi_{y_0})$. On the
other hand, if $\nu(T,\varphi_{y_0})<a$, there exists $t_0<0$ 
such that $\nu(T,\varphi_{y_0},t_0)<a$. Fix $r_0<t_0$. 
The semi-continuity property (3.13) shows that there exists a
neighborhood $\omega$ of $y_0$ such that $\sup_{y\in\omega}~
\nu(T,\varphi_y,r_0)<a$. For all $y\in\omega$, we get
$$V(y,r)\ge-C-a\int_r^{r_0}\chi'(t)dt=-C+a(r-r_0),$$
and this implies $U_a(y)\ge-C-a r_0$.\qed}

\theorem{6.12}{If $Y$ is connected and if $E_c\ne Y$, then $E_c$
is a closed complete pluripolar subset of $Y$, i.e. there exists
a continuous plurisubharmonic function $w:Y\lra[-\infty,+\infty[$ such that
$E_c=w^{-1}(-\infty)$.}

\proof{We first observe that the family $(U_a)$ is increasing in $a$,
that $U_a=-\infty$ on $E_c$ for all $a<c$ and that
$\sup_{a<c}U_a(y)>-\infty$ if $y\in Y\ssm E_c$
(apply Lemma~6.10). For any integer $k\ge 1$, let $w_k\in\ci(Y)$ be a
plurisubharmonic regularization of $U_{c-1/k}$ such that
$w_k\ge U_{c-1/k}$ on $Y$ and $w_k\le -2^k$ on\break
$E_c\cap Y_k$ where $Y_k=\{y\in Y\,;\,d(y,\partial Y)\ge 1/k\}$. Then
Lemma~6.10~(a) shows that the family
$(w_k)$ is uniformly bounded below on every compact subset of
$Y\ssm E_c$. We can also choose $w_k$ uniformly
bounded above on every compact subset of $Y$ because
$U_{c-1/k}\le U_c$. The function
$$w=\sum_{k=1}^{+\infty}~2^{-k}w_k$$
satifies our requirements.\qed}
\medskip
\pritem{$\bullet$} {\bf Third step}: 
{\sl estimation of the singularities of the potentials} $U_a$.

\lemma{6.13}{Let $y_0\in Y$ be a given point, $L$ a compact
neighborhood of $y_0$, $K\subset X$ a compact subset and $r_0$
a real number $<-1$ such that
    $$\{(x,y)\in X\times L;\varphi(x,y)\le r_0\}\subset K\times L.$$
Assume that $e^{\varphi(x,y)}$ is locally H\"older continuous in $y$
and that
$$|f(x,y_1)-f(x,y_2)|\le M |y_1-y_2|^\gamma$$
for all $(x,y_1,y_2)\in K\times L\times L$. Then, for all 
$\varepsilon\in{}]0,1[$, there exists a real number
$\eta(\varepsilon)>0$ such that all $y\in Y$ with 
$|y-y_0|<\eta(\varepsilon)$ satisfy
    $$U_a(y)\le\rho(y)+\big((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\big)
        \Big(\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\Big).$$}

\proof{First, we try to estimate $\nu(T,\varphi_y,r)$ when $y\in L$ is
near $y_0$. Set
$$\left\{ 
\eqalign{\psi(x)&=(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2\cr
         \psi(x)&=\max\bigl(\varphi_y(x),(1-\varepsilon)\varphi_{y_0}(x)+
                  \varepsilon r-\varepsilon/2\bigr)\cr
         \psi(x)&=\varphi_y(x)\cr}
\eqalign{&{~\rm if~}\cr
         &{~\rm if~}\cr
         &{~\rm if~}\cr}
\eqalign{&\varphi_{y_0}(x)\le r-1\cr
   r-1\le&\varphi_{y_0}(x)\le r\cr
     r\le&\varphi_{y_0}(x)\le r_0\cr}\right.$$
and verify that this definition is coherent when $|y-y_0|$ is small enough.
By hypothesis
$$|e^{\varphi_y(x)}-e^{\varphi_{y_0}(x)}|\le M|y-y_0|^\gamma.$$
This inequality implies
$$\eqalign{\varphi_y(x)&\le\varphi_{y_0}(x)+\log\bigl(1+M|y-y_0|^\gamma 
                          e^{-\varphi_{y_0}(x)}\bigr)\cr
           \varphi_y(x)&\ge\varphi_{y_0}(x)+\log\bigl(1-M|y-y_0|^\gamma
                          e^{-\varphi_{y_0}(x)}\bigr).\cr}$$
In particular, for $\varphi_{y_0}(x)=r$, we have
$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2=r-\varepsilon/2$,
thus
      $$\varphi_y(x)\ge r+\log(1-M|y-y_0|^\gamma e^{-r}).$$
Similarly, for $\varphi_{y_0}(x)=r-1$, we have
$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2=
r-1+\varepsilon/2$, thus
      $$\varphi_y(x)\le r-1+\log(1+M|y-y_0|^\gamma e^{1-r}).$$
The definition of $\psi$ is thus coherent as soon as
$M|y-y_0|^\gamma e^{1-r}\le\varepsilon/2$ , i.e.
      $$\gamma\log|y-y_0|+\log{{2eM}\over\varepsilon}\le r.$$
In this case $\psi$ coincides with $\varphi_y$ on a neighborhood of 
$\{\psi=r\}$ , and with
         $$(1-\varepsilon)\varphi_{y_0}(x)+\varepsilon r-\varepsilon/2$$
on a neighborhood of the polar set $\psi^{-1}(-\infty)$. By Stokes'
formula applied to $\nu(T,\psi,r)$, we infer
  $$\nu(T,\varphi_y,r)=\nu(T,\psi,r)\ge\nu(T,\psi)=
                 (1-\varepsilon)^p\nu(T,\varphi_{y_0}).$$
From (6.11) we get $V(y,r)\le r\nu(T,\varphi_y,r)$, hence
$$\leqalignno{
     U_a(y)&\le\rho(y)+V(y,r)-ar\le 
                   \rho(y)+r\big(\nu(T,\varphi_y,r)-a\big),\cr
     U_a(y)&\le\rho(y)+r\bigl((1-\varepsilon)^p\nu(T,\varphi_{y_0})-a\bigr).
                   &(6.14)\cr}$$
Suppose $\gamma\log|y-y_0|+\log(2eM/\varepsilon)\le r_0$ , i.e. 
$|y-y_0|\le(\varepsilon/2eM)^{1/\gamma}e^{r_0/\gamma}\,$; 
one can then choose $r=\gamma\log|y-y_0|+\log(2eM/\varepsilon)$, and by
$(6.14)$ this yields the inequality asserted in Lemma~6.13.\qed}
\medskip
\pritem{$\bullet$} {\bf Fourth step}:
{\sl application of H\"ormander's $L^2$ estimates.}

The end of the proof rests upon the following crucial result, known as
the H\"ormander-Bombieri-Skoda theorem (cf. [H\"o], [Bo] and [Sk1,Sk2]).

\theorem{6.15}{Let $u$ be a plurisubharmonic function on a pseudoconvex open
set $\Omega\subset\bbbc^n$. For every point $z_0\in\Omega$ such that $e^{-u}$
is integrable in a neighborhood of $z_0$, there exists a holomorphic
function $F$ on $\Omega$ such that $F(z_0)=1$ and
$$\int_\Omega{|F(z)|^2e^{-u(z)}\over(1+|z|^2)^{n+\varepsilon}}d\lambda(z)
<+\infty.$$}

\corollary{6.16}{Let $u$ be a plurisubharmonic function on a complex 
mani\-fold~$Y$. The set of points in a neighborhood of which $e^{-u}$ 
is not integrable is an analytic subset of $Y$.}

\proof{The result is local, so we may assume that $Y$ is a ball
in $\bbbc^n$. Then the set of non integrability points of $e^{-u}$ is
the intersection of all hypersurfaces $F^{-1}(0)$ defined by the
holomorphic functions $F$ such that
$\int_Y|F|^2e^{-u}d\lambda<+\infty$.
Indeed $F$ must vanish at any non integrability point, and on the other
hand Th.~6.15 shows that one can choose $F(z_0)=1$ at any
integrability point $z_0.$\qed}
\medskip
\pproof{End of proof of Th.~6.4}{The main idea in what follows is due to
Kiselman [Ki2]. For $a,b>0$, we let $Z_{a,b}$ be the set of points in a
neighborhood
of which $\exp(-U_a/b)$ is not integrable. Then $Z_{a,b}$ is analytic,
and as the family $(U_a)$ is increasing in $a$, we have
$Z_{a',b'}\supset Z_{a'',b''}$
if $a'\le a''$, $b'\le b''$.

Let $y_0\in Y$ be a given point. If $y_0\notin E_c$, then $\nu
(T,\varphi_{y_0})<c$ by definition of $E_c$. Choose $a$ such that $\nu
(T,\varphi_{y_0})<a<c$.
Lemma 6.10 (a) implies that $U_a$ is bounded below in a neighborhood of
$y_0$, thus $\exp(-U_a/b)$ is integrable and $y_0\notin Z_{a,b}$ for all
$b>0$.

On the other hand, if $y_0\in E_c$ and if $a<c$, then Lemma~6.13 implies
for all $\varepsilon>0$ that
$$U_a(y)\le(1-\varepsilon)(c-a)\gamma\log|y-y_0|+C(\varepsilon)$$
on a neighborhood of $y_0$. Hence $\exp(-U_a/b)$ is non integrable at
$y_0$ as soon as $b<(c-a)\gamma/2m$, where $m=\dim Y$. We obtain
therefore
$$E_c=\bigcap_{{\scriptstyle a<c}\atop{\scriptstyle b<(c-a)\gamma/2m}}
Z_{a,b}.$$
This proves that $E_c$ is an analytic subset of $Y$.\qed}
\medskip
Finally, we use corollary~6.5 to derive an important decomposition
formula for currents, which is again due to [Siu]. We first begin
by two simple observations.

\lemma{6.17}{If $T$ is a closed
positive current of bidimension $(p,p)$ and $A$ is an irreducible
analytic set in $X$, we set
$$m_A=\inf\{x\in A\,;\,\nu(T,x)\}.$$
Then $\nu(T,x)=m_A$ for all $x\in A\ssm\bigcup A'_j$, where
$(A'_j)$ is a countable family of proper analytic subsets of $A$.
We say that $m_A$ is the generic Lelong number of $T$ along $A$.}

\proof{By definition of $m_A$ and $E_c(T)$, we have $\nu(T,x)\ge m_A$ for
every $x\in A$ and
$$\nu(T,x)=m_A~~~~\hbox{\rm on}~~A\ssm\bigcup_{c\in\bbbq,\,c>m_A}A\cap E_c(T).$$
However, for $c>m_A$, the intersection $A\cap E_c(T)$ is a proper analytic
subset of~$A$.\qed}

\statement{(6.18) Proposition}{Let $T$ be a closed positive current of
bidimension $(p,p)$ and let $A$ be an irreducible $p$-dimensional
analytic subset of $X$. Then $\bbbone_AT=m_A[A]$, in particular
$T-m_A[A]$ is positive.}

\proof{As the question is local and as
a closed positive current of bidimension $(p,p)$ cannot
carry any mass on a $(p-1)$-dimensional analytic subset, it is
enough to work in a neighborhood of a regular point $x_0\in A$.
Hence, by choosing suitable coordinates, we can suppose that $X$
is an open set in $\bbbc^n$ and that $A$ is the intersection of
$X$ with a $p$-dimensional linear subspace. Then, for every point
$a\in A$, the inequality $\nu(T,a)\ge m_A$ implies
$$\sigma_T\big(B(a,r)\big)\ge m_A\,\pi^pr^{2p}/p!=
m_A\sigma_{[A]}\big(B(a,r)\big)$$
for all $r$ such that $B(a,r)\subset X$. Now, set $\Theta=T-m_A[A]$
and $\beta=dd^c|z|^2$.
Our inequality says that $\int\bbbone_{B(a,r)}\Theta\wedge\beta^p\ge 0$.
If we integrate this with respect to some positive continuous
function $f$ with compact support in $A$, we get
$\int_X g_r\Theta\wedge\beta^p\ge 0$ where
$$g_r(z)=\int_A\bbbone_{B(a,r)}(z)\,f(a)\,d\lambda(a)
=\int_{a\in A\cap B(z,r)}f(a)\,d\lambda(a).$$
Here $g_r$ is continuous on $\bbbc^n$, and as $r$ tends to $0$ the function
$g_r(z)/(\pi^pr^{2p}/p!)$ converges to $f$ on $A$ and to $0$
on $X\ssm A$, with a global
uniform bound. Hence we obtain $\int\bbbone_Af\,\Theta\wedge\beta^p\ge 0$.
Since this inequality is true for all continuous functions $f\ge 0$ with
compact support in $A$, we conclude that the measure
$\bbbone_A\Theta\wedge\beta^p$ is positive. By a linear change of coordinates,
we see that
$$\bbbone_A\Theta\wedge\Big(dd^c\sum_{1\le j\le n}
\lambda_j|u_j|^2\Big)^n\ge 0$$
for every basis $(u_1\ld u_n)$ of linear forms and for all
coefficients~\hbox{$\lambda_j>0$.}
Take $\lambda_1=\ldots=\lambda_p=1$ and let the other
$\lambda_j$ tend to~$0$. Then we get
\hbox{$\bbbone_A\Theta\wedge\ii du_1\wedge d\ovl u_1\wedge\ldots\wedge
du_p\wedge d\ovl u_p\ge 0$.} This implies $\bbbone_A\Theta\ge 0$,
or equivalently $\bbbone_AT\ge m_A[A]$. By a result of Skoda [Sk3],
we know that $\bbbone_AT$ is a closed positive current, thus
$\bbbone_AT=\lambda[A]$ with $\lambda\ge 0$. We have just seen
that $\lambda\ge m_A$. On the other hand, $T\ge\bbbone_AT=\lambda[A]$
clearly implies $m_A\ge\lambda$.\qed}

\statement{(6.19) Siu's decomposition formula}{If $T$ is a closed
positive current of bidimension~$(p,p)$, there is a unique
decomposition of $T$ as a (possibly finite) weakly convergent series
$$T=\sum_{j\ge 1}\lambda_j[A_j]+R,~~~~\lambda_j>0,$$
where $[A_j]$ is the current of integration over an irreducible
$p$-dimensional ana\-lytic set $A_j\subset X$ and where $R$ is a closed
positive current with the property that $\dim E_c(R)<p$ for
every~$c>0$.}

\pproof{Uniqueness}{If $T$ has such a decomposition, the
$p$-dimensional components of $E_c(T)$ are $(A_j)_{\lambda_j\ge c}$,
for $\nu(T,x)=\sum\lambda_j\nu([A_j],x)+\nu(R,x)$ is non zero
only on $\bigcup A_j\cup\bigcup E_c(R)$, and is equal to
$\lambda_j$ generically on $A_j$ $\big($more precisely,
$\nu(T,x)=\lambda_j$ at every regular
point of $A_j$ which does not belong to any intersection $A_j\cup A_k$,
$k\ne j$ or to $\bigcup E_c(R)\big)$. In particular $A_j$ and
$\lambda_j$ are unique.}

\pproof{Existence}{Let $(A_j)_{j\ge 1}$ be the countable collection of
$p$-dimensional components occurring in one of the sets $E_c(T)$,
$c\in\bbbq_+^\star$, and let $\lambda_j>0$ be the generic Lelong
number of $T$ along $A_j$. Then Prop.~6.18 shows by induction on $N$
that $R_N=T-\sum_{1\le j\le N}\lambda_j[A_j]$ is positive. As $R_N$
is a decreasing sequence, there must be a
limit $R=\lim_{N\to+\infty}R_N$ in the weak topology. Thus we have
the asserted decomposition. By construction, $R$ has zero generic
Lelong number along $A_j$, so $\dim E_c(R)<p$ for every $c>0$.\qed}

It is very important to note that some components of lower dimension
can actually occur in $E_c(R)$, but they cannot be subtracted because
$R$ has bidimension $(p,p)$. A typical case is the case of a
bidimension $(n-1,n-1)$ current $T=dd^cu$ with
$u=\log(|F_j|^{\gamma_1}+\ldots|F_N|^{\gamma_N})$ and
$F_j\in\oc(X)$. In general $\bigcup E_c(T)=\bigcap F_j^{-1}(0)$ has
dimension~$<n-1$. In that case, an important formula due to King
plays the role of (6.19). We state it in a somewhat more general form
than its original version~[Kg].

\statement{(6.20) King's formula}{Let $F_1\ld F_N$ be holomorphic
functions on a complex manifold~$X$, such that the zero variety
$Z=\bigcap F_j^{-1}(0)$ has codimension~$\ge p$, and set
$u=\log\sum|F_j|^{\gamma_j}$ with arbitrary coefficients $\gamma_j>0$.
Let $(Z_k)_{k\ge1}$ be the irreducible components of $Z$ of codimension $p$
exactly. Then there exist multiplicities $\lambda_k>0$ such that
$$(dd^cu)^p=\sum_{k\ge 1}\lambda_k[Z_k]+R,$$
where $R$ is a closed positive current such that $\bbbone_ZR=0$
and $\codim E_c(R)>p$ for every $c>0$. Moreover the multiplicities
$\lambda_k$ are integers if $\gamma_1\ld \gamma_N$ are integers,
and $\lambda_k=\gamma_1\ldots\gamma_p$ if $\gamma_1\le\ldots\le\gamma_N$
and some partial Jacobian determinant of $(F_1\ld F_p)$ of order $p$ does
not vanish identically along~$Z_k$.}

\proof{Observe that $(dd^cu)^p$ is well defined thanks to Cor.~2.11.
The comparison theorem~5.9 applied with $\varphi(z)=\log|z-x|$,
$v_1=\ldots=v_p=u$, $u_1=\ldots=u_p=\varphi$ and $T=1$ shows that the
Lelong number of $(dd^cu)^p$ is equal to $0$ at every point of~$X\ssm Z$.
Hence $E_c((dd^cu)^p)$ is contained in $Z$ and its $(n-p)$-dimensional
components are members of the family $(Z_k)$. The asserted decomposition
follows from Siu's formula~6.19. We must have
$\bbbone_{Z_k}R=0$ for all irreducible components of~$Z$: when
$\codim Z_k>p$ this is automatically true, and when $\codim Z_k=p$ this
follows from (6.18) and the fact that $\codim E_c(R)>p$. If
$\det(\partial F_j/\partial z_k)_{1\le j,k\le p}\ne 0$ at some point
$x_0\in Z_k$, then $(Z,x_0)=(Z_k,x_0)$ is a smooth germ defined by the
equations $F_1=\ldots=F_p=0$. If we denote
$v=\log\sum_{j\le p}|F_j|^{\gamma_j}$ with $\gamma_1\le\ldots\le\gamma_N$,
then $u\sim v$ near $Z_k$ and
Th.~5.9 implies $\nu((dd^cu)^p,x)=\nu((dd^cv)^p,x)$ for all $x\in Z_k$
near $x_0$. On the other hand, if $G:=(F_1\ld F_p):X\to\bbbc^p$, Cor.~5.4 gives 
$$(dd^cv)^p=G^\star\Big(dd^c\log\sum_{1\le j\le p}|z_j|^{\gamma_j}\Big)^p=
\gamma_1\ldots\gamma_p\,G^\star\delta_0=\gamma_1\ldots\gamma_p\,[Z_k]$$
near~$x_0$. This implies that the generic Lelong number of $(dd^cu)^p$
along $Z_k$ is $\lambda_k=\gamma_1\ldots\gamma_p$. The integrality of
$\lambda_k$ when $\gamma_1\ld\gamma_N$ are integers will be proved in
the next section.\qed}

\titleb{7. Transformation of Lelong Numbers by Direct Images}
Let $F:X\to Y$ be a holomorphic map between complex manifolds of respective
dimensions $\dim X=n$, $\dim Y=m$, and let $T$ be a closed positive current
of bidimension $(p,p)$ on~$X$. If $F_{\restriction\supp\,T}$ is proper, the
direct image $F_\star T$ is defined by
$$\langle F_\star T,\alpha\rangle=\langle T,F^\star\alpha\rangle
\leqno(7.1)$$
for every test form $\alpha$ of bidegree $(p,p)$ on $Y$. This makes sense
because $\supp\,T\cap F^{-1}(\supp\,\alpha)$ is compact. It is easily seen
that $F_\star T$
is a closed positive current of bidimension $(p,p)$ on $Y$. 

\example{7.2}{Let $T=[A]$ where $A$ is a $p$-dimensional irreducible analytic
set in $X$ such that $F_{\restriction A}$ is proper. We know by Remmert's
theorem [Re1,2] that $F(A)$ is an analytic set in~$Y$. Two cases may occur.
Either $F_{\restriction A}$ is generically finite and $F$ induces
an \'etale covering $A\ssm F^{-1}(Z)\lra F(A)\ssm Z$ for some nowhere dense
analytic subset $Z\subset F(A)$, or $F_{\restriction A}$ has generic fibers
of positive dimension and $\dim F(A)<\dim A$. In the first case, let
$s<+\infty$ be the covering degree. Then for every test form $\alpha$ of
bidegree $(p,p)$ on $Y$ we get
$$\langle F_\star[A],\alpha\rangle=\int_A F^\star\alpha=\int_{A\ssm F^{-1}(Z)}
F^\star\alpha=s\int_{F(A)\ssm Z}\alpha=s\,\langle[F(A)],\alpha\rangle$$
because $Z$ and $F^{-1}(Z)$ are negligible sets. Hence $F_\star[A]=
s[F(A)]$. On the other hand, if $\dim F(A)<\dim A=p$, the restriction
of $\alpha$ to $F(A)_\reg$ is zero, and therefore so is this the
restriction of $F^\star\alpha$ to $A_\reg$. Hence $F_\star[A]=0$.\qed}

Now, let $\psi$ be a continuous plurisubharmonic function on $Y$ which is
semi-exhaustive on $F(\supp\,T)$ (this set certainly contains
$\supp F_\star T$). Since $F_{\restriction\supp\,T}$ is proper, it follows
that $\psi\circ F$ is semi-exhaustive on $\supp\,T$, for
$$\supp\,T\cap\{\psi\circ F<R\}=F^{-1}\big(F(\supp\,T)\cap\{\psi<R\}\big).$$

\statement{(7.3) Proposition}{If $F(\supp\,T)\cap\{\psi<R\}\compact Y$,
we have
$$\nu(F_\star T,\psi,r)=\nu(T,\psi\circ F,r)~~~~\hbox{\rm for all}~r<R,$$
in particular $\nu(F_\star T,\psi)=\nu(T,\psi\circ F)$.}

Here, we do not necessarily assume that $X$ or $Y$ are Stein; we thus
replace $\psi$ with $\psi_{\gge s}=\max\{\psi,s\}$, $s<r$, in the definition of
$\nu(F_\star T,\psi,r)$ and $\nu(T,\psi\circ F,r)$.

\proof{The first equality can be written
$$\int_Y F_\star T\wedge\bbbone_{\{\psi<r\}}(dd^c\psi_{\gge s})^p=
\int_X T\wedge(\bbbone_{\{\psi<r\}}\circ F)(dd^c\psi_{\gge s}\circ F)^p.$$
This follows almost immediately from the adjunction formula (7.1) when
$\psi$ is smooth and when we write $\bbbone_{\{\psi<R\}}=\lim\uparrow
g_k$ for some sequence of smooth functions $g_k$. In general, we write
$\psi_{\gge s}$ as a decreasing limit of smooth plurisubharmonic
functions and we apply our monotone continuity theorems (if~$Y$ is not Stein,
Richberg's theorem shows that we can obtain a decreasing sequence of almost
plurisubharmonic approximations such that the negative part of $dd^c$ converges
uniformly to~$0\,$; this is good enough to apply the monotone continuity
theorem; note that the integration is made on compact subsets, thanks to the
semi-exhaustivity assumption on $\psi$).\qed}

It follows from this that understanding the transformation of Lelong numbers
under direct images is equivalent to understanding the effect of $F$ on the
weight. We are mostly interested in computing the ordinary Lelong numbers
$\nu(F_\star T,y)$ associated with the weight $\psi(w)=\log|w-y|$ in some
local coordinates $(w_1\ld w_m)$ on $Y$ near~$y$. Then Prop.~7.3 gives
$$\leqalignno{
\nu(F_\star T,y)&=\nu(T,\log|F-y|)~~~~\hbox{\rm with}&(7.4)\cr
\log|F(z)-y|&={1\over 2}\log\sum|F_j(z)-y_j|^2,~~~~F_j=w_j\circ F.\cr}$$
We are going to show that $\nu(T,\log|F-y|)$ is bounded below by a linear
combination of the Lelong numbers of $T$ at points $x$ in the fiber
$F^{-1}(y)$, with suitable multiplicities attached to $F$ at these points.
These multiplicities can be seen as generalizations of the notion of
multiplicity of an analytic map introduced by W.~Stoll~[St].

\definition{7.5}{Let $x\in X$ and $y=F(x)$. Suppose that the codimension
of the fiber $F^{-1}(y)$ at $x$ is $\ge p$. Then we set
$\mu_p(F,x)=\nu\big((dd^c\log|F-y|)^p,x\big)$.}

Observe that $(dd^c\log|F-y|)^p$ is well defined thanks to Cor.~2.10.
The second comparison theorem~5.9 immediately shows that $\mu_p(F,x)$
is independent of the choice of local coordinates on~$Y$ (and also on $X$,
since Lelong nombers do not depend on coordinates). By definition,
$\mu_p(F,x)$ is the mass carried by $\{x\}$ of the measure
$$(dd^c\log|F(z)-y|)^p\wedge(dd^c\log|z-x|)^{n-p}.$$
We are going to give of more geometric interpretation of this
multiplicity, from which it will follow that $\mu_p(F,x)$ is always a
positive integer (in particular, the proof of (6.20) will be complete).

\example{7.6}{For $p=n=\dim X$, the assumption $\codim_x F^{-1}(y)\ge p$
means that the germ of map $F:(X,x)\lra(Y,y)$ is finite. Let $U_x$ be a
neighborhood of $x$ such that $\ovl U_x\cap F^{-1}(y)=\{x\}$, let $W_y$
be a neighborhood of $y$ disjoint from $F(\partial U_x)$ and let
$V_x=U_x\cap F^{-1}(W_y)$. Then $F:V_x\to W_y$ is proper and finite,
and we have $F_\star[V_x]=s\,[F(V_x)]$ where $s$ is the local covering
degree of $F:V_x\to F(V_x)$ at~$x$. Therefore
$$\eqalign{
\mu_n(F,x)&=\int_{\{x\}}\big(dd^c\log|F-y|\big)^n=\nu\big([V_x],\log|F-y|\big)
=\nu\big(F_\star[V_x],y\big)\cr
&=s\,\nu\big(F(V_x),y\big).\cr}$$
In the particular case when $\dim Y=\dim X$, we have $(F(V_x),y)=(Y,y)$, so
$\mu_n(F,x)=s$. In general, it is a well known fact that the ideal
generated by $(F_1-y_1\ld F_m-y_m)$ in $\oc_{X,x}$ has the same integral
closure as the ideal generated by $n$ generic linear combinations of the
generators, that is, for a generic choice of coordinates $w'=(w_1\ld w_n)$,
$w''=(w_{n+1}\ld w_m)$ on $(Y,y)$, we have $|F(z)-y|\le C|w'\circ F(z)|$
(this is a simple consequence of Lemma~5.7 applied to $A=F(V_x)$).
Hence for $p=n$, the comparison theorem~5.1 gives
$$\mu_n(F,x)=\mu_n(w'\circ F,x)=\hbox{\rm local covering degree of}~
w'\circ F~~{\rm at}~x,$$
for a generic choice of coordinates $(w',w'')$ on $(Y,y)$.\qed}

\noindent{\bf (7.7) Geometric interpretation of $\mu_p(F,x)$.} A formal
application of Crofton's formula 5.12 shows, after a translation, that there
is a small ball $B(x,r_0)$ on which
$$\leqalignno{
(dd^c\log|F(z)-y|)^p\wedge(dd^c\log|z-x|)^{n-p}&{}=\cr
\int_{S\in G(p,n)}
(dd^c\log|F(z)-y|)^p&{}\wedge[x+S]\,dv(S).&(7.7\,{\rm a})\cr}$$
For a rigorous proof of (7.7$\,$a), we replace $\log|F(z)-y|$ by the smooth
function ${1\over2}\log(|F(z)-y|^2+\varepsilon^2)$ and let $\varepsilon$
tend to~$0$ on both sides. By~(2.3) (resp. by (2.10)), the wedge product
$(dd^c\log|F(z)-y|)^p\wedge[x+S]$ is well defined on a small ball $B(x,r_0)$
as soon as $x+S$ does not intersect $F^{-1}(y)\cap\partial B(x,r_0)$
(resp. intersects $F^{-1}(y)\cap B(x,r_0)$ at finitely many points);
thanks to the assumption $\codim(F^{-1}(y),x)\ge p$, Sard's theorem shows
that this is the case for all $S$ outside a negligible closed subset $E$
in~$G(p,n)$ (resp. by Bertini, an analytic subset $A$ in $G(p,n)$ with
$A\subset E$). Fatou's lemma then implies that the inequality
$\ge$ holds in (7.7$\,$a). To get equality, we observe that we have
bounded convergence on all complements $G(p,n)\ssm V(E)$ of
neighborhoods $V(E)$ of~$E$. However the mass of
$\int_{V(E)}[x+S]\,dv(S)$ in $B(x,r_0)$ is proportional to
$v(V(E))$ and therefore tends to $0$ when $V(E)$ is small;
this is sufficient to complete the proof, since Prop.~2.6~(b) gives
$$\int_{z\in\ovl B(x,r_0)}\big(dd^c\log(|F(z)-y|^2+\varepsilon^2)\big)^p
\wedge\int_{S\in V(E)}[x+S]\,dv(S)\le C\,v(V(E))$$
with a constant $C$ independent of~$\varepsilon$. By evaluating
(7.7$\,$a) on $\{x\}$, we get
$$\mu_p(F,x)=\int_{S\in G(p,n)\ssm A}
\nu\big((dd^c\log|F_{\restriction x+S}-z|)^p,x\big)\,dv(S).
\leqno(7.7\,{\rm b})$$
Let us choose a linear parametrization $g_S:\bbbc^p\to S$ depending
analytically on local coordinates of $S$ in~$G(p,n)$. Then
Theorem~6.4 with $T=[\bbbc^p]$ and $\varphi(z,S)=\log|F\circ g_S(z)-y|$
shows that 
$$\nu\big((dd^c\log|F_{\restriction x+S}-z|)^p,x\big)=
\nu\big([\bbbc^p],\log|F\circ g_S(z)-y|\big)$$
is Zariski upper semicontinuous in $S$ on $G(p,n)\ssm A$. However,
(7.6) shows that these numbers are integers, so
$S\mapsto\nu\big((dd^c\log|F_{\restriction x+S}-z|)^p,x\big)$ must be
constant on a Zariski open subset in~$G(p,n)$. By (7.7$\,$b), we obtain
$$\mu_p(F,x)=\mu_p(F_{\restriction x+S},x)=\hbox{\rm local degree of}~
w'\circ F_{\restriction x+S}~~\hbox{\rm at}~x\leqno(7.7\,{\rm c})$$
for generic subspaces $S\in G(p,n)$ and generic coordinates
$w'=(w_1\ld w_p)$, $w''=(w_{p+1}\ld w_m)$ on~$(Y,y)$.\qed

\example{7.8}{Let $F:\bbbc^n\lra\bbbc^n$ be defined by
$$F(z_1\ld z_n)=(z_1^{s_1}\ld z_n^{s_n}),~~~~s_1\le\ldots\le s_n.$$
We claim that $\mu_p(F,0)=s_1\ldots s_p$. In fact, for a generic
$p$-dimensional subspace $S\subset\bbbc^n$ such that $z_1\ld z_p$
are coordinates on $S$ and $z_{p+1}\ld z_n$ are linear forms in
$z_1\ld z_p$, and for generic coordinates $w'=(w_1\ld w_p)$,
$w''=(w_{p+1}\ld w_n)$ on $\bbbc^n$, we can
rearrange $w'$ by linear combinations so that
$w_j\circ F_{\restriction S}$ is a linear combination of
$(z_j^{s_j}\ld z_n^{s_n})$ and has non zero coefficient in $z_j^{s_j}$
as a polynomial in $(z_j\ld z_p)$.
It is then an exercise to show that $w'\circ F_{\restriction S}$ has
covering degree $s_1\ldots s_p$ at~$0$ [$\,$compute inductively
the roots $z_n$, $z_{n-1}\ld z_j$ of $w_j\circ F_{\restriction S}(z)=a_j$
and use (5.6) to show that the $s_j$ values of $z_j$
lie near~$0$ when $(a_1\ld a_p)$ are small$\,$].\qed}

We are now ready to prove the main result of this section, which des\-cribes
the behaviour of Lelong numbers under proper morphisms. A~similar weaker
result was already proved in [De2] ($\mu_p(F,x)$ was not optimal).

\theorem{7.9}{Let $T$ be a closed positive current of bidimension
$(p,p)$ on $X$ and let $F:X\lra Y$ be an analytic map such that the
restriction $F_{\restriction\supp\,T}$ is proper. Let $I(y)$ be the
set of points $x\in\supp\,T\cap F^{-1}(y)$ such that $x$ is equal to
its connected component in $\supp\,T\cap F^{-1}(y)$ and
$\codim(F^{-1}(y),x)\ge p$. Then we have
$$\nu(F_\star T,y)\ge\sum_{x\in I(y)}\mu_p(F,x)\,\nu(T,x).$$}

In particular, we have $\nu(F_\star T,y)\ge\sum_{x\in I(y)}\nu(T,x)$.
This inequality no longer holds if the summation is extended to
all points $x\in\supp\,T\cap F^{-1}(Y)$ and if this set contains
positive dimensional connected components: for example, if $F:X\lra Y$ is
the blow-up of $Y$ at a point and $E$ is the exceptional divisor, then $T=[E]$
has direct image $F_\star[E]=0$ thanks to~(7.2).

\proof{We proceed in three steps.
\smallskip
\noindent{\sl Step 1. Reduction to the case of a single point $x$ in the
fiber.} It is sufficient to prove the inequality when the summation is
taken over an arbitrary finite subset $\{x_1\ld x_N\}$ of~$I(y)$.
As $x_j$ is equal to its connected component in $\supp\,T\cap F^{-1}(y)$,
it has a fondamental system of relative open-closed
neighborhoods, hence there are disjoint neighborhoods $U_j$ of $x_j$
such that $\partial U_j$ does not intersect $\supp\,T\cap F^{-1}(y)$. Then
the image $F(\partial U_j\cap\supp\,T)$ is a closed set
which does not contain~$y$. Let $W$ be a neighborhood of $y$ disjoint from
all sets $F(\partial U_j\cap\supp\,T)$, and let $V_j=U_j\cap F^{-1}(W)$.
It is clear that $V_j$ is a neighborhood of $x_j$ and that
$F_{\restriction V_j}:V_j\to W$ has a proper
restriction to $\supp\,T\cap V_j$. Moreover, we obviously have
$F_\star T\ge \sum_j (F_{\restriction V_j})_\star T$ on~$W$.
Therefore, it is enough to check the inequality
$\nu(F_\star T,y)\ge\mu_p(F,x)\,\nu(T,x)$ for a single point $x\in I(y)$,
in the case when $X\subset\bbbc^n$, $Y\subset\bbbc^m$ are open
subsets and~$x=y=0$.
\smallskip
\noindent{\sl Step 2. Reduction to the case when $F$ is finite.}
By (7.4), we have
$$\eqalign{
\nu(F_\star T,0)&=\inf_{V\ni 0}\int_V T\wedge(dd^c\log|F|)^p\cr
&=\inf_{V\ni 0}\lim_{\varepsilon\to 0}\int_V T\wedge
\big(dd^c\log(|F|+\varepsilon|z|^N)\big)^p,\cr}$$
and the integrals are well defined as soon as $\partial V$ does
not intersect the set $\supp\,T\cap F^{-1}(0)$ (may be after replacing
$\log|F|$ by $\max\{\log|F|,s\}$ with $s\ll 0$). For every $V$ and
$\varepsilon$, the last integral is larger than $\nu(G_\star T,0)$ where
$G$ is the finite morphism defined by
$$G:X\lra Y\times\bbbc^n,~~~~(z_1\ld z_n)\longmapsto
(F_1(z)\ld F_m(z),z_1^N\ld z_n^N).$$
We claim that for $N$ large enough we have $\mu_p(F,0)=\mu_p(G,0)$.
In fact, $x\in I(y)$ implies by definition $\codim(F^{-1}(0),0)\ge p$.
Hence, if $S=\{u_1=\ldots=u_{n-p}=0\}$ is a generic $p$-dimensional
subspace of $\bbbc^n$, the germ of variety $F^{-1}(0)\cap S$ defined by
$(F_1\ld F_m,u_1\ld u_{n-p})$ is~$\{0\}$. Hilbert's Nullstellensatz
implies that some powers of $z_1\ld z_n$ are in the ideal $(F_j,u_k)$.
Therefore $|F(z)|+|u(z)|\ge C|z|^a$ near $0$ for some integer $a$ independent
of $S$ (to see this, take coefficients of the $u_k$'s as additional variables);
in particular $|F(z)|\ge C|z|^a$ for $z\in S$ near~$0$. The comparison
theorem~5.1 then shows that $\mu_p(F,0)=\mu_p(G,0)$ for $N\ge a$.\break
If we are able to prove that $\nu(G_\star T,0)\ge\mu_p(G,0)\nu(T,0)$ in
case $G$ is finite, the obvious inequality
$\nu(F_\star T,0)\ge\nu(G_\star T,0)$ concludes the proof.

\noindent{\sl Step 3. Proof of the inequality
$\nu(F_\star T,y)\ge\mu_p(F,x)\,\nu(T,x)$ when $F$ is finite and
$F^{-1}(y)=x$.} Then $\varphi(z)=\log|F(z)-y|$ has a single isolated
pole at $x$ and we have $\mu_p(F,x)=\nu((dd^c\varphi)^p,x)$. It is
therefore sufficient to apply the following Proposition.\qed}

\statement{(7.10) Proposition}{Let $\varphi$ be a semi-exhaustive continuous
plurisubharmonic function on $X$ with a single isolated pole at~$x$. Then
$$\nu(T,\varphi)\ge\nu(T,x)\,\nu((dd^c\varphi)^p,x).$$}

\proof{Since the question is local, we can suppose that $X$ is the ball
$B(0,r_0)$ in $\bbbc^n$ and $x=0$. Set $X'=B(0,r_1)$ with $r_1<r_0$
and $\Phi(z,g)=\varphi\circ g(z)$ for $g\in{\rm Gl}_n(\bbbc)$. Then
there is a small neighborhood $\Omega$ of the unitary group
$U(n)\subset{\rm Gl}_n(\bbbc)$ such that $\Phi$ is
plurisubharmonic on $X'\times\Omega$ and semi-exhaustive with respect to~$X'$.
Theorem~6.4 implies that the map $g\mapsto\nu(T,\varphi\circ g)$ is Zariski
upper semi-continuous on~$\Omega$. In particular, we must have
$\nu(T,\varphi\circ g)\le\nu(T,\varphi)$ for all $g\in\Omega\ssm A$ in the
complement of a complex analytic set~$A$. Since ${\rm Gl}_n(\bbbc)$ is the
complexification of~$U(n)$, the intersection $U(n)\cap A$ must be a nowhere
dense real analytic subset of~$U(n)$. Therefore, if $dv$ is the Haar measure
of mass $1$ on~$U(n)$, we have
$$\leqalignno{
\nu(T,\varphi)&\ge\int_{g\in U(n)}\nu(T,\varphi\circ g)\,dv(g)\cr
&=\lim_{r\to 0}\int_{g\in U(n)}dv(g)
\int_{B(0,r)}T\wedge(dd^c\varphi\circ g)^p.&(7.11)\cr}$$
Since $\int_{g\in U(n)}(dd^c\varphi\circ g)^pdv(g)$ is a unitary
invariant $(p,p)$-form on~$B$, Lemma 5.11 implies
$$\int_{g\in U(n)}(dd^c\varphi\circ g)^pdv(g)=\big(dd^c\chi(\log|z|)\big)^p$$
where $\chi$ is a convex increasing function. The Lelong number at $0$ of the
left hand side is equal to $\nu((dd^c\varphi)^p,0)$, and must be equal to the
Lelong number of the right hand side, which is $\lim_{t\to-\infty}\chi'(t)^p$
(to see this, use either Formula~(3.5) or Th.~5.9). Thanks to the last
equality, Formulas (7.11) and (3.5) imply
$$\eqalignno{
\nu(T,\varphi)&\ge\lim_{r\to 0}
\int_{B(0,r)}T\wedge\big(dd^c\chi(\log|z|)\big)^p\cr
&=\lim_{r\to 0}\chi'(\log r-0)^p\nu(T,0,r)
\ge\nu((dd^c\varphi)^p,0)\,\nu(T,0).&\sq\cr}$$}

Another interesting question is to know whether it is possible to get
inequalities in the opposite direction, i.e. to find upper bounds for
$\nu(F_\star T,y)$ in terms of the Lelong numbers $\nu(T,x)$.
The example $T=[\Gamma]$ with the curve $\Gamma:
t\mapsto(t^a,t^{a+1},t)$ in $\bbbc^3$ and $F:\bbbc^3\to\bbbc^2$,
$(z_1,z_2,z_3)\mapsto(z_1,z_2)$, for which $\nu(T,0)=1$ and
$\nu(F_\star T,0)=a$, shows that this may be possible only when $F$
is finite. In this case, we have:

\theorem{7.12}{Let $F:X\to Y$ be a proper and finite analytic map and let
$T$ be a closed positive current of bidimension $(p,p)$ on~$X$. Then
$$\nu(F_\star T,y)\le\sum_{x\in\supp\,T\cap F^{-1}(y)}\ovl\mu_p(F,x)\,
\nu(T,x)\leqno({\rm a})$$
where $\ovl\mu_p(F,x)$ is the multiplicity defined as follows: if
$H:(X,x)\to(\bbbc^n,0)$ is a germ of finite map, we set
$$\leqalignno{
\sigma(H,x)&=\inf\big\{\alpha>0\,;\,\exists C>0,\,|H(z)|\ge C|z-x|^\alpha~
\hbox{\rm near}~x\big\},&({\rm b})\cr
\ovl\mu_p(F,x)&=\inf_G{\sigma(G\circ F,x)^p\over\mu_p(G,0)},&({\rm c})\cr}$$
where $G$ runs over all germs of maps $(Y,y)\lra(\bbbc^n,0)$ such that
$G\circ F$ is finite.}

\proof{If $F^{-1}(y)=\{x_1\ld x_N\}$, there is a neighborhood $W$ of $y$
and disjoint neighborhoods $V_j$ of $x_j$ such that $F^{-1}(W)=\bigcup V_j$.
Then $F_\star T=\sum(F_{\restriction V_j})_\star T$ on $W$, so it is
enough to consider the case when $F^{-1}(y)$ consists of a single point~$x$.
Therefore, we assume that $F:V\to W$ is proper and finite, where
$V$, $W$ are neighborhoods of $0$ in $\bbbc^n$, $\bbbc^m$ and $F^{-1}(0)=\{0\}$.
Let $G:(\bbbc^m,0)\lra(\bbbc^n,0)$ be a germ of map such that $G\circ F$
is finite. Hilbert's Nullstellensatz shows that there exists $\alpha>0$ and
$C>0$ such that $|G\circ F(z)|\ge C|z|^\alpha$ near~$0$. Then the comparison
theorem~5.1 implies
$$\nu(G_\star F_\star T,0)=\nu(T,\log|G\circ F|)\le\alpha^p\nu(T,\log|z|)=
\alpha^p\nu(T,0).$$
On the other hand, Th.~7.9 applied to $\Theta=F_\star T$ on $W$ gives
$$\nu(G_\star F_\star T,0)\ge\mu_p(G,0)\,\nu(F_\star T,0).$$
Therefore
$$\nu(F_\star T,0)\le{\alpha^p\over\mu_p(G,0)}\nu(T,0).$$
The infimum of all possible values of $\alpha$ is by definition
$\sigma(G\circ F,0)$, thus by taking the infimum over $G$ we obtain
$$\nu(F_\star T,0)\le\ovl\mu_p(F,0)\,\nu(T,0).\eqno\sq$$}

\example{7.13}{Let $F(z_1\ld z_n)=(z_1^{s_1}\ld z_n^{s_n})$,
$s_1\le\ldots\le s_n$ as in~7.8. Then we have
$$\mu_p(F,0)=s_1\ldots s_p,~~~~~~\ovl\mu_p(F,0)=s_{n-p+1}\ldots s_n.$$
To see this, let $s$ be the lowest common multiple of $s_1\ld s_n$
and let $G(z_1\ld z_n)=(z_1^{s/s_1}\ld z_n^{s/s_n})$. Clearly
$\mu_p(G,0)=(s/s_{n-p+1})\ldots(s/s_n)$ and $\sigma(G\circ F,0)=s$, so
we get by definition $\ovl\mu_p(F,0)\le s_{n-p+1}\ldots s_n$.
Finally, if $T=[A]$ is the current of integration over the $p$-dimensional
subspace $A=\{z_1=\ldots=z_{n-p}=0\}$, then $F_\star[A]=
s_{n-p+1}\ldots s_n\,[A]$ because $F_{\restriction A}$
has covering degree $s_{n-p+1}\ldots s_n$. Theorem~7.12 shows that
we must have $s_{n-p+1}\ldots s_n\le\ovl\mu_p(F,0)$, QED.
If $\lambda_1\le\ldots\le\lambda_n$ are positive real numbers and
$s_j$ is taken to be the integer part of $k\lambda_j$ as $k$
tends to $+\infty$, Theorems~7.9 and 7.12 imply in the limit
the following:}

\corollary{7.14}{For $0<\lambda_1\le\ldots\le\lambda_n$,
Kiselman's directional Lelong numbers satisfy the inequalities
$$\lambda_1\ldots\lambda_p\,\nu(T,x)\le\nu(T,x,\lambda)\le
\lambda_{n-p+1}\ldots\lambda_n\,\nu(T,x).\eqno\sq$$}

\remark{7.15}{It would be interesting to have a direct geometric
interpretation of~$\ovl\mu_p(F,x)$. In fact, we do not even know whether
$\ovl\mu_p(F,x)$ is always an integer.}

\titleb{8. A Schwarz Lemma. Application to Number Theory}
In this section, we show how Jensen's formula and Lelong numbers can be
used to prove a fairly general Schwarz lemma relating growth and zeros
of entire functions in~$\bbbc^n$.
In order to simplify notations, we denote by $|F|_r$ the supremum of
the modulus of a function $F$ on the ball of center $0$ and radius~$r$.
Then, following [De1], we present some applications with a more
arithmetical flavour.

\statement{(8.1) Schwarz lemma}{Let $P_1\ld P_N\in\bbbc[z_1\ld z_n]$ be
polynomials of degree~$\delta$, such that their homogeneous parts of
degree $\delta$ do not vanish simultaneously except at~$0$. Then there
is a constant $C\ge 2$ such that for all entire functions
$F\in\oc(\bbbc^n)$ and all $R\ge r\ge 1$ we have
$$\log|F|_r\le\log|F|_R-\delta^{1-n}\nu([Z_F],\log|P|)\,\log{R\over Cr}$$
where $Z_F$ is the zero divisor of $F$ and $P=(P_1\ld P_N):\bbbc^n\lra
\bbbc^N$. Moreover
$$\nu([Z_F],\log|P|)\ge\sum_{w\in P^{-1}(0)}{\rm ord}(F,w)\,\mu_{n-1}(P,w)$$
where ${\rm ord}(F,w)$ denotes the vanishing order of $F$ at $w$ and
$\mu_{n-1}(P,w)$ is the $(n-1)$-multiplicity of $P$ at $w$, as defined
in $(7.5)$ and $(7.7)$.}

\proof{Our assumptions imply that $P$ is a proper and finite map. The last
inequality is then just a formal consequence
of formula (7.4) and Th.~7.9 applied to~$T=[Z_F]$. Let $Q_j$ be
the homogeneous part of degree $\delta$ in~$P_j$.
For~$z_0\in B(0,r)$, we introduce the weight functions
$$\varphi(z)=\log|P(z)|,~~~~~~\psi(z)=\log|Q(z-z_0)|.$$
Since $Q^{-1}(0)=\{0\}$ by hypothesis, the homogeneity of $Q$ shows
that there are constants $C_1,C_2>0$ such that
$$C_1|z|^\delta\le|Q(z)|\le C_2|z|^\delta~~~~\hbox{\rm on}~~\bbbc^n.
\leqno(8.2)$$
The homogeneity also implies $(dd^c\psi)^n=\delta^n\,\delta_{z_0}$.
We apply the Lelong Jensen formula~4.5 to the measures $\mu_{\psi,s}$
associated with $\psi$ and to~$V=\log|F|$. This gives
$$\mu_{\psi,s}(\log|F|)-\delta^n\log|F(z_0)|=\int_{-\infty}^sdt\int_{\{\psi<t\}}
[Z_F]\wedge(dd^c\psi)^{n-1}.\leqno(8.3)$$
By (4.2), $\mu_{\psi,s}$ has total mass $\delta^n$ and has support in
$$\{\psi(z)=s\}=\{Q(z-z_0)=e^s\}\subset B\big(0,r+(e^s/C_1)^{1/\delta}\big).$$
Note that the inequality in the Schwarz lemma is obvious if $R\le Cr$, so
we can assume $R\ge Cr\ge 2r$. We take $s=\delta\log(R/2)+\log C_1\,$; then
$$\{\psi(z)=s\}\subset B(0,r+R/2)\subset B(0,R).$$
In particular, we get $\mu_{\psi,s}(\log|F|)\le \delta^n\log|F|_R$ and
formula~(8.3) gives
$$\log|F|_R-\log|F(z_0)|\ge\delta^{-n}\int_{s_0}^sdt\int_{\{\psi<t\}}
[Z_F]\wedge(dd^c\psi)^{n-1}\leqno(8.4)$$
for any real number $s_0<s$. The proof will be complete if we are able to
compare the integral in (8.4) to the corresponding integral with $\varphi$
in place of~$\psi$. The argument for this is quite similar to the proof
of the comparison theorem, if we observe that $\psi\sim\varphi$ at
infinity. We introduce the auxiliary function
$$w=\cases{
\max\{\psi,(1-\varepsilon)\varphi+\varepsilon t-\varepsilon\}
&on $\{\psi\ge t-2\}$,\cr
(1-\varepsilon)\varphi+\varepsilon t-\varepsilon
&on $\{\psi\le t-2\}$,\cr}$$
with a constant $\varepsilon$ to be determined later, such that
$(1-\varepsilon)\varphi+\varepsilon t-\varepsilon>\psi$ near $\{\psi=t-2\}$ and
$(1-\varepsilon)\varphi+\varepsilon t-\varepsilon<\psi$ near $\{\psi=t\}$.
Then Stokes' theorem implies
$$\leqalignno{
\int_{\{\psi<t\}}&[Z_F]\wedge(dd^c\psi)^{n-1}=
\int_{\{\psi<t\}}[Z_F]\wedge(dd^cw)^{n-1}\cr
&\ge(1-\varepsilon)^{n-1}\int_{\{\psi<t-2\}}[Z_F]\wedge(dd^c\varphi)^{n-1}
\ge(1-\varepsilon)^{n-1}\nu([Z_F],\log|P|).&(8.5)\cr}$$
By (8.2) and our hypothesis $|z_0|<r$, the condition $\psi(z)=t$ implies
$$\eqalign{
|Q(z-z_0)|=e^t~~&\Longrightarrow~~
e^{t/\delta}/C_1^{1/\delta}\le|z-z_0|\le e^{t/\delta}/C_2^{1/\delta},\cr
|P(z)-Q(z-z_0)|&\le C_3(1+|z_0|)(1+|z|+|z_0|)^{\delta-1}\le
C_4 r(r+e^{t/\delta})^{\delta-1},\cr
\Big|{P(z)\over Q(z-z_0)}-1\Big|&\le C_4 re^{-t/\delta}
(re^{-t/\delta}+1)^{\delta-1}\le 2^{\delta-1}C_4 re^{-t/\delta},\cr}$$
provided that $t\ge \delta\log r$. Hence for $\psi(z)=t\ge s_0\ge
\delta\log(2^\delta C_4r)$, we get
$$|\varphi(z)-\psi(z)|=\Big|\log{|P(z)|\over|Q(z-z_0)|}\Big|
\le C_5 re^{-t/\delta}.$$
Now, we have
$$\big[(1-\varepsilon)\varphi+\varepsilon t-\varepsilon\big]-\psi=
(1-\varepsilon)(\varphi-\psi)+\varepsilon(t-1-\psi),$$
so this difference is $<C_5re^{-t/\delta}-\varepsilon$ on $\{\psi=t\}$
and $>-C_5re^{(2-t)/\delta}+\varepsilon$ on $\{\psi=t-2\}$. Hence it is
sufficient to take $\varepsilon=C_5re^{(2-t)/\delta}$. This number has to be
$<1$, so we take $t\ge s_0\ge 2+\delta\log(C_5r)$. Moreover, (8.5) actually
holds only if $P^{-1}(0)\subset\{\psi<t-2\}$, so by (8.2) it is enough to take
$t\ge s_0\ge 2+\log(C_2(r+C_6)^\delta)$ where $C_6$ is such that
$P^{-1}(0)\subset\ovl B(0,C_6)$. Finally, we see that we can choose
$$s=\delta\log R-C_7,~~~~~s_0=\delta\log r+C_8,$$
and inequalities (8.4), (8.5) together imply
$$\log|F|_R-\log|F(z_0)|\ge\delta^{-n}\Big(\int_{s_0}^s
(1-C_5re^{(2-t)/\delta})^{n-1}\,dt\Big)\nu([Z_F],\log|P|).$$
The integral is bounded below by
$$\int_{C_8}^{\delta\log(R/r)-C_7}
(1-C_9e^{-t/\delta})\,dt\ge\delta\log(R/Cr).$$
This concludes the proof, by taking the infimum when $z_0$ runs
over $B(0,r)$.\qed}

\corollary{8.6}{Let $S$ be a finite subset of $\bbbc^n$ and let $\delta$
be the minimal degree of algebraic hypersurfaces containing~$S$. Then
there is a constant $C\ge 2$ such that for all $F\in\oc(\bbbc^n)$ and all
$R\ge r\ge 1$ we have
$$\log|F|_r\le\log|F|_R-{\rm ord}(F,S){\delta+n(n-1)/2\over n!}
\log{R\over Cr}$$
where ${\rm ord}(F,S)=\min_{w\in S}{\rm ord}(F,w)$.}

\proof{In view of the Schwarz Lemma~8.1, we only have to select
suitable polynomials $P_1\ld P_N$. The vector space $\bbbc[z_1\ld
z_n]_{<\delta}$ of polynomials of degree $<\delta$ in $\bbbc^n$ has
dimension
$$m(\delta)={\delta+n-1\choose n}=
{\delta(\delta+1)\ldots(\delta+n-1)\over n!}.$$
By definition of $\delta$, the linear forms
$$\bbbc[z_1\ld z_n]_{<\delta}\lra\bbbc,~~~~P\longmapsto P(w),~~w\in S$$
vanish simultaneously only when~$P=0$. Hence we can find $m=m(\delta)$
points $w_1\ld w_m\in S$ such that the linear forms $P\mapsto P(w_j)$
define a basis of~$\bbbc[z_1\ld z_n]_{<\delta}^\star$. This means that there is
a unique polynomial $P\in\bbbc[z_1\ld z_n]_{<\delta}$ which takes given values
$P(w_j)$ for $1\le j\le m$. In~particular, for every multiindex $\alpha$,
$|\alpha|=\delta$, there is a unique polynomial \hbox{$R_\alpha\in
\bbbc[z_1\ld z_n]_{<\delta}$} such that $R_\alpha(w_j)=w_j^\alpha$. Then the
polynomials $P_\alpha(z)=z^\alpha-R_\alpha(z)$ have degree~$\delta$,
vanish at all points $w_j$ and their homogeneous parts of maximum degree
$Q_\alpha(z)=z^\alpha$ do not vanish simultaneously except at~$0$.
We simply use the fact that $\mu_{n-1}(P,w_j)\ge 1$ to get
$$\nu([Z_F],\log|P|)\ge\sum_{w\in P^{-1}(0)}{\rm ord}(F,w)\ge
m(\delta)\,{\rm ord}(F,S).$$
Theorem 8.1 then gives the desired inequality, because $m(\delta)$ is a
polynomial with positive coefficients and leading terms 
$(\delta^n+n(n-1)/2\,\delta^{n-1}+\ldots)/n!$\qed}

Let $S$ be a finite subset of~$\bbbc^n$. According to Waldschmidt~[Wa1],
we introduce for every integer $t>0$ a number $\omega_t(S)$ equal to
the minimal degree of polynomials $P\in\bbbc[z_1\ld z_n]$ which vanish
at order $\ge t$ at every point of~$S$. The obvious subadditivity
property
$$\omega_{t_1+t_2}(S)\le\omega_{t_1}(S)+\omega_{t_2}(S)$$
easily shows that
$$\Omega(S):=\inf_{t>0}{\omega_t(S)\over t}=
\lim_{t\to+\infty}{\omega_t(S)\over t}.$$
We call $\omega_1(S)$ the {\sl degree} of $S$ (minimal degree of algebraic
hypersurfaces containing~$S$) and $\Omega(S)$ the {\sl singular degree}
of~$S$. If we apply Cor.~8.6 to a polynomial $F$ vanishing at order
$t$ on $S$ and fix $r=1$, we get
$$\log|F|_R\ge t{\delta+n(n-1)/2\over n!}\log{R\over C}+\log|F|_1$$
with $\delta=\omega_1(S)$, in particular
$${\rm deg}\,F\ge t{\omega_1(S)+n(n-1)/2\over n!}.$$
The minimum of ${\rm deg}\,F$ over all such $F$ is by
definition~$\omega_t(S)$. If we divide by $t$ and take the infimum
over~$t$, we get the interesting inequality
$${\omega_t(S)\over t}\ge\Omega(S)\ge{\omega_1(S)+n(n-1)/2\over n!}.
\leqno(8.7)$$

\remark{8.8}{The constant ${\omega_1(S)+n(n-1)/2\over n!}$ in (8.6) and
(8.7) is optimal for $n=1,2$ but not for~$n\ge3$. It can be shown by means
of H\"ormander's $L^2$ estimates (see [Wa2]) that for every $\varepsilon>0$
the Schwarz Lemma holds with coefficient $\Omega(S)-\varepsilon$ in
Cor.~8.6$\,$:
$$\log|F|_r\le\log|F|_R-{\rm ord}(F,S)(\Omega(S)-\varepsilon)\log{R\over
C_\varepsilon r},$$
and that $\Omega(S)\ge(\omega_u(S)+1)/(u+n-1)$ for every $u\ge 1$
(this last inequality is due to Esnault-Viehweg [E-V], who used
deep tools of algebraic geometry; Azhari [Az] reproved it recently by
means of H\"ormander's $L^2$ estimates). Rather simple examples
(see [De1]) lead to the conjecture
$$\Omega(S)\ge{\omega_u(S)+n-1\over u+n-1}~~~~\hbox{for every}~~u\ge 1.$$
The special case $u=1$ of the conjecture is due to Chudnovsky [Ch].}

Finally, let us mention that Cor.~8.6 contains Bombieri's theorem [Bo]
on algebraic values of meromorphic maps satisfying algebraic
differential equations.
Recall that an entire function $F\in\oc(\bbbc^n)$ is said to be of
order $\le\rho$ if for every $\varepsilon>0$ there
is a constant $C_\varepsilon$ such that $|F(z)|\le C_\varepsilon\exp(
|z|^{\rho+\varepsilon})$. A meromorphic function is said to be of order
$\le\rho$ if it can be written $G/H$ where $G$, $H$ are entire functions
of order $\le\rho$. 

\statement{(8.9) Theorem {\rm(Bombieri)}}{Let $F_1\ld F_N$ be meromorphic
functions on $\bbbc^n$, such that $F_1\ld F_d$, $n<d\le N$, are algebraically
independent over $\bbbq$ and have finite orders $\rho_1\ld \rho_d$.
Let $K$ be a number field of degree~\hbox{$[K:\bbbq]$}.
Suppose that the ring $K[f_1\ld f_N]$ is stable under all derivations
$d/dz_1\ld d/dz_n$. Then the set $S$ of points $z\in\bbbc^n$, distinct from
the poles of the $F_j$'s, such that $(F_1(z)\ld F_N(z))\in K^N$ is
contained in an algebraic hypersurface whose degree $\delta$ satisfies
$${\delta+n(n-1)/2\over n!}\le{\rho_1+\ldots+\rho_d\over d-n}[K:\bbbq].$$}

\proof{If the set $S$ is not contained in any algebraic hypersurface of
degree $<\delta$, the linear algebra argument used in the proof of
Cor.~8.6 shows that we can find $m=m(\delta)$ points $w_1\ld w_m\in S$
which are not located on any algebraic hypersurface of degree~$<\delta$.
Let $H_1\ld H_d$ be the denominators of $F_1\ld F_d$. The standard
arithmetical methods of transcendental number theory allow us to
construct a sequence of entire functions in the following way: we set
$$G=P(F_1\ld F_d)(H_1\ldots H_d)^s$$
where $P$ is a polynomial of degree $\le s$ in each variable with integer
coefficients. The polynomials $P$ are chosen so that $G$ vanishes at a very
high order at each point~$w_j$. This amounts to solving a linear system
whose unknowns are the coefficients of $P$ and whose coefficients
are polynomials in the derivatives of the $F_j$'s (hence lying
in the number field $K$). Careful estimates of size and denominators
and a use of the Dirichlet-Siegel box principle leads to the
following lemma (cf. for example Waldschmidt~[Wa2]).}

\lemma{8.10}{For every $\varepsilon>0$, there exist constants $C_1,C_2>0$,
$r\ge 1$ and an infinite sequence $G_t$, $t\in T\subset\bbbn$
$($depending on $m$ and on the choice of the points $w_j)$, such that
\smallskip
\item{\rm(a)} $G_t$ vanishes  at order $\ge t$ at all points $w_1\ld w_m\,;$
\smallskip
\item{\rm(b)} $|G_t|_r\ge (C_1t)^{-t\,[K:\bbbq]}\,;$
\smallskip
\item{\rm(c)} $|G_t|_{R(t)}\le C_2^t$~~ where
$R(t)=(t^{d-n}/\log t)^{1/(\rho_1+\ldots+\rho_d+\varepsilon)}$.
\smallskip}

An application of Cor.~8.6 to $F=G_t$ and $R=R(t)$ gives the desired
bound for the degree $\delta$ as $t$ tends to~$+\infty$ and
$\varepsilon$ tends to~$0$. If $\delta_0$ is the largest integer which
satisfies the inequality of Th.~8.9, we get a contradiction if we take
$\delta=\delta_0+1$. This shows that $S$ must be contained in an
algebraic hypersurface of degree $\delta\le\delta_0$.\qed

\titleb{9. Global Intersection Class and Self-intersection}
Let $X$ be a compact complex $n$-dimensional manifold. With every closed
current $\Theta$ of degree $k$ (or of bidegree $(p,q)$ with $p+q=k$),
we can associate a De Rham cohomology class $\{\Theta\}\in
H^k_{DR}(X,\bbbr)$. In fact, it is well known that De Rham cohomology
can be computed either by the complex of smooth differential forms or by
the complex of currents: both complexes of sheaves are fine resolutions
of the locally constant sheaf~$\bbbr$. Moreover, the assignment
$\Theta\mapsto\{\Theta\}$ is continuous with respect to the weak topology
of currents; this can be easily seen e.g. by Poincar\'e duality. In
particular, with every analytic cycle $A=\sum\lambda_jA_j$ of pure dimension
$p$ in $X$ is associated a De Rham cohomology class
$$\{A\}=\Big\{\sum\lambda_j[A_j]\Big\}\in H^{2n-2p}_{DR}(X,\bbbr).
\leqno(9.1)$$
When the coefficients $\lambda_j$ are integers, the class $\{A\}$ lies in
the image of $H^{2n-2p}(X,\bbbz)\,$: this is for instance a consequence
of the fact that every analytic set can be triangulated.

The wedge product of smooth differential forms defines a ring structure
on De Rham cohomology. Given two currents $\Theta_1$, $\Theta_2$ on~$X$,
there is a well defined {\sl intersection class} $\{\Theta_1\}
\cdot\{\Theta_2\}$ in the cohomology ring, even when $\Theta_1\wedge\Theta_2$
is not defined pointwise as a current. Especially, when $\deg\,\Theta_1+
\deg\,\Theta_2=\dim_\bbbr X$, the top degree class
$\{\Theta_1\}\cdot\{\Theta_2\}$ can be
considered as a number after integration over~$X$. These simple
observations show in fact that wedge products of closed positive currents
cannot be defined in a reasonable way without further assumptions:
if $X$ is the blow-up of some other manifold at one point and
$E\simeq\bbbp^{n-1}$ is the exceptional divisor, then
$\oc(E)_{\restriction E}\simeq\oc(-1)$ and so
$\{E\}^n=\int_Ec_1(\oc(E))^{n-1} =(-1)^{n-1}\,$; thus $\{E\}^2<0$ if
$X$ is a surface$\,$! The same example shows that, in general, a closed
positive $(1,1)$-current $T$ cannot be approximated in the weak
topology by smooth closed positive currents: a necessary condition for
this is that $\{T\}^p\cdot\{Y\}\ge 0$ for every $p$-dimensional
subvariety $Y\subset X$. However, a result proved in [De7] which we
shall now recall shows that $T$ can be approximated by closed real
currents with small negative part controlled by the curvature of~$X$.
This result allows us to compute self-intersections
by taking weak limits of products in which the original currents
have been replaced by their regularizations. This technique will be
applied here to get a fairly general self-intersection inequality
for closed positive currents of bidegree~$(1,1)$.

  We say that a bidimension $(p,p)$ current $T$ 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 $dd^c\varphi$ is almost positive; conversely, if a
locally integrable function $\varphi$ is such that $dd^c\varphi$ is
almost positive, then $\varphi$ is equal a.e. to an almost psh
function. If $T$ is closed and almost positive, the Lelong numbers 
$\nu(T,x)$ are well defined, since the negative part always contributes
to zero. We refer to [De7] for a proof of the following basic
approximation theorem:

\theorem{9.2}{Let $T$ be a closed almost positive $(1,1)$-current and
let $\alpha$ be a smooth real $(1,1)$-form in the the same
$dd^c$-cohomology class as $T$, i.e. $T=\alpha+dd^c\psi$ 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 curvature form satisfies
$$c\big(\oc_{TX}(1)\big)+\pi^\star u\ge 0$$
with $\pi:P(T^\star X)\to X$ 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
$T_{c,k}=\alpha+dd^c\psi_{c,k}$ such that $\psi_{c,k}$
is smooth on $X\ssm E_c(T)$ and decreases to $\psi$ as $k$ tends to~$+\infty$
$($in particular, $T_{c,k}$ is smooth on $X\ssm E_c(T)$ and converges weakly
to $T$ on~$X)$, and
$$T_{c,k}\ge\gamma-\lambda_{c,k}u-\varepsilon_k\omega$$
where
\item{\rm(i)}$\lambda_{c,k}(x)$ is a decreasing sequence of continuous functions
on $X$ such that $\lim_{k\to+\infty}\lambda_{c,k}(x)=\min\big(\nu(T,x),c\big)$
at every point,
\item{\rm(ii)}$\lim_{k\to+\infty}\varepsilon_k=0$,
\item{\rm(iii)}$\nu(T_{c,k},x)=\big(\nu(T,x)-c\big)_+$
at every point $x\in X$.\qed}

Here $\oc_{TX}(1)$ is the canonical line bundle associated with $TX$ over
the hyperplane bundle~$P(T^\star X)$. Observe that the theorem gives in
particular approximants $T_{c,k}$ which are smooth everywhere on $X$ if
$c$ is taken such that $c>\max_{x\in X}\nu(T,x)$. The equality
in~(iii) means that the procedure kills all Lelong numbers that are
$\le c$ and shifts all others downwards by $c$. Hence Th.~9.2 is an
analogue over manifolds of Kiselman's procedure [Ki1,2] for killing Lelong
numbers of a plurisubharmonic functions on an open subset of~$\bbbc^n$.

\corollary{9.3}{Let $\Theta$ be a closed almost positive current of
bidimension $(p,p)$ and let $\alpha_1\ld\alpha_q$ be closed almost
positive $(1,1)$-currents such that $\alpha_1\wedge\ldots\wedge\alpha_q
\wedge \Theta$ is well defined by application of criteria~$2.3$ or $2.5$,
when $\alpha_j$ is written locally as~$\alpha_j=dd^cu_j$. Then
$$\{\alpha_1\wedge\ldots\wedge\alpha_q\wedge \Theta\}=
\{\alpha_1\}\cdots\{\alpha_q\}\cdot\{\Theta\}.$$}

\proof{Theorem~9.2 and the monotone continuity
theorems of \S~2 show that
$$\alpha_1\wedge\ldots\wedge\alpha_q\wedge \Theta=\lim_{k\to+\infty}
\alpha_1^k\wedge\ldots\wedge\alpha_q^k\wedge \Theta$$
where $\alpha_j^k\in\{\alpha_j\}$ is smooth. Since the result is
by definition true for smooth forms, we conclude by the weak
continuity of cohomology class assignment.\qed}

Now, let $X$ be a compact K\"ahler manifold equipped with a K\"ahler
metric~$\omega$. The {\sl degree} of a closed positive current $\Theta$
with respect to $\omega$ is by definition
$$\deg_\omega\Theta=\int_X\Theta\wedge\omega^p,~~~~
\hbox{\rm bidim}\,\Theta=(p,p).\leqno(9.4)$$
In particular, the degree of a $p$-dimensional analytic set $A\subset X$
is its volume $\int_A\omega^p$ with respect to~$\omega$. We are
interested in the following problem.

\statement{(9.5) Problem}{Let $T$ be a closed positive $(1,1)$-current
on~$X$. Is it possible to derive a bound for the codimension~$p$
components in the Lelong upperlevel sets $E_c(T)$ in terms of the
cohomology class $\{T\}\in H^2_{DR}(X,\bbbr)\,{\rm ?}$}

We introduce 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
$\codim E_c(T)=p$ with at least some component of codimension $p$
when $c\in{}]b_p,b_{p+1}]$. Let $(Z_{p,k})_{k\ge 1}$ be the $p$-codimensional
components occurring in any of the sets $E_c(T)$ for $c\in{}]b_p,b_{p+1}]$,
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~$Z_{p,k}$. Then we have
the following self-intersection inequality.

\theorem{9.6}{Suppose that $X$ is K\"ahler and that $\oc_{TX}(1)$ has
a hermitian metric such that $c\big(\oc_{TX}(1)\big)+\pi^\star u\ge 0$,
where $u$ is a smooth closed semi\-positive $(1,1)$-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.}

By neglecting the second term in the right hand side and taking the
wedge product with $\omega^{n-p}$, we get the following interesting
consequence:

\corollary{9.7}{If $\omega$ is a
K\"ahler metric on $X$ and if $\{u\}$ is a semipositive cohomology class such
that $c_1\big(\oc_{TX}(1)\big)+\pi^\star\{u\}$ is semipositive, the degrees of
the components $Z_{p,k}$ with respect to $\omega$ satisfy the estimate
$$\eqalign{
\sum_{k=1}^{+\infty}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)
&\int_X[Z_{p,k}]\wedge\omega^{n-p}\cr
&\le\big(\{T\}+b_1\{u\}\big)\cdots\big(\{T\}+b_p\{u\}\big)
\cdot\{\omega\}^{n-p}.\cr}$$}

By a semipositive cohomology class of type $(1,1)$, we mean a class
in the closure of the K\"ahler cone of~$X$. As a special case, if $D$
is an effective divisor and $T=[D]$, we get a bound for
the degrees of the $p$-codimensional singular strata of $D$ in terms of
a polynomial of degree $p$ in the cohomology class~$\{D\}$; the
multiplicities $(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)$ are then positive
integers.\break The case when $X$ is $\pb^n$ or a homogeneous manifold
is especially simple: then $TX$ is generated by sections and
we can take $u=0\,$; the bound is thus simply $\{D\}^p\cdot
\{\omega\}^{n-p}$; the same is true more generally as soon as $\oc_{TX}(1)$
is semipositive. The main idea of the proof
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 it becomes possible to define the wedge product
$T_1\wedge\ldots\wedge T_p$ by means of Th.~2.5. Here are the details:

\pproof{Proof of 9.6}{We argue by induction on~$p$. For $p=1$,
Siu's decomposition formula shows that
$$T=\sum\nu_{1,k}[Z_{1,k}]+R,$$
and 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
$T_{c,k}=\alpha+dd^c\psi_{c,k}$
produced by Th.~9.2 is such that $\codim L(\psi_{c,k})=\codim E_c(T)\ge p$.
Hence Cor.~2.10 shows that
$$\Theta_{p,c,k}=\Theta_{p-1}\wedge(T_{c,k}+c\,u+\varepsilon_k\omega)$$
is well defined. If $\varepsilon_k$ tends to zero slowly enough,
$T_{c,k}+c\,u+\varepsilon_k\omega$ is positive by (9.2$\,$i), so
$\Theta_{p,c,k}\ge 0$. Moreover, the
cohomology class of $\Theta_{p,c,k}$ is $\{\Theta_{p-1}\}\cdot
(\{T\}+c\{u\}+\varepsilon_k\{\omega\})$, converging to
$\{\Theta_{p-1}\}\cdot(\{T\}+c\{u\})$. Since the mass
$\int_X\Theta_{p,c,k}\wedge\omega^{n-p}$ remains uniformly bounded,
the family $(\Theta_{p,c,k})_{c\in{}]b_p,b_p+1],k\ge 1}$ is relatively
compact in the weak topology. 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 (3.12), (5.10) and (9.2$\,$iii). Hence by induction
we 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 $(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)$. This already
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 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 $\lim (dd^c\psi'_{c,k})_{\rm abc}
=(dd^c\psi)_{\rm abc}$ almost everywhere (see Lemma~10.12 in [De6]).\qed}
\vskip40pt
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\vskip15pt
\eightpoint
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\noindent\tenpoint
(May 6, 1991, revised on May 18, 1992)
\end

