%Article J.-P. Demailly ``A numerical criterion for very ample line bundles"

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\centerline{\bftwelve A numerical criterion for very ample line bundles}
\vskip10pt
{\noindent\hangindent4cm\hangafter-1
{\bf Jean-Pierre Demailly

}}

{\noindent\hangindent4cm\hangafter-4{\sl
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
Laboratoire associ\'e au C.N.R.S.
n\kern-2pt{\rm\char'27}188,\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\vskip20pt\noindent
{\bf Abstract}\pointir{\sl Let $X$ be a projective algebraic manifold of
dimension $n$ and let $L$ be an ample line bundle over $X$. We give 
a numerical criterion ensuring that the adjoint bundle $K_X+L$ is very 
ample. The sufficient conditions are expressed in terms of lower bounds 
for the intersection numbers $L^p\cdot Y$ over subvarieties $Y$ of~$X$. 
In the case of surfaces, our criterion gives universal bounds and
is only slightly weaker than I.~Reider's criterion. When $\dim X\ge 3$
and $\codim Y\ge 2$, the lower bounds for $L^p\cdot Y$ involve a 
numerical constant which depends on the geometry of~$X$. By means of
an iteration process, it is finally shown that $2K_X+mL$ is very ample
for $m\ge 12\,n^n$. Our approach is mostly analytic and based on a 
combination of H\"ormander's $L^2$~estimates for the operator $\op$, 
Lelong number theory and the Aubin-Calabi-Yau theorem.}
\vskip20pt

\noindent{\bf Table of contents}\vskip5pt
{\eightpoint\noindent
1. Introduction\dotfill p. 2\break
2. Singular hermitian metrics on holomorphic line bundles\dotfill p. 5\break
3. Basic results on Lelong numbers\dotfill p. 6\break
4. $L^2$ estimates and existence of holomorphic sections\dotfill p. 8\break
5. Aubin-Calabi-Yau theorem and convexity inequalities\dotfill p. 12\break
6. Mass concentration in the Monge-Amp\`ere equation\dotfill p. 14\break
7. Choice of the logarithmic singularities\dotfill p. 17\break
8. Upper bound for the 1-codimensional polar components\dotfill p. 19\break
9. Approximation of closed positive (1,1)-currents by divisors\dotfill p. 21\break
10. Self-intersection inequality for closed positive currents\dotfill p. 25\break
11. Proof of the criterion in arbitrary dimension\dotfill p. 33\break
12. Universal bounds for very ample line bundles\dotfill p. 38\break
References\dotfill p. 43
}\vskip20pt
\noindent
{\bf Key words:} Very ample line bundle, numerical criterion,
plurisubharmonic function, closed positive current, Lelong number,
Monge-Amp\`ere equation, Aubin-Calabi-Yau theorem, intersection theory.
\vskip10pt
\noindent
{\bf A.M.S. Classification:} 32J25, 32L10, 32C30
\vfill\eject

\titre 1. Introduction|

Let $L$ be a holomorphic line bundle over a projective algebraic
manifold $X$ of dimension $n$. We denote the canonical line bundle of
$X$ by $K_X$ and use an additive notation for the group ${\rm Pic}(X)
=H^1(X,{\cal O}^\star)$. The original motivation of this work was to
study the following tantalizing conjecture of Fujita [Fu~88]:
if~$L\in{\rm Pic}(X)$ is ample, then $K_X+(n+2)L$ is very ample; the
constant $n+2$ would then be optimal since $K_X+(n+1)L=\oc_X$ is not
very ample when $X=\pb^n$ and $L=\oc(1)$. Although such a sharp result
seems at present out of reach, a consequence of our results will be
that $2K_X+mL$ is  always very ample for $L$ ample and $m$ larger than
some universal  constant depending only on~$n$.

Questions of this sort play a very important role in the classification
theory of projective varieties. In his pioneering work [Bo~73], Bombieri 
proved the existence of pluricanonical embeddings of low degree for 
surfaces of general type. More recently, for an ample line bundle $L$ 
over an algebraic surface $S$, I.~Reider [Rei~88] obtained a sharp 
numerical criterion ensuring that the adjoint line bundle $K_S+L$ is 
spanned or very ample; in particular, $K_S+3L$ is always spanned and
$K_S+4L$ very ample. Reider's method
was further developed by Catanese [Ca~88], Sakai [Sa~88] and
Beltrametti-Francia-Sommese [BFS~89], who studied
the existence of higher order embeddings via $s$-jets. Reider's approach
is based on the construction of rank two vector bundles associated to some
$0$-cycles in special position with respect to the linear system
$|K_S+L|$ and a use of Bogomolov's inequality for stable vector bundles.
Unfortunately, these methods do not apply in dimension $\ge 3$ and no
similar general result was available. In a somewhat different
context, Fujita [Fu~87] proved that $K_X+(n+2)L$ is always ample.
This result is obtained via Mori's theory of extremal rays [Mo~82]
and the cone theorem of Kawamata (cf. [Ka~84], [KMM~87]), but the 
arguments are purely numerical and give apparently no insight on the 
very ample property. 

Our purpose here is to explain a completely different analytic
approach which is applicable in arbitrary dimension.
Let us first recall a few usual notations that will be used constantly
in the sequel:
$$L_1\cdots L_p\cdot Y=\int_Y c_1(L_1)\wedge\ldots\wedge c_1(L_p)\leqno(1.1)$$
denotes the intersection product of $p$ line bundles $L_1\ld L_p$ 
over a $p$-dimensional subvariety $Y\subset X$. In case $L_1=\ldots=L_p$
we write instead $L^p\cdot Y$ and in case $Y=X$ we omit $Y$ in the
notation. Similar notations will be used for divisors. Recall that a 
line bundle (or a $\rb$-divisor) $L$ over $X$ is said 
to be numerically effective, {\sl nef} for short, if $L\cdot C\ge 0$ 
for every curve $C\subset X\,$; in this case $L$ is said to be {\sl big}
if $L^n>0$.  More generally, a vector bundle $E$ is said to be 
nef if the associated line bundle $\oc_E(1)$ is nef 
over $P(E^\star)=$ projective space of hyperplanes in $E\,$; any vector
bundle $E$ such that some symmetric power $S^mE$ is spanned by its global
sections is nef. In this context, we shall prove:\eject

\tha Main Theorem|Let $X$ be a projective $n$-fold and let $L$ be a big nef
line bundle over $X$. Suppose that there is a number $a\ge 0$ such
that $TX\otimes\oc(aL)$ is nef. Then $K_X+L$ is spanned at each point of a given
subset $\Xi$ of $X$ $($resp. separates all points in $\Xi$,
resp. generates $s$-jets at any point of $\Xi\,)$ provided that
$L^n>\sigma_0$ with $\sigma_0=n^n$ $($resp. $\sigma_0=2n^n$, resp.
$\sigma_0=(n+s)^n\,)$, and that there exists a sequence $0=\beta_1<\ldots<
\beta_n\le 1$ such that any subvariety $Y\subset X$ of codimension
$p=1,2\ld n-1$ intersecting $\Xi$ satisfies
$$L^{n-p}\cdot Y>(\beta_{p+1}-\beta_1)^{-1}\ldots(\beta_{p+1}-
\beta_p)^{-1}\sum_{0\le j\le p-1}S_j^p(\beta)\,a^j\sigma_{p-j}$$
with $S^p_0(\beta)=1$, $S^p_j(\beta)={}$ elementary symmetric function
of degree $j$ in $\beta_1\ld\beta_p$ and
$$\sigma_p=\Big(1-\Big(1-{\sigma_0\over L^n}\Big)^{p/n}\Big)L^n,~~~~
\sigma_p\in{}]\sigma_0p/n,\sigma_0[\,.$$\fintha

The expression ``separation of points" used here includes infinitesimal
separation, that is, generation of $1$-jets at each point (the constant
$\sigma_0=(n+1)^n$ corresponding to $s=1$ can therefore be replaced by
the smaller value $2n^n$).
In fact, our proof also gives sufficient conditions for the generation
of jets corresponding to arbitrary \hbox{$0$-dimensional} subschemes
$(\Xi,\oc_\Xi)$ of $X$, simply by changing the value of 
$\sigma_0\,$; for example, if $(\Xi,\oc_\Xi)$ is a local complete
intersection, the constant $\sigma_0$ can be taken equal to 
$n^n\,h^0(\Xi,\oc_\Xi)\,$; unfortunately, this value is in general far 
from being optimal. Notice that the number $a$ involved in the hypothesis
on $TX$ need not be an integer nor even a rational number: the
hypothesis then simply means that any real divisor associated to
$\oc_{TX}(1)+a\,\pi^\star L$ is nef over~$P(T^\star X)$.

As the notation is rather complicated, it is certainly worth examining
the particular case of surfaces and $3$-folds. 
If $X$ is a surface, we have $\sigma_0=4$ (resp. $\sigma_0=8$, 
resp. $\sigma_0=(2+s)^2$), and we take $\beta_1=0$, $\beta_2=1$. This gives
only two conditions, namely
$$L^2>\sigma_0,~~~~L\cdot C>\sigma_1\leqno(1.2)$$
for every curve $C$ intersecting $\Xi$. In that case, the proof shows that 
the assumption on the existence of $a$ is unnecessary. These bounds are not
very far from those obtained with Reider's method, although they are not 
exactly as sharp. If $X$ is a $3$-fold,
we have $\sigma_0=27$ (resp. $\sigma_0=54$, resp. $\sigma_0=(3+s)^3$),
and we take $\beta_1=0<\beta_2=\beta<\beta_3=1$. Therefore our condition
is that there exists $\beta\in{}]0,1[$ such that
$$L^3>\sigma_0,~~~~L^2\cdot S>\beta^{-1}\sigma_1,~~~~
L\cdot C>(1-\beta)^{-1}(\sigma_2+\beta\,a\sigma_1)\leqno(1.3)$$
for every curve $C$ or surface $S$ intersecting $\Xi$.

In general, we measure the ``amount of ampleness" of a nef line bundle $L$
on a subset $\Xi\subset X$ by the number
$$\mu_\Xi(L)=\min_{1\le p\le n}~~\min_{\dim Y=p,\,Y\cap\Xi\ne\emptyset}~~
(L^p\cdot Y)^{1/p},\leqno(1.4)$$
where $Y$ runs over all $p$-dimensional subvarieties of $X$ intersecting~$\Xi$.
The Nakai-Moishezon criterion tells us that $L$ is ample if and only
if~$\mu_X(L)>0$. An effective version of this criterion can be easily
deduced from the Main Theorem: in fact, a suitable choice of the constants
$\beta_p$ in terms of $a$, $\sigma_0$ and $\mu_\Xi(L)$ yields:

\tha Corollary 1|Let $L$ be a big nef line bundle over $X$ such that
$TX\otimes\oc(aL)$ is nef for some $a\ge 0$, and let $\Xi$ be an arbitrary
subset of~$X$. Then the line bundle $K_X+mL$ spans
$($resp. separates points, resp. generates $s$-jets$)$ on $\Xi$
as soon as
$$m>{1\over\mu_\Xi(L)}\max\!\Big\{\!B_n\sigma_0,
(B_n\sigma_0)^{\textstyle{1\over n-k}}\,
(a\mu_\Xi(L))^{{\textstyle{k\over n-k}}\big({\textstyle{1\over n-1}{+}
{1\over n-2}{+}\ldots{+}{1\over k+1}}\big)}\!\Big\}_{1\le k\le n-2}$$
with $\sigma_0=n^n$ $($resp. $\sigma_0=2n^n$, resp. $\sigma_0=(n+s)^n)$ and
with a constant $B_n<2.005$ depending only on~$n$ $($table $(11.11)$
contains the first values of~$B_n)$.\fintha

When $L$ is ample, the number $a$ always exists and we have $\mu_\Xi(L)
\ge 1$ for any choice of $\Xi$. We thus get an explicit lower bound
$m_0$ depending only on~$n,a$ such that $K_X+mL$ is spanned or very
ample for $m\ge m_0$. Unfortunately, these lower bounds are rather far
from Fujita's expected conditions $m\ge n+1$ and $m\ge n+2$ respectively.
Observe however that the lower bound for $L^n$ in the Main Theorem is
optimal: if $X=\pb^n$ and $L=\oc(1)$, then $K_X=\oc(-n-1)$ so $K_X+nL$
is not spanned, although $(nL)^n=n^n=\sigma_0$. Similarly $K_X+(n+s)L$ 
does not generate $s$-jets, although
$\big((n+s)L\big)^n=(n+s)^n=\sigma_0$. When $X\subset\pb^{n+1}$ is the
$n$-dimensional quadric and $L=\oc_X(1)$, then $K_X+nL=\oc_X$ is not
very ample, although $(nL)^n= 2n^n=\sigma_0$.

Another unsatisfactory feature is that our bounds depend on the geometry
of $X$ through the number~$a$, while the case of curves or surfaces
suggests that they should not. In fact, our proof uses a rather
delicate self-intersection inequality for closed positive currents,
and this inequality (which is essentially optimal) depends in a crucial
way on a bound for the ``negative part" of~$TX$. It follows that new
ideas of a different nature are certainly necessary to get universal
bounds for the very ampleness of $K_X+L$. However, an elementary argument
shows that $TX\otimes\oc(K_X+nF)$ is always nef when $F$ is very ample
(see lemma~12.1). This observation combined with an iteration of
the Main Theorem finally leads to a universal result. Corollary~2 below
extends in particular Bombieri's result on pluricanonical embeddings
of surfaces of general type to arbitrary dimensions (at least when $K_X$ is
supposed to be ample, see 12.10 and~12.11), and can be seen as an effective
version of Matsusaka's theorem ([Ma~72], [KoM~83])$\,$:

\tha Corollary 2|If $L$ is an ample line bundle over~$X$, then $2K_X+mL$
is very ample, resp. generates $s$-jets, when $(m-1)\mu_X(L)+s>
2C_n\sigma_0$ with a constant $C_n<3$ depending only on~$n$
$($see table $(12.3))$. In particular, $2K_X+mL$
is very ample for $m>4C_nn^n$ and generates higher $s$-jets for
$m>2C_n\sigma_0$.\fintha

Our approach is based on three rather powerful analytic tools. 
First, we use H\"ormander's $L^2$ estimates for the operator
$\overline\partial$ with singular plurisubharmonic weights 
to prove a general abstract existence theorem for sections of
$K_X+L$ with prescribed jets at finitely many points; the idea is 
similar to that of the H\"ormander-Bombieri-Skoda theorem, but
following an idea of A.~Nadel [Na~89], we consider plurisubharmonic 
functions with logarithmic poles
associated to an arbitrary ideal in $\oc_{X,x}$ (see Corollary~4.6).
We refer to [De~90] for further results relating ample or nef line bundles
to singular hermitian metrics. The second tool is the Aubin-Calabi-Yau theorem.
This fundamental result allows us to solve the Monge-Amp\`ere equation
$(\omega+{i\over\pi}\partial\op\psi)^n=f$ where 
$\omega={i\over 2\pi}c(L)$ is the curvature form of~$L$, and the right hand
side $f$ is an arbitrary positive $(n,n)$-form with $\int_X f=L^n$.
We let $f$ converge to a linear combination of Dirac measures and show
that the solution $\psi$ produces in the limit a singular weight on $L$
with logarithmic poles. In order to control the poles and singularities, 
we use in an essential way a convexity inequality due to Hovanski [Hov~79]
and Teissier [Te~79,82], which can be seen 
as a generalized version of the Hodge index theorem for surfaces.
Finally we invoke in several occasions the theory of closed positive
currents and Lelong numbers (see [Le~57,69]). In particular, the 
generalized Lelong numbers introduced in [De~87] are used as a substitute
of the intersection theory of algebraic cycles in our analytic context. 
The self-intersection inequality~10.7 can be seen as a generalization to
currents (and in any dimension) of the classical upper bound $d(d-1)/2$ 
for the number of multiple points of a plane curve of degree~$d$. It~actually
gives a bound for the sum of degrees of the irreducible components in the 
sublevel sets of Lelong numbers of a closed positive $(1,1)$-current $T$ 
with integral cohomology class $\{T\}$, in terms of an explicit polynomial
in $\{T\}$.

A major part of this work has been done during the fall 1989, while the
author was visiting Bayreuth University under the support of the 
DFG-Forschungsschwerpunktprogramm``Komplexe Mannigfaltigkeiten".
The author expresses special thanks to Michael Schneider and Thomas
Peternell for very stimulating discussions.

\titre 2. Singular hermitian metrics on holomorphic line bundles|

Let $L$ be a holomorphic line bundle over a projective algebraic
manifold $X$ and $n=\dim X$.  If $L$ is equipped with a hermitian
metric, we denote by $c(L)={i\over 2\pi}\nabla^2$ the Chern curvature form,
which is a closed real $(1,1)$-form representing the first Chern class
$c_1(L)\in H^2(X,\zb)$.  It is well known that $L$ is ample if and only
if $L$ has a smooth hermitian metric such that $c(L)$ is positive
definite at every point. 

However, we are also interested in singular metrics, because they often
give additional information about the existence of sections of high
multiples~$mL$. By definition, a {\sl singular metric} on $L$ is a metric 
which is given in any trivialization $\tau:L_{\restriction\Omega}
\buildo\simeq\over\longrightarrow\Omega\times \cb$ by
$$\|\xi\|=|\tau(\xi)|\,e^{-\varphi (x)},~~~~x\in \Omega ,~\xi\in L_x
\leqno(2.1)$$
where $\varphi \in L^1_{\rm loc}(\Omega)$ is a weight function. Then
the curvature of $L$ is given by the $(1,1)$-current $c(L)={i\over\pi}
\partial\op\varphi$ on $\Omega$. For example, to any divisor 
$D=\sum\lambda_jD_j$ with coefficients $\lambda_j\in\zb$ is associated
the invertible sheaf $\oc(D)$ of meromorphic functions $f$ such that
${\rm div}(f)+D\ge 0\,$; the corresponding line bundle can be
equipped with the singular metric defined by $||f||=|f|$. If $g_j$ is a
generator of the ideal of $D_j$ on an open set $\Omega\subset X$, then 
$\tau(f)=f\prod g_j^{\lambda_j}$ defines a trivialization of $\oc(D)$
over $\Omega$, thus our singular metric is associated to the weight
$\varphi=\sum\lambda_j\log|g_j|$. By the Lelong-Poincar\'e equation,
we find
$$c\big(\oc(D)\big)={i\over\pi}\partial\op\varphi=[D],\leqno(2.2)$$
where $[D]=\sum\lambda_j[D_j]$ denotes the current of integration over $D$.

In the sequel, all singular metrics are supposed to have positive
curvature in the sense of currents (cf. [Le~57]); i.e., the weight
functions $\varphi$ are supposed to be plurisubharmonic. Let us recall 
some results of [De~90]: consider the real Neron-Severi space $NS_\rb(X)=
\big(H^2(X,\zb)\cap H^{1,1}(X)\big)\otimes\rb$ of algebraic cohomology 
classes of degree~$2$, and let $\Gamma_+\subset NS_\rb(X)$ (resp.
$\Gamma_a\subset\Gamma_+$), be the closed convex cone generated by 
cohomology classes of effective (resp. ample) divisors $D\,$; 
denote by $\Gamma^\circ_+$ (resp. $\Gamma^\circ_a$)
the interior of $\Gamma_+$ (resp. $\Gamma_a$). Then, if $\omega$ is
a K\"ahler metric on $X$ and $\varepsilon>0$, we have the following 
equivalent properties:
$$\leqalignno{
&\qquad c_1(L)\in\Gamma_+\Llra L~\hbox{\rm has a singular metric with}~
c(L)\ge 0\,;&(2.3)\cr
&\qquad c_1(L)\in\Gamma^\circ_+\Llra\exists\varepsilon,~L~
\hbox{\rm has a singular metric with}~
c(L)\ge \varepsilon\omega\Llra \kappa(L)=n\,;&(2.4)\cr
&\qquad c_1(L)\in\Gamma_a\Llra\forall\varepsilon,~L~
\hbox{\rm has a smooth metric with}~
c(L)\ge-\varepsilon\omega\Llra L~\hbox{is nef}\,;&(2.5)\cr
&\qquad c_1(L)\in\Gamma^\circ_a\Llra\exists\varepsilon,~L~
\hbox{\rm has a smooth metric with}~
c(L)\ge \varepsilon\omega\Llra L~\hbox{is ample}.&(2.6)\cr}$$
The notation $\kappa(L)$ stands for the Kodaira dimension of $L$, that
is by definition, the supremum of the generic rank of the rational maps to
projective space defined by the non zero sections in $H^0(X,mL)$ for $m\ge 1$
(if any), and $\kappa(L)=-\infty$ otherwise; alternatively, $\kappa(L)$ is the 
smallest constant such that $h^0(X,mL)\le O(m^{\kappa(L)})$. The only 
thing that will be needed here is the fact that
a big nef line bundle satisfies the equivalent properties in (2.4); we 
shall briefly sketch the proof of this. If $L$ is nef, the Hilbert polynomial
of $\chi(X,mL)$ has leading coefficient $L^n/n!\ge 0$, and it is well known
that $h^j(X,mL)=O(m^{n-1})$, thus $h^0(X,mL)=L^n/n!\,m^n+O(m^{n-1})$. Hence
$L$ is big if and only if $\kappa(L)=n$. Let $A$ be an ample divisor. 
Then $H^0(X,mL-A)$ is the kernel
of $H^0(X,mL)\to H^0(A,mL_{\restriction A})$, and the target has dimension 
$\le Cm^{n-1}$. When $\kappa(L)=n$ we get $H^0(X,mL-A)\ne 0$ for $m$ large,
so there is an effective divisor $E$ such that $mL\simeq A+E$. Now, 
$pL+A$ is ample for every $p\ge 0$, so $pL+A$ has a smooth metric
with $c(pL+A)\ge\varepsilon_p\omega$, and the isomorphism $(m+p)L\simeq pL+A+E$
gives a metric on $L$ such that
$$c(L)=(m+p)^{-1}(c(pL+A)+[E])\ge (m+p)^{-1}\varepsilon_p\,\omega.\leqno(2.7)$$
Observe that the singular part $(m+p)^{-1}[E]$ can be chosen as small 
as desired by taking $p$ large.

\titre 3. Basic results on Lelong numbers|

These results will be needed in the sequel as an analytic analogue 
of some standard facts in the intersection theory of algebraic cycles.
They are developed in more details in [De~82a,85,87] (cf. Lelong
[Le~57,69] for an earlier presentation). We first recall a few definitions.
Let $T$ be a closed positive current of bidimension $(p,p)$, that is,
of bidegree $(n-p,n-p)$, on an open set $\Omega\subset\cb^n$. 
The Lelong number of $T$ at a point $x\in\Omega$ is defined by 
$\nu(T,x)=\lim_{r\to 0{\scriptscriptstyle+}}\nu(T,x,r)$ where
$$\nu(T,x,r)={1\over(2\pi r^2)^p}\int_{B(x,r)}T(z)\wedge
(i\partial\op|z|^2)^p\leqno(3.1)$$
measures the ratio of the mass of $T$ in the ball $B(x,r)$ to the 
area of the ball of radius $r$ in $\cb^p$; this ratio is an increasing
function of $r$ (cf. [Le~69]), and the limit $\nu(T,x)$ does not
depend on the choice of coordinates. In the case where $T$ is a
current of integration $[A]$ over an analytic subvariety $A$, the
Lelong number $\nu([A],x)$ coincides with the multiplicity of $A$ at $x$
(Thie's theorem [Th~67]).

More generally, let $\varphi$ be a continuous plurisubharmonic function
with an isolated $-\infty$ pole at $x$, e.g. a function of the form
$\varphi(z)=\log\sum_{1\le j\le N}|g_j(z)|^{\gamma_j}$, $\gamma_j>0$,
where $(g_1\ld g_N)$ is an ideal of germs of holomorphic functions in 
$\oc_x$ with $g^{-1}(0)=\{x\}$. According to [De~87], the generalized
Lelong number $\nu(T,\varphi)$ of $T$ with respect to the
weight $\varphi$ is the limit when $t$ tends to $-\infty$ of
$$\nu(T,\varphi,t)=\int_{\varphi(z)<t}T\wedge\Big({i\over\pi}
\partial\op\varphi\Big)^p.\leqno(3.2)$$
Because of the logarithmic singularity of $\varphi$, the integral is not
well defined a priori. In fact, we can use Bedford and Taylor's 
definition of the Monge-Amp\`ere operator for locally bounded 
plurisubharmonic functions (see \S~10) and set 
$$\int_{\varphi(z)<t}T\wedge
(i\partial\op\varphi)^p=\int_{\varphi(z)<t}T(z)\wedge
\big(i\partial\op\max(\varphi(z),s)\big)^p,~~~~s<t\,;\leqno(3.3)$$
observe that the right hand side is independent of $s$ by Stokes' 
formula. The relation with ordinary Lelong numbers
comes from the equality
$$\nu(T,x,r)=\nu(T,\varphi,\log r),~~~~
\varphi(z)=\log|z-x|,\leqno(3.4)$$
in particular $\nu(T,x)=\nu(T,\log|\bu-x|)$. This equality is in turn a
consequence of the following general formula, applied to $\chi(t)=e^{2t}$
and $t=\log r\,$:
$$\int_{\varphi(z)<t}T\wedge\big(i\partial\op\chi\circ\varphi\big)^p=
\chi'(t-0)^p\int_{\varphi(z)<t}T\wedge\big(i\partial\op\varphi)^p,
\leqno(3.5)$$
where $\chi$ is an arbitrary convex increasing function. To prove the
formula, we use a regularization and thus suppose that $T$,
$\varphi$ and $\chi$ are smooth, and that $t$ is a non critical value of
$\varphi$. Then Stokes' formula shows that the integrals on the left and on
the right of (3.5) are equal respectively to
$$\int_{\varphi(z)=t}T\wedge\big(i\partial\op\chi\circ\varphi\big)^{p-1}
\wedge i\op(\chi\circ\varphi),~~~~
\int_{\varphi(z)=t}T\wedge\big(i\partial\op\varphi\big)^{p-1}
\wedge i\op\varphi,$$
and the differential form of bidegree $(p-1,p)$ appearing in the integrand
of the first integral is equal to
$(\chi'\circ\varphi)^p\,(i\partial\op\varphi)^{p-1}\wedge i\op\varphi$.
The expected formula follows. 

It is shown in [De~87] that $\nu(T,\varphi)$ depends only on the
asympotic behaviour of $\varphi$ near the pole $x$, namely the Lelong
number remains unchanged for a weight $\psi$ such that $\lim_{z\to x}
\psi(z)/\varphi(z)=1$. More generally, if $\limsup_{z\to x}
\psi(z)/\varphi(z)=\lambda$, then
$$\nu(T,\psi)\le\lambda^p\,\nu(T,\varphi).\leqno(3.6)$$
Finally, let $F$ is a proper holomorphic map from a neighborhood of $x$
onto a neighborhood of $y$ in $\cb^n$, and let $\psi$ be a 
continuous plurisubharmonic function with an isolated pole at $y$.
The definition of the direct image $F_\star T$ by adjunction of
$F^\star$ easily shows that for $t<t_0$ sufficiently small
$$\nu(F_\star T,\psi,t)=\nu(T,\psi\circ F,t),~~~~
\nu(F_\star T,\psi)=\nu(T,\psi\circ F).\leqno(3.7)$$

For any closed positive current $T$ of bidimension $(p,p)$ on a complex
manifold $X$ and any positive number $c$, we let $E_c(T)$ be the set of
points $z\in X$ where $\nu(T,z)\ge c$. By a theorem of [Siu~74], all
sublevel sets $E_c(T)$ are closed analytic subsets of $X$ of dimension
at most $p$. Moreover $T$ can be written as a convergent series of
closed positive currents
$$T=\sum_{k=1}^{+\infty}\lambda_k\,[Z_k]+R,\leqno(3.8)$$
where $[Z_k]$ is a current of integration over an irreducible
analytic set of dimension~$p$, and $R$ is a residual current 
with the property that $\dim E_c(R)<p$ for every $c>0$.
This decomposition is locally and globally unique: the sets $Z_k$
are precisely the $p$-dimensional components occurring in the
sublevel sets $E_c(T)$, and $\lambda_k=\min_{x\in Z_k}\nu(T,x)$ is
the generic Lelong number of $T$ along $Z_k$.

The Lelong number of a plurisubharmonic function $w$ on $X$
can also be defined by
$$\nu(w,x)=\liminf_{z\to x}{w(z)\over\log|z-x|}~,\leqno(3.9)$$
where $z=(z_1\ld z_n)$ are local coordinates near~$x$, and $|~~|$ denotes
an arbitrary norm on $\cb^n$. It is well known that $\nu(w,x)$ is equal
to the Lelong number $\nu(T,x)$ of the associated positive
$(1,1)$-current $T={i\over\pi}\partial\op w$. Accordingly, we set 
$E_c(w)=E_c(T)$. 

\titre 4. $L^2$ estimates and existence of holomorphic sections|

We first state the basic existence theorem of H\"ormander for solutions 
of $\ov\partial$ equations, in the form that is most convenient to us. 

\tha Proposition 4.1|Suppose that $X$ is a Stein or compact projective
manifold equipped with a K\"ahler metric $\omega$.  Let $L$ be a line
bundle with a hermitian metric associated to singular plurisubharmonic
weight functions $\psi$ such that $c(L)\ge\varepsilon\omega$ for some 
$\varepsilon>0$. For every
$q\ge 1$ and every $(n,q)$ form $v$ with values in $L$ such that 
$\op v=0$ and $\int_X|v|^2e^{-2\psi}dV_\omega<+\infty$, there is a 
$(n,q-1)$-form $u$ with values in $L$ such that $\op u=v$ and
$$\int_X|u|^2e^{-2\psi}dV_\omega\le{1\over2\pi q\varepsilon}
\int_X|v|^2e^{-2\psi}dV_\omega.$$\fintha

Here $dV_\omega$ stands for the K\"ahler volume element $\omega^n/n!$,
and $|u|^2e^{-2\psi}$ denotes somewhat abusively the pointwise norm of
$u(z)$ at each point $z\in X$, although $\psi$ is only defined on
an open set in $X$. The operator $\op$ is taken in the sense of
distribution theory.

\remarque Proof|The result is standard when $X$ is Stein and $L$ is the
trivial bundle (see [AV~65] and [H\"o~66]).  In general, there exists a
hypersurface $H\subset X$ such that $X\ssm H$ is Stein and $L$
is trivial over $X\ssm H$.  We then solve the equation $\op
u=v$ over $X\ssm H$ and observe that the solution extends to
$X$ thanks to the $L^2$ estimate (cf. [De~82b], lemma 6.9).\finpr

We will also use the concept of {\sl multiplier ideal sheaf} introduced
by A.~Nadel [Na~89]. The main idea actually goes back to the fundamental
works of Bombieri [Bo~70] and H.~Skoda [Sk~72a]. Let $\varphi$ be a
plurisubharmonic function on $X\,$; to $\varphi$ is associated the ideal
subsheaf $\ic(\varphi)\subset\oc_X$ of germs of holomorphic functions $f\in
\oc_{X,x}$ such that $|f|^2e^{-2\varphi}$ is integrable with respect to the
Lebesgue measure in some local coordinates near~$x$. The zero variety
$V\ic(\varphi)$ is thus the set of
points in a neighborhood of which $e^{-2\varphi}$ is non integrable. This
zero variety is closely related to the Lelong sublevel sets
$E_c(\varphi)$. Indeed, if  $\nu(\varphi,x)=\gamma$, the convexity
properties of plurisubharmonic functions show that
$$\varphi(z)\le\gamma\log|z-x|+O(1)~~~~\hbox{\rm at}~~x,$$
hence there exists a constant $C>0$ such that $e^{-2\varphi(z)}\ge 
C|z-x|^{-2\gamma}$ in a neighborhood of $x$. We easily infer that 
$$\nu(\varphi,x)\ge n+s\Longrightarrow\ic(\varphi)_x\subset\mc_{X,x}^{s+1},
\leqno(4.2)$$
where $\mc_{X,x}$ is the maximal ideal
of $\oc_{X,x}$. In the opposite direction, it is well known that 
$\nu(\varphi,x)<1$ implies the integrability of $e^{-2\varphi}$ 
in a neighborhood of $x$ (cf. Skoda [Sk~72a]), that is,
$\ic(\varphi)_x=\oc_{X,x}$. In particular, the zero variety
$V\ic(\varphi)$ of $\ic(\varphi)$ satisfies
$$E_n(\varphi)\subset V\ic(\varphi)\subset E_1(\varphi).
\leqno(4.3)$$ 

\tha Lemma 4.4 {\rm([Na~89])}|For any plurisubharmonic function
$\varphi$ on~$X$, the sheaf
$\ic(\varphi)$ is a coherent sheaf of ideals over $X$.\fintha

\remarque Proof|Since the result is local, we may assume that $X$ is the unit
ball in $\cb^n$. Let $E$ be the set of all holomorphic functions $f$ 
on $X$ such that $\int_X|f|^2e^{-2\varphi}\,d\lambda<+\infty$. By the
strong noetherian property of coherent sheaves, the set $E$
generates a coherent ideal sheaf $\jc\subset\oc_X$. It is clear that
$\jc\subset\ic(\varphi)$; in order to prove the equality,
we need only check that $\jc_x+\ic(\varphi)_x\cap
\mc^{s+1}_{X,x}=\ic(\varphi)_x$ for every integer $s$, in
view of the Krull lemma. Let $f\in\ic(\varphi)_x$ be defined 
in a neighborhood $V$ of $x$ and $\theta$ be a cut-off function
with support in $V$ such that $\theta=1$ in a neighborhood of $x$.
We solve the equation $\op u=\op(\theta f)$ by means of H\"ormander's
$L^2$ estimates~4.1, with $L$ equal to the trivial line bundle
and with the strictly plurisubharmonic weight 
$$\psi(z)=\varphi(z)+(n+s)\log|z-x|+|z|^2.$$ 
We get a solution $u$ such that
$\int_X |u|^2e^{-2\varphi}|z-x|^{-2(n+s)}d\lambda<\infty$, thus
$F=\theta f-u$ is holomorphic, $F\in E$ and 
$f_x-F_x=u_x\in\ic(\varphi)_x\cap\mc_{X,x}^{s+1}$.
This proves our contention.\finpr

Now, suppose that $X$ is a projective $n$-fold equipped with a K\"ahler 
metric $\omega$. Let $L$ be a line bundle over $X$ with a singular metric 
of curvature $T=c(L)\ge 0$. All sublevel sets $E_c(T)$ are algebraic subsets
of $X$, and if $\varphi$ is the weight representing the metric in an open
set $\Omega\subset X$, then $E_c(\varphi)=E_c(T)\cap\Omega$. The
ideal sheaf $\ic(\varphi)$ is independent of the choice of the trivialization
and so it is the restriction to $\Omega$ of a
global coherent sheaf on $X$ which we shall still call $\ic(\varphi)$ by
abuse of notation. In this context, we have the following interesting
vanishing theorem, which can be seen as a generalization of the
Kawamata-Viehweg vanishing theorem [Ka~82], [Vi~82].

\tha Theorem 4.5 {\rm([Na~89])}|Let $L$ be a line bundle over $X$ with 
$\kappa(L)=n$. Assume that $L$ is equipped with a singular metric 
of weight $\varphi$ such that $c(L)\ge\varepsilon\omega$ for some
$\varepsilon>0$. Then $H^q\big(X,\oc(K_X+L)\otimes\ic(\varphi)\big)=0$ 
for all $q\ge 1$.\fintha

\remarque Proof|Let ${\cal F}^q$ be the sheaf of germs of $(n,q)$-forms 
$u$ with values in $L$ and with measurable coefficients, such that
both $|u|^2e^{-2\varphi}$ and $|\op u|^2e^{-2\varphi}$ are locally
integrable. The $\op$ operator defines a complex of sheaves
$({\cal F}^\bu,\op)$ which is a resolution of the sheaf
$\oc(K_X+L)\otimes\ic(\varphi)$: indeed, the kernel of $\op$ in degree
$0$ consists of all germs of holomorphic $n$-forms with values in $L$
which satisfy the integrability condition; hence the coefficient
function lies in $\ic(\varphi)$; the exactness in degree $q\ge 1$
follows from proposition~4.1 applied on arbitrary small balls.
Each sheaf ${\cal F}^q$ is a ${\cal C}^\infty$-module, so ${\cal F}^\bu$ 
is a fine resolution. Moreover, $H^q\big(\Gamma(X,{\cal F}^\bu)\big)=0$
for $q\ge 1$ by proposition~4.1 applied globally on $X$. The theorem
follows.\finpr 

\tha Corollary 4.6|Let $L$ be a big nef line bundle over $X$.
Assume that $L$ is equipped with a singular metric 
of weight $\varphi$ such that $c(L)\ge 0$ and let 
$x_1\ld x_N$ be isolated points in the zero variety $V\ic(\varphi)$. 
Then for every $\varepsilon>0$, there is a surjective map
$$H^0(X,K_X+L)\lraww\bigoplus_{1\le j\le N}\oc(K_X+L)_{x_j}\otimes
\big(\oc_X/\ic((1-\varepsilon)\varphi)\big)_{x_j}.$$\fintha
\vskip-12pt
\remarque Proof|This result can be seen as a generalization of the
H\"ormander-Bombieri-Skoda theorem ([Bo~70], [Sk~72a,75]); it could
be proved directly by using H\"ormander's $L^2$ estimates and
cut-off functions.
If $c(L)\ge\delta\omega$ for some $\delta>0$, we apply theorem~4.5
to obtain the vanishing of the first $H^1$ group in the long exact
sequence of cohomology associated to
$$0\la\ic(\varphi)\la\oc_X\la\oc_X/\ic(\varphi)\la 0,$$
twisted by $\oc(K_X+L)$. The asserted surjectivity property follows
immediately; as $\ic(\varphi)\subset\ic\big((1-\varepsilon)\varphi\big)$,
we see in that case that we can even take
$\varepsilon=0$ and drop the nef assumption on~$L$. If
$c(L)\ge 0$ merely, we try to modify the metric so as to obtain a
positive lower bound for the curvature.
By (2.7) there is a singular metric on $L$ associated to a weight $\psi$
with ${i\over\pi}\partial\op\psi\ge\delta\omega$, $\delta>0$, and with a 
singularity of $\psi$ so small that $e^{-2\psi}\in L^1_\loc$.
Replace the metric on $L$ by the metric associated to the weight
$\varphi_\varepsilon=(1-\varepsilon)\varphi+\varepsilon\psi$. 
Then $e^{-2\varphi_\varepsilon}=(e^{-2\varphi})^{1-\varepsilon}
(e^{-2\psi})^{\varepsilon}$ is integrable on any open set where
$e^{-2\varphi}$ is integrable, so $V\ic(\varphi_\varepsilon)\subset
V\ic(\varphi)$ and the points $x_j$ are still isolated in
$V\ic(\varphi_\varepsilon)$. Moreover $\ic(\varphi_\varepsilon)\subset
\ic\big((1-\varepsilon)\varphi\big)$, for $\psi$ is locally bounded
above, and $c(L)_\varepsilon={i\over\pi}\partial\op\varphi_\varepsilon
\ge\varepsilon\delta\,\omega$. We are thus reduced to the first case.\finpr

\remarque Example 4.7|Suppose that $\nu(\varphi,x)>n+s$ and that $x$
is an isolated point in $E_1(\varphi)$. Then $\ic((1-\varepsilon)
\varphi)_x\subset\mc_{X,x}^{s+1}$ for $\varepsilon$ small enough,
and $x$ is isolated in $V\ic(\varphi)$ by (4.2), (4.3).
We infer that $H^0(X,K_X+L)\lraww J^s_x(K_X+L)$ is surjective
onto $s$-jets of sections at $x$.

\remarque Example 4.8|Suppose that $(z_1\ld z_n)$ are local coordinates
centered at $x$ and that
$$\varphi(z)\le\gamma\log(|z_1|+\ldots+|z_{n-1}|+|z_n|^2)+O(1),~~~~
\gamma>n.$$
Then $\ic((1-\varepsilon)\varphi)\subset(z_1\ld z_{n-1},z_n^2)$ for
$\varepsilon$ small. To check this, observe that for any $\delta>0$
the Parseval-Bessel formula gives
$$\int_{|z|<\delta}{\big|\sum a_\alpha z^\alpha\big|^2\,d\lambda(z)\over
(|z_1|+\ldots+|z_{n-1}|+|z_n|^2)^{2n}}=
\int_{|z|<\delta}{\sum |a_\alpha|^2 |z^\alpha|^2\,d\lambda(z)\over
(|z_1|+\ldots+|z_{n-1}|+|z_n|^2)^{2n}}\,;$$
the integral is divergent unless the coefficients $a_0$ and $a_{(0\ld 0,1)}$
vanish. Indeed, using polar coordinates $z_n=re^{i\theta}$ and setting 
$z'=(z_1\ld z_{n-1})$, we get
$$\eqalign{
\int_{|z|<\delta}{|z_n|^2\,d\lambda(z)\over
(|z_1|+\ldots+|z_{n-1}|+|z_n|^2)^{2n}}
&\ge 2\pi\int_{|z'|<\delta/2}d\lambda(z')\int_0^{\delta/2}
      {r^3dr\over(|z'|+r^2)^{2n}}\cr
&\ge C\int_{|z'|<\delta/2}{d\lambda(z')\over|z'|^{2n-2}}=+\infty.\cr}$$
Thus, if $x$ is isolated in $E_1(\varphi)$, we are able to prescribe 
the value of the section at~$x$ and its derivative $\partial/
\partial z_n$ along the direction $z'=0$.

\remarque Remark 4.9|More generally, it is interesting to consider
logarithmic poles of the form
$$\varphi(z)=\gamma\log\Big(\sum_{1\le j\le N}|g_j(z)|\Big)+O(1),$$
where $\jc=(g_1\ld g_N)\subset \mc_{X,x}$ is an arbitrary ideal with
isolated zero $\{x\}$. However, in this case, we do not know what is the
general rule relating the ideal $\ic(\gamma\log|g|)_x$ to the ideal $\jc$.
Observe that $\ic(\gamma\log|g|)_x$ only depends on the integral closure
$\ov\jc=\{$germs $f$ such that $|f|\le C\sum|g_j|\}$. It is almost
obvious by definition that $\ic(\varphi)$ itself is always integrally
closed. Let $\wt\jc=(\wt g_1\ld\wt g_n)\subset\jc$ be the ideal generated
by $n$ generic linear combinations $\wt g_k$ of $g_1\ld g_N$. Then $\wt\jc$
and $\jc$ have the same integral closure and we have $\sum|\wt g_k|\ge
C'\sum|g_j|$ with some $C'>0\,$; indeed these ideals have the same
multiplicity by a result of Serre [Se~57], and this implies the equality of
their integral closures thanks to a result of D.~Rees~[Ree~61]. The ideal 
$\ic(\gamma\log|\wt g|)_x$ associated to $\wt\jc$ thus coincides with 
$\ic(\gamma\log|g|)_x$. We see that there is no loss of generality considering 
only ideals generated by exactly $n$ generators (as we shall do in 
\S\S~6,7). Finally, the proof of the Brian\c con-Skoda theorem [BSk~74] 
shows that
$$\ic(\gamma\log|g|)_x=\ic(\gamma\log|\wt g|)_x\subset\wt\jc\subset\jc~~~~
\hbox{\rm when}~~\gamma>n.\leqno(4.10)$$ 
In fact, (4.10) is a straightforward consequence of Skoda's division theorem
[Sk~72b], applied to the elements of $\ic(\gamma\log|\wt g|)$.

\titre 5. Aubin-Calabi-Yau theorem and convexity inequalities|

The above results can be applied to construct sections of a given 
line bundle, provided we are able to produce singular metrics with
logarithmic poles . For this, we use in several essential ways the well-known
theorem of Aubin-Yau on the Calabi conjecture.  What we need is the following
existence result about solutions of Monge-Amp\`ere equations.

\tha Lemma 5.1 {\rm [Yau~78], see also [Au~78])}|Let $X$ be a compact complex
$n$-dimen\-sional manifold with a smooth K\"ahler metric $\omega$. Then
for any smooth volume form $f>0$ with $\int_X f=\int_X \omega^n$, there
exists a K\"ahler metric $\wt\omega$ in the same K\"ahler class as
$\omega$ such that $\wt\omega^n=f$.\finpr\fintha

The method for constructing singular metrics from the Aubin-Calabi-Yau theorem
will be explained in detail in \S~6. Before, we need a useful
convexity inequality due to Hovanski [Hov~79] and Teissier [Te~79,82],
which is a natural generalization of the usual Hodge
index theorem for surfaces. This inequality is reproved along similar
lines in [BBS~89], where it is applied to the study of projective
$n$-folds of log-general type. For the sake of completeness, we include
here a different and slightly simpler proof, based on
Yau's theorem~5.1 instead of the Hodge index theorem. Our proof also
has the (relatively minor) advantage of working over arbitrary 
K\"ahler manifolds.

\tha Proposition 5.2|In any dimension $n\,:$

\noindent{\rm(a)}~~if $\alpha_1\ld\alpha_n$ are semipositive
$(1,1)$-forms on $\cb^n$, then
$$\alpha_1\wedge\alpha_2\wedge\ldots\wedge\alpha_n\ge
(\alpha_1^n)^{1/n}(\alpha_2^n)^{1/n}\ldots(\alpha_n^n)^{1/n}.$$
{\rm(b)}~~if $u_1\ld u_n$ are semipositive cohomology classes of type
$(1,1)$ on a K\"ahler manifold $X$ of dimension $n$, then
$$u_1\cdot u_2\cdots u_n\ge(u_1^n)^{1/n}(u_2^n)^{1/n}\ldots(u_n^n)^{1/n}.$$
\fintha\vskip-10pt

By a semipositive cohomology class of type $(1,1)$, we mean a class in 
the closed convex cone of $H^{1,1}(X,\rb)$ generated by K\"ahler classes.
For instance, inequality (b) can be applied to $u_j=c_1(L_j)$ when 
$L_1\ld L_n$ are nef line bundles over a projective manifold.

\remarque Proof|Observe that (a) is a pointwise inequality between
$(n,n)$-forms whereas (b) is an inequality of a global nature for the cup
product intersection form. We first show that (a) holds when only two of 
the forms $\alpha_j$ are distinct, namely that
$$\alpha^p\wedge\beta^{n-p}\ge(\alpha^n)^{p/n}(\beta^n)^{(n-p)/n}$$
for all $\alpha,\beta\ge 0$. By a density argument, we may suppose
$\alpha,\beta>0$. Then there is a simultaneous orthogonal basis in which 
$$\alpha=i\sum_{1\le j\le n}\lambda_j\,dz_j\wedge d\ov z_j,~~~~
\beta=i\sum_{1\le j\le n}dz_j\wedge d\ov z_j$$
with $\lambda_j>0$, and (a) is equivalent to
$$p!(n-p)!\sum_{j_1<\ldots<j_p}\lambda_{j_1}\ldots\lambda_{j_p}\ge
n!\,(\lambda_1\ldots\lambda_n)^{p/n}.$$
As both sides are homogeneous of degree $p$ in $(\lambda_j)$, we may 
assume $\lambda_1\ldots\lambda_n=1$.
Then our inequality follows from the inequality between the arithmetic and
geometric means of the numbers $\lambda_{j_1}\ldots\lambda_{j_p}$.
Next, we show that statements (a) and (b) are equivalent in any
dimension $n$.

\noindent(a) $\Longrightarrow$ (b).  By density, we may suppose that
$u_1\ld u_n$ are K\"ahler classes.  Fix a positive $(n,n)$ form
$f$ such that $\int_X f=1$.  Then lemma~5.1 implies that there is a
K\"ahler metric $\alpha_j$ representing $u_j$ such that $\alpha_j^n=u_j^n f$. 
Inequality (a) combined with an integration over $X$ yields
$$u_1\cdots u_n=\int_X\alpha_1\wedge\ldots\wedge\alpha_n\ge
(u_1^n)^{1/n}\ldots(u_n^n)^{1/n}\int_X f.$$
(b) $\Longrightarrow$ (a). The forms $\alpha_1\ld\alpha_n$ 
can be considered as constant $(1,1)$-forms on any complex torus 
$X=\cb^n/\Gamma$. Inequality (b) applied to the associated cohomology
classes $u_j\in H^{1,1}(X,\rb)$ is then equivalent to (a).

Finally we prove (a) by induction on $n$, assuming the result already
proved in dimension $n-1$. We may suppose that $\alpha_n$ is positive
definite, say \hbox{$\alpha_n=i\sum dz_j\wedge d\ov z_j$} in a suitable
basis. Denote by $u_1\ld u_n$ the associated cohomology classes on the
abelian variety $X=\cb^n/\zb[i]^n$. Then $u_n$ has integral periods, 
so some multiple of $u_n$ is the first Chern class of a very ample line
bundle $\oc(D)$ where $D$ is a smooth irreducible divisor in $X$.
Without loss of generality, we may suppose $u_n=c_1(\oc(D))$. Thus
$$u_1\cdots u_{n-1}\cdot u_n=u_{1\restriction D}\cdots u_{n-1\restriction D}$$
and by the induction hypothesis we get
$$u_1\cdots u_n\ge(u_{1\restriction D}^{n-1})^{1/(n-1)}\ldots 
(u_{n-1\restriction D}^{n-1})^{1/(n-1)}.$$  
However $u_{j\restriction D}^{n-1}=u_j^{n-1}\cdot u_n\ge
(u_j^n)^{(n-1)/n}(u_n^n)^{1/n}$, since (a) and (b) are equivalent and
(a) is already proved in the case of two forms. (b) follows in
dimension~$n$, and therefore (a) holds in $\cb^n$.\finpr

\remarque Remark {\rm 5.3}|In case $\alpha_j$ (resp.  $u_j$) are
positive definite, the equality holds in 5.2 (a,b) if and only if
$\alpha_1\ld\alpha_n$ (resp.  $u_1\ld u_n$) are proportional.  In our
inductive proof, the restriction morphism $H^{1,1}(X,\rb)\la
H^{1,1}(D,\rb)$ is injective for $n\ge 3$ by the hard Lefschetz theorem,
hence it is enough to consider the case of $\alpha^p\wedge\beta^{n-p}$. 
The equality between arithmetic and geometric means occurs only when all
numbers $\lambda_{j_1}\ldots\lambda_{j_p}$ are equal, so all $\lambda_j$
must be equal and $\alpha=\lambda_1\beta$, as desired.  
More generally, there is an inequality
$$\leqalignno{
\alpha_1\wedge\ldots\wedge\alpha_p&\wedge\beta_1\wedge\ldots\wedge
\beta_{n-p}\ge\cr
&\ge(\alpha_1^p\wedge\beta_1\wedge\ldots\wedge\beta_{n-p})^{1/p}
\ldots(\alpha_p^p\wedge\beta_1\wedge\ldots\wedge\beta_{n-p})^{1/p}&(5.4)\cr}$$
for all $(1,1)$-forms $\alpha_j$, $\beta_k\ge 0$. Once again, inequality
(5.4) is easier to be proved with cohomology classes rather than forms. By a
density argument, we may suppose that all forms $\beta_j$ are positive
definite and have coefficients in $\qb[i]$. Let
$u_1\ld u_p$ be the cohomology classes of type $(1,1)$ associated to
$\alpha_1\ld\alpha_p$ on $X=\cb^n/\zb[i]^n$. The cohomology
class of $\beta_1$ is a rational multiple of the first Chern class 
of a very ample line bundle $\oc(Y_1)$, where $Y_1$ is a smooth
irreducible divisor in $X$, that of $\beta_{2\restriction Y_1}$ is a multiple
of such a divisor $Y_2$ in $Y_1$, and by induction the cohomology class
of $\beta_1\wedge\ldots\wedge\beta_{n-p}$ is equal to a multiple
of the cohomology class of a connected $p$-dimensional submanifold 
$Y\subset X$. Then (5.4) is equivalent to the already known inequality
$$u_{1\restriction Y}\cdots u_{p\restriction Y}\ge
(u_{1\restriction Y}^p)^{1/p}\ldots(u_{p\restriction Y}^p)^{1/p}.~~\finpr$$

\titre 6. Mass concentration in the Monge-Amp\`ere equation|

In this crucial section, we show how the Aubin-Calabi-Yau theorem can
be applied to construct singular metrics on ample (or more generally
big and nef) line bundles. We first suppose that $L$ is an ample line
bundle over a projective $n$-fold $X$ and that $L$ is equipped with a
smooth metric of positive curvature. We consider the K\"ahler metric
$\omega={i\over 2\pi}c(L)$. Any form $\wt\omega$ in the K\"ahler class
of $\omega$ can be written as $\wt\omega=\omega+{i\over\pi}\partial\op\psi$,
i.e. is the curvature form of $L$ after multiplication of the original 
metric by a smooth weight function $e^{-\psi}$. By lemma~5.1, the 
Monge-Amp\`ere equation
$$\Big(\omega+{i\over\pi}\partial\op\psi\Big)^n=f\leqno(6.1)$$
can be solved for $\psi$, whenever $f$ is a smooth $(n,n)$-form with
$f>0$ and $\int_X f=L^n$.  In order to produce logarithmic poles at given
points $x_1\ld x_N\in X$, the main idea is to let $f$ converge to a
Dirac measure at $x_j\,$; then $\wt\omega$ will be shown to converge 
to a closed positive $(1,1)$-current with non zero Lelong number at $x_j$. 

Let $(z_1\ld z_n)$ be local coordinates centered at $x_j$, defined on 
some neighborhood $V_j\simeq\{|z|<R_j\}$. Let $g_j=(g_{j,1}\ld g_{j,n})$
be arbitrary holomorphic functions on $V_j$ such that
$g_j^{-1}(0)=\{x_j\}$, and let
$$\log|g_j|=\log\Big(\sum_{1\le k\le n}|g_{j,k}|^2\Big)^{1/2}.
\leqno(6.2)$$
Then $\log|g_j|$ has an isolated logarithmic pole at $x_j$ and
$({i\over\pi}\partial\op\log|g_j|)^n=\rho_j\,\delta_{x_j}$, where $\rho_j$
is the degree of the covering map $g_j:(\cb^n,x_j)\la(\cb^n,0)$. Indeed
$\partial\op\log|g_j|=g_j^\star\partial\op\log|z|$ has rank $(n-1)$ on
$V_j\ssm\{x_j\}$, and formula (3.5) with $\chi(t)=e^{2t}$ gives
$$\eqalign{
\int_{|g_j(z)|<r}\Big({i\over\pi}\partial\op\log|g_j|\Big)^n
&={1\over(2\pi r^2)^n}\int_{|g_j(z)|<r}g_j^\star(i\partial\op|w|^2)^n\cr
&={\rho_j\over(2\pi r^2)^n}\int_{|w|<r}(i\partial\op|w|^2)^n=\rho_j\cr}$$
for every $r>0$ small enough. Now, let $\chi:\rb\to\rb$ be a smooth convex
increasing function such that $\chi(t)=t$ for $t\ge 0$ and $\chi(t)=-1/2$
for $t\le -1$. We set
$$\alpha_{j,\varepsilon}={i\over\pi}\partial\op
\big(\chi(\log|g_j|/\varepsilon)\big).\leqno(6.3)$$
Then $\alpha_{j,\varepsilon}$ is a smooth positive $(1,1)$-form, and
$\alpha_{j,\varepsilon}={i\over\pi}\partial\op\log|g_j|$ over the set of
points $z\in V_j$ such that $|g_j(z)|>\varepsilon$. It follows that 
$\alpha_{j,\varepsilon}^n$ has support in the compact set
$|g_j(z)|\le\varepsilon$, and Stokes' formula gives
$$\int_{V_j}\alpha_{j,\varepsilon}^n=
\int_{V_j}\Big({i\over\pi}\partial\op\log|g_j|\Big)^n=\rho_j.\leqno(6.4)$$
Hence $\alpha_{j,\varepsilon}^n$ converges weakly to the Dirac measure 
$\rho_j\delta_{x_j}$ as $\varepsilon$ tends to $0$. For all positive 
numbers $\tau_j>0$ such that $\sigma=\sum\rho_j\tau_j^n<L^n$, lemma~5.1
gives a solution of the Monge-Amp\`ere equation
$$\omega_\varepsilon^n=\sum_{1\le j\le N}\tau_j^n\,\alpha_{j,\varepsilon}^n+
\Big(1-{\sigma\over L^n}\Big)\omega^n~~~~\hbox{\rm with}~~
\omega_\varepsilon=\omega+{i\over\pi}\partial\op\psi_\varepsilon,\leqno(6.5)$$
since the right-hand side of the first equation is $>0$ and has the 
correct integral value $L^n$ over $X$. The solution $\psi_\varepsilon$ is
merely determined up to a constant. If $\gamma$ is an arbitrary
K\"ahler metric on~$X$, we can normalize $\psi_\varepsilon$ in such a
way that  $\int_X\psi_\varepsilon\,\gamma^n=0$. 

\tha Lemma 6.6|There is a sequence $\varepsilon_\nu$ converging to zero 
such that $\psi_{\varepsilon_\nu}$ has a limit $\psi$ in $L^1(X)$
and such that the sequence of $(1,1)$-forms
$\smash{\omega_{\varepsilon_\nu}}$ converges weakly towards a closed 
positive current $T$ of type~$(1,1)$. Moreover, the cohomology class 
of $T$ is equal to $c_1(L)$ and $T=\omega+{i\over\pi}\partial\op\psi$.
\fintha

\remarque Proof|The integral $\int_X \omega_\varepsilon\wedge\gamma^{n-1}=
L\cdot\{\gamma\}^{n-1}$ remains bounded, so we can find a sequence
$\varepsilon_\nu$ converging to zero such that the subsequence
$\omega_{\varepsilon_\nu}$ converges weakly towards a closed positive 
current $T$ of bidegree $(1,1)$. The cohomology class of a current is
continuous with respect to the weak topology (this can be seen by 
Poincar\'e duality). The cohomology class of $T$ is thus equal to 
$c_1(L)$. The function $\psi_\varepsilon$ satisfies the equation
${1\over\pi}\Delta\psi_\varepsilon={\rm tr}_\gamma(\omega_\varepsilon-\omega)$
where $\Delta$ is the Laplace operator associated to~$\gamma$. Our 
normalization of $\psi_\varepsilon$ implies
$$\psi_\varepsilon=\pi\,G\,{\rm tr}_\gamma(\omega_\varepsilon-\omega),$$
where $G$ is the Green operator of $\Delta$. As $G$ is
a compact operator from the Banach space of bounded Borel measures into 
$L^1(X)$, we infer that some subsequence $(\psi_{\varepsilon_\nu})$ of
our initial subsequence converges to a limit $\psi$ in $L^1(X)$. By the
weak continuity of $\partial\op$, we get
$T=\lim(\omega+{i\over\pi}\partial\op\psi_{\varepsilon_\nu})=
\omega+{i\over\pi}\partial\op\psi$.\finpr\vskip7pt

Let $\Omega\subset X$ be an open coordinate patch such that $L$ is
trivial on a neighborhood of $\ov\Omega$, and let $e^{-h}$ be the weight
representing the initial hermitian metric on $L_{\restriction\ov\Omega}$.
Then ${i\over\pi}\partial\op h=\omega$ and ${i\over\pi}\partial\op
(h+\psi_\varepsilon)=\omega_\varepsilon$, so the function 
$\varphi_\varepsilon=h+\psi_\varepsilon$ defines a plurisubharmonic weight
on $L_{\restriction\Omega}$, as well as its limit $\varphi=h+\psi$.
By the continuity of $G$, we also infer from the proof of lemma~6.6
that the family $(\psi_\varepsilon)$ is bounded in $L^1(X)$. The usual 
properties of subharmonic functions then show that there is a uniform 
constant $C$ such that $\varphi_\varepsilon\le C$ on $\ov\Omega$. We use
this and equation (6.5) to prove that the limit $\varphi$ has logarithmic
poles at all points $x_j\in\Omega$, thanks to Bedford and Taylor's 
maximum principle for solutions of Monge-Amp\`ere equations~[BT~76]:

\tha Lemma 6.7|Let $u,v$ be smooth $($or continuous$)$ plurisubharmonic
functions on $\ov\Omega$, where $\Omega$ is a bounded open set in $\cb^n$. If 
$$u_{\restriction\partial\Omega}\ge v_{\restriction\partial\Omega}~~~
\hbox{\rm and}~~~(i\partial\op u)^n\le(i\partial\op v)^n~~~\hbox{\rm on}~~
\Omega,$$
then $u\ge v$ on $\Omega$.\finpr\fintha

In the application of lemma 6.7, we suppose that $\Omega$ is a neighborhood
of $x_j$ and take
$$u=\tau_j\big(\chi(\log|g_j|/\varepsilon)+
\log\varepsilon\big)+C_1,~~~~v=\varphi_\varepsilon,$$
where $C_1$ is a large constant. Then for $\varepsilon>0$ small enough
$$\eqalign{
u_{\restriction\partial\Omega}&=\tau_j\log|g_j|+C_1,~~~~
v_{\restriction\partial\Omega}\le C,\cr
\Big({i\over\pi}\partial\op v\Big)^n&=\omega_\varepsilon^n\ge\tau_j^n\,
\alpha_{j,\varepsilon}^n=\Big({i\over\pi}\partial\op u\Big)^n
~~~~\hbox{\rm on}~~\Omega.\cr}$$
For $C_1$ sufficiently large, we infer $u\ge v$ on $\Omega$, hence
$$\varphi_\varepsilon\le\tau_j\log(|g_j|+\varepsilon)+C_2~~~~
\hbox{\rm on}~~\Omega.$$

\tha Corollary 6.8| The plurisubharmonic weight $\varphi=h+\psi$ on 
$\smash{L_{\restriction\Omega}}$ associated to the limit function 
$\psi=\lim\smash{\psi_{\varepsilon_\nu}}$ satisfies
${i\over\pi}\partial\op\varphi=T$. Moreover, $\varphi$
has logarithmic poles at all points $x_j\in\Omega$ and
$$\varphi(z)\le\tau_j\log|g_j(z)|+O(1)~~~~\hbox{\rm at}~~x_j.\finpr$$
\fintha

{\bf Case of a big nef line bundle}. All our arguments were
developed under the assumption that $L$ is ample, but if $L$ is only nef
and big, we can proceed in the following way. Let $A$ be a fixed ample
line bundle with smooth curvature form $\gamma=c(A)>0$. As $mL+A$ is ample
for any $m\ge 1$, by~5.1 there exists a smooth hermitian metric on $L$ 
depending on $m$, such that $\omega_m=c(L)_m+{1\over m}c(A)>0$ and 
$$\omega_m^n={(L+{1\over m}A)^n\over A^n}\,\gamma^n.\leqno(6.9)$$
However, a priori we cannot control the asymptotic behaviour of $\omega_m$
when $m$ tends to infinity, so we introduce the
sequence of non necessarily positive $(1,1)$-forms $\omega'_m=c(L)_1+
{1\over m}c(A)\in\{\omega_m\}$, which is uniformly bounded in $C^\infty(X)$ 
and converges to $c(L)_1$. Then we solve the Monge-Amp\`ere equation
$$\omega_{m,\varepsilon}^n=\sum_{1\le j\le N}\tau_j^n\,\alpha_{j,\varepsilon}^n
+\Big(1-{\sigma\over(L+{1\over m}A)^n}\Big)\omega_m^n\leqno(6.10)$$
with $\omega_{m,\varepsilon}=\omega'_m+{i\over\pi}\partial\op
\psi_{m,\varepsilon}$ and some smooth function $\psi_{m,\varepsilon}$ such
that\break $\int_X\psi_{m,\varepsilon}\gamma^n=0\,$; this is again possible
by Yau's theorem~5.1. The numerical condition needed on $\sigma$ to solve
(6.10) is obviously satisfied for all $m$ if we suppose
$$\sigma=\sum\rho_j\tau_j^n<L^n<\Big(L+{1\over m}A\Big)^n.$$
The same arguments as before show that there exist a
convergent subsequence $\lim_{\nu\to+\infty}\psi_{m_\nu,\varepsilon_\nu}=
\psi$ in $L^1(X)$ and a closed positive $(1,1)$-current \hbox{$T=\lim
\omega_{m_\nu,\varepsilon_\nu}$}
$=\,c(L)_1+{i\over\pi}\partial\op\psi\in c_1(L)$
such that Corollary~6.8 is still valid; in this case, $h$ is taken to be the
weight function corresponding to $c(L)_1$. Everything thus works as in the
ample case.

\titre 7. Choice of the logarithmic singularities|

Let us assume (with the notation of \S~6) that each point $x_j$ is 
isolated in $E_1(\varphi)$. Then we conclude by (4.3) and Corollary~4.6 
that there is a surjective map
$$H^0(X,K_X+L)\lraww\bigoplus_{1\le j\le N}\oc(K_X+L)_{x_j}\otimes
\big(\oc_X/\ic((1-\varepsilon)\varphi)\big)_{x_j}.\leqno(7.1)$$
However, finding sufficient conditions ensuring that $x_j$ is isolated in
$E_1(\varphi)=E_1(T)$ is a harder question. Therefore, we postpone
this task to the next section and explain instead how to choose the
logarithmic poles $\log|g_j|$ and the constants $\tau_j$ to
obtain specified ideals and jets of sections at each point $x_j$.

Suppose that an ideal $\jc_j\subset\mc_{X,x_j}$ is given at~$x_j$,
in other words, that we are given a $0$-dimensional subscheme $(\Xi,\oc_\Xi)$
with $\Xi=\{x_1\ld x_N\}$ and $\oc_{\Xi,x_j}=\oc_{X,x_j}/\jc_j$. We want to
find sufficient conditions for the surjectivity of the restriction map
$$H^0(X,K_X+L)\la H^0\big(\Xi,\oc_\Xi(K_X+L)\big)=
\bigoplus_{1\le j\le N}\oc(K_X+L)_{x_j}\otimes\oc_{X,x_j}/\ic_j.$$
By (7.1), we need only find a germ of map $g_j:(X,x_j)\to(\cb^n,0)$
and a constant $\tau_{j,0}$ such that $\ic(\tau_{j,0}\log|g_j|)\subset\jc_j$.
For $\tau_j>\tau_{j,0}$ and $\varepsilon$ small enough, corollary~6.8 then 
implies $\ic\big((1-\varepsilon)\varphi\big)\subset\jc_j$. Thus we have to
choose $\sigma$ slightly larger than $\sigma_0=\sum\rho_j\tau_{j,0}^n$ where
$\rho_j$ is the degree of the covering map $g_j\,$;
this is possible only if $L^n>\sigma_0$. Let us discuss some specific cases.

{\bf Spannedness.} To obtain that $K_X+L$ spans at $x\in X$, we consider a 
single point $x_1=x$ and take $\jc_1=\mc_{X,x}$, $g_1(z)=(z_1\ld z_n)$,
$\tau_{1,0}=n$ and $\sigma_0=\tau_{1,0}^n=n^n$. Then $\ic(\tau_{1,0}
\log|g_1|)\subset\mc_{X,x}$, as desired.

{\bf Separation of points.} To obtain the separation of two points 
$x_1\ne x_2$ in $X$ by sections of $K_X+L$, we make the same choices as 
above at $x_1,x_2$ and get $\sigma_0=\tau_{1,0}^n+\tau_{2,0}^n=2n^n$.  
If $x_1,x_2$ are ``infinitely near" in some direction $\xi\in TX$,
we choose coordinates $(z_1\ld z_n)$ centered at $x=x_1=x_2$ so that 
$\partial/\partial z_n=\xi$, and we
set $\jc_1=(z_1\ld z_{n-1},z_n^2)$. By example~4.8, we can choose
$g_1(z)=(z_1\ld z_{n-1},z_n^2)$ and $\tau_{1,0}=n$. Then the degree of 
$g_1$ is $\rho_1=2$ and we find again $\sigma_0=\rho_1\tau_{1,0}^n=2n^n$. 

{\bf Generation of $s$-jets}. Instead of just considering jets at
one point, we wish to look at several points simultaneously which may
come into coincidence. Such a concern appears also in 
the work of Beltrametti-Sommese [BSo~90], where an extensive study of the
surface case is made. The relevant definition is as follows.

\tha Definition 7.2|We say that $L$ generates $s$-jets on a given 
subset $\Xi\subset X$ if $H^0(X,L)\lraww \bigoplus J^{s_j}_{x_j}L$ is onto
for any choice of points $x_1\ld x_N\in\Xi$ and integers $s_1\ld s_N$ with
$\sum(s_j+1)=s+1$. We say that $L$ is $s$-jet ample if the above property
holds for $\Xi=X$.\fintha

With this terminology, $L$ is $0$-jet ample if and only if $L$ is
spanned and $1$-jet ample if and only if $L$ is very ample. In
order that $K_X+L$ generates $s$-jets on $\Xi$, we take 
$x_1\ld x_N\in\Xi$ arbitrary, $g_j(z)=(z_1\ld z_n)$ at each $x_j$  and 
$\tau_{j,0}=n+s_j$. Therefore $\sigma_0=\max\sum(n+s_j)^n$ over all 
decompositions $s+1=\sum(s_j+1)$. In fact, if we set $t_j=s_j+1$, the
following lemma gives $\sigma_0=(n+s)^n$, that is, the maximum is reached
when only one point occurs.\finpr

\tha Lemma 7.3|Let $t_1\ld t_N\in{}[1,+\infty[$. Then
$$\sum_{1\le j\le N}(n-1+t_j)^n\le\Big(n-1+\sum_{1\le j\le N}t_j\Big)^n.$$
\fintha

\remarque Proof|The right hand side is a polynomial with nonnegative
coefficients and the coefficient of a monomial $t_j^k$ involving exactly
one variable is the same as in the left hand side (however, the constant
term is smaller). Thus the difference
is increasing in all variables and we need only consider the case
$t_1=\ldots=t_N=1$. This case follows from the obvious inequality
$$n^nN=n^n+{n\choose 1}n^{n-1}(N-1)\le(n+N-1)^n.\finpr$$

\tha Corollary 7.4|Let $L$ be a big nef line bundle. A sufficient 
condition for spannedness $($resp. separation of points, $s$-jet ampleness)
of $K_X+L$ on a given set $\Xi$ is $L^n>\sigma_0$ with
$$\sigma_0=n^n,~~~~\hbox{\sl resp.}~~\sigma_0=2n^n,~~~~\hbox{\sl resp.}~~
\sigma_0=(n+s)^n,$$
provided that the solution $\omega_\varepsilon$ of $(6.5)$ $($resp. the
solution $\omega_{m,\varepsilon}=\omega'_m+{i\over\pi}\partial\op
\psi_{m,\varepsilon}$ of $(6.10))$, always has a subsequence converging 
to a current $T$ for which all points $x\in\Xi\cap E_1(T)$ are isolated
in $E_1(T)$.\fintha

{\bf Case of an arbitrary 0-dimensional subscheme.} Let
$\jc_j=(h_{j,k})_{1\le k\le N}$ be an arbitrary ideal in $\oc_{X,x_j}$ with
$V\jc_j=\{x_j\}$. By remark~4.9, we
can take $g_j=(g_{j,1}\ld g_{j,n})$ to be $n$ generic elements of $\ic_j$ and
$\tau_{j,0}=n$. Indeed, property (4.10) then shows that $\ic(\gamma\log|g_j|)
\subset\jc_j$ for $\gamma>n$. In this case, we find $\sigma_0=n^n\sum\rho_j$.
Unfortunately, this value is in general very far from being optimal:
for instance, we would get $\sigma_0=n^n(s+1)^n$ instead of $(n+s)^n$
in the case of $s$-jets. If $(\Xi,\oc_\Xi)$ is a local complete intersection,
that is, if each $\jc_j$ has $N=n$ generators, we simply take $g_j=(h_{j,1}\ld
h_{j,n})$. Thus we obtain $\rho_j=\dim\oc_{X,x_j}/\jc_j$ and
$$\sigma_0=n^n h^0(\Xi,\oc_\Xi).\leqno(7.5)$$

\titre 8. Upper bound for the 1-codimensional polar components|

The goal of this section is to give a rather simple derivation of numerical
conditions ensuring that $\codim(E_1(T),x)\ge 2$ at a given point $x$.
In particular, we will obtain a criterion for very ample line bundles
over surfaces. Although these results are only formal consequences of
those obtained in the next two sections, we feel preferable to
indicate first the basic ideas in a simple case.

Let $L$ again denote an ample line bundle over a projective algebraic
manifold~$X$ and keep the same notation as in \S~6. Siu's
decomposition formula (3.8) applied to $T=\lim\omega_{\varepsilon_\nu}$ 
gives
$$T=\sum_{k=1}^{+\infty}\lambda_k\,[H_k]+R,\leqno(8.1)$$
where $[H_k]$ is the current of integration over an irreducible
hypersurface $H_k$ and $\codim E_c(R)\ge 2$ for every $c>0$. As we
would like $E_1(T)$ to have isolated points at $x_j$, 
a difficulty may come from the singular points of high multiplicities
in the hypersurfaces $H_k$. We thus need to find upper bounds for 
the coefficients $\lambda_k$. The convexity inequality~5.2 can be 
used for this purpose to obtain a lower bound of the mass of $R\,:$

\tha Proposition 8.2|We have
$$\sum_{k=1}^{+\infty}\lambda_k\,L^{n-1}\cdot H_k\le
\sigma_1=\big(1-(1-\sigma/L^n)^{1/n}\big)L^n.$$\fintha

\remarque Proof|As $\int_X T\wedge\omega^{n-1}=L^n$ and
$\int_X [H_k]\wedge\omega^{n-1}=L^{n-1}\cdot H_k$, we need only prove that 
$$\int_X R\wedge\omega^{n-1}\ge(1-\sigma/L^n)^{1/n}L^n.\leqno(8.3)$$
Let $\theta$ be a smooth function on $X$ such that $0\le\theta\le 1$,~
$\theta=1$ in a neighborhood of $\bigcup_{1\le k\le N}H_k$ and
$\int_X \theta\,\omega^n<\varepsilon_0$, where $\varepsilon_0>0$ is
an arbitrarily small number. This is possible because
$\bigcup_{1\le k\le N}H_k$ is a closed set of zero 
Lebesgue measure in~$X$. Then
$$\eqalign{
\int_X(1-\theta)\,T\wedge\omega^{n-1}
&\ge\liminf_{\varepsilon\to 0}
\int_X(1-\theta)\,\omega_\varepsilon \wedge\omega^{n-1}\cr
&\ge\liminf_{\varepsilon\to 0}\int_X(1-\theta)\,(\omega_\varepsilon^n)^{1/n}
(\omega^n)^{1-1/n}\cr
&\ge\big(1-\sigma/L^n)^{1/n}\int_X(1-\theta)\,\omega^n\cr}$$
by the convexity inequality and equation (6.5). By our choice of $\theta$
we have $(1-\theta)\,T\le\sum_{k>N}\lambda_k\,[H_k]+R\,$, so
$$\int_X\Big(\sum_{k>N}\lambda_k\,[H_k]+R\Big)\wedge\omega^{n-1}
\ge\big(1-\sigma/L^n)^{1/n}(L^n-\varepsilon_0).$$
Since $N$ and $\varepsilon_0$ were arbitrary, we get the expected
inequality~(8.3).\finpr

If $L$ is big and nef, the same result  can be obtained by replacing $\omega$ 
with $\omega_{k_0}$ and $\omega_\varepsilon$ with $\omega_{k,\varepsilon}$
in the above inequalities ($\varepsilon\to 0$, $k\to+\infty$) and by letting
$k_0$ tend to $+\infty$ at the end. Now suppose that for any hypersurface 
$H$ in $X$ passing through a given point $x$ we have
$$L^{n-1}\cdot H>\big(1-(1-\sigma_0/L^n)^{1/n}\big)L^n.\leqno(8.4)$$
We can choose $\sigma=\sigma_0+\varepsilon$ such that inequality
(8.4) is still valid with $\sigma$ instead of~$\sigma_0$, and then all 
hypersurfaces $H_k$ passing through $x$ have coefficients $\lambda_k<1$
in (8.1). Thus
$$E_1(T)\subset E_1(R)\cup\bigcup_k H_{k,{\rm sing}}
\cup\bigcup_{k\ne l}(H_k\cap H_l)
\cup\bigcup_{\lambda_k<1}\big(H_k\cap E_{1-\lambda_k}(R)\big)
\cup\bigcup_{\lambda_k\ge 1}H_k,$$
because the contribution of $[H_k]$ to the Lelong number of $T$ is
equal to $1$ at a regular point. As all terms in the union have
codimension $\ge 2$ except the last ones which do not contain $x$,
condition (8.4) ensures that $\codim\big(E_1(T),x\big)\ge 2$.\break 
In the case of surfaces, we can therefore apply corollary 7.4 to obtain:

\tha Corollary 8.5|Let $X$ be a smooth algebraic surface, and let $L$
be a big nef line bundle over $X$. Then on a given subset $\Xi\subset X$
$$\table{~~~
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl&~K_X+L&\verl~~\hbox{\rm is spanned}~\,
&\verl~\,\hbox{\rm separates points}~
&\verl~~\hbox{\rm generates $s$-jets}~~&\verl\cr
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl&\verl\qquad\quad L^2>~~~&\verl\quad\quad~\,4&\verl~~~8~~~\verl~~~9~~~
\verl~~\,12&\verl\quad\quad~(2+s)^2&\verl\cr
&\ptstep~~\smash{\raise-3pt\hbox{\rm when}}~~
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl&\verl~~~\,\forall C,\,~~~L\cdot C>~~
&\verl\quad\quad~\,2&\verl~~~6~~~\verl~~~5~~~\verl~~~4
&\verl\quad~~2+3s+s^2&\verl\cr
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr}$$
for all curves $C\subset X$ intersecting $\Xi$.
In particular, if $L$ is ample, $K_X+mL$ is always globally spanned 
for $m\ge 3$ and very ample for $m\ge 5$.\fintha

\remarque Proof|For $s$-jets, we have $\sigma_0=(2+s)^2$, so we find the
condition
$$L^2>(2+s)^2,~~~~L\cdot C>\big(1-(1-(2+s)^2/L^2)^{1/2}\big)L^2.$$
The last constant decreases with $L^2$ and is thus at most equal to
the value obtained when $L^2=(2+s)^2+1\,$; its integral part is precisely
$2+3s+s^2$.\finpr

The above lower bounds on $L^2$ are sharp but not those for $L\cdot C$.
Reider's method shows in fact that $K_X+mL$ is very ample as soon as $m\ge 4$.
In the higher dimensional case, a major difficulty is to ensure that
the germs $\big(E_1(T),x_j\big)$ do not contain any analytic set of
dimension $1,2\ld n-2$. This cannot be done without considering
``self-intersections" of $T$ and prescribing suitable bounds for all
intermediate intersection numbers $L^p\cdot Y$.

\titre 9. Approximation of closed positive (1,1)-currents by divisors|

Let $L$ be a line bundle with $c_1(L)\in\Gamma_+$ and let $T=c(L)\ge 0$
be the curvature current of some singular metric on $L$. Our goal is
to approximate $T$ in the weak topology by divisors which have roughly the
same Lelong numbers as~$T$. The existence of weak approximations by divisors
has already been proved in [Le~72] for currents defined on a pseudoconvex 
open set $\Omega\subset\cb^n$ with $H^2(\Omega,\rb)=0$,
and in [De~82c] in the situation considered here.
However, the result of [De~82c] is less precise than what we actually
need and moreover the proof contains a small gap; a complete proof
will therefore be included here.

\tha Proposition 9.1|For any $T=c(L)\ge 0$ and any ample line bundle $F$,
there is a sequence of non zero sections $h_s\in H^0(X,p_sF+q_sL)$ with
$p_s,q_s>0$, $\lim q_s=+\infty$ and $\lim p_s/q_s=0$, such that the divisors
$D_s={1\over q_s}\div(h_s)$ satisfy $T=\lim D_s$ in the weak topology
and $\,\sup_{x\in X}|\nu(D_s,x)-\nu(T,x)|\to 0$ as $s\to+\infty$.
\fintha

\remarque Remark 9.2|The proof will actually show, with very slight
modifications, that Prop.~9.1 also holds when $X$ is a Stein manifold
and $L$ is an arbitrary holomorphic line bundle. The last assertion concerning
Lelong numbers implies that there is a sequence $\varepsilon_s>0$ converging
to~$0$ such that $E_c(T)=\bigcap_{s\ge 1}E_{c-\varepsilon_s}(D_s)$. When $D$
is an effective divisor, given locally as the divisor of a holomorphic
function~$h$, then $E_c(D)$ is the set of points $x\in X$ such that the
derivatives $h^{(\alpha)}(x)=0$ for all multi-indices $\alpha$
with~$|\alpha|<c$. This gives a new proof of Siu's result\break [Siu~74] that
$E_c(T)$ is an analytic set, at least in the case of bidegree $(1,1)$-currents
(in fact the case of an arbitrary bidegree is easily reduced to the
$(1,1)$ case by a standard argument due to P.~Lelong). Proposition~9.1
is therefore already non trivial locally.

\remarque Proof|We first use H\"ormander's $L^2$ estimates to construct 
a suitable family of holomorphic sections and combine this with some ideas
of [Le~72] in a second step. Select a smooth metric with positive curvature
on~$F$, choose $\omega=c(F)>0$ as a K\"ahler metric on $X$ and fix some 
large integer $m$ (how large $m$ must be will be specified later).
For all $s\ge 1$ we define
$$w_s(z)=\sup_{1\le j\le N}{1\over s}\log||f_j(z)||,$$
where $(f_1\ld f_N)$ is an orthonormal basis of the space of sections of
\hbox{$\oc(mF+sL)$} with finite global $L^2$ norm $\int_X||f||^2dV_\omega$. Let
$e_F$ and $e_L$ be non vanishing holomorphic sections of $F$ and $L$ on a
trivializing open set $\Omega$, and let
$e^{-\psi}=||e_F||$, $e^{-\varphi}=||e_L||$ be the corresponding weights.
If $f$ is a section of $\oc(mF+sL)$ and if we still denote by $f$ the
associated complex valued function on $\Omega$ with respect to the
holomorphic frame $e_F^m\otimes e_L^s$, we have
$||f(z)||=|f(z)|e^{-m\psi(z)-s\varphi(z)}\,$; here $\varphi$ is
plurisubharmonic, $\psi$ is smooth and
strictly plurisubharmonic, and $T={i\over\pi}\partial\op\varphi$,~
$\omega={i\over\pi}\partial\op\psi$. In $\Omega$, we can write
$$w_s(z)=\sup_{1\le j\le N}{1\over s}\log|f_j(z)|-\varphi(z)-
{m\over s}\psi(z).$$
In particular $T_s:={i\over\pi}\partial\op w_s+T+{m\over s}\omega$ is a 
closed positive current belonging to the cohomology class $c_1(L)+
{m\over s}c_1(F)$.

{\bf Step 1}. We check that $T_s$ converges to $T$ as $s$ tends to $+\infty$
and that $T_s$ satisfies the inequalities
$$\nu(T,x)-{n\over s}\le\nu(T_s,x)\le\nu(T,x)$$
at every point $x\in X$. Note that $T_s$ is defined on $\Omega$ by
$T_s={i\over\pi}\partial\op v_{s,\Omega}$ with 
$$v_{s,\Omega}(z)=\sup_{1\le j\le N}{1\over s}\log|f_j(z)|,~~~~
\int_\Omega|f_j|^2e^{-2m\psi-2s\varphi}dV_\omega\le 1.$$
We suppose here that $\Omega$ is a coordinate open set with analytic 
coordinates $(z_1\ld z_n)$. Take $z\in\Omega'\compact\Omega$ and
$r\le r_0={1\over 2}d(\Omega',\partial\Omega)$. By the $L^2$ estimate and the
mean value inequality for subharmonic functions, we obtain
$$|f_j(z)|^2\le{C_1\over r^{2n}}\int_{|\zeta-z|<r}
|f_j(\zeta)|^2d\lambda(\zeta)\le
{C_2\over r^{2n}}\sup_{|\zeta-z|<r}e^{2s\varphi(\zeta)}$$
with constants $C_1$, $C_2$ independent of $s$ and $r$ (the smooth function 
$\psi$ is bounded on any compact subset of $\Omega$). Hence we infer
$$v_{s,\Omega}(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over 2s}
\log{C_2\over r^{2n}}.\leqno(9.3)$$
If we choose for example $r=1/s$ and use the upper semi-continuity of
$\varphi$, we infer $\limsup_{s\to+\infty}v_{s,\Omega}\le\varphi$.
Moreover, if $\gamma=\nu(\varphi,x)=\nu(T,x)$, then
$\varphi(\zeta)\le\gamma\log|\zeta-x|+O(1)$ near $x$. By taking $r=|z-x|$ 
in (9.3), we find
$$\eqalign{
&v_{s,\Omega}(z)\le\sup_{|\zeta-x|<2r}\varphi(\zeta)-{n\over s}\log r+O(1)
\le\Big(\gamma-{n\over s}\Big)\log|z-x|+O(1),\cr
&\nu(T_s,x)=\nu(v_{s,\Omega},x)\ge\Big(\gamma-{n\over s}\Big)_+\ge
\nu(T,x)-{n\over s}.\cr}$$
In the opposite direction, the inequalities require deeper arguments since 
we actually have to construct sections in $H^0(X,mF+sL)$. Assume that
$\Omega$ is chosen isomorphic to a bounded pseudoconvex open set in $\cb^n$.
By the Ohsawa-Takegoshi $L^2$ extension theorem [Oh~88], for every point
$x\in\Omega$, there is a holomorphic function $g$ on $\Omega$ such that
$g(x)=e^{s\varphi(x)}$ and 
$$\int_\Omega|g(z)|^2e^{-2s\varphi(z)}d\lambda(z)\le C_3,$$ 
where $C_3$ depends only on $n$ and $\diam(\Omega)$. For $x\in\Omega'$, set
$$\sigma(z)=\theta\big(|z-x|/r\big)\,g(z)\,e_F(z)^m\otimes e_L(z)^s,~~~~
r=\min\big(1,2^{-1}d(\Omega',\partial\Omega)\big),$$
where $\theta:\rb\to[0,1]$ is a cut-off function such that
$\theta(t)=1$ for $t<1/2$ and $\theta(t)=0$ for $t\ge 1$. We solve the
global equation $\op u=v$ on $X$ with $v=\op\sigma$, after multiplication
of the metric of $mF+sL$ with the weight
$$e^{-2n\rho_x(z)},~~~~\rho_x(z)=\theta\big(|z-x|/r\big)\,\log|z-x|\le 0.$$
The $(0,1)$-form $v$ can be considered as a $(n,1)$-form with values in
the line bundle $\oc(-K_X+mF+sL)$ and the resulting curvature form of this
bundle is
$${\rm Ricci}(\omega)+m\omega+sT+n{i\over\pi}\partial\op\rho_x.$$
Here the first two summands are smooth, $i\partial\op\rho_x$ is 
smooth on $X\ssm\{x\}$ and $\ge 0$ on $B(x,r/2)$, and $T$ is a positive
current. Hence by choosing $m$ large enough, we can suppose that this 
curvature form is $\ge\omega$, uniformly for $x\in\Omega'$. By
proposition~4.1, we get a
solution $u$ on $X$ such that
$$\int_X||u||^2e^{-2n\rho_x}dV_\omega\le C_4
\int_{r/2<|z-x|<r}|g|^2e^{-2m\psi-2s\varphi-2n\rho_x}dV_\omega\le C_5\,;$$
to get the estimate, we observe that $v$ has support in the corona
$r/2<|z-x|<r$ and that $\rho_x$ is bounded there. Thanks to the logarithmic
pole of $\rho_x$, we infer that $u(x)=0$. Moreover
$$\int_\Omega||\sigma||^2dV_\omega\le
\int_{\Omega'+B(0,r/2)}|g|^2e^{-2m\psi-2s\varphi}dV_\omega\le C_6,$$
hence $f=\sigma-u\in H^0(X,mF+sL)$ satisfies $\int_X||f||^2dV_\omega\le C_7$ and
$$||f(x)||=||\sigma(x)||=||g(x)||\,||e_F(x)||^m||e_L(x)||^s=
||e_F(x)||^m=e^{-m\psi(x)}.$$
In our orthonormal basis $(f_j)$, we can write $f=\sum\lambda_jf_j$
with $\sum|\lambda_j|^2\le C_7$. Therefore
$$\eqalign{
&e^{-m\psi(x)}=||f(x)||\le\sum|\lambda_j|\,\sup||f_j(x)||\le
\sqrt{C_7N}\,e^{sw_s(x)},\cr
&w_s(x)\ge{1\over s}\log(C_7N)^{-1/2}||f(x)||\ge
-{1\over s}\Big(\log(C_7N)^{1/2}+m\,\psi(x)\Big)\cr}$$
where $N=\dim H^0(X,mF+sL)=O(s^n)$. By adding $\varphi+{m\over s}\psi$, we
get $v_{s,\Omega}\ge\varphi-C_8s^{-1}\log s$. Thus $\lim_{s\to+\infty}
v_{s,\Omega}=\varphi$ everywhere, $T_s={i\over\pi}\partial\op v_{s,\Omega}$
converges weakly to $T={i\over\pi}\partial\op\varphi$, and
$$\nu(T_s,x)=\nu(v_{s,\Omega},x)\le\nu(\varphi,x)=\nu(T,x).$$
Note that $\nu(v_{s,\Omega},x)={1\over s}\min\ord_x(f_j)$ where
$\ord_x(f_j)$ is the vanishing order of $f_j$ at $x$, so our initial 
lower bound for $\nu(T_s,x)$ combined with the last inequality gives
$$\nu(T,x)-{n\over s}\le{1\over s}\min\ord_x(f_j)\le\nu(T,x).\leqno(9.4)$$

{\bf Step 2:} {\sl Construction of the divisors $D_s$}.

Select sections $(g_1\ld g_N)\in H^0(X,m_0F)$ with $m_0$ so large that
$m_0F$ is very ample, and set 
$$h_{k,s}=f_1^kg_1+\ldots+f_N^kg_N\in H^0\big(X,(m_0+km)F+ksL\big).$$
For almost every $N$-tuple $(g_1\ld g_N)$, lemma~9.5 below and the weak
continuity of $\partial\op$ show that
$$D_{k,s}={1\over ks}{i\over\pi}\partial\op\log|h_{k,s}|=
{1\over ks}\,{\rm div}(h_{k,s})$$
converges weakly to $T_s={i\over\pi}\partial\op v_{s,\Omega}$ as $k$
tends to $+\infty$, and that
$$\nu(T_s,x)\le\nu\Big({1\over ks}D_{k,s},x\Big)\le\nu(T,x)+{1\over ks}.$$
This, together with the first step, implies the proposition for some
subsequence $D_s=D_{k(s),s}$. We even obtain the more explicit inequality
$$\nu(T,x)-{n\over s}\le\nu\Big({1\over ks}D_{k,s},x\Big)\le\nu(T,x)+
{1\over ks}.\finpr$$

\tha Lemma 9.5|Let $\Omega$ be an open subset in $\cb^n$ and let
$f_1\ld f_N\in H^0(\Omega,\oc_\Omega)$ be non zero functions. 
Let $G\subset H^0(\Omega,\oc_\Omega)$ be a finite dimensional subspace
whose elements generate all $1$-jets at any point of $\Omega$. 
Finally, set $v=\sup\log|f_j|$ and
$$h_k=f_1^kg_1+\ldots+f_N^kg_N,~~~~g_j\in G\ssm\{0\}.$$
Then for all $(g_1\ld g_N)$ in $(G\ssm\{0\})^N$ except a set of measure~$0$,
the sequence ${1\over k}\log|h_k|$ converges to $v$ in $L^1_{\rm loc}(\Omega)$
and
$$\nu(v,x)\le\nu\Big({1\over k}\log|h_k|\Big)\le\nu(v,x)+{1\over k},
~~~~\forall x\in X,~~\forall k\ge 1.$$\fintha

\remarque Proof|The sequence ${1\over k}\log|h_k|$ is locally uniformly
bounded above and we have
$$\lim_{k\to+\infty}{1\over k}\log\big|h_k(z)\big|=v(z)$$
at every point $z$ where all absolute values $|f_j(z)|$ are distinct
and all $g_j(z)$ are nonzero. This is a set of full measure in $\Omega$
because the sets $\{|f_j|^2=|f_l|^2,\,j\ne l\}$ and $\{g_j=0\}$ are real
analytic and thus of zero measure (without loss of generality, we may
assume that $\Omega$ is connected and that the $f_j$'s are not pairwise
proportional). The well-known uniform integrability properties of 
plurisubharmonic functions then show that ${1\over k}\log|h_k|$ converges
to $v$ in $L^1_{\rm loc}(\Omega)$. It is easy to see that $\nu(v,x)$ is the
minimum of the vanishing orders ${\rm ord}_x(f_j)$, hence
$$\nu(\log|h_k|,x)={\rm ord}_x(h_k)\ge k\,\nu(v,x).$$
In the opposite direction, consider the set ${\cal E}_k$ of all $(N+1)$-tuples
$$(x,g_1\ld g_N)\in\Omega\times G^N$$
for which $\nu(\log|h_k|,x)\ge k\,\nu(v,x)+2$. Then ${\cal E}_k$ is a
constructible set in $\Omega\times G^N$: it has a locally finite
stratification by analytic sets, since
$${\cal E}_k=\bigcup_{s\ge 0}\Big(\bigcup_{j,\,|\alpha|=s}
\big\{x\,;\,D^\alpha f_j(x)\ne 0\big\}\times G^N\Big)\cap
\bigcap_{|\beta|\le ks+1}\big\{(x,g_l)\,;\,D^\beta h_k(x)=0\big\}.$$
The fiber ${\cal E}_k\cap(\{x\}\times G^N)$ over a point $x\in\Omega$
where $\nu(v,x)=\min{\rm ord}_x(f_j)=s$ is the vector space of $N$-tuples
$(g_j)\in G^N$ satisfying the equations $D^\beta
\big(\sum f_j^kg_j(x)\big)=0$, $|\beta|\le ks+1$. However, if
${\rm ord}_x(f_j)=s$, the linear map
$$(0\ld 0,g_j,0\ld 0)\longmapsto
\big(D^\beta(f_j^kg_j(x))\big)_{|\beta|\le ks+1}$$
has rank $n+1$, because it factorizes into an injective map
$J^1_xg_j\mapsto J^{ks+1}_x(f_j^kg_j)$. It follows that the fiber
${\cal E}_k\cap(\{x\}\times G^N)$ has codimension at least $n+1$.
Therefore
$$\dim {\cal E}_k\le\dim(\Omega\times G^N)-(n+1)=\dim G^N-1$$
and the projection of ${\cal E}_k$ on $G^N$ has measure zero by Sard's
theorem. By definition of ${\cal E}_k$, any choice of $(g_1\ld g_N)\in
G^N\ssm\bigcup_{k\ge 1} {\rm pr}({\cal E}_k)$ produces functions $h_k$
such that  $\nu(\log|h_k|,x)\le k\,\nu(v,x)+1$ on~$\Omega$.\finpr

\titre 10. Self-intersection inequality for closed positive currents|

Let $L$ be a nef line bundle over a projective algebraic manifold
$X$ and let $T=c(L)\ge 0$ be the curvature current of any singular
metric on $L$. We want to derive a bound for the codimension $p$
components in the sublevel sets $E_c(T)$
in terms of the $p$-th power $\{T\}^p$ of the cohomology class of~$T$.
The difficulty is that, in general, $T^p$ does not make sense as 
a current. However, products of currents can be defined in some
special circumstances. Let $M$ be an arbitrary complex manifold and
\hbox{$n=\dim_\cb X$.} Suppose given a closed positive current of bidegree
$(p,p)$ on~$M$ and a {\sl locally bounded} plurisubharmonic function $\psi$
on~$M$. According to Bedford-Taylor [BT~82], the product $\Theta\wedge 
i\partial\op\psi$ can then be defined by
$$\Theta\wedge i\partial\op\psi=i\partial\op\big(\psi\,\Theta).\leqno(10.1)$$
Here $\Theta$ is a differential form with measure coefficients,
so its product by the locally bounded Borel function $\psi$ is a well
defined current of order $0$, and the derivative $\partial\op$ 
can be taken in the sense of distribution theory. The resulting
current $\Theta\wedge i\partial\op\psi$ is again positive, as is easily 
seen by taking the weak limit with a sequence of smooth approximations
of $\psi$. More generally, if $\psi_1\ld\psi_m$ are locally bounded
plurisubharmonic functions, the product $\Theta\wedge i\partial\op\psi_1
\wedge\ldots\wedge i\partial\op\psi_m$ is well defined by induction on~$m$.
Various examples (cf. [Ki~84]) show that such products cannot be
defined in a reasonable way for arbitrary plurisubharmonic
functions~$\psi_j$. However, functions with $-\infty$ poles can be 
admitted if the polar set is sufficiently small.

\tha Proposition 10.2|Let $\psi$ be a plurisubharmonic function on $M$
such that $\psi$ is locally bounded on $M\ssm A$, where $A$
is an analytic subset of $M$ of codimension $\ge p+1$ at each point.
Then $\Theta\wedge i\partial\op\psi$ can be defined in
such a way that $\Theta\wedge i\partial\op\psi=\lim_{\nu\to+\infty}
\Theta\wedge i\partial\op\psi_\nu$ in the weak topology of currents, 
for any decreasing sequence $(\psi_\nu)_{\nu\ge 1}$ of plurisubharmonic 
functions converging to~$\psi$. Moreover, at every point $x\in X$ we have
$$\nu\Big(\Theta\wedge{i\over\pi}\partial\op\psi,x\Big)\ge\nu(\Theta,x)\,
\nu(\psi,x).$$\fintha

\remarque Proof|When $\psi$ is locally bounded everywhere, we have 
$\,\lim\psi_\nu\,\Theta=\psi\,\Theta$ by the monotone convergence 
theorem and we can apply the continuity of $\partial\op$ with
respect to the weak topology to conclude that
$\Theta\wedge i\partial\op\psi=\lim_{\nu\to+\infty}
\Theta\wedge i\partial\op\psi_\nu$.

First assume that $A$ is discrete. Since our results are local, we 
may suppose that $M$ is a ball $B(0,R)\subset\cb^n$ and that $A=\{0\}$. 
For every $s\le 0$, the function $\psi^{\gge s}=\max(\psi,s)$
is locally bounded on $M$, so the product $\Theta\wedge 
i\partial\op\psi^{\gge s}$ is well defined. For $|s|$ large, the function
$\smash{\psi^{\gge s}}$ differs from $\psi$ only in a small neighborhood 
of the origin, at which $\psi$ may have a $-\infty$ pole. Let $\gamma$
be a $(n-p-1,n-p-1)$-form with constant coefficients and set
$s(r)=\liminf_{|z|\to r-0}\psi(z)$. By Stokes' formula, we see that
$$\int_{B(0,r)}\Theta\wedge i\partial\op\psi^{\gge s}\wedge\gamma\leqno(10.3)$$
does not depend on $s$ when $s<s(r)$, for the difference of two such 
integrals involves the $\partial\op$ of a current with compact support in
$B(0,r)$. Taking $\gamma=(i\partial\op|z|^2)^{n-p-1}$, we see that the current
$\Theta\wedge i\partial\op\psi$ has finite mass on $B(0,r)\ssm\{0\}$
and we can define $\langle\oneb_{\{0\}}(\Theta\wedge i\partial\op\psi),
\gamma\rangle$ to be the limit of the integrals $(10.3)$ as $r$ tends to
zero and $s<s(r)$. In this case, the weak convergence statement is easily
deduced from the locally bounded case discussed above.
 
In the case where $\codim A\ge p+1$, we use a slicing technique to reduce the 
situation to the discrete case. Set $q=n-p-1$. There are linear
coordinates $(z_1\ld z_n)$ centered at any singular point of $A$,
such that $0$ is an isolated point of $A\cap\big(\{0\}\times\cb^{p+1}\big)$.
Then there are small balls $B'=B(0,r')$ in $\cb^q$, $B''=B(0,r'')$ in
$\cb^{p+1}$ such that $A\cap(\smash{\ov B}'\times\partial B'')=
\emptyset$, and the projection map
$$\pi:\cb^n\la\cb^q,~~~~z=(z_1\ld z_n)\longmapsto z'=(z_1\ld z_q)$$
defines a finite proper mapping $A\cap(B'\times B'')\la B'$.
These properties are preserved if we slightly change the direction of
projection. Take sufficiently many projections $\pi_m$ associated
to coordinate systems $(z^m_1\ld z^m_n)$, $1\le m\le N$, such that the 
family of $(q,q)$-forms
$$i\,dz^m_1\wedge d\ov z^m_1\wedge\ldots\wedge i\,dz^m_q\wedge d\ov z^m_q$$
defines a basis of the space of $(q,q)$-forms. Expressing any
compactly supported smooth $(q,q)$-form in such a basis, we see
that we need only define
$$\leqalignno{
&\int_{B'\times B''}\Theta\wedge i\partial\op\psi\wedge f(z',z'')\,
i\,dz_1\wedge d\ov z_1\wedge\ldots\wedge i\,dz_q\wedge d\ov z_q=&(10.4)\cr
~~\int_{B'}\Big\{&\int_{B''}f(z',\bu)\,
\Theta(z',\bu)\wedge i\partial\op\psi(z',\bu)\Big\}
i\,dz_1\wedge d\ov z_1\wedge\ldots\wedge i\,dz_q\wedge d\ov z_q\cr}$$
where $f$ is a test function with compact support in $B'\times B''$,
and $\Theta(z',\bu)$ denotes the slice of $\Theta$ on the fiber
$\{z'\}\times B''$ of the projection $\pi:\cb^n\to\cb^q$ (see e.g.
Federer [Fe~69]). Here $\Theta(z',\bu)$ is defined for almost every
$z'\in B'$ and is again a closed positive current of bidegree $(p,p)$
on $B''$. The right hand side of (10.4) makes sense since all fibers 
$(\{z'\}\times B'')\cap A$ are discrete and the double integral is
convergent (this will be explained in a moment).
The weak convergence statement can be derived from the discrete case
by~(10.4) and the bounded convergence theorem. Indeed, the boundedness 
condition is checked as follows: observe that the functions $\psi_1\ge
\ldots\ge\psi_\nu\ge\psi$ are uniformly bounded below on some cylinder 
$$K_{\delta,\varepsilon}=\smash{\ov B}\big((1-\delta)r'\big)\times
\Big(\smash{\ov B}(r'')\ssm\smash{\ov B}\big((1-\varepsilon)r''\big)\Big)$$
disjoint from $A$, with $\varepsilon\ll\delta\ll 1$ so small that
${\rm Supp}\,f\subset\smash{\ov B}\big((1-\delta)r'\big)\times
\smash{\ov B}\big((1-\varepsilon)r''\big)\,$; for all 
$z'\in\smash{\ov B}((1-\delta)r')$, the Chern-Levine-Nirenberg inequality 
[CLN~69] shows that
$$\int_{B((1-\varepsilon)r'')}\!\!\!\Theta(z',\bu)\wedge
i\partial\op\psi_\nu^{\gge s}(z',\bu)\le C_\varepsilon 
\max_{K_{\delta,\varepsilon}}\big(\psi_1^+,|s|\big)
\int_{B((1-\varepsilon/2)r'')}\!\!\!
\Theta(z',\bu)\wedge i\partial\op|z''|^2$$
$\big[$Proof: introduce a cut-off function $\chi_\varepsilon(z'')$ equal 
to $1$ near $\smash{\ov B}((1-\varepsilon)r'')$ with support in
$\smash{\ov B}((1-\varepsilon/2)r'')$, integrate by parts and write
$|i\partial\op\chi_\varepsilon|\le C_\varepsilon i\partial\op|z|^2\,\big]$;
for $s$ sufficiently large (independent of $\nu$), the left hand integral
does not depend on $s$ and is equal by definition to the corresponding 
integral involving $\psi_\nu$; the right hand side, of course, has a bounded 
integral over $B((1-\delta)r')$ because we integrate $\Theta$ against
a smooth form. The same argument with $\psi$ instead of $\psi_\nu$ shows
that the right hand side of (10.4) is convergent.

It only remains to prove the final statement concerning Lelong numbers.
Assume that $M=B(0,r)$ and $x=0$. By definition
$$\nu\Big(\Theta\wedge{i\over\pi}\partial\op\psi,x\Big)=
\lim_{r\to 0}\int_{|z|\le r}\Theta\wedge{i\over\pi}\partial\op\psi
\wedge\Big({i\over\pi}\partial\op\log|z|\Big)^{n-p-1}.$$
Set $\gamma=\nu(\psi,x)$ and
$$\psi_\nu(z)=\max\big(\psi(z),(\gamma-\varepsilon)\log|z|-\nu\big)$$
with $0<\varepsilon<\gamma$ (if $\gamma=0$, there is nothing to prove).
Then $\psi_\nu$ decreases to $\psi$ and
$$\int_{|z|\le r}\kern-7.5pt\Theta\wedge{i\over\pi}\partial\op\psi
\wedge\Big({i\over\pi}\partial\op\log|z|\Big)^{n-p-1}\ge
\limsup_{\nu\to+\infty}\int_{|z|\le r}\kern-7.5pt\Theta\wedge{i\over\pi}
\partial\op\psi_\nu\wedge\Big({i\over\pi}\partial\op\log|z|\Big)^{n-p-1}$$
by the weak convergence of $\Theta\wedge i\partial\op\psi_\nu\,$; here 
$({i\over\pi}\partial\op\log|z|)^{n-p-1}$ is not
smooth on $\ov B(0,r)$, but the integrals remain unchanged if we replace
$\log|z|$ by $\chi(\log|z|/r)$ with a smooth convex function $\chi$ such
that $\chi(t)=t$ for $t\ge -1$ and $\chi(t)=0$ for $t\le -2$. Now, we have
$\psi(z)\le\gamma\log|z|+C$ near $0$, so $\psi_\nu(z)$ coincides with
$(\gamma-\varepsilon)\log|z|-\nu$ on a small ball $B(0,r_\nu)\subset
B(0,r)$ and we infer
$$\eqalign{
\int_{|z|\le r}\Theta\wedge{i\over\pi}\partial\op\psi_\nu\wedge
\Big({i\over\pi}\partial\op\log|z|\Big)^{n-p-1}&\ge(\gamma-\varepsilon)
\int_{|z|\le r_\nu}\Theta\wedge\Big({i\over\pi}\partial\op\log|z|\Big)^{n-p}\cr
&\ge(\gamma-\varepsilon)\nu(\Theta,x).\cr}$$
As $r\in{}]0,R[$ and $\varepsilon\in{}]0,\gamma[$ were arbitrary, the
desired inequality follows.\finpr

\tha Corollary 10.5|For $1\le j\le p$, let $T_j={i\over\pi}
\partial\op\psi_j$ be closed positive \hbox{$(1,1)$-currents} 
on a complex manifold $M$. Suppose that there are analytic sets 
$A_2\supset\ldots\supset A_p$ in $M$ with
$\codim A_j\ge j$ at every point such that each $\psi_j$, $j\ge 2$, 
is locally bounded on $M\ssm A_j$. Let $\{A_{p,k}\}_{k\ge 1}$ be the
irreducible components of $A_p$ of codimension $p$ exactly and let
$\nu_{j,k}=\min_{x\in A_{p,k}}\nu(T_j,x)$ be the generic Lelong number
of $T_j$ $($or $\psi_j)$ along $A_{p,k}$. Then $T_1\wedge\ldots\wedge T_p$
is well-defined and
$$T_1\wedge\ldots\wedge T_p\ge\sum_{k=1}^{+\infty}\nu_{1,k}\ldots
\nu_{p,k}\,[A_{p,k}].$$\fintha

\remarque Proof|By induction on $p$, Proposition 10.2 shows that 
$T_1\wedge\ldots\wedge T_p$ is well defined. Moreover, we get
$$\nu(T_1\wedge\ldots\wedge T_p,x)\ge
\nu(T_1,x)\ldots\nu(T_p,x)\ge\nu_{1,k}\ldots\nu_{p,k}$$
at every point $x\in A_{p,k}$. The desired inequality is a consequence
of Siu's decomposition theorem (3.8).\finpr

Now, let $X$ be a projective $n$-fold and let $T$ be a closed positive
$(1,1)$-current on~$X$. By the Lebesgue decomposition theorem, we can write
$T=T_{\rm abc}+T_{\rm sing}$ where $T_{\rm abc}$ has absolutely continuous
coefficients with respect to the Lebesgue measure and the coefficients of
$T_{\rm sing}$ are singular measures. In general, $T_{\rm abc}$ and
$T_{\rm sing}$ are positive but non closed. We fix an arbitrary set
$\Xi\subset X$ and for $p=1,2\ld n,n+1$ we set
$$b_p=b_p(T,\Xi)=\inf\big\{c>0\,;\,
\codim\big(E_c(T),x\big)\ge p,~\forall x\in \Xi\big\},\leqno(10.6)$$
with the convention that a germ has codimension $>n$ if and only if it
is empty. Then $0=b_1\le b_2\le\ldots\le b_n\le b_{n+1}$ with
$b_{n+1}=\max_{x\in \Xi}\nu(T,x)<+\infty$, and for $c\in{}]b_p,b_{p+1}]$,
$E_c(T)$ has codimension $\ge p$ at every point of $\Xi$ and has at least
one component of codimension~$p$ exactly which intersects $\Xi$. We call
$b_1,b_2,\ldots$ the {\sl jumping values} of the Lelong numbers of $T$
over~$\Xi$. Our goal is to prove the following fundamental inequality
for the Lelong sublevel sets $E_c(T)$,
when $T$ is the curvature current $c(L)$ of a line bundle (this
restriction is unnecessary but the general case is more involved,
see [De~92] for a general proof).

\tha Theorem 10.7|Suppose that there is a semipositive line bundle $G$
over $X$ and a constant $a\ge 0$ such that $\oc_{TX}(1)+a\,\pi^\star G$ 
is nef; set $u=a\,c(G)\ge 0$ with any smooth semipositive metric on~$G$.
Let $T=c(L)\ge 0$ be the curvature current of a nef line bundle $L$,
let $\Xi\subset X$ be an arbitrary subset and $b_p=b_p(T,\Xi)$. Denote
by $\{Z_{p,k}\}_{k\ge 1}$ the irreducible components of 
codimension $p$ in $\smash{\bigcup_{c>b_p}E_c(T)}$ which intersect
$\Xi$ and let $\nu_{p,k}\in{}]b_p,b_{p+1}]$
be the generic Lelong number of $T$ along~$Z_{p,k}$.
Then the De Rham cohomology class
\hbox{$(\{T\}+b_1\{u\})\cdots(\{T\}+b_p\{u\})$} can be represented by a
closed positive current $\Theta_p$ of bidegree $(p,p)$ such that
$$\Theta_p\ge\sum_{k\ge 1}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)\,[Z_{p,k}]
+(T_{\rm abc}+b_1u)\wedge\ldots\wedge(T_{\rm abc}+b_pu).$$
The same is true for $\Xi=X$ if we only suppose $c_1(L)\in\Gamma_+$
instead of $L$ nef.\fintha

Here $\bigwedge(T_{\rm abc}+b_ju)$ is computed pointwise as a $(p,p)$-form.
It follows in particular from our inequality that $T_{\rm abc}^p$ has
locally integrable coefficients for all $p$. Let $\omega$ be a K\"ahler 
metric on $X$. If we take the wedge product of the fundamental 
inequality~10.7 by $\omega^{n-p}$, integrate over $X$ and neglect 
$T_{\rm abc}$ in the right hand side, we get:

\tha Corollary 10.8|With the notation of theorem~$10.7$, the degrees with
respect to $\omega$ of the
\hbox{$p$-codimensional} components $Z_{p,k}$ of $\bigcup_{c>b_p}E_c(T)$
intersecting $\Xi$ satisfy
$$\eqalign{
\sum_{k=1}^{+\infty}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)
&\int_X[Z_{p,k}]\wedge\omega^{n-p}\cr
&\le\big(\{T\}+b_1\{u\}\big)\cdots\big(\{T\}+b_p\{u\}\big)
\cdot\{\omega\}^{n-p}.\finpr\cr}$$\fintha

In particular, if $D$ is a nef divisor and if $L=\oc(D)$ is equipped with
the singular metric such that $T=c(L)=[D]$, 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 case
$X=\pb^n$ is of course especially simple: Since $T\pb^n$ is ample,
we can take~$u=0$, and then the bound is simply $\{D\}^p\cdot
\{\omega\}^{n-p}$; the same is true more generally as soon as $TX$ is nef.
It is natural to try to find an interpretation of the
$(p,p)$-form involving $T_{\rm abc}$ in the general inequality.
Unfortunately this $(p,p)$-form is not closed and so it does not
correspond to an intrinsic cohomology class that would have
a simple counterpart in algebraic geometry. Nevertheless, the additional
Monge-Amp\`ere mass provided by $T_{\rm abc}$ is absolutely crucial 
for the purely algebraic application which we shall make in the next
section. Our intuition is that the additional $(p,p)$-form must be
understood as an {\sl excess of self-intersection}, measuring
asymptotically the amount of freedom a divisor in the linear system
$H^0(X,sL)$ can keep while being moved through the fixed singular strata
prescribed by $sT$, when $s$ tends to infinity.

\remarque Proof of Theorem 10.7|By the first step of the proof of theorem~9.1
and by (9.4), there is a positive line bundle $F$ with the following property:
for every $s\ge 0$ there exist sections $f_i\in H^0(X,F+sL)$, $1\le i\le N(s)$,
with
$$\nu(T,x)-{n\over s}\le{1\over s}\min\ord_x(f_i)\le\nu(T,x),~~~~
\forall x\in X.$$
The main idea is to decrease the Lelong numbers by replacing each section 
$f_i$ by some of its high order derivatives, or rather by some jet section. 
In this way, the polar components with low generic Lelong number disappear,
and we can decrease the dimension so as to be able to take intersections of 
currents (thanks to proposition~10.2 or corollary~10.5). Of course, introducing
jet sections also introduces symmetric powers of the cotangent bundle; this is
the reason why the curvature of $TX$ plays an important role in the inequality.

{\bf First step:} {\sl killing Lelong numbers in the singular metric of $L$.}

Consider the $m$-jet section $J^mf_i$ with values in the vector bundle
$E_m=J^m\oc(F+sL)$ of $m$-jets. First suppose that $a$ is rational.
There are exact sequences
$$0\la S^mT^\star X\otimes\oc(F+sL)\la E_m\la E_{m-1}\la 0,$$
and $S^mTX\otimes\oc(maG)$ is nef by our assumptions. By induction on~$m$
we easily infer that
$$E_m^\star\otimes\oc(2F+sL+maG)$$
is ample (in an exact sequence of vector bundles with ample extreme terms, so
is the middle term). Hence there is a symmetric power of order $q$ with
$qa\in\nb$ such that
$$S^qE_m^\star\otimes\oc(2qF+qsL+qmaG)$$
is generated by holomorphic sections $g_j$. We use the pairing of $S^qE_m$
and $S^qE_m^\star$ to get sections
$$S^q(J^mf_i).g_j\in H^0\big(X,\oc(2qF+qsL+qmaG)\big).$$
By means of these sections, for each pair $(s,m)$ we define a new singular
metric $||~~||_{s,m}$ on $L$ such that
$$||\xi||_{s,m}={||\xi||\over\sum_{i,j}||S^q(J^mf_i).g_j||^{1/qs}},~~~~
\xi\in L,$$
where $||~~||$ denotes the original singular metric on $L$ as well as the
induced metric on $\oc(2qF+qsL+qmaG)$; here the metric of $F$ (resp.~$G$) is
smooth and has positive (resp. semipositive) curvature. Denote by $\varphi$ the
weight of the original metric on $L$, by $\varphi_{s,m}$ the new one, and by
$\psi_F$, $\psi_G$ the weights of $F$, $G$ on some trivializing open set
$\Omega\subset X$. Then
$$\varphi_{s,m}={1\over qs}\log\sum_{i,j}\big|S^q(J^mf_i).g_j\big|-
{2\over s}\psi_F-{m\over s}a\psi_G,\leqno(10.9)$$
because $e^{-\varphi}$ appears in the numerator and
$\exp(-2q\psi_F-qs\varphi-qma\psi_G)^{{1\over qs}}$ in the denominator
of $||\xi||_{s,m}$. As $\psi_F$, $\psi_G$ are smooth and the $g_j$'s do not
vanish simultaneously, we get
$$\nu(\varphi_{s,m},x)={1\over s}\min_i{\rm ord}_x(J^mf_i)=
{1\over s}\big(\min_i{\rm ord}_x(f_i)-m\big)_+.$$
Hence we have the inequality
$$\Big(\nu(T,x)-{m+n\over s}\Big)_+\le\nu(\varphi_{s,m},x)\le
\Big(\nu(T,x)-{m\over s}\Big)_+\,;\leqno(10.10)$$
that is, we have been able to construct a new curvature current
${i\over\pi}\partial\op\varphi_{s,m}$ on $L$ in which all the Lelong
numbers that were $\le m/s$ have been killed. Unfortunately the curvature
is no longer $\ge 0$, but by (10.9) we have
$${i\over\pi}\partial\op\varphi_{s,m}\ge-{2\over s}c(F)-{m\over s}a\,c(G)=
-{2\over s}\omega-{m\over s}u,\leqno(10.11)$$
where $\omega=c(F)>0$. Only the term ${2\over s}\omega$ can be made
arbitrarily small. Now, for each $s$, select an integer $m$ such that 
$b_p<m/s\le b_p+{1\over s}$. By (10.9) and (10.10), we see that $\varphi_{s,m}$
is locally bounded on $X\ssm E_{m/s}(T)$, and the definition of $b_p$ implies
that $E_{m/s}(T)$ has codimension $\ge p$ in a neighborhood of~$\Xi$. 

{\bf Second step:} {\sl Construction of the $p$-th intersection current
$\Theta_p$.}

By induction on $p$, we suppose that $\Theta_{p-1}$ has already been
constructed ($\Theta_1=T$ satisfies the requirements for $p=1$).
By proposition~10.2, the wedge product $\Theta_{p-1}\wedge{i\over\pi}
\partial\op\varphi_{s,m}$ is well defined in a neighborhood of $\Xi$.
However, this is not satisfactory when $\Xi\ne X$, because we need a
current defined everywhere on~$X$. This is the reason why we have to 
assume $L$ nef when $\Xi\ne X$. Under this assumption, there is for 
each $s$ a smooth metric on $L$, associated to some weight $\rho_s$ on 
the trivializing open set $\Omega$, such that ${i\over\pi}\partial\op
\rho_s\ge-{1\over s}\omega$. We introduce the weight
$$\psi_{s,m,A,B}=\sup\big(\varphi\,,\,\varphi_{s,m}-A\,,\,\rho_s-B\big),$$
where $A,B>0$ are large constants. This weight corresponds to the
singular metric on $L$ given by
$$||\xi||_{s,m,A,B}=\inf\big(||\xi||\,,\,e^A||\xi||_{s,m}\,,\,
e^B||\xi||_{\rho_s}\big).$$
Clearly $\psi_{s,m,A,B}$ converges to $\varphi$ as $A,B$ tend to $+\infty$,
and $\psi_{s,m,A,B}$ is locally bounded; therefore the curvature current
$T_{s,m,A,B}={i\over\pi}\partial\op\psi_{s,m,A,B}$ converges weakly to
$T={i\over\pi}\partial\op\varphi$ as $A,B$ tend to $+\infty$. Moreover,
the assumed lower bound on ${i\over\pi}\partial\op\rho_s$ combined with
(10.11) implies
$$T_{s,m,A,B}\ge-{2\over s}\omega-{m\over s}u\,;$$
this is easily seen by adding ${2\over s}\psi_F+{m\over s}a\psi_G$ to each
term in the supremum formula defining $\psi_{s,m,A,B}$. Now,
the positive $(p,p)$-current
$$\Theta_{p,s,m,A,B}=\Theta_{p-1}\wedge\Big(T_{s,m,A,B}
+{2\over s}\omega+{m\over s}u\Big)$$
is well defined over $X$ since $\psi_{s,m,A,B}$ is locally bounded. Its
cohomology class is independent of $A,B$ and converges to
$\{c_1(L)\}\cdot\big(c_1(L)+b_pu\big)$ when $s$ tends to $+\infty$
(by the choice of $m$ made at the end of the first step, we have
$\lim m/s=b_p$). Hence the family $(\Theta_{p,s,m,A,B})$ is weakly
compact. First extract a weak limit $\Theta_{p,s,m,A}$ by taking some 
subsequence $B_\nu\to+\infty$. By proposition~10.2 we see that in
a neighborhood of $\Xi$
$$\Theta_{p,s,m,A}=\lim_{B\to+\infty}\Theta_{p,s,m,A,B}=
\Theta_{p-1}\wedge\Big(T_{s,m,A}+{2\over s}\omega+{m\over s}u\Big),$$
where
$$T_{s,m,A}={i\over\pi}\partial\op\psi_{s,m,A}\,,~~~~
\psi_{s,m,A}=\sup(\varphi,\varphi_{s,m}-A).$$
Indeed the codimension of the set of poles of $\psi_{s,m,A}$ is at least $p$
in a neighborhood of $\Xi$. Now, by (10.10), we have
$$\nu(\psi_{s,m,A},x)\ge\min\big(\nu(\varphi,x),\nu(\varphi_{s,m},x)\big)
\ge\Big(\nu(T,x)-{m+n\over s}\Big)_+.$$
Proposition 10.2 shows that
$$\nu(\Theta_{p,s,m,A},x)\ge\nu(\Theta_{p-1},x)\Big(\nu(T,x)-
{m+n\over s}\Big)_+~~~~\hbox{\rm near}~~\Xi.$$
By induction on $p$, we conclude that the generic Lelong number of 
$\Theta_{p,s,m,A}$ along $Z_{p,k}$ is at least equal to
$$(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_{p-1})\Big(\nu(T,x)-{m+n\over s}\Big)_+.$$
In fact, $Z_{p,k}$ meets $\Xi$ at some point $x$, and therefore the inequality
holds at least on a neighborhood of $x$ in $Z_{p,k}$. Siu's decomposition
formula (3.8) yields
$$\Theta_{p,s,m,A}\ge(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_{p-1})
\Big(\nu(T,x)-{m+n\over s}\Big)_+[Z_{p,k}].$$
Now, extract a weak limit $\Theta_{p,s,m}$ for some subsequence
$A_\nu\to+\infty$ and then a weak limit $\Theta_p$ for some subsequence
$m_\nu/s_\nu\to b_p$ with $s_\nu\to+\infty$. We obtain a current $\Theta_p$
such that $\{\Theta_p\}=\{\Theta_{p-1}\}\cdot\big(c_1(L)+b_p\{u\}\big)$ and
$$\Theta_p\ge(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)\,[Z_{p,k}].$$
It only remains to show by induction on $p$ that 
$$\Theta_{p,{\rm abc}}\ge(T_{\rm abc}+b_1u)\wedge\ldots\wedge
(T_{\rm abc}+b_pu).$$
As the coefficients of $[Z_{p,k}]$ are singular with respect to the Lebesgue 
measure, $\Theta_p$ will actually be larger than the sum. By construction, 
there exists a subsequence $(s_\nu,m_\nu,A_\nu,B_\nu)$ such that
$$\eqalign{
\Theta_p&=\lim\Theta_{p-1}\wedge\Big({i\over\pi}\partial\op
\psi_{s_\nu,m_\nu,A_\nu,B_\nu}+{2\over s_\nu}\omega+
{m_\nu\over s_\nu}u\Big),\cr
\psi_{s_\nu,m_\nu,A_\nu,B_\nu}&=\sup\big(\varphi\,,\,\varphi_{s_\nu,m_\nu}-
A_\nu\,,\,\rho_{s_\nu}-B_\nu\big).\cr}$$
The desired lower bound follows from lemma 10.12 below. At the beginning
of the proof, $a$ was supposed to be rational, but this extra assumption
can be removed as above by extracting a weak limit $\Theta_{p,a_\nu}\to
\Theta_p$ with a sequence of rational numbers decreasing to $a\in\rb_+$.
If $\Xi=X$, everything works even if we omit the term $\rho_k-B$ in the
definition of $\psi_{k,m,A,B}\,$: we can start directly with
$\psi_{k,m,A}$ because its polar set has codimension $\ge p$ on the whole
space $X$. Hence the nef assumption on $L$ is not necessary.\finpr

\tha Lemma 10.12|Let $\Omega\subset\cb^n$ be an open subset and let $\varphi$
be an arbitrary plurisubharmonic function on $\Omega$. Set 
$\varphi_\nu=\max(\varphi,\psi_\nu)$ where $\psi_\nu$ is a decreasing 
sequence of plurisubharmonic functions converging to $-\infty$, each
$\psi_\nu$ being locally bounded in $\Omega$ $($or perhaps only
in the complement of an analytic subset of codimension~$\ge p)$.
Let $\Theta$ be a closed positive current of bidegree $(p-1,p-1)$.\break
If $\Theta\wedge i\partial\op\varphi_\nu$ converges to a weak 
limit~$\Theta'$, then
$$\Theta'_{\rm abc}\ge\Theta_{\rm abc}\wedge(i\partial\op\varphi)_{\rm abc}.$$
\fintha

\remarque Proof|Let $(\rho_\varepsilon)$ (resp. $(\wt\rho_\varepsilon)$)
be a family of regularizing kernels on $\cb^n$ (resp. on $\rb^2$), and
let $\max_\varepsilon(x,y)=({\max}\star\wt\rho_\varepsilon)(x,y)$ be
a regularized max function. For $\varepsilon>0$ small enough, the function
$$\varphi_{\nu,\varepsilon}={\textstyle\max_\varepsilon}
(\varphi\star\rho_\varepsilon,\psi_\nu\star\rho_\varepsilon)$$
is plurisubharmonic and well defined on any preassigned open set
$\Omega'\compact\Omega$. As $\varphi_{\nu,\varepsilon}$ decreases to
$\varphi_\nu$ when $\varepsilon$ decreases to $0$, proposition~10.2
shows that
$$\lim_{\varepsilon\to 0}\Theta\wedge i\partial\op\varphi_{\nu,\varepsilon}
=\Theta\wedge i\partial\op\varphi_\nu$$
in the weak topology. Let $(\beta_j)$ be a sequence of test forms
which is dense in the space of test forms of bidegree $(n-p,n-p)$ and
contains strongly positive forms with arbitrary large compact support
in $\Omega$. Select $\varepsilon_\nu>0$ so small that
$$\langle\Theta\wedge i\partial\op\varphi_{\nu,\varepsilon_\nu}
-\Theta\wedge i\partial\op\varphi_\nu\,,\,\beta_j\rangle\le{1\over\nu}
~~~~\hbox{\rm for}~~j\le\nu.$$
Then the sequence $\Theta\wedge i\partial\op\varphi_{\nu,\varepsilon_\nu}$
is locally uniformly bounded in mass and converges weakly to the same limit
$\Theta'$ as $\Theta\wedge i\partial\op\varphi_\nu$. Moreover, at every 
point $x\in\Omega$ such that $\varphi(x)>-\infty$, we have
$\varphi_{\nu,\varepsilon_\nu}(x)\ge\varphi(x)>\psi_\nu\star
\rho_{\varepsilon_\nu}(x)+1$ for $\nu$ large, because
$\lim_{\nu\to-\infty}\psi_\nu=-\infty$ locally uniformly. Hence 
$\varphi_{\nu,\varepsilon_\nu}=\varphi\star\rho_{\varepsilon_\nu}$
on a neighborhood of $x$ (which may depend on $\nu$) and
$i\partial\op\varphi_{\nu,\varepsilon_\nu}(x)=(i\partial\op\varphi)
\star\rho_{\varepsilon_\nu}(x)$ for $\nu\ge\nu(x)$. By the Lebesgue
density theorem, if $\mu$ is a measure of absolutely continuous part
$\mu_{\rm abc}$, the sequence $\mu\star\rho_{\varepsilon_\nu}(x)$ converges
to $\mu_{\rm abc}(x)$ at almost every point. Therefore
$\lim i\partial\op\varphi_{\nu,\varepsilon_\nu}(x)=
(i\partial\op\varphi)_{\rm abc}(x)$ almost everywhere For any strongly positive test 
form $\alpha=i\alpha_1\wedge\ov\alpha_1\wedge\ldots\wedge i\alpha_{n-p}
\wedge\ov\alpha_{n-p}$ of bidegree $(n-p,n-p)$ on $\Omega$, we get
$$\eqalign{
\int_\Omega\Theta'\wedge\alpha&=\lim_{\nu\to+\infty}\int_\Omega\Theta
\wedge i\partial\op\varphi_{\nu,\varepsilon_\nu}\wedge\alpha\cr
&\ge\liminf_{\nu\to+\infty}\int_\Omega\Theta_{\rm abc}
\wedge i\partial\op\varphi_{\nu,\varepsilon_\nu}\wedge\alpha
\ge\int_\Omega\Theta_{\rm abc}\wedge i\partial\op\varphi_{\rm abc}
\wedge\alpha.\cr}$$
Indeed, the first inequality holds because $i\partial\op
\varphi_{\nu,\varepsilon_\nu}$ is smooth, and the last one results 
from Fatou's lemma. This implies 
$\Theta'_{\rm abc}\ge\Theta_{\rm abc}\wedge(i\partial\op\varphi)_{\rm abc}$
and lemma~10.12 follows.\finpr

\titre 11. Proof of the criterion in arbitrary dimension|

We return here to the point where we arrived at the end of \S~6,
and apply our self-intersection inequality~10.7 to complete the proof of
the Main Theorem. First suppose, with the notation of \S~6,
that $L$ is an ample line bundle over~$X$. The idea is to apply 
inequality~10.7 to the $(1,1)$-current $T=\lim\omega_{\varepsilon_\nu}$ 
produced by equation~(6.5), and to integrate the inequality with respect 
to the K\"ahler form $\omega=c(L)$. Before doing this, we need to
estimate the excess of intersection in terms of $T_{{\rm abc}}^n$.

\tha Proposition 11.1|The absolutely continuous part $T_{{\rm abc}}$ 
of $T$ satisfies
$$T_{{\rm abc}}^n\ge\Big(1-{\sigma\over L^n}\Big)\omega^n~~~~
\hbox{\sl a.e. on}~~X.$$\fintha

\remarque Proof|The result is local, so we can work in an open set 
$\Omega$ which is relatively compact in a coordinate patch of $X$. 
Let $\rho_\delta$ be a family of
smoothing kernels. By a well known lemma (see e.g. [BT~76], 
proposition~5.1), the operator 
$A\mapsto(\det A)^{1/n}$ is concave on the cone of nonnegative
hermitian $n\times n$ matrices, hence we get
$$\big[\big(\omega_\varepsilon\star\rho_\delta(x)\big)^n\big]^{1/n}\ge
(\omega_\varepsilon^n)^{1/n}\star\rho_\delta(x)\ge
\Big(1-{\sigma\over L^n}\Big)^{1/n}(\omega^n)^{1/n}\star\rho_\delta(x),$$
thanks to equation (6.5). As $\varepsilon_\nu$ tends to $0$,
$\omega_{\varepsilon_\nu}\star\rho_\delta$ converges to
$T\star\rho_\delta$ in the strong topology of $C^\infty(\Omega)$, thus
$$\big((T\star\rho_\delta)^n\big)^{1/n}\ge
\Big(1-{\sigma\over L^n}\Big)^{1/n}(\omega^n)^{1/n}\star\rho_\delta~~~~
\hbox{\rm on}~~\Omega.$$
Now, take the limit as $\delta$ goes to $0$. By the Lebesgue density
theorem $T\star\rho_\delta(x)$ converges almost everywhere
to $T_{{\rm abc}}(x)$ on $\Omega$, so we are done.\finpr\vskip7pt

According to the notation used in \S~10, we consider an arbitrary
subset $\Xi\subset X$ and introduce the jumping values
$$b_p=\inf\big\{c>0\,;\,\codim\big(E_c(T),x\big)\ge p,~\forall x\in\Xi\big\}.$$
By proposition~11.1 and inequality 5.2(a), we have
$$T_{{\rm abc}}^j\wedge\omega^{n-j}\ge
\Big(1-{\sigma\over L^n}\Big)^{j/n}\omega^n.\leqno(11.2)$$
Now, suppose that $\oc_{TX}(1)+a\,\pi^\star L$ is nef for some
constant $a\ge 0$. We can then apply theorem~10.7 with $u=a\,\omega$ and
$\{\Theta_p\}=(1+b_1a)\cdots(1+b_pa)\{\omega^p\}\,$; by taking the 
wedge product of $\Theta_p$ with $\omega^{n-p}$, we get
$$\eqalign{
(1+b_1a)\ldots(1+b_pa)\int_X\omega^n&\ge
\sum_{k\ge 1}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)
\int_X[Z_{p,k}]\wedge\omega^{n-p}\cr
&{}+\int_X(T_{{\rm abc}}+b_1a\,\omega)\wedge\ldots\wedge
(T_{{\rm abc}}+b_pa\,\omega)\wedge\omega^{n-p}.\cr}$$
Combining this inequality with (11.2) for $T_{{\rm abc}}^{p-j}$ yields
$$\eqalign{
(1+b_1a)\ldots(1+b_pa)\,L^n&\ge
\sum_{k\ge 1}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)\,L^{n-p}\cdot Z_{p,k}\cr
&{}+\sum_{0\le j\le p}S_j^p(b)\,a^j\Big(1-{\sigma\over L^n}\Big)^{(p-j)/n}
L^n,\cr}$$
where $S_j^p(b)$, $1\le j\le p$, denotes the elementary symmetric function
of degree $j$ in $b_1\ld b_p$ and $S_0^p(b)=1$. As $\prod(1+b_ja)=\sum
S_j^p(b)a^j$, we get
$$\leqalignno{
~~~~\smash{\sum_{k\ge 1}}(\nu_{p,k}-b_1)\ldots(\nu_{p,k}-b_p)\,
&L^{n-p}\cdot Z_{p,k}\le\cr
&\sum_{0\le j\le p}S_j^p(b)\,a^j\Big(1-\Big(1-{\sigma\over L^n}
\Big)^{(p-j)/n}\Big)L^n.&(11.3)\cr}$$
If $L$ is only supposed to be big and nef, we follow essentially the same
arguments and replace $\omega$ in all our inequalities by 
$\omega_m=c(L)_m+{1\over m}c(A)$ with $A$ ample (see section~6).
Note that all $(n,n)$-forms $\omega_m^n$ were defined to be proportional to 
$\gamma^n=c(A)^n$, so inequality~11.1 becomes in the limit
$$T_{{\rm abc}}^n\ge\Big(1-{\sigma\over L^n}\Big)
{L^n\over A^n}\gamma^n=\Big(1-{\sigma\over L^n}\Big)
{L^n\over(L+{1\over m}A)^n}\omega_m^n.$$
The intersection inequality (11.3) is the expected generalization of
proposition~8.2 in arbitrary codimension. In this inequality, $\nu_{p,k}$
is the generic Lelong number of $T$ along~$Z_{p,k}$, and $Z_{p,k}$ runs over
all \hbox{$p$-codimensional} components $Y$ of $\bigcup_{c>b_p}E_c(T)$
intersecting $\Xi\,$; by definition of $b_j$ we have $\max_k\nu_{p,k}=b_{p+1}$.
Hence we obtain:

\tha Theorem 11.4|Let $L$ be a big nef line bundle such that
$TX\otimes\oc(aL)$ is nef, and let $T\in c_1(L)$
be the positive curvature current obtained by concentrating the Monge-Amp\`ere
mass $L^n$ into a finite sum of Dirac measures with total mass~$\sigma$, plus
some smooth positive density spread over $X$ $($equations $(6.5)$ and $(6.10))$.
Then the jumping values $b_p$ of the Lelong number of $T$ over an arbitrary
subset $\Xi\subset X$ satisfy the inductive inequalities
$$(b_{p+1}-b_1)\ldots(b_{p+1}-b_p)\le{1\over\min_Y L^{n-p}\cdot Y}
\sum_{0\le j\le p-1}S_j^p(b)\,a^j\sigma_{p-j}\,,\leqno(11.5)$$
where $\sigma_j=\big(1-(1-\sigma/L^n)^{j/n}\big)L^n$, and where $Y$
runs over all \hbox{$p$-codimensional} subvarieties of $X$
intersecting~$\Xi$.\fintha

Observe that $\sigma_j$ is increasing in $j\,$; in particular
$\sigma_j<\sigma_n=\sigma$ for $j\le n-1$. Moreover, the convexity of the 
exponential function shows that 
$t\mapsto{1\over t}\big(1-(1-\sigma/L^n)^t\big)L^n$ is decreasing, thus
$\sigma_j>\sigma_p j/p$ for $j<p\,$; in particular $\sigma_j>\sigma j/n$
for $j\le n-1$. We are now in a position to prove the following general
result, which contains the Main Theorem as a special case:

\tha Theorem 11.6|On a projective $n$-fold~$X$, let
$g_j:(X,x_j)\to(\cb^n,0)$ be germs of finite holomorphic maps with
covering degree $\rho_j$. Let $\jc_1\subset\oc_{X,x_1},\,\ldots\,$,
$\jc_N\subset\oc_{X,x_N}$ be the associated ideals
$\ic\big(\tau_{j,0}\log|g_j|\big)_{x_j}$ and let
$$\sigma_0=\sum\rho_j\tau_{j,0}^n,~~~~
\sigma_p=\big(1-(1-\sigma_0/L^n)^{p/n}\Big)L^n,~~1\le p\le n-1,$$
where $L$ is a big nef line bundle such that $L^n>\sigma_0$. Suppose that
$\oc_TX(1)+a\,\pi^\star L$ is nef over
$P(T^\star X)$ and that there is a sequence $0=\beta_1<\ldots<\beta_n\le 1$
with
$$L^{n-p}\cdot Y>(\beta_{p+1}-\beta_1)^{-1}\ldots(\beta_{p+1}-
\beta_p)^{-1}\sum_{0\le j\le p-1}S_j^p(\beta)\,a^j\sigma_{p-j}$$
for every subvariety $Y\subset X$ of codimension $p=1,2\ld n-1$
passing through one of the points $x_j$. Then there is a surjective map
$$H^0(X,K_X+L)\lraww\bigoplus_{1\le j\le N}\oc(K_X+L)_{x_j}\otimes
\big(\oc_{X,x_j}/\jc_j\big).$$\fintha

\remarque Proof|Select $\tau_j>\tau_{j,0}$ so that $L^{n-p}\cdot Y$ still
satisfies the above lower bound with the corresponding value
$\sigma>\sigma_0$. Then apply theorem~11.4 with $\Xi=\{x_1\ld x_N\}$.
Inequality (11.5) shows inductively that $b_p<\beta_p$
for $p\ge 2$, so $b_n<1$ and we get $\codim\big(E_1(T),x_j\big)=n$ at
each point $x_j$. Thanks to (4.3), corollaries 4.6 and 6.8 imply the
desired surjectivity property.\finpr

\remarque Proof of Corollary 1|This is only a matter of straightforward
calcula\-tions, but adjusting the constants $\beta_p$ to get
optimal exponents for $\sigma_0$, $a$ and $\mu_\Xi(L)$ in the lower 
bound of $m$ requires some care. By the convexity argument already explained,
we have $\sigma_{p-j}\le(p-j)\sigma_1\le p(1-1/p)^j\sigma_1$. As $\beta_1=0$,
we find
$$\eqalign{
\sum_{0\le j\le p-1}S_j^p(\beta)\,a^j\sigma_{p-j}
&\le p\sigma_1\sum_{0\le j\le p-1}S_j^p(\beta)\,\Big({p-1\over p}a\Big)^j\cr
&=p\sigma_1\Big(1+\beta_2{p-1\over p}a\Big)\ldots\Big(1+\beta_p{p-1\over p}a
\Big).\cr}$$
When we replace $L$ by $mL$, the constant $a$ is replaced by $a/m$, and
by definition of $\mu=\mu_\Xi(L)$ we have $(mL)^k\cdot Y\ge(m\mu)^k$.
Hence theorem~11.6 yields the sufficient condition
$$(m\mu)^{n-p}\ge\prod_{j\le p}(\beta_{p+1}-\beta_j)^{-1}p\sigma_1
\Big(1+\beta_2{p-1\over p}{a\over m}\Big)\ldots\Big(1+\beta_p{p-1\over p}
{a\over m}\Big)\leqno(11.7)$$
with $0=\beta_1<\ldots<\beta_n<1$.
When $p=1$ we get $(m\mu)^{n-1}\ge\beta_2^{-1}\sigma_1$,
and when $p=n-1$ the inequality implies
$m\mu>(n-1)\sigma_1>(1-1/n)\sigma_0$. We suppose
in fact $m\mu\ge\lambda\sigma_0$ where $\lambda>1-1/n$ is a constant
which will be adjusted later to an optimal value; in particular
$m\mu\ge\lambda n^n$. We will choose
$\beta_j/\beta_{j+1}$ so small that
$\prod_{j\le p}(\beta_{p+1}-\beta_j)\ge U_p^{-1}\beta_{p+1}^p$ with
a constant $U_p$ slightly larger than $1$. We are thus led to define
$\beta_p$ inductively by the formula
$$\beta_{p+1}^p=U_p{p\sigma_1\over(m\mu)^{n-p}}
\Big(1+\beta_2{p-1\over p}{a\over m}\Big)\ldots\Big(1+\beta_p
{p-1\over p}{a\over m}\Big),~~~~1\le p\le n-1,\leqno(11.8)$$
and $m$ has to be taken so large that $\beta_n<1\,$; suppose that
this is the case. The first step is
to determine admissible constants $1<U_1<\ldots<U_{n-1}$.
For $j\le p$, (11.8) implies
$${1\over j-1}\Big({\beta_j\over m\mu}\Big)^{j-1}\le
{1\over p-1}\Big({\beta_p\over m\mu}\Big)^{p-1}
\Longrightarrow
\beta_j\le\Big({j-1\over n-1}\Big)^{\textstyle{1\over j-1}}
\big(m\mu\big)^{\textstyle{-{n-j\over j-1}}}$$
by taking $p=n$ in the first inequality. In general, for $j\le p$ we get
$$\leqalignno{
\Big({\beta_j\over\beta_p}\Big)^{p-1}&\le
{j-1\over p-1}\Big({\beta_j\over m\mu}\Big)^{p-j}\le
{j-1\over p-1}\Big({j-1\over n-1}\Big)^{\textstyle{p-j\over j-1}}
(m\mu)^{\textstyle-{(p-j)\big(1+{n-j\over j-1}\big)}},\cr
{\beta_j\over\beta_p}&\le
{(j-1)^{\textstyle{1\over j-1}}\over(p-1)^{\textstyle{1\over p-1}}
(n-1)^{\textstyle{{1\over j-1}-{1\over p-1}}}}
(m\mu)^{\textstyle-{(n-1)\big({1\over j-1}-{1\over p-1}\big)}}
\le{\gamma_j\over\gamma_p}&(11.9)\cr}$$
with
$$\gamma_j=\Big({j-1\over n-1}\Big)^{\textstyle{1\over j-1}}
(\lambda n^n)^{\textstyle -(n-1)\big({1\over j-1}-{1\over n-1}\big)}.$$
The sequence $(\gamma_j)$ is strictly increasing and satisfies
$\gamma_j/\gamma_{j+1}<1/n$. Thus we can take
$$U_p=\prod_{2\le j\le p}\big(1-\gamma_j/\gamma_{p+1}\big)^{-1}.$$
Let $k$ be the largest integer in $\{1\ld n-1\}$ such that
$\beta_ka/m\le 1$ and let
$t=\max\big(\beta_ka/m,\gamma_k/\gamma_{k+1}\big)\in{}
[\gamma_k/\gamma_{k+1},1]$. Inequality (11.9) implies
$$\Big(1+\beta_j{p-1\over p}{a\over m}\Big)\le\cases{\displaystyle
\Big(1+{p-1\over p}{\gamma_j\over\gamma_k}\,t\Big)&for~ $2\le j\le k$,\cr
\displaystyle\Big({p-1\over p}+{\gamma_k\over\gamma_j}\,t^{-1}\Big)
\beta_j{a\over m}&for~ $k+1\le j\le p$.\cr}$$
The product of all factors $(\ldots t^{\pm 1})$ over $j=2\ld p$ is a
polynomial with positive coefficients in $\rb[t,t^{-1}]$, hence is a
convex function of~$t$. Therefore, the product is at most equal to
the maximum of its values for $t=1$ or $t=\gamma_k/\gamma_{k+1}$:
$$\eqalign{
\max\Bigg\{\prod_{2\le j\le k}\Big(1+{p-1\over p}{\gamma_j\over
\gamma_k}\Big)&\prod_{k+1\le j\le p}
\Big({p-1\over p}+{\gamma_k\over\gamma_j}\Big)\,,\cr
&\prod_{2\le j\le k}\Big(1+{p-1\over p}{\gamma_j\over\gamma_{k+1}}\Big)
\prod_{k+1\le j\le p}\Big({p-1\over p}+
{\gamma_{k+1}\over\gamma_j}\Big)\Bigg\}.\cr}$$
We have $(p-1)/p+\gamma_k/\gamma_j\le 1-1/p+1/n\le 1$ for $j>k$,
moreover the sequence $\gamma_j/\gamma_{j+1}$ is increasing.
If we introduce the increasing sequence
$$V_p=\prod_{2\le j\le p}\Big(1+{p-1\over p}{\gamma_j\over
\gamma_p}\Big),$$
it is then easy to check that the above maximum is bounded by
$V_p$ for $p\ge k$. Therefore (11.8) gives
$$\beta_{p+1}^p\le U_pV_p{p\sigma_1\over(m\mu)^{n-p}}\,
\beta_{k+1}\ldots\beta_p\Big({a\over m}\Big)^{p-k}~~~~
\hbox{\rm for}~~p\ge k.$$
As $U_pV_p\,p\le W_n=U_{n-1}V_{n-1}(n-1)$, by induction these inequalities
yield
$$\beta_{p+1}\le\Big({W_n\sigma_1\over(m\mu)^{n-k}}\Big)^{1/k}
(a\mu)^{\textstyle{1\over p}+{1\over p-1}+\ldots+{1\over k+1}}~~~~
\hbox{\rm for}~~p\ge k,$$
where the exponent of $a\mu$ is understood to be $0$
if~$p=k$. Finally, we have $(mL)^n\ge (m\mu)^n\ge(\lambda\sigma_0)^n$
by definition of $\mu$ and by our initial hypothesis $m\mu\ge\lambda
\sigma_0$, hence $\sigma_0/(mL)^n\le\lambda^{-n}\sigma_0^{1-n}\le
\lambda^{-n}n^{-n(n-1)}$. It follows again from a convexity argument that
$$\sigma_1/\sigma_0\le\Big(1-\big(1-\lambda^{-n}n^{-n(n-1)}\big)^{1/n}\Big)
\lambda^nn^{n(n-1)}=T_n.$$
Hence $W_n\sigma_1\le B_n\sigma_0$ with $B_n=W_nT_n=(n-1)U_{n-1}V_{n-1}T_n$.
Therefore, sufficient conditions in order that $\beta_n<1$ are:
$$\leqalignno{
m\mu&>B_n\sigma_0~~~~\hbox{\rm for}~~k=n-1,&(11.10)\cr
(m\mu)^{n-k}&>B_n\sigma_0\,(a\mu)^{k\big({\textstyle{1\over n-1}+
{1\over n-2}+\ldots+{1\over k+1}}\big)}&(11.10_k)\cr}$$
for $k\in\{1\ld n-2\}$. These conditions are equivalent to the inequality
stated in corollary~1. Observe that our constant $B_n$ depends on $\lambda$.
The initial hypothesis $m\mu\ge\lambda\sigma_0$ will be automatically
satisfied if we adjust $\lambda$ so that $B_n(\lambda)=\lambda$; this
is always possible because $B_n(\lambda)$ is decreasing in $\lambda$
and $B_n(\lambda)>1-1/n$. With this choice, a numerical calculation
shows that $B_n<2.005$ for all $n$ and $\lim_{n\to +\infty}B_n=2$.
For small values of $n$, we find (with rounding by above):
$$\table{
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl~\,n&\verl~~~2&\verl~~~3&\verl~~~4&\verl~~~5&\verl~~~6
&\verl~~~7&\verl~~~8&\verl~~~9&\verl~~10&\verl~~11&\verl~~12&\verl\cr
\smash{(11.11)\quad}
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl~B_n~&\verl\,0.625\,&\verl\,1.019\,&\verl\,1.309\,&\verl\,1.485\,
&\verl\,1.603\,&\verl\,1.687\,&\verl\,1.748\,&\verl\,1.794\,
&\verl\,1.830\,&\verl\,1.858\,&\verl\,1.879\,&\verl\cr
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr}$$
and $B_n<2$ for $n\le 46$.

\titre 12. Universal bounds for very ample line bundles|

Let $X$ be an ample line bundle over a projective $n$-fold~$X$. In order to
find universal conditions for $K_X+L$ to be very ample, our main theorem
would require a universal value $a$ depending only on $n=\dim_\cb X$ such that
$TX\otimes\oc(aL)$ is always nef. However, this is clearly impossible as the
example of curves already show: if~$X$ is a curve of genus $g$ and $L$ has
degree $1$, then $TX\otimes\oc(aL)$ is nef if and only if $a\ge 2g-2$. In
general, it is an interesting unsolved question to know whether such a
value $a$ can be found explicitly in terms of geometric invariants of~$X$
(Chern classes, $\ldots$). Here, these difficulties can be avoided by means
of the following simple lemma.

\tha Lemma 12.1|Let $F$ be a very ample line bundle over $X$. Then the
vector bundle $TX\otimes\oc(K_X+nF)$ is nef and generated by global
sections.\fintha

\remarque Proof|By the very ample assumption, the $1$-jet bundle $J^1F$
is generated by its sections. Consider the exact sequence
$$0\la T^\star X\otimes F\la J^1F\la F\la 0$$
where rank$(J^1F)=n+1$ and det$(J^1F)=K_X+(n+1)F$. The $n$-th exterior
power $\bigwedge^n(J^1F)$ is also generated by sections and there is a
surjective morphism
$${\textstyle\bigwedge^n}(J^1F)=(J^1F)^\star\otimes{\rm det}(J^1F)\la
(TX\otimes F^\star)\otimes{\rm det}(J^1F)=TX\otimes\oc(K_X+nF).$$
Hence $TX\otimes\oc(K_X+nF)$ is generated by sections and, in particular,
it is nef.\finpr\vskip5pt

The next idea consists in the following iteration trick: lemma~12.1
suggests that a universal lower bound for the nefness of $TX\otimes\oc(aL')$
can be achieved with $L'=K_X+L$ if $L$ is sufficiently ample. Then it follows
from the Main Theorem that $K_X+L'=2K_X+L$ is very ample under suitable
numerical conditions. Lemma~12.1 applied with $F=2K_X+L$ shows that
$TX\otimes\oc\big((2n+1)K_X+nL\big)$ is nef, and thus $TX\otimes\oc\big(
(2n+1)L''\big)$ is nef with $L''=K_X+{1\over 2}L\le L'$. Hence we see
that the Main Theorem can be iterated. The special value $a=2n+1$ will
play an important role.

\tha Lemma 12.2|Let $L'$ be an ample line bundle over $X$. Suppose that
$TX\otimes\oc\big((2n+1)L'\big)$ is nef. Then $K_X+L'$ is very ample,
resp. generates $s$-jets, as soon as $\mu_X(L')>C_n\sigma_0$ with
the corresponding value of $\sigma_0$ and with a constant $C_n<3$
depending only on~$n$.\fintha

\remarque Proof|If $\mu=\mu_X(L')$, $a=2n+1$ and $\sigma_0=2n^n$
$\big($resp. $\sigma_0=(n+s)^n\big)$, then the first arguments in the
proof of corollary~1 give the sufficient condition
$$\mu^{n-p}>\prod_{2\le j\le p}(\beta_{p+1}-\beta_j)^{-1}p\sigma_1
\Big(1+{p-1\over p}\beta_2(2n+1)\Big)\ldots\Big(1+{p-1\over p}
\beta_p(2n+1)\Big)$$
with $0=\beta_1<\ldots<\beta_n=1$. We suppose
$\mu\ge\lambda\sigma_0$ (in particular $\mu\ge 2\lambda n^n$)
and choose
$$\beta_p=\big(\alpha^{n-p+1}(2n^n)^{n-p}\big)^{1/(p-1)},~~~~2\le p\le n-1$$
with suitable constants $\lambda$, $\alpha$ to be determined later.
In analogy with the proof of corollary~1, we introduce the constants
$$\eqalign{
U'_p&=\prod_{2\le j\le p}\Big(1-{\beta_j\over\beta_{p+1}}\Big)^{-1},~~~~
V'_p=\prod_{2\le j\le p}\Big(1+{p-1\over p}\beta_j(2n+1)\Big),\cr
T'_n&=\Big(1-\big(1-\lambda^{-n}(2n^n)^{-(n-1)}\big)^{1/n}\Big)
\lambda^n(2n^n)^{n-1}.\cr}$$
We have $\sigma_1\le T'_n\sigma_0$ and our conditions become
$$\mu^{n-p}>U'_p\beta_{p+1}^ppT'_n\sigma_0V'_p
=\cases{
pU'_pV'_pT'_n\alpha^{n-p}(2n^n)^{n-p-1}\sigma_0&for $p\le n-2$,\cr
(n-1)U'_{n-1}V'_{n-1}T'_n\sigma_0&for $p=n-1$.\cr}$$
As $\sigma_0\ge 2n^n$, a sufficient condition is
$$\mu>\max\big\{(n-1)U'_{n-1}V'_{n-1}T'_n\,,\,
\alpha(pU'_pV'_pT'_n)^{1/(n-p)}\big\}_{1\le p\le n-2}\sigma_0.$$
We adjust $\lambda$ and $\alpha$ so that
$$(n-1)U'_{n-1}V'_{n-1}T'_n=\alpha\max_{1\le p\le n-2}(pU'_pV'_pT'_n)^{1/(n-p)}
=\lambda,$$
and we take this common value to be our constant $C_n$. A numerical
calculation gives $C_n<3$ for all $n$ and $\lim_{n\to +\infty}C_n=3$.
The first values are given by
$$\table{
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl~\,n&\verl~~~2&\verl~~~3&\verl~~~4&\verl~~~5&\verl~~~6
&\verl~~~7&\verl~~~8&\verl~~~9&\verl~~10&\verl~~11&\verl~~12&\verl\cr
\smash{(12.3)\quad}
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl~C_n~&\verl\,0.563\,&\verl\,0.737\,&\verl\,0.995\,&\verl\,1.201\,
&\verl\,1.370\,&\verl\,1.510\,&\verl\,1.629\,&\verl\,1.730\,&\verl\,1.817\,
&\verl\,1.893\,&\verl\,1.959\,&\verl\cr
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr}$$
Hence Lemma 12.2 is proved.\finpr

\tha Lemma 12.4|Let $F$ be a line bundle which generates $s$-jets
at every point. Then $F^p\cdot Y\ge s^p$ for every $p$-dimensional
subvariety $Y\subset X$.\fintha

\remarque Proof|Fix an arbitrary point $x\in Y$. Then consider the
singular metric on $F$ given by
$$||\xi||^2={|\xi|^2\over\sum|u_j(z)|^2},$$
where $(u_1\ld u_N)$ is a basis of $H^0(X,F\otimes\mc_x^s)$. By our
assumption, these sections have an isolated common zero of order $s$
at~$x$. Hence $F$ possesses a singular metric such that the weight
$\varphi={1\over 2}\log\sum|u_j|^2$ is plurisubharmonic and has an
isolated logarithmic pole of Lelong number $s$ at~$x$. By the comparison
inequality (3.6) with $\psi(z)=\log|z-x|$, we get
$$F^p\cdot Y\ge\int_{B(x,\varepsilon)}[Y]\wedge
\Big({i\over\pi}\partial\op\varphi\Big)^p\ge s^p\nu([Y],\psi)=s^p\nu(Y,x)
\ge s^p.\finpr$$

\remarque Proof of Corollary 2|As $L$ is ample, there exists an integer
$q$ (possibly very large) such that
$$\cases{
K_X+qL\phantom{\big)}&is ample,\cr
TX\otimes\oc\big((2n+1)(K_X+qL)\big)&is nef,\cr
\mu_X(K_X+qL)>C_n\sigma_0.&\cr}\leqno(12.5)$$
By lemma 12.2 applied to $L'=K_X+qL$, we find that $F=K_X+L'=
2K_X+qL$ is very ample and generates $s$-jets. In particular
$K_X+{q\over 2}L$ is an ample $\qb$-divisor, and for any $p$-dimensional
subvariety $Y\subset X$ we have
$$\eqalign{
\big(K_X+(q-1)L\big)^p\cdot Y
&=\Big({1\over 2}F+(q/2-1)L\Big)^p\cdot Y\cr
&=\sum_{0\le k\le p}{p\choose k}2^{k-p}(q/2-1)^kF^{p-k}\cdot L^k\cdot Y.\cr}$$
By the convexity inequality 5.2~(b) and lemma 12.4 we get
$$F^{p-k}\cdot L^k\cdot Y\ge (F^p\cdot Y)^{1-k/p}(L^p\cdot Y)^{k/p}\ge
s^{p-k}\big(\mu_X(L)\big)^k.$$
Hence $\big(K_X+(q-1)L\big)^p\cdot Y\ge\big((q/2-1)\mu_X(L)+s/2\big)^p$
and
$$\mu_X\big(K_X+(q-1)L\big)\ge{1\over 2}\big((q-2)\mu_X(L)+s\big).$$
Moreover, lemma~12.1 applied to $F$ shows that
$$TX\otimes\oc(K_X+nF)=TX\otimes\oc\big((2n+1)K_X+nqL\big)$$
is nef. As $nq/(2n+1)\le q/2\le q-1$ for $q\ge 2$, we find that all
properties (12.5) except perhaps the last one remain valid with $q-1$
in place of $q\,$:
$$\cases{
K_X+(q-1)L\phantom{\big)}&is ample,\cr
TX\otimes\oc\big((2n+1)(K_X+(q-1)L)\big)&is nef,\cr
\mu_X\big(K_X+(q-1)L\big)\ge{1\over 2}\big((q-2)\mu_X(L)+s\big).&\cr}
\leqno(12.6)$$
By induction we conclude that (12.6) is still true for the smallest
integer $q-1=m$ such that
$$(q-2)\mu_X(L)+s=(m-1)\mu_X(L)+s>2 C_n\sigma_0.$$
For this value of $m$ lemma~12.2 implies that $2K_X+mL$ is very ample, resp.
generates $s$-jets.\finpr

\remarque Remark 12.7|If $G$ is a nef line bundle, the proof of corollary~2
can be applied without modification to show that $2K_X+mL+G$ is very ample,
resp. generates $s$-jets, for $(m-1)\mu_X(L)+s>2C_n\sigma_0\,$:
indeed, adding $G$ can only increase the numbers $\mu_X(K_X+qL+G)$
occurring in the induction.

\remarque Remark 12.8|The condition $(m-1)\mu_X(L)+s>C_n\sigma_0$
is never satisfied for $m=1$. However, it is still possible to obtain a
sufficient condition in order that $2K_X+L$ generates $s$-jets.
Indeed, the last step of the iteration shows that $2K_X+2L$ generates
$s'$-jets and that $\mu_X(K_X+L)\ge s'/2$ if $\mu_X(L)+s'>2C_n(n+s')^n$.
Choose $s'>2C_n(n+s)^n$. Then $\mu_X(K_X+L)>C_n(n+s)^n$ and
we can perform another iteration to conclude
that $2K_X+L$ generates $s$-jets. Of course, the corresponding lower bound
for $\mu_X(L)$ is extremely large, of the order of magnitude of
$(2C_n)^{n+1}(n+s)^{n^2}$.

\remarque Remark 12.9|A numerical computation of $4C_nn^n$ in corollary~2
gives the following bounds for $2K_X+mL$ to be very ample when
$L$ is ample:
$$\table{\kern80pt
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl~~~n&\verl~\,2&\verl~\,3&\verl~~~4&\verl~~~~5&\verl~~~~\,6&\verl\cr
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr
&\verl~m\ge~&\verl~10~&\verl~80~&\verl~1019~&\verl~15010~
&\verl~255537~&\verl\cr
&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\hrulefill&\cr}$$
\vskip4pt

We now list a few immediate consequences of our results, in
connection with some classical questions of algebraic geometry.

\tha Corollary~12.10|Let $X$ be a projective $n$-fold of general type
with $K_X$ ample. Then $mK_X$ is very ample for $m\ge 12\,n^n$.\fintha

\tha Corollary~12.11|Let $X$ be a Fano $n$-fold, that is, a $n$-fold
such that $-K_X$ is ample. Then $-mK_X$ is very ample for
$m\ge 12\,n^n$.\fintha

Corollaries 12.10 and 12.11 follow easily from corollary~2 applied to
\hbox{$L=\pm K_X\,$:} then we obtain that \hbox{$2K_X+mL$} is very ample
for $m>4C_nn^n$, and a numerical check shows that \hbox{$4C_nn^n+2<12\,n^n$}
for all $n$. Hence we get pluricanonical embeddings $\Phi:X\to\pb^N$ such that
$\Phi^\star\oc(1)=\pm mK_X$ with $m=12\,n^n$. The image $Y=\Phi(X)$ has degree
$$\hbox{\rm deg}(Y)=\int_Y c_1\big(\oc(1)\big)^n=
\int_X c_1\big(\pm mK_X\big)^n=m^n|K_X^n|.$$
It can be easily reproved from this that there are only finitely
many deformation types of Fano $n$-folds, as well as of $n$-folds
of general type with $K_X$ ample, corresponding to a given
discriminant $|K_X^n|\,$; such results were already known by the
fundamental finiteness theorem of [Ma~72] and [KoM~72]. In the Fano case,
it is conjectured that there is a universal bound $(-K_X)^n\le A_n\,$:
if such a universal bound could be proved, it would become possible to
obtain an explicit upper bound for the number of deformation types of
Fano $n$-folds in any dimension~$n$.\footnote{(${}^\star$)}{{\sl Added
after proof\/}: such bounds have been obtained recently by
J.$\,$Koll\'ar-Y.$\,$Miyaoka-S.$\,$Mori and independently by
F.$\,$Campana.}\finpr\vskip4pt

Finally, let $L$ be an ample line bundle over an arbitrary projective
$n$-fold~$X$. It follows from Mori's theory that
$K_X+(n+1)L$ is always nef (see [Fu~87]). If $K_X+tL$ is nef for some
integer $t\ge 0$, Fujita conjectures that $m(K_X+tL)$ is
spanned for every positive integer $m>n+1-t$. Although such
a sharp result seems very hard to prove, our results
allow us to prove that some explicit multiple of $K_X+(t+\varepsilon)L$
is very ample for every $\varepsilon>0$ rational.

\tha Corollary 12.12|If $L$ is an ample line bundle such that $K_X+tL$ is
nef for some integer $t\ge 0$, the line bundle
$m\big(K_X+(t+\varepsilon)L\big)$ is very ample for every $\varepsilon>0$
and every integer $m>0$ such that $m\varepsilon\in\nb$ and
$m\varepsilon>8C_nn^n-2t-1$.\fintha

\remarque Proof|First suppose that $m=2p$ is even. Then either
$p\varepsilon$  or $p\varepsilon-1/2$ is an integer.
Apply corollary~2 to the ample line bundle
$$\eqalign{
L'&=(p-1)\big(K_X+tL\big)+(p\varepsilon+t)L,~~~~\hbox{\rm resp.}\cr
L'&=(p-1)\big(K_X+tL\big)+(p\varepsilon+t-1/2)L.\cr}$$
In the first case, we find $\mu_X(L')\ge(p\varepsilon+t)\mu_X(L)\ge
p\varepsilon+t$, hence $2K_X+2L'=2p\big(K_X+(t+\varepsilon)L\big)$
is very ample when $\mu_X(L')+1\ge p\varepsilon+t+1>4C_nn^n\,$;
in the second case, we get the condition $p\varepsilon+t+1/2
>4C_nn^n$ and we apply remark~12.7 to
conclude that $2K_X+2L'+L$ is very ample. When $m=2p+1$ is odd,
we argue in the same way with
$$\eqalign{
L'&=(p-1)\big(K_X+tL\big)+\big((2p+1)\varepsilon/2+t\big)L,~~~~
\hbox{\rm resp.}\cr
L'&=(p-1)\big(K_X+tL\big)+\big((2p+1)\varepsilon/2+t-1/2\big)L,\cr}$$
and conclude that
$2K_X+2L'+\big(K_X+tL\big)$ or $2K_X+2L'+\big(K_X+(t+1)L\big)$
is very ample when $(2p+1)\varepsilon/2+t+1/2>4C_nn^n$.\finpr
\vfill\eject

\centerline{\bf References}
\vskip15pt
\eightpoint
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\vskip30pt
\tenpoint\noindent(December 11, 1990, revised on September 11, 1991)

\end


