%
% Pseudoconvex-concave duality and regularization of currents
%
% Jean-Pierre Demailly,
% Universit\'e de Grenoble I, Institut Fourier
% 38402 Saint-Martin d'H\`eres, France
%
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\title{Pseudoconvex-concave duality} 
\title{and regularization of currents}
\bigskip

\centerline{\twelvebf Jean-Pierre Demailly}\medskip
\centerline{\twelverm Universit\'e de Grenoble I, Institut Fourier}
\titlerunning{J.-P.\ Demailly, Pseudoconvex-concave duality and 
regularization of currents}
\vskip10pt
\line{\hfill \it to the memory of Michael Schneider}
\vskip10pt

\noindent{\bf Abstract} 

{\eightpoint
The goal of this work is to investigate some
basic properties of Finsler metrics on holomorphic vector bundles, in
the perspective of obtaining geometric versions of the Serre duality
theorem. We establish a duality framework under which pseudoconvexity
and pseudoconcavity properties get exchanged -- up to some technical
restrictions. These duality properties are shown to be related to
several geometric problems, such as the conjecture of Hartshorne and
Schneider, asserting that the complement of a $q$-codimensional
algebraic subvariety with ample normal bundle is $q$-convex. In full
generality, a functorial construction of Finsler metrics on symmetric
powers of a Finslerian vector bundle is obtained.  The construction 
preserves positivity of curvature, as expected from the fact that tensor
products of ample vector bundles are ample. From this, a new shorter
and more geometric proof of a basic regularization theorem for
closed $(1,1)$ currents is derived. The technique is based on the
construction of a mollifier operator for plurisubharmonic functions,
depending on the choice of a Finsler metric on the cotangent bundle
and its symmetric powers.}
\vskip8pt

\noindent{\bf Contents}
\vskip3pt

{\eightpoint
\sectref{0}
\sectref{1}
\sectref{2}
\sectref{3}
\sectref{4}
\sectref{5}
\sectref{6}
\sectref{7}
\sectref{8}
\sectref{9}
\sectref{10}
\sectref{11}}
\vskip8pt

\noindent
{\bf Key words:} Serre duality, pseudoconvexity, pseudoconcavity,
Finsler metric, symmetric power, Chern curvature,
Hartshorne-Schneider conjecture, plurisubharmonic function, positive
current, regularization of currents, Legendre transform, Lelong number,
Ohsawa-Takegoshi theorem, Skoda $L^2$ estimates 
\vskip8pt

\noindent
{\bf A.M.S. Classification 1991:} 14B05, 14J45, 32C17, 32J25, 32S05
\vfill\eject

\section{\S0. Introduction}

The goal of the present paper is to investigate some duality
properties connecting pseudoconvexity and pseudoconcavity. Our
ultimate perspective would be a geometric duality theory parallel to
Serre duality, in the sense that Serre duality would be the underlying
cohomological theory. Although similar ideas have already been used by
several authors in various contexts, e.g.\ for the study of direct
images of sheaves (Ramis-Ruget-Verdier [RRV71]), or in connection with
the study of Fantappie transforms and lineal convexity (see Kiselman's
recent work [Kis97]), or in the study of Monge-Amp\`ere equations
(Lempert [Lem85]), we feel that the ``convex-concave'' duality theory
still suffers from a severe lack of understanding.

Our main concern is about {\it Finsler metrics} on holomorphic
vector bundles. As is well known, a holomorphic vector bundle $E$
on a compact complex manifold is ample in the sense of Hartshorne
[Har66] if and only if its dual $E^\star$ admits a strictly pseudoconvex
tubular neighborhood of $0$, that is, if and only if $E^\star$ has
a strictly plurisubharmonic smooth Finsler metric. In that case, we expect
$E$ itself to have a tubular neighborhood of the zero section such that
the Levi form of the boundary has everywhere signature $(r-1,n)$, where
$r$ is the rank of $E$ and $n=\dim X$; in other words, $E$ has a Finsler 
metric whose Levi form has signature $(r,n)$. This is indeed the case if $E$
is positive in the sense of Griffiths, that is, if the above plurisuharmonic
Finsler metric on $E^\star$ can be chosen to be {\it hermitian}; 
more generally, Sommese [Som78, 79, 82] has observed that everything works 
well if the Finsler metric is fiberwise convex (in the ordinary sense). 
The Kodaira-Serre vanishing theorem tells us that strict pseudoconvexity
of $E^\star$ implies that the cohomology of high symmetric powers
$S^mE$ is concentrated in degree $0$, while the Andreotti-Grauert vanishing
theorem tells us that $(r,n)$ convexity-concavity of $E$ implies
that the cohomology of $S^mE^\star$ is concentrated in degree $n$.
Of course, both properties are connected from a cohomological view
point by the Serre duality theorem, but the related geometric picture
seems to be far more involved. A still deeper unsolved question
is Griffiths' conjecture on the equivalence of ampleness and 
positivity of curvature for hermitian metrics [Gri69].

One of the difficulties is that ``linear'' duality between $E$ and
$E^\star$ is not sufficient to produce the expected biduality properties
relating convexity on one side and concavity on the other side.  What
seems to be needed rather, is a duality between large
symmetric powers $S^mE$ and $S^mE^\star$, asymptotically as $m$ goes
to infinity (``polynomial duality''). Although we have not been able
to find a completely satisfactory framework for such a theory, one of
our results is that there is a functorial and natural construction
which assigns Finsler metrics on all symmetric powers $S^mE$, whenever
a Finsler metric on $E$ is given.  The assignment has the desired
property that the Finsler metrics on $S^mE$ are plurisubharmonic if
the Finsler metric on $E$ was. The construction uses ``polynomial
duality'' in an essential way, although it does not produce good
metrics on the dual bundles $S^mE^\star$.

Several interesting questions depend on the solution to these
problems.  R.~Hartshorne [Har70] raised the question whether the
complement of an algebraic subvariety $Y$ with ample normal bundle
$N_Y$ in a projective algebraic variety $X$ is $q$-convex in the sense
of Andreotti-Grauert, with $q=\codim Y$. Michael Schneider [Sch73]
proved the result in the case the normal bundle is positive is the
sense of Griffiths, thus yielding strong support for Hartshorne's
conjecture. As a consequence of Sommese's observation, Schneider's
result extends the case if $N_Y^\star$ has a strictly pseudoconvex and
fiberwise convex neighborhood of the zero section, which is the case
for instance if $N_Y$ is ample and globally generated.

Other related questions which we treat in detail are the approximation
of closed positive $(1,1)$-currents and the attenuation of their
singularities.  In general, a closed positive current $T$ cannot be
approximated (even in the weak topology) by smooth closed positive
currents. A cohomological obstruction lies in the fact that $T$ 
may have negative intersection numbergs $\{T\}^p\cdot Y$ with some 
subvarieties $Y\subset X$. This is the case for instance if $T=[E]$
is the current of integration on a the exceptional curve of a
blown-up surface and $Y=E$. However, as we showed in previous papers
([Dem82, Dem92, 94]), the approximation is possible if we allow the
regularization $T_\varepsilon$ to have a small negative part.
The main point is to control this negative part accurately, in term
of the global geometry of the ambient geometry~$X$. It turns out
that more or less optimal bounds can be described in terms of the
convexity of a Finsler metric on the tangent bundle $T_X$. Again,
a relatively easy proof can be obtained for the case of a hermitian
metric ([Dem82, 94]), but the general Finsler case (as solved in [Dem 92])
still required very tricky analytic techniques. We give here an
easier and more natural method based on the use of ``symmetric products'' 
of Finsler metrics.

Many of the ideas presented here have matured over a long period of time,
for a large part through discussion and joint research with Thomas 
Peternell and Michael Schneider. Especially, the earlier results 
[Dem92] concerning smoothing of currents were strongly motivated
by techniques needed in our joint work [DPS94]. I~would like here
to express my deep memory of Michael Schneider, and my gratitude for
his very beneficial mathematical influence.

\section{\S1. Pseudoconvex Finsler metrics and ample vector bundles}

Let $X$ be a complex manifold and $E$ a holomorphic vector bundle over $X$.
We set $n=\dim_\bC X$ and $r=\rank E$. Following S.~Kobayashi [Kob75],
we introduce

\claim 1.1. Definition|A $($positive definite$)$ Finsler metric on $E$ is
a positive complex homogeneous function $\xi\mapsto\Vert\xi\Vert_x$ defined
on each fiber $E_x$, that is, such that $\Vert\lambda\xi\Vert_x=
|\lambda|\Vert\xi\Vert_x$ for each $\lambda\in\bC$ and $\xi\in E_x$,
and $\Vert\xi\Vert_x>0$ for $\xi\ne 0$.
\endclaim

We will in general assume some regularity, e.g.\ continuity of
the function $(x,\xi)\mapsto\Vert\xi\Vert_x$ on the total space $E$ of the
bundle. We say that the metric is smooth if it is smooth on
$E\ssm\{0\}$. The logarithmic indicatrix of the Finsler metric is by
definition the function
$$
\chi(x,\xi)=\log\Vert\xi\Vert_x.\leqno(1.2)
$$
We will say in addition that the Finsler metric is {\sl convex} if
the function $\xi\mapsto\Vert\xi\Vert_x$ is convex on each fiber $E_x$
(viewed as a real vector space). A Finsler metric is convex if and only
if it derives from a norm (hermitian norms are of course of a special
interest in this respect); however, we will
have to deal as well with non convex Finsler metrics. 

The interest in Finsler metrics essentially arises from the following
well-known characterization of ample vector bundles ([Kod54], [Gra58],
[Kob75]).

\claim 1.3. Theorem|Let $E$ be a vector bundle on a compact complex 
manifold~$X$. The following properties are equivalent.
\smallskip
\item{\rm(1)} $E$ is ample $($in the sense of Hartshorne [Ha66]$)$.
\smallskip
\item{\rm(2)} $\cO_{P(E)}(1)$ is an ample line bundle on the 
projectivized bundle $P(E)$ $($of hyperplanes of $E)$.
\smallskip
\item{\rm(3)} $\cO_{P(E)}(1)$ carries a smooth hermitian metric of
positive Chern curvature form.
\smallskip
\item{\rm(4)} $E^\star$ carries a smooth Finsler metric which
is strictly plurisubharmonic on the total space $E^\star\ssm\{0\}$.
\smallskip
\item{\rm(5)} $E^\star$ admits a smoothly bounded strictly pseudoconvex
tubular neighborhood $U$ of the zero section.
\vskip0pt
\endclaim

Actually, the equivalence of (1), (2) is a purely algebraic fact, while the
equivalence of (2) and (3) is a consequence of the Kodaira embedding theorem.
The equivalence of (3) and (4) just comes from the observation
that a Finsler metric on $E^\star$ can be viewed also as a 
{\sl hermitian metric} $h^\star$ on the line bundle $\cO_{P(E)}(-1)$ 
(as the total space of $\cO_{P(E)}(-1)$ coincides with
the blow-up of $E^\star$ along the zero section), and from the obvious
identity
$$
(\pi_{P(E)})^\star\Theta_{h^\star}(\cO_{P(E)}(-1))
=-{i\over 2\pi}\ddbar\chi^\star,
$$
where $\Theta_{h^\star}(\cO_{P(E)}(-1))$ denotes the Chern curvature form
of $h^\star=e^{\chi^\star}$, and $\pi_{P(E)}:E^\star\ssm\{0\}\to P(E)$ the
canonical projection. Finally, if we have a Finsler metric as in (4), then
$U_{\varepsilon}=\{\xi^\star\,;\;\Vert\xi^\star\Vert^\star<\varepsilon\}$
is a fundamental system of strictly pseudoconvex neighborhood of the
zero section of $E^\star$. Conversely, given such a neighborhood $U$,
we can make it complex homogeneous by replacing $U$ with 
$U^\star=\bigcap_{|\lambda|\ge 1}\lambda U$. Then $U^\star$ is the unit
ball bundle of a continuous strictly plurisubharmonic Finsler metric on
$E^\star$ (which can further be made smooth thanks to Richberg's 
regularization theorem [Ric68], or by the much more precise results 
of [Dem92], which will be reproved in a simpler way in section~9).

\claim 1.4. Remark|{\rm It is unknown whether the ampleness of $E$ 
implies the existence of a {\sl convex} strictly plurisubharmonic
Finsler metric on $E^\star$. Sommese [Som78] observed that this is
the case if $E$ is ample and generated by sections. In fact, if there
are sections $\sigma_j\in H^0(X,E)$ generating $E$, then
$$
h_0(\xi^\star)=\Big(\sum_j|\sigma_j(x)\cdot\xi^\star|^2\Big)^{1/2}
$$
defines a weakly plurisubharmonic and strictly convex (actually hermitian)
metric on $E^\star$. On the other hand, the ampleness implies the
existence of a strictly plurisubharmonic Finsler metric $h_1$, thus
$(1-\varepsilon)h_0+\varepsilon h_1$ is {\sl strictly plurisubharmonic} and
{\sl strictly convex} for $\varepsilon$ small enough. Griffiths conjectured
that ampleness of $E$ might even be equivalent to the existence of a 
{\sl hermitian metric} with positive curvature, thus to the existence
of a hermitian strictly plurisubharmonic metric on~$E^\star$. Not much is
known about this conjecture, except that it holds true if $r=\rank E=1$
(Kodaira) and $n=\dim X=1$ (Umemura [Um73], see also Campana-Flenner [CaF90]).
Our feeling is that the general case should depend on deep facts of
gauge theory (some sort of vector bundle version of the Calabi-Yau theorem 
would be needed).}
\endclaim

\section{\S2. Linearly dual Finsler metrics}

Given a Finsler metric $\Vert\kesp\Vert$ on a holomorphic vector bundle
$E$, one gets a dual (or rather {\sl linearly dual}) Finsler metric 
$\Vert\kesp\Vert^\star$ on $E^\star$ by putting
$$
\Vert\xi^\star\Vert_x^\star=\sup_{\xi\in E_x\ssm\{0\}}
{|\xi\cdot\xi^\star|\over\Vert\xi\Vert_x},\qquad\xi^\star\in E^\star_x.
\leqno(2.1)
$$
Equivalently, in terms of the logarithmic indicatrix, we have
$$
\chi^\star(x,\xi^\star)=\sup_{\xi\in E_x\ssm\{0\}}
\log|\xi\cdot\xi^\star|-\chi(x,\xi),\qquad \xi^\star\in E^\star_x.
\leqno(2.2)
$$
It is clear that the linearly dual metric $\Vert\kesp\Vert^\star$ is always 
convex, and therefore the biduality formula 
$\Vert\kesp\Vert^{\star\star}=\Vert\kesp\Vert$
holds true if and only if $\Vert\kesp\Vert$ is convex. 

A basic observation made by Sommese [Som78] is that the
pseudoconvexity of a Finsler metric is related to some sort of
pseudoconcavity of the dual metric, provided that the given metric
is fiberwise convex. We will reprove it briefly in order to
prepare the reader to the general case (which requires {\sl
polynomial} duality, and not only {\sl linear} duality). We first need
a definition.

\claim 2.3. Definition|Let $E$ be equipped with a smooth Finsler metric
of logarithmic indicatrix $\chi(x,\xi)=\log\Vert\xi\Vert_x$.
We say that $\Vert\kesp\Vert$ has transversal Levi signature $(r,n)$ 
$($where $r=\rank E$ and $n=\dim X)$ if, at every point $(x,\xi)\in E\ssm
\{0\}$, the Levi form $i\ddbar(e^\chi)$ is positive definite along the
fiber $E_x$ and negative definite on some $n$-dimen\-sional subspace
$W\subset T_{E,(x,\xi)}$ which is transversal to the fiber~$E_x$.
\endclaim

This property can also be described geometrically as follows.

\claim 2.4. Proposition|The Finsler metric $\Vert\kesp\Vert$ on $E$
has transversal Levi signature $(r,n)$ if and only if it is fiberwise
strictly pseudoconvex, and through every point $(x_0,\xi_0)$ of the
unit sphere bundle \hbox{$\Vert\xi\Vert=1$}
passes a germ of complex $n$-dimensional submanifold $M_0$ which is
entirely contained in the unit ball bundle $\{\Vert\xi\Vert\le 1\}$
and has $($strict$)$ contact order $2$ at~$(x_0,\xi_0)$.
\endclaim

\proof\ {\sl(Sketch)}. If the geometric property (2.4) is satisfied,
we simply take $W=T_{M_0,(x_0,\xi_0)}$. Conversely, if $i\ddbar e^\chi$
has signature $(r,n)$ as in 2.3, then $i\ddbar\chi$ has signature
$(r-1,n)$ (with one zero eigenvalue in the radial direction, since
$\chi$ is $\log$ homogeneous). The Levi form of the hypersurface
$\chi=0$ thus has signature $(r-1,n)$ as well, and we can take
the negative eigenspace $W\subset T_{E,(x_0,\xi_0)}$ to be tangent to
that hypersurface. The germ $M_0$ is then taken to be the graph
of a germ of holomorphic section $\sigma:(X,x_0)\to E$ tangent to $W$,
with the second order jet of $\sigma$ adjusted in such a way
that $\chi(x,\sigma(x))\le-\varepsilon|x-x_0|^2$ $\big($as 
$\partial\chi(x_0)\ne 0$, one can eliminate the holomorphic and
antiholomorphic parts in the second order jet of
$\chi(x,\sigma(x))\big)$.\qed

One has the following basic result, the main part of which (a) is due 
to A.~Sommese [Som78].

\claim 2.5. Theorem|Let $E$ be equipped with a smooth Finsler metric
of logarithmic indicatrix $\chi(x,\xi)=\log\Vert\xi\Vert_x$. Assume that
the metric is $($fiberwise$)$ strictly convex. 
\smallskip
\item{\rm(a)} If the metric $\Vert\kesp\Vert$ is strictly plurisubharmonic
on $E\ssm\{0\}$, then the dual metric $\Vert\kesp\Vert^\star
=e^{\chi^\star}$ has transversal Levi signature $(r,n)$
on~$E^\star\ssm\{0\}$.
\smallskip
\item{\rm(b)}
In the opposite direction, if $\Vert\kesp\Vert$ has transversal Levi 
signature $(r,n)$, then $\Vert\kesp\Vert^\star$ is strictly plurisubharmonic 
on~$E^\star\ssm\{0\}$.
\vskip0pt
\endclaim

\claim 2.6. Remark|{\rm Theorem 2.5 still holds under the following more
general, but more technical hypothesis, in place of the strict convexity 
hypothesis: 
\smallskip
\noindent
(H) {\sl For every point $(x,[\xi^\star])\in P(E_x)$, the supremum 
$$
\chi^\star(x,\xi^\star)=\sup_{\xi\in E_x\ssm\{0\}}
\log|\xi\cdot\xi^\star|-\chi(x,\xi),\qquad \xi^\star\in E^\star_x
$$
is reached on a unique line $[\xi]=f(x,[\xi^\star])\in P(E_x)$, where
$[\xi]$ is a non critical maximum point along $P(E_x)$.}
\smallskip
Notice that the supremum is always reached in at least one
element $[\xi]\in P(E_x)$, just by compactness. The assumption that there
is a unique such point $[\xi]=f(x,[\xi^\star])$ which is non critical
ensures that $f$ is smooth by the implicit function theorem, hence
$\chi^\star$ will be also smooth.

The uniqueness assumption is indeed satisfied if the Finsler metric 
of $E$ is {\sl strictly convex}.  Indeed, if the maximum is reached for two
non colinear vectors $\xi_0$, $\xi_1$ and if we adjust $\xi_0$ and $\xi_1$
by taking multiples such that $\xi_0\cdot \xi^\star= \xi_1\cdot
\xi^\star=1$, then again $\xi_t\cdot \xi^\star=1$ for all
$\xi_t=(1-t)\xi_0+t\xi_1\in{}]\xi_0,\xi_1[$, while the strict
convexity implies
$\chi(x,\xi_t)<\chi(x,\xi_0)=\chi(x,\xi_1)$, contradiction.
We see as well that the maximum must be a non critical point, and that
the Finsler metric $\Vert\kesp\Vert^\star$ is strictly convex. Thus,
in this case, $\Vert\kesp\Vert$ is strictly plurisubharmonic if and only if
$\Vert\kesp\Vert^\star$ has transversal Levi signature $(r,n)$.}
\endclaim

\claim 2.7. Remark|{\rm In Theorem 2.5, the extra convexity assumption
(or its weaker counterpart (H)) is certainly needed. In fact, if the
conclusions were true without any further assumption, the linear
bidual of a continuous plurisubharmonic Finsler metric would still be
plurisubharmonic (since we can approximate locally such metrics by
smooth strictly plurisubharmonic ones). This would imply that the
convex hull of a pseudoconvex circled tubular neighborhood is
pseudoconvex. However, if we equip the trivial rank two vector bundle
$\bC\times\bC^2$ over $\bC$ with the plurisubharmonic Finsler metric
$$
\Vert\xi\Vert_x=\max\big(|\xi_1|,|\xi_2|,|x|\sqrt{|\xi_1|\,|\xi_2|}\big),
$$
a trivial computation shows that the convex hull is associated with the
metric
$$
\Vert\xi\Vert'_x=\max\Big(|\xi_1|,|\xi_2|,{|x|^2\over 1+|x|^2}
(|\xi_1|+|\xi_2|)\Big)
$$
which is not plurisubharmonic in~$x$.\qed}
\endclaim

\proof\ {\sl of Theorem 2.5.} (a) First observe that 
$\exp(\chi^\star)=\Vert\kesp\Vert^\star$ is convex, and
even strictly convex since the assumptions are not affected by small 
smooth $C^\infty$ or $C^2$ perturbations on $\chi$. Thus $i\ddbar
\exp(\chi^\star)$ has at least $r$ positive eigenvalues eigenvalues
along the vertical directions of $E^\star\to X$.

Let $f:P(E^\star)\to P(E)$ be defined as in condition 2.5~(H), and let 
$\wt f:E\ssm\{0\}^\star\to E\ssm\{0\}$ be a lifting of $f$. One can get
such a global lifting $\wt f$ by setting e.g.\ 
$\wt f(x,\xi^\star)\cdot\xi^\star=1$, so that $\wt f$ is uniquely defined.
By definition of $\chi^\star$ and $f$, we have
$$
\chi^\star(x,\xi^\star)=\log|\wt f(x,\xi^\star)\cdot\xi^\star|-
\chi(x,\wt f(x,\xi^\star))
$$
in a neighborhood of~$(x_0,\xi_0^\star)$. Fix a local trivialization
$E_{|U}\simeq U\times V$ xhere $V\simeq\bC^r$ and view $\wt f$ as a map 
$\wt f:E^\star_{|U}\simeq U\times V^\star\to V$ defined in a neighborhood
of~$(x_0,\xi_0^\star)$. As $\dim E=n+r$ and $\dim V=r$, the kernel of
the $\dbar$-differential
$$
\dbar\wt f_{(x_0,\xi_0^\star)}:T_{E^\star,(x_0,\xi_0^\star)}\to V
$$
is a complex subspace $W_0\subset T_{E^\star,(x_0,\xi_0^\star)}$ of dimension
$p\ge n$. By definition of~$W_0$, there is a germ of $p$-dimensional
submanifold $M\subset E^\star$ with $T_{M,(x_0,\xi_0^\star)}=W_0$, and a
germ of holomorphic function $g:M\to V$ such that 
$$
\wt f(x,\xi^\star)=g(x,\xi^\star)+O(|x-x_0|^2+|\xi^\star-\xi^\star_0|^2)
\quad\hbox{on~$M$}.
$$
This implies
$$
\chi^\star(x,\xi^\star)=\log|g(x,\xi^\star)\cdot\xi^\star|-
\chi(x,g(x,\xi^\star))+
O(|x-x_0|^3+|\xi^\star-\xi^\star_0|^3)\quad\hbox{on~$M$}.
$$
In fact, since $\xi_0=\wt f(x_0,\xi_0^\star)$ is a stationary point for
$\xi\mapsto\log|\xi\cdot\xi^\star|-\chi(x,\xi)$, the partial derivative 
in $\xi$ is $O(|x-x_0|+|\xi-\xi_0|)$, and a substitution
of $\xi=\wt f(x,\xi^\star)$ by $\xi_1=g(x,\xi^\star)$ introduces an error 
$$
O(|x-x_0|+|\xi-\xi_0|+|\xi_1-\xi_0|)\,|\xi-\xi_1|=
O(|x-x_0|^3+|\xi^\star-\xi^\star_0|^3)
$$ 
at most. Therefore
$$
i\ddbar\chi^\star(x,\xi^\star)=-i\ddbar\chi(x,g(x,\xi^\star))<0
\quad
\hbox{in restriction to $W_0=T_{M,(x_0,\xi_0^\star)}$}.
$$
This shows that $i\ddbar\chi^\star$ has at least $p\ge n$ negative
eigenvalues. As there are already $r$ negative eigenvalues, the only 
possibility is that $p=n$. 
\medskip\noindent
(b) The assumption on $(E,\chi)$ means that for every 
$(x_0,\xi_0)\in E\ssm\{0\}$, there is a germ of holomorphic section 
$\sigma:X\to E$ such that $-\chi(x,\sigma(x))$ is strictly 
plurisubharmonic and $\sigma(x_0)=\xi_0$. Fix $\xi_0^\star\in E^\star_{x_0}
\ssm\{0\}$ and take $\xi_0\in E_{x_0}\ssm\{0\}$ to be the unique point
where the maximum defining $\chi^\star$ is reached. Then we infer
that $\chi^\star(x,\xi^\star)\ge\log|\xi^\star\cdot\sigma(x)|-
\chi(x,\sigma(x))$, with equality at $(x_0,\xi_0^\star)$. An obvious
application of the mean value inequality then shows that $\chi^\star$
is plurisubharmonic and that $i\ddbar\chi^\star$ is strictly positive
in all directions of $T_{E^\star}$, except the radial vertical 
direction.\qed

\section{\S3. A characterization of signature (r,n) concavity}

Let $E$ be a holomorphic vector bundle equipped with a smooth Finsler metric
which satisfies the concavity properties exhibited by Theorem 2.5. We then
have the following results about supremum of plurisubharmonic functions.

\claim 3.1. Theorem|Assume that the Finsler metric $\Vert\kesp\Vert_E$ on
$E$ has transversal Levi signature $(r,n)$. Then, for every plurisubharmonic 
function $(x,\xi)\mapsto u(x,\xi)$ on the total space $E$, the function
$$
M_u(x,t)=\sup_{\Vert\xi\Vert_E\le|e^t|}u(x,\xi)
$$
is plurisubharmonic on $X\times\bC$.
\endclaim

\proof. Let us first consider the restriction $x\mapsto M_u(x,0)$,
and pick a point $x_0$ in $X$. Let $\xi_0\in E$, $\Vert\xi_0\Vert_E=1$ be 
a point such that $M_u(x_0,0)=u(x_0,\xi_0)$. By Proposition~2.4, there
a germ of holomorphic section $\sigma:(X,x_0)\to E$ such that 
$\sigma(x_0)=\xi_0$, whose graph is contained in the unit ball bundle
$\Vert\xi\Vert_E\le 1$. Thus $M_u(x,0)\ge u(x,\sigma(x))$ and
$u(x_0,\sigma(x_0))=M_u(x_0,0)$. This implies that $M_u(x,0)$ satisfies 
the mean value inequality at~$x_0$. As $x_0$ is arbitrary, we conclude
that $x\mapsto M_u(x,0)$ is plurisubharmonic. The plurisubharmonicity 
in $(x,t)$ follows by considering the pull-back of $E^\star$ to 
$X\times\bC$ by the projection $(x,t)\mapsto x$, equipped with the 
Finsler metric $|e^{-t}|\,\Vert\xi\Vert_E$ at point~$(x,t)$.
We again have osculating holomorphic sections contained in the
unit ball bundle $\Vert\xi\Vert_E\le |e^t|$, and the conclusion
follows as before.\qed

\noindent
We now turn ourselves to the ``converse'' result:

\claim 3.2. Theorem|Let $\Vert\kesp\Vert_E$ be a smooth
Finsler metric on $E$ which is fiberwise strictly plurisubharmonic on
all fibers~$E_x$. Assume that $X$ is Stein and that
$$
M_u(x,t)=\sup_{\Vert\xi\Vert_E\le|e^t|}u(x,\xi)
$$
is plurisubharmonic on $X\times\bC$ for every
plurisubharmonic function $(x,\xi)\mapsto u(x,\xi)$ on the total space $E$.
Then the Levi form of $\Vert\kesp\Vert_E$ has at least $n$ seminegative
eigenvalues, in other words $\Vert\kesp\Vert_E$ is, locally over $X$, a 
limit of smooth Finsler metrics of transversal Levi signature $(r,n)$.
\endclaim

\proof. Once we know that there are at least $n$ seminegative eigenvalues,
we can produce metrics of signature $(r,n)$ by considering
$$
(x,\xi)\mapsto\Vert\xi\Vert_E\,e^{-\varepsilon|x|^2},\qquad
\varepsilon>0
$$
in any coordinate patch, whence the final assertion. Now, assume that
the Levi form of $\Vert\kesp\Vert_E$ has at least $(r+1)$ positive
eigenvalues at some point $(x_0,\xi_0)\in E$. Then the direct sum of 
positive eigenspaces in $T_{E,(x_0,\xi_0)}$ projects to a positive
dimensional subspace in~$T_{X,x_0}$. Consider a germ of smooth complex
curve $\Gamma\subset X$ passing through $x_0$, such that its tangent at
$x_0$ is contained in that subspace. Then (after shrinking $\Gamma$ if
necessary) the restriction of the metric $\Vert\kesp\Vert_E$ to 
$E_{|\Gamma}$ is strictly plurisubharmonic. By the well-known properties 
of strictly pseudoconvex domains the unit ball bundle $\Vert\xi\Vert_E<1$
admits a peak function $u$ at $(x_0,\xi_0)$, that is, there is a smooth 
strictly plurisuharmonic function $u$ on $E_{|\Gamma}$ which is equal to 
$0$ at $(x_0,\xi_0)$ and strictly negative on the set 
$\{(x,\xi)\ne(x_0,\xi_0)\,;\,\Vert\xi\Vert\le1\}$. As $u$ is smooth, we 
can extend it to $E_{|B}$, where $B=B(x_0,\delta)$ is a small ball 
centered at $x_0$. As $X$ is Stein, we can even extend it to $E$, possibly
after shrinking $B$. Now $M_u(x,0)$ is equal to $0$ at $x_0$ and strictly
negative elsewhere on the curve $\Gamma$. This contradicts the maximum
principle and shows that $M_u$ cannot be plurisubharmonic. Hence the
assumption was absurd and the Levi form of $\Vert\kesp\Vert_E$ has at
least $n$ seminegative eigenvalues.\qed

\section{\S4. A conjecture of Hartshorne and Schneider on 
complements of algebraic subvarieties}
\sectionrunning{\S4. A conjecture of Hartshorne and Schneider}

Our study is closely connected to the following interesting 
(and unsolved) conjecture of R.~Hartshorne, which was first
partially confirmed by Michael Schneider [Sch73] 
in the case of a Griffiths positive normal bundle.

\claim 4.1. Conjecture|If $X$ is a projective $n$-dimensional manifold
and $Y\subset X$ is a complex submanifold of codimension $q$ with
ample normal bundle $N_Y$, then $X\ssm Y$ is $q$-convex in the sense
of Andreotti-Grauert. In other words, $X\ssm Y$ has a smooth
exhaustion function whose Levi form has at least $n-q+1$ positive
eigenvalues on a neighborhood of $Y$.
\endclaim

Using Sommese's result 2.5~(a), one can settle the following special
case of the conjecture.

\claim 4.2. Proposition (Sommese)|In addition to the hypotheses in the
conjecture, assume that $N_Y^\star$ has a {\rm strictly convex}
plurisubharmonic Finsler metric $($this is the case for instance if
$N_Y$ is generated by global sections$)$. Then $X\ssm Y$ is $q$-convex.
\endclaim

\proof. By adding $\varepsilon$ times a strictly plurisubharmonic Finsler
metric on $N_Y^\star$ (which exists thanks to the assumption that $N_Y$
is ample), we can even assume that the metric on $N_Y^\star$ is strictly 
convex {\sl and} strictly plurisubharmonic. Then the dual metric
on $N_Y$ has a Levi form of signature $(q,n-q)$. Let $\wt X\to X$ be the
blow-up of $X$ with center $Y$, and $\wt Y=P(N_Y^\star)$ the exceptional
divisor. Then, by Theorem 2.5, the Finsler metric on $N_Y$ corresponds
to a hermitian metric on 
$$
\cO_{P(N_Y^\star)}(-1)\simeq N_{\wt Y}=
\cO_{\wt X}({\wt Y})_{|\wt Y},
$$ 
whose curvature form has signature $(q-1,n-q)$ on $\wt Y$. Take an
arbitrary smooth extension of that metric to a metric of $\cO_{\wt X}
({\wt Y})$ on $\wt X$.  After multiplying the metric by a factor
of the form $\exp(C\,d(z,\wt Y)^2)$ in a neighborhood of $\wt Y$
(where $C\gg 0$ and $d(z,\wt Y)$ is the riemannian distance to $\wt Y$
with respect to some metric), we can achieve that the curvature of
$\cO_{\wt X}({\wt Y})$ acquires an additional negative eigenvalue in
the normal direction to $\wt Y$. In this way, the curvature form of
$\cO_{\wt X}({\wt Y})$ has signature $(q-1,n-q+1)$ in a neighborhood
of $\wt Y$. We let $\sigma_{\wt Y}\in H^0(\wt X,\cO_{\wt X}({\wt Y}))$
be the canonical section of divisor ${\wt Y}$. An exhaustion of $X\ssm
Y=\wt X\ssm\wt Y$ with the required properties is obtained by putting
$\psi(z)= -\log\Vert\sigma_{\wt Y}(z)\Vert$.\qed

\section{\S5. Symmetric and tensor products of Finsler metrics}

Let $E$ be a holomorphic vector bundle of rank~$r$. In the sequel, we 
consider the $m$-th symmetric product
$S^{m_1}E\times S^{m_2}E\to S^{m_1+m_2}E$ and the $m$-th symmetric 
power $E\to S^mE$, $\xi\mapsto\xi^m$, which we view as the result
of taking products of polynomials on~$E^\star$. We also use the
duality pairing 
$S^mE^\star\times S^mE\to \bC$, denoted by $(\theta_1,\theta_2^\star)\mapsto
\theta_1\cdot \theta_2^\star$. In multi-index notation, we have
$$
(e)^\alpha\cdot(e^\star)^\beta=\delta_{\alpha\beta}{\alpha!\over(|\alpha|)!},
$$
where $(e_j)_{1\le j\le r}$ is a basis of $E$, $(e^\star_j)_{1\le j\le r}$ 
the dual basis in~$E^\star$,
$1\le j\le r$, and \hbox{$(e)^\alpha=e_1^{\alpha_1}\ldots e_r^{\alpha_r}$}
[Caution: this formula implies that $\theta_1^p\cdot
\theta_2^{\star p}\ne(\theta_1\cdot\theta_2^\star)^p$ for general
elements $\theta_1\in S^mE$, $\theta_2\in S^mE^\star$, although
this is true if $m=1$].

Whilst the linear dual $\Vert\kesp\Vert_{E^\star}$ of a Finsler metric 
$\Vert\kesp\Vert_E$ is not well behaved if $\Vert\kesp\Vert_E$ is not convex,
we will see that (positive) symmetric powers and tensor powers can
always be equipped with natural well behaved Finsler metrics.
For an element $\theta^\star \in S^mE^\star$,
viewed as a homogeneous polynomial of degree $m$ on~$E$, we set
$$
\Vert\theta^\star\Vert_{S^mE^\star,L^\infty_{1,m}}
=\sup_{\xi\in E\ssm\{0\}}{|\theta^\star\cdot\xi^m|\over\Vert\xi\Vert_E^m}
=\sup_{\Vert \xi\Vert_E\le 1}|\theta^\star\cdot\xi^m|.
\leqno(5.1)
$$
[In the notation $L^\infty_{1,m}$, the upper index $\infty$ refers to the fact 
that we use $\sup$ norms, while the lower indices $1$ refers to the fact that
$\theta^\star$ appears with exponent $1$, and $\xi^m$ with exponent~$m$.]
This definition  just reduces to the definition of the dual metric in 
the case $m=1$, and thus need not be better behaved than the dual metric
from the view point of curvature. On the other hand, 
for all $\theta_i^\star\in S^{m_i}E^\star$, $i=1,2$, it satisfies the
submultiplicative law
$$
\Vert\theta_1^\star\,\theta_2^\star\Vert_{S^{m_1+m_2}E^\star,
L^\infty_{1,m_1+m_2}}\le
\Vert\theta_1^\star\Vert_{S^{m_1}E^\star,L^\infty_{1,m_1}}
\Vert\theta_2^\star\Vert_{S^{m_2}E^\star,L^\infty_{1,m_2}}.
$$
On the ``positive side'',
i.e.\ for $\tau\in S^mE$, we define a sequence of metrics 
$\Vert\kesp\Vert_{S^mE,L^\infty_{p,1}}$ on $S^mE$, $p\ge 1$, and their 
``limit'' $\Vert\kesp\Vert_{S^mE,L^\infty_{\infty,1}}$ by putting
$$
\leqalignno{
\Vert\tau\Vert_{S^mE,L^\infty_{p,1}}&=\sup_{\theta^\star\in 
S^{mp}E^\star\ssm\{0\}}
\Big({|\tau^p \cdot \theta^\star|\over\Vert\theta^\star
\Vert_{S^{pm}E^\star,L^\infty_{1,pm}}}\Big)^{1/p}&(5.2)\cr
&=\sup_{\Vert\theta^\star\Vert_{S^{mp}E^\star,L^\infty_{1,pm}}
\le 1}|\tau^p\cdot\theta^\star|^{1/p},\cr
\Vert\tau\Vert_{S^mE,L^\infty_{\infty,1}}
&=\limsup_{p\to+\infty}~~\Vert\tau\Vert_{S^mE,L^\infty_{p,1}}.
&(5.3)\cr}
$$
In the case $m=1$, we have of course $S^1E=E$, but neither
$\Vert\kesp\Vert_{S^1E,L^\infty_{p,1}}$ nor
$\Vert\kesp\Vert_{S^1E,L^\infty_{\infty,1}}$ necessarily coincide with the
original metric $\Vert\kesp\Vert_E$. In fact, by definition, it is
easily seen that the unit ball bundle
$\Vert\xi\Vert_{S^1E,L^\infty_{\infty,1}} \le 1$ is just the (fiberwise)
{\sl polynomial hull} of the ball bundle $\Vert\xi\Vert_E\le 1$. In
particular, $\Vert\kesp\Vert_{S^1E,L^\infty_{\infty,1}}$ and
$\Vert\kesp\Vert_E$ do coincide if and only if $\Vert\kesp\Vert_E$ is
plurisubharmonic on all fibers~$E_x$, which is certainly the case if
$\Vert\kesp\Vert_E$ is globally plurisubharmonic on~$E$ [By contrast,
the unit ball bundle $\Vert\xi\Vert_{S^1E,L^\infty_{1,1}}\le 1$ is the
convex hull of $\Vert\xi\Vert_E\le 1$, and need not be pseudoconvex
even if the latter is; see Remark~2.7]. Our first observation is

\claim 5.4. Proposition|The $L^\infty$ metric
$\Vert\kesp\Vert_{S^mE,L^\infty_{\infty,1}}$ is always well defined and
non degenerate $($in the sense that the $\limsup$ is finite and non
zero for $\tau\ne0)$, and it defines a continuous Finsler metric on
$S^mE$.
\endclaim

\proof. If in (5.2) we restrict $\theta^\star$ to be of the form
$\theta^\star=(\xi^\star)^{mp}$, then 
$$
\tau^p\cdot\theta^\star=(\tau\cdot\xi^{\star m})^p,
\qquad
\Vert(\xi^\star)^{mp}\Vert_{S^{mp}E^\star,L^\infty_{1,mp}}=
\Vert\xi^\star\Vert_{E^\star}^{mp},
$$
where $\Vert\kesp\Vert_{E^\star}$ is the linear dual of $\Vert\kesp\Vert_E$.
From this we infer
$$
\Vert\tau\Vert_{S^mE,L^\infty_{p,1}}\ge
\sup_{\Vert\xi^\star\Vert_{E^\star}\le 1}|\tau\cdot\xi^{\star m}|
$$
for all $p=1,2,\ldots,\infty$, in particular 
$\Vert\tau\Vert_{S^mE,L^\infty_{\infty,1}}$ is 
non degenerate. In the other direction, we have to show that 
$\Vert\tau\Vert_{S^mE,L^\infty_{\infty,1}}$ is finite.
We first make an explicit calculation
when $\Vert\kesp\Vert_E$ is a hermitian norm. We may assume $E=\bC^r$
with its standard hermitian norm. Then, writing $\theta^\star\cdot\xi^m=
\sum_{|\alpha|=m}c_\alpha\xi^\alpha$ in multi-index notation, we get
$$
\Vert\theta^\star\Vert_{S^mE^\star,L^\infty_{1,m}}^2=\sup_{\Vert\xi\Vert=1}
\Big|\sum_{|\alpha|=m}c_\alpha\xi^\alpha\Big|^2\ge
\sup_{t_1+\cdots+t_r=1}~~\sum_{|\alpha|=m}|c_\alpha|^2t^\alpha.
$$
This is obtained by integrating over the $n$-torus $\xi_j=t_j^{1/2}e^{iu_j}$, 
$0\le u_j<2\pi$ (with $t=(t_j)$ fixed, $\sum t_j=1$), and applying 
Parseval's formula. We can now replace the right hand supremum by the average
over the $(n-1)$-simplex $\sum t_j=1$. A~short computation yields
$$
\Vert\theta^\star\Vert_{S^mE^\star,L^\infty_{1,m}}^2\ge
{(r-1)!\over(m+1)(m+2)\ldots(m+r-1)}\sum_{|\alpha|=m}
|c_\alpha|^2{\alpha!\over(|\alpha|)!}.
$$
However $\sum_{|\alpha|=m}|c_\alpha|^2{\alpha!\over(|\alpha|)!}$ is just the
hermitian norm on $S^mE^\star$ induced by the inclusion $S^mE^\star\subset 
(E^\star)^{\otimes m}$. The dual norm is the hermitian norm on~$S^mE$. 
From this, we infer
$$
\eqalign{
\Vert\tau\Vert_{S^mE,L^\infty_{p,1}}
&\le
\Big({(mp+1)(mp+2)\ldots(mp+r-1)\over(r-1)!}\Big)^{1/2p}
\Vert\tau^p\Vert_{S^{mp}E,{\rm herm}}^{1/p},\cr
\Vert\tau\Vert_{S^mE,L^\infty_{\infty,1}}
&\le\Vert\tau\Vert_{S^mE,{\rm herm}}\cr}
$$
[using the obvious fact that hermitian norms are submultiplicative], whence
the finiteness of $\Vert\tau\Vert_{S^mE,L^\infty_{\infty,1}}$.
Finally, given any two Finsler metrics ${}_1\Vert\kesp\Vert_E$ and
${}_2\Vert\kesp\Vert_E$ such that
${}_1\Vert\kesp\Vert_E\le{}_2\Vert\kesp\Vert_E$, it is clear that
${}_1\Vert\tau\Vert_{S^mE,L^\infty_{\infty,1}}\le
{}_2\Vert\tau\Vert_{S^mE,L^\infty_{\infty,1}}$.
By comparing a given Finsler norm $\Vert\kesp\Vert_E={}_1\Vert\kesp\Vert_E$
with a hermitian norm ${}_2\Vert\kesp\Vert_E$, we conclude
that the metric $\Vert\kesp\Vert_{S^mE,L^\infty_{\infty,1}}$ must be finite.
Moreover, comparing the metrics $\Vert\kesp\Vert_E$ at nearby points, we
see that $\Vert\kesp\Vert_{S^mE,L^\infty_{\infty,1}}$
varies continuously (and that it depends continuously on 
$\Vert\kesp\Vert_E$).\qed

Our next observation is that the $L^\infty$ metrics on the negative 
symmetric powers $S^mE^\star$ can be replaced by $L^2$ metrics without
changing the final metric $\Vert\kesp\Vert_{S^mE,L^\infty_{\infty,1}}$ on 
$S^mE$. To see this, fix an arbitrary smooth positive volume form 
$dV$ on $P(E_x^\star)$, with (say) $\int_{P(E_x^\star)}dV=1$.
We can view any element $\theta^\star\in S^mE^\star$ as a section of
$H^0(P(E_x^\star),\cO_{P(E_x^\star)}(m))$. Let 
$\Vert\theta^\star\Vert_{\cO(m)}^2$ be the pointwise norm on 
$\cO_{P(E^\star)}(m)$ induced by $\Vert\kesp\Vert_E$, and let
$d\sigma$ be the area measure on the unit sphere bundle
$\Sigma(E)$ induced by $dV$. We then set
$$
\Vert\theta^\star\Vert_{S^mE^\star,L^2_{1,m}}^2
=\int_{P(E_x^\star)}\Vert\theta^\star\Vert_{\cO(m)}^2dV
=\int_{\xi\in\Sigma(E_x^\star)}|\theta^\star\cdot\xi^m|^2d\sigma(\xi).
$$
Clearly
$$
\Vert\theta^\star\Vert_{S^mE^\star,L^2_{1,m}}\le
\Vert\theta^\star\Vert_{S^mE^\star,L^\infty_{1,m}}.
$$
On the other hand, there exists a constant $C$ such that
$$
\Vert\theta^\star\Vert^2_{S^mE^\star,L^\infty_{1,m}}\le
C\,m^{r-1}\Vert\theta^\star\Vert^2_{S^mE^\star,L^2_{1,m}}.\leqno(5.5)
$$
This is seen by applying the mean value inequality for subharmonic 
functions, on balls of radius${}\sim 1/\sqrt{m}$ centered at arbitrary
points in~$P(E_x)$.
In fact, in a suitable local trivialization of $\cO_{P(E^\star)}$
near a point $[\xi_0]\in P(E_x^\star)$, we can write 
$\Vert\theta^\star\Vert^2_{\cO(m)}=|\theta_0^\star|^2e^{-m\psi}$ where 
$\theta_0^\star$ is the holomorphic function representing $\theta^\star$, and
$\psi$ is the weight of the metric on $\cO_{P(E^\star)}(1)$. We let
$\ell$ be the holomorphic part in the first jet of $\psi$ at $[\xi_0]$,
and apply the mean value inequality to 
$$
|\theta_0^\star e^{-m\ell}|^2e^{-m(\psi-2\Re\ell)}.
$$
As $\psi-2\Re\ell$ vanishes at second order at $[\xi_0]$, its maximum on
a ball of radius $1/\sqrt{m}$ is $O(1/m)$. Hence, up to a constant independent
of $m$, we can replace $|\theta_0^\star e^{-m\ell}|^2e^{-m(\psi-2\Re\ell)}$
by the subharmonic function $|\theta_0^\star e^{-m\ell}|^2$. Inequality
(5.5) then follows from the mean value inequality on the ball 
$B([\xi_0],1/\sqrt{m})$ [noticing that the volume of this ball
is${}\sim{1/m^{r-1}}$]. Now (5.5) shows that the replacement of 
$\Vert\theta^\star\Vert_{S^{pm}E^\star,L^\infty_{1,pm}}$ by
$\Vert\theta^\star\Vert_{S^{pm}E^\star,L^2_{1,pm}}$ in (5.2) and (5.3)
does not affect the limit as $p$ tends to~$+\infty$.

If $\Vert\kesp\Vert_E$ is (globally) plurisubharmonic, we can even use more
global $L^2$ metrics without changing the limit. Take a small Stein 
open subset $U\compact X$ and fix a K\"ahler metric $\omega$ on 
$P(E^\star_{|U})$. To any section $\sigma\in H^0(\pi^{-1}(U),
\cO_{P(E^\star)}(m))=H^0(U,S^mE^\star)$, we associate the $L^2$ norm
$$
\Vert\sigma\Vert^2_{S^mE^\star,L^2_{1,m}(U)}=
\int_{\pi^{-1}(U)}|\sigma|^2dV_\omega,
$$
where $\pi:P(E^\star)\to X$ is the canonical projection. In this way, 
we obtain a Hilbert space 
$$
\cH_{E,m}(U)=\big\{\sigma\,;~\Vert\sigma\Vert^2_{S^mE^\star,L^2_1(U)}
<+\infty\big\}\subset H^0(\pi^{-1}(U),\cO_{P(E^\star)}(m)),
$$
and associated (non hermitian!) metrics
$$
\eqalign{
\Vert\tau\Vert_{S^mE_x,L^2_{p,1}(U)}
&=\sup_{\sigma\in\cH_{E,mp}(U),\,\Vert\sigma\Vert\le 1}
|\sigma(x)\cdot\tau^p|^{1/p},\cr
\Vert\tau\Vert_{S^mE_x,L^2_{\infty,1}(U)}
&=\limsup_{p\to+\infty}~~\Vert\tau\Vert_{S^mE_x,L^2_{p,1}(U)},\qquad
\tau\in S^mE^\star_x.
\cr}
$$
As these metrics are obtained by taking sups of plurisubharmonic functions
($(x,\tau)\mapsto\sigma(x)\cdot\tau^p$ is holomorphic on the total space 
of $S^mE$), it is clear that the corresponding metrics are plurisubharmonic
on $S^mE$. Furthermore, an argument entirely similar to the one used for 
(5.5) shows that
$$
\Vert\sigma(x)\Vert^2_{S^mE^\star_x,L^2_{1,m}}\le C\,m^n\,
\Vert\sigma\Vert^2_{S^mE^\star,L^2_{1,m}(U)}\qquad
\forall x\in U'\compact U.
$$
In order to get this, we apply the mean value inequality on
balls of radius $1/\sqrt{m}$ centered at points of the fiber 
$P(E^\star_x)$ and transversal to that fiber,
in combination with Fubini's theorem. In the other direction, the 
Ohsawa-Takegoshi $L^2$ extension theorem ([OhT87], [Ohs88], [Man93]),
shows that every element $\theta^\star\in S^mE^\star_x$, viewed as a
section of $\cO_{P(E^\star_x)}(m)$, can be extended to a section
$\sigma\in H^0(\pi^{-1}(U),\cO_{P(E^\star)}(m))$ such that
$$
\Vert\sigma\Vert_{S^mE^\star,L^2_{1,m}(U)}\le 
C'\Vert\theta^\star\Vert_{S^mE^\star_x,L^2_{1,m}},
$$
where $C'$ does not depend on $x\in U$. For this, we use the fundamental 
assumption that $\Vert\kesp\Vert_E$ is plurisubharmonic (and take profit of 
the fact that $\cO_{P(E^\star)}(1)$ is relatively ample to get enough
positivity in the curvature estimates: write e.g.\
$\cO_{P(E^\star)}(m)=\cO_{P(E^\star)}(m-m_0)\otimes\cO_{P(E^\star)}(m_0)$,
keep the original metric on the first factor $\cO_{P(E^\star)}(m-m_0)$, 
and put a metric with uniformly positive curvature on the second
factor). From this, we conclude that $\Vert\tau\Vert_{S^mE_x,
L^2_{\infty,1}(U)}$ coincides with the metric defined in (5.3). Since this
metric depends {\sl in fine} only on $\Vert\kesp\Vert_E$, we will simply
denote it by $\Vert\kesp\Vert_{S^mE}$. We have thus proven:

\claim 5.6. Theorem|If $\Vert\kesp\Vert_E$ is $($strictly$)$ 
plurisubharmonic on $E$, then $\Vert\kesp\Vert_{S^mE}$ is 
$($strictly$)$ plurisubharmonic on $S^mE$.
\endclaim

The case of strict plurisubharmonicity can be handled by more or less
obvious perturbation arguments and will not be detailed here. As a
consequence, we get the Finsler metric analogue of the fact that a 
direct sum or tensor product of ample vector bundles is ample.

\claim 5.7. Corollary|If $E$, $F$ are holomorphic vector bundles,
and $\Vert\kesp\Vert_E$, $\Vert\kesp\Vert_F$ are $($strictly$)$ 
plurisubharmonic Finsler metrics on $E$, $F$, there exist naturally defined
$($strictly$)$ plurisubharmonic Finsler metrics $\Vert\kesp\Vert_{E\oplus F}$,
$\Vert\kesp\Vert_{E\otimes F}$ on $E\oplus F$, $E\otimes F$ respectively.
\endclaim

\proof. In the case of the direct sum, we simply set 
$\Vert\xi\oplus\eta\Vert_{E\oplus F}=\Vert\xi\Vert_E+\Vert\eta\Vert_F$.
The logarithmic indicatrix is given by
$$
\chi_{E\oplus F}(x,\xi,\eta)=
\log\big(\exp(\chi_E(x,\xi))+\exp(\chi_F(x,\eta))\big),
$$
and it is clear from there that $\chi_{E\oplus F}$ is plurisubharmonic.
Now, we observe that $S^2(E\oplus F)=S^2E\oplus S^2F\oplus (E\otimes F)$.
Hence $E\otimes F$ can be viewed as a subbundle of $S^2(E\oplus F)$. To
get the required Finsler metric on $E\otimes F$, we just apply Theorem 5.5
to $S^2(E\oplus F)$ and take the induced metric on $E\otimes F$.\qed

\claim 5.8. Remark|{\rm It would be interesting to know whether good
Finsler metrics could be defined as well on the dual symmetric powers
$S^mE^\star$. One natural candidate would be to use the already
defined metrics $\Vert\kesp\Vert_{S^mE}$ and to set
$$
\eqalign{
\Vert\tau^\star\Vert_{S^mE^\star,L^\infty_{p,1}}
&=\sup_{\Vert\theta\Vert_{S^{pm}E}\le 1}|\tau^{\star p}\cdot\theta|^{1/p},\cr
\Vert\tau^\star\Vert_{S^mE^\star,L^\infty_{\infty,1}}
&=\limsup_{p\to+\infty}~~\Vert\tau^\star\Vert_{S^mE^\star,L^\infty_{p,1}}.\cr}
$$
However, we do not know how to handle these ``bidually defined'' Finsler 
metrics, and the natural question whether
$\Vert\kesp\Vert_{S^mE^\star,L^\infty_{\infty,1}}$ has transversal signature
$(\dim S^mE^\star,n)$ probably has a negative answer if $\Vert\kesp\Vert_E$
is not convex (although this might be ``asymptotically true'' as
$m$ tends to~$+\infty$).}
\endclaim

\claim 5.9. Relation to cohomology vanishing and duality theorems|{\rm If
$\Vert\kesp\Vert_{E^\star}$ is smooth and strictly plurisubharmonic, then
$E$ is ample, thus its symmetric powers $S^mE$ have a lot of sections
and the Kodaira-Serre vanishing theorem holds true, i.e.\
$$
H^q(X,S^mE\otimes\cF)=0,\qquad q\ne 0,
$$
for every coherent sheaf~$\cF$ and $m\ge m_0(\cF)$ large enough. In a parallel
way, if $\Vert\kesp\Vert_E$ has a metric of signature $(r,n)$, then the
line bundle $\cO_{P(E^\star)}(1)$ has a hermitian metric such that the
curvature has signature $(r-1,n)$ over $P(E^\star)$. From this, by the
standard Bochner technique, we conclude that
$$
H^q(P(E^\star),\cO_{P(E^\star)}(m)\otimes\cG)=0,\qquad q\ne n,
$$
for every locally free sheaf $\cG$ on $P(E^\star)$ and $m\ge m_0(\cG)$. 
The  Leray spectral sequence shows that
$$
H^q(X,S^mE^\star\otimes\cF)=H^q(P(E^\star),\cO_{P(E^\star)}(m)
\otimes\pi^\star\cF),
$$
thus we have vanishing of this group as well is $\cF$ is locally free and
$q\ne n$, $m\ge m_0(\cF)$. The Serre duality theorem connects the two facts 
via an isomorphism
$$
H^q(X,S^mE^\star\otimes\cF)^\star=
H^{n-q}(X,S^mE\otimes\cF^\star\otimes K_X).
$$
What we are looking for, in some sense, is a ``Finsler metric version'' of
the Serre duality theorem. Up to our knowledge, the duality works well only 
for convex Finsler metrics (and also asymptotically, for high symmetric 
powers $S^mE$ which carry positively curved hermitian metrics).\qed
}
\endclaim

\section{\S6. A trick on Taylor series}

Let $\pi:E\to X$ be a holomorphic vector bundle, such that $E^\star$ is
equipped with a continuous plurisubharmonic Finsler metric 
$\Vert\xi^\star\Vert_{E^\star}=\exp(\chi^\star(x,\xi^\star))$.
Thanks to section 5, we are able to define plurisubharmonic Finsler
metrics $\Vert\kesp\Vert_{S^mE^\star}$ on all symmetric powers of $E^\star$.
Our goal is to use these metrics in order to define plurisubharmonic
sup functionals for holomorphic or plurisubharmonic functions.
We first start with the simpler case when $\Vert\kesp\Vert_{E^\star}$ is
convex.

\claim 6.1. Theorem|Assume that $\Vert\kesp\Vert_{E^\star}$ is 
plurisubharmonic and convex, and let $\Vert\xi\Vert_E=
\exp(\chi(x,\xi))$ be the $($linearly$)$ dual metric. Then, for 
every plurisubharmonic function $(x,\xi)\mapsto
u(x,\xi)$ on the total space $E$, the function
$$
M^\chi_u(x,t)=\sup_{\Vert\xi\Vert_E\le|e^t|}u(x,\xi)
$$
is plurisubharmonic on $X\times\bC$.
\endclaim

\proof. This is a local result on $X$, so we can assume that $X$ is
an open set $\Omega\subset\bC^n$ and that $E=\Omega\times\bC^r$ is
trivial. By the standard approximation techniques, we can approximate
$\Vert\kesp\Vert_{E^\star}$ by smooth strictly convex and strictly 
plurisubharmonic metrics ${}_\varepsilon\Vert\kesp\Vert_{E^\star}\ge
\Vert\kesp\Vert_{E^\star}$ which decrease to $\Vert\kesp\Vert_{E^\star}$
as $\varepsilon$ decreases to~$0$.
We then get a decreasing family $\mathop{\lim{\downarrow}}
_{(\varepsilon\to 0)}{}_\varepsilon M_u(x,t)=M_u(x,t)$. It is thus
enough to treat the case of smooth strictly convex and strictly 
plurisubharmonic metrics $\Vert\kesp\Vert_{E^\star}$. In that
case, $\Vert\kesp\Vert_E$ has a Levi form of signature $(r,n)$
and we conclude by Theorem~3.1.\qed

Unfortunately, in the general case when $\Vert\kesp\Vert_{E^\star}$ is 
not convex, this simple approach does not work [in the sense that
$M_u$ is not always plurisubharmonic]. We circumvent this difficulty
by using instead the well-known trick of Taylor expansions, and replacing
the sup with a more sophisticated evaluation of norms.
If $f$ is a holomorphic function on the total space of $E$,
the Taylor expansion of $f$ along the fibers of $E$ can be written as
$$
f(x,\xi)=\sum_{m=0}^{+\infty}a_m(x)\cdot\xi^m,\qquad \xi\in E_x,
$$
where $a_m$ is a section in $H^0(X,S^mE^\star)$. In that case, we set
$$
\wh M^\chi_f(x,t)=\sum_{m=0}^{+\infty}\Vert a_m(x)\Vert_{S^mE^\star}|e^{mt}|.
\leqno(6.2)
$$
This is by definition a plurisubharmonic function on $X\times \bC$.
In fact, $\log\wh M^\chi_f(x,t)$ is a plurisubharmonic function as well. 
As we will see in the following lemma, $\wh M^\chi_f$ will play essentially
the same role as $M_{|f|}$ could have played.

\claim 6.3. Lemma|Fix a hermitian metric 
$\Vert\kesp\Vert_{E^\star,{\rm herm}}\ge\Vert\kesp\Vert_{E^\star}$, and
let $\Vert\kesp\Vert_{E,{\rm herm}}$ be the dual metric.
Then there is an inequality
$$
\sup_{\Vert\xi\Vert_E\le|e^t|}|f(x,\xi)|\le \wh M^\chi_f(x,t)\le 
\Big(1+{1\over\varepsilon}\Big)^r
\sup_{\Vert\xi\Vert_{E,{\rm herm}}\le (1+\varepsilon)|e^t|}|f(x,\xi)|.
$$
\endclaim

\proof. The left hand inequality is obtained by expanding
$$
f(x,\xi)\le\sum_{m=0}^{+\infty}|a_m(x)\cdot\xi^m|
\le\sum_{m=0}^{+\infty}\Vert a_m(x)\Vert_{S^mE^\star}\Vert\xi\Vert_E^m,
$$
thanks to the fact that $a_m(x)^p\cdot\xi^{mp}=(a_m(x)\cdot\xi^m)^p$.
In the other direction, we have 
$$
\eqalign{
\Vert a_m(x)\Vert_{S^mE^\star}
&\le\Vert a_m(x)\Vert_{S^mE^\star,{\rm herm}}\cr
&\le{(m+1)\ldots(m+r-1)\over(r-1)!}
\sup_{\Vert\xi\Vert_{E,{\rm herm}}\le 1}|a_m(x)\cdot\xi^m|\cr}
$$
thanks to the inequalities obtained in the proof of Proposition 5.4. Now, the
standard Cauchy inequalities imply
$$
\sup_{\Vert\xi\Vert_{E,{\rm herm}}=1}|a_m(x)\cdot\xi^m|
\le {1\over R^m}\sup_{\Vert\xi\Vert_{E,{\rm herm}}=R}|f(x,\xi)|.
$$
Combining all the above with $R>|e^t|$, we get
$$
\eqalign{
\wh M^\chi_f(x,t)
&\le\sup_{\Vert\xi\Vert_{E,{\rm herm}}\le R}|f(x,\xi)|
\sum_{m=0}^{+\infty}{|e^{mt}|\over R^m}{(m+1)\ldots(m+r-1)\over (r-1)!}\cr
&\le{1\over \big(1-{|e^t|\over R}\big)^r}
\sup_{\Vert\xi\Vert_{E,{\rm herm}}\le R}|f(x,\xi)|
\le \Big(1+{1\over\varepsilon}\Big)^r
\sup_{\Vert\xi\Vert_{E,{\rm herm}}\le (1+\varepsilon)|e^t|}|f(x,\xi)|.\cr}
$$
The lemma is proved.\qed

\claim 6.4. Remark|{\rm It is clear that the sup functional $M^\chi_{|f|}$ is 
submultiplicative, i.e.\
$$
M^\chi_{|fg|}(x,t)\le M^\chi_{|f|}(x,t)\,M^\chi_{|g|}(x,t).
$$ 
However, the analogous property for $\wh M^\chi_f$ would require to know 
whether 
$$
\Vert a\cdot b\Vert_{S^{m_1+m_2}E^\star}\le
\Vert a\Vert_{S^{m_1}E^\star}\,\Vert b\Vert_{S^{m_2}E^\star},
$$
(or a similar inequality with a constant $C$ independent of $m_1$, $m_2$).
It is not clear whether such a property is true, since the precise
asymptotic behaviour of the metrics $\Vert\kesp\Vert_{S^mE^\star}$
is hard to understand. In order to circumvent this problem, we 
select a non increasing sequence of real numbers $\rho_m\in{}]0,1]$
with $\rho_0=1$, such that
$$
\rho_{m_1+m_2}\Vert a\cdot b\Vert_{S^{m_1+m_2}E^\star}\le
\rho_{m_1}\rho_{m_2}\Vert a\Vert_{S^{m_1}E^\star}\Vert b\Vert_{S^{m_2}E^\star}
\leqno(6.5)
$$
for all $m_1$, $m_2$. One can easily find such a sequence $\rho=(\rho_m)$
by induction on~$m$, taking $\rho_m/\rho_{m-1}$ small enough. Then
$$
\wh M^{\chi,\rho}_f(x,t):=\sum_{m=0}^{+\infty}
\rho_m\Vert a_m(x)\Vert_{S^mE^\star}|e^{mt}|
\leqno(6.6)
$$
obviously satisfies the submultiplicative property. On the other hand,
we lose the left hand inequality in Lemma 6.3. This unsatisfactory 
feature will create additional difficulties which we can only
solve at the expense of using deeper analytic techniques.\qed}
\endclaim

We are mostly interested in the case when $E=T_X$ is the tangent
bundle, and assume that a plurisubharmonic Finsler metric
$\Vert\xi^\star\Vert_{T^\star_X}=\exp(\chi^\star(x,\xi^\star))$ is 
given. Locally, on a small coordinate open set 
$U\compact U_0\subset X$ associated with a holomorphic chart 
$$
\tau:U_0\to \tau(U_0)\subset\bC^n,
$$
we have a corresponding trivialization 
$\tau':T_{X|U}\simeq \tau(U)\times\bC^n$. Given a holomorphic function 
$f$ in a neighborhood of $\ol U$, we consider the holomorphic 
function such that $F(x,\xi)=f(\alpha(x,\xi))$ where
$\alpha(x,\xi)=\tau^{-1}(\tau(x)+\tau'(x)\xi))$. It is 
defined on a sufficiently small ball bundle
$B_\varepsilon(T_{X|U})=\{(x,\xi)\in T_{X|U}\,;\;
\Vert\xi\Vert<\varepsilon\}$, $\varepsilon>0$. Thus
$$
\wt M^{\chi,\rho}_f(x,t):=\wh M^{\chi,\rho}_F(x,t)
\leqno(6.7)
$$
makes sense for $|e^t|<c\varepsilon$, $c>0$. Again, by construction,
this is a plurisubharmonic function of $(x,t)$ on $U\times
\{|e^t|<c\varepsilon\}$. This function will be used as a replacement
of the sup of $f$ on the Finsler ball $\alpha(x,B(0,|e^t|))\subset X$ 
(which we unfortunately know nothing about). However, the definition
is not coordinate invariant, and we have to investigate the effect of
coordinate changes.

\claim 6.8. Lemma|Consider two holomorphic coordinate
coordinate charts $\tau_j$ on a neighborhood of $\ol U$, $j=1,2$,
and the corresponding maps 
$$
\alpha_j:B_\varepsilon(T_{X|U})\to U_0,\qquad
\alpha_j(x,\xi)=\tau_j^{-1}(\tau_j(x)+\tau_j'(x)\xi)).
$$
Let $F_j=f\circ\alpha_j$, $j=1,\,2$, and let $\delta>0$ be fixed.
Then there is a choice of a decreasing sequence $\rho=(\rho_m)$
such that
$$
\wh M^{\chi,\rho}_{F_2}(x,t)\le(1+\delta)\wh M^{\chi,\rho}_{F_1}(x,t),
$$
where $\rho$ depends on $U$, $\tau_1$, $\tau_2$, but not
on~$f$ $($here $|e^t|$ is suppose to be chosen small enough
so that both sides are defined$)$. Any sequence $\rho$ with
$\rho_m/\rho_{m-1}$ smaller that a given suitable sequence
of small numbers works.
\endclaim

In other words, if the sequence $(\rho_m)$ decays sufficiently fast,
the functional $\wt M^{\chi,\rho}_f(x,t)$ defined above can be chosen
to be ``almost'' coordinate invariant.

\proof. It is easy to check by the implicit function theorem that there 
exists a (uniquely) defined map $w:T_X\to T_X$, defined near the zero
section and tangent to the identity at~$0$, such that
$$
\alpha_2(x,\xi)=\alpha_1(x,w(x,\xi)),\qquad w(x,\xi)=\xi+O(\xi^2).
$$
Hence, if we write 
$$
f\circ\alpha_j(x,\xi)=\sum_{m=0}^{+\infty}a_{m,j}(x)\cdot\xi^m,
\qquad j=1,\,2,
$$
the series corresponding to index $j=2$ is obtained from the
$j=1$ series by substituting $\xi\mapsto w(x,\xi)$. It follows that
$$
a_{m,2}(x)=a_{m,1}(x)+\sum_{\mu<m}L_{m,\mu}(x)\cdot a_{\mu,1}(x)
$$
where $L_{m,\mu}:S^\mu T^\star_X\to S^mT^\star_X$ are certain holomorphic
linear maps depending only on the chart mappings $\tau_1$, $\tau_2$.
If $\rho_m/\rho_{m-1}$ is small enough, the contribution given by
$\rho_m\sum_{\mu<m}L_{m,\mu}(x)\cdot a_{\mu,1}(x)$ is negligible
compared to the $\rho_\mu\Vert a_{\mu,1}(x)\Vert_{S^\mu T^\star_X}$.
The lemma follows.\qed

\section{\S7. Approximation of plurisubharmonic functions by
logarithms of holomorphic functions}
\sectionrunning{\S7. Approximation of plurisubharmonic functions by
holomorphic functions}

The next step is to extend the $\wt M_f^{\chi,\rho}$ functional to
plurisubharmonic functions defined on a complex manifold, when the
cotangent bundle $T^\star_X$ is equipped with a Finsler metric. The
simplest way to do this is to approximate such functions by logarithms
of holomorphic functions, by means of the Ohsawa-Takegoshi $L^2$
extension theorem ([OhT87], [Ohs88], [Man93]). We reproduce here some
of the techniques introduced in [Dem92], but with substantial
improvements. The procedure is still local and not completely
canonical, so we will have later to apply a gluing procedure.

\claim 7.1. Theorem|Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$U\subset\bC^n$. For every $p>0$, let
$\cH_{p\varphi}(U)$ be the Hilbert space of holomorphic functions $f$
on $U$ such that $\int_U|f|^2e^{-2p\varphi}d\lambda<+\infty$ and
let \hbox{$\varphi_p={1\over 2p}\log\sum|\sigma_\ell|^2$} where 
$(\sigma_\ell)$ is an orthonormal basis of~$\cH_{p\varphi}(U)$. 
Then there are constants \hbox{$C_1,C_2>0$} independent of $p$ such that
\smallskip
\item{\rm (i)}$\displaystyle\varphi(z)-{C_1\over p}\le
\varphi_p(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over p}
\log{C_2\over r^n}$\hfill\break
for every $z\in U$ and $r<d(z,\partial U)$. In particular,
$\varphi_p$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$U$ when $p\to+\infty$ and
\smallskip
\item{\rm (ii)}$\displaystyle\nu(\varphi,z)-{n\over p}\le\nu(\varphi_p,z)\le
\nu(\varphi,z)$~ for every $z\in U$.
\endclaim

\proof. Note that $\sum|\sigma_\ell(z)|^2$ is the square of the norm
of the evaluation linear form $f\mapsto f(z)$ on $\cH_{p\varphi}(U)$. As
$\varphi$ is locally bounded above, the $L^2$ topology is actually stronger
than the topology of uniform convergence on compact subsets of~$U$.
It follows that the series $\sum|\sigma_\ell|^2$ converges uniformly on
$U$ and that its sum is real analytic. Moreover, we have
$$
\varphi_p(z)=\sup_{f\in B_p(1)}{1\over p}\log|f(z)|
\leqno(7.2)
$$
where $B_p(1)$ is the unit ball of $\cH_{p\varphi}(U)$. For
$r<d(z,\partial U)$, the mean value inequality applied to the
plurisubharmonic function $|f|^2$ implies
$$
\eqalign{
|f(z)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|\zeta-z|<r}
|f(\zeta)|^2d\lambda(\zeta)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2p\sup_{|\zeta-z|<r}\varphi(\zeta)\Big)
\int_U|f|^2e^{-2p\varphi}d\lambda.\cr}
$$
If we take the supremum over all $f\in B_p(1)$ we get
$$
\varphi_p(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over 2p}
\log{1\over\pi^nr^{2n}/n!}
$$
and the second inequality in (i) is proved. Conversely, the
Ohsawa-Takegoshi extension theorem ([OhT87], [Ohs88], [Man93]) applied
to the $0$-dimensional subvariety $\{z\}\subset U$ shows that for any
$a\in\bC$ there is a holomorphic function $f$ on $U$ such that
$f(z)=a$ and
$$
\int_U|f|^2e^{-2p\varphi}d\lambda\le C_3|a|^2e^{-2p\varphi(z)},
$$
where $C_3$ only depends on $n$ and~${\rm diam}\,U$. We fix $a$
such that the right hand side is~$1$. This gives the other
inequality
$$\varphi_p(z)\ge{1\over p}\log|a|=\varphi(z)-{\log C_3\over 2p}.$$
The above inequality implies $\nu(\varphi_p,z)\le\nu(\varphi,z)$. In the
opposite direction, we find
$$\sup_{|x-z|<r}\varphi_p(x)\le\sup_{|\zeta-z|<2r}\varphi(\zeta)+
{1\over p}\log{C_2\over r^n}.$$
Divide by $\log r$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a plurisubharmonic function over $B(x,r)$ 
tends to the Lelong number at~$x$. Thus we obtain
$$\nu(\varphi_p,x)\ge\nu(\varphi,x)-{n\over p}.\eqno\square$$

Another important fact is that the approximations $\varphi_p$ do no depend
much on the open set~$U$, and they have a good dependence on $\varphi$
under small perturbations. In fact, let $U',U''\subset U$ be Stein open
subsets, and let $\varphi'$, $\varphi''$ be plurisubharmonic functions
on $U'$, $U''$ such that $|\varphi'-\varphi''|\le \varepsilon$ 
on~$U'\cap U''$. If $f'$ is a function in the unit ball of 
$\cH_{p\varphi}(U')$, then
$$
\int_{U'\cap U''}|f'|^2e^{-2p\varphi''}d\lambda\le e^{2p\varepsilon}
$$
by the hypothesis on $\varphi'-\varphi''$. For every $x_0\in U'$, we 
can find a function $f''\in\cH_{p\varphi''}(U)$ such that 
$f''(x_0)=f'(x_0)$ and
$$
\int_{U''}|f''|^2e^{-2p\varphi''}d\lambda\le
{C\over (d(x_0,\complement U))^{2n+2}}e^{2p\varepsilon}
\int_{U'}|f'|^2e^{-2p\varphi'}d\lambda.
$$
This is done as usual, by solving the equation $\dbar g =
\dbar(\theta f')$ with a cut-off function $\theta$ supported in the
ball $B(x_0,\delta/2)$ and equal to $1$ on $B(x_0,\delta/4)$,
$\delta=d(x_0,\complement U)$, with the weight 
$2p\varphi(z)+2n\log|z-x_0|\,$; the desired function is then 
$f''=\theta f'-g$.
By readjusting $f''$ by a  constant so that $f''$ is in the unit sphere,
and by taking the sup of $\log|f'(x_0)|$ and $\log|f''(x_0)|$ for all 
$f'$ and $f''$ in the unit ball of their respective Hilbert spaces, we 
conclude that
$$
\varphi_p'(x)\le\varphi''_p(x)+\varepsilon+{1\over 2p}\log{C\over 
d(x,\complement(U'\cap U''))^{2n+2}}
\qquad\hbox{on $U'\cap U''$},
$$
with some constant $C>0$ depending only on the pair $(U',U'')$.
By symmetry, we get
$$
|\varphi_p'(x)-\varphi''_p(x)|
\le\varepsilon+{1\over 2p}\log{C\over d(x,\complement(U'\cap U''))^{2n+2}}
\qquad\hbox{on $U'\cap U''$}.
\leqno(7.3)
$$

The next idea would be to take Taylor series much in the same way as we
did in \S$\,$6, and look e.g.\ at
$$
\Phi^{\chi,\rho,p}(x,t)=
\sup_{f\in B_p(1)}{1\over p}\log\wt M_f^{\chi,\rho}(x,t).
$$
The main problem with this approach occurs when we want to check the effect
of a change of coordinate patch. We then want to compare the jets with
those of the functions $f$ obtained on another coordinate patch, say up
to an order $Cp$ for $C\gg 0$ large. The comparison would be easy (by the
usual H\"ormander-Bombieri $\dbar$-technique, as we did for the $0$-jets
in (7.3)) for jets of small order in comparison to $p$, but going to such 
high orders introduces intolerable distortion in the required bounds. A 
solution to this problem is to introduce further approximations of 
$\varphi_p$ for which we have better control on the jets. This can be
done by using Skoda's $L^2$-estimates for surjective bundle morphisms
[Sko72b, 78]. This approach was already used in [Dem92], but in a less
effective fashion.

Let $K_p^U:\varphi\mapsto\varphi_p$ be the transformation defined above.
This transformation has the effect of converting the singularities of
$\varphi$, which are a priori arbitrary, into logarithmic 
analytic singularities (and, as a side effect, the multiplicities 
get discretized, with values in ${1\over p}\bN$). We simply iterate
the process twice, and look at
$$
\varphi_{p,q}=K_{pq}^U(K_p^U(\varphi))
\leqno(7.4)
$$
for large integers $q\gg p\gg 1$. In other words,
$$
\varphi_p(z)={1\over 2p}\log\sum|\sigma_\ell(z)|^2,\qquad
\varphi_{p,q}(z)={1\over 2pq}\log\sum|\wt\sigma_\ell(z)|^2
$$
where $\sigma=(\sigma_\ell)_{\ell\in\bN}$ and 
$\wt\sigma=(\wt\sigma_\ell)_{\ell\in\bN}$ 
are Hilbert bases of the $L^2$ spaces
$$
\cH_{p\varphi}(U)=\Big\{
\int_U|f|^2e^{-2p\varphi}d\lambda<+\infty\Big\},\qquad\kern-2pt
\cH_{pq\varphi_p}(U)=\Big\{
\int_U|f|^2|\sigma|^{-2q}d\lambda<+\infty\Big\}.
$$
Theorem 7.1 shows that we still have essentially the same estimates
for $\varphi_{p,q}$ as we had for $\varphi_p$, namely
$$
\leqalignno{
&\varphi(z)-{C_1\over p}\le
\varphi_{p,q}(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+\Big({1\over p}+{1\over
pq}\Big)\log{C_2\over r^n}
&(7.5~{\rm i})\cr\cr
&\nu(\varphi,z)-n\Big({1\over p}+{1\over pq}\Big)\le
\nu(\varphi_{p,q},z)\le \nu(\varphi,z).
&(7.5~{\rm ii})\cr}
$$
The major improvement is that we can now compare the jets when $U$ varies, 
even when we allow a small perturbation on $\varphi$ as well.

\claim 7.6. Proposition|Suppose that we have plurisubharmonic functions 
$\varphi',\,\varphi''$ defined on bounded Stein open sets
\hbox{$U',\, U''\compact\bC^n$}, with $|\varphi'-\varphi''|\le
\varepsilon$ on $U'\cap U''$. Let $\sigma'=(\sigma'_\ell)_{\ell\in\bN}$,
$\sigma''=(\sigma''_\ell)_{\ell\in\bN}$ be the associated Hilbert bases
of $\cH_{p\varphi'}(U')$, $\cH_{p\varphi''}(U'')$, and 
$\wt\sigma'=(\wt\sigma'_\ell)_{\ell\in\bN}$, 
$\wt\sigma''=(\wt\sigma''_\ell)_{\ell\in\bN}$ the bases
of $\cH_{pq\varphi'_p}(U')$, $\cH_{pq\varphi''_p}(U'')$.
Fix a Stein open set $W\compact U'\cap U''$
and a holomorphic function $f''$ on $U''$ such that
$$
\int_{U''}|f''|^2|\sigma''|^{-2q}d\lambda\le 1,\qquad q>n+1.
$$
\item{\rm (i)} One can write $f''=\sum_{L\in\bN^m}g_L(\sigma')^L$ on $W$ 
with $m=q-n-1$ and
$$
\int_{W}\sum_L|g_L|^2|\sigma'|^{-2(n+1)}d\lambda\le
C^{2q}e^{2pq\varepsilon}
$$
with a constant $C>1$ depending only on $d(W,\complement(U'\cap U''))$.
\vskip12pt
\vbox{
\item{\rm (ii)} There are holomorphic functions $h_\ell$ on $W$ such that
$f''=\sum h_\ell\wt\sigma_\ell'$ on $W$, and
$$
\sup_W\sum_\ell|h_\ell|^2\le C_1(p)C^{2q}e^{2pq\varepsilon}
$$
where $C$ is as in {\rm(i)} and $C_1(p)$ depends on $p$ $($and $U',U'',W$ 
as well\/$)$.
\vskip0pt}
\endclaim

\proof. (i) Thanks to (7.3), we have 
$|\sigma'|\ge C^{-1}e^{-p\varepsilon}|\sigma''|$ on $W$
for some constant $C>1$ depending only only on $d(W,\complement(U'\cap U''))$.
Therefore
$$
\int_{W}|f|^2|\sigma'|^{-2q}d\lambda\le C^{2q}e^{2pq\varepsilon}.
$$
Let us apply Skoda's $L^2$ division theorem (Corollary 10.6)
with $r=n$, $m=q-n-1$, $\alpha=1$, on the Stein open set $W$. 
Our assertion (i) follows, after absorbing the extra constant $(q-n)$ 
in~$C^{2q}$.

\noindent
(ii) We first apply (i) on a Stein open set $W_1$ such that
$W\compact W_1\compact U'\cap U''$, and write in this way
$f''=\sum_L g_L(\sigma')^L$ with the $L^2$ estimate as in (i).
By Nadel [Nad90] (see also [Dem93]), the ideal sheaf $\cI$ of 
holomorphic functions $v$ on $U'$ such that
$$
\int_{U'}|v|^2|\sigma'|^{-2(n+1)}d\lambda<+\infty
$$
is coherent and locally generated by its global $L^2$ sections (of course,
this ideal depends on the $\sigma'_\ell$, hence on $p$ and~$\varphi'$).
It follows that we can find finitely many holomorphic functions
$v_1,\ldots,v_N$, $N=N(p)$, such that
$$
\int_{U'}|v_j|^2|\sigma'|^{-2(n+1)}d\lambda=1
$$
and $\cI(W_1)=\sum v_j\cO(W_1)$. As the topology given
by the $L^2$ norm on the $L^2$ sections of $\cI(W_1)$ is stronger than 
the Fr\'echet topology of uniform convergence on compact subsets, and as
we have a Fr\'echet epimorphism $\cO(W_1)^{\oplus N}\to\cI(W_1)$,
$(a_1,\ldots,a_N)\mapsto \sum a_jv_j$, the open mapping theorem shows that
we can write $g=\sum a_jv_j$ with
$$
\sup_W\sum_{j=1}^N|a_j|^2\le A(p)\int_{W_1}|g|^2
|\sigma'|^{-2(n+1)}d\lambda
$$
for every holomorphic function $g$ on $W_1$ for which the right hand side
is finite [the constant $A(p)$ depends on $\cI$, hence on $p\,$]. In particular,
we can write $g_L=\sum_j a_{j,L}v_j$ with
$$
\sup_W\sum_{j,L}|a_{j,L}|^2\le A(p)\int_{W_1}\sum_L|g_L|^2
|\sigma'|^{-2(n+1)}d\lambda\le A(p)C^{2q}e^{2pq\varepsilon}.
$$
We find
$$
f''=\sum_L g_L(\sigma')^L=\sum_{j,L}a_{j,L}g_j(\sigma')^L,
$$
and as $L$ runs over all multiindices of length $m=q-n-1$ we get
$$
\int_{U'}\sum_{j,L}|g_j(\sigma')^L|^2|\sigma'|^{-2q}d\lambda=
\int_{U'}\sum_{j}|g_j|^2|\sigma'|^{-2(n+1)}d\lambda=N=N(p).
$$
We can therefore express the function $g_j(\sigma')^L$ in terms of the
Hilbert basis $(\wt\sigma'_\ell)$
$$
g_j(\sigma')^L=\sum_\ell b_{j,L,\ell}\wt\sigma'_\ell,\qquad
b_{j,L,\ell}\in\bC,\quad \sum_{j,L,\ell}|b_{j,L,\ell}|^2=N(p).
$$
Summing up everything, we obtain
$$
f''=\sum_{j,L,\ell}a_{j,L}b_{j,L,\ell}\wt\sigma'_\ell
=\sum_\ell h_\ell\wt\sigma'_\ell,\qquad
h_\ell=\sum_{j,L}a_{j,L}b_{j,L,\ell}.
$$
The Cauchy-Schwarz inequality implies
$$\sup_W\sum_\ell|h_\ell|^2\le
\sup_W\sum_{j,L}|a_{j,L}|^2\sum_{j,L,\ell}|b_{j,L,\ell}|^2
\le N(p)A(p)C^{2q}e^{2pq\varepsilon},
$$
as desired.\qed
\medskip

Now, assume that $X$ is a complex manifold such that $T^\star_X$ is
equipped with a plurisubharmonic Finsler metric. As all constructions 
to be used are local, we may suppose that we are in a small
coordinate open subset $U_0\compact X$ or, equivalently,
in a Stein open set $U_0\compact\bC^n$, with $\varphi$ being defined 
on~$U_0$. We fix Stein open sets $U\compact U_1\compact U_0$ and
select a sequence $\rho=(\rho_m)$ satisfying property (6.5) on each fiber 
$T^\star_{U,x}$. Finally, for $(x,t)\in U\times\bC$, we set
$$
\Phi^{\chi,\rho,p,q}(x,t):=
\sup_{f\in B_{p,q}(1)}{1\over pq}\log\wh M_F^{\chi,\rho}(x,t)+{C_0\over p},
\qquad C_0\gg 0,
\leqno(7.7)
$$
where $f$ runs over the unit ball $B_{p,q}(1)$ of $\cH_{pq\varphi_p}(U_1)$
and $F(x,\xi)=f(x+\xi)$. Then $\Phi^{\chi,\rho,p,q}(x,t)$ is well defined
on $U\times\{\Re t<-A\}$ for $A\ge 0$ sufficiently large. Thanks to Lemma 
6.8, the choice of coordinates on $U_0$ is essentially irrelevant when 
we compute $\wh M_F^{\chi,\rho}(x,t)$, provided that $\rho$ decays fast
enough. Moreover, a change of coordinate $\tau:U_0\mapsto \tau(U_0)$ has 
the effect of replacing  $\varphi$ by $\varphi^\tau=\varphi\circ\tau^{-1}$ 
and $\varphi_p$ by $\varphi^\tau_p=\varphi_p\circ\tau^{-1}+O(1/p)$, since
the only change occurring in the definition of 
$\cH_{p\varphi^\tau}(\tau(U_1))$
is the replacement of the Lebesgue volume form $d\lambda$
by $\tau^\star d\lambda$, which affects the $L^2$
norm by at most a constant. Similarly, the $L^2$ norm of
$\cH_{pq\varphi_p^\tau}(\tau(U_1))$ gets modified by an irrelevant
multiplicative factor $\exp(O(q))$, inducing a negligible error 
term $O(1/p)$ in~(7.7). If $C_0\ge 0$ is large enough, (7.2) combined
with 7.5~(i) implies that
$$
\varphi(z)\le \varphi_{p,q}(z)+{C_0\over p}=\sup_f{1\over pq}\log|f(x)|
+{C_0\over p}\le \sup_{|\zeta-z|<r}\varphi(\zeta)+\Big({1\over p}+{1\over
pq}\Big)\log{C\over r^n}
$$
for some $C>0$, where $f$ runs over 
$B_{p,q}(1)\subset\cH_{pq\varphi_p}(U_1)$.
Lemma 6.3 applied with $r=|e^t|$ then gives
$$
\leqalignno{
&&(7.8~{\rm i})\cr
\Phi^{\chi,\rho,p,q}(x,t)&\le
\sup_{|z-x|\le C'|e^t|}\varphi_{p,q}(z)+{C_0\over p}\le
\sup_{|z-x|\le C|e^t|}
\varphi(z)-n\Big({1\over p}+{1\over pq}\Big)\Re t+{C\over p},\cr
&&(7.8~{\rm ii})\cr
\Phi^{\chi,\rho,p,q}(x,t)&\ge
\sup_{|z-x|\le c_{\chi,\rho,p,q,\varphi,U}|e^t|}\varphi_{p,q}(z)+{C_0\over p}
\ge\sup_{|z-x|\le c_{\chi,\rho,p,q,\varphi,U}|e^t|}\varphi(z)
\ge\varphi(x)\cr
}
$$
where $C$, $C'$ are universal constants, and $c_{\chi,\rho,p,q,\varphi,U}$
depends on all given data, but is independent of~$x$. The last inequality 
is a simple consequence of the fact that the Taylor series 
$\wh M^{\chi,\rho}_f(x,t)=\sum_{m\ge 0}\rho_m\Vert
a_m(x)\Vert |e^{mt}|$ are never identically zero, hence their behavior
as $|e^t|\to 0$ is the same as for the series $\sum_{m\le N}
\rho_m\Vert a_m(x)\Vert |e^{mt}|$, truncated at some rank 
$N=N_{p,q,\varphi,U}$. The constant $c_{\chi,\rho,p,q,\varphi,U}$ then
essentially depends only on $\inf_{m\le N}\rho_m$. The upper and lower
bound provided by (7.8) imply in particular
$$
\leqalignno{
&\lim_{\Re t\to -\infty}{\Phi^{\chi,\rho,p,q}(x,t)\over\Re t}=
\lim_{\Re t\to -\infty}{\sup_{|z-x|\le|e^t|}\varphi_{p,q}(z)\over\Re t}=
\nu(\varphi_{p,q},x),&(7.9~{\rm i})\cr
&\lim_{\Re t\to -\infty}\Big|{\Phi^{\chi,\rho,p,q}(x,t)\over\Re t}-
{\sup_{|z-x|\le|e^t|}\varphi_{p,q}(z)\over\Re t}\Big|=0,&(7.9~{\rm ii})\cr}
$$
where the second limit is uniform on $U$ [For
this, we use the convexity of $\Re t\mapsto\sup_{|z-t|\le |e^t|}\varphi(z)$
to check that the constants $C$ in $\sup_{|z-t|\le C|e^t|}\varphi(z)$
are irrelevant]. For future reference, we also note
$$
\leqno\hbox{(7.10)~~The functions $\Phi^{\chi,\rho,p,q}(x,t)$ are continuous
on $U\times\{\Re t<-A\}$.}
$$
This is an immediate consequence of the fact that the unit
ball $B_{p,q}(1)$ of $\cH_{pq\varphi_p}(U_1)$ is a
normal family of holomorphic functions. We now investigate the effect
of a perturbation on~$\varphi$.

\claim 7.11. Proposition|Let $U'\compact U'_1\compact U_0$, $U''\compact
U''_1\compact U_0$ and let $\varphi',\,\varphi''$ be plurisubharmonic 
functions on $U'_1$, $U''_1$ such that $|\varphi''-\varphi'-\Re g|\le
\varepsilon$ on~$U'_1\cap U''_1$, for some holomorphic function
$g\in\cO(U'_1\cap U''_1)$. There are constants $C_2(p)$ and $C_3$ 
$($depending also on $\varphi'$, $\varphi''$, $g$, $U'$, $U'')$ such that
$$
\big|\Phi^{\prime\prime\,\chi,\rho,p,q}(x,t)-
\Phi^{\prime\,\chi,\rho,p,q}(x,t)-\Re g(x)\big|
\le 2\varepsilon+{C_2(p)\over q}+{C_3\over p}
$$
for all $x\in U'\cap U''$ and $|e^t|<r_0(\varepsilon)$ small enough.
\endclaim

\proof. We first treat the simpler case when $g=0$. By 7.6~(ii), every 
function $f''\in B''_{p,q}(1)\subset
\cH_{pq\varphi_p}(U''_1)$ can be written
$$
f''=\sum_{\ell\in\bN}h_l\wt\sigma'_\ell,\qquad
\sup_W\sum_\ell|h_\ell|^2\le C_1(p)C^{2q}e^{2pq\varepsilon}
$$
on any relatively compact neighborhood $W$ of $\ol{U'\cap U''}$ in
$U'_1\cap U''_1$. Fix a small polydisk $\ol D(r)\subset\bC^n$ such
that $\ol{U'\cap U''}+\ol D(r) \subset U'_1\cap U''_1$, and expand
$$
h_\ell(x)=\sum_{\alpha\in\bN^n}a_{\ell,\alpha}(x-x_0)^\alpha
$$
as a power series at each point $x_0\in U'\cap U''$. By integrating 
$\sum|h_\ell|^2$ over the polycircle $\prod\partial D(x_{0,j},r_j)$, we find
$$
\sum_{\ell\in\bN,\alpha\in\bN^n}|a_{\ell,\alpha}|^2|r^\alpha|^2
\le C_1(p)C^{2q}e^{2pq\varepsilon}.
\leqno(7.12)
$$
By substituting $h_\ell$ with its Taylor expansion in the definition of
$f''$, we find
$$
f''(x)=\sum_{\alpha\in\bN^n}p_\alpha(x)\, w_\alpha(x)
$$
where
$$
p_\alpha(x)=(x-x_0)^\alpha,\qquad
w_\alpha(x)=\sum_{\ell\in\bN}a_{\ell,\alpha}\wt\sigma'_\ell.
$$
The $L^2$ norm of $w_\alpha$ in $\cH_{pq\varphi'_p}(U'_1)$ is
$\big(\sum_\ell|a_{\ell,\alpha}|^2\big)^{1/2}$, hence
by definition
$$
{C_0\over p}+{1\over pq}\log {\wh M^{\chi,\rho}_{w_\alpha}(x_0,t)\over 
\big(\sum_\ell|a_{\ell,\alpha}|^2\big)^{1/2}
}\le \Phi'_{p,q}(x_0,t).
$$
On the other hand, if $|e^t|\ll \Vert r\Vert$, Lemma 6.3 implies that 
$$
\wh M^{\chi,\rho}_{p_\alpha}(x_0,t)\le \sup_{D(x_0,r/3)}|(x-x_0)^\alpha|\le
2^{-\alpha}r^\alpha
$$
From this, we infer
$$
\eqalign{
\wh M^{\chi,\rho}_{f''}(x_0,t)
&\le \sum_{\alpha\in\bN^n}\wh M^{\chi,\rho}_{p_\alpha}(x_0,t)
\wh M^{\chi,\rho}_{w_\alpha}(x_0,t)\cr
&\le\sum_{\alpha\in\bN^n}2^{-\alpha}r^\alpha 
\Big(\sum_\ell|a_{\ell,\alpha}|^2\Big)^{1/2}
\exp\big(pq(\Phi'_{p,q}(x_0,t)-C_0/p)\big)\cr
&\le\sum_{\alpha\in\bN^n}2^{-\alpha}
\big( C_1(p)C^{2q}e^{2pq\varepsilon}\big)^{1/2}
\exp\big(pq(\Phi'_{p,q}(x_0,t)-C_0/p)\big)
\cr}
$$
thanks to (7.12). By taking ${1\over pq}\log(\ldots)$ and passing to the
$\sup$ over all $f''$, we get
$$
\Phi''_{p,q}(x_0,t)\le {1\over pq}\log\big(2^n
\big( C_1(p)C^{2q}e^{2pq\varepsilon}\big)^{1/2}\big) + \Phi'_{p,q}(x_0,t).
$$
Proposition 7.11 is thus proved for the case $g=0$, even with $\varepsilon$
instead of $2\varepsilon$ in the final estimate. In case $g$ is non zero,
we observe that the replacement of $\varphi'$ by $\varphi'+\Re g$ yields 
isomorphisms of Hilbert spaces
$$
\matrix{
&\llap{$\cH_{p\varphi'}(U'_1)$}
\longrightarrow\cH_{p(\varphi'+\Re g)}(U'_1),\quad
\hfill& f\longmapsto e^{pg}f,\hfill\cr
\noalign{\vskip 5pt}
\phantom{\cH_{pq\varphi'_p}(U'_1)}
&\llap{$\cH_{pq\varphi'_p}(U'_1)$}\longrightarrow
\cH_{pq(\varphi'_p+\Re g)}(U'_1),\quad\hfill
&f\longmapsto e^{pqg}f.\hfill\cr}
$$
The only difference occurring in the proof is that we get
$$
f''=e^{pq g}\sum_{\ell\in\bN}h_l\wt\sigma'_\ell
$$
instead of $f''=\sum_{\ell\in\bN}h_l\wt\sigma'_\ell$. In the upper bound
for $\wh M^{\chi,\rho}_{f''}(x_0,t)$, this introduces an extra term
$\wh M^{\chi,\rho}_{e^{pqg}}(x_0,t)$, which we evaluate as
$\exp(pq(\Re g(x_0) +O(|e^t|)))$ thanks to Lemma 6.3. The general 
estimate follows, possibly with an additional $\varepsilon$ error
when $|e^t|$ is small enough.\qed
\medskip

The final step in the construction is to ``glue'' together the
functions $\Phi^{\chi,\rho,p,q}(x,t)$, $(p,q)\in\bN^2$. We choose a
fast increasing sequence $p\mapsto q(p)$, in such a way that
${C_2(p)\over q(p)}\le {1\over p}$, where $C_2(p)$ is the constant
occurring in Proposition~7.11. We now define
$$
\leqalignno{
\wt M^{\chi,\rho,s}_\varphi(x,t)&{}:=\Phi^{\chi,\rho,s}(x,t)\cr
&{}:=\sup_{p\ge s}\Big(\Phi^{\chi,\rho,p,q(p)}(x,t-\log p)+{\log p\over p}
+n\Big({1\over p}+{1\over p\,q(p)}\Big)\Re t\Big)&(7.13)\cr}
$$
[The terms in $\log p$ are there only for a minor technical reason, to make
sure that $\wt M^{\chi,\rho,s}_\varphi(x,t)$ is a continuous function].
In this way, we achieve the expected goals, namely:

\claim 7.14. Proposition|Let $\varphi$ be a plurisubharmonic function
defined on a bounded Stein open set $U_0\compact \bC^n$ such that 
$T^\star_{U_0}$ is equipped with a plurisubharmonic smooth Finsler
metric, and let $U\compact U_0$. Then there is a functional
$\wt M^{\chi,\rho,s}$ $($associated with the choice of a
sequence $q(p)$ which may have to be adjusted when $\varphi$ varies,
but can be taken fixed if $\varphi$ remains in a bounded set of
$L^1(U_0))$, such that the functions 
$\Phi^{\chi,\rho,s}(x,t)=\wt M^{\chi,\rho,s}_\varphi(x,t)$ satisfy the
following properties:
\smallskip
\item{\rm(i)} The functions $\Phi^{\chi,\rho,s}(x,t)$ are defined on 
$U\times\{\Re t<-A\}$ for $A>0$ large, and are locally bounded continuous
plurisubharmonic functions depending only on $\Re t\,;$ moreover,
$p\mapsto \Phi^{\chi,\rho,s}(x,t)$ is a decreasing family of functions.
\smallskip
\item{\rm(ii)} ~~$\displaystyle
\varphi(x)\le \Phi^{\chi,\rho,s}(x,t)\le \sup_{\Vert z-x\Vert\le Cs^{-1}|e^t|}
\varphi(z)+C\,{\log s\over s}$~~ for some $C\gg 0$.
\smallskip
\item{\rm(iii)} ~~$\displaystyle
\lim_{\Re t\to -\infty}
\Big|{\Phi^{\chi,\rho,s}(x,t)\over\Re t}-{\sup_{\Vert z-x\Vert\le |e^t|}
\varphi(z)\over\Re t}\Big|=0$\medskip
\item{}uniformly on every compact subset of $U$, in particular
$$
\lim_{\Re t\to -\infty}{\Phi^{\chi,\rho,s}(x,t)\over\Re t}=\nu(\varphi,x)
$$
for every $x\in U$.
\smallskip
\item{\rm(iv)} For every holomorphic change of coordinates
$\tau:U_0\to \tau(U_0)$, the sequence $\rho=(\rho_m)$ can be
chosen $($depending only on $\tau)$ such that for some constant $C>0$ 
we have
$$
\big|\wt M^{\chi,\rho,s}_{\varphi\circ\tau^{-1}}(\tau(x),t)-
\wt M^{\chi,\rho,s}_{\varphi}(x,t)\big|\le {C\over s}\qquad
\hbox{for all $x\in U$},
$$ 
when $T^\star_{\tau(U_0)}$ is equipped with the induced Finsler metric.
\smallskip
\item{\rm(v)} Let $U'\compact U'_1\compact U_0$, $U''\compact U''_1
\compact U_0$ be Stein open subsets, and let $\varphi'$, $\varphi''$ be 
plurisubharmonic functions on $U'_1$, $U''_1$ such that 
$|\varphi'-\varphi|\le 1$ on $U'_1$, $|\varphi''-\varphi|\le 1$ on $U''_1$
and $|\varphi''-\varphi'-\Re g|\le\varepsilon$ on $U'_1\cap U''_1$ for some
holomorphic function $g\in\cO(U'_1\cap U''_1)$. Then
$$
\big|\wt M^{\chi,\rho,s}_{\varphi''}(x,t)-
\wt M^{\chi,\rho,s}_{\varphi'}(x,t)\big|\le 2\varepsilon+{C\over s}\qquad
\hbox{for all $x\in U'\cap U''$ and $|e^t|<r_0(\varepsilon)$},
$$
where $C=C(\varphi,U',U'')$.\vskip0pt
\endclaim

\proof. All properties are almost immediate consequences of the properties 
already obtained for $\Phi^{\chi,\rho,p,q}$, simply by taking the supremum.
We check e.g.\ the continuity of $\Phi^{\chi,\rho,s}$, inequality (ii) and
the second statement of (iii). In fact, (7.8~i,ii) imply
$$
\eqalign{
\Phi^{\chi,\rho,p,q(p)}(x,t-\log p)+{\log p\over p}
&{}+n\Big({1\over p}+{1\over p\,q(p)}\Big)\Re t\cr
&{}\le\sup_{\Vert z-x\Vert\le Cp^{-1}|e^t|}
\varphi(z)+C\,{\log p\over p}\cr
&{}\ge\varphi(x)+ {\log p\over p}
+n\Big({1\over p}+{1\over p\,q(p)}\Big)\Re t,\cr}
$$
and (7.14~ii) follows from this. Moreover, the function
$\Phi^{\chi,\rho,p,q(p)}(x,t-\log p)+\cdots$ converges to $\varphi(x)$ 
as $p\to+\infty$, while its terms get${}>\varphi(x)$ for $p$
large, thanks to the lower bound. It follows that the $\sup$ in 
(7.13) is locally finite, therefore $\Phi^{\chi,\rho,s}$ is continuous.
To prove (iii), we first observe that the right hand inequality in (i) gives
$$
\lim_{\Re t\to -\infty}{\Phi^{\chi,\rho,s}(x,t)\over\Re t}
\ge \lim_{\Re t\to -\infty}{\sup_{|z-x|\le|e^t|}\varphi(z)\over\Re t}
=\nu(\varphi,x).
$$
In the other direction, the definition of $\Phi^{\chi,\rho,s}(x,t)$ combined
with (7.8~ii) implies 
$$
\eqalignno{
\Phi^{\chi,\rho,s}(x,t)&\ge \Phi^{\rho,p,q(p)}(x,t-\log p)+n\Big({1\over p}+
{1\over p\,q(p)}\Big)\Re t\cr
&\ge \sup_{|z-x|\le p^{-1}c_{\rho,p,q(p),\varphi,U}|e^t|}\varphi(z)
+n\Big({1\over p}+{1\over p\,q(p)}\Big)\Re t\cr}
$$
for all $p\ge s$, hence
$$
\lim_{\Re t\to -\infty}{\Phi^{\chi,\rho,s}(x,t)\over\Re t} \le
\lim_{\Re t\to -\infty}{\sup_{|z-x|\le|e^t|}\varphi(z)\over\Re t}
+n\Big({1\over p}+{1\over p\,q(p)}\Big).
$$
We get the desired conclusion by letting $p\to+\infty$.\qed

\section{\S8. A variant of Kiselman's Legendre transform}

To begin with, let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set $U\compact\bC^n$. Consider the trivial 
vector bundle $T_U=U\times\bC^n$,
and assume that $T^\star_U$ is equipped with a smooth Finsler metric
${}_{\chi^\star}\Vert\xi^\star\Vert^\star=
\exp(\chi^\star(z,\xi^\star))$ for $\xi^\star\in T^\star_{U,z}$. 
We assume that the curvature of the Finsler
metric \hbox{$\Vert\xi^\star\Vert^\star_z=e^{\chi^\star(z,\xi^\star)}$}
on $T^\star_U$ satisfies 
$$
{i\over\pi}\ddbar\chi^\star(z,\xi^\star)+\pi_U^\star u(z)\ge 0
\leqno(8.1)
$$
for some nonnegative continuous $(1,1)$-form $u$ on~$U$, where 
$\pi_U:T^\star_U\to U$ is the projection. If $\chi^\star=\log h^\star$
is a hermitian metric on $T^\star_X$, we let $h$ be the dual
metric on $T_X$ and set
$$
\Phi^h_\infty(z,w)=\sup_{{}_h\Vert\xi\Vert_z\le |e^w|}\varphi(z+\xi).
\leqno(8.2)^h
$$
By Theorem 6.1, this definition works equally well when $h^\star$ is
a {\it fiberwise convex} Finsler metric.
Clearly $\Phi^h_\infty(z,w)$ depends only on the real part $\Re w$ of $w$
and is defined on the open set $\Omega$ of points $(z,w)\in U\times\bC$
such that $\Re w<\log d_z(z,\partial U)$, where $d_z$ denotes
euclidean distance with respect to ${}_h\Vert\kesp\Vert_z$. Now, we would
like to extend this to the case of a general Finsler metric, without
any convexity assumption. As a replacement for the ``sup formula'' 
$(8.2)^h$, we set
$$
\Phi^{\chi,\rho,s}_\infty(z,w)=
\wt M^{\chi,\rho,s}_{\varphi}(z,w)\leqno(8.2)^\chi
$$
where $\wt M^{\chi,\rho,s}$ denotes the functional associated with~$\chi$,
as in \S$\,$7. Here, however, $\chi(z,\xi^\star)$ need not be 
plurisubharmonic. This is not a real difficulty, since the definition
of the $\wh M^{\chi,\rho}$ functional in (6.6) shows that
$$
\wt M^{\chi,\rho,s}_{\varphi}(z,w)=
\wt M^{\chi_v,\rho,s}_{\varphi}(z,w-v(z))
$$
for any smooth function $v$ on $U$ such that $\chi_v(z,\xi^\star):=
\chi(z,\xi^\star)+v(z)$ is plurisubharmonic on~$U$ (and such a function 
always exists by our assumption (8.1)).
One of our main concern is to investigate singularities of~$\varphi$ and 
these singularities are reflected in the way $\Phi^h_\infty(z,w)$
and $\Phi^{\chi,\rho,s}_\infty$ decay to $-\infty$ as $\Re w$ goes 
to~$-\infty$. In this perspective, Proposition 7.14~(iii) shows
that considering $\Phi^{\chi,\rho,s}_\infty(z,w)$ instead of 
$\Phi^h_\infty(z,w)$ does not make any difference.
Moreover $\Phi^{\chi,\rho,s}_\infty(z,w)$ and $\Phi^h_\infty(z,w)$ are
both convex increasing function of~$\Re w$. For $(z,w)\in\Omega$ and
$c>0$, we introduce the (generalized) {\sl Legendre transform}
$$
\leqalignno{
\Phi^h_c(z,w)&={}~\inf_{t\le 0}~\Phi^h_\infty(z,w+t)-ct,&(8.3)^h\cr
\Phi^{\chi,\rho,s}_c(z,w)&=
{}~\inf_{t\le 0}~\Phi^{\chi,\rho,s}_\infty(z,w+t)-ct.
&(8.3)^\chi\cr}
$$
It is easy to see that these functions are increasing in $c$ and that
$$
\lim_{c\to 0}\Phi^h_c(z,w)=\varphi(z),\qquad
\lim_{c\to +\infty}\Phi^h_c=\Phi^h_\infty.
\leqno(8.4)^h
$$
The analogue for $\Phi^{\chi,\rho,s}_c$ is
$$
\leqalignno{
\lim_{c\to 0}{}&\Phi^{\chi,\rho,s}_c(z,w)=
\lim_{\Re t\to-\infty}\Phi^{\chi,\rho,s}_\infty(z,\Re t)\in
\big[\varphi(z),\varphi(z)+C\log s/s\big],&(8.4)^\chi\cr
\lim_{c\to +\infty}{}&\Phi^{\chi,\rho,s}_c=\Phi^{\chi,\rho,s}_\infty.\cr}
$$
When ${}_h\Vert\xi\Vert_z$ is taken to be a constant metric, we know by
Kiselman [Kis78] that $\Phi^h_\infty$ and $\Phi^h_c$ are plurisubharmonic
functions of the pair~$(z,w)$, and that the Lelong numbers of 
$\Phi^h_c(\bu,w)$ are given by
$$
\nu\big(\Phi^h_c(\bu,w),z\big)=\big(\nu(\varphi,z)-c\big)_+,\qquad
\forall(z,w)\in\Omega.\leqno(8.5)^h
$$
Since $(8.5)^h$ depends only on the maps $z\mapsto \Phi^h_\infty(z,w)$
with $w$ fixed, the equality is still valid when $h$ is a variable
hermitian metric, and Proposition 7.14~(iii) even shows that the analogous
property for $\Phi^{\chi,\rho,s}_c$ is true:
$$
\nu\big(\Phi^{\chi,\rho,s}_c(\bu,w),z\big)=\big(\nu(\varphi,z)-c\big)_+,\qquad
\forall(z,w)\in\Omega.\leqno(8.5)^\chi
$$
As usual we denote by
$$
E_c(\varphi)=\big\{z\in U\,;\;\nu(\varphi,x)\ge c\big\}
\leqno(8.6)
$$ 
the Lelong sublevel sets of~$\varphi$.
From now on, we omit the superscripts in the notation $\Phi^h_c$ or
$\Phi^{\chi,\rho,s}_c$ since all properties are the same in both cases.
In general, $\Phi_\infty$ is continuous on $\Omega$ and its right derivative
$\partial\Phi_\infty(z,w)/\partial\Re w_+$ is upper semicontinuous;
indeed, this partial derivative is the decreasing limit of
$\big(\Phi_\infty(z,w+t)-\Phi_\infty(z,w)\big)/t$ as~$t\downarrow
0_+$.  It follows that $\Phi_c$ is continuous on
$\Omega\ssm(E_c(\varphi)\times\bC)$: in fact, we have
$\nu(\varphi,z)=\lim_{t\to-\infty}\partial\Phi_\infty(z,t)/ \partial
t_+<c$ on every compact set $K\subset\Omega\ssm(E_c(\varphi)
\times\bC)$, so by the upper semicontinuity there is a constant $t_0$
such that $\partial\Phi_\infty(z,w+t)/\partial t_+<c$ for $(z,w)\in K$
and~$t<t_0$.  Therefore
$$
\Phi_c(z,w)=\inf_{t_0\le t\le 0}\Phi_\infty(z,w+t)-ct\qquad
\hbox{\rm on}~K,$$
and this infimum with compact range is continuous.
Our next goal is to investigate the plurisubharmonicity of $\Phi_c$.

\claim 8.7. Proposition|Assume the curvature of the Finsler
metric \hbox{$\Vert\xi^\star\Vert^\star_z=e^{\chi^\star(z,\xi^\star)}$}
on $E^\star$ satisfies 
$$
{i\over\pi}\ddbar\chi^\star(z,\xi^\star)+\pi_X^\star u(z)\ge 0
$$
for some nonnegative continuous $(1,1)$-form $u$ on~$X$, where 
$\pi_X:E^\star\to X$ is the projection. Then $\Phi_c=\Phi_c^{\chi,\rho,s}$
$[$and likewise $\Phi_c=\Phi^h_c]$ enjoys the following properties.
\item{\rm(i)}For all $\eta\ge 0$, we have
$$\Phi_c(z,w-\eta)\ge\Phi_c(z,w)-\min\Big\{{\partial\Phi_\infty(z,w)\over
\partial\Re w_-}\,,\,c\Big\}\eta\,;$$
\item{\rm(ii)}For $(\zeta,\eta)\in T_U\times\bC$ and $c\in{}]0,+\infty]$,
the Hessian of $\Phi_c$ satisfies
$${i\over\pi}\ddbar(\Phi_c)_{(z,w)}(\zeta,\eta)
\ge -\min\Big\{{\partial\Phi_\infty(z,w)\over\partial\Re w_+}\,,\,
c\Big\}\,u_z(\zeta).$$
\endclaim

\proof. (i) For $\eta\ge 0$ and $t\le 0$, the convexity of $\Phi_\infty(z,w)$
in $\Re w$ implies
$$
\Phi_\infty(z,w+t-\eta)\ge\Phi_\infty(z,w+t)-\eta\,
{\partial\Phi_\infty(z,w+t)\over\partial\Re w_-}
$$
As $\partial\Phi_\infty(z,w)/\partial\Re w_-$ is increasing in $\Re w$,
the infimum of both sides minus $ct$ gives
$$\Phi_c(z,w-\eta)\ge\Phi_c(z,w)-\eta{\partial\Phi_\infty(z,w)\over\partial
\Re w_-}.$$
On the other hand, the change of variables $t=t'+\eta$ yields
$$\Phi_c(z,w-\eta)\ge\inf_{t'\le-\eta}\Phi_\infty(z,w+t')-c(t'+\eta)\ge
\Phi_c(z,w)-c\eta.$$
Property (i) follows.
\smallskip

\noindent(ii) Fix $(z_0,w_0)\in\Omega$ and a semipositive quadratic function
$v(z)$ on $\bC^n$ such that 
\hbox{${i\over\pi}\ddbar v(0)>u_{z_0}$}. Then the inequality 
${i\over\pi}\ddbar v(z-z_0)>u(z)$ still holds on a neighborhood 
$U_0$ of~$z_0$,
and the Finsler metric $\Vert\xi^\star\Vert^\star_ze^{v(z-z_0)}$ is
plurisubharmonic on this neighborhood. From this, we conclude by
Lemma 7.7 that the associated function
$$
h(z,w):=\Phi_\infty(z,w+v(z-z_0))
$$
is plurisubharmonic on $U_0$. Its Legendre transform
$$
h_c(z,w)=\inf_{t\le 0}h(z,w-t)=\Phi_c(z,w+v(z-z_0))
$$
is again plurisubharmonic. For small $(\zeta,\eta)\in
T_U\times\bC$, the mean value inequality yields
$$\eqalign{\int_0^{2\pi}\Phi_c(z_0+e^{i\theta}\zeta,w_0+e^{i\theta}\eta)\,
{d\theta\over 2\pi}
&=\int_0^{2\pi}h_c\big(z_0+e^{i\theta}\zeta,
w_0+e^{i\theta}\eta-v(\zeta)\big)\,{d\theta\over 2\pi}\cr
&\ge h_c\big(z_0,w_0-v(\zeta)\big)=\Phi_c\big(z_0,w_0-v(\zeta)\big)\cr
&\ge\Phi_c(z_0,w_0)-\min\Big\{{\partial\Phi_\infty(z_0,w_0)\over
\partial\Re w_+}\,,\,c\Big\}v(\zeta)\cr}
$$
[the last inequality follows from~(i)]. For $A>\partial\Phi_\infty
(z_0,w_0)/\partial{\Re}\,w_+$, we still have $A>\partial\Phi_\infty(z,w)
/\partial{\Re}\,w_+$ in a neighborhood of $(z_0,w_0)$  by the upper 
semicontinuity, and we conclude that the function
$\Phi_c(z,w)+\min\{A,c\}v(z)$ satisfies the mean value
inequality near~$(z_0,w_0)$. Hence $\Phi_c(z,w)+\min\{A,c\}v(z)$
is plurisubharmonic near $(z_0,w_0)$. Since this is still true as $A$ tends to
$\partial\Phi_\infty(z,w)/\partial{\Re}\,w_+$ and ${i\over\pi}\ddbar v$
tends to $u_{z_0}$, the proof of (ii) is complete.\qed

\section{\S9. Regularization of closed positive (1,1)-currents}

The next step is to describe a gluing process for the construction of
global regularizations of almost plurisubharmonic functions. We
suppose that $T^\star_X$ is equipped with a Finsler metric
\hbox{$\Vert\xi^\star\Vert^\star_x=e^{\chi^\star(x,\xi^\star)}$}
satisfying
$$
{i\over\pi}\ddbar\chi^\star(x,\xi^\star)+\pi_X^\star u(x)\ge 0,
$$ 
where $u$ is a smooth semipositive $(1,1)$-form on~$X$.  Notice that
${i\over\pi}\ddbar\chi^\star(z,\xi^\star)$ is just the Chern curvature
of the induced hermitian metric on $\cO_{TX}(1)$. An {\sl almost
positive} $(1,1)$-current is by definition a real $(1,1)$-current
such that $T\ge\gamma$ for some real $(1,1)$-form $\gamma$ with
locally bounded coefficients. An {\sl almost psh function} is a
function $\psi$ which can be written locally as $\psi=\varphi+w$ where
$\varphi$ is plurisubharmonic and $w$ smooth. With these definitions,
${i\over\pi}\ddbar\psi$ is almost positive if and only if $\psi$
is almost psh.

The following theorem was proved in [Dem92] with a rather long and
tricky proof. We present here a shorter and better approach using
our modified Kiselman-Legendre transforms.

\claim 9.1. Theorem|Let $T$ be a closed almost positive $(1,1)$-current
and let $\alpha$ be a smooth real $(1,1)$-form in the same
$\ddbar$-cohomology class as $T$, i.e. $T=\alpha+{i\over\pi}\ddbar
\psi$ where $\psi$ is an almost psh function.
Let $\gamma$ be a continuous real $(1,1)$-form such that $T\ge\gamma$.
Suppose that $\cO_{TX}(1)$ is equipped with a smooth hermitian
metric such that the Chern curvature form satisfies
$$\Theta\big(\cO_{TX}(1)\big)+\pi^\star_X u\ge 0$$
with $\pi_X:P(T^\star X)\to X$ and with some nonnegative smooth $(1,1)$-form
$u$ on~$X$. Fix a hermitian metric $\omega$ on~$X$. Then for every
$c>0$, there is a sequence of closed almost positive $(1,1)$-currents
$T_{c,k}=\alpha+{i\over\pi}\ddbar\psi_{c,k}$ such that $\psi_{c,k}$
is smooth on $X\ssm E_c(T)$ and decreases to $\psi$ as $k$ tends to~$+\infty$
$($in particular, the current $T_{c,k}$ is smooth on $X\ssm E_c(T)$ and
converges weakly to $T$ on~$X)$, and such that
\item{\rm(i)}$T_{c,k}\ge\gamma-\min\{\lambda_k,c\}u-\varepsilon_k\omega$~~
where:
\item{\rm(ii)}$\lambda_k(x)$ is a decreasing sequence of continuous functions
on $X$ such that $\lim_{k\to+\infty}\lambda_k(x)=\nu(T,x)$ at every point,
\item{\rm(iii)}$\varepsilon_k$ is positive decreasing and
$\lim_{k\to+\infty}\varepsilon_k=0$,
\item{\rm(iv)}$\nu(T_{c,k},x)=\big(\nu(T,x)-c\big)_+$
at every point $x\in X$.
\endclaim

\proof. We first show that we indeed can write $T=\alpha+{i\over\pi}
\ddbar\psi$ with $\alpha$ smooth. Let $(U^0_j)$ be a finite covering
of $X$ by coordinate balls and $(\theta_j)$ a partition of unity
subordinate to~$(U^0_j)$. If $T$ is written locally
$T={i\over\pi}\ddbar\psi_j$ with $\psi_j$ defined on $U^0_j$, then
$\psi=\sum\theta_j\psi_j$ has the property that
$\alpha:=T-{i\over\pi}\ddbar\psi$ is smooth. This is an easy
consequence of the fact that $\psi_k-\psi_j$ is plurisubharmonic,
hence smooth, on $U^0_j\cap U^0_k$, writing $T$ as
${i\over\pi}\ddbar\psi_k$ over $U^0_k$.  By replacing $T$ with
$T-\alpha$ and $\gamma$ with $\gamma-\alpha$, we can assume that
$\alpha=0$ (in other words, Theorem 9.1 essentially deals only with
the singular part of $T$).

We can therefore assume that $T={i\over\pi}\ddbar\psi$, where $\psi$ is 
an almost plurisubharmonic function on~$X$ such that
$T\ge\gamma$ for some continuous $(1,1)$-form $\gamma$.
We select a finite covering ${\Cal W}=(W_\nu)$ of $X$ by open
coordinate charts. Given $\delta>0$, we take in each $W_\nu$ a
maximal family of points with (coordinate) distance to the boundary
$\ge 3\delta$ and mutual distance $\ge\delta$. In this way, we get for 
$\delta>0$ small a finite covering of $X$ by open balls $U_j$ of
radius~$\delta$, such that the concentric ball $U^0_j$ of radius
$2\delta$ is relatively compact in the corresponding chart~$W_\nu$. Let
$\tau_j:U^0_j\to B^0_j:=B(a_j,2\delta)$ be the isomorphism given by the
coordinates of~$W_\nu$ and
$$
\matrix{
&B_j\compact B^1_j\compact B^0_j,\hfill\quad
&B_j=B(a_j,\delta),\hfill&B^1_j=B(a_j,\sqrt 2\,\delta),\hfill
&B^0_j=B(a_j,2\delta),\hfill\cr
&\ks U_j\compact U^1_j\compact U^0_j,\quad\hfill
&\ks U_j=\tau_j^{-1}(B_j),\quad\hfill&\ks U^1_j=\tau_j^{-1}(B^1_j),\quad\hfill
&\ks U^0_j=\tau_j^{-1}(B^0_j).\hfill\cr}
$$
Let $\varepsilon(\delta)$ be a modulus of continuity for $\gamma$ on the 
sets $U^0_j$, such that $\lim_{\delta\to 0}\varepsilon(\delta)=0$ and 
$\gamma_x-\gamma_{x'}\le{1\over 2}\varepsilon(\delta)\,\omega_x$ for all
$x,x'\in U^0_j$. We denote by $\gamma_j$ the $(1,1)$-form with constant
coefficients on $B^0_j$ such that $\tau_j^\star\gamma_j$
coincides with $\gamma-\varepsilon(\delta)\,\omega$ at
$\tau_j^{-1}(a_j)$. Then we have
$$0\le\gamma-\tau_j^\star\gamma_j\le 2\varepsilon(\delta)\,\omega\qquad
\hbox{\rm on}~~U_j\leqno(9.2)$$
for $\delta>0$ small. We set $\psi_j=\psi\circ\tau_j^{-1}$ on
$B^0_j$ and let $\wt\gamma_j$ be the homogeneous quadratic
function in $z-a_j$ such that ${i\over\pi}\ddbar\wt\gamma_j=
\gamma_j$ on $B^0_j$. Finally, we set
$$\varphi_j(z)=\psi_j(z)-\wt\gamma_j(z)\qquad
\hbox{\rm on}~B^0_j.
\leqno(9.3)$$
It is clear that $\varphi_j$ is plurisubharmonic, since
$${i\over\pi}\ddbar(\varphi_j\circ\tau_j)=T-\tau_j^\star\gamma_j
\ge\gamma-\tau_j^\star\gamma_j\ge 0.$$
We combine $(8.2)^\chi$ and $(8.3)^\chi$ to define ``regularized'' functions
$$\leqalignno{
\Phi_{j,c}^{\chi,\rho,s}(z,w)
&=\inf_{t\le 0}\wt M^{\chi,\rho,s}_{\varphi_j}(z,w),
\qquad z\in B^1_j,
&(9.4)\cr
\Psi_{j,c}^{\chi,\rho,s}(z,w)
&=\Phi_{j,c}^{\chi,\rho,s}(z,w)+\wt\gamma_j(z)-
\varepsilon(\delta)^{1/2}|z-a_j|^2,
\qquad z\in B^1_j,\phantom{\Big)}
&(9.5)\cr
\Psi^{\chi,\rho,s}_c(x,w)&=\sup_{U^1_j\ni x}\Psi^{\chi,\rho,s}_{j,c}
(\tau_j(x),w),\qquad
x\in X,
\phantom{\Big)}&(9.6)\cr}
$$
for $\Re w<-A$, with $A\gg 0$. We have to check that the gluing procedure 
used in the definition of $\Psi^{\chi,\rho,s}_c$ does not introduce
discontinuities when $x$ passes through a boundary~$\partial U^1_j$. 
For this, we must compare $\Psi^{\chi,\rho,s}_{j,c}(\tau_j(x),w)$ and
$\Psi^{\chi,\rho,s}_{k,c}(\tau_k(x),w)$ on overlapping open sets
$U^1_j$, $U^1_k$. The comparison involves two points:

\noindent
$\bu$  effect of replacing $\psi_j$ with $\psi_j-\wt\gamma_j$,\hfil\break
$\bu$ effect of coordinate changes.

Let us first assume for simplicity that $U^1_j$ and $U^1_k$
are contained in the same coordinate patch $W_\nu$ (in such a way that
$\tau_j=\tau_k$ on $U^1_j\cap U^1_k$, therefore in this case, we do not 
have to worry about coordinate changes). Then $\psi_j=\psi_k$ on 
$B^1_j\cap B^1_k$, and therefore $\varphi_k-\varphi_j=
\wt\gamma_j-\wt\gamma_k$ is a quadratic
function whose Levi form is $O(\varepsilon(\delta))$, by the assumption
on the modulus of continuity of~$\gamma$. This quadratic function 
can be written as
$$
\wt\gamma_j(z)-\wt\gamma_k(z)=\Re g_{jk}(z)+q_{jk}(z-z^0_{jk}),
$$
the sum of an affine pluriharmonic part $\Re g_{jk}$
and a quadratic term $q_{jk}(z-z^0_{jk})$ which takes 
$O(\varepsilon(\delta)\delta^2)$ values (since $\diam B^1_j\cap B^1_k
\le\delta$). Therefore we have
$$
|\varphi_k-\varphi_j-\Re g_{jk}|\le C\varepsilon(\delta)\delta^2.
$$
By 7.14~(v), we conclude that
$$
\big|\Phi_{k,c}^{\chi,\rho,s}(z,w)-\Phi_{j,c}^{\chi,\rho,s}(z,w)
-\Re g_{jk}(z)\big|\le 2C\varepsilon(\delta)\delta^2+{C'\over s}
$$
for some constants $C$, $C'$, hence
$$
\big|\big(\Phi_{k,c}^{\chi,\rho,s}(z,w)+\wt\gamma_k(z)\big)-
\big(\Phi_{j,c}^{\chi,\rho,s}(z,w)+\wt\gamma_j(z)\big)\big|
\le 3C\varepsilon(\delta)\delta^2+{C'\over s}.
$$
Now, in case $U^1_j$ and $U^1_k$ are not equipped with the same coordinates,
7.14~(iv) shows that an extra error term $C/s$ is introduced by the
change of coordinates $\tau_{jk}=\tau_j\circ\tau_k^{-1}$, and also
possibly a further $O(\delta^3)$ term due to the fact that 
$\wt\gamma_j\circ\tau_{jk}$ differs from a quadratic function by terms
of order $3$ or more in the $\tau_k$-coordinates. Combining everything
together, we get
$$
\eqalign{
\big|\big(\Phi_{k,c}^{\chi,\rho,s}(\tau_k(z),w)&{}+\wt\gamma_k(\tau_k(z))\big)-
\big(\Phi_{j,c}^{\chi,\rho,s}(\tau_j(z),w)+\wt\gamma_j(\tau_j(z))\big)\big|\cr
&\le C''\Big(\varepsilon(\delta)\delta^2+\delta^3+{1\over s}\Big)
\le C'''\varepsilon(\delta)\delta^2\cr}
$$
if we choose $s\ge 1/(\varepsilon(\delta)\delta^2)$. We assume from
now on that $s$ is chosen in this way. For
$x\in\partial U^1_j=\tau_j^{-1}(S(a_j,\sqrt{2}\delta))$, formula (9.5)
yields
$$
\Psi_{j,c}^{\chi,\rho,s}(\tau_j(z),w)=
\Phi_{j,c}^{\chi,\rho,s}(\tau_j(z),w)+\wt\gamma_j(\tau_j(z))-
\varepsilon(\delta)^{1/2}2\delta^2,
$$
whereas there exists $k$ such that $x\in U_k=\tau_k^{-1}(B(a_k,\delta))$, 
hence
$$
\Psi_{k,c}^{\chi,\rho,s}(\tau_k(z),w)\ge
\Phi_{k,c}^{\chi,\rho,s}(\tau_j(z),w)+\wt\gamma_k(\tau_k(z))-
\varepsilon(\delta)^{1/2}\delta^2.
$$
We infer from this
$$
\Psi_{k,c}^{\chi,\rho,s}(\tau_k(z),w)-\Psi_{j,c}^{\chi,\rho,s}(\tau_j(z),w)
\ge \varepsilon(\delta)^{1/2}\delta^2 -C'''\varepsilon(\delta)\delta^2>0
$$
for $\delta$ small enough. This shows that formula (9.6) makes sense
for $\delta$ small. Formulas (9.2) and (9.5) show that
$$
{i\over\pi}\ddbar_z\Psi_{j,c}^{\chi,\rho,s}(\tau_j(z),w)\ge
{i\over\pi}\ddbar_z\Phi_{j,c}^{\chi,\rho,s}(\tau_j(z),w)+\gamma-
C\varepsilon(\delta)^{1/2}\omega
\leqno(9.7)
$$
for some constant $C>0$. The sequence of approximations $\psi_{c,k}$
needed in the theorem is obtained by taking sequences $\delta_k\downarrow 0$,
$s_k\ge 1/(\varepsilon(\delta_k)\delta_k^2)$ and $A_k\uparrow +\infty$,
and putting
$$
\wt\psi_{c,k}(z)=\Psi^{\chi,\rho,s_k}_c(z,-A_k)+{1\over k}
$$
where $\Psi^{\chi,\rho,s_k}_c$ is constructed as above by means of an 
open covering ${\Cal U}_k$ of $X$ with balls of radii${}\sim\delta_k$. 
By (9.7) and Proposition 8.7~ii), we find
$$
{i\over\pi}\ddbar\wt\psi_{c,k}\ge-\min\Big({\partial\Phi^{\chi,\rho,s_k}_\infty
\over\partial \Re w_-}(z,-A_k),c\Big)u-C\varepsilon(\delta_k)^{1/2}\omega.
$$
As $\displaystyle\lim_{\Re w\to-\infty}{\partial\Phi^{\chi,\rho,s}
\over\partial\Re w_-}(z,w)=\nu(\varphi,z)=\nu(\psi,z)$, a suitable
choice of $A_k$ ensures that
$$
\wt\lambda_k(z):={\partial\Phi^{\chi,\rho,s_k}_\infty
\over\partial \Re w_-}(z,-A_k)\to\nu(\psi,z)\qquad
\hbox{as $k\to+\infty$}.
$$ 
Furthermore, an appropriate 
choice of the sequences $\delta_k$, $s_k$, $A_k$ guarantees that the 
sequence $\wt\psi_{c,k}$ is non increasing. [The only point we have to mind
about is the effect of a change of the open covering, as the radius
$\delta_k$ of the covering balls decreases to~$0$. However, Proposition
7.14~(iv,$\,$v) shows that the effect can be made negligible with respect to
${1\over k}-{1\over k+1}$, and then everything is ok]. We can ensure as
well that $\lambda_k$ is decreasing, by replacing if necessary
$\wt\lambda_k$ with
$$
\lambda_k(z)=\sup_{\ell\ge k}\wt\lambda_\ell(z).
$$
Finally, the functions $\wt\psi_{c,k}$ that we got are (a priori) just
known to be continuous on $X\ssm E_c(T)$, thanks to Proposition
7.14~(i) and the discussion before Proposition~8.7. Again, Richberg's
approximation theorem [Ric68] shows that we can replace
$\wt\psi_{c,k}$ with a smooth approximation $\psi_{c,k}$ on $X\ssm
E_c(T)$, with $|\wt\psi_{c,k}-\psi_{c,k}|$ arbitrarily small in
uniform norm, and at the expense of losing an extra error term
$\varepsilon_k\omega$ in the lower bound for
${i\over\pi}\ddbar\psi_{c,k}$. Theorem 9.1 is proved.\qed

\section{\S10. Appendix: basic results on $L^2$ estimates}
We state here the basic $L^2$ existence theorems used in the above
sections, concerning $\dbar$ equations or holomorphic functions. The
first of these is the intrinsic manifold version of H\"ormander's $L^2$
estimates [H\"or65,$\,$66], based on the Bochner-Kodaira-Nakano technique
(see also Andreotti-Vesentini [AV65]).

\claim 10.1. Theorem|Let $L$ be a holomorphic line bundle on a weakly
pseudoconvex $n$-dimensional manifold~$X$ equipped with a K\"ahler
metric~$\omega$. Suppose that $L$ has a smooth hermitian metric
whose curvature form satisfies
$$
2\pi\,\Theta(L)+i\ddbar\varphi\ge A\,\omega
$$
where $\varphi$ is an almost psh function and $A$ a positive continuous
function on~$X$. Then for every form $v$ of type $(n,q)$, $q\ge 1$, with
values in $L$, such that $\dbar v=0$ and
$$\int_X {1\over A}|v|^2e^{-\varphi}dV_\omega<+\infty,$$
there exists a form $u$ of type $(n,q-1)$ with values in $L$
such that $\dbar u=v$ and
$$\int_X|u|^2e^{-\varphi}dV_\omega\le{1\over q}
\int_X {1\over A}|v|^2e^{-\varphi}dV_\omega.$$
\endclaim

A {\sl weakly pseudoconvex manifold} is by definition a complex
manifold possessing a smooth weakly pseudoconvex exhaustion function
(examples: Stein manifolds, compact manifolds, the total space of a
Griffiths weakly negative vector bundle, $\ldots$). Suppose that
$\varphi$ has Lelong number $\nu(\varphi,x)=0$ at a given point~$x$.
Then for every $m$ the weight $e^{-m\varphi}$ is integrable in a small
neighborhood $V$ of~$x$ (see [Sko72a]). Let $\theta$ be a cut-off
function equal to $1$ near $x$, with support in~$V$. Let $z$ be
coordinates and let $e$ be a local frame of $L$ on~$V$. For $\varepsilon$
small enough, the curvature form
$$2\pi\,\Theta(L)+i\ddbar\big(\varphi(z)+2\varepsilon\theta(z)\log|z-x|\big)$$
is still positive definite. We apply A.1 to the bundle $L^m$
equipped with the corresponding weight $m(\varphi(z)+2\varepsilon
\theta(z)\log|z-x|)$, and solve the equation $\dbar u=v$ for the $(n,1)$-form
$v=\dbar(\theta(z)P(z)dz_1\wedge\ldots\wedge dz_n\otimes e^m)$ associated to
an arbitrary polynomial~$P$. The $L^2$ estimate shows that the solution $u$
has to vanish at order $\ge q+1$ at~$x$ where $q=[m\varepsilon]-n$,
hence
$$\theta(z)P(z)dz_1\wedge\ldots\wedge dz_n\otimes e^m-u(z)$$
is a holomorphic section of $K_X\otimes L^m$ with prescribed jet of
order $q$ at~$x$.

\claim 10.2. Corollary|Suppose that $2\pi\,\Theta(L)+i\ddbar\varphi\ge
\delta\,\omega$ for some $\delta>0$. Let $x\in X$ be such that
$\nu(\varphi,x)=0$. Then there exists $\varepsilon>0$ such that the
sections in $H^0(X,K_X\otimes L^m)$ generate all jets of order
$\le m\varepsilon$ at $x$ for $m$ large.\qed
\endclaim

We now state the basic $L^2$ extension theorem which was 
needed in several occasions. A detailed proof can be found in [OhT87], [Ohs88]
and [Man93] (see also our lecture notes [Dem96], Theorem 13.6). Only the 
case $q=0$ (dealing with holomorphic sections) does play a role in this work.

\claim 10.3. Ohsawa-Takegoshi theorem|Let $X$ be a weakly pseudoconvex 
$n$-dimensional complex manifold
equipped with a K\"ahler metric $\omega$, let $L$ $($resp.\ $E)$ be
a hermitian holomorphic line bundle $($resp.\ a hermitian holomorphic 
vector bundle of rank $r$ over~$X)$, and $s$ a global holomorphic 
section of~$E$. Assume that $s$ is generically transverse to the zero
section, and let
$$
Y=\big\{x\in X\,;\;s(x)=0, \Lambda^r ds(x)\not= 0\big\},\qquad
p=\dim Y=n-r.
$$
Moreover, assume that the $(1,1)$-form 
$\ii\Theta(L)+r\,\ii\,\ddbar\log|s|^2$ is semipositive and that there is
a continuous function $\alpha\ge 1$ such that the following two inequalities
hold everywhere on $X:$
\smallskip
\item{\rm a)}
$\displaystyle\ii\Theta(L)+r\,\ii\,\ddbar\log|s|^2\ge\alpha^{-1}
{\{\ii\Theta(E)s,s\}\over|s|^2}\,,$
\smallskip
\item{\rm b)} $|s|\le e^{-\alpha}$.
\smallskip\noindent
Then for every smooth $\dbar$-closed \hbox{$(0,q)$-form $f$} over $Y$ with 
values in the line bundle $\Lambda^nT_X^\star\otimes L$ 
$($restricted to~$Y)$, such that
$\int_Y|f|^2|\Lambda^r(ds)|^{-2}dV_\omega<+\infty$, 
there  exists a $\dbar$-closed $(0,q)$-form $F$ over $X$ with values 
in $\Lambda^nT^\star_X\otimes L$, such that $F$ is smooth over 
$X\ssm\{s=\Lambda^r(ds)=0\}$, satisfies $F_{\restriction Y}=f$ and
$$
\int_X{|F|^2\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}\,,
$$
where $C_r$ is a numerical constant depending only on~$r$.
\endclaim

\claim 10.4. Corollary|Let $Y$ be a pure dimensional closed complex
submanifold of~$\bC^n$, let $\Omega$ be a bounded pseudoconvex open set
and let $\varphi$ be a plurisubharmonic function on~$\Omega$. Then for
any holomorphic function $f$ on $Y\cap\Omega$ with
$$\int_{Y\cap\Omega}|f|^2e^{-\varphi}dV_Y<+\infty,$$
there exists a holomorphic extension F to $\Omega$ such that
$$\int_{\Omega}|F|^2e^{-\varphi}dV\le
A\int_{Y\cap\Omega}|f|^2e^{-\varphi}dV_Y<+\infty.$$
Here $A$ depends only on $Y$ and on the diameter of~$\Omega$.
\endclaim

Finally, a crucial application of Skoda's $L^2$ estimates [Sko72b, 78] for
ideals of holomorphic functions was made in section~5:

\claim 10.5. Theorem|Let $\varphi$ be a plurisubharmonic function
on a pseudoconvex open set~$\Omega\subset\bC^n$ and let
$\sigma_1\ld \sigma_N$ be holomorphic functions on~$\Omega$
$($the sequence $\sigma_j$ can be infinite$)$.
Set $r=\min\{N-1,n\}$ and $|\sigma|^2=\sum|\sigma_j|^2$.
Then, for every holomorphic function $f$ on $\Omega$ such that
$$\int_\Omega|f|^2|\sigma|^{-2(r+1+\alpha)}e^{-\varphi}dV<+\infty,~~~~
\alpha>0,$$
there exist holomorphic functions $g_1\ld g_N$ on $\Omega$ such that
$f=\sum_{1\le j\le N}g_j\sigma_j$ and
$$\int_\Omega|g|^2|\sigma|^{-2(r+\alpha)}e^{-\varphi}dV\le
{\alpha+1\over\alpha}
\int_\Omega|f|^2|\sigma|^{-2(r+1+\alpha)}e^{-\varphi}dV<+\infty.$$
\endclaim

\claim 10.6. Corollary|With the same notations, suppose that
$$\int_\Omega|f|^2|\sigma|^{-2(r+m+\alpha)}e^{-\varphi}dV<+\infty$$ for
some $\alpha>0$ and some integer $m\ge 1$. Then there exist holomorphic
functions $g_L$ for all $L=(\ell_1\ld\ell_m)\in\{1\ld N\}^m$ such that
$$\eqalign{
f=\sum_L g_L\sigma^L~~~~&\hbox{\rm with}~~\sigma^L=
\sigma_{\ell_1}\sigma_{\ell_2}\ldots\sigma_{\ell_m},\cr
\int_\Omega\sum_L|g_L|^2|\sigma|^{-2(r+\alpha)}e^{-\varphi}dV&\le
{\alpha+m\over\alpha}
\int_\Omega|f|^2|\sigma|^{-2(r+m+\alpha)}e^{-\varphi}dV<+\infty.\cr}$$
\endclaim

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

\bigskip\null
\section{References}
\medskip

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}
\vskip20pt

\noindent
Jean-Pierre Demailly\\
Universit\'e de Grenoble I\\
D\'epartement de Math\'ematiques\\
Institut Fourier, UMR 5582 du CNRS\\
38402 Saint-Martin d'H\`eres Cedex, France

\noindent
e-mail: {\tt demailly@ujf-grenoble.fr}
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\noindent
(version of September 22, 1998, printed on \today)
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\noindent{\bf R\'esum\'e} 

{\eightpoint
Le but de ce travail est d'\'etudier les propri\'et\'es de base des
m\'etriques de Finsler sur les fibr\'es vectoriels holomorphes, dans
la perspective d'obtenir des versions g\'eom\'etriques du th\'eor\`eme
de dualit\'e de Serre.  Nous mettons en place un contexte de dualit\'e
dans lequel les propri\'et\'es de pseudoconvexit\'e et de
pseudoconcavit\'e se trouvent \'echang\'ees -- \`a quelques
restrictions techniques pr\`es. Nous montrons que ces propri\'et\'es
de dualit\'e sont li\'ees \`a plusieurs probl\`emes de nature
g\'eom\'etrique, tels que la conjecture de Hartshorne et Schneider,
affirmant que le compl\'ementaire d'une sous-vari\'et\'e alg\'ebrique
de codimension $q$ dont le fibr\'e normal est ample est $q$-convexe.
En toute g\'en\'eralit\'e, nous obtenons une construction fonctorielle
de m\'etriques de Finsler sur les puissances sym\'etriques d'un
fibr\'e vectoriel finsl\'erien. La construction pr\'eserve la
positivit\'e de la courbure, comme le laisse esp\'erer le fait qu'un
produit tensoriel de fibr\'es vectoriels ample soit ample. De l\`a,
nous d\'eduisons une preuve plus courte et plus g\'eometrique d'un
th\'eor\`eme fondamental de r\'egularisation des courants ferm\'es de
type $(1,1)$. La technique repose sur la construction d'un op\'erateur
r\'egularisant pour les fonctions plurisubharmoniques, qui lui m\^eme
met en jeu le choix d'une m\'etrique de Finsler sur le fibr\'e
cotangent et ses puissances sym\'etriques.
}

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{\bf Key words:} Serre duality, pseudoconvexity, pseudoconcavity,
Finsler metric, symmetric power, Chern curvature,
Hartshorne-Schneider conjecture, plurisubharmonic function, positive
current, regularization of currents, Legendre transform, Lelong number,
Ohsawa-Takegoshi theorem, Skoda $L^2$ estimates 
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\noindent
{\bf Mots-Cl\'es:} dualit\'e de Serre, pseudoconvexit\'e, pseudoconcavit\'e,
m\'etrique de Finsler, puissance sym\'etrique, courbure de Chern,
conjecture de Hartshorne-Schneider, fonction plurisubharmonique,
courant positif, r\'egularisation des courants, transform\'ee de Legendre,
nombre de Lelong, th\'eor\`eme de Ohsawa-Takegoshi, estimations $L^2$ de
Skoda
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\noindent
{\bf A.M.S. Classification 1991:} 14B05, 14J45, 32C17, 32J25, 32S05

