%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Bergman bundles and the plurigenera
% conjecture for polarized
% families of Kähler manifolds
%
% Jean-Pierre Demailly (Universit\'e Grenoble Alpes, Institut Fourier)
%
% August 2018
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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%%% Main File

\def\abbreviatedauthorname{J.-P.\ Demailly}

\twolines\twolines
\title{Bergman bundles and the}\vskip3pt
\title{plurigenera conjecture for polarized}\vskip3pt
\title{families of K\"ahler manifolds}\vskip15pt
\titlerunning{Bergman bundles and invariance of plurigenera}

\author{Jean-Pierre Demailly}
\address{Institut Fourier, Universit\'e Grenoble Alpes*}
\footnote{}{\eightrm\baselineskip=9pt\strut\kern-14.5pt*
This work is supported by the
European Research Council project ``Algebraic and K\"ahler Geometry''\\
\hbox{(ERC-ALKAGE, grant No. 670846 from September 2015)}\vskip-9pt}
\vskip2cm

{\eightpoint\baselineskip=10pt
\noindent
{\bf Abstract}. We introduce the concept of Bergman bundle attached
to a hermitian manifold $X$ -- assuming the manifold $X$ to be compact,
although the results are local for a large part. The Bergman bundle
is some sort of infinite dimensional very ample Hilbert bundle
whose fibers isomorphic to the standard Hardy space on the complex
unit ball; however the bundle is locally trivial only in the real analytic
category, and its complex structure is strongly twisted. We compute
the Chern curvature of the Bergman bundle, and show that it is strictly
positive. As an application, we investigate the conjecture on
the invariance of plurigenera in the general situation
of polarized families of compact K\"ahler manifolds -- a long standing
and still unsolved conjecture of Yum-Tong Siu.\par
}

\section{0}{Introduction}

\claim 0.1. Conjecture|Let $\pi:\cX\to S$ be a proper holomorphic
map defining a family of smooth compact K\"ahler manifolds over an
irreducible base~$S$. Assume that $\pi$ admits local polari\-zations,
i.e.\ every point $s_0\in S$
has a neighborhood $V$ such that $\pi^{-1}(V)$ carries a closed smooth
$(1,1)$-form $\omega$ such that $\omega_{|X_t}$ is positive definite
on~$X_t:=\pi^{-1}(t)$. Then the plurige\-nera $p_m(X_t)=h^0(X_t,mK_{X_t})$
of fibers are independent of~$t$ for all $m\ge 0$.
\endclaim

\section{1}{Exponential map and tubular neighborhoods}

Let $X$ be a compact $n$-dimensional complex manifold and $Y\subset X$ a
smooth totally real submanifold, i.e.\ such that $T_Y\cap JT_Y=\{0\}$ for
the complex structure $J$ on~$X$. By a well known result of Grauert,
such a $Y$ always admits a fundamental system of Stein tubular neighborhoods
$U\subset X$ (this would be even true when $X$ is noncompact, but we only
need the compact case here). In fact, if $(\Omega_\alpha)$ is a finite
covering of $X$ such that $Y\cap\Omega_\alpha$ is a smooth complete
intersection $\{z\in\Omega_\alpha\,;\;x_{\alpha,j}(z)=0\}$, $1\leq j\leq q$
(where $q=\codim_\bR Y\geq n$), then
one can take $U=U_\varepsilon=\{\varphi(z)<\varepsilon\}$ where
$$
\varphi(z)=\sum_\alpha \theta_\alpha(z)\sum_{1\leq j\leq q}(x_{\alpha,j}(z))^2
\geq 0
\leqno(1.1)
$$
where $(\theta_\alpha)$ is a partition of unity subordinate to
$(\Omega_\alpha)$. The reason is that $\varphi$ is strictly plurisubharmonic
near~$Y$, as
$$
i\ddbar\varphi_{|Y}=
2i\sum_\alpha \theta_\alpha(z)\sum_{1\leq j\leq q}\partial x_{\alpha,j}
\wedge \dbar x_{\alpha,j}
$$
and $(\partial x_{\alpha,j})_j$ has rank $n$ at every point of $Y$, by the
assumption that $Y$ is totally real.

Now, let $\overline X$ be the complex conjugate manifold associated with
the integrable almost complex structure $(X,-J)$ (in other words,
$\cO_{\overline X}=\overline{\cO_X}$); we denote by $x\mapsto\overline x$
the identity map $\Id:X\to\overline X$ to stress that it is
conjugate holomorphic. The underlying real analytic manifold
$X^\bR$ can be embedded diagonally in $X\times\overline X$ by the
diagonal map $\delta:x\mapsto(x,\overline x)$, and the image
$\delta(X^\bR)$ is a totally real submanifold of $X\times \overline X$.
In fact, if $(z_{\alpha,j})_{1\leq j\leq n}$ is a holomorphic coordinate
system relative to a finite open covering $(\Omega_\alpha)$ of $X$,
then the $\overline z_{\alpha,j}$ define holomorphic coordinates
on $\overline X$ relative to $\overline\Omega_\alpha$, and
the ``diagonal'' $\delta(X^\bR)$ is the totally real submanifold
of pairs $(z,w)$ such that $w_{\alpha,j}=\overline z_{\alpha,j}$ for
all $\alpha,j$. In that case, we can take Stein tubular neighborhoods
of the form $U_\varepsilon=\{\varphi<\varepsilon\}$ where
$$
\varphi(z,w)=\sum_\alpha \theta_\alpha(z)\theta_\alpha(w)
\sum_{1\leq j\leq q}|\overline w_{\alpha,j}-z_{\alpha,j}|^2.
\leqno(1.2)
$$
Here, the strict plurisubharmonicity of $\varphi$ near $\delta(X^\bR)$ is
obvious from the fact that
$$
|w_{\alpha,j}-\overline z_{\alpha,j}|^2=
|z_{\alpha,j}|^2+|w_{\alpha,j}|^2-2\Re(z_{\alpha,j}w_{\alpha,j}).
$$
For $\varepsilon>0$ small, the first projection $\pr_1:U_\varepsilon\to X$
gives a complex fibration whose fibers are $C^\infty$-diffeomorphic to
balls, but they need not be biholomorphic to complex balls in general.
In order to achieve this property, we proceed in the following way. Pick a
real analytic hermitian metric $\gamma$ on $X\,$; take e.g.\ the $(1,1)$-part
$\gamma=g^{(1,1)}={1\over 2}(g+J^*g)$ of the Riemannian metric obtained
as the pull-back $g=\delta^*(\sum_j i df_j\wedge d\overline f_j)$,
where the $(f_j)_{1\leq j\leq N}$ provide a holomorphic immersion of the
Stein neighborhood $U_\varepsilon$ into~$\bC^N$. Let $\exp:T_X\to X$,
$(z,\xi)\mapsto \exp_z(\xi)$ be the exponential map associated with
the metric $\gamma$, in such a way that $\bR\ni t\mapsto\exp_z(t\xi)$ are
geodesics ${D\over dt}({du\over dt})=0$ for the
the Chern connection $D$ on $T_X$ (see e.g.\ [Dem??], [??]). Then $\exp$ is real analytic, and we have Taylor expansions
$$
\exp_z(\xi)=\sum_{\alpha,\beta\in\bN^n}a_{\alpha\beta}(z)\xi^\alpha\overline
\xi^\beta,\quad\xi\in T_{X,z}
$$
with real analytic coefficients $a_{\alpha\beta}$, where
$\exp_z(\xi)=z+\xi+O(|\xi|^2)$ in
local coordinates. The real analyticity means that these expansions are
convergent on a neighborhood $|\xi|_\gamma<\varepsilon_0$ of the zero
section of~$T_X$. We define the fiber-holomorphic part of the exponential
map to be
$$
\exph:T_X\to X,\quad(z,\xi)\mapsto\exph_z(\xi)=
\sum_{\alpha\in\bN^n}a_{\alpha\,0}(z)\xi^\alpha.\leqno(1.3)
$$
It is uniquely defined, is convergent on the same tubular neighborhood
$\{|\xi|_\gamma<\varepsilon_0\}$, has the property that
$\xi\mapsto\exph_z(\xi)$ is holomorphic for $z\in X$ fixed, and
satisfies again
$\exph_z(\xi)=z+\xi+O(\xi^2)$ in coordinates. By the implicit function,
theorem, the map $(z,\xi)\mapsto (z,\exph_z(\xi))$ is a real analytic
diffeomorphism from a neighborhood of the zero section of $T_X$ onto
a neighborhood $V$ of
the diagonal in $X\times X$. Therefore, we get an inverse real
analytic mapping $X\times X\supset V\to T_X$, which we denote by
$(z,w)\mapsto (z,\xi)$, $\xi=\logh_z(w)$, such that
$w\mapsto\logh_z(w)$ is holomorphic on $V\cap(\{z\}\times X)$, and
$\logh_z(w)=w-z+O((w-z)^2)$ in coordinates. The tubular neighborhood
$$
U_{\gamma,\varepsilon}=\{(z,w)\in X\times \overline X\,;~
|\logh_z(\overline w)|_\gamma<\varepsilon\}
$$
is Stein for $\varepsilon>0$ small; in fact, if $p\in X$ and
$(z_1,\ldots,z_n)$ is a holomorphic coordinate system centered at $p$
such that $\gamma_p=i\sum dz_j\wedge d\overline z_j$, then
$|\logh_z(\overline w)|_\gamma^2=|\overline w-z|^2+O(|\overline w-z|^3)$, hence
$i\ddbar|\logh_z(\overline w)|_\gamma^2>0$ at $(p,\overline p)\in
X\times\overline X$. By construction, the fiber $\pr_1^{-1}(z)$ of
$\pr_1:U_{\gamma,\varepsilon}\to X$ is biholomorphic to the
$\varepsilon$-ball of the complex vector space $T_{X,z}$ equipped with
the hermitian metric $\gamma_z$. In this way, we get a locally
trivial real analytic bundle $\pr_1:U_{\gamma,\varepsilon}$ whose fibers
are complex balls; it is important to notice, however, that this ball
bundle need not -- and in fact, will never -- be holomorphically locally trivial.

\section{2}{Curvature of Bergman bundles}

\subsection{2.A}{Bergman version of the Dolbeault complex}

Let $X$ be a $n$-dimensional compact complex manifold equipped with a
real analytic hermitian metric~$\gamma$,
$U_\varepsilon=U_{\gamma,\varepsilon}\subset X\times\overline X$
the ball bundle considered in \S1 and
$$
p=(\pr_1)_{|U_\varepsilon}:U_\varepsilon\to X,\qquad
\overline p=(\pr_2)_{|U_\varepsilon}:U_\varepsilon\to\overline X
$$
the natural projections. Our goal is
to compute the curvature of what we call the ``Bergman
direct image sheaf''
$$
\cB_\varepsilon=p^{L^2}_*(\overline p^*\cO(K_{\overline X})),
\leqno(2.1)
$$
whose space of sections over an open subset $V\subset X$ is
defined to be $\cB_\varepsilon(V)={}$holomorphic sections $f$ of
$\overline p^*\cO(K_{\overline X})$ on $p^{-1}(V)$ that are
in $L^2(p^{-1}(K))$  for all compact subsets $K\compact V$,
i.e.
$$
\int_{p^{-1}(K)}i^{n^2}f\wedge \overline f\wedge \gamma^n
<+\infty,\quad\forall K\compact V.
\leqno(2.2)
$$
Then $\cB_\varepsilon$ is an $\cO_X$-module, and by the
Ohsawa-Takegoshi extension theorem
applied to the subvariety $p^{-1}(z)\subset U_\varepsilon$,
its fiber $B_{\varepsilon,z}=\cB_{\varepsilon,z}/\gm_z\cB_{\varepsilon,z}$ is
isomorphic to the Hilbert space
$\cH^2(\bB_n)$ of $L^2$ holomorphic $n$-forms on $p^{-1}(z)\simeq\bB_n$.
In fact, if we use orthonormal coordinates $(w_1,\ldots,w_n)$
provided by $\exph$ acting on the hermitian space
$(T_{X,z},\gamma_z)$ and centered at $\overline z$, we get
a biholomorphism $\bB_n\to p^{-1}(z)$ given by the
homothety $\eta_\varepsilon:w\mapsto \varepsilon w$,
and a corresponding isomorphism
$$
\leqalignno{
&B_{\varepsilon,z}\longrightarrow \cH^2(\bB_n),\qquad f\longmapsto
g=\eta_\varepsilon^*f,~~\hbox{i.e.\ with $I=\{1,\ldots,n\}$},&(2.3)\cr
&f_I(w)\,dw_1\wedge\ldots\wedge dw_n\longmapsto
\varepsilon^n\,f_I(\varepsilon w)\,dw_1\wedge\ldots\wedge dw_n,\quad
w\in\bB_n,&(2.3')\cr
&\Vert g\Vert^2=\int_{\bB_n}2^{-n}i^{n^2}g\wedge\overline g,\quad
g=g(w)\,dw_1\wedge\ldots\wedge dw_n\in\cH^2(\bB_n).&(2.3'')\cr}
$$
As a consequence, $B_\varepsilon\to X$ can be seen as a locally trivial
(infinite dimensional) real analytic bundle of typical fiber $\cH^2(\bB_n)$,
and we can equip $B_{\varepsilon,z}$ with the natural $L^2$ metric obtained
by declaring (2.3) to be an isometry. In this way, we get a real
analytic hermitian metric on the Bergman bundle~$B_\varepsilon\to X$.

Similarly, we can consider a ``Bergman version'' of the Dolbeault
complex, by introducing a sheaf $\cF_\varepsilon^q$ over $X$ of
$(n,q)$-forms which can be written locally over small
open sets $V\subset X$ as
$$
f(z,w)=\sum_{|J|=q}f_J(z,w)\,dw_1\wedge\ldots\wedge dw_n\wedge
d\overline z_J,\quad
(z,w)\in U_\varepsilon\cap(V\times\overline X),
\leqno(2.4)
$$
where the $f_J(z,w)$ are $C^\infty$ smooth functions on
$U_\varepsilon\cap(V\times\overline X)$
such that $f_J(z,w)$ is holomorphic in $w$ and both
$f$ and $\dbar f=\dbar_zf$ are in $L^2(p^{-1}(K))$ for all
compact subsets $K\compact V$. By construction, the usual $\dbar$
operator yields a complex of sheaves $(\cF^\bullet,\dbar\,)$ and
the kernel $\Ker\dbar:\cF^0\to\cF^1$ coincides with $\cB_\varepsilon$.
We are going to see that $B_\varepsilon$ can somehow be seen as an
infinite dimensional very ample vector bundle. This is already illustrated
by the following result.

\claim 2.5. Proposition|Assume here that $\varepsilon>0$ is taken so small
that $\psi(z,w):=|\logh_z(w)|^2$ is strictly plurisubharmonic up to the
boundary on the compact set $\overline U_\varepsilon
\subset X\times\overline X$. Then the complex of sheaves
$(\cF^\bullet,\dbar)$ is a resolution of $\cB_\varepsilon$ by soft
sheaves over~$X$ $($actually, by $\cC^\infty_X$-modules$\,)$,
and for every holomorphic vector bundle
$E\to X$ and every $q\geq 1$ we have
$$
H^q(X,\cB_\varepsilon\otimes\cO(E))=H^q\big(\Gamma(X,
\cF^q_\varepsilon\otimes\cO(E)),\dbar\big)=0.
$$
Moreover the fibers $B_{\varepsilon,z}\otimes E_z$ are always generated
by global sections of $H^0(X,\cB_\varepsilon\otimes\cO(E))$.
\endclaim

\proof{Proof} By construction, we can equip $U_\varepsilon$ with the
the associated K\"ahler metric $\omega=i\ddbar\psi$ which is smooth
and strictly positive on $\overline U_\varepsilon$.
We can then take an arbitrary smooth hermitian metric $h_E$ on $E$ and
multiply it by $e^{-C\psi_\varepsilon}$ to obtain a bundle with positive
arbitrarily large curvature tensor. The exactness of $\cF^\bullet$ and
cohomology vanishing then follow from
the standard H\"ormander $L^2$ estimates applied either 
locally on $p^{-1}(V)$ for small Stein open sets $V\subset X$,
or globally on $U_\varepsilon$. The global generation of fibers is
an immediate consequence of the Ohsawa-Takegoshi $L^2$ extension theorem.\qed
\medskip

It would not be very hard to show that the same result holds for an
arbitrary coherent sheaf $\cE$ instead of a locally free sheaf $\cO(E)$,
the reason being that $p^*\cE$ admits a resolution by (finite dimensional)
locally free sheaves $\cO_{U_{\varepsilon'}}^{\oplus N}$ on a Stein
neighborhood $U_{\varepsilon'}$ of $\overline U_\varepsilon$.

\subsection{2.B}{Curvature tensor of the Bergman bundle
on ($\bC^n$, std)}

In the model situation $X=\bC^n$ with its standard hermitian metric,
we consider the tubular neighborhood
$$
U_\varepsilon:=\{(z,w)\in\bC^n\times\bC^n\,;\;
|\overline w-z|<\varepsilon\}
\leqno(2.6)
$$
and the projections
$$
p=(\pr_1)_{|U_\varepsilon}:U_\varepsilon\to X=\bC^n,~~
(z,w)\mapsto z,\quad
\overline p=(\pr_2)_{|U_\varepsilon}:U_\varepsilon\to X=\bC^n,~~
(z,w)\mapsto w
$$
If one insists on working on a compact complex manifold, the geometry
is locally identical to that of a complex torus $X=\bC^n/\Lambda$ equipped
with a constant hermitian metric $\gamma$. 

\claim 2.7. Remark|{\rm Again, we have to insist
that the Bergman bundle $B_\varepsilon$ is not holomorphically locally
trivial, even in the above situation where we have invariance by translation.
In the category of real analytic bundles, we get a trivialization from
$\cB_\varepsilon(V)$ onto the topological tensor product of
$\cO_X(V)$ by the Hilbert space $\cH^2(\bB_n)$, namely
$$
\tau:\cB_\varepsilon(V)\mathop{\longrightarrow}\limits^{\simeq}
\cO_X(V)\mathop{\widehat\otimes}\cH^2(\bB_n),\quad f\longmapsto\tau(f)=g,
\quad g(z,w):=f(z,\varepsilon w+\overline z),~~w\in\bB_n.
$$
Complex structures of these bundles are defined by the
$(0,1)$-connections $\dbar_z$ of the associated Dolbeault complexes,
but obviously $\dbar_zf$~and $\dbar_zg$ do not match. In fact,
if we write
$g(z,w)=u(z,w)\,dw_1\wedge\ldots\wedge dw_n\in
C^\infty(V)\mathop{\widehat\otimes}\cH^2(\bB_n)
$ (i.e.\ $u(z,w)$ is a smooth
function that is holomorphic in $w$), then we get
$$
\eqalign{
&f(z,w)=g(z,(w-\overline z)/\varepsilon)=
\varepsilon^{-n}u(z,(w{-}\overline z)/\varepsilon)\,
dw_1\wedge\ldots\wedge dw_n,\cr
&\dbar_zf(z,w)=\varepsilon^{-n}\Big(
\dbar_zu(z,(w{-}\overline z)/\varepsilon)-\varepsilon^{-1}\!\!
\sum_{1\leq j\leq n}{\partial u\over\partial w_j}
(z,(w{-}\overline z)/\varepsilon)\,d\overline z_j\Big)\wedge
dw_1\wedge\ldots\wedge dw_n.\cr}
$$
Therefore the trivialization $\tau_*:f\mapsto u$ yields at the level of
$\dbar$-connections an identification
$$
\tau_*:\dbar_zf\mathop{\longmapsto}\limits^\simeq
\dbar_zu+Au
$$
where the ``connection matrix''
$A\in \Gamma(V,\Lambda^{0,1}T^*_X\otimes_\bC
\End(\cO_X(V)\mathop{\widehat\otimes}\cH^2(\bB_n))$
is the linear operator $A=\Id_{\cO_X(V)}\mathop{\widehat\otimes}\Psi$ induced
by the unbounded Hilbert space operator 
$$
\Psi:\cH^2(\bB_n)\to\cH^2(\bB_n)\otimes_\bC\Lambda^{0,1}T^*_X,\quad
u\mapsto\Psi u=
-\varepsilon^{-1}\sum_{1\leq j\leq n}{\partial u\over \partial w_j}\,
d\overline z_j.
$$
We see that the holomorphic structure of $B_\varepsilon$ is given
by a $(0,1)$-connection that differs by the matrix $A$ from the trivial
$(0,1)$-connection, and as $A$ is unbounded, there is no way we can
make it trivial by a real analytic gauge change with values in Lie
algebra of continuous endomorphisms of $\cH^2(\bB_n)$.\qed
}
\endclaim

We are now going to compute the curvature tensor of the Bergman bundle
$B_\varepsilon$. For the sake of simplicity, we identify here
$\cH^2(\bB_n)$ to the Hardy space of $L^2$ holomorphic functions via
$u\mapsto g=u(w)\,dw_1\wedge\ldots\wedge dw_n$. After rescaling,
we can also assume $\varepsilon=1$, and at least in a first step,
we perform our calculations on $B_1$ rather than~$B_\varepsilon$.
Let us write $w^\alpha=\prod_{1\leq j\leq n}w_j^{\alpha_j}$ for a multiindex
$\alpha=(\alpha_1,\ldots,\alpha_n)\in\bN^n$, and denote by $\lambda$ the
Lebesgue measure on~$\bC^n$. A~straightforward calculation gives
$$
\int_{\bB_n}|w^\alpha|^2d\lambda(w)=\pi^n{\alpha_1!\ldots\alpha_n!\over(|\alpha|+n)!},
\quad |\alpha|=\alpha_1+\cdots+\alpha_n.
$$
In fact, by using polar coordinates $w_j=r_je^{i\theta_j}$ and
writing $t_j=r_j^2$, we get
$$
\int_{\bB_n}|w^\alpha|^2d\lambda(w)=(2\pi)^n\int_{r_1^2+\cdots+r_n^2<1}
r^{2\alpha}\,r_1dr_1\ldots r_ndr_n=\pi^n I(\alpha)
$$
with
$$
I(\alpha)=\pi^n\int_{t_1+\cdots+t_n<1}
t^\alpha\,dt_1\ldots dt_n.
$$
Now, an induction on $n$ together with the Fubini formula gives
$$
\eqalign{
I(\alpha)&=\int_0^1t_n^{\alpha_n}dt_n
\int_{t_1+\cdots+t_{n-1}<1-t_n}(t')^{\alpha'}dt_1\ldots dt_{n-1}\cr
&=I(\alpha')\int_0^1(1-t_n)^{\alpha_1+\cdots+\alpha_{n-1}+n-1}t_n^{\alpha_n}
dt_n\cr}
$$
where $t'=(t_1,\ldots,t_{n-1})$ and $\alpha'=(\alpha_1,\ldots,\alpha_{n-1})$.
As $\int_0^1x^a(1-x)^bdt={a!b!\over(a+b+1)!}$, we get inductively
$$
I(\alpha)={(|\alpha'|+n-1)!\alpha_n!\over(|\alpha|+n)!}I(\alpha')~~~
\Rightarrow~~~I(\alpha)={\alpha_1!\ldots\alpha_n!\over(|\alpha|+n)!}.
$$
This implies that a Hilbert (orthonormal) basis of $\cO\cap L^2(\bB_n)$ is
$$
e_\alpha(w)=\pi^{-n/2}
\sqrt{{(|\alpha|+n)!\over\alpha_1!\ldots\alpha_n!}}\,w^\alpha.
\leqno(2.8)
$$
As a consequence, and quite classically, the Bergman kernel of the unit
ball $\bB_n\subset\bC^n$ is
$$
K_n(w)=\sum_{\alpha\in\bN^n}|e_\alpha(w)|^2
=\pi^{-n}\sum_{\alpha\in\bN^n}
{(|\alpha|+n)!\over\alpha_1!\ldots\alpha_n!}\,|w^\alpha|^2
=n!\pi^{-n}(1-|w|^2)^{-n-1}.\leqno(2.9)
$$
If we come back to $U_\varepsilon$ for $\varepsilon>0$ not necessarily equal
to $1$ (and do not omit any more the trivial $n$-form
$dw_1\wedge\ldots\wedge dw_n$), we have
to use a rescaling $(z,w)\mapsto(\varepsilon^{-1}z,\varepsilon^{-1}w)$.
This gives for the Hilbert bundle $B_\varepsilon$ a real analytic
orthonormal frame
$$
e_\alpha(z,w)=\pi^{-n/2}\varepsilon^{-|\alpha|-n}
\sqrt{{(|\alpha|+n)!\over\alpha_1!\,...\,\alpha_n!}}\;
(w-\overline z)^\alpha\;dw_1\wedge\ldots\wedge dw_n
\leqno(2.10)
$$
A germ of holomorphic section $\sigma\in\cO(B_\varepsilon)$ near
$z=0$ (say) is thus given by a convergent power series
$$
\sigma(z,w)=\sum_{\alpha\in \bN^n}\xi_\alpha(z)\,e_\alpha(z,w)
$$
such that the functions $\xi_\alpha$ are real analytic on a neighborhood of $0$
and satisfy the following two conditions:
$$\leqalignno{
&|\sigma(z)|_h^2:=\sum_{\alpha\in \bN^n}|\xi_\alpha(z)|^2~~\hbox{is uniformly convergent},
&(2.11)\cr
&\dbar_{z_k}\sigma(z,w)=\sum_{\alpha\in \bN^n}
\dbar_{z_k}\xi_\alpha(z)\,e_\alpha(z,w)+
\xi_\alpha(z)\,\dbar_{z_k}e_\alpha(z,w)\equiv 0.&(2.12)\cr}
$$
Let $c_k=(0,\ldots,1,\ldots, 0)$ be the canonical basis of
the $\bZ$-module $\bZ^n$. A straightforward calculation from (2.10) yields
$$
\dbar_{z_k}e_\alpha(z,w)=-\varepsilon^{-1}
\sqrt{\alpha_k(|\alpha|+n)}\; e_{\alpha-c_k}(z,w).
$$
We have the slight problem that the coefficients are unbounded as
$|\alpha|\to+\infty$, and therefore the two terms occurring in
(2.12) need not form convergent series when taken separately. However
if we take $\sigma\in\cO(B_{\rho\varepsilon})$ in a slightly bigger
tubular neighborhood ($\rho>1$), the $L^2$ condition implies that
$\sum_\alpha(\rho')^{2|\alpha|}|\xi_\alpha|^2$ is uniformly convergent
for every $\rho'\in{}]1,\rho[\,$,
and this is more than enough to ensure convergence, since the growth of
$\alpha\mapsto\sqrt{\alpha_k(|\alpha|+n)}$ is at most linear; we can even
iterate as many derivatives as we want. For a smooth section
$\sigma\in C^\infty(B_{\rho\varepsilon})$, the coefficients
$\xi_\alpha$ are smooth, with $\sum\rho^{2|\alpha|}
|\partial_z^\beta\dbar_z^\gamma\xi_\alpha|^2$
convergent for all $\beta,\gamma$, and we get
$$
\eqalign{
\dbar_{z_k}\sigma(z,w)
&=\sum_{\alpha\in\bN^n}\dbar_{z_k}\xi_\alpha(z)\,e_\alpha(z,w)+
\xi_\alpha(z)\,\dbar_{z_k}e_\alpha(z,w)\cr
&=\sum_{\alpha\in\bN^n}\dbar_{z_k}\xi_\alpha(z)\,e_\alpha(z,w)-
\varepsilon^{-1}\sqrt{\alpha_k(|\alpha|+n)}\;
\xi_\alpha(z)\;e_{\alpha-c_k}(z,w)\cr
&=\sum_{\alpha\in\bN^n}\big(\dbar_{z_k}\xi_\alpha(z)
-\varepsilon^{-1}\sqrt{(\alpha_k+1)(|\alpha|+n+1)}\;
\xi_{\alpha+c_k}(z)\big)\;e_\alpha(z,w),\cr}
$$
after replacing $\alpha$ by $\alpha+c_k$ in the terms containing
$\varepsilon^{-1}$. The $(0,1)$-part $\nabla_h^{0,1}$ of the Chern connection
$\nabla_h$ of $(B_\varepsilon,h)$ with respect to the orthonormal
frame~$(e_\alpha)$ is thus given by
$$
\nabla_h^{0,1}\sigma
=\sum_{\alpha\in\bN^n}\Big(\dbar\xi_\alpha
-\sum_k\varepsilon^{-1}\sqrt{(\alpha_k+1)(|\alpha|+n+1)}\;
\xi_{\alpha+c_k}\;d\overline z_k\Big)\otimes e_\alpha.\leqno(2.13)
$$
The $(1,0)$-part can be derived from the identity
$\partial|\sigma|_h^2=\langle\nabla_h^{1,0}\sigma,\sigma\rangle_h+
\langle\sigma,\nabla_h^{0,1}\sigma\rangle_h$.
However
$$
\eqalign{
\partial_{z_j}|\sigma|_h^2
&=\partial_{z_j}\sum_{\alpha\in\bN^n}\xi_\alpha\overline\xi_\alpha=
\sum_{\alpha\in\bN^n}(\partial_{z_j}\xi_\alpha)\;\overline\xi_\alpha+
\xi_\alpha\;(\overline{\dbar_{z_j}\xi_\alpha}\,)\cr
&=\sum_{\alpha\in\bN^n}
\Big(~\partial_{z_j}\xi_\alpha+\varepsilon^{-1}
\sqrt{\alpha_j(|\alpha|+n)}\;
\xi_{\alpha-c_j}\Big)\,\overline\xi_\alpha\cr
&+\sum_{\alpha\in\bN^n}\xi_\alpha\Big(~\overline{\dbar_{z_j}\xi_\alpha
-\varepsilon^{-1}\sqrt{(\alpha_j+1)(|\alpha|+n+1)}\;
\xi_{\alpha+c_j}}~\Big).\cr}
$$
For $\sigma\in C^\infty(B_{\rho\varepsilon})$, it follows from there that 
$$
\nabla_h^{1,0}\sigma=
\sum_{\alpha\in\bN^n}\Big(\partial\xi_\alpha+\varepsilon^{-1}\sum_j
\sqrt{\alpha_j(|\alpha|+n)}\;\xi_{\alpha-c_j}dz_j\Big)\otimes e_\alpha.
\leqno(2.14)
$$
Finally, to find the curvature tensor of $(B_\varepsilon,h)$, we only
have to compute the $(1,1)$-form $(\nabla_h^{1,0}\nabla_h^{0,1}+
\nabla_h^{0,1}\nabla_h^{1,0})\sigma$ and take the terms that contain
no differentiation at all, especially in view of the usual identity
$\partial\dbar+\dbar\partial=0$ and the fact that we also have here
$(\nabla_h^{1,0})^2=0$, $(\nabla_h^{0,1})^2=0$. As
$(\alpha-c_j)_k=\alpha_k-\delta_{jk}$ and $(\alpha+c_k)_j=
\alpha_j+\delta_{jk}$, we are left with
$$
\eqalign{
&\big(\nabla_h^{1,0}\nabla_h^{0,1}+\nabla_h^{0,1}\nabla_h^{1,0}\big)\sigma\cr
&~~{}=
-\,\varepsilon^{-2}\sum_{\alpha\in\bN^n}\sum_{j,k}
\sqrt{\alpha_j(|\alpha|{+}n)}\;
\sqrt{(\alpha_k{-}\delta_{jk}{+}1)(|\alpha|{+}n)}\;
\xi_{\alpha-c_j+c_k}\;dz_j\wedge d\overline z_k\otimes e_\alpha\cr
&~~{\phantom{{}={}}}{+}\,\varepsilon^{-2}\sum_{\alpha\in\bN^n}\sum_{j,k}
\sqrt{(\alpha_j{+}\delta_{jk})(|\alpha|{+}n{+}1)}\;
\sqrt{(\alpha_k{+}1\rlap{\hbox{$\phantom{\alpha_j}$}})(|\alpha|{+}n{+}1)}\;
\xi_{\alpha-c_j+c_k}\;dz_j\wedge d\overline z_k\otimes e_\alpha.\cr
&~~{}=
-\,\varepsilon^{-2}\sum_{\alpha\in\bN^n}\sum_{j,k}
\sqrt{(\alpha_j{-}\delta_{jk})(\alpha_k{-}\delta_{jk})}\;(|\alpha|+n-1)\;
\xi_{\alpha-c_j}\;dz_j\wedge d\overline z_k\otimes e_{\alpha-c_k}\cr
&~~{\phantom{{}={}}}{+}\,\varepsilon^{-2}\sum_{\alpha\in\bN^n}\sum_{j,k}
\sqrt{\alpha_j\alpha_k}\;(|\alpha|+n)\;
\xi_{\alpha-c_j}\;dz_j\wedge d\overline z_k\otimes e_{\alpha-c_k}.\cr
&~~{}=\phantom{{}-\,}
\varepsilon^{-2}\sum_{\alpha\in\bN^n}\sum_{j,k}
\sqrt{\alpha_j\alpha_k}\;\xi_{\alpha-c_j}\;
dz_j\wedge d\overline z_k\otimes e_{\alpha-c_k}\cr
&~~{\phantom{{}={}}}{+}\,\varepsilon^{-2}\sum_{\alpha\in\bN^n}
\sum_{j}(|\alpha|+n-1)\;
\xi_{\alpha-c_j}\;dz_j\wedge d\overline z_j\otimes e_{\alpha-c_j},\cr}
$$
where the last summation comes from the subtraction of the diagonal terms
$j=k$. By changing $\alpha$ into $\alpha+c_j$ in that summation,
we obtain the following expression of the curvature tensor
of $(B_\varepsilon,h)$.

\claim 2.15. Theorem|The curvature tensor of the Bergman bundle
$(B_\varepsilon,h)$ is given by
$$
\langle\Theta_{B_\varepsilon,h}\sigma(v,Jv),\sigma\rangle_h
=\varepsilon^{-2}\sum_{\alpha\in\bN^n}\Bigg(\bigg|
\sum_j\sqrt{\alpha_j}\;\xi_{\alpha-c_j}v_j\bigg|^2
+\sum_{j}(|\alpha|+n)\;|\xi_\alpha|^2|v_j|^2\Bigg)
$$
for every $\sigma=\sum_\alpha\xi_\alpha e_\alpha\in B_{\rho\varepsilon}$,
$\rho>1$, and every tangent vector $v=\sum v_j\;\partial/\partial z_j$.
\endclaim

The above curvature hermitian tensor is positive definite, and even
positive definite unbounded if we view it as a hermitian form
on $T_X\otimes B_\varepsilon$ rather than on $T_X\otimes B_{\rho\varepsilon}$.
This is not so surprising since the connection matrix was already an
unbounded operator. Philosophically, the very ampleness of~$B_\varepsilon$
was also a strong indication that the curvature should have been positive.
Observe that we have in fact
$$
\leqalignno{
\varepsilon^{-2}\sum_{\alpha\in\bN^n}
\sum_{j}&(|\alpha|+n)\;|\xi_\alpha|^2|v_j|^2\cr
&\leq\langle\Theta_{B_\varepsilon,h}\sigma(v,Jv),\sigma\rangle_h
\leq 2\varepsilon^{-2}\sum_{\alpha\in\bN^n}
\sum_{j}(|\alpha|+n)\;|\xi_\alpha|^2|v_j|^2,&(2.16)\cr}
$$
thanks to the Cauchy-Schwarz inequality
$$
\eqalign{
\sum_{\alpha\in\bN^n}\bigg|\sum_j\sqrt{\alpha_j}\;\xi_{\alpha-c_j}v_j\bigg|^2
&\leq\sum_\ell|v_\ell|^2\sum_{\alpha\in\bN^n}\sum_j\alpha_j|\xi_{\alpha-c_j}|^2
=\sum_\ell|v_\ell|^2\sum_j\sum_{\alpha\in\bN^n}\alpha_j|\xi_{\alpha-c_j}|^2\cr
&=\sum_\ell|v_\ell|^2\sum_j\sum_{\alpha\in\bN^n}(\alpha_j+1)|\xi_\alpha|^2
=\sum_\ell|v_\ell|^2\sum_{\alpha\in\bN^n}(|\alpha|+n)|\xi_\alpha|^2.\cr}
$$

\subsection{2.C}{Curvature of Bergman bundles on compact
hermitian manifolds}

We consider here the general situation of a compact hermitian manifold
$(X,\gamma)$ described in \S1, where $\gamma$ is real analytic and
$\exph$ is the associated partially holomorphic exponential map.
Fix a point $x_0\in X$, and use a holomorphic system of coordinates
$(z_1,\ldots,z_n)$ centered at $x_0$, provided
by $\exph_{x_0}:T_{X,x_0}\supset V\to X$. If we take $\gamma_{x_0}$
orthonormal coordinates on $T_{X,x_0}$, then by construction
the fiber of $p:U_\varepsilon\to X$ over $x_0$ is the
standard $\varepsilon$-ball in the coordinates $(w_j)=(\overline z_j)$.
Let $T_X\to V\times\bC^n$ be the trivialization of $T_X$ in the
coordinates~$(z_j)$, and
$$
X\times X\to T_X,\quad
(z,w)\mapsto\xi=\logh(z,w)
$$
the expression of $\logh$ near $(x_0,\overline x_0)$,
that is, near $(z,w)=(0,0)$. By our choice of coordinates, we have
$\logh(0,w)=w$ and of course $\logh(z,z)=0$, hence
we get a real analytic expansion of the form
$$
\eqalign{
\logh(z,w)=w-z&+\sum z_ja_j(w)+\sum\overline z_ja'_j(w)\cr
&+\sum z_jz_kb_{jk}(w)
+\sum \overline z_j\overline z_kb'_{jk}(w)
+\sum z_j\overline z_kc_{jk}(w)+O(|z|^3)\cr}
$$
with holomorphic coefficients $a_j$, $a'_j$, $b_{jk}$, $b'_{jk}$, $c_{jk}$
such that $a_j(0)=a'_j(0)=0$. In fact by [Dem??], we always have
$da'_j(0)=0$, and if $\gamma$ is K\"ahler, the equality $da_j(0)=0$
also holds; we will not use these properties  here. In coordinates, we
then have locally near $(0,0)\in\bC^n\times\bC^n$
$$
U_{\varepsilon,z}=\big\{(z,w)\in\bC^n\times\bC^n\,;\;
|\Phi(z,w)|<\varepsilon\big\}
$$
where $\Phi(z,w)=\overline{\logh(z,\overline w)}$ has a similar expansion
$$
\leqalignno{
\Phi(z,w)=w-\overline z&+\sum z_ja_j(w)+\sum\overline z_ja'_j(w)\cr
&+\sum z_jz_kb_{jk}(w)
+\sum \overline z_j\overline z_kb'_{jk}(w)
+\sum z_j\overline z_kc_{jk}(w)+O(|z|^3)&(2.17)\cr}
$$
(when going from $\logh$ to $\Phi$, the coefficients $a_j$, $a'_j$ and
$b_j$, $b'_j$ get twisted, but we do not care and keep the same
notation for $\Phi$, as we will not refer to $\logh$ any more).
In this situation, the Hilbert bundle $B_\varepsilon$
has a real analytic normal
frame given by $\wt e_\alpha=\Phi^*e_\alpha$ where
$$
e_\alpha(w)=\pi^{-n/2}\varepsilon^{-|\alpha|-n}
\sqrt{{(|\alpha|+n)!\over\alpha_1!\,...\,\alpha_n!}}\;
w^\alpha\;dw_1\wedge\ldots\wedge dw_n
\leqno(2.18)
$$
and the pull-back $\Phi^*e_\alpha$ is taken with respect to $w\mapsto\Phi(z,w)$
($z$ being considered as a parameter). For a local section
$\sigma=\sum_\alpha\xi_\alpha\wt e_\alpha\in C^\infty(B_{\rho\varepsilon})$
we find
$$
\dbar_{z_k}\sigma(z,w)=
\sum_{\alpha\in\bN^n}\dbar_{z_k}\xi_\alpha(z)\,\wt e_\alpha(z,w)+
\xi_\alpha(z)\,\dbar_{z_k}\wt e_\alpha(z,w)
$$
Near $z=0$, by taking the derivative of $\Phi^*e_\alpha(z,w)$, we find
$$
\eqalign{
\dbar_{z_k}\wt e_\alpha(z,w)=
&-\varepsilon^{-1}\sqrt{\alpha_k(|\alpha|+n)}\;\wt e_{\alpha-c_k}(z,w)\cr
&+\varepsilon^{-1}\sum_m
\sqrt{\alpha_m(|\alpha|+n)}\;
\bigg(a'_{k,m}(w)+\sum_jz_jc_{jk,m}(w)\bigg)\;
\wt e_{\alpha-c_m}(z,w)\cr
&+\sum_m \bigg(
{\partial a'_{k,m}\over\partial w_m}(w)+
\sum_jz_j{\partial c_{jk,m}\over\partial w_m}(w)\bigg)
\;\wt e_\alpha(z,w)
+O(\overline z,|z|^2),\cr}
$$
where the last  sum comes from the expansion of
$dw_1\wedge\ldots\wedge dw_n$, and $a'_{k,m}$, $c_{jk,m}$ are
the $m$-th components of~$a'_k$ and $c_{jk}$. This
gives two additional terms in comparison to the translation invariant case,
but these terms are ``small'' in the sense that the first one vanishes
at $(z,w)=(0,0)$ and the second one does not involve~$\varepsilon^{-1}$.
If $\nabla_{h,0}^{0,1}$ is the \hbox{$\dbar$-connection} associated with the
standard tubular neighborhood $|\overline w-z|<\varepsilon$, we thus find
in terms of the local trivialization $\sigma\simeq\xi=\sum\xi_\alpha
\wt\varepsilon_\alpha$ an expression of the form
$$
\nabla_h^{0,1}\sigma\simeq\nabla_{h,0}^{0,1}\xi+A^{0,1}\xi,
$$
where
$$
\eqalign{
A^{0,1}\bigg(\sum_\alpha&\xi_\alpha\wt e_\alpha\bigg)
=\sum_{\alpha\in\bN^n}\sum_k\cr
&\xi_\alpha\Bigg(\varepsilon^{-1}\sum_m
\sqrt{\alpha_m(|\alpha|+n)}\;
\bigg(a'_{k,m}(w)+\sum_jz_jc_{jk,m}(w)\bigg)\;
d\overline z_k\otimes\wt e_{\alpha-c_m}\cr
&\qquad{}+\sum_m \bigg(
{\partial a'_{k,m}\over\partial w_m}(w)+
\sum_jz_j{\partial c_{jk,m}\over\partial w_m}(w)\bigg)
\;d\overline z_k\otimes\wt e_\alpha\Bigg)+O(\overline z,|z|^2).\cr}
$$
The corresponding $(1,0)$-parts satisfy
$$
\nabla_h^{1,0}\sigma\simeq\nabla_{h,0}^{1,0}\xi+A^{1,0}\xi,\qquad
A^{1,0}=-(A^{0,1})^*,
$$
and the corresponding curvature tensors are related by
$$
\Theta_{\beta_\varepsilon,h}=
\Theta_{\beta_\varepsilon,h,0}+\partial A^{0,1}+\dbar A^{1,0}+
A^{1,0}\wedge A^{0,1}+A^{0,1}\wedge A^{1,0}.\leqno(2.19)
$$
At $z=0$ we have
$$
\eqalign{
A^{0,1}\xi
=\sum_{\alpha\in\bN^n}\sum_k\xi_\alpha\Bigg(\varepsilon^{-1}
&\sum_{m}
\sqrt{\alpha_m(|\alpha|+n)}\;a'_{k,m}(w)\;
d\overline z_k\otimes\wt e_{\alpha-c_m}\cr
+&\sum_m{\partial a'_{k,m}\over\partial w_m}(w)\;
d\overline z_k\otimes\wt e_\alpha\Bigg),\cr
\partial A^{0,1}\xi
=\sum_{\alpha\in\bN^n}\sum_k\xi_\alpha\Bigg(\varepsilon^{-1}
&\sum_{j,m}
\sqrt{\alpha_m(|\alpha|+n)}\;c_{jk,m}(w)\;dz_j\wedge
d\overline z_k\otimes\wt e_{\alpha-c_m}\cr
+&\sum_{j,m}{\partial c_{jk,m}\over\partial w_m}(w)\;
dz_j\wedge d\overline z_k\otimes\wt e_\alpha\Bigg),\cr}
$$
and $A^{1,0}$, $\dbar A^{1,0}$ are, up to the sign, the adjoint endomorphisms
of $A^{0,1}$ and $\partial A^{0,1}$. The unboundedness comes from the fact
that we have unbounded factors $\sqrt{(\alpha_m+1)(|\alpha|+n+1)}\,$;
it is worth noticing that multiplication by a holomorphic factor $u(w)$ is
a continuous operator on the fibers $B_{\varepsilon,z}$, whose norm remains
bounded as $\varepsilon\to 0$. In this setting, it can be seen that
the only term in (2.19) that is (a priori) not small with respect to the
main term $\Theta_{B_\varepsilon,h,0}$ is the term involving
$\varepsilon^{-2}$ in $A^{1,0}\wedge A^{0,1}+A^{0,1}\wedge A^{1,0}$.
When expressed in the quadratic form
$\langle\Theta_{B_\varepsilon,h}\xi,\xi\rangle$, the other terms are indeed
$O(\varepsilon^{-1}\sum(|\alpha|+n)|\xi_\alpha|^2)$ or smaller by the
Cauchy-Schwarz inequality. As a consequence of (2.16), they are bounded by
$O(\varepsilon)\langle\Theta_{B_\varepsilon,h,0}\xi,\xi\rangle$ and are thus
essentially ``negligible''. We now come
to the ``annoying term'' and compute it more or less explicitly by expanding
$a'_{k,m}(w)$ into a power series $\sum_\mu c_{k,m,\mu}w^\mu$,
$|\mu|\geq 1$. The relevant term in $A^{0,1}$ is
$$
\varepsilon^{-1}\sum_{k,m}
\sum_{\mu\in\bN^n\ssm\{0\}}{1\over \mu!}\partial_w^\mu
a'_{km}(0)\; d\overline z_k\otimes W^\mu D_m
$$
where $D_m$ and
$W^\mu=W_1^{\mu_1}\ldots W_n^{\mu_n}$ are
operators on the Hilbert space $\cH^2(B_{\varepsilon,0})$, defined by
$$
D_m\wt e_\alpha=\sqrt{\alpha_m(|\alpha|+n)}\;\wt e_{\alpha-c_m},\quad
W_m(f)=w_m f.
$$
The corresponding term in $A^{1,0}$ is the opposite of the adjoint, namely
$$
-\varepsilon^{-1}\sum_{j,\ell}
\sum_{\lambda\in\bN^n\ssm\{0\}}{1\over \lambda!}\partial_w^\lambda a'_{k\ell}(0)\;
dz_j\otimes D^*_\ell W^{*\lambda}
$$
and the annoying term in $A^{1,0}\wedge A^{0,1}+A^{0,1}\wedge A^{1,0}$ is
$$
\leqalignno{
R=&-\varepsilon^{-2}\sum_{j,k,\ell,m}
\sum_{\lambda,\mu\in\bN^n\ssm\{0\}}
{1\over\lambda!}\partial_w^\lambda a'_{k\ell}(0)\;
{1\over\mu!}\partial_w^\mu a'_{km}(0)\;
dz_j\wedge d\overline z_k\otimes{}&(2.20)\cr
&\kern150pt{}
\Big(D^*_\ell W^{*\lambda}W^\mu D_m-
W^\mu D_mD^*_\ell W^{*\lambda}\Big).\cr}
$$
We have here
$\Vert W_m\Vert\leq\varepsilon$ ($W_m$ is multiplication by $w_m$, and we
have $|w_m|\leq\varepsilon$ on $B_{\varepsilon,0}$). The operators
$D^*_\ell$ and $D_m$ are unbounded, but the important point
is that the commutators have substantially better continuity
than what could be expected a priori. We have for instance
$$
\eqalign{
&W_m(\wt e_\alpha)=\varepsilon\sqrt{{\alpha_m+1\over|\alpha|+n+1}}\;\wt e_{\alpha+c_m},\quad
W_\ell^*(\wt e_\alpha)=\varepsilon\sqrt{{\alpha_\ell\over|\alpha|+n}}\;
\wt e_{\alpha-c_\ell},\cr
&[W^*_\ell,W_m](\wt e_\alpha)=\varepsilon^2\bigg(
{\sqrt{(\alpha_\ell+\delta_{\ell m})(\alpha_m+1)}\over|\alpha|+n+1}-
{\sqrt{\alpha_\ell(\alpha_m+1-\delta_{\ell m})}\over|\alpha|+n}\,\bigg)\;\wt e_{\alpha-c_\ell+c_m},\cr}
$$
and it is easy to see that the coefficient between large braces is bounded
for $\ell\neq m$ by $\sqrt{\alpha_\ell(\alpha_m+1)}/((|\alpha|+n)(|\alpha|+n+1))
\leq(|\alpha|+n)^{-1}$, and for $\ell=m$ we have as well
$$
\bigg|{(\alpha_\ell+1)(|\alpha|+n)-\alpha_\ell(|\alpha|+n+1)\over
(|\alpha|+n)(|\alpha|+n+1)}\bigg|\leq(|\alpha|+n)^{-1}.
$$
Therefore $\Vert [W^*_\ell,W_m](\wt e_\alpha)\Vert\leq
\varepsilon^2(|\alpha|+n)^{-1}$. We obtain similarly
$$
\eqalign{
&D_\ell^*(\wt e_\alpha)=\sqrt{(\alpha_\ell+1)(|\alpha|+n+1)}\;
\wt e_{\alpha+c_\ell},\cr
&[D^*_\ell,D_m](\wt e_\alpha)=\Big(
\sqrt{(\alpha_\ell+1-\delta_{\ell m})\alpha_m}\;(|\alpha|+n)\cr
&\kern100pt{}
-\sqrt{(\alpha_\ell+1)(\alpha_m+\delta_{\ell m})}\;(|\alpha|+n+1)\,\Big)\;
\wt e_{\alpha+c_\ell-c_m}\cr}
$$
and a coefficient between braces controlled by $2(|\alpha|+n)$, thus
$\Vert [D^*_\ell,D_m](\wt e_\alpha)\Vert\leq 2(|\alpha|+n)$. Finally
$$
\eqalign{
&[W^*_\ell,D_m](\wt e_\alpha)=\varepsilon\Bigg(
\sqrt{{(\alpha_\ell-\delta_{\ell m})\alpha_m(|\alpha|+n)\over
|\alpha|+n-1}}%%\cr&\kern100pt{}
-\sqrt{{\alpha_\ell(\alpha_m-\delta_{\ell m})(|\alpha|+n-1)\over
|\alpha|+n}}\;\Bigg)\;
\wt e_{\alpha-c_\ell-c_m}\cr}
$$
with a coefficient between braces less than $1$, thus
$\Vert [W^*_\ell,D_m](\wt e_\alpha)\Vert\leq\varepsilon$. By
adjunction, the same is true for $[D^*_\ell,W_m]$. Now, we observe
that both $D^*_\ell W^{*\lambda}W^\mu D_m(\wt e_\alpha)$ and
$W^\mu D_mD^*_\ell W^{*\lambda}(\wt e_\alpha)$ are multiples
of $\wt e_{\alpha+c_\ell-c_m-\lambda+\mu}$. By considering
the second product $W^\mu D_mD^*_\ell W^{*\lambda}$ and
permuting successively $D_mD^*_\ell$, $D_mW^{*\lambda}$,
$W^\mu D^*_\ell$, $W^\mu W^{*\lambda}$, the diffe\-rence with
$D^*_\ell W^{*\lambda}W^\mu D_m$ is expressed
as a sum of $1+|\lambda|+|\mu|+|\lambda||\mu|$ terms involving commutators.
Since $\Vert W_\ell(\wt e_\alpha)\Vert\leq\varepsilon$,
$\Vert D^*_\ell(\wt e_\alpha)\Vert\leq|\alpha|+n+1$ and
$\Vert D_m(\wt e_\alpha)\Vert\leq |\alpha|+n$, the bounds for the resulting
coefficients are $\varepsilon^{|\lambda|+|\mu|}$ multiplied (in this order) by
$$
2((|\alpha|-|\lambda|)_++n),~~
(|\alpha|-|\lambda|)_++n~\hbox{($|\lambda|$ times)},~~
|\alpha|-|\lambda|+|\mu|)_++n~\hbox{($|\mu|$ times)},~~
$$
and a sum of $|\lambda||\mu|$ products
$$
{((|\alpha|-|\lambda|+|\mu|)_++n)(|\alpha|+n)\over(|\alpha|-|\lambda|)_++n}
\quad\hbox{(the worst bound)}.
$$
In total, if $r>0$ is the radius of convergence of the functions
$a'_{k\ell}(w)$ and $r'<r$, (2.20) implies for any
$\xi=\sum_\alpha\xi_\alpha\wt e_\alpha$
a bound of the form
$$
\eqalign{
\big|\langle R(\xi),\xi\rangle\big|
\leq\sum_{\alpha\in\bN^n}\sum_{\lambda,\mu\in\bN^n\ssm\{0\}}
C&\Big({\varepsilon\over r'}\Big)^{|\lambda|+|\mu|}
(2+|\lambda|+|\mu|+|\lambda||\mu|)\cr
&{((|\alpha|-|\lambda|+|\mu|)_++n)(|\alpha|+n)\over
(|\alpha|-|\lambda|)_++n}|\xi_\alpha||\xi_{\alpha-\lambda+\mu}|.\cr}
$$
We claim that for $\varepsilon\in{}]0,r'[$, the coefficient of
$|\xi_\alpha||\xi_{\alpha-\lambda+\mu}|$ is less than
$$
C'\varepsilon\sqrt{(|\alpha|+n)((|\alpha|-|\lambda|+|\mu|)_++n)}.\leqno(2.21)
$$
In fact $|\lambda|+|\mu|\ge 2$, and for $\delta>0$ arbitrary we have
$2+|\lambda|\leq C''(1+\delta)^{|\lambda|-1/2}$ for some $C''=C''(\delta)>0$,
and a similar inequality for $|\mu|$, thus
$$
C\Big({\varepsilon\over r'}\Big)^{|\lambda|+|\mu|}
(2+|\lambda|+|\mu|+|\lambda||\mu|)\leq
C\,C^{\prime\prime\,2}\;{\varepsilon\over r'}
\Big({(1+\delta)\varepsilon\over r'}\Big)^{|\lambda|+|\mu|-1}.
$$
Now, we split the summation between $|\lambda|+|\mu|\leq(|\alpha|+n)/2$ and
$|\lambda|+|\mu|>(|\alpha|+n)/2$. In the first case the bound (2.21) holds
since
$$
(|\alpha|-|\lambda|)_++n\geq\max\Big({1\over 3}(
(|\alpha|-|\lambda|+|\mu|)_++n,{1\over 2}(|\alpha|+n)\Big).
$$
In the second case, (2.21) also holds, as we then have
$$
{((|\alpha|-|\lambda|+|\mu|)_++n)(|\alpha|+n)\over
(|\alpha|-|\lambda|)_++n}\leq{6\over n}(|\lambda|+|\mu|)^2\leq
C'''\;(1+\delta)^{|\lambda|+|\mu|-1}.
$$
We thus see that (2.21) holds for $\varepsilon\leq r/4$. Under that assumption, the Cauchy-Schwarz inequality together with (2.21) implies
$$
\big|\langle R(\xi),\xi\rangle\big|\leq C'\varepsilon\sum_{\alpha\in\bN^n}
(|\alpha|+n)|\xi_\alpha|^2,\leqno(2.22)
$$
which is what we wanted.

\section{3}{Invariance of plurigenera for polarized K\"ahler families}

Let $\cX\to S$ be a polarized family of compact K\"ahler manifolds.
The goal of this section is to investigate the conjecture asserting
that the plurigenera $p_m(X_t)=h^0(X_t,mK_{X_t})$ of fibers are
independent of~$t$ for all $m\ge 0$. The above conjecture has first been
proved by Y.T.~Siu [Siu98] in the case of projective varieties of
general type (in which case the proof has been translated in a purely
algebraic form by Y.~Kawamata [Kaw99]), and then by [Siu00] and P{\u
a}un [Pau04] in the case of arbitrary projective varieties; in the
nonrestricted projective case, no algebraic proof of the result is
known. Our ambition is to extend the techniques to the K\"ahler
context. This requires substantial
modifications of the proof developed by Siu and P{\u a}un, since
it involved in a crucial manner the use of an auxiliary ample line
bundle. We replace here the ample line bundle by a use of the
Hilbert bundles studied in the previous section.

\claim 3.1. Conjecture|Let $\pi:\cX\to S$ be a proper holomorphic
map defining a family of smooth compact K\"ahler manifolds over an
irreducible base~$S$. Assume that $\pi$ admits local polari\-zations,
i.e.\ every point $s_0\in S$
has a neighborhood $U$ such that $\pi^{-1}(U)$ carries a closed smooth
$(1,1)$-form $\omega$ such that $\omega_{|X_t}$ is positive definite
on~$X_t:=\pi^{-1}(t)$. Then the plurige\-nera $p_m(X_t)=h^0(X_t,mK_{X_t})$
of fibers are independent of~$t$ for all $m\ge 0$.
\endclaim

The above statement would follow directly from the following more technical
result.

\claim 3.2. Conjecture {\rm (generalized version of the Claudon-P\u{a}un
theorem)}|
Let $\pi: \cX \to \Delta$ be a polarized family of compact K\"ahler
manifolds over a disc~$\Delta\subset\bC$, and
let $(\cL_j,h_j)_{0\le j\le N-1}$ be $($singular$)$ hermitian
line bundles with semi-positive curvature currents $i\Theta_{\cL_j,h_j}\ge 0$
on~$\cX$. Assume that
\item{\rm (a)} the restriction of $h_j$ to the central fiber $X_0$ is well
defined $($i.e.\ not identically~$+\infty)$.
\smallskip
\item{\rm (b)} the multiplier ideal sheaf
$\cI(h_{j|X_0})$ is trivial for $1\le j\le N-1$.
\smallskip
\noindent
Then any section $\sigma$ of
$\cO(mK_{\cX}+\sum \cL_j)_{|X_0}\otimes\cI(h_{0|X_0})$ over the central
fiber $X_0$ extends into a section $\widetilde\sigma$ of
$\cO(mK_{\cX}+\sum \cL_j)$ over a certain neighborhood $\cX'=\pi^{-1}(\Delta')$
of $X_0$, where $\Delta'\subset\Delta$ is a sufficienty small disc
centered at~$0$.
\endclaim

\claim 3.3. Remark|{\rm A standard cohomological argument shows that we
can in fact take $\cX'=\cX$ in the conclusion of Theorem 3.2, because
the direct image sheaf $\cE=\pi_*\cO(mK_{\cX}+\sum \cL_j)$
is coherent, and the restriction
$\cE\to\cE\otimes(\cO_\Delta/\gm_0\cO_\Delta)$ induces
a surjective map at the $H^0$ level on the Stein space $\Delta$, so
we can extend $\widetilde\sigma$ mod $\pi^*\gm_0$ to $\cX$.}
\endclaim

\proof{Conjecture $3.2$ implies Conjecture $3.1$} The
invariance of plurigenera is in fact
just obtained as the special case of Conjecture~3.2 when all line
bundles $\cL_j$ and their metrics $h_j$ are trivial. Since the
dimension $t\mapsto
h^0(X_t,mK_{X_t})$ is always upper semicontinuous and since Conjecture 3.2
implies the lower semicontinuity, we conclude that the dimension
must be constant along analytic discs, hence along the irreducible base $S$,
by joining any two points through a chain of analytic discs.\qed

\claim 3.4. Lemma|For $\varepsilon\leq\varepsilon_0$ small enough, one
can find a Stein open subset $\cU'_\varepsilon\subset
\cX'\times\overline X_0$ , where $\cX'=\pi^{-1}(\Delta')\to\Delta'$
is the restriction of $\pi:\cX\to\Delta$ to a disc
$\Delta'\compact\Delta$ centered at~$0$, of sufficiently small radius
$r'\leq r_0(\varepsilon)$, such that the projection
$\pr_1:\cU'_\varepsilon\to \cX'$ is a complex ball bundle over
$\cX'$ that is locally trivial real analytically.
\endclaim

\proof{Proof} The argument is very similar to what we did in \S1, applied
to the central fiber $X_0$ only. Let $\gamma_0$ be a
real analytic hermitian metric on~$X_0$ and
$\exph:T_{X_0}\to X_0$ be the corresponding real analytic
and fiber-holomorphic exponential map associated with $\gamma_0$, as
in \S1. By construction,
$Y=\cX\times \overline X_0$ has a tangent bundle $T_Y$ that admits
a direct sum decomposition $T_Y=\pr_1^*T_\cX\oplus\pr_2^*T_{\overline X_0}$
into holomorphic subbundles. Let $\Phi_t:X_0\to X_t$, $t\in\Delta$, be
a real analytic family of diffeomorphisms such that $\Phi_0=\Id$ (by
the Ehresmann lemma, such a family can be obtained by taking the flow
of a real analytic vector field $\tau$ on $\cX$ that lifts $d/dt\in T_\Delta$
on a neighborhood of $\overline{\Delta'}$).
We consider the graph $\Gamma'$ of $(\Phi_t)$, namely the real
analytic set $\Gamma'\subset\cX'\times\overline X_0$
of points of the form $(\Phi_t(x),\overline x)$, $x\in X_0$,
$t\in\Delta'$, and take
$$
\cU'_\varepsilon=\big\{(\Phi_t(x),\overline{\exph_x(\xi)}\,)\,;~
(\Phi_t(x),\overline x)\in\Gamma',~\xi\in T_{X_0,x},
~|\xi|_\gamma^2<\varepsilon^2-|t|^2)\}.
$$
Let $\logh:X_0\times X_0\to T_{X_0}$ be the inverse of $\exph$ near the
diagonal. Then $\cU'_\varepsilon$ is defined as the set of points
$(w,\overline z)\in\cX'\times \overline X_0$ such that
$$
t=\pi(w)~\hbox{satisfies $|t|<r'={}$radius}~\Delta',\quad
|\logh(\Phi_t^{-1}(w),\overline z)|_{\gamma_0}^2+|t|^2<\varepsilon^2.
$$
Now, the function
$$
\psi(w,z)=|\logh(\Phi_t^{-1}(w),\overline z)|_{\gamma_0}^2+|t|^2
=|\logh(\Phi_{\pi(w)}^{-1}(w),\overline z)|_{\gamma_0}^2+|\pi(w)|^2
\leqno(3.5)
$$
is strictly plurisubharmonic along its zero variety $t=0$, $w=\overline z$
(equal to the diagonal of \hbox{$X_0\times \overline X_0$}, therefore it must
be strictly plurisubharmonic on the domain defined by conditions (3.5)
(let us recall that $\pi:\cX\to\Delta$ is holomorphic).
As a consequence  $\cU'_\varepsilon$ is stein for
\hbox{$\varepsilon\leq\varepsilon_0$} small enough,
and the inequality $r'\leq r_0(\varepsilon):=\varepsilon/2$ guarantees
that the fibers of \hbox{$p=\pr_1:\cU'_\varepsilon\to \cX'$} are nonempty balls.
The other stated properties can be checked exactly as in~\S1.\qed

In order to study Conjecture~3.2, we first state a technical 
extension theo\-rem needed for the proof, which is a special case
of the well-known and extremely powerful Ohsawa-Takegoshi theorem
[OT87], see also [Oh??], [Dem??].

\claim 3.6. Proposition|Let $\pi:\cZ\to\Delta$ be a smooth and proper morphism
from a $($non compact$\,)$ K\"ahler manifold $\cZ$ to a disc
$\Delta\subset\bC$ and let $(\cL,h)$ be
a  $($singular$)$ hermitian line bundle with semi-positive
curvature current $\ii\Theta_{\cL,h}\ge 0$ on~$\cZ$.
Let $\omega$ be a global K\"ahler metric on $\cZ$, and let $dV_\cZ$,
$dV_{Z_0}$ the respective induced volume elements on $\cZ$ and
$Z_0=\pi^{-1}(0)$. Assume that $h_{Z_0}$ is well defined $($i.e.\ almost
everywhere finite$)$. Then any holomorphic section $s$ of
$\cO(K_\cZ+\cL)\otimes\cI(h_{|Z_0})$ extends into a section $\wt s$ over $\cZ$
satisfying an $L^2$ estimate
$$
\int_{\cZ}\Vert \wt s\Vert^2_{\omega\otimes h}dV_\cZ\le C_0
\int_{Z_0}\Vert s\Vert^2_{\omega\otimes h}dV_{Z_0},
$$
where $C_0\ge 0$ is some universal constant $($depending on $\dim\cZ$ and $\diam\Delta$, but otherwise independent of $\cZ$, $\cL$, $\ldots\,)$.
\endclaim

\claim 3.7. Remark|{\rm The assumptions of Proposition 3.6 imply that $\cZ$ is
holomorphically convex and complete K\"ahler, thus the technique
of [Dem??] does apply to yield the result.}
\endclaim

\proof{Attempt of proof of Conjecture $3.2$} Let $p=\pr_1:\cU'_\varepsilon\to
\cX'$ be as in Lemma~3.4, and $q=\pr_2:\cU'_\varepsilon\to\overline X_0$. 
We take $\varepsilon<\varepsilon_0$ and
use on $\cZ:=\cU'_\varepsilon$ a K\"ahler metric $\omega_0$ defined
on the Stein manifold $\cU'_{\varepsilon_0}$. On can define e.g.\ $\omega_0$
as the $i\ddbar$ of a strictly plurisubharmonic exhaustion function
on $\cU'_{\varepsilon_0}$, but we can also take the restriction 
of $\pr_1^*\omega+\pr_2^*\smash{\overline\omega_{|\overline X_0}}$
where $\omega$ is the K\"ahler metric on the total space $\cX$,
and $\overline\omega={}{-}\,\omega$ the corresponding K\"ahler metric on
the conjugate space $\overline\cX$.
\medskip

\noindent {\it First step: construction of a sequence of extensions
on $\cZ=\cU'_\varepsilon$ via the Ohsawa-Takegoshi extension theorem.}

The strategy is to apply iteratively the special case 3.6 of the
Ohsawa-Takegoshi extension theorem on the total space of the fibration
$$
\pi'=\pi\circ p:\cZ=\cU'_\varepsilon\to\cX'\to\Delta',
$$
and to extend sections of ad hoc pull-backs $p^*\cG$
from the zero fiber $Z_0=\cU'_\varepsilon\cap(X_0\times\overline X_0)$
to the whole of $\cZ=\cU'_\varepsilon$. We write $h_j=e^{-\varphi_j}$ in terms
of local plurisubharmonic weights, and define
inductively a sequence of line bundles $\cG_m$ by putting $\cG_0=\cO_X$ and
$$
\cG_m=\cG_{m-1}+K_\cX+\cL_r\qquad\hbox{if $m=Nq+r$,~~$0\le r\le N-1$}.
$$
By construction we have 
$$
\eqalign{
\cG_m&=mK_\cX+\cL_1+\cdot\cdot\cdot+\cL_m,\kern28pt\hbox{for $1\le m\le N-1$}\,,\cr
\cG_{m+N}-\cG_m=\cG_N&=NK_\cX+\cL_0+\cdot\cdot\cdot+\cL_{N-1}\,,\quad
\hbox{for all $m\geq 0$.}\cr}
$$
The game is to construct inductively families of sections, say
$\{\wt f_j^{(m)}\}_{j=1,\ldots, J(m)}$, of $p^*\cG_m$ over~$\cZ$, together
with ad hoc $L^2$ estimates, in such a way that
{\plainitemindent=9mm
\smallskip
\item{\rm (3.8)} for $m=0,\ldots,N-1$, $p^*\cG_m$ is generated by $L^2$
sections $\{\wt f_j^{(m)}\}_{j=1,\ldots, J(m)}\;$ on $\cU'_{\varepsilon_0}\,$;
\smallskip
\item{\rm (3.9)} we have the $m$-periodicity relations
$J(m+N)=J(m)$ and $\wt f_j^{(m)}$ is an extension of
$f_j^{(m)}:=(p^*\sigma)^qf_j^{(r)}$ over $\cZ$ for $m=Nq+r$,
where $f_j^{(r)}:=\wt f^{(r)}_{j|Z_0}$, $0\le r\le N-1$.
\smallskip}
\noindent
Property (3.8) can certainly be achieved since $\cU'_{\varepsilon_0}$ is Stein,
and for $m=0$ we can take $J(0)=1$ and $\widetilde f^{(0)}_1=1$.
Now, by induction, we equip $p^*\cG_{m-1}$ with the
tautological metric
$|\xi|^2/\sum|\wt f_j^{(m-1)}(x)|^2$, and
$$
\widetilde\cG_m:=p^*\cG_m-K_\cZ=p^*\cG_m-(p^*K_\cX+q^*K_{\overline X_0})=
p^*(\cG_{m-1}+\cL_r)-q^*K_{\overline X_0}
$$
with that metric multiplied by $p^*h_r=e^{-p^*\varphi_r}$ and a fixed smooth
metric $e^{-\psi}$ of positive curvature on
$(-q^*K_{\overline X_0})_{|\cU'_{\varepsilon_0}}$
(remember that $\cU'_{\varepsilon_0}$ is Stein!).
It is clear that these metrics have semi-positive curvature currents on $\cZ$
(by adjusting $\psi$, we could even take them to be strictly positive
if we wanted).
In this setting, we apply the Ohsawa-Takegoshi theorem to the line bundle
\hbox{$K_\cZ+\widetilde\cG_m=p^*\cG_m$}, and extend in this way
$f_j^{(m)}$ into a section $\wt f_j^{(m)}$ over~$\cZ$. By construction
the pointwise norm of that section
in $p^*\cG_{m|X_0}$ in a local trivialization of the bundles involved is
the ratio
$$
{|f_j^{(m)}|^2\over\sum_\ell|f_\ell^{(m-1)}|^2}e^{-p^*\varphi_r-\psi},
$$
up to some fixed smooth positive factor depending only on the metric induced
by $\omega_0$ on~$K_\cZ$. However, by the induction relations, we have
$$
{\sum_j|f_j^{(m)}|^2\over\sum_\ell|f_\ell^{(m-1)}|^2}e^{-p^*\varphi_r}=
\cases{
\displaystyle
{\sum_j|f_j^{(r)}|^2\over\sum_\ell|f_\ell^{(r-1)}|^2}e^{-p^*\varphi_r}
&\hbox{for $m=Nq+r$, $0<r\le N-1$},\cr
\displaystyle
{\sum_j|f_j^{(0)}|^2\over\sum_\ell|f_\ell^{(N-1)}|^2}
|p^*\sigma|^2e^{-p^*\varphi_0}
&\hbox{for $m\equiv 0\mod N$, $m>0$}.\cr}
$$
Since the sections $\{f_j^{(r)}\}_{0\leq r<N}$ generate their line bundle on
$\cU_{\varepsilon_0}\supset\overline{\cU'_\varepsilon}$, the ratios
involved are positive functions without zeroes and poles, hence smooth and
bounded [possibly after shrinking a little bit the base disc $\Delta'$,
as is permitted].
On the other hand, assumption 3.2~(b) and the fact that $\sigma$ has
coefficients in the multiplier ideal sheaf $\cI(h_{0|X_0})$ tell us that
$\smash{e^{-p^*\varphi_r}}$, $1\le r<m$ and
$|p^*\sigma|^2\smash{e^{-p^*\varphi_0}}$ are locally integrable on~$Z_0$.
It follows that there is a constant $C_1\ge 0$ such that
$$
\int_{Z_0}
{\sum_j|f_j^{(m)}|^2\over\sum_\ell|f_\ell^{(m-1)}|^2}e^{-p^*\varphi_r-\psi}
dV_{\omega_0} \le C_1
$$
for all $m\ge 1$ (of course, the integral certainly involves finitely many
trivializations of the bundles involved, whereas the integrand expression is
just local in each chart). Inductively, the $L^2$ extension theorem produces
sections $\wt f_j^{(m)}$ of $p^*\cG_m$ over $\cZ$ such that
$$
\int_{\cZ}
{\sum_j|\wt f_j^{(m)}|^2\over\sum_\ell|\wt f_\ell^{(m-1)}|^2}e^{-p^*\varphi_r-\psi}
dV_{\omega_0} \le C_2=C_0C_1.
$$

\noindent
{\it Second step: applying the H\"older inequality.} Put
$k=Nq(k)+r(k)$ with $0\leq r(k)<N$, and
take $m=Nq(m)$ to be a multiple of~$N$. 
The H\"older inequality $|\int\prod_{1\leq k\leq m}u_kd\mu|\leq
\prod_{1\leq k\leq m}(\int|u_k|^md\mu)^{1/m}$ applied to the measure
$\mu=dV_{\omega_0}$ and to the product of functions
$$
\Bigg({\sum_j|\wt f_j^{(m)}|^2\over\sum_\ell|\wt f_\ell^{(0)}|^2}\Bigg)^{1/m}
e^{-{1\over N}p^*(\varphi_0+\ldots+\varphi_{N-1})-\psi}
=\prod_{1\leq k\leq m}
\Bigg({\sum_j|\wt f_j^{(k)}|^2\over\sum_\ell|\wt f_\ell^{(k-1)}|^2}
e^{-p^*\varphi_{r(k)}-\psi}\Bigg)^{1/m}
$$
in which $\sum_\ell|\wt f_\ell^{(0)}|^2=|\wt f_1^{(0)}|^2=1$ and
$\sum_j|\wt f_j^{(m)}|^2=|\wt f_1^{(m)}|^2$, implies that
$$
\int_{\cZ}\big|\wt f_1^{(m)}\big|^{2/m}
e^{-{1\over N}p^*(\varphi_0+\ldots+\varphi_{N-1})-\psi}dV_{\omega_0}\le C_2.
\leqno(3.10)
$$
As the functions
$\smash{\varphi_{r(k)}}$ and $\psi$ are locally bounded from above,
we infer from there the weaker inequality
$$
\int_{\cZ}\big|\wt f_1^{(m)}\big|^{2/m}dV_{\omega_0}\le C_3.
\leqno(3.10')
$$
The last inequality is to be understood as an inequality that holds in fact
only locally over~$\cX'$, on sets of the form $p^{-1}(V)$, where $V\compact\cX'$
are small coordinate open sets where our line bundles are trivial, so that
the section $\wt f_1^{(m)}$ of $q(m)\,p^*(NK_\cX+\sum \cL_j)$ can be viewed
as a holomorphic function on $p^{-1}(V)$.\medskip

\noindent
{\it Third step: construction of a singular hermitian metric on
$NK_\cX+\sum\cL_j$}. The rough idea is to extract a weak limit of the
$m$-th root occurring in (3.10), $(3.10')$, combined with an integration
on the fibers of $p:\cZ=\cU'_\varepsilon\to\cX'$, to get a singular
hermitian metric on~\hbox{$NK_\cX+\sum \cL_j$}. This is the crucial
step in the proof, and the
place where the K\"ahler setup requires new arguments; especially, the
integration on fibers makes the weak limit argument much less obvious
than in the projective setup, and requires at least the results of \S2
on Bergman bundles.

\claim 3.11. Proposition|Assume that the sections $\widetilde f^{(m)}_1$
have been constructed on a slightly bigger tubular neighborhood
$\cU''_{\rho''\varepsilon}\to\cX'':=\pi^{-1}(\Delta'')$ with $\rho''>1$,
and $\Delta''$ of radius $r''>r'$ $[$this~condition can of course
always be achieved, since we have some flexibility on the choice of the
tubular neighborhoods$\,]$. Then there exists a subsequence
$m\in M_0\subset\bN$ such that the limit
$$
\theta(z)=\lim_{m\in M_0\atop m\to+\infty}{1\over m}\log\int_{w\in \cU'_{\varepsilon,z}}
|\wt f_1^{(m)}(z,w)\big|^2dV_{\omega_0}(w),\quad z\in\cX'
$$
exists almost everywhere on~$\cX'$, and $H=e^{-N\theta}$ defines a
singular hermitian metric on $p^*(NK_\cX+\sum\cL_j)$.
i.e.\ $i\ddbar\theta\geq 0$. Moreover, it satisfies the estimates
\smallskip
\item{\rm(a)}$|\sigma|^2_H=|\sigma|^2e^{-N\theta}=1$ on $X_0\subset \cX'\;;$
\smallskip
\item{\rm(b)} 
$\displaystyle
\int_{\cX'}e^{-\theta}e^{-{1\over N}(\varphi_0+\ldots+\varphi_{N-1})}
dV_{\omega}<\infty\;:$
\smallskip
\item{\rm(c)} if $\rho\in{}]1,\rho''[$, there is a constant $C_4=C_4(\rho)>0$
such that for  every $\beta>0$
$$
i\ddbar e^{\beta\theta}\geq-\varepsilon^{-2}(\log\rho)^{-1}e^{C_4\beta}\omega.
$$
\endclaim

\proof{Proof} First notice that the choice of the K\"ahler metric $\omega_0$
on $\cX\times\overline X_0$ is irrelevant in the definition of $\theta$
(the $L^2$ integrals may eventually change by bounded multiplicative factors,
which get killed as $m\to+\infty$). The mean value inequality
for plurisubharmonic functions applied on
small balls of $\cU''_{\rho''\varepsilon}$ centered at points
$(z,w)\in\cU'_{\rho'\varepsilon}$, $\rho'\in{}]\rho,\rho''[$, implies that
we have a uniform upper bound ${1\over m}\log|\wt f_1^{(m)}|^2\leq
C_5=C_5(\rho')$
on $\cU'_{\rho'\varepsilon}$ (that is, in fact, locally over trivializing
open sets of the line bundles in $\cX'$, but we can arrange that there
are only finitely many of these). By construction
$\wt f_1^{(m)}=(p^*\sigma)^{q(m)}$ on $Z_0=p^{-1}(X_0)$, thus
${1\over m}\log|\wt f_1^{(m)}|^2={1\over N}p^*\log|\sigma|^2$ on~$Z_0$
and $\theta={1\over N}\log|\sigma|^2$ on $X_0$. Therefore the limit exists
at least on $X_0$ and estimate (a) is satisfied.

Let us consider now the Bergman bundle $B_\varepsilon\to\cX'$, and let us
write locally over $\cX'$
$$
\wt f_1^{(m)}(z,w)=\sum_{\alpha\in\bN^n}\xi_\alpha(z)\,e_\alpha(z,w)
\otimes g(z)^{q(m)},\quad z\in\cX',~
w\in\cU'_{\varepsilon,z}
$$
in terms of an orthonormal frame $(e_\alpha)_{\alpha\in\bN^n}$ of
$\cU'_\varepsilon$ as defined in \S2, where $g$ is a local
generator of $\cO_\cX(NK_\cX+\sum\cL_j)$. The fact that we have 
uniform convergence of the series on $\cU'_{\rho'\varepsilon}$ and
a uniform upper bound $e^{C_5m}$ implies for the holomorphic
power series in $w$ a uniform bound of the form
$$
\sum_{\alpha\in\bN^n}\rho^{2|\alpha|}|\xi_\alpha(z)|^2\leq e^{C_6m},\quad
\hbox{for some $C_6=C_6(\rho)$}.\leqno(3.12)
$$
For a suitable smooth (and
essentially irrelevant) correcting factor $\wt\psi(z,w)$ on $\cX''$, we have
a uniform bound
$$
\theta_m(z):={1\over m}\log\int_{w\in \cU'_{\varepsilon,z}}
|\wt f_1^{(m)}(z,w)\big|^2e^{-\wt\psi(z,w)}dV_{\omega_0}(w)
={1\over m}\log\sum_{\alpha\in\bN^n}|\xi_\alpha(z)|^2\leq C_5
$$
where the summation $\sum_{\alpha\in\bN^n}|\xi_\alpha(z)|^2$ is real analytic
and $\theta_m(z)\to{1\over N}\log|\sigma(z)|^2$ uniformly on~$X_0$.
The next idea is to estimate the Hessian form of $z\mapsto e^{\beta\theta_m(z)}$
for every~$\beta>0$ fixed. We have of course
$$
e^{\beta\theta_m(z)}=\Bigg(\sum_{\alpha\in\bN^n}|\xi_\alpha(z)|^2\Bigg)^{\beta/m}
=\Vert\xi(z)\Vert^{2\beta/m}=\langle\xi(z),\xi(z)\rangle^{\beta/m}
$$
where $\xi$ is nothing else than the expression of the section $\wt f^{(m)}_1$
is the (real analytic) trivia\-lization of the Bergman bundle $B_\varepsilon$,
and $\langle{\scriptstyle\bullet},{\scriptstyle\bullet}\rangle$ the natural
hermitian metric on~$B_\varepsilon$.
Now, a standard calculation with respect to the Bergman connection $\nabla
=\nabla^{1,0}+\nabla^{0,1}$ of $B_\varepsilon$ yields $\nabla^{0,1}\xi=0$,
hence
$$
\leqalignno{
i\ddbar e^{\beta\theta_m(z)}
&={\beta\over m}\Vert\xi\Vert^{2\beta/m-2}\Bigg(
i\langle\nabla^{1,0}\xi,\kern-1pt\nabla^{1,0}\xi\rangle-
\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle-
\Big(1\kern1pt{-}\kern1pt{\beta\over m}\Big)
{i\langle\nabla^{1,0}\xi,\xi\rangle\wedge
\overline{\langle\nabla^{1,0}\xi,\xi\rangle}\over
\Vert\xi\Vert^2}\Bigg)\kern-5pt\cr
&\geq-{\beta\over m}\Vert\xi\Vert^{2\beta/m-2}
\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle
&(3.13)\cr}
$$
by the Cauchy-Schwarz inequality. On the other hand, the curvature bound
obtained in \S2 yields
$$
\leqalignno{
\big|\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle\big|_{\omega}
&\leq 2\varepsilon^{-2}\sum_{\alpha\in\bN^n}(|\alpha|+n)|\xi_\alpha|^2\cr
&\leq 2\varepsilon^{-2}
\Bigg(\sum_{\alpha\in\bN^n}|\xi_\alpha|^2\Bigg)^{1-\beta/m}
\Bigg(\sum_{\alpha\in\bN^n}(|\alpha|+n)^{m/\beta}|\xi_\alpha|^2\Bigg)^{\beta/m}
&(3.14)\cr}
$$
by the discrete H\"older inequality. We compare the last summation to
(3.12) by taking the maximum of $t\mapsto(t+n)^{m/\beta}\rho^{-2t}$ which
is reached
for $t_0+n={m\over 2\beta}(\log\rho)^{-1}$ by an elementary calculation.
This gives
$$
\leqalignno{
\Bigg(\sum_{\alpha\in\bN^n}(|\alpha|+n)^{m/\beta}|\xi_\alpha|^2\Bigg)^{\beta/m}
&\leq (t_0+n)\rho^{-t_0\beta/m}
\Bigg(\sum_{\alpha\in\bN^n}\rho^{2|\alpha|}|\xi_\alpha(z)|^2\Bigg)^{\beta/m}\cr
&\leq {m\over 2\beta}(\log\rho)^{-1}\rho^{n\beta/m}e^{C_6\beta-1/2}.&(3.15)\cr}
$$
A combination of the last three estimates (3.13--3.15) gives
$$
i\ddbar e^{\beta\theta_m(z)}\geq-\varepsilon^{-2}(\log\rho)^{-1}
\rho^{n\beta/m}e^{C_6\beta-1/2}\omega.
\leqno(3.16)
$$
The above lower bound is uniform with respect to $m$ as $m\to+\infty$, and we
see that $e^{\beta\theta_m}$ is quasi plurisubharmonic.
By well known facts of pluripotential theory, there exists an upper
semicontinuous regularization
$$
\theta=\Big(\limsup_{m\to+\infty}\theta_m\Big)^*
$$
and a subsequence $m\in M_0\subset\bN$ such that
$$
\theta=\limsup_{m\in M_0\atop m\to+\infty}\theta_m~~\hbox{almost
everywhere on $\cX'$},
$$
and satisfying in the limit the Hessian estimate
$$
i\ddbar e^{\beta\theta}\geq-\varepsilon^{-2}(\log\rho)^{-1}
e^{C_6\beta-1/2}\omega\leqno(3.17)
$$
for every $\beta>0$. Property (c) is proved.\qed

\claim 3.18. What to do~?|{\rm Our hope (possibly after modifying the sections
$f_j^{(m)}$ in an adequate manner) is that one can gain a factor
converging to zero in estimate (3.16), and thus in (3.17). This would
prove that $e^{\beta\theta}$ is
plurisubharmonic for every $\beta>0$, hence the weight
$\theta=\lim_{\beta\to 0}{1\over \beta}(e^{\beta\theta}-1)$ would also
be plurisubharmonic.}
\endclaim

\noindent
{\it Fourth step: applying Ohsawa-Takegoshi once again with the singular
hermitian metric produced in the third step.}

Assuming that 3.18 holds, we have proved that $NK_\cX+\sum \cL_j$
possesses a hermitian metric
$H=e^{-N\theta}$ such that $\Vert\sigma\Vert_H\le 1$ oon $X_0$ and
$\Theta_H\ge 0$ on~$\cX'$.
In~order to conclude, we equip the bundle
$$
\cE=(N-1)K_\cX+\sum \cL_j
$$
with the metric $\eta=H^{1-1/N}\prod h_j^{1/N}$, and
$NK_\cX+\sum \cL_j=K_\cX+\cE$ with the metric $\omega\otimes\eta$.
It is important here that $\cX$ possesses a global K\"ahler polarization
$\omega$, otherwise the required estimates would not be valid.
Clearly $\eta$ has a semi-positive curvature current on $\cX'$ and in a
local trivialization we have
$$
\Vert\sigma\Vert^2_{\omega\otimes\eta}\le
C|\sigma|^2\exp\Big(-(N-1)\theta-{1\over N}\sum \varphi_j\Big)
\le C \Big(|\sigma|^2\prod e^{-\varphi_j}\Big)^{1/N}
$$
on $X_0$. Since $|\sigma|^2e^{-\varphi_0}$ and $e^{-\varphi_r}$, $r>0$ are
all locally integrable, we see that $\Vert\sigma\Vert^2_{\omega\otimes\eta}$
is also locally integrable on $X_0$ by the H\"older inequality. A new (and
final) application of the
$L^2$ extension theorem to the hermitian line bundle $(\cE,\eta)$ implies
that $\sigma$ can be
extended to~$\cX'$. Conjecture 3.2 would then be proved.\qed
\endproof


\bigskip\noindent
Jean-Pierre Demailly\\
Universit\'e Grenoble Alpes\\
Institut Fourier, 100 rue des Maths\\
38610 Gi\`eres, France\\
{\tt jean-pierre.demailly@univ-grenoble-alpes.fr}

\bigskip\noindent
(version of August 5, 2018, printed on \today, \timeofday)
\end
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