\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 $V$ such that $\pi^{-1}(V)$ carries a closed smooth
$(1,1)$-form $\omega$ for which $\omega_{|X_t}$ is positive definite
on~$X_t:=\pi^{-1}(t)$, $t\in V$. 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, except
for the fact that $\cX$ is no longer compact, but this is not a problem
since $\cX\to\Delta$ is proper, and we can always shrink a little bit
$\Delta$ to achieve uniform bounds (would they be needed). Let $\gamma$ 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 $Z_0=(\pi')^{-1}(0)=p^{-1}(X_0)$
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 $\cG_m$ on $\cX'$
by putting $\cG_0=\cO_{\cX'}$ 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
