
The $\rho$-derivative of $\theta_{m,\rho}$ may be difficult to estimate, thus
we replace (4.14) by an integrated form and consider
$$
\wt\theta_{m,\rho,\rho'}(z)={2\over \rho^2-\rho^{\prime\,2}}
\int_{\rho'}^\rho \theta_{m,u}(z)\;u\,du.
$$
Then we obtain
$$
i\ddbar\;\wt\theta_{m,\rho,\rho'}
\geq-{2+O(\rho\varepsilon)\over \varepsilon^2(\rho^2-\rho^{\prime\,2})}
(\theta_{m,\rho}-\theta_{m,\rho'})\;\omega.
\leqno(4.15)
$$


However,  formula (2.18) shows that
$$
\partial_{w_k}\wt e_\alpha(z,w)=\varepsilon^{-1}\;
\sqrt{\alpha_k(|\alpha|+n)}\;\wt e_{\alpha-c_k}(z,w),
$$
thus
$$
\sum_k\Vert\partial_{w_k}\xi_\beta\Vert^2=\varepsilon^{-2}
\sum_k\sum_{\alpha\in\bN^{n+1}}\alpha_k(|\alpha|+n)\;|\xi_{\alpha\beta}|^2
=\varepsilon^{-2}
\sum_{\alpha\in\bN^{n+1}}|\alpha|(|\alpha|+n)\;|\xi_{\alpha\beta}|^2
$$
and
$$
\varepsilon^{-2}
\sum_{\alpha\in\bN^{n+1}}(|\alpha|+n)^2\;|\xi_{\alpha\beta}|^2\leq
n^2\,\Vert\xi_\beta\Vert^2+2\sum_k\Vert\partial_{w_k}\xi_\beta\Vert^2
\leq C_8\bigg(\Vert\xi_\beta\Vert^2+\sum_k\Vert\tau_k\xi_\beta\Vert^2\bigg).
$$
Since $\tau_k\xi_\beta=\xi_{\beta+c_k}$, we have
$$
\sum_{|\beta|=\ell}\sum_k\Vert\tau_k\xi_\beta\Vert^2\leq\nu
\sum_{|\beta'|=\ell+1}\Vert\xi_{\beta'}\Vert^2,
$$
therefore
$$
\varepsilon^{-2}\sum_{\beta\in\bN^\nu}c^{|\beta|}
\sum_{\alpha\in\bN^{n+1}}(|\alpha|+n)^2|\xi_{\alpha\beta}|^2
\leq C_8\big(\Vert\xi\Vert^2+\nu c^{-1}\Vert\xi\Vert^2\big)
\leq {C_9\over (\rho-1)^2\varepsilon^2}\;\Vert\xi\Vert^2,
$$
if we recall the value of $c=(8C_6^2)^{-1}(\rho-1)\varepsilon^2$. By (4.16)
we get
$$
\big|\langle i\Theta_{B_\varepsilon\mathop{\widehat\otimes}\ell^2(\bN^{n+1})}
\xi,\xi\rangle\big|_{\omega}\leq
{C_{10}\over(\rho-1)\varepsilon^2}\;\Vert\xi\Vert^2
$$
and in fine, by (4.15), we have
$$
i\ddbar\theta_m\geq-{C_{10}\over m(\rho-1)\varepsilon^2}\;\omega.
\leqno(4.17)
$$
Here the constant $C_{10}$ can be seen to depend only on the geometry of
$(T_\cX,\omega)$. On the other hand, estimate (4.14) is equivalent to a uniform
upper bound
$$
\int_{z\in\cX'}\theta_m\;dV_{\omega_0}\leq
C_{11}+4(n+1)\;\log{1\over(\rho-1)\varepsilon}
$$

The above lower bound is uniform with respect to $m$ as $m\to+\infty$, and we
see that $e^{\lambda\theta_m}$ 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
$$
i\ddbar e^{\lambda\theta}\geq-\varepsilon^{-2}(\log\rho)^{-1}
e^{C_6\lambda-1/2}\omega\leqno(4.17)
$$
for every $\lambda>0$. Property (c) is proved.\qed

---------------------------------




and a uniform upper bound $\theta_m\leq C_9$ on $\cX'$.
Here the double summation 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
$z\mapsto e^{\lambda\theta_m(z)}$
for every~$\lambda>0$ fixed. We have of course
$$
e^{\lambda\theta_m(z)}=\Bigg(\sum_{\alpha\in\bN^n}|\xi_{m,\alpha}(z)|^2\Bigg)^{\lambda/m}
=\Vert\xi(z)\Vert^{2\lambda/m}=\langle\xi(z),\xi(z)\rangle^{\lambda/m}
$$
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
$$
\leqalignno{
i\ddbar e^{\lambda\theta_m(z)}
&={\lambda\over m}\Vert\xi\Vert^{2\lambda/m-2}\Bigg(
i\langle\nabla^{1,0}\xi,\kern-1pt\nabla^{1,0}\xi\rangle-
\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle-
\Big(1\kern1pt{-}\kern1pt{\lambda\over m}\Big)
{i\langle\nabla^{1,0}\xi,\xi\rangle\wedge
\overline{\langle\nabla^{1,0}\xi,\xi\rangle}\over
\Vert\xi\Vert^2}\Bigg)\kern-5pt\cr
&\geq-{\lambda\over m}\Vert\xi\Vert^{2\lambda/m-2}
\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle
&(3.13)\cr}
$$
by the Cauchy-Schwarz inequality. On the other hand, the curvature bound
obtained in \S2 yields
$$
\leqalignno{
\big|\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle\big|_{\omega}
&\leq 2\varepsilon^{-2}\sum_{\alpha\in\bN^n}(|\alpha|+n)|\xi_{m,\alpha}|^2\cr
&\leq 2\varepsilon^{-2}
\Bigg(\sum_{\alpha\in\bN^n}|\xi_{m,\alpha}|^2\Bigg)^{1-\lambda/m}
\Bigg(\sum_{\alpha\in\bN^n}(|\alpha|+n)^{m/\lambda}|\xi_{m,\alpha}|^2\Bigg)^{\lambda/m}
&(3.14)\cr}
$$
by the discrete H\"older inequality. We compare the last summation to
(3.12) by taking the maximum of $t\mapsto(t+n)^{m/\lambda}\rho^{-2t}$ which
is reached
for $t_0+n={m\over 2\lambda}(\log\rho)^{-1}$ by an elementary calculation.
This gives
$$
\leqalignno{
\Bigg(\sum_{\alpha\in\bN^n}(|\alpha|+n)^{m/\lambda}|\xi_{m,\alpha}|^2\Bigg)^{\lambda/m}
&\leq (t_0+n)\rho^{-t_0\lambda/m}
\Bigg(\sum_{\alpha\in\bN^n}\rho^{2|\alpha|}|\xi_{m,\alpha}(z)|^2\Bigg)^{\lambda/m}\cr
&\leq {m\over 2\lambda}(\log\rho)^{-1}\rho^{n\lambda/m}e^{C_6\lambda-1/2}.&(3.15)\cr}
$$
A combination of the last three estimates (3.13--3.15) gives
$$
i\ddbar e^{\lambda\theta_m(z)}\geq-\varepsilon^{-2}(\log\rho)^{-1}
\rho^{n\lambda/m}e^{C_6\lambda-1/2}\omega.
\leqno(3.16)
$$
The above lower bound is uniform with respect to $m$ as $m\to+\infty$, and we
see that $e^{\lambda\theta_m}$ 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
$$
i\ddbar e^{\lambda\theta}\geq-\varepsilon^{-2}(\log\rho)^{-1}
e^{C_6\lambda-1/2}\omega\leqno(3.17)
$$
for every $\lambda>0$. 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 $e^{\lambda\theta}$ is
plurisubharmonic for every $\lambda>0$, hence the weight
$\theta=\lim_{\lambda\to 0}{1\over \lambda}(e^{\lambda\theta}-1)$ 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~$\cX'$. Theorem 3.2 is proved.\qed
\endproof

\section{4}{Invariance of plurigenera for polarized K\"ahler families}

The goal of this section is to prove that for every polarized family
$\cX\to S$ of compact K\"ahler manifolds, the plurigenera
$p_m(X_t)=h^0(X_t,mK_{X_t})$ of fibers are independent of~$t$ for all $m\ge 0$.
This result 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 [Siu02] and P{\u a}un [Pau07]
in the case of arbitrary projective varieties; in the nonrestricted
projective case, no algebraic proof of the result is known. We extend
here the result to the K\"ahler context. This requires substantial
modifications of the proof, since
the technique of Siu and P{\u a}un involved in a crucial manner the use
of an auxiliary ample line bundle. We replace it here by a use of
the Hilbert bundles studied in the previous section.

\claim 4.1. Theorem|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 follows directly from the following more technical
result.

\claim 4.2. Theorem {\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 4.3. Remark|{\rm A standard cohomological argument shows that we
can in fact take $\cX'=\cX$ in the conclusion of Theorem 4.2, 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{Proof of Theorem $4.1$} The
invariance of plurigenera is in fact
just obtained as the special case of Theorem~4.2 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 Theorem~4.2
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 4.4. Lemma|Let $\cX'=\pi^{-1}(\Delta')\to\Delta'$ be the restriction
of $\pi:\cX\to\Delta$ to a disc $\Delta'\compact\Delta$ centered at~$0$,
of radius $r'\leq r$. For $\varepsilon\leq\varepsilon_0=\varepsilon_0(r')$
small enough, one can find a Stein open subset $\cU'_\varepsilon\subset
\cX'\times\overline\cX$ , where
 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 arguments are very similar to those of \S1, except
for the fact that $\cX$ is no longer compact, but this is not a problem
since $\cX\to\Delta$ is proper, and since we can always shrink $\Delta$
a little bit to achieve uniform bounds (would they be needed).
Let $\gamma$ be a real analytic hermitian metric on~$\cX$, and
$\exph:T_{\cX}\to\cX$ be the corresponding real analytic
and fiber-holomorphic exponential map associated with $\gamma$, as
in \S1.  The map $\exph$ is no longer everywhere defined, but if
we restrict it to the $\varepsilon$-tubular neighborhood
of the zero section in $T_{\cX'}$, we get for $\varepsilon>0$ small enough
a real analytic diffeomorphism $(z,\xi)\mapsto(z,\exph_z(\xi))$ onto a tubular
neighborhood of the diagonal of $\cX'\times\cX'$. The rest of the
proof is identical to what we did in~\S1, taking
$$
\cU'_\varepsilon=\big\{(z,w)\in\cX'\times\overline\cX\,;\;
|\logh_z(\overline w)|_\gamma<\varepsilon\big\}.\lreqno{(4.5)}{\square}
$$

In order to study Theorem~4.2, 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
[OhT87], see also [Ohs88], [Dem00].

\claim 4.6. 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 4.7. Remark|{\rm The assumptions of Proposition 4.6 imply that $\cZ$ is
holomorphically convex and complete K\"ahler, thus the technique
of [Dem82] does apply to yield the result.}
\endclaim

\proof{Proof of Theorem $4.2$} Let $p=\pr_1:\cU'_\varepsilon\to
\cX'$ be as in Lemma~4.4, and $q=\pr_2:\cU'_\varepsilon\to\overline\cX$. 
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 $\pr_1^*\omega+\pr_2^*\smash{\overline\omega_{|\overline\cX}}$
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 4.6 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 $Z_0=\pi^{\prime\,-1}(0)=p^{-1}(X_0)$
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 $\cG_m$ 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 (4.8)} 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 (4.9)} 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 (4.8) 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
$|\xi|^2/\sum_\ell|\wt f_\ell^{(m-1)}(x)|^2$, and
$$
\widetilde\cG_m:=p^*\cG_m-K_\cZ=p^*\cG_m-(p^*K_{\cX'}+q^*K_{\overline\cX})=
p^*(\cG_{m-1}+\cL_r)-q^*K_{\overline\cX}
$$
with that metric multiplied by $p^*h_r=e^{-p^*\varphi_r}$ and a fixed smooth
metric $e^{-\psi}$ of positive curvature on
$(-q^*K_{\overline\cX})_{|\cU'_{\varepsilon_0}}$
(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 $p^*\cG_{m|Z_0}$ 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 4.2~(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
$$
\int_{\cZ}\big|\wt f_1^{(m)}\big|^{2/m}
e^{-{1\over N}p^*(\varphi_0+\ldots+\varphi_{N-1})-\psi}dV_{\omega_0}\le C_2.
\leqno(4.10)
$$
As the functions
$\smash{\varphi_{r(k)}}$ and $\psi$ are locally bounded from above,
we infer from this the weaker inequality
$$
\int_{\cZ}\big|\wt f_1^{(m)}\big|^{2/m}dV_{\omega_0}\le C_2.
\leqno(4.10')
$$
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 ``vertical'' holomorphic vector fields.}
For technical reasons, we need to differentiate the sections
$\wt f_1^{(m)}$ along the fibers of $p:\cZ\to\cX'$, i.e.\ along the
extra factor $\overline\cX$. Since our sections take values
in pull-backs $q(m)\,p^*(NK_{\cX'}+\sum \cL_j)$ (which are trivial along
the fibers of $p$, namely $\varepsilon$-balls in $\overline\cX$),
such ``vertical'' derivatives make sense
and do not require the use of a connection. Since $\cZ$ is Stein
and $q^*T_{\overline\cX}$ is a holomorphic subbundle of $T_\cZ$,
we can find global holomorphic vector fields
$\tau_1,\ldots,\tau_{\nu}\in H^0(\cZ,q^*T_{\overline\cX})$ that
generate the subbundle $q^*T_{\overline\cX}$ over the whole of $\cZ\,$;
a standard Whitney type argument shows that one needs only $\nu:=\dim_\bC\cZ+
\rank q^*T_{\overline\cX}=3(n+1)$ such vector fields to achieve the 
global generation property, but we do not need the precise value of
$\nu$~here. For
any finite sequence $\beta=(\beta_1,\ldots,\beta_\ell)\in\{1,\ldots,\nu\}^\ell$
of length $|\beta|=\ell$ and any holomorphic section $f\in H^0(\cZ,p^*\cG)$
of the pull-pack of a line or vector bundle $\cG\to\cX'$, we define
$$
\tau^\beta f=\tau_{\beta_1}\ldots\tau_{\beta_\ell} f
\leqno(4.11)
$$
to be the iterated derivative of $f$ with respect to the vector
fields $\tau_j$. This is again a holomorphic section of $p^*\cG$ on~$\cZ$.
\medskip

\noindent
{\it Fourth 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 (4.10), $(4.10')$, 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 the results of \S2
on Bergman bundles. The estimates do not work directly for
the given sections $\wt f_1^{(m)}$, and we have to add a suitable
combination of the derivatives to obtain good estimates.

\claim 4.12. Proposition|Assume that the sections $\widetilde f^{(m)}_1$
and vector fields $\tau_j$
have been constructed on a slightly bigger tubular neighborhood
$\cU'_{\rho\varepsilon}\to\cX'$ with $\rho>1$, $[$this~condition can of course
always be achieved, since we have a lot of flexibility on the choice of the
tubular neighborhoods, by taking e.g.
$\varepsilon\leq {1\over 2}\varepsilon_0(r')\,]$. Then there exists $c>0$ and
a subsequence $m\in M_0\subset\bN$ such that the limit
$$
\theta(z)=\lim_{m\in M_0\atop m\to+\infty}{1\over m}
\log\int_{w\in \cU'_{\varepsilon,z}}
\sum_{\beta\in\bN^\nu} {c^{|\beta|}\over(|\beta|!)^2}\;
|\tau^\beta\wt f_1^{(m)}(z,w)\big|^2dV_{\omega_0}(w),\quad z\in\cX'
$$
exists almost everywhere on~$\cX'$, and $H=e^{-N\theta}$ defines a
singular hermitian metric of positive curvature current 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 $\cX'\times\overline\cX$ is irrelevant in the definition of $\theta$
(the $L^2$ integrals may eventually change by bounded multiplicative factors,
which get killed as $m\to+\infty$). We use estimate $(4.10')$ on
$\cZ=\cU'_{\rho\varepsilon}$, together
with the mean value inequality for plurisubharmonic functions, applied on
balls of the fibers $\cU'_{\rho\varepsilon,z}=p^{-1}(z)$ centered at points
$(z,w)\in\cU'_\varepsilon$ and of radius $(\rho-1)\varepsilon/2$ in $w$.
Let us take $\rho'={\rho+1\over 2}\in{}]1,\rho[$.
If we view the fibers as standard hermitian balls in the hermitian vector space
$(T_{\overline\cX},\omega_0)$, we obtain a uniform upper bound
$$
\sup_{\cU'_{\rho'\varepsilon,z}}|\wt f_1^{(m)}|^{2/m}\leq
{C_5\over((\rho-1)\varepsilon)^{2(n+1)}}
\int_{\cU'_{\rho\varepsilon,z}}|\wt f_1^{(m)}|^{2/m}dV_{\omega_0},
\quad\forall z\in\cX'.
$$
Here our sections can be seen as functions only locally over trivializing
open sets of the line bundles in $\cX'$, but we can arrange that there
are only finitely many of these. Now, Cauchy inequalities applied on
the fibers give an upper bound
$$
\sup_{\cU'_{\varepsilon},z}{1\over|\beta|!}|\tau^\beta g|\leq
\bigg({C_6\over(\rho'-1)\varepsilon)}\bigg)^{|\beta|}
\sup_{\cU'_{\rho'\varepsilon}}|g|,\quad C_6>0
\leqno(4.13)
$$
for every holomorphic function $g$. By integrating over $z\in \cX'$, a
combination of these estimates implies 
$$
\int_{(z,w)\in\cU'_\varepsilon}
\sum_{\beta\in\bN^\nu}{c^{|\beta|}\over(|\beta|!)^2}\big|\tau^\beta\wt f_1^{(m)}\big|^2\;dV_{\omega_0}\leq\sum_{\beta\in\bN^\nu}\bigg(
{c\,C_6^2\over(\rho'-1)^2\varepsilon^2}\bigg)^{|\beta|}
{C_5^m\over((\rho-1)\varepsilon)^{2m(n+1)}}.
$$
The choice $c=(8C_6^2)^{-1}(\rho-1)^2\varepsilon^2$ yields
$$
\int_{(z,w)\in\cU'_\varepsilon}
\sum_{\beta\in\bN^\nu}{c^{|\beta|}\over(|\beta|!)^2}\big|\tau^\beta\wt f_1^{(m)}\big|^2\;dV_{\omega_0}\leq
{2^\nu\;C_5^m\over((\rho-1)\varepsilon)^{2m(n+1)}},
$$
and the Jensen inequality $\int g\,d\mu\leq\log\int e^gd\mu$ for the probability
measures $\mu_z={1\over\Vol(\cU'_{\varepsilon,z})}dV_{\omega_0}$ on the fibers
implies by Fubini
$$
\int_{z\in\cX'}{1\over m}\log\int_{w\in\cU'_{\varepsilon,z}}\;
\sum_{\beta\in\bN^\nu}{c^{|\beta|}\over(|\beta|!)^2}\big|\tau^\beta\wt f_1^{(m)}\big|^2\;dV_{\omega_0}\leq
C_7+4(n+1)\;\log{1\over(\rho-1)\varepsilon}
\leqno(4.14)
$$
(the factor $4(n+1)$ could be replaced by $2(1+1/m)(n+1)$, but this is
irrelevant). By construction
$\wt f_1^{(m)}=(p^*\sigma)^{q(m)}$ on $Z_0=p^{-1}(X_0)$  and
all higher derivatives $\tau^\beta\wt f_1^{(m)}$ vanish on $Z_0$,
thus we are left with the sum
${1\over m}\log|\wt f_1^{(m)}|^2={1\over N}p^*\log|\sigma|^2$ 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.

Now, let us consider now the Bergman bundle $B_\varepsilon\to\cX'$ where
$\dim\cX'=n+1$, and write locally over $\cX'$
$$
{1\over|\beta|!}\;
\tau^\beta\wt f_1^{(m)}(z,w)=\sum_{\alpha\in\bN^{n+1}}\xi_{\alpha\beta}(z)\,e_\alpha(z,w)
\otimes g(z)^{q(m)},\quad z\in\cX',~
w\in\cU'_{\varepsilon,z}
$$
in terms of an orthonormal frame $(e_\alpha)_{\alpha\in\bN^{n+1}}$ of
$B_\varepsilon$ as defined in \S2, where $g$ is a local
generator of $\cO_\cX(NK_{\cX'}+\sum\cL_j)$. For a suitable smooth (and
essentially irrelevant) correcting factor $\wt\psi(z,w)$ on
$\cU'_{\varepsilon_0}$ depending only
on $|dw_1\wedge\ldots\wedge dw_n|_{\omega_0}^2$, we have an equality
$$
\eqalign{
\theta_m(z):&={1\over m}\log\int_{w\in \cU'_{\varepsilon,z}}
\sum_{\beta\in\bN^\nu}{c^{|\beta|}\over(|\beta|!)^2}\;
|\tau^\beta\wt f_1^{(m)}(z,w)\big|^2e^{-\wt\psi(z,w)}dV_{\omega_0}(w)\cr
&={1\over m}\log\bigg(
\sum_{\beta\in\bN^\nu} c^{|\beta|}\sum_{\alpha\in\bN^{n+1}}|\xi_{\alpha\beta}(z)|^2\bigg)
={1\over m}\log\bigg(
\sum_{\beta\in\bN^\nu}c^{|\beta|}\Vert\xi_\beta(z)\Vert^2\bigg)
\cr}
$$
where $\xi_\beta=(\xi_{\alpha\beta})_{\alpha\in\bN^{n+1}}$ is nothing else than
the expression of the section $\tau^\beta\wt f^{(m)}_1$
is the (real analytic) trivia\-lization of the Bergman bundle $B_\varepsilon$,
equipped with its natural hermitian metric
$\langle{\scriptstyle\bullet},{\scriptstyle\bullet}\rangle$, and
$\xi=(\xi_\beta)_{\beta\in\bN^\nu}\in B_\varepsilon\mathop{\widehat\otimes}
\ell^2(\bN^{n+1})$ equipped with the metric
$$
\Vert\xi\Vert^2=\sum_{\beta\in\bN^\nu}c^{|\beta|}\Vert\xi_\beta(z)\Vert^2
=\sum_{\beta\in\bN^\nu}c^{|\beta|}\sum_{\alpha\in\bN^{n+1}}|\xi_{\alpha\beta}(z)|^2.
$$
The inequalities (4.13) show that the series converge uniformly over the
whole of~$\cX'$.
A~standard calculation with respect to the Bergman connection $\nabla
=\nabla^{1,0}+\nabla^{0,1}$ of $B_\varepsilon$ and the induced connection
on $B_\varepsilon\mathop{\widehat\otimes}\ell^2(\bN^{n+1})$
yields $\nabla^{0,1}\xi=0$, hence
$$
\leqalignno{
i\ddbar\theta_m
&={1\over m}\;{1\over\Vert\xi\Vert^2}\Bigg(
i\langle\nabla^{1,0}\xi,\kern-1pt\nabla^{1,0}\xi\rangle-
\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle-
{i\langle\nabla^{1,0}\xi,\xi\rangle\wedge
\overline{\langle\nabla^{1,0}\xi,\xi\rangle}\over
\Vert\xi\Vert^2}\Bigg)\kern-5pt\cr
&\geq-{1\over m}\;{\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle\over
\Vert\xi\Vert^2}&(4.15)\cr}
$$
by the Cauchy-Schwarz inequality. On the other hand, the curvature bound
obtained in \S2 yields
$$
\leqalignno{
\big|\langle i&\Theta_{B_\varepsilon\mathop{\widehat\otimes}\ell^2(\bN^{n+1})}
\xi,\xi\rangle\big|_{\omega}
\leq (2+O(\varepsilon))\varepsilon^{-2}\sum_{\beta\in\bN^\nu}c^{|\beta|}
\sum_{\alpha\in\bN^{n+1}}(|\alpha|+n)|\xi_{\alpha\beta}|^2\cr
&\kern20pt{}
\leq (2+O(\varepsilon))\varepsilon^{-2}
\Bigg(\sum_{\beta\in\bN^\nu}c^{|\beta|}
\sum_{\alpha\in\bN^{n+1}}|\xi_{\alpha\beta}|^2\Bigg)^{1/2}
\Bigg(\sum_{\beta\in\bN^\nu}c^{|\beta|}
\sum_{\alpha\in\bN^{n+1}}(|\alpha|+n)^2|\xi_{\alpha\beta}|^2\Bigg)^{1/2}
&(4.16)\cr}
$$
by the Cauchy-Schwarz inequality. However, formula (2.18) shows that
$$
\partial_{w_k}\wt e_\alpha(z,w)=\varepsilon^{-1}\;
\sqrt{\alpha_k(|\alpha|+n)}\;\wt e_{\alpha-c_k}(z,w),
$$
thus
$$
\sum_k\Vert\partial_{w_k}\xi_\beta\Vert^2=\varepsilon^{-2}
\sum_k\sum_{\alpha\in\bN^{n+1}}\alpha_k(|\alpha|+n)\;|\xi_{\alpha\beta}|^2
=\varepsilon^{-2}
\sum_{\alpha\in\bN^{n+1}}|\alpha|(|\alpha|+n)\;|\xi_{\alpha\beta}|^2
$$
and
$$
\varepsilon^{-2}
\sum_{\alpha\in\bN^{n+1}}(|\alpha|+n)^2\;|\xi_{\alpha\beta}|^2\leq
n^2\,\Vert\xi_\beta\Vert^2+2\sum_k\Vert\partial_{w_k}\xi_\beta\Vert^2
\leq C_8\bigg(\Vert\xi_\beta\Vert^2+\sum_k\Vert\tau_k\xi_\beta\Vert^2\bigg).
$$
Since $\tau_k\xi_\beta=\xi_{\beta+c_k}$, we have
$$
\sum_{|\beta|=\ell}\sum_k\Vert\tau_k\xi_\beta\Vert^2\leq\nu
\sum_{|\beta'|=\ell+1}\Vert\xi_{\beta'}\Vert^2,
$$
therefore
$$
\varepsilon^{-2}\sum_{\beta\in\bN^\nu}c^{|\beta|}
\sum_{\alpha\in\bN^{n+1}}(|\alpha|+n)^2|\xi_{\alpha\beta}|^2
\leq C_8\big(\Vert\xi\Vert^2+\nu c^{-1}\Vert\xi\Vert^2\big)
\leq {C_9\over (\rho-1)^2\varepsilon^2}\;\Vert\xi\Vert^2,
$$
if we recall the value of $c=(8C_6^2)^{-1}(\rho-1)\varepsilon^2$. By (4.16)
we get
$$
\big|\langle i\Theta_{B_\varepsilon\mathop{\widehat\otimes}\ell^2(\bN^{n+1})}
\xi,\xi\rangle\big|_{\omega}\leq
{C_{10}\over(\rho-1)\varepsilon^2}\;\Vert\xi\Vert^2
$$
and in fine, by (4.15), we have
$$
i\ddbar\theta_m\geq-{C_{10}\over m(\rho-1)\varepsilon^2}\;\omega.
\leqno(4.17)
$$
Here the constant $C_{10}$ can be seen to depend only on the geometry of
$(T_\cX,\omega)$. On the other hand, estimate (4.14) is equivalent to a uniform
upper bound
$$
\int_{z\in\cX'}\theta_m\;dV_{\omega_0}\leq
C_{11}+4(n+1)\;\log{1\over(\rho-1)\varepsilon}
$$

The above lower bound is uniform with respect to $m$ as $m\to+\infty$, and we
see that $e^{\lambda\theta_m}$ 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
$$
i\ddbar e^{\lambda\theta}\geq-\varepsilon^{-2}(\log\rho)^{-1}
e^{C_6\lambda-1/2}\omega\leqno(4.17)
$$
for every $\lambda>0$. Property (c) is proved.\qed

\claim 4.18. 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 (4.16), and thus in (4.17). This would
prove that $e^{\lambda\theta}$ is
plurisubharmonic for every $\lambda>0$, hence the weight
$\theta=\lim_{\lambda\to 0}{1\over \lambda}(e^{\lambda\theta}-1)$ 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 4.18 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~$\cX'$. Theorem 4.2 is proved.\qed
\endproof


---------------------------------




and a uniform upper bound $\theta_m\leq C_9$ on $\cX'$.
Here the double summation 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
$z\mapsto e^{\lambda\theta_m(z)}$
for every~$\lambda>0$ fixed. We have of course
$$
e^{\lambda\theta_m(z)}=\Bigg(\sum_{\alpha\in\bN^n}|\xi_{m,\alpha}(z)|^2\Bigg)^{\lambda/m}
=\Vert\xi(z)\Vert^{2\lambda/m}=\langle\xi(z),\xi(z)\rangle^{\lambda/m}
$$
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
$$
\leqalignno{
i\ddbar e^{\lambda\theta_m(z)}
&={\lambda\over m}\Vert\xi\Vert^{2\lambda/m-2}\Bigg(
i\langle\nabla^{1,0}\xi,\kern-1pt\nabla^{1,0}\xi\rangle-
\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle-
\Big(1\kern1pt{-}\kern1pt{\lambda\over m}\Big)
{i\langle\nabla^{1,0}\xi,\xi\rangle\wedge
\overline{\langle\nabla^{1,0}\xi,\xi\rangle}\over
\Vert\xi\Vert^2}\Bigg)\kern-5pt\cr
&\geq-{\lambda\over m}\Vert\xi\Vert^{2\lambda/m-2}
\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle
&(4.13)\cr}
$$
by the Cauchy-Schwarz inequality. On the other hand, the curvature bound
obtained in \S2 yields
$$
\leqalignno{
\big|\langle i\Theta_{B_\varepsilon}\xi,\xi\rangle\big|_{\omega}
&\leq 2\varepsilon^{-2}\sum_{\alpha\in\bN^n}(|\alpha|+n)|\xi_{m,\alpha}|^2\cr
&\leq 2\varepsilon^{-2}
\Bigg(\sum_{\alpha\in\bN^n}|\xi_{m,\alpha}|^2\Bigg)^{1-\lambda/m}
\Bigg(\sum_{\alpha\in\bN^n}(|\alpha|+n)^{m/\lambda}|\xi_{m,\alpha}|^2\Bigg)^{\lambda/m}
&(4.14)\cr}
$$
by the discrete H\"older inequality. We compare the last summation to
(4.12) by taking the maximum of $t\mapsto(t+n)^{m/\lambda}\rho^{-2t}$ which
is reached
for $t_0+n={m\over 2\lambda}(\log\rho)^{-1}$ by an elementary calculation.
This gives
$$
\leqalignno{
\Bigg(\sum_{\alpha\in\bN^n}(|\alpha|+n)^{m/\lambda}|\xi_{m,\alpha}|^2\Bigg)^{\lambda/m}
&\leq (t_0+n)\rho^{-t_0\lambda/m}
\Bigg(\sum_{\alpha\in\bN^n}\rho^{2|\alpha|}|\xi_{m,\alpha}(z)|^2\Bigg)^{\lambda/m}\cr
&\leq {m\over 2\lambda}(\log\rho)^{-1}\rho^{n\lambda/m}e^{C_6\lambda-1/2}.&(4.15)\cr}
$$
A combination of the last three estimates (4.13--4.15) gives
$$
i\ddbar e^{\lambda\theta_m(z)}\geq-\varepsilon^{-2}(\log\rho)^{-1}
\rho^{n\lambda/m}e^{C_6\lambda-1/2}\omega.
\leqno(4.16)
$$
The above lower bound is uniform with respect to $m$ as $m\to+\infty$, and we
see that $e^{\lambda\theta_m}$ 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
$$
i\ddbar e^{\lambda\theta}\geq-\varepsilon^{-2}(\log\rho)^{-1}
e^{C_6\lambda-1/2}\omega\leqno(4.17)
$$
for every $\lambda>0$. Property (c) is proved.\qed

\claim 4.18. 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 (4.16), and thus in (4.17). This would
prove that $e^{\lambda\theta}$ is
plurisubharmonic for every $\lambda>0$, hence the weight
$\theta=\lim_{\lambda\to 0}{1\over \lambda}(e^{\lambda\theta}-1)$ 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 4.18 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~$\cX'$. Theorem 4.2 is proved.\qed
\endproof
\vskip30pt



\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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