\magnification=1200
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\hsize=12.5cm   
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\parindent=0.6cm
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\def\eightpointbf{%
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\def\twelvepointbf{%
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% main item macros

\newdimen\srdim \srdim=\hsize
\newdimen\irdim \irdim=\hsize
\def\NOSECTREF#1{\noindent\hbox to \srdim{\null\dotfill ???(#1)}}
\def\SECTREF#1{\noindent\hbox to \srdim{\csname REF\romannumeral#1\endcsname}}
\def\INDREF#1{\noindent\hbox to \irdim{\csname IND\romannumeral#1\endcsname}}
\newlinechar=`\^^J
\def\openauxfile{
  \immediate\openin1\jobname.
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%  \message{^^JCAUTION\string: you MUST run TeX a second time^^J}
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  \else
  \input \jobname.aux
%  \message{^^JCAUTION\string: if the file has just been modified you may 
%    have to run TeX twice^^J}
  \let\sectref=\SECTREF \let\indref=\INDREF
  \fi
% \message{to get correct page numbers displayed in Contents or Index 
%   Tables^^J}
  \immediate\openout1=\jobname.aux
  \let\END=\end \def\end{\immediate\closeout1\END}}
        
\newbox\titlebox   \setbox\titlebox\hbox{\hfil}
\newbox\sectionbox \setbox\sectionbox\hbox{\hfil}
\def\folio{\ifnum\pageno=1 \hfil \else \ifodd\pageno
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\footline={\hfil}
\headline={\folio}           

\def\blankline{\phantom{}\hfil\vskip0pt}
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\def\title#1{\noindent\hfil$\smash{\hbox{\seventeenpointbf #1}}$\hfil
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\def\titleleft#1{\noindent$\smash{\hbox{\seventeenpointbf #1}}$\hfil
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\newdimen\itemindent \itemindent=\parindent
\def\setitemindent#1{\setbox0=\hbox{#1~}\itemindent=\wd0}
\def\item#1{\par\noindent\hangindent\itemindent%
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\def\itemitem#1{\par\noindent\hangindent2\itemindent%
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\def\itemitemitem#1{\par\noindent\hangindent3\itemindent%
            \kern2\itemindent\rlap{#1}\kern\itemindent\ignorespaces}

\def\qtq#1{\quad\text{#1}\quad}

\long\def\claim#1|#2\endclaim{\par\vskip 5pt\noindent 
{\tenpointbf #1.}\ {\sl #2}\par\vskip 5pt}

\def\proof{\noindent{\sl Proof}}

\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or July\or August\or
September\or October\or November\or December\fi \space\number\day,
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\newcount\@tempcnta \newcount\@tempcntb 
\def\timeofday{{%
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\def\bibitem#1&#2&#3&#4&%
{\hangindent=1.66cm\hangafter=1
\noindent\rlap{\hbox{\eightpointbf #1}}\kern1.66cm{\rm #2}{\sl #3}{\rm #4.}} 

% blackboard symbols
\def\bB{{\Bbb B}}
\def\bC{{\Bbb C}}
\def\bG{{\Bbb G}}
\def\bH{{\Bbb H}}
\def\bN{{\Bbb N}}
\def\bP{{\Bbb P}}
\def\bQ{{\Bbb Q}}
\def\bR{{\Bbb R}}
\def\bT{{\Bbb T}}
\def\bU{{\Bbb U}}
\def\bZ{{\Bbb Z}}
\def\bone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}

% gothic symbols
\def\gm{{\euf m}}
\def\gS{{\euf S}}

% calligraphic symbols
\def\cC{{\Cal C}}
\def\cE{{\Cal E}}
\def\cF{{\Cal F}}
\def\cO{{\Cal O}}
\def\cH{{\Cal H}}
\def\cI{{\Cal I}}
\def\cJ{{\Cal J}}
\def\cP{{\Cal P}}
\def\cS{{\Cal S}}
\def\cZ{{\Cal Z}}

% special symbols
\def\ampersand{\&}
\def\ld{,\ldots,}
\def\bu{{\scriptstyle\bullet}}
\def\QED{\phantom{$\quad$}\hfill QED\medskip}
\def\square{{\hfill \hbox{
\vrule height 1.453ex  width 0.093ex  depth 0ex
\vrule height 1.5ex  width 1.3ex  depth -1.407ex\kern-0.1ex
\vrule height 1.453ex  width 0.093ex  depth 0ex\kern-1.35ex
\vrule height 0.093ex  width 1.3ex  depth 0ex}}}
\def\qed{\phantom{$\quad$}\hfill$\square$\medskip}
\def\hexnbr#1{\ifnum#1<10 \number#1\else
 \ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else
 \ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi}
\def\msatype{\hexnbr\msafam}
\def\msbtype{\hexnbr\msbfam}
\mathchardef\restriction="3\msatype16   \let\restr\restriction
\mathchardef\compact="3\msatype62
\mathchardef\smallsetminus="2\msbtype72   \let\ssm\smallsetminus
\mathchardef\subsetneq="3\msbtype28
\mathchardef\supsetneq="3\msbtype29
\mathchardef\leqslant="3\msatype36   \let\le\leqslant
\mathchardef\geqslant="3\msatype3E   \let\ge\geqslant
\mathchardef\ltimes="2\msbtype6E
\mathchardef\rtimes="2\msbtype6F

% hats and tildes and over/underlines
\let\ol=\overline
\let\ul=\underline
\let\wt=\widetilde
\let\wh=\widehat
\let\text=\hbox
\def\build#1|#2|#3|{\mathrel{\mathop{\null#1}\limits^{#2}_{#3}}}

% mathematical operators
\def\Div{\mathop{\rm Div}\nolimits}
\def\lc{\mathop{\rm lc}\nolimits}
\def\Re{\mathop{\rm Re}\nolimits}
\def\Im{\mathop{\rm Im}\nolimits}
\def\Id{\mathop{\rm Id}\nolimits}
\def\Tr{\mathop{\rm Tr}\nolimits}
\def\Ker{\mathop{\rm Ker}\nolimits}
\def\End{\mathop{\rm End}\nolimits}
\def\Hom{\mathop{\rm Hom}\nolimits}
\def\Aut{\mathop{\rm Aut}\nolimits}
\def\GL{\mathop{\rm GL}\nolimits}
\def\PGL{\mathop{\rm PGL}\nolimits}
\def\Pic{\mathop{\rm Pic}\nolimits}
\def\Proj{\mathop{\rm Proj}\nolimits}
\def\Supp{\mathop{\rm Supp}\nolimits}
\def\Ricci{\mathop{\rm Ricci}\nolimits}
\def\Vect{\mathop{\rm Vect}\nolimits}
\def\codim{\mathop{\rm codim}\nolimits}
\def\rank{\mathop{\rm rank}\nolimits}
\def\deg{\mathop{\rm deg}\nolimits}
\def\ord{\mathop{\rm ord}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits} 
\def\diam{\mathop{\rm diam}\nolimits}
\def\Vol{\mathop{\rm Vol}\nolimits}
\def\MA{\mathop{\rm MA}\nolimits}
\def\PSH{\mathop{\rm PSH}\nolimits}
\def\pr{\mathop{\rm pr}\nolimits}
\def\exph{\mathop{\rm exph}\nolimits}
\def\logh{\mathop{\rm logh}\nolimits}
\def\del{{\partial}}
\def\dbar{{\overline\partial}}
\def\ddbar{{\partial\overline\partial}}

% subscript operands
\def\alg{{\rm alg}}
\def\red{{\rm red}}
\def\reg{{\rm reg}}
\def\sing{{\rm sing}}
\def\loc{{\rm loc}}
\def\can{{\rm can}}
\def\ee{{\rm e}}
\def\ii{{\rm i}}
\def\Const{{\rm Const}}
\def\bfe{{\bf e}}

% figures inserted as PostScript files
%\special{header=/home/demailly/grlib.ps}
\long\def\InsertFig#1 #2 #3 #4\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" 
(/home/demailly/psinputs/grlib.ps) run
#3}}#4$}}
\long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}}

% main text

\proof. We first assume for simplicity than $\Delta=0$. We modify
the approximation procedure used in section~1 by putting
$\wt\rho_\delta\psi(z)=\wt\Psi(z,\delta)$ and
$$\wt\Psi(z,w)=\log\int_{\zeta\in T_{X,z}}
\exp\big(\psi(\exph_z(w\zeta))\big)\,
\chi(|\zeta|^2)\,d\zeta,\qquad
(z,w)\in X\times\bC,\leqno(4.4)
$$
where $\exph_z(\zeta)$ and $\chi$ are defined as before. Here we take
$h_\Delta=h_\infty$ to be the metric on $K_X$ associated with the
given hermitian metric $\omega$ on~$T_X$, and we consider $\alpha$-psh
functions $\psi$ with respect to $\alpha=\Theta_{K_X,h_\infty}$ such that
$$
\int_X e^\psi\,h_\infty^{-1}=\int_X e^{\psi(z)}\,dV_\omega(z)\le 1,
\qquad dV_\omega={\omega^n\over n!}.
$$
The main point of introducing (4.4) as the regularization process is that
the integral $\int_Xe^\psi\,dV_\omega$ is almost preserved by our 
transform. In fact, if we change variables by putting $u=\exph_z(\zeta)$,
then in a neighborhood of the diagonal of $X\times X$ we have an inverse map
$\logh:X\times X\to T_X$ such that $\exph_z(\logh(z,u))=u$ and we then find
for $w$ small enough
$$
\wt\Psi(z,w)=\log\int_X e^{\psi(u)}\chi\bigg({|\logh(z,u)|^2\over |w|^2}\bigg)
{1\over |w|^{2n}}dV_\omega(\logh(z,u)),
$$
hence by Fubini
$$
\int_X e^{\wt\Psi(z,w)}dV_\omega(z)=
\int_{u\in X} K_w(z,u)e^{\psi(u)}dV_\omega(u)
$$
where $K_w$ is a kernel on $X\times X$ such that
$$
K_w(z,u)=
\int_{z\in X} {1\over |w|^{2n}}
\chi\bigg({|\logh(z,u)|^2\over |w|^2}\bigg)
{ dV_\omega(\logh(z,u))\over dV_\omega(u)}
dV_\omega(z).
$$
This kernel extends as a smooth function of all three variables $(z,u,w)$
for $w$ in a sufficient small disk $D(0,\varepsilon)$, and we have moreover
$K_0(z,u)=1$ and $K_w=K_{|w|}$, hence $K_w=1+O(|w|^2)$ with a uniform
bound on the compact manifold~$X\times X$. This shows that we only need
correcting $\smash{\wt\Psi}$ by the addition of a term $-C_0|w|^2$ to enforce
the upper bound $\int_X \smash{e^{\tilde\Psi(z,w)}}dV_\omega(z)\le 1$. 

\claim (4.5) Lemma|The regularization $\wt\Psi(z,w)$ satisfies 
essentially the same estimates as $\Psi(z,w)$,
namely there are constants $A=\sup_{|\zeta|\le 1,|\theta|\le 1}\{-c_{jk\ell m}
\zeta_j\ol\zeta_k\theta_\ell\ol\theta_m\}\ge 0$ and $K>0$ such that 
$$
\leqalignno{
&~~~~[0,\delta_0]\ni t\mapsto\wt\Psi(z,t)+Kt^2\quad
\hbox{is increasing,}&(4.6)\cr
\noalign{\vskip5pt}
&~~~~\alpha(z)+dd^c\wt\Psi(z,w)\ge -A\wt\lambda(z,|w|)|dz|^2
-K\big(|w|^2|dz|^2+|dz||dw|+|dw|^2),&(4.7)\cr
&~~~~\wt\lambda(z,t)={d\over d\log t}(\wt\Psi(z,t)+Kt^2)
\build\longrightarrow~~||t\to 0_+|\nu(\psi,z)\qquad
\hbox{$($Lelong number$)$.}&(4.8)\cr}
$$
\endclaim

\proof\ {\it of lemma}. The calculations made in [Dem94] (see Proposition
3.8 there) show that there is a smooth $(1,0)$ vector field $\tau$
(resp.\ $(1,1)$-vector field $V)$ depending linearly on
$(\theta,\eta)\in T_{X,z}\times\bC$ (resp.\ depending 
hermitian-quadratically on $(\theta,\eta)$) such that the Hessian form
of $\Psi(z,w)$ is given by
$$
\ddbar\Psi_{(z,w)}[\theta,\eta]^2=\int_{\zeta\in T_{X,x}}\kern-12pt\big(
\ddbar\psi_{\exph_z(w\zeta)}\cdot(\tau\wedge\overline\tau+|w|^2V)
+O(|w|^N)[\theta,\eta]^2\big)\,\chi(|\zeta|^2)\,d\zeta
$$
where $N$ is an arbitrary large integer and the error term $O(|w|^N...)$
is negligible when $N\ge 3$. A similar (substantially simpler) 
calculation yields
$$
\partial\Psi_{(z,w)}[\theta,\eta]=\int_{\zeta\in T_{X,x}}\kern-12pt\big(
\partial\psi_{\exph_z(w\zeta)}\cdot\tau
+O(|w|^N)[\theta,\eta]\big)\,\chi(|\zeta|^2)\,d\zeta.
$$
We replace $\psi$ by $e^\psi$ in these formulas and compute the $\ddbar$ of
$\wt\Psi(z,w)$ as defined in~(4.4). Modulo error terms 
which are omitted for the sake of legibility, we then get
$$
\eqalign{
\ddbar\wt\Psi_{(z,w)}&[\theta,\eta]^2
={\int_{T_X}e^\psi(\ddbar\psi+\partial\psi\wedge\dbar\psi)_{\exph_z(w\zeta)}
\cdot(\tau\wedge\overline\tau+|w|^2V)\,\chi(|\zeta|^2)\,d\zeta
\over \int_{T_X}e^{\psi(\exph_z(w\zeta))}\,\chi(|\zeta|^2)\,d\zeta}\cr
&-{\int_{T_X}e^\psi\partial\psi_{\exph_z(w\zeta)}
\cdot\tau\,\chi(|\zeta|^2)\,d\zeta
\wedge\overline{\int_{T_X}e^\psi\partial\psi_{\exph_z(w\zeta)}
\cdot\tau\,\chi(|\zeta|^2)d\zeta}
\over\Big(\int_{T_X}e^{\psi(\exph_z(w\zeta))}\,\chi(|\zeta|^2)\,d\zeta
\Big)^2}~.\cr}
$$
By the Cauchy-Schwarz inequality, the second term in the right hand side is
bounded the positive term involving $\partial\psi\wedge\dbar\psi\cdot
\tau\wedge\overline\tau$. This implies
$$
\eqalign{
\ddbar&\wt\Psi_{(z,w)}[\theta,\eta]^2
\ge{\int_{T_X}e^\psi\ddbar\psi_{\exph_z(w\zeta)}
\cdot\tau\wedge\overline\tau\,\chi(|\zeta|^2)\,d\zeta
\over \int_{T_X}e^{\psi(\exph_z(w\zeta))}\,\chi(|\zeta|^2)\,d\zeta}\cr
&+{\int_{T_X}e^\psi
(\ddbar\psi+\partial\psi\wedge\dbar\psi)_{\exph_z(w\zeta)}
\cdot(|w|^2V+O(|w|^N)[\theta,\eta]^2)\,\chi(|\zeta|^2)\,d\zeta
\over \int_{T_X}e^{\psi(\exph_z(w\zeta))}\,\chi(|\zeta|^2)\,d\zeta}~.\cr}
$$

\end
