% This file is a solution template for:
% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.
% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
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% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
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\title[\ \kern-190pt\rlap{\blank{J.-P.\ Demailly (Grenoble), KSCV12, Gyeongju, July 7, 2017}}\kern180pt\rlap{\blank{Extension of cohomology classes on
nonreduced subspaces}}\kern181.5pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{General extension theorem for cohomology classes
on non reduced analytic subspaces}

%% \subtitle{Presentation Subtitle} % (optional)

\author[] % (optional, use only with lots of authors)
{Jean-Pierre Demailly}

\institute[]{Institut Fourier, Université de Grenoble Alpes\ \ \&\ \ Académie des Sciences de Paris}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[]% (optional)
{\strut\kern-10pt
The 12th Korean Conference on Several Complex Variables \hbox{KSCV12\kern-9pt}\\
\alert{in honor of Kang-Tae Kim on the occasion of his 60th birthday}\\
The Kolon Hotel in Gyeong-Ju, South Korea, July 3--7, 2017}

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\begin{document}
%%\def\pause{}

% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
%%\AtBeginSubsection[]
%%{
%% \begin{frame}<beamer>
%%    \frametitle{Outline}
%%    \tableofcontents[currentsection,currentsubsection]
%%  \end{frame}
%%}


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% the following command: 

%\beamerdefaultoverlayspecification{<+->}

\begin{frame}
  \pgfdeclareimage[height=2cm]{uga-logo}{logo_UGA}
  \pgfuseimage{uga-logo}
  \pgfdeclareimage[height=2cm]{acad-logo}{academie_logo2}
  \pgfuseimage{acad-logo}
  \vskip-12pt
  \titlepage
\end{frame}

%%\begin{frame}
%%  \frametitle{Outline}
%%  \tableofcontents
%% You might wish to add the option [pausesections]
%%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

\begin{frame}
\frametitle{References}

This is a joint work with \alert{Junyan Cao \&\ Shin-ichi Matsumura}\vskip4pt

\claim{J.-P. Demailly},
\textit{Extension of holomorphic functions defined on non reduced analytic subvarieties, }
arXiv:1510.05230v1, 
Advanced Lectures in Mathematics Volume 35.1, 
the legacy of Bernhard Riemann after one hundred and fifty years, 2015. 
\vskip5pt

\claim{J.Y. Cao, J.-P. Demailly, S-i. Matsumura},
\textit{A general extension theorem for cohomology classes 
on non reduced analytic subspaces},
arXiv:1703.00292v2, Science China Mathematics, {\bf 60} (2017) 949--962
\vskip5pt

\claim{T. Ohsawa},
\textit{On a curvature condition that implies a cohomology injectivity theorem 
of Koll\'ar-Skoda type,} 
Publ. Res. Inst. Math. Sci. {\bf{41}} (2005), no. 3, 565--577.
\vskip5pt

\claim{T. Ohsawa, K. Takegoshi},
\textit{On the extension of $L^2$holomorphic functions,} 
Math.\ Zeitschrift {\bf 195} (1987), 197--204.
\end{frame}

\begin{frame}
  \frametitle{The extension problem}
  \vskip-6pt
Let $(X,\omega)$ be a \alert{possibly noncompact} $n$-dimensional 
K\"ahler \hbox{manifold,\kern-10pt}
$\cJ\subset\cO_X$ a coherent ideal sheaf, $Y=V(\cJ)$ its zero variety and
$$
\cO_Y=\cO_X/\cJ.
$$
Here $Y$ may be non reduced, i.e.\
\alert{$\cO_Y$ may have nilpotent elements}.\pause\vskip5pt

Also, let $(L,h_L)$ be a hermitian holomorphic line bundle on $X$, and
\alert{$$\Theta_{L,h_L}=i\,\ddbar\log h_L^{-1}$$}%%
its curvature current (we allow singular metrics, $h_L=e^{-\varphi}$,
$\varphi\in L^1_{\rm loc}$, $\Theta_{L,h_L}$ being computed in the
sense of currents).

\begin{block}{Question} Under which conditions on $X$, $Y=V(\cJ)$,
$(L,h_L)$~~ is\vskip4pt
\alert{\centerline{$H^q(X, K_X{\otimes}L)\to  H^q(Y, (K_X{\otimes}L)_{|Y})=
H^q(X, \cO_X(K_X{\otimes}L){\otimes}\cO_X/\cJ)$}}\vskip4pt
a surjective restriction morphism?
\end{block}
\end{frame}

\begin{frame}
\frametitle{First (too restrictive) answer}
Consider the exact sequence
$$0\to\cJ\to\cO_X\to\cO_X/\cJ\to0$$
twisted by $\cO_X(K_X\otimes L)$,\pause\ and the corresponding
long exact sequence of cohomology groups\vskip6pt
$\displaystyle
\cdots~{}\to
H^q(X, K_X\otimes L)\to H^q(X, \cO_X(K_X\otimes L)\otimes\cO_X/\cJ)$\\
$\displaystyle\strut\kern5.5cm{}\to
\alert{H^{q+1}(X, \cO_X(K_X{\otimes}L)\otimes\cJ)}~\cdots$\pause\vskip6pt
It is therefore enough to have
$$
H^{q+1}(X, \cO_X(K_X{\otimes}L)\otimes\cJ)=0.
$$
In order to kill $H^{q+1}$ it is enough to make a \alert{strict positivity}
(ampleness) assumption, by the Kodaira-Nakano / Nadel vanishing
theorems.\pause\
\alert{But we do not want to make such strong assumptions!}
\end{frame}

\begin{frame}
\frametitle{Assumptions (1)}
\vskip-4pt
We assume $X$ to be \alert{holomorphically convex}. By
the Cartan-Remmert theorem, this is the case iff $X$ admits a\\
\alert{proper holomorphic map $p:X\to S$} only a Stein
complex space~$S$.
\pause
\begin{block}{Observation : cohomology is then always Hausdorff}
Let $X$ be a holomorphically convex
complex space and $\cF$ a coherent analytic sheaf over $X$.
Then all cohomology groups \alert{$H^q(X,\cF)$ are Hausdorff} with respect
to their natural topology $($local uniform
convergence of holomorphic \v{C}ech cochains$)$
\end{block}
\pause
{\bf Proof.} $H^q(X,\cF)\simeq H^0(S,R^qp_*\cF)$ is a Fréchet space.
\pause
\begin{block}{Corollary} To solve an equation $\dbar u=v$ on a
holomorphically convex manifold~$X$, it is enough to solve
it approximately:\vskip4pt
\alert{\centerline{$\dbar u_\varepsilon=v+w_\varepsilon,\qquad
w_\varepsilon\to 0~~\hbox{as $\varepsilon\to 0$}$}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Assumptions (2)}
\vskip-5pt  
We assume that the subvariety $Y\subset X$ is defined by\vskip6pt
\alert{\centerline{$Y=V(\cI(e^{-\psi})),\qquad
\cO_Y:=\cO_X/\cI(e^{-\psi})$}}\vskip6pt
where $\psi$ is a quasi-psh function
with \emph{analytic singularities}, i.e.\
locally on a neighborhood $V$ of an arbitrary point $x_0\in X$ we have
\vskip6pt
\alert{\centerline{$
\psi(z)=c\log\sum|g_j(z)|^2+v(z),\quad g_j{\in}\cO_X(V),~c{>}0,~
v{\in} C^\infty(V),$}}\vskip6pt\pause
and $\cI(e^{-\psi})\subset\cO_X$ is the \alert{multiplier ideal sheaf}
\vskip6pt
\alert{\centerline{$
\cI(e^{-\psi})_{x_0}=\big\{f\in\cO_{X,x_0}\,;\;\exists U\ni x_0\,,\;
\int_U|f|^2e^{-\psi}d\lambda<+\infty\big\}$}}
\vskip3pt\pause
\begin{block}{Claim {\bf (Nadel)}}
$\cI(e^{-\psi})$ is a coherent ideal sheaf.
\end{block}
\pause
Moreover $\cI(e^{-\psi})$ is always an integrally closed ideal.
\vskip6pt\pause
{\bf Typical choice:} \alert{$\psi(z)=c\log|s(z)|_{h_E}^2,~c>0,~
s\in H^0(X,E)$.}
\end{frame}

\begin{frame}
\frametitle{Log resolution / reduction to the divisorial case}
The simplest case is when $Y=\sum m_jY_j$ is
an effective simple normal crossing divisor and $\cO_Y=\cO_X/\cO_X(-Y)$.
We can then take
\alert{$$
\psi(z)=\sum c_j\log|\sigma_{Y_j}|^2_{h_j},~~~c_j>0,~\lfloor c_j\rfloor=m_j,
$$}%%
for some smooth hermitian metric $h_j$ on $\cO_X(Y_j)$. Then
\alert{$$\textstyle
\cI(e^{-\psi})=\cO_X\big(-\sum m_jY_j\big),\quad~~
i\ddbar\psi=\sum c_j(2\pi [Y_j]  - \Theta_{\cO(Y_j),h_j})
$$}\pause%%
The case of a higher codimensional multiplier ideal scheme $\cI(e^{-\psi})$
can easily be reduced to the divisorial case by using a suitable
log resolution (a composition of blow ups, thanks to Hironaka's
desingularization theorem).
\end{frame}
  
\begin{frame}
\frametitle{Main results}
\vskip-7pt  
\begin{block}{Theorem {\bf(JY. Cao, D-- , S-i. Matsumura, January 2017)}}
Take $(X,\omega)$ to be \alert{Kähler and holomorphically convex},\\
and let $(L,h_L)$ be a hermitian line bundle such that
\vskip4pt
\alert{\centerline{$\kern-5pt(**)\qquad
\Theta_{L,h_L}+(1+\alpha\delta)i\ddbar \psi \geq 0\qquad\text{in the sense
of currents}$}}\vskip4pt
for some $\delta(x)>0$ continuous and $\alpha=0,1$.
Then:\pause\\ the morphism induced by the natural inclusion 
$\cI(h_Le^{-\psi}) \to \cI(h_L)$\vskip4pt
\alert{\centerline{$H^{q}(X, K_X \otimes L \otimes \cI(h_Le^{-\psi})) 
\to H^{q}(X, K_X \otimes L \otimes \cI(h_L))$}}\vskip4pt
is injective for every $q\geq 0$,\pause\ in other words,
the sheaf morphism $\cI(h) \to  \cI(h_L)/\cI(h_Le^{-\psi})$ yields
a surjection\vskip4pt
\alert{\centerline{$
H^{q}(X, K_X \otimes L \otimes \cI(h_L)) \to 
H^{q}(X, K_X \otimes L \otimes \cI(h_L)/\cI(h_Le^{-\psi})).
$}}
\end{block}
\vskip-5pt\pause
\begin{block}{Corollary {\bf (take $h_L$ smooth${}\Rightarrow\cI(h_L)=\cO_X$,
and $Y=V(\cI(e^{-\psi})$)}} 
If $h_L$ is smooth, $\cO_Y=\cO_X/\cI(e^{-\psi})$ and \alert{$h_L,\,\psi$
satisfy $(**)$}, \hbox{then\kern-15pt}
\alert{$H^{q}(X, K_X \otimes L)\to H^{q}(Y, (K_X \otimes L)_{|Y})$
is surjective.}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Comments / algebraic consequences}
\vskip-4pt  
The exact sequence
\alert{$0\to\cI(h_Le^{-\psi})\to
\cI(h_L)\to\cI(h_L)/\cI(h_Le^{-\psi})\to 0\kern-20pt$}\\
implies that both injectivity and surjectivity hold when
\vskip5pt
\centerline{$H^q(X, K_X\otimes L\otimes\cI(h_Le^{-\psi})=0$,}\vskip5pt
and for this it is enough to have a \alert{strict curvature assumption}
\vskip5pt
\centerline{$(*{*}*)\quad\qquad
\alert{\Theta_{L,h_L}+i\ddbar \psi \geq \delta\omega>0}\qquad\text{in the sense
of currents.}\quad{}$}\pause

\begin{block}{Corollary (purely algebraic)}
Assume that $X$ is projective (or that one has a projective
morphism $X\to S$ over an affine algebraic base~$S$). Let $Y=\sum m_jY_j$
be an effective divisor and $\cO_Y=\cO_X/\cO_X(-Y)$. If (as $\bQ$-divisors)
\vskip5pt
\alert{$(**)\kern40pt
L-(1+\delta)\sum c_jY_j = G_\delta+U_\delta,~~\lfloor c_j\rfloor=m_j$}\vskip5pt
with $\delta=0$ or $\delta_0\in\bQ_+^*$, $G_\delta$ semiample and
$U_\delta\in{\rm Pic}^0(X)$, \hbox{then\kern-10pt}\vskip5pt
\alert{\centerline{$H^{q}(X, K_X \otimes L)\to H^{q}(Y, (K_X \otimes L)_{|Y})$}}
\vskip0pt
is surjective.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Possible use for abundance by induction on $\dim X$ ?}
For a line bundle $L$, one defines the Kodaira-Iitaka dimension
$\kappa(L)=\limsup_{m\to+\infty} \log\dim H^0(X,L^{\otimes m})/\log m$
and the numerical dimension ${\rm nd}(L)={}$maximum power of non
zero positive intersection of a positive current $T\in c_1(L)$, if
$L$ is pseudo-effective, and ${\rm nd}(L)=-\infty$ otherwise. They always
satisfy
\alert{$$
-\infty\leq \kappa(L)\leq{\rm nd}(L)\leq n=\dim X.
$$}\pause%%
One says that $L$ is \alert{abundant} if $\kappa(L)={\rm nd}(L)$.
The~\alert{abundance conjecture} states that $K_X$ is always abundant
if $X$ is nonsingular and projective (or even compact Kähler). More
generally, it is expected that $K_X+\Delta$ is abundant for every
effective klt \hbox{$\bQ$-divisor~$\Delta$.\kern-10pt}\pause\vskip5pt

When $X$ is not uniruled, i.e.\ $K_X$ is pseudoeffective, one can ask
whether the following \alert{generalized abundance property} holds true:
let $L$ be a line bundle such that $L{-}\varepsilon K_X$ is pseudoeffective,
\hbox{$0<\varepsilon\ll 1$;\kern-15pt}\break does there exist
$G\in\Pic^0(X)$ such that $L+G$ is abundant~?
\end{frame}

\begin{frame}
\frametitle{Twisted Bochner-Kodaira-Nakano inequality (Ohsawa-Takegoshi)}
Let $(X,\omega)$ be a K\"ahler manifold and let $\eta,\,\lambda>0$ be
smooth functions on~$X$.\\ For every compacted supported section
$u\in\cC^\infty_c(X,\Lambda^{p,q}T^*_X\otimes L)$ with values in a
hermitian line bundle $(L,h_L)$, one has
\alert{
\begin{eqnarray*}
\|(\eta+\lambda)^{{1\over2}}\dbar^*u\|^2\kern-20pt
&&{}+\|\eta^{{1\over2}}\dbar u\|^2+\|\lambda^{{1\over2}}\partial u\|^2+
2\|\lambda^{-{1\over2}}\partial\eta\wedge u\|^2\\
&&\ge\int_X\langle B^{p,q}_{L,h_L,\omega,\eta,\lambda}u,u\rangle dV_{X,\omega}
\end{eqnarray*}}%%
where $dV_{X,\omega}=\frac{1}{n!}\omega^n$ is the Kähler volume element
and $B^{p,q}_{L,h_L,\omega,\eta,\lambda}$ is the Hermitian operator on
$\Lambda^{p,q}T^*_X\otimes L$ such that
\alert{$$
B^{p,q}_{L,h_L,\omega,\eta,\lambda}=
[\eta\,i\Theta_L-i\,\ddbar\eta-i\lambda^{-1}\partial\eta\wedge
\dbar\eta~,~\Lambda_\omega].
$$}
\end{frame}
  
\begin{frame}
\frametitle{Approximate solutions to $\dbar$-equations}
\vskip-4pt
\begin{block}{Main $L^2$ estimate}
Let $(X,\omega)$ be a K\"ahler manifold possessing a complete K\"ahler metric
let $(E,h_E)$ be a Hermitian vector bundle over~$X$.
Assume that $B=B^{n,q}_{E,h,\omega,\eta,\lambda}$
satisfies \alert{$B+\varepsilon\Id>0$} for some $\varepsilon>0$ $($so that $B$
can be just semi-positive or even slightly negative$)$.\pause\\
Take a section
$v\in L^2(X,\Lambda^{n,q}T^*_X\otimes E)$ such that $\dbar v=0$ and\vskip6pt
\alert{\centerline{$\displaystyle M(\varepsilon):=
\int_X\langle (B+\varepsilon\Id)^{-1}v,v\rangle\,dV_{X,\omega}<+\infty.$}}
\pause\vskip7pt
\strut\kern-2pt Then there exists an approximate solution
\alert{\hbox{$u_\varepsilon\,{\in}\,
L^2(X,\Lambda^{n,q-1}T^*_X\,{\otimes}\,E)$\kern-6pt}}
and a
\alert{correction term $w_\varepsilon\in L^2(X,\Lambda^{n,q}T^*_X\otimes E)$}
such that\vskip7pt
\strut\kern40pt\alert{$\dbar u_\varepsilon=v+w_\varepsilon$}~~~~
and\vskip3pt
\alert{\centerline{$\displaystyle
\int_X(\eta+\lambda)^{-1}|u_\varepsilon|^2\,dV_{X,\omega}+
\frac{1}{\varepsilon}\int_X|w_\varepsilon|^2\,dV_{X,\omega}\le M(\varepsilon).
\phantom{\Bigg|}$}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Proof: setting up the relevant $\dbar$ equation (1)}
\vskip-5pt
Every cohomology class in\vskip7pt
\centerline{$
H^q(X,\cO_X(K_X\otimes L)\otimes\cI(h_L)/\cI(h_Le^{-\psi}))
$}\vskip7pt
is represented by a holomorphic \v{C}ech $q$-cocycle with respect to
a Stein covering $\cU=(U_i)$, say $(c_{i_0\ldots i_q})$,\vskip5pt
\alert{\centerline{$
c_{i_0\ldots i_q}\in H^0\big(U_{i_0}\cap\ldots\cap U_{i_q},
\cO_X(K_X\otimes L)\otimes\cI(h_L)/\cI(h_Le^{-\psi})\big).
$}}\pause\vskip5pt
By the standard sheaf theoretic isomorphism with Dolbeault cohomology, this 
class is represented by a smooth $(n,q)$-form\vskip4pt
\alert{\centerline{$\displaystyle
f=\sum_{i_0,\ldots,i_q}c_{i_0\ldots i_q}\rho_{i_0}
\dbar\rho_{i_1}\wedge\ldots\dbar\rho_{i_q}
$}}\vskip3pt
by means of a partition of unity $(\rho_i)$ subordinate to $(U_i)$. This form is
to be interpreted as a form on the (non reduced) analytic subvariety $Y$ associated with the colon ideal sheaf $\cJ=\cI(he^{-\psi}):
\cI(h)$ and the structure sheaf $\cO_Y=
\cO_X/\cJ$.
\end{frame}

\begin{frame}
\frametitle{Proof: setting up the relevant $\dbar$ equation (2)}
\vskip-3pt
We get an extension of $f$ as a smooth (no longer
$\dbar$-closed) $(n,q)$-form on $X$ by taking\vskip4pt
\centerline{$\displaystyle
\widetilde{f}=\sum_{i_0,\ldots,i_q}\widetilde{c}_{i_0\ldots i_q}\rho_{i_0}
\dbar\rho_{i_1}\wedge\ldots\dbar\rho_{i_q}$}\vskip3pt
where $\widetilde{c}_{i_0\ldots i_q}={}$extension of
$c_{i_0\ldots i_q}$ from $U_{i_0}{\cap}{\ldots}{\cap}U_{i_q}{\cap}Y$ to
\hbox{$U_{i_0}{\cap}{\ldots}{\cap}U_{i_q}$\kern-20pt}\\\pause
\vskip4pt
\hbox{\pgfdeclareimage[height=3cm]{tubular}{tubular}
\pgfuseimage{tubular}}
\crush{\LabelTeX 0 10 \RGBColor{1 0 0}{$Y$}\ELTX}%%
\crush{\LabelTeX 0 30.6
 \RGBColor{0 0.7 0}{$\{x{\in}X\,/\;t{<}\psi(x){<}t{+}1\}$}\ELTX}%%
\crush{\LabelTeX 78.7 25 $1$\ELTX}%%
\crush{\LabelTeX 78.7 9 $0$\ELTX}%%
\crush{\LabelTeX 94.7 20 \RGBColor{0 0 1}{$\theta(s)$}\ELTX}%%
\crush{\LabelTeX 100 9 $1$\ELTX}%%
\crush{\LabelTeX 107 9 $s$\ELTX}%%
\pause\vskip-15pt
Now, truncate $\widetilde{f}$ as $\theta(\psi-t){\cdot}\widetilde{f}$ on
the green hollow tubular neighborhood, and solve
an approximate \hbox{$\dbar$-equation\kern-30pt}\vskip4pt
$(*)\kern60pt\dbar u_{t,\varepsilon}=
\dbar(\theta(\psi-t){\cdot}\widetilde{f})+w_{t,\varepsilon}$
\end{frame}

\begin{frame}
\frametitle{Proof: setting up the relevant $\dbar$ equation (3)}
\vskip-4pt
Here we have\vskip4pt
\alert{\centerline{$
\dbar(\theta(\psi-t)\cdot\widetilde{f})=
\theta'(\psi-t)\dbar\psi\wedge\widetilde{f}+
\theta(\psi-t)\cdot\dbar\widetilde{f}
$}}\vskip4pt
where the first term vanishes near $Y$ and the second one is $L^2$ with
respect to $h_Le^{-\psi}$ (as the image of $\dbar\widetilde{f}$ in
$\cI(h_L)/\cI(h_Le^{-\psi})$ is \hbox{$\dbar f=0$).\kern-10pt}
\vskip5pt\pause
With ad hoc ``twisting functions'' $\eta=\eta_t:=1-\delta\chi_t(\psi)$,
$\lambda:=\pi(1+\delta^2\psi^2)$ and a suitable adjustment
$\varepsilon=e^{(1+\beta)t}$, $\beta\ll 1$, we can find approximate 
$L^2$ solutions of the $\dbar$-equation such that\vskip-18pt
$$\dbar u_{t,\varepsilon}=\dbar(\theta(\psi -t){\cdot}\widetilde{f})
+w_{t,\varepsilon} \text{ }, \qquad
\int_X |u_{t,\varepsilon}|_{\omega,h_L} ^2 e^{-\psi} dV_{X, \omega} < +\infty$$
\vskip-8pt
and
\vskip-16pt
$$ \lim_{t\rightarrow -\infty} \int_X |w_{t,\varepsilon}|_{\omega, h_L} ^2
e^{-\psi} d V_{X,\omega} =0.$$\pause
The estimate on $u_{t,\varepsilon}$ with respect to the weight $h_Le^{-\psi}$
shows that
\alert{$\theta(\psi-t)\cdot\widetilde{f}-u_{t,\varepsilon}$} is an approximate
extension of~$f$.\hfill$\square$
\end{frame}

\begin{frame}
\frametitle{Can one get estimates for the extension ?}
\vskip-6pt  
The answer is \alert{yes if $\psi$ is log canonical}, namely
$\cI(e^{-(1-\varepsilon)\psi})=\cO_X$ for all $\varepsilon>0$.\pause
\begin{block}{Ohsawa's residue measure} If $\psi$ is log canonical,
one can also associate in a natural way a measure $dV_{Y^\circ,\omega}[\psi]$
on the set $Y^\circ$ of regular points of $Y$ as follows. If
$g\in\cC_c(Y^\circ)$ is a compactly supported continuous function on
$Y^\circ$ and $\widetilde g$ a compactly supported extension of $g$ to
$X$, one sets\vskip2pt
\alert{\centerline{$\displaystyle
\int_{Y^\circ}g\,dV_{Y^\circ,\omega}[\psi]=
\lim_{t\to-\infty}\int_{\{x\in X\,,\;t<\psi(x)<t+1\}}
\widetilde ge^{-\psi}\,dV_{X,\omega}
$}}\vskip-18pt$\strut$
\end{block}\vskip-6pt\pause
\begin{block}{Theorem} If $\psi$ is lc and the curvature hypothesis is
satisfied, for any $f$ in
$H^0(Y,K_X\otimes L\otimes\cI(h_L)/\cI(h_Le^{-\psi}))$ s.t.
\hbox{$\int_{Y^\circ}|f|^2_{\omega,h_L}dV_{Y^\circ,\omega}[\psi]<+\infty$,
\kern-20pt}\break
there exists $\smash{\widetilde f}\in H^0(X,K_X\otimes L\otimes\cI(h_L))$
which extends $f$, such that\vskip2pt
\alert{\centerline{$\displaystyle
\int_X(1+\delta^2\psi^2)^{-1}e^{-\psi}
|\widetilde f|_{\omega,h_L}^2dV_{X,\omega}\leq
\frac{34}{\delta}\int_{Y^\circ}|f|^2_{\omega,h_L}dV_{Y^\circ,\omega}[\psi].
$}}\vskip-17pt$\strut$
\end{block}
\end{frame}

\begin{frame}
\frametitle{Can one get estimates for the extension ? (sequel)}
\vskip-6pt  
If $\psi$ is not log canonical, consider the ``last jumps'' $m_{p-1}<m_p\leq 1$
such that $\cI(h_Le^{-m_{p-1}\psi})\supsetneq
\cI(h_Le^{-m_p\psi})=\cI(h_Le^{-\psi})$ and \hbox{assume\kern-20pt}\vskip6pt
\alert{\centerline{$\displaystyle
f\in H^0(Y,K_X\otimes L\otimes\cI(h_Le^{-m_{p-1}\psi})/\cI(h_Le^{-m_p\psi})),
$}}\vskip6pt
i.e., $f$ vanishes just a little bit less than prescribed by the sheaf
$\cI(h_Le^{-\psi}))$. Then there is still a corresponding residue measure:
\pause\vskip-3pt
\begin{block}{Higher multiplicity residue measure} If $f$ is as above,
and $\widetilde f$ is a local extension, 
one can associate a higher multiplicity residue measure
$|f|^2dV_{Y^\circ,\omega}[\psi]$ (formal notation) as follows. If
$g\in\cC_c(Y^\circ)$ and $\widetilde g$ a compactly supported extension
of $g$ to $X$, one sets\vskip4pt
\alert{\centerline{$\displaystyle
\int_{Y^\circ}g\,|f|^2dV_{Y^\circ,\omega}[\psi]=
\lim_{t\to-\infty}\int_{\{x\in X\,,\;t<\psi(x)<t+1\}}
\widetilde g\,|\widetilde f|^2e^{-m_p\psi}\,dV_{X,\omega}
$}}\vskip-18pt$\strut$
\end{block}\pause
Then a global extension
$\widetilde f\in H^0(X,K_X\otimes L\otimes\cI(h_Le^{-m_{p-1}\psi}))$ exists,
that satisfies the expected $L^2$ estimate.
\end{frame}

\begin{frame}
\frametitle{Special case / limitations of the $L^2$ estimates}
\vskip-5pt
In the special case when $\psi$ is given by $\psi(z)=r\log|s(z)|^2_{h_E}$
for a section $s\in H^0(X,E)$ generically transverse to the
zero section of a rank $r$ vector vector $E$ on $X$, the subvariety
$Y=s^{-1}(0)$ has codimension $r$, and one can check easily that
\alert{$$
dV_{Y^\circ,\omega}[\psi]=\frac{dV_{Y^\circ,\omega}}
{|\Lambda^r(ds)|_{\omega,h_E}^2}.
$$}\pause%%
Thus one sees that the residue measure takes into account in a very precise
manner the singularities of $Y$. It may happen that
$dV_{Y^\circ,\omega}[\psi]$ has infinite mass near the
singularities of $Y$, as is the case when $Y$ is
a simple normal crossing divisor.\pause\vskip5pt

Therefore, sections
$s\in H^0(Y,(K_X\otimes L)_{|Y}$ may not be $L^2$ with respect to
$dV_{Y^\circ,\omega}[\psi])$, and the $L^2$ estimate of the approximate
extension can blow up as $\varepsilon\to 0$. The surprising fact is
this is however sufficient to prove the qualitative extension
theorem, but without any effective $L^2$ estimate in the limit.
\end{frame}

\begin{frame}
\frametitle{The end}
\strut\vskip-3mm  
\hbox{\strut\kern5mm\pgfdeclareimage[height=6.5cm]{CalabiYau}{CalabiYau}
\strut\kern2cm\pgfuseimage{CalabiYau}}\vskip-7cm
\centerline{\huge\bf Happy birthday Kang-Tae !}\vskip5.5cm\pause
Thanks and congratulations to the organizers!\\
Jisoo Byun, Hong Rae Cho, Sung-Yeon Kim,\\
Kang-Hyurk Lee, Jong-Do Park
\end{frame}

\end{document}