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\title[\RGBColor{1 1 1}{\
A sharp lower bound for the log canonical threshold}]
% (optional, use only with long paper titles)
{A sharp lower bound for the log canonical threshold
of an isolated plurisubharmonic singularity}

\author{Jean-Pierre Demailly / Ph\d{a}m Ho\`ang Hi\d{\^e}p}
\institute[Grenoble I]{Institut Fourier, Universit\'e de Grenoble I, France
\vskip5pt
GAEL XX +}

\date[]{June 28, 2012}
%\date{February 10, 2012}

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

\begin{frame}
  \pgfdeclareimage[height=2cm]{acad-logo}{academie_logo2}
  \pgfuseimage{acad-logo}
  \pgfdeclareimage[height=2cm]{ujf-logo}{logo_ujf}
  \pgfuseimage{ujf-logo}
  \titlepage
\end{frame}

\begin{frame}
\frametitle{Singularities of plurisubharmonic (psh) functions}

\claim{Goal}: study \alert{local singularities} of a plurisubharmonic 
function $\varphi$ on a neighborhood of a point in ${\mathbb C}^n$.
\vskip4pt
$\varphi:X\to[-\infty,+\infty[$~~upper semicont. / mean value
inequality.\kern-15pt
\vskip3pt
Poles : $\varphi^{-1}(-\infty)$ (not always closed, sometimes fractal)
\vskip3pt
\pause
\claim{Algebraic setting}:
\vskip3pt
\centerline{$\varphi(z)=\frac{1}{2}\log(|g_1|^2+\ldots+|g_N|^2)$}
\vskip3pt
associated to some ideal $\cJ=(g_1,\ldots,g_N)\subset{\mathcal O}_{X,p}$ 
of holo- 
morphic (algebraic) functions on some \hbox{complex~variety~$X$.\kern-15pt}

\vskip3pt
\pause
\claim{More generally}: consider a sequence $(\cJ_k)_{k\in{\mathbb N}}$ of such
ideals, with 
$$\cJ_k\cJ_\ell\subset\cJ_{k+\ell}$$
Try to understand ``$\lim (\cJ_k)^{1/k}$'' (Lazarsfeld, Ein, 
Musta\c{t}\v{a}...)
\end{frame}

\begin{frame}
\frametitle{Lelong numbers and log canonical thresholds}
The easiest way of measuring singularities of psh functions is by
using \alert{Lelong numbers}: 
$$
\nu(\varphi,p)=\liminf_{z\to p}\frac{\varphi(z)}{\log|z-p|}.
$$
\claim{Example}$\,$:\vskip3pt
\hbox{$\varphi(z)=\frac{1}{2}\log(|g_1|^2+\ldots+|g_N|^2)\Rightarrow
\nu(\varphi,p)=\min\,$ord${}_p(g_j)\in{\mathbb N}$.}
\vskip6pt
\pause

Another useful invariant is the \alert{log canonical threshold}.

\begin{block}{Definition}
Let $X$ be a complex manifold, $p\in X$, and $\varphi$ be
a plurisubharmonic function defined on $X$.
\pause
The \alert{log canonical threshold} or \emph{complex singularity exponent} of $\varphi$ at $p$ is defined by
\vskip5pt
\alert{\centerline{$
c_p(\varphi)=\sup\big\{c\ge 0\,:\,e^{-2c\varphi} \text{ is $L^1$ on a neighborhood of $p$}\big\}.$}}
\end{block}
\end{frame}
%
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\begin{frame}
\frametitle{log canonical threshold of coherent ideals}
For simplicity we will take here $p=0$ and denote 
$$c(\varphi)=c_0(\varphi).$$\pause
The log canonical threshold is a subtle invariant$\,$!
\vskip5pt
\claim{Calculation in the case of analytic singularities$\,$:} take
$$
\varphi(z)=\frac{1}{2}\log(|g_1|^2+\ldots+|g_N|^2),~~~
\cJ=(g_1,\ldots,g_N).$$
Then by Hironaka, $\exists$ modification $\mu:\widetilde X\to X$
such that 
\alert{$$\mu^*\cJ=(g_1\circ\mu,\ldots,g_N\circ\mu)=\cO(-\sum a_jE_j)$$}
for some normal crossing divisor. \pause
Let \alert{$\cO(\sum b_jE_j)$ be the divisor of Jac$(\mu)$}. \pause We have
\alert{$$
c(\varphi)=\min_{E_j,\,\mu(E_j)\ni 0}\frac{1+b_j}{a_j}\in{\mathbb Q}_+^*.
$$}
\end{frame}
%
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\begin{frame}
\frametitle{Proof of the formula for the log canonical threshold}
In fact, we have to find the supremum of $c>0$ such that
\alert{
$$
I=\int_{V\ni 0}\frac{d\lambda(z)}{\big(|g_1|^2+\ldots+|g_N|^2\big)^c}<+\infty.
$$}
Let us perform the change of variable $z=\mu(w)$. Then
$$d\lambda(z)=|{\rm Jac}(\mu)(w)|^2\sim \Big|\prod w_j^{b_j}\Big|^2\,
d\lambda(w)$$
with respect to coordinates on the blow-up $\widetilde V$ of $V$, and
$$
I\sim\int_{\widetilde V}\frac{\big|\prod w_j^{b_j}\big|^2\,
d\lambda(w)}{\big|\prod w_j^{a_j}\big|^{2c}}
$$
so convergence occurs if $ca_j-b_j<1$ for all $j$.
\end{frame}
%
\begin{frame}
\frametitle{Notations and basic facts}
\begin{itemize}
\item A domain $\Omega\subset\mathbb{C}^n$ is called \emph{hyperconvex} if \hbox{$\exists\psi \in \PSH(\Omega)$,\kern-20pt}\break
$\psi\leq 0$, such that
$\{z: \psi(z)<c\} \Subset \Omega$ for all $c<0$.
\vskip1pt
\pause
\item $\displaystyle
\Eo(\Omega){=}\left\{\!\varphi{\in}\PSH\cap L^{\infty}(\Omega):\!\!\lim_{z\to\partial\Omega}\!\varphi(z){=}0,\int_\Omega\!\!(dd^c\varphi)^n{<}{+}\infty\right\}$\kern-20pt
\pause
\item
$\displaystyle
\F (\Omega)=\Big\{\varphi\in \PSH(\Omega):\ \exists\ \Eo(\Omega)\ni\varphi_p\searrow\varphi,\text{ and}$
${}\kern5cm\displaystyle
\sup_{p\geq 1}\int_\Omega(dd^c\varphi_p)^n<+\infty\Big\},$
\item
$\displaystyle\E(X)=\{\varphi\in\PSH(X)
~\mbox{locally in $\F(\Omega)$ mod $C^\infty(\Omega)$}\}$
\end{itemize}
\vskip-5pt
\pause
\begin{block}{Theorem {\rm (U. Cegrell)}}
\alert{$\E(X)$ is the largest subclass of psh functions} defined on a complex manifold $X$ for which the complex Monge-Amp\`ere operator is 
\alert{locally well-defined}.
\end{block}
\end{frame}

%
\begin{frame}
\frametitle{Intermediate Lelong numbers}
Set here $d^c=\frac{i}{2\pi}(\overline\partial -\partial)$ so that
$dd^c=\frac{i}{\pi}\partial\overline\partial$.
\pause
If $\varphi\in\E(\Omega)$ and $0\in\Omega$, the products $(dd^c\varphi)^j$
are well defined and one can consider the Lelong numbers
$$e_{j} (\varphi)=\nu\big((dd^c\varphi)^j,0\big).$$
In other words
\alert{$$
e_{j} (\varphi) = \int_{\{0\}} (dd^c\varphi)^j\wedge (dd^c\log\| z\|)^{n-j}\,.
$$}
One has $e_0(\varphi)=1$ and $e_1(\varphi)=\nu(\varphi,0)$ (usual Lelong
number).
\pause
When
$$
\varphi(z)=\frac{1}{2}\log(|g_1|^2+\ldots+|g_N|^2),
$$
one has $e_j(\varphi)\in{\mathbb N}$.
\end{frame}
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\begin{frame}
\frametitle{The main result}
\begin{block}{Main Theorem (Demailly \& Ph\d{a}m)}
Let $\varphi\in\E(\Omega)$.
If $e_1(\varphi)=0$, then $c(\varphi)=\infty$.\\
\pause
Otherwise, we have
\alert{\centerline{$\displaystyle
c(\varphi)\geq\sum_{j=0}^{n-1}\frac {e_j(\varphi)} {e_{j+1}(\varphi)}.$}}
The lower bound improves a classical result of H.\ Skoda (1972), according
to which
\centerline{$\displaystyle
\frac{1}{e_1(\varphi)} \le c(\varphi) \le \frac{n}{e_1(\varphi)}.
$}
\end{block}
\pause\vskip3pt
Remark: The above theorem is optimal, with equality for
\alert{$$
\varphi(z)=\log(|z_1|^{a_1}+\ldots+|z_n|^{a_n}),~~0<a_1\leq a_2\leq\ldots
\leq a_n.
$$}
Then $\displaystyle e_j(\varphi)=a_1\ldots a_j,~~
c(\varphi)=\frac{1}{a_1}+\ldots+\frac{1}{a_n}$.
\end{frame}

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\begin{frame}
\frametitle{Geometric applications}
The log canonical threshold has a lot of applications. It is essentially
a local version of Tian's invariant, which determines a sufficient condition
for the \alert{existence of K\"ahler-Einstein metrics}.
\pause

Another important application is to \alert{birational rigidity}.

\begin{block}{Theorem {\rm (Pukhlikov 1998, Corti 2000, de Fernex 2011)}}
Let $X$ be a smooth hypersurface of degree $d$
in $\mathbb{CP}^{n+1}$. Then if $d=n+1$, 
\alert{{\rm Bir}$(X)\simeq{}${\rm Aut}$(X)$}
\end{block}

It was first shown by Manin-Iskovskih in the early 70's  that a 
3-dim quartic in $\mathbb{CP}^4$ ($n=3$, $d=4$) is not rational.
\pause

\begin{block}{Question} For $3\le d\le n+1$, when is it true that
{\rm Bir}$(X)\simeq{}${\rm Aut}$(X)$ (birational rigidity) ?
\end{block}
\end{frame}

%
\begin{frame}
\frametitle{Lemma~1}

\begin{block}{Lemma~1}
Let $\varphi\in\E(\Omega)$ and $0\in\Omega$. Then we have that
\[
e_{j}(\varphi )^2\leq e_{j-1} (\varphi ) e_{j+1} (\varphi ),
\]
for all $j=1,\ldots,n-1.$
\end{block}

In other words $j\mapsto\log e_j(\varphi)$ is convex, thus we have
$e_j(\varphi)\ge e_1(\varphi)^j$ and the ratios
$e_{j+1}(\varphi)/e_j(\varphi)$ are increasing.

\begin{block}{Corollary} If $e_1(\varphi)=\nu(\varphi,0)=0$, then
$e_{j}(\varphi)=0$ for $j=1,2,\ldots,n-1$.
\end{block}

A hard conjecture by V.\ Guedj and A.\ Rashkovskii~($\sim 1998)\kern-10pt$
states that $\varphi\in\E(\Omega)$, $e_1(\varphi)=0$ also 
implies $e_n(\varphi)=0$.
\end{frame}
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\begin{frame}
\frametitle{Proof of Lemma~1}
Without loss generality we can assume that $\Omega$ is the unit ball and $\varphi\in\Eo(\Omega)$.
\pause
For $h,\,\psi\in\Eo(\Omega)$ an integration by parts and the Cauchy-Schwarz
inequality yield
\begin{multline*}
\left[\int_\Omega -h (dd^c\varphi)^j\wedge (dd^c\psi)^{n-j}\right]^2\\
=\Bigg[\int_\Omega d\varphi\wedge d^c\psi \wedge (dd^c\varphi)^{j-1} \wedge (dd^c\psi)^{n-j-1}\wedge dd^ch\Bigg]^2\\
\leq\int_\Omega d\psi\wedge d^c\psi \wedge (dd^c\varphi)^{j-1}\wedge (dd^c\psi)^{n-j-1}\wedge dd^ch\\ \int_\Omega d \varphi\wedge d^c\varphi \wedge (dd^c\varphi)^{j-1}\wedge (dd^c\psi)^{n-j-1}\wedge dd^ch\\
=\int_\Omega -h (dd^c\varphi)^{j-1}\wedge (dd^c\psi)^{n-j+1}\int_\Omega -h (dd^c\varphi)^{j+1}\wedge (dd^c\psi)^{n-j-1}\ ,
\end{multline*}
\end{frame}
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\begin{frame}
\frametitle{Proof of Lemma~1, continued}
Now, as $p\to+\infty$, take
\[
h(z)=h_p(z)=\max\Big(-1,\frac{1}{p}\log\Vert z\Vert\Big)\nearrow \left\{
\begin{array}{ccc}
0 &\text{ if } z\in\Omega\backslash\{0\}\\
-1 &\text{ if } z=0.\hfill
\end{array} \right.
\]
\pause
By the monotone convergence theorem we get in the limit that
\begin{multline*}
\left[\int_{ \{0\} } (dd^c\varphi)^j\wedge (dd^c\psi)^{n-j}\right]^2
\leq\int_{ \{0\} } (dd^c\varphi)^{j-1}\wedge (dd^c\psi)^{n-j+1}\\ \int_{ \{0\} }
(dd^c\varphi)^{j+1}\wedge (dd^c\psi)^{n-j-1}.
\end{multline*}
For $\psi(z)=\ln \Vert z\Vert$, this is the desired estimate.\qed
\end{frame}
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\begin{frame}
\frametitle{Lemma~2}

\begin{block}{Lemma~2}
Let $\varphi,\psi\in\E(\Omega)$ be such that $\varphi \leq \psi$ (i.e $\varphi$ is ''more singular'' than $\psi$).
\pause
Then we have
\[
\sum_{j=0}^{n-1} \frac { e_{j} (\varphi ) } { e_{j+1} (\varphi ) } \leq \sum_{j=0}^{n-1} \frac { e_{j} (\psi ) } { e_{j+1} (\psi ) }\, .
\]
\end{block}

The argument if based on the monotonicity of Lelong numbers with
respect to the relation $\varphi\le\psi$, and on the monotonicity of
the right hand side in the relevant range of values.
\end{frame}
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\begin{frame}
\frametitle{Proof of Lemma~2}
Set
\alert{\[
D=\{t{=}(t_1,...,t_n){\in}[0,{+}\infty)^n:t_1^2{\leq}t_2, t_{j}^2{\leq}
t_{j-1}t_{j+1},\forall j=2,...,n-1\}.
\]}
\pause
Then $D$ is a convex set in $\REP{n}$, as can be checked by a straightforward application of
the Cauchy-Schwarz inequality.
\pause
Next, consider the function $f: \text{int } D\to [0,+\infty)$
defined by
\alert{\[
f(t_1,\ldots,t_n) = \frac 1 {t_1}+\frac {t_1} {t_2}\ldots+\frac {t_{n-1}} {t_n}\, .\]}
\pause
We have 
\[
\frac {\partial f} {\partial t_j} (t) = -\frac { t_{j-1} } { t_j^2 } + \frac { 1 } { t_{j+1} }\leq 0,\qquad \forall t\in D\, .
\]
\end{frame}
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\begin{frame}
\frametitle{Proof of Lemma~2, continued}
For $a,b\in\text{int } D$ such that $a_j\geq b_j$, $j=1,\ldots,n$, the function
\[
[0,1]\ni \lambda\to f( b+ \lambda(a-b) )
\]
is decreasing.
\pause
Hence,
\[
f(a)\leq f(b) \qquad \text{for all } a,b\in\text{int } D,\; a_j\geq b_j,\; j=1,\ldots,n\ .
 \]
\pause
On the other hand, the hypothesis $\varphi\leq\psi$ implies that $e_j (\varphi) \geq e_j (\psi)$, $j=1,\ldots,n$, by the comparison principle.
Therefore we have that
\[
f(e_1 (\varphi),\ldots,e_n (\varphi))\leq f(e_1 (\psi),\ldots,e_n (\psi))\, .
\]
\qed
\end{frame}
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\begin{frame}
\frametitle{Proof of the Main Theorem}
It will be convenient here to introduce Kiselman's refined Lelong number.

\bigskip

\begin{block}{Definition}
Let $\varphi\in \PSH (\Omega)$. Then the function defined by
\[
\nu_{\varphi}(x)=\lim_{t\to -\infty} \frac {\max\left\{\varphi (z): |z_1|=e^{x_1t},\ldots,|z_n|=e^{x_nt}\right\}} {t}
\]
is called the \alert{refined Lelong number} of $\varphi$ at $0$.
\end{block}
\pause
\bigskip

The refined Lelong number of $\varphi$ at $0$ is increasing in each variable $x_j$, and concave on $\REP{n}_+$.

\end{frame}
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\begin{frame}
\frametitle{Proof of the Main Theorem}

The proof is divided into the following steps:

\begin{itemize}\itemsep5mm

\item\alert{Proof of the theorem in the toric case}, i.e. $\varphi (z_1,\ldots,z_n) =
\varphi (|z_1|,\ldots,|z_n|)$ depends only on $|z_j|$
and therefore we can without
loss of generality assume that $\Omega = \Delta^n$ is the unit polydisk.
\pause

\item\alert{Reduction to the case of plurisubharmonic functions with analytic singularity}, i.e. $\varphi = \log (|f_1|^2+\ldots+|f_N|^2)$, where $f_1,\ldots,f_N$ are germs of holomorphic functions at $0$.
\pause

\item\alert{Reduction to the case of monomial ideals}, i.e. for $\varphi = \log (|f_1|^2+\ldots.+|f_N|^2)$, where $f_1,\ldots,f_N$ are germs of monomial elements at $0$.
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Proof of the theorem in the toric case}
Set
\[
\Sigma=\left\{ x=(x_1,\ldots,x_n)\in\RE_+^n:\ \sum\limits_{j=1}^n x_j = 1 \right\}\, .
\]
\pause
We choose $x^0=(x_1^0,\ldots,x_n^0)\in\Sigma$ such that
\[
\nu_{\varphi} (x^0) = \max\{\nu_{\varphi} (x):\ x\in S\}.
\]
\pause
By Theorem~5.8 in [Kis94] we have the following formula
\alert{\[
c(\varphi) = \frac 1 { \nu_\varphi (x^0) }\, .
\]}
\end{frame}
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\begin{frame}
\frametitle{Proof of the theorem in the toric case, continued}
Set~~ $\displaystyle 
\zeta (x) = \nu_\varphi(x^0) \min\left(\frac { x_1 } {x_1^0},\ldots,\frac { x_n } {x_n^0}\right),\qquad \forall x\in\Sigma\, .$\\
\pause
Then $\zeta$ is the smallest nonnegative concave increasing function on $\Sigma$ such that $\zeta (x^0)=\nu_\varphi (x^0)$, hence $\zeta\le \nu_\varphi$.
\pause
This implies that
\begin{multline*}
\varphi (z_1,\ldots,z_n) \leq -\nu_{\varphi} (-\ln |z_1|,\ldots,-\ln |z_n|)\\
\leq -\zeta (-\ln |z_1|,\ldots,-\ln |z_n|)\\
\leq \nu_{\varphi} (x^0)\max\left( \frac { \ln |z_1| } {x_1^0},\ldots,\frac { \ln |z_n| } {x_n^0} \right):=\psi (z_1,\ldots,z_n).
\end{multline*}
\pause
By Lemma~2 we get that
\[
f(e_1 (\varphi),...,e_n (\varphi))\leq f(e_1 (\psi),...,e_n (\psi))=c(\psi)=\frac{1}{\nu_\varphi(x^0)}=c(\varphi)\,.
\]
\end{frame}

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\begin{frame}
\frametitle{Reduction to the case of plurisubharmonic functions with analytic singularity}
Let $\mathcal H_{m\varphi}(\Omega)$ be the Hilbert space of holomorphic functions $f$ on $\Omega$ such that
\[
\int_\Omega|f|^2e^{-2m\varphi}dV<+\infty\, ,
\]
\pause
and let $\psi_m=\frac{1}{2m}\log\sum|g_{m,k}|^2$ where $\{g_{m,k}\}_{k\geq 1}$ be an orthonormal basis for $\mathcal H_{m\varphi}(\Omega)$.
\pause
Using $\bar\partial$-equation with $L^2$-estimates (D-Koll\'ar), there are 
constants $C_1,C_2>0$ independent of $m$ such that
\alert{\[
\varphi(z)-\frac{C_1}{m}\le
\psi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+\frac{1}{m}\log\frac{C_2}{r^n}
\]}
for every $z\in\Omega$ and $r<d(z,\partial\Omega)$.
\end{frame}
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\begin{frame}
\frametitle{Reduction to the case of plurisubharmonic functions with analytic singularity, continued}
and
\[
\nu(\varphi)-\frac{n}{m}\le\nu(\psi_m)\le\nu(\varphi),\qquad
\frac 1 {c(\varphi)}-\frac{1}{m}\le\frac{1}{c(\psi_m)}\le \frac{1}{c(\varphi)}.
\]
\pause
By Lemma~2, we have that
\[
f(e_1 (\varphi),\ldots,e_n (\varphi))\leq f(e_1 (\psi_m),\ldots,e_n (\psi_m)),\qquad \forall m\geq 1.
\]
\pause
The above inequalities show that in order to prove the lower bound of $c(\varphi)$ in the Main Theorem, we only need prove
 it for $c(\psi_m)$ and then let
$m\to\infty$.
\end{frame}
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\begin{frame}
\frametitle{Reduction to the case of monomial ideals}

For $j=0,\ldots,n$ set
\[
\mathcal J=(f_1,\ldots,f_N),\; c(\mathcal J) = c(\varphi), \text{ and } e_j(\mathcal J) = e_j(\varphi)\, .
\]
\pause
Now, by fixing a multiplicative order on the monomials
\[
z^\alpha=z_1^{\alpha_1}\ldots z_n^{\alpha_n}
\]
it is well known that one can construct a flat family $(\mathcal J_s)_{s\in\C}$
of ideals of $\cO_{\mathcal{C}^n,0}$ depending on a complex parameter $s\in\C$, such that $\cJ_0$ is a monomial ideal, $\cJ_1=\cJ$ and
\[
\dim(\cO_{\mathcal{C}^n,0}/\cJ_s^t)=\dim(\cO_{\mathcal{C}^n,0}/\cJ^t)\; \text{ for all } s,t\in\mathbb{N}\, .
\]
\pause
In fact $\cJ_0$ is just the initial ideal associated to $\cJ$ with respect to the monomial order.
\end{frame}
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\begin{frame}
\frametitle{Reduction to the case of monomial ideals, continued}

Moreover, we can arrange by a generic rotation of
coordinates $\CEP{p}\subset\CEP{n}$ that the family of ideals
$\cJ_{s|\mathbb{C}^p}$ is also flat,
\pause
and that the dimensions
\[
\dim\left(\cO_{\mathbb C^p,0}/(\cJ_{s|\mathbb{C}^p})^t\right)=\dim\left(\cO_{\mathbb{C}^p,0}/(\cJ_{|\mathbb{C}^p})^t\right)
\]
\pause
compute the intermediate multiplicities
\[
e_p(\cJ_s)=\lim_{t\to+\infty}\frac{p!}{t^p}\dim\left(\cO_{\mathbb{C}^p,0}/(\cJ_{s|\mathbb{C}^p})^t\right)=e_p(\cJ),
\]
\pause
in particular, $e_p(\cJ_0)=e_p(\cJ)$ for all~$p$.
\pause
The semicontinuity property of the log canonical threshold implies that 
$c(\cJ_0)\leq c(\cJ_s)=c(\cJ)$ for all $s$, so the lower bound is valid for $c(\cJ)$ if it is valid for $c(\cJ_0)$.
\end{frame}
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\begin{frame}
\frametitle{About the continuity of Monge-Ampère operators}
\begin{block}{Conjecture} Let $\varphi\in\E(\Omega)$ and $\Omega\ni 0$.
Then the analytic approximations  $\psi_m$ satisfy $e_j(\psi_m)\to
e_j(\varphi)$ as $m\to+\infty$, in other words, we have ``strong continuity''
of Monge-Ampère operators and higher Lelong numbers with respect
to Bergman kernel approximation.
\end{block}
\pause

In the 2-dimensional case, $e_2(\varphi)$ can be computed as follows
(at least when $\varphi\in\E(\omega)$ has analytic singularities).
\pause

Let \alert{$\mu:\widetilde\Omega\to \Omega$
be the blow-up of $\Omega$ at~$0$}. Take local coordinates $(w_1,w_2)$
on $\widetilde\Omega$ so that the exceptional divisor $E$ can be
written $w_1=0$.
\end{frame}

\begin{frame}
\frametitle{About the continuity of Monge-Ampère operators (II)}
With $\gamma=\nu(\varphi,0)$, we get that\vskip-11pt
\alert{$$\widetilde\varphi(w)=\varphi\circ\mu(w)-\gamma\log|w_1|$$}
is psh with generic Lelong numbers equal to $0$
along $E$, and therefore there are at most countably many points
$x_\ell\in E$ at which \hbox{$\gamma_\ell=\nu(\widetilde\varphi,x_\ell)>0$}.
Set $\Theta=dd^c\varphi$, $\widetilde\Theta=dd^c\widetilde\varphi=\mu^*\Theta
-\gamma[E]$. Since $E^2=-1$ in cohomology, we have
$\{\widetilde\Theta\}^2=\{\mu^*\Theta\}^2-\gamma^2$ in 
$H^2(E,{\mathbb R})$, hence
\vskip-21pt
\alert{%
$$\int_{\{0\}}(dd^c\varphi)^2=\gamma^2+\int_E(dd^c\widetilde\varphi)^2.
\leqno(*)$$}\vskip-15pt
\pause
If $\widetilde\varphi$ only has ordinary logarithmic poles at the $x_\ell$'s, 
then $\int_{E} (dd^c\widetilde\varphi)^2=\sum \gamma_\ell^2$, but in general the situation is more complicated. Let us blow-up any of the points $x_\ell$ and repeat the process $k$ times.
\end{frame}

\begin{frame}
\frametitle{About the continuity of Monge-Ampère operators (III)}
We set $\ell=\ell_1$ in what follows, as this was the 
first step, and at step $k=0$ we omit any indices as $0$ is the only point
we have to blow-up to start with.
We then get inductively $(k+1)$-iterated blow-ups depending on multi-indices
$\ell=(\ell_1,\ldots,\ell_k)=(\ell',\ell_k)$ with
$\ell'=(\ell_1,\ldots,\ell_{k-1})$,
\alert{$$
\mu_\ell:\widetilde\Omega_\ell\to \widetilde\Omega_{\ell'}, ~~~k\ge 1,~~
\mu_\emptyset=\mu:\widetilde\Omega_\emptyset=\widetilde\Omega\to\Omega,~~
\gamma_\emptyset=\gamma
$$}%
and exceptional divisors $E_\ell\subset\widetilde\Omega_\ell$ lying over points
$x_\ell\in E_{\ell'}\subset\widetilde\Omega_{\ell'}$, where
\alert{%
\begin{eqnarray*}
&&\gamma_\ell = \nu(\widetilde\varphi_{\ell'},x_\ell)>0,\\
&&\widetilde\varphi_\ell(w)=
\widetilde\varphi_{\ell'}\circ\mu_\ell(w)-
\gamma_\ell\log|w_1^{(\ell)}|,\\
&&\hbox{($w_1^{(\ell)}=0$ an equation of $E_\ell$
in the relevant chart).}
\end{eqnarray*}}
\end{frame}

\begin{frame}
\frametitle{About the continuity of Monge-Ampère operators (IV)}
Formula $(*)$ implies\vskip-23pt
\alert{$$
e_2(\varphi)\ge\sum_{k=0}^{+\infty}~\sum_{\ell\in{\mathbb N}^k}~
\gamma_\ell^2\leqno(**)
$$}%
with equality when $\varphi$ has an analytic singularity at $0$.
We conjecture that $(**)$ is always an equality whenever
\hbox{$\varphi\in\E(\Omega).\kern-20pt$}\vskip3pt
\pause

This would imply the Guedj-Rashkovskii conjecture.\vskip3pt
\pause

Notice that the currents $\Theta_\ell=dd^c\widetilde\varphi_\ell$
satisfy inductively $\Theta_\ell=\mu_\ell^*\Theta_{\ell'}-\gamma_\ell[E_\ell]$, hence the cohomology class of $\Theta_\ell$ restricted to $E_\ell$ is equal
to $\gamma_\ell$ times the fundamental generator of $E_\ell$.
As a consequence we have
\alert{$$
\sum\nolimits_{\ell_{k+1}\in{\mathbb N}} \gamma_{\ell,\ell_{k+1}}\le \gamma_\ell,
$$}%
in particular $\gamma_\ell=0$ for all $\ell\in{\mathbb N}^k$ if 
$\gamma=\nu(\varphi,0)=0$.
\end{frame}

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\begin{frame}[allowframebreaks]{References}

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\end{frame}

\end{document}