% 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>.
%
% 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-192pt\rlap{\blank{J.-P.\ Demailly (Grenoble), S\'eminaire Bourbaki, March 19, 2016}}\kern181pt\rlap{\blank{Variational approach for complex Monge-Amp\`ere equations}}\kern184pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Variational approach for\\
complex Monge-Amp\`ere equations\\
and geometric applications}

%% \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)
{March 19, 2016\\
S\'eminaire Bourbaki\\
Institut Henri Poincar\'e, Paris\\}

%%\subject{Talks}
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% mathematical operators
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\newcommand\ord{\mathop{\mathrm{ord}}\nolimits}
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\def\bibitem[#1]#2#3{\medskip{\bf[#1]} #3}

\begin{document}

% 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}
%%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}

\begin{frame}
  \pgfdeclareimage[height=2cm]{uga-logo}{logo_UGA}
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  \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.

%% \section*{Basic concepts}
%% \def\pause{}

\begin{frame}
\frametitle{Abstract and goals}
\strut\vskip-20pt
$\bullet$ Recent work by \alert{Berman, Berndtsson, Boucksom, Eyssidieux,\\
Guedj, Jonsson, Zeriahi} (among others) leads to a new variational
approach for the solution of
Monge-Amp\`ere equations on compact K\"ahler manifolds.\pause
\vskip5pt
$\bullet$ The method can be made independent of the previous PDE 
technicalities of Yau's approach.\pause
\vskip5pt
$\bullet$  It is based on the study of certain functionals 
(Ding-Tian, Mabuchi) on the space of K\"ahler
metrics, and their geodesic convexity due to X.X.\ Chen and
Berman-Berndtsson in its full generality.\pause
\vskip5pt
$\bullet$ Applications
include the existence and uniqueness of K\"ahler-Einstein metrics on
$\bQ$-Fano varieties with log terminal singularities, 
and a new proof by Berman-Boucksom-Jonsson of a uniform version of the 
Yau-Tian-Donaldson conjecture solved around 2013 by
Chen-Donaldson-Sun.
\end{frame}

\begin{frame}
\frametitle{K\"ahler-Einstein metrics}
To a~K\"ahler metric on a compact complex $n$ fold $X$
$$\omega=i\sum_{1\le j,k\le n}\omega_{jk}(z)\,dz_j\wedge d\overline
z_k,~~d\omega=0$$
one associates its \alert{Ricci curvature form}
$$
\Ricci(\omega)=\Theta_{\Lambda^nT_X,\Lambda^n\omega}=
-dd^c\log\det(\omega_{jk})$$
where $d^c=\frac{1}{4i\pi}(\partial-\dbar)$, $dd^c=\frac{i}{2\pi}\ddbar$.
\pause
The~K\"ahler metric $\omega$ is said to be \alert{K\"ahler-Einstein} if
$$
\alert{\Ricci(\omega)=\lambda\omega}\quad
\hbox{for some $\lambda\in\bR$}.\leqno(*)
$$\pause
This requires $\lambda\omega\in c_1(X)$, hence $(*)$ can be solved only when
$c_1(X)$ is positive definite, negative definite or zero, and after rescaling 
$\omega$ by a constant, one can always assume that~$\lambda\in\{0,1,-1\}$.
\end{frame}

\begin{frame}
\frametitle{K\"ahler-Einstein $\Longleftrightarrow$ Monge-Amp\`ere equation (1)}

Fix a reference K\"ahler metric $\omega_0$ and put 
$\omega=\omega_0+dd^c\varphi$. The KE condition $(*)$ is equivalent to
\alert{$$
(\omega_0+dd^c\varphi)^n=e^{-\lambda\varphi+f}\omega_0^n.\leqno(**)
$$}\pause
$\bullet$ When $\lambda=-1$ and $c_1(X)<0$, i.e.\ $c_1(K_X)>0$, Aubin 
has shown in 1978 that  there is always a unique solution, hence a
unique K\"ahler metric $\omega\in c_1(K_X)$ such that 
\alert{$$\Ricci(\omega)=-\omega.$$}\pause
This is a very natural generalization of the existence of constant 
curvature metrics on complex algebraic curves, implied by Poincar\'e's 
uniformization theorem in dimension~1.

\end{frame}

\begin{frame}
\frametitle{K\"ahler-Einstein $\Longleftrightarrow$ Monge-Amp\`ere equation (2)}
$\bullet$ For $\lambda=0$ and $c_1(X)=0$, a celebrated result of 
Yau (solution of the Calabi conjecture, 1978) states
that there exists a unique metric $\omega=\omega_0+dd^c\varphi$ in the
given cohomology class $\{\omega_0\}$ such that $\Ricci(\omega)=0$.\pause
Moreover, the 
Monge-Amp\`ere equation 
\alert{$$(\omega_0+dd^c\varphi)^n =e^f\omega_0^n$$}
has a unique solution whenever
$\smash{\int_Xe^f\omega_0^n=\int_X\omega_0^n}$.\\
\pause
Equivalently, the Ricci curvature form
can be prescribed to be equal any given smooth closed $(1,1)$-form
\alert{$$
\Ricci(\omega)=\rho,
$$}
provided that $\rho\in c_1(X)$. 
\end{frame}

\begin{frame}
\frametitle{The case of Fano manifolds}
For $\lambda=+1$, the equation to solve is\vskip-20pt
\alert{$$
(\omega_0+dd^c\varphi)^n=e^{-\varphi+f}\omega_0^n.\leqno(**)
$$}\vskip-20pt
This is possible only if $-K_X$ (${}=\Lambda^nT_X$) is ample. One then says
that \alert{$X$ is a Fano manifold}.

\pause
When solutions exist, it~is known by Bando Mabuchi (1987) that they
are unique up to the action of the identity component $\Aut^0(X)$ in
the complex Lie group of biholomorphisms of~$X$.

\begin{block}{Berman-Boucksom-Jonsson 2015}
Let $X$ be a Fano manifold with finite automorphism group. 
Then $X$ admits a K\"ahler-Einstein metric if and only if
it is \alert{uniformly K-stable}.
\end{block}

Recently, Chen, Donaldson and Sun got this result under the more general
assumption that $X$ is \alert{$K$-stable} (Bourbaki/Ph.~Eyssidieux, 
january 2015).
\end{frame}

\begin{frame}
\frametitle{The case of log Fano varieties}

\begin{block}{Definition}
A \alert{log Fano pair} is a klt pair $(X,\Delta)$ such that $X$ is 
projective and the $\bQ$-divisor $A=-(K_X+\Delta)$ is ample. 
\end{block}
\pause

Here $X$ is a normal compact complex variety $X$ and $\Delta$
an effective $\bQ$-divisor such that $K_X+\Delta$ is $\bQ$-Cartier. \pause
By Hironaka, there exists a \alert{log resolution} $\pi:\tilde X\to X$ of
$(X,\Delta)$, i.e.\ a modification of $X$ over the complement of the
singular loci of $X$ and $\Delta$, such that the pull-back of $\Delta$
and of $X_{\rm sing}$ consists of simple normal crossing (snc) divisors in
$\tilde X$. \pause One writes
$$
\textstyle
\pi^*(K_X+\Delta)=K_{\tilde X}+E,\qquad E=\sum_j a_j E_j
$$
for some $\bQ$-divisor $E$ whose push-forward to $X$ is $\Delta$ (since
$X_{\rm sing}$ has codimension $2$, the components $E_j$ that lie over
$X_{\rm sing}$ yield $\pi_*E_j=0$). 
The coefficient $-a_j\in\bQ$ is known as the \alert{discrepancy} of
$(X,\Delta)$ along $E_j$. 
\end{frame}

\begin{frame}
\frametitle{The klt condition (``Kawamata Log Terminal'')}

\begin{block}{Definition}
$(X,\Delta)$ is klt if $a_j<1$ for all $j$.
\end{block}

Let $r$ be a positive integer such that
$r(K_X+\Delta)$ is Cartier, and $\sigma$ a local generator of
$\cO(r(K_X+\Delta))$ on some open set $U\subset X$. Then the $(n,n)$
form
$$
|\sigma|^{2/r}:=i^{n^2}\,\sigma^{1/r}\wedge\overline{\sigma^{1/r}}
$$
is a volume form with poles along $S=\Supp\Delta\cup X_{\rm sing}$.\pause

By the change of variable formula, the local integrability can be 
checked by pulling
back $\sigma$ to $\tilde X$, in which case it is easily seen that 
the integrability occurs if and only if $a_j<1$ for all~$j$, i.e. when
$(X,\Delta)$ is klt.\pause

In local coordinates
\alert{$$|\sigma|^{2/r}\sim\frac{\hbox{volume form}}{\prod |z_j|^{2a_j}}.$$}
\end{frame}

\begin{frame}
\frametitle{Singular Monge-Ampère equation}
\strut\vskip-20pt
By definition
\alert{$(X,\Delta)$ log Fano $\Longrightarrow$ 
$c_1(X,\Delta)\ni{}\omega_0$ K\"ahler}.
\pause

Every form $\omega=\omega_0+dd^c\psi\in\{\omega_0\}$ can be seen
as the curvature form of a smooth hermitian metric $h$ on
$\cO(-(K_X+\Delta))$, whose weight is $\phi=u_0+\psi$ where $u_0$
is a local potential of~$\omega_0$, hence
$$
\omega=\omega_0+dd^c\psi=dd^c\phi
$$
where $\phi$ is understood as the weight of a global metric 
formally denoted $h=e^{-\phi}$ on the $\bQ$-line bundle $\cO(-(K_X+\Delta))$.
\pause

The inverse $e^\phi$ is a hermitian metric on $\cO(K_X+\Delta)$. If 
$\sigma$ is a local generator of
$\cO(r(K_X+\Delta))$, the product $|\sigma|^{2/r}e^\phi=e^{\psi+u_0}$
is (locally) a smooth positive function whenever $\varphi$ is smooth, hence
$$
e^{-\phi}=|\sigma|^{2/r}e^{-(\psi+u_0)}
$$
is an integrable volume form on $X$ with poles 
along \alert{$S:=\Supp\Delta\cup\{\hbox{singularities}\}$}. 
The KE condition can be rewritten
\alert{$$
(dd^c\phi)^n=c\,e^{-\phi}\quad\hbox{on $X\smallsetminus S$}
\Leftrightarrow \Ricci(\omega)=\omega+[\Delta].
$$}
\end{frame}

\begin{frame}
\frametitle{The space of K\"ahler metrics}
Let $A\in\smash{H^{1,1}_{\ddbar}}(X,\bR)$ be a K\"ahler 
$\ddbar$-cohomology class, and let 
$$\alert{\omega_0=\alpha+dd^c\psi_0=dd^c\phi_0\in A}$$
be a K\"ahler metric.\pause

Here we are mostly interested in the Fano case
$A=-K_X$ and the log Fano case \hbox{$A=-(K_X+\Delta)$}.  Let
\hbox{$V=\int_X\omega_0^n=A^n$} be the volume of~$\omega_0$.

\begin{block}{Definition} The space $\cK_A$ of K\"ahler metrics
(resp.\ $\cP_A$ of K\"ahler potentials) is 
the set of K\"ahler metrics $\omega$ (resp.\ functions $\psi$) such that
\vskip3pt
\centerline{\alert{$\omega=\omega_0+dd^c\psi>0.$}}
\vskip3pt
Here $\phi=u_0+\psi$ is thought intrinsically as a hermitian metric 
\hbox{$h=e^{-\phi}$} 
on~$A$ with strictly plurisubharmonic (psh) weight $\phi$.
\end{block}\pause

Clearly $\cK_A\simeq\cP_A/\bR$.
\end{frame}

\begin{frame}
\frametitle{The Riemannian structure on $\cP_A$}
The basic operator of interest on
$\cP_A$ is the \alert{Monge-Amp\`ere \hbox{operator\kern-15pt}}
\alert{$$
\cP_A\to \cM_+,\qquad
\MA(\phi)=(dd^c\phi)^n=(\omega_0+dd^c\psi)^n
$$}
According to Mabuchi the space 
$\cP_A$ can be seen as some sort of infinite dimensional Riemannian 
manifold: a~``tangent vector'' to $\cP_A$ is 
an infinitesimal variation $\delta\phi\in C^\infty(X,\bR)$ of $\phi$ (or $\psi$),
and the infinitesimal Riemannian metric at a point $h=e^{-\phi}$ 
is given by
\alert{$$
\Vert\delta\phi\Vert_2^2=\frac{1}{V}\int_X(\delta\phi)^2\MA(\phi).
$$}\pause
X.X.~Chen and his collaborators have studied the metric and geometric
properties of the space $\cP_A$, showing in particular that it is a
path metric space (a non trivial assertion in this infinite
dimensional setting) of nonpositive curvature in the sense of
Alexandrov. A key step has been to produce almost
$C^{1,1}$-geodesics which minimize the geodesic distance.
\end{frame}

\begin{frame}
\frametitle{Basic functionals (1)}
Given $\phi_0,\phi\in\cP_A$, one defines:\vskip3pt

\noindent$\bullet$
The \alert{Monge-Amp\`ere functional}\vskip-19pt
$$
\leqalignno{
E_{\phi_0}(\phi)&=
\frac{1}{(n+1)V}\sum_{j=0}^n\int_X(\phi-\phi_0)(dd^c\phi)^j
\wedge(dd^c\phi_0)^{n-j}\cr
&=\frac{1}{(n+1)V}\sum_{j=0}^n\int_X
\psi(\omega_0+dd^c\psi)^j\wedge\omega_0^{n-j},~~\psi=\phi-\phi_0.&(*{*}*)\cr}
$$\pause
It is a \alert{primitive} of the Monge-Amp\`ere operator in the sense
that $dE_{\phi_0}(\phi)=\frac{1}{V}\MA(\phi)$, i.e.\ for any path 
$[T,T']\ni t\mapsto\phi_t$, one has
\alert{
$$\frac{d}{dt}E_{\phi_0}(\phi_t)=\frac{1}{V}\int_X\dot\phi_t\MA(\phi_t)\quad
\hbox{where $\displaystyle\dot\phi_t=\frac{d}{dt}\phi_t.$}$$}
This is easily checked by a differentiation under the integral sign.
\end{frame}

\begin{frame}
\frametitle{Basic functionals (2)}
As a consequence $E$ satisfies the \alert{cocycle relation}
\alert{$$
E_{\phi_0}(\phi_1)+E_{\phi_1}(\phi_2)=E_{\phi_0}(\phi_2),
$$}
so its dependence on $\phi_0$ is only up to a constant.\pause

Finally, if $\phi_t$ depends linearly on $t$, one has 
$\ddot\phi_t=\frac{d^2}{dt^2}
\phi_t=0$ and
a further differentiation of $(*{*}*)$ yields
$$\eqalign{
\frac{d^2}{dt^2}E_{\phi_0}(\phi_t)&=\frac{n}{V}\int_X\dot\phi_t\,dd^c\dot\phi_t
\wedge(dd^c\phi_t)^{n-1}\cr
&=-\frac{n}{V}\int_X d\dot\phi_t\wedge d^c\dot\phi_t
\wedge(dd^c\phi_t)^{n-1}\le 0.\cr}$$
It follows that $E_{\phi_0}$ is \alert{concave} on $\cP_A$.
\pause

\end{frame}

\begin{frame}
\frametitle{The $J$ and $J^*$ functionals}
\noindent$\bullet$ The concavity of $E$ implies the nonnegativity of
$J_{\phi_0}(\phi):=dE_{\phi_0}(\phi_0)\cdot(\phi-\phi_0)-E_{\phi_0}(\phi)$,
This quantity is called the Aubin \alert{$J$-energy} functional 
$$
J_{\phi_0}(\phi)=V^{-1}\int_X(\phi-\phi_0)(dd^c\phi_0)^n-E_{\phi_0}(\phi)\ge 0.
$$\pause

\noindent$\bullet$ By exchanging the roles of $\phi$, $\phi_0$ and putting
$J^*_{\phi_0}(\phi)=J_\phi(\phi_0)\ge 0$, the cocycle relation for $E$ yields
$E_\phi(-\phi_0)=-E_{\phi_0}(\phi)$. The \alert{transposed 
$J$-energy functional} is
$$
\eqalign{
J^*_{\phi_0}(\phi):&=
E_{\phi_0}(\phi)-V^{-1}\int_X(\phi-\phi_0)(dd^c\phi)^n\cr
&=E_{\phi_0}(\phi)-V^{-1}\int_X\psi(\omega_0+dd^c\psi)^n\ge 0,~~
\psi=\phi-\phi_0.\cr}
$$
\end{frame}

\begin{frame}
\frametitle{The symmetric $I$ functional}
\noindent$\bullet$ 
The \alert{$I$-functional} is the symmetric functional defined by 
$$
\leqalignno{
&I_{\phi_0}(\phi)=I_{\phi}(\phi_0):=-\frac{1}{V}\int_X(\phi-\phi_0)
\big(\MA(\phi)-\MA(\phi_0)\big)\cr
&=\sum_{j=0}^{n-1}V^{-1}\int_X d(\phi-\phi_0){\wedge}d^c(\phi-\phi_0)
{\wedge}(dd^c\phi)^j{\wedge}(dd^c\phi_0)^{n-1-j}\ge 0.\cr}
$$\pause
In fact $I_{\phi_0}(\phi)=J_{\phi_0}(\phi)+J^*_{\phi_0}(\phi)$, and one 
can also write
$$
I_{\phi_0}(\phi)=V^{-1}\left(
\int_X\psi\,\omega_0^n-\int_X\psi(\omega_0+dd^c\psi)^n\right).
$$\pause
It satisfies the \alert{quasi-triangle inequality}: $\exists c_n>0$ s.t.
$$
I_{\phi_0}(\phi) \leq c_n\big(I_{\phi_0}(\phi_1)  +I_{\phi_1}(\phi)\big). 
\quad\hbox{
$\forall$ $\phi_0,\,\phi_1,\,\phi \in \cP_A$.}$$
\end{frame}

\begin{frame}
\frametitle{The Ding and Mabuchi functionals (1)}

\noindent$\bullet$ In the Fano or log Fano setting, the 
\alert{Ding functional} is defined by 
$$
D_{\phi_0}=L-L(\phi_0)-E_{\phi_0},\quad
\hbox{where}~~ L(\phi)=-\log\int_X e^{-\phi}.
$$\pause
Recall: $e^{-\phi}$ is integrable by the klt condition. 
\medskip

\noindent
$\bullet$ Given probability measures  $\mu,\nu$ on a space $X$, the 
\alert{relative entropy} $\Entr_\mu(\nu)\in[0,+\infty]$ of $\nu$ with
respect to $\mu$ is defined as the integral
\alert{$$
\Entr_\mu(\nu):=\int_X\log\left(\frac{d\nu}{d\mu}\right)d\nu,
$$}
if $\nu$ is absolutely continuous w.r.t.\ $\mu\,$;
$\Entr_\mu(\nu)=+\infty$ other\-wise.\\
\claim{Pinsker inequality}: for all proba measures $\mu,\nu$ one has
$$\Entr_\mu(\nu)\ge \frac{1}{2}\Vert\mu-\nu\Vert^2\ge 0.$$
In particular, $\mu=\nu\Longleftrightarrow \Entr_\mu(\nu)=0$.
\end{frame}

\begin{frame}
\frametitle{The Ding and Mabuchi functionals (2)}

In the Fano or log Fano situation, 
the \alert{\hbox{entropy} functional} $H_{\phi_0}(\phi)$ is defined to
be the entropy of the proba\-bility measure $\smash{\frac{1}{V}}(dd^c\phi)^n$ with
respect to $e^{L(\phi_0)}e^{-\phi_0}$, namely
$$
H_{\phi_0}(\phi)=\int_X\log\left(\frac{(dd^c\phi)^n/V}{e^{L(\phi_0)}e^{-\phi_0}}
\right)\frac{(dd^c\phi)^n}{V}\ge 0.
$$\pause
\vskip6pt

\noindent$\bullet$ The \alert{Mabuchi functional}  is then defined by
$$
M_{\phi_0}=H_{\phi_0}-J^*_{\phi_0}.
$$\pause
One gets the more explicit expression
$$
M_{\phi_0}(\phi)=\int_X\log\left(\frac{e^\phi(dd^c\phi)^n}{V}
\right)\frac{(dd^c\phi)^n}{V}-E_{\phi_0}(\phi)-L(\phi_0).
$$
\end{frame}

\begin{frame}
\frametitle{Comparison properties}

\begin{block}{Observation} If $c$ is a constant, then 
$$E_{\phi_0}(\phi+c)=E_{\phi_0}(\phi)+c\quad\hbox{and}\quad
L(\phi+c)=L(\phi)+c.$$
On the other hand, the functionals 
$I_{\phi_0},J_{\phi_0},J^*_{\phi_0},D_{\phi_0},H_{\phi_0},M_{\phi_0}$ are invariant
by~\hbox{$\phi\mapsto\phi+c$} and therefore descend to the quotient space
$\cK_A=\cP_A/\bR$ of K\"ahler metrics \hbox{$\omega=dd^c\phi\in A$}.
\end{block}\pause
\vskip5pt

\begin{block}{Comparison between $I$, $J$, $J^*$}
The functionals $I$, $J$, $J^*$ are essentially growth equivalent:
$$
\frac{1}{n}J_{\phi}(\phi_0)\le J_{\phi_0}(\phi)\le\frac{n+1}{n}
J_{\phi_0}(\phi)\le I_{\phi_0}(\phi)\le (n+1)J_{\phi_0}(\phi).
$$
\end{block}
\end{frame}

\begin{frame}
\frametitle{Comparison between Ding and Mabuchi \hbox{functionals\kern-15pt}}
\strut\vskip-28pt
\begin{block}{Proposition} Let $(X,\Delta)$ be a log Fano manifold. Then 
$M_{\phi_0}(\phi)\ge D_{\phi_0}(\phi)$ and, in case of equality, 
$\phi$ must be K\"ahler-Einstein.
\end{block}

{\it Proof.} From the definitions one gets 
$$\eqalign{
&\qquad\qquad{}M-D=(H-J^*)(L-L(\phi_0)-E),\cr
&\qquad\qquad{}E_{\phi_0}(\phi)-J^*_{\phi_0}(\phi)=V^{-1}\int_X(\phi-\phi_0)(dd^c\phi)^n,\cr
&M_{\phi_0}(\phi)-D_{\phi_0}(\phi)\cr
&\qquad{}=\int_X\left(\log\left(\frac{(dd^c\phi)^n/V}
{e^{L(\phi_0)}e^{-\phi_0}}\right)+
(\phi-\phi_0)\right)\frac{(dd^c\phi)^n}{V}+L(\phi_0)-L(\phi)\cr
&\qquad{}=\int_X\log\left(\frac{(dd^c\phi)^n/V}{e^{L(\phi)}e^{-\phi}}
\right)\frac{(dd^c\phi)^n}{V}\ge 0.\cr}
$$ 
In case of equality, Pinsker implies KE condition:
\hbox{$\frac{(dd^c\phi)^n}{V}=e^{L(\phi)}e^{-\phi}$\kern-10pt}
\end{frame}

\begin{frame}
\frametitle{Non pluripolar products}
\strut\vskip-20pt
$\bullet$ \claim{Bedford-Taylor Monge-Amp\`ere products}~:
for $u_j\in L^\infty_{\rm loc}$, one sets inductively\vskip-20pt
\alert{$$
dd^cu_1\wedge dd^cu_2\wedge\ldots\wedge dd^cu_k:=
dd^c(u_1\,dd^cu_2\wedge\ldots\wedge dd^cu_k)
$$}\pause
$\bullet$ \claim{Non pluripolar products (Guedj-Zeriahi)}\\
Let $\cP(X,\omega_0)$ be the set of
$\omega_0$-psh potentials, i.e.\ $\phi=\phi_0+\psi$ such
that $dd^c\phi=\omega_0+dd^c\psi\ge 0$. \pause\\ 
The functions
$\psi_\nu:=\max\{\psi,-\nu\}$ are again $\omega_0$-psh and bounded for
all $\nu\in\bN$. The Monge-Amp\`ere measures $(\omega_0+dd^c\psi_\nu)^n$ 
are therefore well-defined in the sense of Bedford-Taylor, and one
defines for any bidegree $(p,p)$ a positive current\vskip-19pt
\alert{$$
\eqalign{
T&=\langle(\omega_1+dd^c\psi_1)\wedge...\wedge(\omega_p+dd^c\psi_p)\rangle
=\lim_{\nu\to+\infty}\cr
&{\bf 1}_{\bigcap\{\psi_j>-\nu\}}
(\omega_1+dd^c\max\{\psi_1,-\nu\})\wedge...\wedge(\omega_p+dd^c
\max\{\psi_p,-\nu\})\cr}
$$}\pause\vskip-12pt
\claim{Basic fact: $T$ is still closed} [Proof uses ideas of Skoda \&\
Sibony].
\end{frame}

\begin{frame}
\frametitle{Space of potentials of finite energy}
One introduces for any $p\in[1,+\infty[$ the space
\alert{$$
\cE^p(X,\omega_0):=\left\{\phi=\phi_0+\psi\,;\;
\int_X|\psi|^p\MA(\omega_0+dd^c\psi)<+\infty\right\},
$$}%
and $\int_X\MA(\omega_0+dd^c\psi)=\int_X\omega_0^n$ (``full non pluripolar
mass''). One says that 
functions $\psi\in\cE^p(X,\omega_0)$ have \emph{finite
$\cE^p$-energy}. \pause
One also denotes by
\alert{$$
\cT^{\,p}(X,\omega_0)\subset\cT^{\,p}_\full(X,\omega_0)
$$} 
the corresponding set of \emph{currents with finite $\cE^p$-energy}, which 
can be identified with the quotient space
$$
\cT^p(X,\omega_0)=\cE^p(X,\omega_0)/\bR\quad\hbox{via}~~\phi\mapsto
dd^c\phi=\omega_0+dd^c\psi.
$$\pause
It is important to note that $\cT^{\,p}(X,\omega_0)$ is \alert{not a
closed subset} of $\cT(X,\omega_0)$ for the weak topology. 
\end{frame}

\begin{frame}
\frametitle{Finite energy extension of the functionals}
\strut\vskip-30pt
\begin{block}{Finite energy extension of the functionals}
All functionals $E,L,I,J,J^*,D,H,M$ have a natural extension
to arguments $\phi,\phi_0\in\cE^1(X,\omega_0)$, and $I,J,J^*,D,H,M$ descend
to~\hbox{$\cT^1(X,\omega_0)=\cE^1(X,\omega_0)/\bR$}.
\end{block}\pause\vskip-8pt

\begin{block}{Theorem (BBGZ)}
The map $T=\omega_0+dd^c\psi\mapsto V^{-1}\langle T^n\rangle$ is
a bijection between $\cT^1(X,\omega_0)$ and the space of
probability measures $\cM^1(X,\omega_0)$ of finite energy.
\end{block}
Here one uses the \alert{Legendre-Fenchel transform}
$$
E_0^*(\mu):=\sup_{\phi=\phi_0+\psi\in\cE^1(X,\omega_0)}
\left(E_0(\psi)-\int_X\psi\,\mu\right)\in[0,+\infty]
$$
where $E_0(\psi)=E_{\phi_0}(\phi_0+\psi)$, 
and $\mu$ has \alert{finite energy} if \hbox{$E_0^*(\mu)<+\infty$.\kern-10pt}
\end{frame}

\begin{frame}
\frametitle{Sufficient conditions for existence of KE metrics}
\strut\vskip-28pt
\begin{block}{Theorem (BBEGZ)}
For a current $\omega=dd^c\phi\in\cT^1(X,A)$, the following conditions are
\hbox{equivalent}.
\begin{itemize}
\item[\just{(i)}] $\omega$ is a K\"ahler-Einstein metric for $(X,\Delta)$. 
\item[\just{(ii)}] The Ding functional reaches its infimum at $\phi\,:$
$D_{\phi_0}(\phi)=\inf_{\cE^1(X,A)/\bR}D_{\phi_0}$.
\item[\just{(iii)}] The Mabuchi functional reaches its infimum at $\phi\,:$
$M_{\phi_0}(\phi)=\inf_{\cE^1(X,A)/\bR}M_{\phi_0}$. 
\end{itemize}
\end{block}\pause\vskip-6pt

\begin{block}{Corollary (BBEGZ)}
Let $X$ be a $\bQ$-Fano variety with log terminal singularities. 
\begin{itemize}
\item[\just{(i)}] The identity component $\Aut^0(X)$ of the
  automorphism group of $X$ acts transitively on the set of
  KE metrics on $X$,
\item[\just{(ii)}] If the Mabuchi functional of $X$ is proper, then
  $\Aut^0(X)=\{1\}$ and $X$ admits a unique K\"ahler-Einstein metric.
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Test configurations}
\begin{block}{Definition}
A test configuration $(\cX,\cA)$ for a ($\bQ$-)polarized
projective variety $(X,A)$ consists of the following data$\,:$
\begin{itemize}
\item[\just{(i)}] a flat and proper morphism $\pi:\cX\to\bC$ of
  algebraic varieties; one denotes by $X_t=\pi^{-1}(t)$ the fiber over
  $t\in\bC$.
\item[\just{(ii)}] a $\bC^*$-action on $\cX$ lifting the canonical
  action on $\bC$;
\item[\just{(iii)}] an isomorphism $X_1\simeq X$.
\item[\just{(iv)}] a $\bC^*$-linearized ample line bundle $\cA$ on
  $\cX\,$; one puts $A_t=\cA_{|X_t}$.
\item[\just{(v)}] an isomorphism $(X_1,A_1)\simeq (X,A)$ extending the
  one in~$\mathrm{(iii)}$.
\end{itemize}
\end{block}\pause

$K$ stability (and uniform $K$-stability) is defined in termes of certain
numerical invariants attached to arbitrary test configurations.
\end{frame}

\begin{frame}
\strut\vskip-27pt
\frametitle{Donaldson-Futaki invariants}
\begin{block}{Donaldson-Futaki invariant}
Let $N_m=h^0(X,mA)$ and $w_m\in\bZ$ be the weight of the
\hbox{$\bC^*$-action} on the determinant $\det H^0(\cX_0,m\cA_0)$. 
Then there is an asymptotic expansion
\alert{$$
\frac{w_m}{mN_m}=F_0+m^{-1}F_1+m^{-2}F_2+\dots~.
$$}
and one defines \alert{$\DF(\cX,\cA):=-2F_1$}.
\end{block}\pause\vskip-6pt

\begin{block}{Definition}
The polarized variety $(X,A)$ is said K-stable if
\alert{$\DF(\cX,\cA)\ge 0$}
for all normal test configurations, with
equality iff $(\cX,\cA)$ is trivial.
\end{block}\pause\vskip-6pt

\begin{block}{Generalized Yau-Tian-Donaldson conjecture}
Let $(X,A)$ be a polarized variety. Then $X$ admits a cscK metric
$($short hand for K\"ahler metric with \alert{constant scalar curvature}$)$
$\omega\in c_1(A)$ if and only if $(X,A)$ is K-stable.
\end{block}

\end{frame}

\begin{frame}
\frametitle{Uniform K-stability}
The Duister\-maat-Heckman measure $\DH_{(X,A)}$ is the proba distribution
measure of the $\bC^*$-action weights:
\alert{$$
\DH_{(X,A)}=\lim_{m\to\infty}\sum_{\lambda\in\bZ}
\frac{\dim H^0(X,mA)_\lambda}{\dim H^0(X,mA)}\,\,\delta_{\lambda/m},
\quad\delta_p:=\hbox{Dirac at $p$},
$$}
where $H^0(X,mA)=\bigoplus_{\lambda\in\bZ}H^0(X,mA)_\lambda$ is the
weight space decomposition.  For each $p\in[1,\infty]$, the $L^p$-norm
$\|(\cX,\cA)\|_p$ of an ample test configuration $(\cX,\cA)$ is
defined as the $L^p$ norm
\alert{$$
\|(\cX,\cA)\|_p=\left(\int_\bR |\lambda-b(\mu)|^p\,d\mu(\lambda)\right)^{1/p},
\quad
b(\mu)=\int_\bR\lambda\,d\mu(\lambda).
$$}\vskip-8pt\pause

\begin{block}{Definition (Sz\'ekelyhidi)}
The polarized variety $(X,A)$ is said to be
\alert{$L^p$-uniformly K-stable} if there exists $\delta>0$ such
that $\DF(\cX,\cA)\ge\delta\,\|(\cX,\cA)\|_p$ for all normal test
configurations. [Note: only possible if $p<\frac{n}{n-1)}$.]
\end{block}

\end{frame}

\begin{frame}
\frametitle{Sufficiency of uniform K-stability} 
\begin{block}{Berman-Boucksom-Jonsson 2015}
Let $X$ be a Fano manifold with finite automorphism group. 
Then $X$ admits a K\"ahler-Einstein metric if and only if
it is \alert{uniformly K-stable} \claim{(in a related and simpler
``non archimedean'' sense)}.
\end{block}\pause

Let $A=-K_X$. A ray $(\phi_t)_{t\ge 0}$ in $\cP_A$ corresponds to an
$S^1$-invariant metric $\Phi$ on the pull-back of $-K_X$ to the
product of $X$ with the punctured unit disc $\bD^*$. The ray is
called \alert{subgeodesic} when $\Phi$ is plurisubharmonic (psh for
short). Denoting by $F$ any of the functionals $M,D$ or $J$, there
is a limit
$$
\lim_{t\to+\infty}\frac{F(\phi_t)}{t}=F^{\NA}(\cX,\cA)
$$
Here $F^{\NA}$ can be seen as the corresponding ``non-Archimedean'' functional.
\end{frame}

\end{document}