% 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
% modify this file as you see fit and even to delete this copyright
% notice. 
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\usepackage{amsmath,amssymb}
\usepackage[latin1]{inputenc}
\usepackage{colortbl}
\usepackage[english]{babel}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.

\title[\ \kern-190pt \blank{Jean-Pierre Demailly (Grenoble), Yamabe Memorial Conference\kern8.5pt
Holom.\ Morse Inequal.\ \&\ the Green-Griffiths-Lang conjecture}]
% (optional, use only with long paper titles)
{Holomorphic Morse Inequalities and the Green-Griffiths-Lang conjecture}

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

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

\institute[]{Institut Fourier, Universit\'e de Grenoble I, France\\
\&\ Acad\'emie 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)
{October 6, 2012\\
Yamabe Memorial Conference\\
University of Minnesota, Minneapolis}

%%\subject{Talks}
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% out. 

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% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

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% mathematical operators
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\special{header=/home/demailly/psinputs/mathdraw/grlib.ps}
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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
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%\beamerdefaultoverlayspecification{<+->}

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

\begin{frame}
  \frametitle{Entire curves\pgn}
  \begin{itemize}
  \item \claim{{\bf Definition.}} By an \alert{entire curve} we mean a non
    constant holomorphic map $f:\bC\to X$ into a complex 
    $n$-dimensional manifold.\pause\\
    $X$ is said to be \alert{(Brody) hyperbolic} if \alert{$\not\!\exists$ 
    such $f:\bC\to X$}.\vskip-14pt
    \pause
  \item
    If $X$ is a \alert{bounded} open subset $\Omega\subset\bC^n$, then there
    are no entire curves $f:\bC\to\Omega$ (\alert{Liouville's theorem}),\\
    \claim{$\Rightarrow$ every bounded open set $\Omega\subset\bC^n$ is
    hyperbolic}
    \pause
  \item
    $X=\overline\bC\ssm\{0,1,\infty\}=\bC\ssm\{0,1\}$
    has no entire curves,\\
    \claim{so it is hyperbolic} (\alert{Picard's theorem})
    \pause
  \item
    A complex torus $X=\bC^n/\Lambda$ ($\Lambda$ lattice) has a lot of
    entire curves. As $\bC$ simply connected, every $f:\bC\to X=\bC^n/\Lambda$
    lifts as $\tilde f:\bC\to\bC^n$,
    \alert{$\tilde f(t)=(\tilde f_1(t),\ldots,\tilde f_n(t))$},
    and $\tilde f_j:\bC\to\bC$ can be arbitrary entire functions.
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Projective algebraic varieties\pgn}
  \begin{itemize}
  \item Consider now the complex projective $n$-space
   \alert{$$
   \bP^n=\bP^n_\bC=(\bC^{n+1}\ssm\{0\})/\bC^*,\qquad
   [z]=[z_0:z_1:\ldots:z_n].$$\vskip-20pt}
  \pause
  \item An entire curve $f:\bC\to \bP^n$ is given by a map
     $$t\longmapsto[f_0(t):f_1(t):\ldots:f_n(t)]$$
   where $f_j:\bC\to\bC$ are holomorphic functions without common zeroes
   (so there are a lot of them).
   \pause
  \vskip7pt
  \item More generally, look at a (complex) 
    \alert{projective manifold}, i.e.
    \alert{$$X^n\subset\bP^N,\qquad X=\{[z]\,;\;P_1(z)=...=P_k(z)=0\}$$}
    where $P_j(z)=P_j(z_0,z_1,\ldots,z_N)$ are homogeneous polynomials
    (of some degree $d_j$), such that $X$ is\\ \alert{non singular}.
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Complex curves (genus 0 and 1)\pgn}
  \pgfdeclareimage[height=4.5cm]{curves1}{curves1}
  \pgfuseimage{curves1}
  \vskip10pt

  Canonical bundle $K_X=\Lambda^n T^*_X$ (here $K_X=T^*_X$)\pause

  \begin{itemize}
  \item $g=0:~X=\bP^1$\kern52pt courbure $T_X>0$ 
  \hbox{\claim{not hyperbolic}\kern-10pt}\pause
  \item $g=1:~X=\bC/(\bZ+\bZ\tau)$\kern8pt courbure $T_X=0$
  \hbox{\claim{not hyperbolic}\kern-15pt}
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Hyperbolicity of curves\pgn}
  \frametitle{Complex curves (genus $g\ge 2$)\pgn}
  \pgfdeclareimage[height=3.9cm]{curves2}{curves2}
  \pgfuseimage{curves2}
  \vskip10pt
  deg$\,K_X=2g-2$
  \vskip4pt
  If $g\ge 2,~~X\simeq \bD/\Gamma~~(T_X<0)~~\Rightarrow~~$ 
  \claim{$X$ is hyperbolic}.\vskip5pt
  \pause
  In fact every curve $f:\bC\to X\simeq \bD/\Gamma$ lifts to
  $\widetilde f:\bC\to \bD$,\\
 \claim{and so must be constant by Liouville}.
 \end{frame}

\begin{frame}
  \frametitle{Kobayashi metric / hyperbolic manifolds\pgn}
  \begin{itemize}
  \item
    For a complex manifold, $n=\dim_\bC X$, one defines
    \alert{the Kobayashi pseudo-metric} :
    $x\in X$, $\xi\in T_X$\\
    $\kappa_x(\xi)=\inf\{\lambda>0\,;\;\exists f:\bD\to X,\;f(0)=x,\;
    \lambda f_*(0)=\xi\}$\\
    On $\bC^n$, $\bP^n$ or complex tori $X=\bC^n/\Lambda$, one has 
    \alert{$\kappa_X\equiv 0.$}\vskip3pt
   \pause
   \item
    $X$ is said to be \alert{hyperbolic in the sense of Kobayashi}
    if the associated integrated pseudo-distance is a distance\\
    (i.e.\ it separates points -- i.e.\ has Hausdorff topology).\vskip3pt
    \pause
    \item
        \claim{{\bf Examples.}} 
        $*$ $X=\Omega/\Gamma$, $\Omega$ bounded symmetric domain.\\
        $*$ any product $X=X_1\times \ldots\times X_s$ where $X_j$ hyperbolic.
    \vskip3pt\pause
    \item 
        \claim{{\bf Theorem (dimension $n$ arbitrary)} 
        {\rm (Kobayashi, 1970)}\\ 
         {\it $T_X$ negatively 
         curved $(T^*_X>0$, i.e.\ ample$)$ $\Rightarrow$ 
         \hbox{$X$ hyperbolic.\kern-10pt}}}\\
        Recall that a holomorphic vector bundle $E$ is \alert{ample}
        iff its symmetric powers $S^mE$ have global sections which
        generate $1$-jets of (germs of) sections at any point $x\in X$.
  \end{itemize}
\end{frame}

\begin{frame}
\frametitle{Ahlfors-Schwarz lemma\pgn}
The proof of the above Kobayashi result depends crucially on:
\vskip3pt
\claim{{\bf Ahlfors-Schwarz lemma.}} Let $\gamma=i\sum\gamma_{jk}
dt_j\wedge d\overline t_k$ be an
almost everywhere positive hermitian form on the ball $B(0,R)\subset\bC^p$,
such that $-{\rm Ricci}(\gamma):=i\,\ddbar\log\det\gamma\ge A\gamma$ in the
sense of currents, for some constant $A>0$ $($this means in particular
that $\det\gamma=\det(\gamma_{jk})$ is such that $\log\det\gamma$ is
plurisubharmonic$)$. Then the  $\gamma$-volume form is controlled by
the Poincar\'e volume form~:
$$
\det(\gamma)\le\Big({p+1\over AR^2}\Big)^p{1\over(1-|t|^2/R^2)^{p+1}}.
$$
\end{frame}

\begin{frame}
\frametitle{Brody theorem\pgn}
 \claim{{\bf Brody reparametrization Lemma.}} {\it Assume that $X$ is 
\alert{compact}, let $\omega$ be a hermitian
metric on~$X$ and $f:\bD\to X$ a holomorphic map. For every
$\varepsilon>0$, there exists a radius $R\ge(1-\varepsilon)\|f'(0)\|_\omega$
and a homographic transformation $\psi$ of the disk $D(0,R)$ onto
$(1-\varepsilon)\bD$ such that $\|(f\circ\psi)'(0)\|_\omega=1$ and
$\|(f\circ\psi)'(t)\|_\omega 
\le(1-|t|^2/R^2)^{-1}$ for every $t\in D(0,R)$.\\
\alert{$\Rightarrow$ if $f'$ unbounded,
$\exists g=\lim f\circ\psi_\nu:\bC\to X$ 
with $\Vert g'\Vert_\omega\le 1$.}}\vskip3pt\pause

 \claim{{\bf Brody theorem (1978).}} {\it If $X$ is \alert{compact} then $X$ is
  Kobayashi hyperbolic if and only if there are no
  entire holomorphic curves $f:\bC\to X$ \alert{$($Brody hyperbolicity$)$}.}
 \vskip4pt\pause
  Hyperbolic varieties are especially interesting for their expected
  diophantine properties :\vskip1pt
  \claim{{\bf Conjecture} {\rm (S.\ Lang, 1986)}} {\it An arithmetic
  projective variety $X$ is hyperbolic iff $X(\bK)$ is finite
  for every number field~$\bK$.}
\end{frame}


\begin{frame}
  \frametitle{Varieties of general type\pgn}
  \begin{itemize}  
    \item
        \claim{{\bf Definition} {\it A non singular projective variety $X$ is
         said to be of \alert{general type} if the growth of
         pluricanonical sections\vskip3pt
         \centerline{$\dim H^0(X,K_X^{\otimes m})\sim cm^n, \qquad K_X=\Lambda^nT^*_X$}\vskip3pt
         is maximal.}}
         \vskip4pt
         (sections locally of the form 
          $f(z)\,(dz_1\wedge\ldots\wedge dz_n)^{\otimes m}$)
         \vskip4pt
         \claim{\bf Example}: A non singular hypersurface 
         $X^n\subset\bP^{n+1}$ of 
         degree $d$ satisfies \alert{$K_X=\cO(d-n-2)$},\\
         $X$ is \alert{of general type iff $d>n+2$.}\vskip4pt
         \pause
    \item
        \claim{{\bf Conjecture CGT.}} {\it If a compact variety $X$
         is hyperbolic, then \alert{it should be of general type}, and
         if $X$ is non singular, then $K_X=\Lambda^nT^*_X$ should be 
         \alert{ample}, i.e.\ \alert{$K_X>0$} $($Kodaira$)$\\
         $($equivalently 
         $\exists$ K\"ahler metric
         $\omega$ such that \alert{Ricci$(\omega)<0)$}.}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Conjectural characterizations of hyperbolicity\pgn}
  \begin{itemize}  
    \item 
        \claim{{\bf Theorem.}} {\it Let $X$ be projective algebraic. 
         Consider the following properties :\\
         {\rm (GT)} Every subvariety $Y$ of $X$ is of \alert{general type}.\\
         {\rm (AH)} $\exists\varepsilon>0$, $\forall C\subset X$ 
         algebraic curve
         $$ 2g(\bar C)-2\ge \varepsilon\deg(C).$$
         \ \kern27pt$(X$ \alert{``algebraically hyperbolic''}$)$\\
         {\rm (HY)} $X$ is \alert{hyperbolic}\\
         {\rm (JC)} $X$ possesses a \alert{jet-metric with negative 
         curvature} on its $k$-jet bundle $X_k$ $[$to be defined later$]$,
         for $k\ge k_0\gg 1$.\vskip3pt
         Then $\hbox{\rm(JC)}\Rightarrow \hbox{\rm(GT)},~\hbox{\rm(AH)},~
         \hbox{\rm(HY)}$,\\
         \ \kern26pt$\hbox{\rm(HY)}\Rightarrow \hbox{\rm(AH)}$,\\ and
         if Conjecture CGT holds,
         $\hbox{\rm(HY)}\Rightarrow \hbox{\rm(GT)}$.}
  \pause
  \item 
     \claim{It is expected that all 4 properties are in fact equivalent
     for projective varieties.}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Green-Griffiths-Lang conjecture\pgn}
  \begin{itemize}  
  \item
    \claim{{\bf Conjecture}} {\rm (Green-Griffiths-Lang = GGL)} {\it
    Let $X$ be a projective variety of general type. Then there exists
    an algebraic variety $Y\subsetneq X$ such that for all non-constant
    holomorphic $f:\bC\to X$ one has $f(\bC)\subset Y$.}\vskip4pt
  \pause
   \item 
   Combining the above conjectures, we get :\vskip1pt
   \claim{{\bf Expected consequence}} {\rm (of CGT + GGL)} Properties:\\ 
   {\rm (HY)} {\it $X$ is \alert{hyperbolic}}\\
   {\rm (GT)} {\it Every subvariety $Y$ of $X$ is of
   \alert{general type}\\
   are equivalent if {\rm CGT + GGL} hold.}\vskip4pt
  \pause
   \item \claim{{\bf Arithmetic counterpart} {\rm (Lang 1987).}} {\it If
   $X$ is a variety of general type defined over a number field and $Y$ is
   the Green-Griffiths locus $($Zariski closure of $\bigcup f(\bC))$,
   then $X(\bK)\ssm Y$ is finite for every number field $\bK$.}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Results obtained so far\pgn}
  \vskip-10pt
  \begin{itemize} 
  \item
  Using ``jet technology'' and \alert{deep results of McQuillan} for 
  curve foliations on surfaces, D.\ -- El Goul proved\\
  \claim{{\bf Theorem} {\rm (solution of Kobayashi conjecture, 1998).}}\\
  {\it A very generic surface $X{\subset}\bP^3$ of 
  \alert{degree${}\ge 21$} is \hbox{hyperbolic.\kern-1cm}\\}
  Independently McQuillan got degree${}\ge 35$.\\
  Recently improved to \alert{degree${}\ge 18$} (P\u{a}un, 2008).
  \vskip3pt
  \claim{For $X\subset \bP^{n+1}$, the optimal bound should be 
  \alert{degree${}\ge 2n+1$ for $n\ge 2$} {\bf(Zaidenberg)}.}
  \vskip3pt
  \item
  \claim{{\bf Generic GGL conjecture for $\dim_\bC X=n$}\\ 
  {\rm (S.~Diverio, J.~Merker, E.~Rousseau, 2009).}}\\
  {\it If $X\subset \bP^{n+1}$ is a \alert{generic $n$-fold of degree 
  $d\ge d_n:=2^{n^5}$},\\
  \alert{$[$also $d_3=593$, $d_4=3203$, $d_5=35355$, $d_6=172925\,]$}}
  then $\exists Y\subsetneq X$ s.t.\ 
  $\forall$ non const.\ $f{:}\,\bC\to X$ 
  \hbox{satisfies $f(\bC)\subset Y$\kern-15pt}\\
  \pause
  \claim{\bf Moreover {\rm (S.~Diverio, S.~Trapani, 2009)}}
  \hbox{\alert{$\codim_\bC Y\ge 2$} $\Rightarrow$\kern-20pt}\\
  generic hypersurface $X\subset\bP^4$ of degree${}\ge 593$ 
  \hbox{is \alert{hyperbolic}.\kern-15pt}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Definition of algebraic differential operators\pgn}

  The main idea in order to attack GGL is to use differential equations.
  Let 
  $$\bC\to X,~~~t\mapsto f(t)=(f_1(t),\ldots,f_n(t))$$
  be a curve written 
  in some local holomorphic coordinates $(z_1,\ldots,z_n)$ on $X$.
  \pause

  Consider \alert{algebraic differential operators}  which can be 
  written locally in multi-index notation
  \begin{eqnarray*}
    P(f_{[k]})&=&P(f',f'',\ldots,f^{(k)})\\
             &=&\sum
    a_{\alpha_1\alpha_2\ldots\alpha_k}(f(t))~f'(t)^{\alpha_1}
    f''(t)^{\alpha_2}\ldots f^{(k)}(t)^{\alpha_k}
    \end{eqnarray*}
    where $a_{\alpha_1\alpha_2\ldots\alpha_k}(z)$ are holomorphic coefficients
    on $X$ and $t\mapsto z=f(t)$ is a curve, $f_{[k]}=
    (f',f'',\ldots,f^{(k)})$ \alert{its $k$-jet}.
    \pause
    Obvious $\bC^*$-action :
    $$\lambda \cdot f(t)=f(\lambda t),~~~
    \alert{(\lambda \cdot f) ^{(k)}(t)=\lambda^kf^{(k)}(\lambda t)}$$
    $\Rightarrow$ \alert{weighted degree} 
    $m=|\alpha_1|+2|\alpha_2|+\ldots+k|\alpha_k|$.\vskip 3pt

\end{frame}

\begin{frame}
  \frametitle{Vanishing theorem for differential operators\pgn}
  \begin{itemize}  
    \item
    \claim{{\bf Definition.}} {\it $E^\GG_{k,m}$ is the sheaf $($bundle$)$ of
    algebraic differential operators of order $k$ and weighted degree $m$.}
   \pause
   \item
   \claim{{\bf Fundamental vanishing theorem}\\
   {\rm [Green-Griffiths 1979], [Demailly 1995],
    \hbox{[Siu-Yeung 1996]\kern-30pt}}}\\
    {\it Let $P\in H^0(X,E^\GG_{k,m}\otimes\cO(-A))$ 
   be a global algebraic differential operator whose coefficients vanish
   on some ample divisor $A$. Then \alert{$\forall f:\bC\to X$, 
   $P(f_{[k]})\equiv 0$}.}\vskip4pt
   \pause
   \item
   \claim{{\it Proof}}. One can assume that $A$ is very ample and 
   intersects $f(\bC)$. 
   Also assume $f'$ bounded (this is not so restrictive by Brody !). Then
   all $f^{(k)}$ are bounded by Cauchy inequality. Hence
   $$\bC\ni t\mapsto P(f',f'',\ldots, f^{(k)})(t)$$
   is a bounded holomorphic function on $\bC$ which vanishes at some point.
   Apply Liouville's theorem !\qed
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Geometric interpretation of vanishing theorem\pgn}
  \begin{itemize}  
    \item
    Let $X_k^{\GG}=J_k(X)^*/\bC^*$ be the \alert{projectivized $k$-jet bundle}
    of $X$ $=$ quotient of non constant $k$-jets by $\bC^*$-action.\\
    Fibers are weighted projective spaces.\vskip1pt
    \claim{{\bf Observation.}} {\it If $\pi_k:X_k^\GG\to X$ is
    canonical projection and $\cO_{X_k^\GG}(1)$ is the \alert{tautological line
    bundle}, then
    $$
    E^\GG_{k,m}=(\pi_k)_*\cO_{X_k^\GG}(m)
    $$}\vskip-15pt\ 
    \pause
    \item
    Saying that $f:\bC\to X$ satisfies the differential equation 
    $P(f_{[k]})=0$ means that 
      $$f_{[k]}(\bC)\subset Z_P$$
    where $Z_P$ is the zero divisor of the section 
      $$\sigma_P\in H^0(X_k^\GG,\cO_{X_k^\GG}(m)\otimes\pi_k^*\cO(-A))$$
    associated with $P$.
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Consequence of fundamental vanishing theorem\pgn}
  \begin{itemize}  
    \item
    \claim{{\bf Consequence of fundamental vanishing theorem.}}{\it \\ 
    If $P_j\in H^0(X,E^\GG_{k,m}\otimes\cO(-A))$ is a basis of sections
    then the image $f(\bC)$ lies in $Y=\pi_k(\bigcap Z_{P_j})$, hence property 
    asserted by the
    GGL conjecture holds true if there are ``enough independent differential
    equations'' so that
      $$Y=\pi_k(\bigcap_j Z_{P_j})\subsetneq X.$$}\vskip-30pt$\strut$
    \pause
    \item
    However, \alert{some differential equations are not very useful}.
    On a surface
    with coordinates $(z_1,z_2)$, a Wronskian equation 
    $f'_1f''_2-f'_2f''_1=0$ tells us
    that $f(\bC)$ sits on a line, but $f''_2(t)=0$ says that the second 
    component is linear affine in time, an essentially \alert{meaningless
    information} which is lost by a change of parameter $t\mapsto\varphi(t)$.
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Invariant differential operators\pgn}
  \begin{itemize}  
    \item
    The $k$-th order Wronskian operator
       $$W_k(f)=f'\wedge f''\wedge \ldots\wedge f^{(k)}$$
    (locally defined in coordinates) has degree $m=\frac{k(k+1)}{2}$ and 
       \alert{$$W_k(f\circ\varphi)=\varphi^{\prime m}W_k(f)\circ \varphi.$$}
    \ \vskip-20pt\ 
    \pause
    \item
    \claim{{\bf Definition.}} {\it A differential operator $P$ of order $k$ and
    degree $m$ is said to be invariant by reparametrization if
      $$P(f\circ\varphi)=\varphi^{\prime m}P(f)\circ \varphi$$
    for any parameter change $t\mapsto\varphi(t)$. Consider their set
      $$E_{k,m}\subset E^\GG_{k,m}~~~\hbox{$($a subbundle$)$}$$}
    \ \vskip-12pt
    (Any polynomial $Q(W_1,W_2,\ldots W_k)$ is invariant, but for $k\ge 3$ there
    are other invariant operators.)
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Category of directed manifolds\pgn}
  \begin{itemize}  
    \item
    \claim{{\bf Goal.}} We are interested in curves $f:\bC\to X$ such that 
    \alert{$f'(\bC)\subset V$}
    where $V$ is a subbundle (or subsheaf) of $T_X$.\vskip6pt
    \pause
    \item
    \claim{{\bf Definition.}} {\it Category of directed manifolds :}
    \vskip2pt
    -- \alert{Objects} : pairs $(X,V)$, $X$ manifold/$\bC$ and 
    \alert{$V\subset \cO(T_X)$}\\
    -- \alert{Arrows} $\psi:(X,V)\to(Y,W)$ holomorphic 
    s.t.~\alert{$\psi_*V\subset W$\kern-20pt}\\
    \pause
    -- \alert{``Absolute case''} $(X,T_X)$\\
    -- \alert{``Relative case''} $(X,T_{X/S})$ where $X\to S$\\
    -- \alert{``Integrable case''} when $[V,V]\subset V$ (foliations)
    \pause
    \vskip6pt
    \item
    \claim{{\bf Fonctor ``1-jet'' :}} $(X,V)\mapsto (\tilde X,\tilde V)$ 
    where :
       \begin{eqnarray*}
       &&\tilde X=P(V)={}\hbox{bundle of projective spaces of lines in $V$}\\
       &&\pi:\tilde X=P(V)\to X,~~~(x,[v])\mapsto x,~~v\in V_x\\
       &&\tilde V_{(x,[v])}=\big\{\xi\in T_{\tilde X,(x,[v])}\,;\;\pi_*\xi\in\bC v
        \subset T_{X,x}\big\}
       \end{eqnarray*}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Semple jet bundles\pgn}
  \begin{itemize} 
    \item
    For every entire curve $f:(\bC,T_\bC)\to(X,V)$ tangent to $V$
    \begin{eqnarray*}
    &&f_{[1]}(t):=(f(t),[f'(t)])\in P(V_{f(t)})\subset \tilde X\\
    &&f_{[1]}:(\bC,T_\bC)\to(\tilde X,\tilde V)~~
    \hbox{\alert{(projectivized 1st-jet)}}
    \end{eqnarray*}\ \vskip-28pt\
    \pause
    \item
    \claim{{\bf Definition.}} {\it Semple jet bundles :}
    \vskip3pt
    -- $(X_k,V_k)=k$-th iteration of fonctor 
       $(X,V)\mapsto(\tilde X,\tilde V)$\\
    -- $f_{[k]}:(\bC,T_\bC)\to(X_k,V_k)$ is the 
    \alert{projectivized $k$-jet of $f$.}
    \vskip7pt
    \pause
    \item
    \claim{{\bf Basic exact sequences}} 
    \begin{eqnarray*}
    &&0\to T_{\tilde X/X}\to\tilde V\build\to^{\pi_\star}_{}\cO_{\tilde X}(-1)
    \to 0\alert{~~~{}\Rightarrow \rk \tilde V=r=\rk V}\\
    &&0\to\cO_{\tilde X}\to \pi^\star V\otimes\cO_{\tilde X}(1)
    \to T_{\tilde X/X}\to 0~~\hbox{\alert{(Euler)}}\\
    \pause
    &&0\to T_{X_k/X_{k-1}}\to V_k\build\to^{(\pi_k)_\star}_{}\cO_{X_k}(-1)
    \to 0\alert{~~~{}\Rightarrow \rk V_k=r}\\
    &&0\to\cO_{X_k}\to \pi_k^\star V_{k-1}\otimes\cO_{X_k}(1)
    \to T_{X_k/X_{k-1}}\to 0~~\hbox{\alert{(Euler)}}
    \end{eqnarray*}

  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Direct image formula\pgn}
  \begin{itemize} 
    \item For $n=\dim X$ and $r=\rk V$,
    get a \alert{tower of $\bP^{r-1}$-bundles}
    $$\pi_{k,0}:X_k\build\to^{\pi_k}_{}X_{k-1}\to\cdots\to X_1
      \build\to^{\pi_1}_{}X_0=X$$
    with \alert{$\dim X_k=n+k(r-1)$, $\rk V_k=r$},\\ 
    and \alert{tautological line bundles $\cO_{X_k}(1)$ on $X_k=P(V_{k-1})$}.
  \pause
  \item
  \claim{{\bf Theorem.}} {\it $X_k$ is a smooth compactification of
  $$X_k^{\GG,\reg}/G_k=J_k^{\GG,\reg}/G_k$$ where $G_k$ is the group of 
  $k$-jets of germs of biholomorphisms of $(\bC,0)$, acting on the right by
  reparametrization:  $(f,\varphi)\mapsto f\circ\varphi$, and 
  $J_k^{\reg}$ is the space of $k$-jets of regular curves.}
  \pause
  \item
  \claim{{\bf Direct image formula.}} {\it \alert{$(\pi_{k,0})_*\cO_{X_k}(m)
  =E_{k,m}V^*={}$} invariant algebraic differential operators 
  $f\mapsto P(f_{[k]})$\\ acting on germs of curves $f:(\bC,T_\bC)\to (X,V)$.}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Algebraic structure of differential rings\pgn}
  \begin{itemize} 
  \item
  Although very interesting, results are currently limited by
  \alert{lack of knowledge on jet bundles and differential operators}
  \item
  \claim{Theorem {\rm(B\'erczi-Kirwan, 2009).}}{\it The ring of germs of 
  invariant 
  differential operators on $(\bC^n,T_{\bC^n})$ at the origin\\
  \ \kern28pt  $\displaystyle\cA_{k,n}=\bigoplus_m E_{k,m}T^*_{\bC^n}~~~
    \hbox{is finitely generated.}$}
  \pause
  \item
  Checked by direct calculations $\forall n$,~$k\le 2$ and $n=2$, 
  \hbox{$k\le 4\,$:\kern-10pt}
  \alert{
  \begin{eqnarray*}
  &&\kern-10pt\cA_{1,n}=\cO[f'_1,\ldots,f'_n]\\
  &&\kern-10pt\cA_{2,n}=\cO[f'_1,\ldots,f'_n,W^{[ij]}],~~~
    W^{[ij]}=f'_if''_j-f'_jf''_i\\
  &&\kern-10pt
    \cA_{3,2}=\cO[f'_1,f'_2,W_1,W_2][W]^2,~~~W_i=f'_iDW-3f''_iW\\
  &&\kern-10pt
    \cA_{4,2}=\cO[f'_1,f'_2,W_{11},W_{22},S][W]^6,~~~W_{ii}=f'_iDW_i-5f''_iW_i
  \end{eqnarray*}}
 where $W=f'_1f''_2-f'_2f''_1$, $S=(W_1DW_2-W_2DW_1)/W$.
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Generalized GGL conjecture\pgn}
  $~$\vskip-20pt
  \begin{itemize} 
  \item
  \claim{{\bf Generalized GGL conjecture.}}
  {\it If $(X,V)$ is directed manifold of
   general type, i.e.\ $\det V^*$ big, then $\exists Y\subsetneq X$ such that
  $\forall f:(\bC,T_\bC)\to(X,V)$ non const., $f(\bC)\subset Y$.}\vskip3pt
  \pause
  \item
  \claim{{\bf Remark.}} Elementary by Ahlfors-Schwarz if 
  \hbox{$r=\rk V=1$.\kern-15pt}\\
  $t\mapsto \log\Vert f'(t)\Vert_{V,h}$ is strictly subharmonic if
  $r=1$ and $(V^*,h^*)$ has${}>0$ curvature in the sense of currents.
  \vskip3pt  \pause
  \item
  \claim{{\bf Strategy.}} \alert{Try some sort of induction on $r=\rk V$}.\\
  First try to get \alert{differential equations} 
  $f_{[k]}(\bC) \subset Z\subsetneq X_k$.\\
  Take \alert{minimal such $k$}. If $k=0$, we are done! 
  Otherwise $k\ge 1$ and $\pi_{k,k-1}(Z)=X_{k-1}$, thus $V'=V_k\cap T_Z$ has 
  rank${}<\rk V_k=r$ and should have again $\det V^{\prime*}$ big $($unless 
  some unprobable geometry situation occurs ?$)$.\vskip4pt
  \pause
  \item
  \claim{{\bf Needed induction step.}} {\it If $(X,V)$ has $\det V^*$ big
  and $Z\subset X_k$ irreducible with $\pi_{k,k-1}(Z)=X_{k-1}$, then
  $(Z,V')$, $V'=V_k\cap T_Z$ has $\cO_{Z_\ell}(1)$ big on $(Z_\ell,V'_\ell)$,
  $\ell\gg 0$.}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Holomorphic Morse inequalities\pgn}
  \claim{{\bf Holomorphic Morse inequalities} {\rm (D-, 1985)}}
  Let $L\to X$ be a holomorphic line bundle on a compact complex
  manifold $X$, $h$ a smooth hermitian metric on $L$ and
   \alert{$$\Theta_{L,h}=\frac{i}{2\pi}\nabla_{L,h}^2=-
   \frac{i}{2\pi}\ddbar\log h$$}
   its curvature form. Then $\forall q=0,1,\ldots,n=\dim_\bC X$
   \alert{$$
   \sum_{j=0}^q(-1)^{q-j}h^j(X,L^{\otimes k})\le
   \frac{k^n}{n!}\int_{X(L,h,\le q)}(-1)^q\Theta_{L,h}^n+o(k^n).
   $$}
   where
   \alert{$$
   X(L,h,q)=\{x\in X\,;\;\Theta_{L,h}(x)~\hbox{has signature}~(n-q,q)\}
   $$}
   ($q$-index set), and
   \alert{$$
   X(L,h,\le q)=\bigcup_{0\le j\le q}X(L,h,\le j)
   $$}
\end{frame}

\begin{frame}
  \frametitle{Holomorphic Morse inequalities (continued) \pgn}
  As a consequence, one gets \claim{an upper bound}
  \alert{$$
  h^0(X,L^{\otimes k})\le
  \frac{k^n}{n!}\int_{X(L,h,0)}\Theta_{L,h}^n+o(k^n)
  $$}
  and a \claim{lower bound}
  \alert{$$
  h^0(X,L^{\otimes k})\ge   h^0(X,L^{\otimes k})-  h^1(X,L^{\otimes k})\ge
  {}\kern3cm~
  $$
  $$
  \ge  \frac{k^n}{n!}\Big(
  \int_{X(L,h,0)}\Theta_{L,h}^n - \int_{X(L,h,1)}|\Theta_{L,h}^n|\Big)
  -o(k^n)
  $$}\pause
  and similar bounds for the higher cohomology groups $H^q$:
  \alert{$$
  h^q(X,L^{\otimes k})\le
  \frac{k^n}{n!}\int_{X(L,h,q)}|\Theta_{L,h}^n|+o(k^n){}\kern4cm~
  $$}
  \alert{$$
  h^q(X,L^{\otimes k})\ge
  \frac{k^n}{n!}\Big(\int_{X(L,h,q)}-\int_{X(L,h,q-1)}-\int_{X(L,h,q+1)}
  |\Theta_{L,h}^n|\Big)-o(k^n)
  $$}
\end{frame}

\begin{frame}
\frametitle{Finsler metric on the $k$-jet bundles\pgn}
 Let $J_kV$ be the bundle of $k$-jets of curves
\hbox{\alert{$f:(\bC,T_\bC)\to(X,V)$}\kern-10pt}
\vskip1.5pt\pause
 Assuming that $V$ is equipped with a hermitian metric $h$,
 one defines a ''weighted Finsler metric'' on $J^kV$ by 
taking \hbox{$p=k!$ and\kern-15pt}
$$
\Psi_{h_k}(f):=\Big(\sum_{1\le s\le k}\varepsilon_s\Vert\nabla^sf(0)
\Vert_{h(x)}^{2p/s}\Big)^{1/p},~~1=\varepsilon_1\gg
\varepsilon_2\gg\cdots\gg\varepsilon_k.
$$\pause%
Letting $\xi_s=\nabla^sf(0)$, this can actually be viewed as a 
metric $h_k$ on $L_k:=\cO_{X_k^\GG}(1)$, with curvature form
$(x,\xi_1,\ldots,\xi_k)\mapsto$\vskip-14pt
\alert{$$
\Theta_{L_k,h_k}=\omega_{{\rm FS},k}(\xi)+{i\over 2\pi}
\sum_{1\le s\le k}{1\over s}{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}}
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}
{\xi_{s\alpha}\overline\xi_{s\beta}\over|\xi_s|^2}\,dz_i\wedge d\overline z_j
$$}%
where \alert{$(c_{ij\alpha\beta})$} are the coefficients of the curvature tensor
\alert{$\Theta_{V^*,h^*}$} and \alert{$\omega_{{\rm FS},k}$ is the vertical
Fubini-Study metric}
on the fibers of $X_k^\GG\to X$. The expression gets simpler by using
polar coordinates $x_s=\vert\xi_s\vert_h^{2p/s}$, 
$u_s=\xi_s/\vert\xi_s\vert_h=\nabla^sf(0)/\vert\nabla^sf(0)\vert$.
\end{frame}

\begin{frame}
\frametitle{Probabilistic interpretation of the curvature\pgn}
In such polar coordinates, one gets the formula
\alert{$$
\Theta_{L_k,h_k}=\omega_{{\rm FS},p,k}(\xi)+
{i\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j
$$}%
where $\omega_{{\rm FS},k}(\xi)$ is positive definite in $\xi$. The other 
terms are a weighted average of the values of the 
curvature tensor $\Theta_{V,h}$ on vectors $u_s$ in the unit sphere
bundle $SV\subset V$. The weighted projective space can be viewed
as a circle quotient of the pseudosphere $\sum|\xi_s|^{2p/s}=1$, so
we can take here $x_s\ge 0$, $\sum x_s=1$. This is essentially a
sum of the form $\sum\frac{1}{s}\gamma(u_s)$ where $u_s$ 
are random points of the sphere, and so as $k\to+\infty$ this
can be estimated by a \alert{``Monte-Carlo'' integral}
$$
\Big(1+\frac{1}{2}+\ldots+\frac{1}{k}\Big)\int_{u\in SV}\gamma(u)\,du.
$$
As $\gamma$ is quadratic here, 
\alert{$\int_{u\in SV}\gamma(u)\,du=\frac{1}{r}\Tr(\gamma)$}.
\end{frame}

\begin{frame}
\frametitle{Main cohomological estimate\pgn}
It follows that the leading term in the
estimate only involves the trace of $\Theta_{V^*,h^*}$, i.e.\ the
curvature of $(\det V^*,\det h^*)$, which can be taken to be${}>0$
if $\det V^*$ is big.
\vskip3pt
\claim{{\bf Corollary} (D-, 2010)} Let $(X,V)$ be a directed manifold, $F\to X$ a
$\bQ$-line bundle, $(V,h)$ and $(F,h_F)$ hermitian.
\hbox{Define\kern-20pt}\vskip3pt
$\displaystyle~\kern5mm
L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big({1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big),
$\vskip3pt
$\displaystyle~\kern5mm
\eta=\Theta_{\det V^*,\det h^*}+\Theta_{F,h_F}.$\vskip3pt
Then for all $q\ge 0$ and all $m\gg k\gg 1$ such that 
$m$ is sufficiently divisible, we have\vskip3pt
\alert{$\displaystyle
h^q(X_k^\GG,\cO(L_k^{\otimes m}))\le{m^{n+kr-1}\over (n{+}kr{-}1)!}
{(\log k)^n\over n!\,(k!)^r}\bigg(
\int_{X(\eta,q)}\!\!\!(-1)^q\eta^n+\frac{C}{\log k}\bigg)\kern-20pt
$\vskip3pt\pause
$\displaystyle
h^0(X_k^\GG,\cO(L_k^{\otimes m}))\ge{m^{n+kr-1}\over (n{+}kr{-}1)!}
{(\log k)^n\over n!\,(k!)^r}\bigg(
\int_{X(\eta,\le 1)}\eta^n-\frac{C}{\log k}\bigg).\kern-20pt
$}%
\end{frame}

\begin{frame}
  \frametitle{Partial solution of the GGL conjecture\pgn}
  Using the above cohomological estimate, we obtain:\vskip3pt
  \claim{{\bf Theorem} (D-, 2010)} Let $(X,V)$ be of general type, i.e.
  $K_V=(\det V)^*$ is a big line bundle. Then there exists $k\ge 1$ and
  an algebraic hypersurface $Z\subsetneq X_k$ such that every entire
  curve $f:(\bC,T_\bC)\mapsto(X,V)$ satisfies $f_{[k]}(\bC)\subset Z$
  (in other words, \alert{$f$ satisfies an algebraic differential equation
   of order $k$}).\vskip5pt\pause

Another important consequence is:\vskip3pt
\claim{{\bf Theorem} (D-, 2012)} A generic hypersurface $X\subset\bP^{n+1}$ of 
degree $d\ge d_n$ with
\alert{$$
d_2=286,\quad d_3=7316,\quad
d_n= \left\lfloor{n^4\over 3}\big(n\log(n\log(24n))\big)^n\right\rfloor
$$}
(for $n\ge 4$) satisfies the Green-Griffiths conjecture.
\end{frame}

\begin{frame}
  \frametitle{A differentiation technique by Yum-Tong Siu\pgn}
  The proof of the last result uses an important idea due to Yum-Tong Siu,
  itself based on ideas of Claire Voisin and Herb Clemens, and then refined
  by M.~P\u{a}un [Pau08], E.~Rousseau [Rou06b] and J.~Merker [Mer09].
 
  The idea consists of studying vector fields on 
  the \alert{relative jet space of the universal family of hypersurfaces of 
  $\bP^{n+1}$}.\vskip3pt
  \pause
  Let $\cX\subset\bP^{n+1}\times\bP^{N_d}$ be the universal hypersurface,
  i.e.  
  $$\cX=\{(z,a)\,;\;
  a=(a_\alpha)~\hbox{s.t.}~P_a(z)=\sum a_\alpha z^\alpha=0\},$$
  let $\Omega\subset\bP^{N_d}$ be the open subset of $a$'s for which 
  \hbox{$X_a=\{P_a(z)=0\}$} is smooth, and let
    $$p:\cX\to\bP^{n+1},~~\pi:\cX_{|\Omega}\to\Omega\subset\bP^{N_d}$$
  be the natural projections.\vskip4pt
\end{frame}

\begin{frame}
  \frametitle{Meromorphic vector fields on jet spaces\pgn}
  Let 
  $$p_k:\cX_k\to\cX\to\bP^{n+1},~~~\pi_k:\cX_k\to\Omega\subset\bP^{N_d}$$
  be the relative Green-Griffiths $k$-jet space of $\cX\to\Omega$. 
  Then J.~Merker [Mer09] has shown that global sections $\eta_j$ of
  \alert{$$\cO(T_{\cX_k})\otimes p_k^*\cO_{\bP^{n+1}}(k^2+2k)\otimes 
   \pi_k^*\cO_{\bP^{N_d}}(1)$$}
  generate the bundle at all points of 
  $\cX_k^{\reg}$ for $k=n=\dim X_a$. From this, it follows that if $P$ 
  is a non zero global section over $\Omega$ of
  $E^\GG_{k,m}T^*_{\cX}\otimes p_k^*\cO_{\bP^{n+1}}(-s)$
  for some $s$, then for a suitable collection of $\eta=(\eta_1,\ldots,
  \eta_m)$, the $m$-th derivatives
    $$D_{\eta_1}\ldots D_{\eta_m}P$$
  yield sections of 
  $H^0\big(\cX,E^\GG_{k,m}T^*_{\cX}\otimes p_k^*\cO_{\bP^{n+1}}(m(k^2+2k)-s)\big)$
  whose joint base locus is contained in $\cX_k^{\sing}$, whence the result.
\end{frame}


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\vskip6pt
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\bibitem[Dem95]{Dem95}
Demailly, J.-P.: \emph{Algebraic Criteria for Kobayashi Hyperbolic Projective Varieties and Jet Differentials}. Algebraic geometry -- Santa Cruz 1995, 285--360,
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\bibitem[Dem10]{Dem10}
Demailly, J.-P.: \emph{Holomorphic Morse inequalities and the 
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\bibitem[Dem12]{Dem12}
Demailly, J.-P.: \emph{Hyperbolic algebraic varieties and holomorphic differential equations}, lecture given at the VIASM Annual Meeting 2012, Hanoi 
26 August 2012, 63p, to appear in Acta Vietnamica

\bibitem[D-EG00]{D-EG00}
Demailly, J.-P., El Goul, J.: \emph{Hyperbolicity of Generic Surfaces of High Degree in Projective $3$-Space}. Amer.\ J.\ Math.\ {\bf 122} (2000), no.\ 3, 515--546.

\bibitem[Div09]{Div09} Diverio, S.: \emph{Existence of global invariant jet
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\bibitem[Lang86]{Lang86}
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\bibitem[Mer08]{Mer08} An algorithm to generate all polynomials in the $k$-jet
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\bibitem[Mer09]{Mer09} Merker, J.: \emph{Low pole order frames on
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\end{document}