% 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. 

\setbeamersize{text margin left=1.5em}
\setbeamersize{text margin right=1.5em}

\newcommand\wider[2][3em]{%
\makebox[\linewidth][c]{%
  \begin{minipage}{\dimexpr\textwidth+#1\relax}
  \raggedright#2
  \end{minipage}%
  }%
}

\catcode`\@=11

\def\normalframenumbering{{\number\c@framenumber/\inserttotalframenumber}}

\def\detailedframenumbering{{\number\c@framenumber/\inserttotalframenumber${}
^{[\ifnum\beamer@slideinframe=\beamer@minimum
\number\beamer@slideinframe\else\advance\beamer@slideinframe by -1{}
\number\beamer@slideinframe\advance\beamer@slideinframe by 1{}\fi{:}
\number\c@page]}$}}

\newcount \c@refinit 
\def\biblioframenumbering{Ref.~\advance\c@page by -\c@refinit \number\c@page
\advance\c@page by \c@refinit${}^{[\number\c@page]}$}
\def\setbibliopages{\c@refinit=\c@page \advance \c@refinit by -1{}
\let\framenumbering=\biblioframenumbering}

\def\ialign{\everycr{}\tabskip\z@skip\halign}
\def\eqalign#1{\null\,\vcenter{\openup\jot\m@th
  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
\def\plainmatrix#1{\null\,\vcenter{\normalbaselines\m@th
    \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr
      \mathstrut\crcr\noalign{\kern-\baselineskip}
      #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}

\catcode`\@=12

\mode<presentation>
% \setbeamertemplate{background canvas}[vertical shading][bottom=red!10,
% top=blue!10]
\usetheme{Warsaw}
\usefonttheme[onlysmall]{structurebold}

\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb}
\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.
\font\sevenrm=cmr10 at 7pt

\definecolor{ColClaim}{rgb}{0,0,0.8}
\definecolor{Alert}{rgb}{0.8,0,0}
\definecolor{Blank}{rgb}{1,1,1}
\def\claim#1{{\color{ColClaim}#1}}
\def\alert#1{{\color{Alert}#1}}
\def\blank#1{{\color{Blank}#1}}
\def\bddK{{{}^b\kern-1pt K}}
\def\bfe{{\bf e}}
\def\bfk{{\bf k}}
\def\Poin{{\hbox{\sevenrm Poincar\'e}}}

\let\framenumbering=\normalframenumbering

\title[\ \kern-190pt\rlap{\blank{J.-P.\ Demailly (Grenoble), Université Paris-Sud, 5 juin 2018}}\kern181pt\rlap{\blank{
Kobayashi conjecture on generic hyperbolicity}}\kern181pt
\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Une preuve effective simple de la\\
conjecture de Kobayashi sur l’hyperbolicité\\
des hypersurfaces algébriques générales}

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

\author[] % (optional, use only with lots of authors)
{\strut\vskip-20pt Jean-Pierre Demailly}

\institute[]{\strut\vskip-20pt
Institut Fourier, Université 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\vskip-20pt Séminaire d'Analyse Harmonique\\
Département de Mathématiques d'Orsay\\
\vskip7pt Université de Paris-Sud, 5 juin 2018}

%%\subject{Talks}
% This is only inserted into the PDF information catalog. Can be left
% out. 

% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

%%\def\\{\hfil\break}
\ifpdf
\font\eightrm=ec-lmr10 at 8pt
\else
\font\eightrm=cmr10 at 8pt
\fi

\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\tors}{\operatorname{torsion}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\reg}{\operatorname{reg}}
\newcommand{\sing}{\operatorname{sing}}

\newcommand{\bB}{{\mathbb B}}
\newcommand{\bC}{{\mathbb C}}
\newcommand{\bD}{{\mathbb D}}
\newcommand{\bG}{{\mathbb G}}
\newcommand{\bK}{{\mathbb K}}
\newcommand{\bN}{{\mathbb N}}
\newcommand{\bP}{{\mathbb P}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bZ}{{\mathbb Z}}

\newcommand{\cA}{{\mathcal A}}
\newcommand{\cC}{{\mathcal C}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cG}{{\mathcal G}}
\newcommand{\cH}{{\mathcal H}}
\newcommand{\cI}{{\mathcal I}}
\newcommand{\cJ}{{\mathcal J}}
\newcommand{\cK}{{\mathcal K}}
\newcommand{\cL}{{\mathcal L}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cV}{{\mathcal V}}
\newcommand{\cX}{{\mathcal X}}

\newcommand{\dbar}{\overline\partial}
\newcommand{\ddbar}{\partial\overline\partial}
\newcommand{\ovl}{\overline}
\newcommand{\wt}{\widetilde}
\newcommand{\lra}{\longrightarrow}
\newcommand{\bul}{{\scriptscriptstyle\bullet}}

% mathematical operators
\renewcommand{\Re}{\mathop{\rm Re}\nolimits}
\renewcommand{\Im}{\mathop{\rm Im}\nolimits}
\newcommand{\Pic}{\mathop{\rm Pic}\nolimits}
\newcommand{\codim}{\mathop{\rm codim}\nolimits}
\newcommand{\Id}{\mathop{\rm Id}\nolimits}
\newcommand{\Sing}{\mathop{\rm Sing}\nolimits}
\newcommand{\Supp}{\mathop{\rm Supp}\nolimits}
\newcommand{\Vol}{\mathop{\rm Vol}\nolimits}
\newcommand{\pr}{\mathop{\rm pr}\nolimits}

\newcommand{\NS}{\mathop{\rm NS}\nolimits}
\newcommand{\GG}{{\mathop{\rm GG}\nolimits}}
\newcommand{\NE}{\mathop{\rm NE}\nolimits}
\newcommand{\ME}{\mathop{\rm ME}\nolimits}
\newcommand{\SME}{\mathop{\rm SME}\nolimits}
\newcommand{\card}{\mathop{\rm card}\nolimits}
\newcommand{\alg}{{\rm alg}}
\newcommand{\nef}{{\rm nef}}
\newcommand{\num}{\nu}
\newcommand{\ssm}{\mathop{\mathbb r}}
\newcommand{\smallvee}{{\scriptscriptstyle\vee}}

% figures inserted as PostScript / PDF files

\ifpdf
\def\RGBColor#1#2{{\pdfliteral{#1 rg}#2\pdfliteral{0 g}}}

\long\def\InsertPSFigure#1 #2 #3 #4\EndFig{\par\advance\psfigurecount by 1%
\pdfximage{\jobname_figures/fig\number\psfigurecount.pdf}%
\setbox0=\hbox{\pdfrefximage\pdflastximage}%
\psfiguredx=#1mm \advance\psfiguredx by 20mm%
\hbox{$\vbox to#2mm{\vfil%
\hbox{$\hskip #1mm\rlap{\smash{\raise-50mm\hbox to #1mm{\strut\kern-\psfiguredx%
\pdfximage width\wd0{\jobname_figures/fig\number\psfigurecount.pdf}%
\pdfrefximage\pdflastximage\kern-\wd0\hfil}}}$}}#4$}}

\long\def\InsertPSFile#1 #2 #3 #4 #5 #6\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{%
\vfil\special{pdf:}}#6$}}

\long\def\InsertImage#1 #2 #3 #4 #5 #6 #7 #8\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{\vfil\hbox{$\rlap{\smash{\pdfximage
\ifnum#3=0 \else width #3mm\fi \ifnum #4=0 \else height #4mm \fi depth 0cm{#7}%
\pdfrefximage\pdflastximage}}$}}#8$}}

\else
\special{header=/home/demailly/psinputs/mathdraw/mdrlib.ps}

\def\RGBColor#1#2{\special{color push rgb #1}#2\special{color pop}}

\long\def\InsertPSFigure#1 #2 #3 #4\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" 
#3}}#4$}}

\long\def\InsertPSFile#1 #2 #3 #4 #5 #6\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{%
\vfil\special{psfile=#5 hscale=#3 vscale=#4}}#6$}}

\long\def\InsertImage#1 #2 #3 #4 #5 #6 #7 #8\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{%
\vfil\special{psfile="`img2eps file #7 height #4 mm width #3 mm gamma #5
angle #6}}#8$}}

\fi

\long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}}

\def\ovl{\overline}
\def\build#1^#2_#3{\mathrel{\mathop{\null#1}\limits^{#2}_{#3}}}
\def\bibitem[#1]#2#3{\medskip{\bf[#1]} #3}

\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}
%%}


% 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}
  \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.

%% \section*{Basic concepts}

\begin{frame}
  \frametitle{Kobayashi hyperbolicity and entire curves}
 \vskip-7pt
 \begin{block}{Kobayashi-Eisenman infinitesimal pseudometrics}
Let $X$ be a complex space, $\dim_\bC X=n$, $\bB_p={}$unit ball in $\bC^p$,
$1\leq p\leq n$ and $\tau_0=\partial/\partial t_1
\wedge\cdots\wedge\partial/\partial t_p\in\Lambda^p\bC^p$. The
\alert{Kobayashi-\kern-15pt}\\
\alert{Eisenman infinitesimal pseudometric $\bfe^p_X$}
is the pseudometric defined on decomposable
$p$-vectors $\xi=\xi_1\wedge\cdots\wedge\xi_p\in\Lambda^pT_{X,x}$, by\vskip5pt
\centerline{$
\bfe^p_X(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\bB_p\to X,\,f(0)=x,\,
\lambda f_\star(\tau_0)=\xi\big\}.$}\vskip5pt\pause
We say that $X$ is \alert{(infinitesimally) $p$-measure hyperbolic} if
\hbox{\alert{$\bfe^p_{X}$ is}\kern-15pt}\\ \alert{everywhere locally
uniformly positive definite} on the tautological line bundle of
the Grassmannian bundle of $p$-subspaces ${\rm Gr}(T_X,p)$.
\end{block}
\pause\vskip-4pt
\begin{block}{Characterization of Kobayashi hyperbolicity {\rm(Brody, 1978)}}
 For a \alert{compact} complex 
 manifold $X$, $\dim_{\bC}X=n$, TFAE:\\
 (i) The \alert{pseudometric $\bfk_X\,{=}\,\bfe^1_x$}
 is everywhere \alert{\hbox{non degenerate$\,$;\kern-15pt}}\\
 (ii) the integrated pseudodistance ${\bf d}_{\bf Kob}$ of $\bfe^1_X$
  \alert{\hbox{is a distance$\,$;\kern-15pt}}\\
 (iii) $X$ \alert{Brody hyperbolic}, i.e.$\not\!\exists$
 entire \hbox{curves $f:\bC\to X$, $f\neq{}$const.\kern-15pt}\\
 \end{block}\vskip-3pt
%% \pause
%% In dim $n=1$, a compact curve $X$ is hyperbolic iff genus $g\ge 2$.
\end{frame}

\begin{frame}
\frametitle{Main conjectures}\vskip-10pt
\begin{block}{Conjecture of General Type (CGT)} 
$\bullet$ A compact variety $X\,/\,\bC$ is
\alert{volume hyperbolic} (w.r.t.\ $\bfe^n_X$) $\Leftrightarrow$\\
$\phantom{\bullet~}X$ is of \alert{general type}, i.e.\ \alert{$K_X$ big}~
[implication $\Leftarrow$ is well \hbox{known].\kern-15pt}\pause\\
$\bullet$ $X$ \alert{Kobayashi (or Brody) 
hyperbolic} should imply \alert{$K_X$ ample}.
\end{block}\vskip-7.6pt
\pause
\begin{block}{Green-Griffiths-Lang Conjecture (GGL)}
    Let $X$ be a projective variety/$\bC$ of general type. Then $\exists
    Y\subsetneq X$ algebraic such that all entire curves
    $f:\bC\to X$ satisfy $f(\bC)\subset Y$.
\end{block}\vskip-7.6pt
\pause
\begin{block}{Arithmetic counterpart (Lang 1987) -- very optimistic ?} 
For $X$ projective defined over a number field $\bK_0$, the
\hbox{exceptional\kern-15pt}\\
locus \alert{$Y={\rm Exc}(X)$} in GGL's conjecture equals
\alert{${\rm Mordel}(X)={}$}\break
\alert{smallest $Y$} such that \alert{$X(\bK)\smallsetminus Y$ is finite},
$\forall\bK$ number field${}\supset\bK_0$.
\end{block}\vskip-7.6pt
\pause
\begin{block}{Consequence of CGT + GGL}
A compact complex manifold $X$ should be Kobayashi hyperbolic iff 
it is projective and  every subvariety $Y$ of $X$ is of \alert{general type}.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Kobayashi conjecture on generic hyperbolicity}
\vskip-9pt
\begin{block}{Kobayashi conjecture (1970)}
$\bullet$ Let $X^n\subset\bP^{n+1}$ be a (very)
generic hypersurface of degree $d\ge d_n$ large enough. Then $X$
is Kobayashi hyperbolic.\pause\\
$\bullet$ By results of Riemann, Poincaré, Zaidenberg, Clemens, Ein, Voisin,
Pacienza, the optimal bound is expected to be \alert{$d_1=4$},\\
\alert{$d_n=2n+1$ for $2\leq n\leq 4$} and \alert{$d_n=2n$ for $n\geq 5$}.
\end{block}\vskip-3pt\pause
Using ``jet technology'' and \alert{deep results of McQuillan} for 
curve foliations on surfaces, the following has been proved:\vskip-20pt{\strut}
\begin{block}{Theorem (Work of McQuillan + D., El Goul, 1998)}
A very generic surface $X^2\subset\bP^3$ of 
\alert{degree $d\ge 21$} is \hbox{hyperbolic}.\\
Independently McQuillan got $d\ge 35$.
\end{block}\vskip-3pt\pause
This has been improved to \alert{$d\ge 18$} (P\u{a}un, 2008).\pause\\
In 2012, Yum-Tong Siu announced a proof of the case of \alert{arbitrary 
dimension~$n$, with a non explicit $d_n$} (and a rather involved proof).
\end{frame}

\begin{frame}
\frametitle{Results on the generic Green-Griffiths conjecture}
\vskip-3pt
By combining an algebraic existence theorem for jet differentials
and Y.T.~Siu's technique of \alert{slanted vector fields}
\hbox{(itself~derived~from\kern-15pt}
ideas of H.~Clemens, L.~Ein and C.~Voisin), the following
\hbox{was proved:\kern-15pt}

\begin{block}{Theorem (S.~Diverio, J.~Merker, 
    E.~Rousseau, 2009, \&\ followers)}
  A generic hypersurface $X^n\subset \bP^{n+1}$ of degree 
\alert{$d\ge d_n:=2^{n^5}$} satisfies the GGL conjecture.\pause\
Bound then improved to {$\alert{d_n=O(e^{n^{1+\varepsilon}})}$}~:
\centerline{$\alert{d_n=9n^n}$
\kern1cm\hbox{\claim{(B\'erczi, 2010, using residue formulas),$\,$~}}}
\centerline{%
$\alert{d_n=(5n)^2\,n^n}\kern0.4cm\hbox{\claim{(Darondeau, 2015, alternative method),}}$}
\centerline{~~
$\alert{d_n=\left\lfloor{n^4\over 3}\big(n\log(n\log(24n))\big)^n\right\rfloor}
\kern0.2cm\hbox{\claim{(D-, 2012), weaker bound,}}$}\break
\claim{but special case of general result for arbitrary projective varieties}.
\end{block}
  \pause
\begin{block}{Theorem (S.~Diverio, S.~Trapani, 2009)} Additionally,
  a generic hypersurface $X^3\subset\bP^4$ of degree \alert{$d\ge 593$}
  is hyperbolic.
\end{block}
\end{frame}


\begin{frame}
\frametitle{Recent proof of the Kobayashi conjecture}
\vskip-4pt
Y.T. Siu's (Abel conf.\ 2002, Invent.\ Math.\ 2015) detailed
\hbox{a strategy\kern-15pt}\break
for the proof of the Kobayashi conjecture. In 2016, Brotbek gave a
more geometric proof, using Wronskian jet differentials.
\begin{block}{Theorem (Brotbek, April 2016)}
Let $Z$ be a projective $n+1$-dimensional projective manifold and
$A\to Z$ a very ample line bundle. Let $\sigma\in H^0(Z,dA)$
be a generic section.$\,$Then for \alert{$d{\gg}1$}
the hypersurface $X_\sigma=\sigma^{-1}(0)$ is
\hbox{\alert{hyperbolic}.\kern-15pt}
\end{block}
\pause%
The initial proof of Brotbek didn't provide effective bounds.
\hbox{Through\kern-20pt}\break
various improvements, Deng Ya got in his PhD thesis
\hbox{(May 2016)\kern-20pt}\break
the explicit bound \alert{$d_n=(n+1)^{n+2}(n+2)^{2n+7}=O(n^{3n+9})$.}
\begin{block}{Theorem (D-, 2018, with a much simplified proof)}
In the above setting, a general hypersurface $X_\sigma\,{=}\,\sigma^{-1}(0)$
is hyperbolic as soon as\vskip-7pt
\centerline{$\strut\kern1cm\alert{d\geq
d_n=\lfloor(en)^{2n+2}/3\rfloor.}$}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Solution of a conjecture of Debarre (2005)}
\vskip-4pt
In the same vein, the following results have also been proved.  
\begin{block}{Solution of Debarre's conjecture (Brotbek-Darondeau
\&\ Xie, 2015)}
Let $Z$ be a projective $(n+c)$-dimensional projective manifold and
$A\to Z$ a very ample line bundle. Let $\sigma_j\in H^0(Z,d_jA)$
be generic sections, $1\le j\le c$. Then, for \alert{$c\geq n$} and
\alert{$d_j\gg 1$ large}, the $n$-dimensional complete intersection
$X_\sigma=\bigcap\sigma_j^{-1}(0)\subset Z$ has an ample cotangent bundle
$T^*_{X_\sigma}$.\pause\\
In particular, \alert{such a generic complete intersection is hyperbolic}.
\end{block}
\pause%
S.Y.\ Xie got the sufficient lower bound $d_j\ge d_{n,c}=N^{N^2}$,
\hbox{$N=n+c$.\kern-15pt}\pause
\vskip5pt

In his PhD thesis, Ya Deng obtained the improved lower bound\vskip4pt
\centerline{\alert{$\displaystyle
d_{n,c}=4\nu(2N-1)^{2\nu+1}+6N-3=O((2N)^{N+3}),\qquad \nu=\lfloor
{\textstyle \frac{N+1}{2}}\rfloor.
$}}\pause\vskip4pt
The proof is obtained by selecting carefully certain special sections
$\sigma_j$ associated with ``lacunary'' polynomials of high degree.
\end{frame}

\begin{frame}
  \frametitle{Category of directed manifolds}
\claim{{\bf Goal.}} More generally, we are interested in curves 
$f:\bC\to X$ such that \alert{$f'(\bC)\subset V$}
where $V$ is a subbundle of~$T_X$, or possibly a singular linear
subspace, i.e.\
a closed irreducible analytic subspace such that $\forall x\in X$, 
$V_x:=V\cap T_{X,x}$ is linear.
\pause
\begin{block}{Definition (Category of directed manifolds)}
 -- \alert{Objects} : pairs $(X,V)$, $X$ manifold/$\bC$ and 
    \alert{$V\subset T_X$}\\
 -- \alert{Morphisms} $\psi:(X,V)\to(Y,W)$ holomorphic 
    s.t.~\alert{$\psi_*V\subset W$\kern-20pt}\\
 \pause
 -- \alert{``Absolute case''} $(X,T_X)$, i.e.\ $V=T_X$\\
 -- \alert{``Relative case''} $(X,T_{X/S})$ where $X\to S$\\
 -- \alert{``Integrable case''} when $[\cO(V),\cO(V)]\subset \cO(V)$ (foliations)
\end{block}\pause%

\begin{block}{Canonical sheaf of a directed manifold $(X,V)$}
When $V$ is nonsingular, i.e.\ a subbundle, one simply sets\vskip4pt
\centerline{\alert{$K_V=\det(V^*)$}\quad\hbox{(as a line bundle)}.}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Canonical sheaf of a singular pair (X,V)}
When $V$ is singular, we first introduce the rank $1$ sheaf \alert{${}^b\cK_V$}
of sections of $\det V^*$ that are \alert{locally bounded} with respect 
to a smooth ambient metric on~$T_X$.\pause\ One can show that ${}^b\cK_V$ is
equal to the integral closure of the image of the natural morphism
\vskip6pt
\centerline{$\displaystyle
\cO(\Lambda^rT_X^*)\to \cO(\Lambda^r V^*)\to \cL_V:={}$invert.~sheaf
$\displaystyle\cO(\Lambda^r V^*)^{**}$}\vskip6pt
that is, if the image is $\cL_V\otimes\cJ_V$, $\cJ_V\subset\cO_X$,
\alert{$${}^b\cK_V=\cL_V\otimes\overline{\cJ}_V,~~~~
\overline{\cJ}_V=\hbox{integral closure of}~\cJ_V.$$}\vskip-14pt\pause

\begin{block}{Consequence} If $\mu:\widetilde X\to X$ is a modification and
$\widetilde X$ is equipped with the pull-back directed structure
$\widetilde V=\overline{\tilde\mu^{-1}(V)}$, then\vskip-10pt
$$\alert{{}^b\cK_V\subset\mu_*({}^b\cK_{\widetilde V})\subset \cL_V}$$\vskip-6pt
and $\mu_*({}^b\cK_{\widetilde V})$ increases with $\mu$.
\end{block}
\end{frame}

\begin{frame}
  \frametitle{Canonical sheaf of a singular pair (X,V)~~[sequel]}
By Noetherianity, one can define a sequence of rank $1$ sheaves
\vskip-10pt
$$\cK^{[m]}_V=\lim_{\mu}\uparrow \mu_*({}^b\cK_{\widetilde V})^{\otimes m},~~~
\mu_*({}^b\cK_V)^{\otimes m}\subset\cK^{[m]}_V\subset \cL_V^{\otimes m}$$\vskip-6pt
which we call the \alert{pluricanonical sheaf sequence} of $(X,V)$.\pause

\begin{block}{Remark}The blow-up $\mu$ for which the limit is attained
may depend on~$m$. We do not know if there is a $\mu$ that works for all
$m$.
\end{block}\pause
This generalizes the concept of \alert{reduced singularities} of foliations,
which is known to work in that form only for surfaces.\pause

\begin{block}{Definition} We say that $(X,V)$ is of \alert{general type} if
\alert{the pluricanonical sheaf sequence $\cK_V^{[\bullet]}$ is big}, i.e.\ 
$H^0(X,\cK^{[m]}_V)$ provides
a generic embedding of $X$ for a suitable $m\gg 1$.
\end{block}
\end{frame}

\begin{frame}
  \frametitle{Definition of algebraic differential operators}
  Let~ \alert{$(\bC,T_\bC)\to (X,V),~~~t\mapsto f(t)=(f_1(t),\ldots,f_n(t))$}
  be a curve written in some local holomorphic coordinates 
  $(z_1,\ldots,z_n)$ on~$X$. It has a local Taylor expansion
  $$f(t)=x+t\xi_1+\ldots+t^k\xi_k+O(t^{k+1}),~~~
    \xi_s=\frac{1}{s!}\nabla^sf(0)$$
  for some connection $\nabla$ on $V$.\vskip3pt
  \pause
  One considers the \alert{Green-Griffiths bundle $E^{\rm GG}_{k,m}V^*$}
  of polynomials of weighted degree $m$ written locally in coordinate charts
  as
  $$P(x\,;\,\xi_1,\ldots,\xi_k)=\sum
    a_{\alpha_1\alpha_2\ldots\alpha_k}(x)\xi_1^{\alpha_1}\ldots\xi_k^{\alpha_k},~~~
    \xi_s\in V.$$\pause
  One can view them as \alert{algebraic differential operators}
  \begin{eqnarray*}
    P(f_{[k]})&=&P(f',f'',\ldots,f^{(k)}),\\
    P(f_{[k]})(t)&=&\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*}
\end{frame}

\begin{frame}
\frametitle{Definition of algebraic differential operators [cont.]}
\vskip-4pt  
Here $t\mapsto z=f(t)$ is a curve, $f_{[k]}=
(f',f'',\ldots,f^{(k)})$ \alert{its $k$-jet}, and
$a_{\alpha_1\alpha_2\ldots\alpha_k}(z)$ are supposed to holomorphic 
functions on~$X$.\vskip6pt
\pause
The reparametrization action : $f\mapsto f\circ\varphi_\lambda$,
$\varphi_\lambda(t)=\lambda t$, $\lambda\in\bC^*$ yields
\alert{$(f \circ \varphi_\lambda)^{(k)}(t)=\lambda^kf^{(k)}(\lambda t)$},
whence a $\bC^*$-action
$$\lambda\cdot(\xi_1,\xi_2,\ldots,\xi_k)=(\lambda\xi_1,
  \lambda^2\xi_2,\ldots,\lambda^k\xi_k).$$
\vskip3pt\pause
$E^{\rm GG}_{k,m}$ is precisely the set of polynomials of weighted
degree $m$, corresponding to coefficients $a_{\alpha_1\ldots\alpha_k}$ with
$m=|\alpha_1|+2|\alpha_2|+\ldots+k|\alpha_k|$.\vskip 3pt\pause
\begin{block}{Direct image formula}
If $J_k^{\rm nc}V$ is the set of non constant $k$-jets, one defines the
\alert{Green-Griffiths} bundle to be \alert{$X_k^\GG=J_k^{\rm nc}V/\bC^*$}
and \alert{$\cO_{X_k^\GG}(1)$} to be the associated tautological
rank $1$ sheaf.
Then we have\vskip4pt
\centerline{\alert{$\displaystyle
\pi_k:X_k^\GG\to X,\qquad E_{k,m}^\GG V^*=(\pi_k)_*\cO_{X_k^\GG}(m)$}}
\end{block}
\end{frame}

\begin{frame}
  \frametitle{Generalized GGL conjecture, strategy of attack}
  \vskip-7pt
  \begin{block}{Generalized GGL conjecture}
  If $(X,V)$ is directed manifold of
   general type, i.e.\ \alert{$\cK_V^{[\bullet]}$ is big}, then 
  \alert{$\exists Y\subsetneq X$} such that
  $\forall f:(\bC,T_\bC)\to(X,V)$, one has
  \alert{$f(\bC)\subset Y.\kern-20pt$}\end{block}\vskip-4pt\pause

  \begin{block}{Fundamental vanishing theorem}
   \claim{\rm [Green-Griffiths 1979], [Demailly 1995],
    \hbox{[Siu-Yeung 1996]\kern-30pt}}\\
   \alert{$\forall P\in H^0(X,E^\GG_{k,m}V^*\otimes\cO(-A))$} : global diff.\ operator on $X$ ($A$~ample divisor), \alert{$\forall f:(\bC,T_\bC)\to (X,V)$}, one has 
\alert{$P(f_{[k]})\equiv 0.\kern-20pt$}\pause\\
$\Longleftrightarrow$ \hbox{\alert{$f_{[k]}(\bC)\,{\subset}\,\sigma^{-1}(0),~
\forall\sigma\,{\in}\,H^0(X_k^\GG,\cO_{X_k^\GG}(m)\otimes\pi_k^*\cO(-A))$}.
\kern-20pt}
\end{block}\vskip-4pt\pause

\begin{block}{Corollary: exploit base locus of algebraic differential
equations}
 \claim{Exceptional locus:}
\alert{${\rm Exc}(X,V)=
\overline{\bigcup_{f}f(\bC)}^{\rm Zar}$},
\hbox{$f:(\bC,T_\bC)\to(X,V)$,\kern-15pt}\\
\claim{Green-Griffiths locus:}
\alert{${\rm GG}(X,V)=\bigcap_k
\pi_k({\rm GG}_k(X,V))$}, where\\
\alert{$
  {\rm GG}_k(X,V)=\bigcap_{\sigma}\sigma^{-1}(0)$},
$\sigma\in H^0(X_k^\GG,\cO_{X_k^\GG}(m)\otimes\pi_k^*\cO(-A))$.\\
Then~~~\alert{${\rm Exc}(X,V)\subset{\rm GG}(X,V)$}.
\end{block}
\end{frame}

\begin{frame}
  \frametitle{Proof of the fundamental vanishing theorem}
  \vskip-4pt
  \claim{\bf Simple case}.
  First assume that $f$ is a \alert{Brody curve}, i.e.\  that
  \alert{$\sup_{t\in\bC}\Vert f'(t)\Vert_\omega<+\infty$}
  for some hermitian metric $\omega$ on~$X$. By raising $P$ to a power, we can
  assume $A$ very ample, and view $P$ as a $\bC$ valued differential operator
  whose coefficients vanish on a very ample divisor $A$.\vskip4pt\pause
  The Cauchy inequalities
  imply that all derivatives $f^{(s)}$ are bounded in any 
  coordinate chart. Hence
  \alert{$u_A(t):=P(f_{[k]})(t)$ is bounded}, and must be
  \alert{constant by Liouville's theorem}.\vskip4pt\pause
  Since $A$ is very ample, we can move $A\in|A|$ such that $A$ hits 
  $f(\bC)\subset X$. Bu then $u_A$ vanishes somewhere and so
  \alert{$u_A\equiv 0$}.
  \vskip4pt
  \claim{\bf General case of a general entire curve $f:(\bC,T_\bC)\to(X,V)$.}
Instead, one makes use of Nevanlinna theory arguments (logarithmic
derivative lemma).\vskip4pt
\claim{\bf Remark}. Generalized GGL conjecture is easy if
rank$\,V=1$.
\end{frame}
    
\begin{frame}
\frametitle{And now ... the Semple jet bundles}
\vskip-3pt  
\wider[2em]{%
  \begin{itemize}
\item
\claim{{\bf Functor ``1-jet'' :}} $(X,V)\mapsto (\tilde X,\tilde V)$ 
where :\vskip-24pt
\begin{eqnarray*}
&&\strut\kern-60pt\tilde X=P(V)={}\hbox{bundle of projective spaces of lines in $V$}\\
\noalign{\vskip-4pt}      
&&\strut\kern-60pt\pi:\tilde X=P(V)\to X,~~~(x,[v])\mapsto x,~~v\in V_x\\
\noalign{\vskip-4pt}         
 &&\strut\kern-60pt\tilde V_{(x,[v])}=\big\{\xi\in T_{\tilde X,(x,[v])}\,;\;\pi_*\xi\in\bC v \subset T_{X,x}\big\}
\end{eqnarray*}\strut\pause\vskip-38pt\strut
\item
\claim{\bf For
every entire curve $f:(\bC,T_\bC)\to(X,V)$ tangent to $V$}\vskip4pt
$f$ lifts as $\bigg\{$\vskip-1.7cm
\begin{eqnarray*}
&&\strut\kern-8pt f_{[1]}(t):=(f(t),[f'(t)])\in P(V_{f(t)})\subset \tilde X\\
\noalign{\vskip-4pt}
&&\strut\kern-8pt f_{[1]}:(\bC,T_\bC)\to(\tilde X,\tilde V)~~
\hbox{\alert{(projectivized 1st-jet)}}
\end{eqnarray*}\vskip-24pt\pause\strut
\item
\claim{{\bf Definition.}} {\it Semple jet bundles :}
\vskip3pt
-- $(X_k,V_k)=k$-th iteration of functor 
   $(X,V)\mapsto(\tilde X,\tilde V)$\\
-- $f_{[k]}:(\bC,T_\bC)\to(X_k,V_k)$ is the 
\alert{projectivized $k$-jet of $f$.}
\vskip5pt
\pause
\item
\claim{{\bf Basic exact sequences}}\vskip-25pt
\begin{eqnarray*}
&&\strut\kern-60pt 0\to T_{X_k/X_{k-1}}\to V_k\build\to^{(\pi_k)_\star}_{}\cO_{X_k}(-1)
\to 0\alert{~~~{}\Rightarrow \rank V_k=r}\\
\noalign{\vskip-4pt}      
&&\strut\kern-60pt 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 for Semple bundles}
\vskip-4pt
For $n=\dim X$ and $r=\rank V$,
one gets a \alert{tower of $\bP^{r-1}$-bundles}\vskip3pt
\centerline{$\pi_{k,0}:X_k\build\to^{\pi_k}_{}X_{k-1}\to\cdots\to X_1
      \build\to^{\pi_1}_{}X_0=X$}\vskip3pt
with \alert{$\dim X_k=n+k(r-1)$, $\rank V_k=r$},\\ 
and \alert{tautological line bundles $\cO_{X_k}(1)$ on $X_k=P(V_{k-1})$}.
\pause
\begin{block}{Theorem} $X_k$ is a smooth compactification of \alert{%
$X_k^{\GG,\reg}/\bG_k=J_k^{\GG,\reg}/\bG_k$}, where $\bG_k$ 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
\end{block}
  
\begin{block}{Direct image formula for invariant differential operators}
\alert{$E_{k,m}V^*:=(\pi_{k,0})_*\cO_{X_k}(m)={}$}
sheaf of algebraic differential operators 
$f\mapsto P(f_{[k]})$ acting on germs of curves $f:(\bC,T_\bC)\to (X,V)$
such that \alert{$P((f\circ\varphi)_{[k]})=
\varphi^{\prime m}P(f_{[k]})\circ\varphi$}.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Strategy of proof of the Kobayashi conjecture
(Brotbek, simplified by D.)}
\vskip-3pt
Let $\pi:\cX\to S$ be family of smooth projective varieties, and let
$\cX_k\to S$ be the \alert{relative Semple tower} of $(\cX,T_{\cX/S})$.\\
If $X_t=\pi^{-1}(t)$, $t\in S$, is the general fiber, then the fiber of
$\cX_k\to S$ is the $k$-stage of the Semple tower $X_{t,k}\to X_t$\pause\\
(the idea is to consider the universal family of hypersurfaces
$X\subset \bP^{n+1}$ of sufficiently high degree $d\gg 1$.)\pause\\

\begin{block}{Basic observation}
Assume that there exists $t_0\in S$ such that we get on
$X_{t_0,k}$ a \alert{nef} ``twisted tautological sheaf''
$\cG_{|X_{t_0,k}}$ where\vskip4pt
\alert{\centerline{$
\cG:=\cO_{\cX_k}(m)\otimes\cI_{k,m}\otimes\pi_{k,0}A^{-1}
$}}\vskip4pt
(in the sense that a log resolution of $\cG$ is nef), and $\cI_{k,m}$
is a suitable ``functorial'' multiplier ideal with
support in the set $\cX_k^{\sing}$ of singular jets.
Then $X_t$ is Kobayashi hyperbolic for general $t\in S$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Simplified proof of the Kobayashi conjecture}
\vskip-3pt
\claim{\bf Proof.} By hypothesis, One can take a resolution
$\mu_{k,m}:\widehat\cX_k\to\cX_k$ of
the ideal $\cI_{k,m}$ as an invertible sheaf $\mu_{k,m}^*\cI_{k,m}$
on $\widehat\cX_{k,m}$, so that
\alert{$\mu_{k,m}^*\cG_{|\widehat X_{t_0,k}}$ is a nef line bundle}.\pause\vskip4pt
Then one can add a small $\bQ$-divisor $\cP_\varepsilon$
that is a
combination of the lower stages $\cO_{\cX_\ell}(m')$, $\ell<k$, and of
the exceptional divisor of $\mu_{k,m}$ so that
\alert{$(\mu_{k,m}^*\cG\otimes\cP_\varepsilon)_{|\widehat X_{t_0,k}}$ is an
ample line bundle}.
\vskip3pt\pause
Since ampleness is a Zariski open property, one concludes that
\alert{$(\mu_{k,m}^*\cG\otimes G_\varepsilon)_{|\widehat X_{t,k}}$} is ample
for general $t\in S$. The fundamental vanishing theorem then implies
that $X_t$ is Kobayashi hyperbolic.~$\square$\vskip4pt\pause
The next idea is to produce a very particular hypersurface $X_{t_0}$
on which there are a lot of non trivial Wronskian operators that
generate the required sheaf\vskip4pt
\centerline{\alert{
$\cG=\cO_{\cX_k}(m)\otimes\cI_{k,m}\otimes\pi_{k,0}A^{-1}$}.}\vskip4pt
Then $\cG_{|X_{k,t_0}}$  is nef and we are done.
\end{frame}

\begin{frame}
\frametitle{Wronskian operators}
\vskip-3pt
Let $L\to X$ be a line bundle, and let\vskip5pt
\centerline{\alert{$s_0,\ldots,s_k\in H^0(X,L)$}}\vskip5pt
be arbitrary sections. One defines Wronskian operators acting
on $f:\bC\to X$, $t\mapsto f(t)$ by $D={d\over dt}$ and
\alert{$$
W(s_0,\ldots,s_k)(f)=\left|
\plainmatrix{
  s_0(f) & s_1(f) & \ldots &s_k(f)\cr
  D(s_0(f)) & D(s_1(f)) & \ldots &D(s_k(f))\cr
  \vdots &  &  &\vdots\cr
  D^k(s_0(f)) & D^k(s_1(f)) & \ldots &D^k(s_k(f))\cr}\right|
$$}\pause
This actually does not depend on the trivialization of $L$ and defines
\vskip4pt
\alert{\centerline{$\displaystyle
W(s_0,\ldots,s_k)\in H^0(X,E_{k,k'}T^*_X\otimes L^{k+1}),~~
k'={k(k+1)\over 2}.
$}}\strut\pause\vskip-4pt
\claim{\bf Problem.} One has to take $L>0$, hence $L^{k+1}>0$~:
seems useless!
\end{frame}

\begin{frame}
\frametitle{Wronskian operators can sometimes be divided~!}
\vskip-3pt
Take e.g.\ $X=\bP^N$, $A=\cO(1)$ very ample, $k\leq N$, $d\geq k$ and
$$s_j(z)=z_j^dq_j(z),~\deg q_j=k~~\Longrightarrow~~
s_j\in H^0(X,A^{d+k}).
$$
{\pause}Then derivatives $D^\ell(s_j\circ f)$ are divisible by $z_j^{d-k}$
for $\ell\leq k$,\pause\\
and (taking $L=A^{d+k}$) we find
\alert{$$\eqalign{
\prod_{0\leq j\leq k}z_j^{-(d-k)}W(s_0,\ldots s_k)
&\in H^0(X,E_{k,k'}T^*_X\otimes A^{(d+k)(k+1)-(d-k)(k+1)})\cr
\noalign{\vskip-11pt}
&=H^0(X,E_{k,k'}T^*_X\otimes A^{2k(k+1)}).\cr}
$$}\pause
Not enough, but the exponent is independent of $d$ and a division
by one more factor $z_j^{d-k}$ would suffice to reach $A^{<0}$,
\hbox{for $d\gg k$.\kern15pt}\pause\vskip4pt
If we take the \claim{Fermat hypersurface $X=\{z_0^d+\ldots+z_N^d=0\}$} and
$k=N-1$, $q_1=\ldots=q_k=q$, then $z_0^d=-\sum_{i>0}z_i^d$ implies that
$W(s_0,\ldots,s_k)=(-1)^kW(s_N,s_1,\ldots,s_k)$ is also divisible
\hbox{by $z_N^{d-k}$,\pause\ so\kern-15pt}
\alert{$$P:=
\prod_{0\leq i\leq k+1}z_i^{-(d-k)}W(s_0,\ldots s_k)
\in H^0(X,E_{k,k'}T^*_X\otimes A^{k(2k+3)-d}).
$$}
\end{frame}

\begin{frame}
\frametitle{Getting more jet differentials from Wronskians}
\vskip-5pt  
A better choice than the Fermat hypersurface is to take
$X=\sigma^{-1}(0)\subset\bP^{n+1}$ with $\sigma\in H^0(\bP^{n+1},\cO(d))$
given by\vskip4pt
\alert{\centerline{$\displaystyle
\sigma\,{}=\!\sum_{0\leq i\leq N}a_i(z)m_i(z)^\delta$,~$a_i\,$``random'',
$\deg a_i\,{=}\,\rho\,{\geq}\,k$,
$\displaystyle m_i(z)\,{=}\prod_{J\ni i}\!\tau_J(z)$,}}\vskip4pt
where the $J$'s run over all subsets $J\subset\{0,1,\ldots,N\}$ with
\hbox{$\card J=n$,\kern-15pt}\\
$\tau_J\in H^0(\bP^{n+1},\cO(1))$ is a sufficiently
general linear section and \hbox{$\delta\gg 1$.\kern-15pt}\vskip4pt\pause
An adequate choice to ensure \claim{smoothness of $X$} is
\alert{$N=n(n+1)$}.\pause\\
Then, for $k\,{\geq}\,N$ and all $J\,{\subset}\,\{0,1,...,N\}$,
$\card J\,{=}\,n$, the \hbox{Wronskians\kern-15pt}
\vskip4pt
\alert{\centerline{$\displaystyle
W_{q,\widehat\tau,k,J}=W(q_1\widehat\tau_1^{\,d-k},...,
q_r\widehat\tau_r^{\,d-k},(a_im_i^\delta)_{i\in\complement J}),~~r=k-N+n$}}
\vskip4pt
with $\deg q_j\,{=}\,k$ are divisible by
$(\widehat\tau_j^{\,d-2k})_{1\leq j\leq n}$ and
$(m_i^{\delta-k})_{i\in\complement J}~~\Rightarrow$
\vskip5pt
\alert{\centerline{
$P_{q,\widehat\tau,k,J}:=\prod_{i\in\complement J}
m_i^{-(\delta-k)}\prod_j\widehat\tau_j^{\,d-2k}W_{k,r}\in
H^0(X,E_{k,k'}T^*_X\otimes A^{c_n})
$}}\vskip5pt
where $c_n\,{=}\,k(k{+}1)\deg m_j\,{=}\,O((en)^{n+5/2})$.\pause\
As $a_im_i^\delta\,{=}\,-\sum_{j\neq i}a_j m_j^\delta$\\ on~$X$, we infer the
divisibility of $P_{q,\widehat\tau,k,J}$ by the extra factor
$\tau_J^{\delta-k}$.
\end{frame}

\begin{frame}
\frametitle{Conclusion: analyzing base loci of Wronskians}
\vskip-7pt  
We need \alert{$\delta>k+c_n$} to reach a negative exponent $A^{<0}$
\vskip5pt
\centerline{$\Rightarrow\alert{d\geq d_n=O((en)^{2n+2})}.$}\pause
\begin{block}{A Bertini type lemma}
For $k\geq n^3+n^2+1$, the $k$-jets of the coefficients
$a_j$ are general enough, the simplified Wronskians
${\widetilde P}_{q,\widehat\tau,k,J}$ \alert{generate the universal Wronskian
ideal} $\cI_{k,k'}$ outside of the hyperplane sections $\tau_J^{-1}(0)$.
\end{block}
The proof is achieved by induction on $\dim X$, taking
\hbox{$X'=\tau_J^{-1}(0).~\square$\kern-20pt}
\vskip5pt\pause
To generalize further, one needs stronger existence theorems
\hbox{for jets.\kern-15pt}

\begin{block}{General existence theorem for jet differentials (D-, 2010)}
  Let $(X,V)$ be of general type, such that
  ${}^b\cK_V^{\otimes p}$ is a \alert{big} rank $1$ sheaf. Then
  \alert{$\exists$ many global sections $P$, $m{\gg}k{\gg}1$} $\Rightarrow$
  \alert{$\exists$~alg.\ hypersurface $Z\subsetneq X_k^\GG$} s.t.\ all
  entire $f:(\bC,T_\bC)\mapsto(X,V)$ satisfy
  \hbox{\alert{$f_{[k]}(\bC)\subset Z$}.}
\end{block}
\end{frame}


\begin{frame}
\frametitle{1${}^{\rm st}$ step: take a Finsler metric on $k$-jet bundles}
\vskip-7pt
 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-16pt}\vskip-21pt%
$$\alert{
\Psi_{h_k}(f):=\Big(\sum_{1\le s\le k}\varepsilon_s\Vert\nabla^sf(0)
\Vert_{h(x)}^{2p/s}\Big)^{1/p},~~1=\varepsilon_1\gg
\varepsilon_2\gg\cdots\gg\varepsilon_k.}
$$\vskip-7pt\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-21pt
\alert{$$
\Theta_{L_k,h_k}=\omega_{{\rm FS},k}(\xi)+{i\over 2\pi}
\sum_{1\le s\le k}{1\over s}{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}}
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}
{\xi_{s\alpha}\overline\xi_{s\beta}\over|\xi_s|^2}\,dz_i\wedge d\overline z_j
$$}\vskip-13pt%
where \alert{$(c_{ij\alpha\beta})$} are the coefficients of the curvature tensor
\hbox{\alert{$\Theta_{V^*,h^*}$}~and\kern-10pt} 
\alert{$\omega_{{\rm FS},k}$ is the vertical Fubini-Study metric}
on the fibers of \hbox{$X_k^\GG\to X$.\kern-10pt}\vskip3pt\pause
The expression gets simpler by using polar coordinates\vskip3pt
\centerline{\alert{$x_s=\vert\xi_s\vert_h^{2p/s}$,~~~
$u_s=\xi_s/\vert\xi_s\vert_h=\nabla^sf(0)/\vert\nabla^sf(0)\vert$.}}
\end{frame}

\begin{frame}
\frametitle{\strut\kern-10pt
2${}^{\rm nd}$ step: probabilistic interpretation of the
curvature}
\vskip-6pt  
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 $SV\subset V$.\pause\\
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{3${}^{\rm rd}$ step: getting the main cohomology estimates}
\vskip-5pt  
$\Rightarrow$ the leading term only involves the trace of $\Theta_{V^*,h^*}$, 
i.e.\ the
curvature of $(\det V^*,\det h^*)$, that can be taken${}>0$
if $\det V^*$ is big.
\begin{block}{Corollary of holomorphic Morse inequalities (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 upper and lower bounds\
[$q=0$ most useful!]\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
$\displaystyle
h^q(X_k^\GG,\cO(L_k^{\otimes m}))\ge{m^{n+kr-1}\over (n{+}kr{-}1)!}
{(\log k)^n\over n!\,(k!)^r}\bigg(
\int_{X(\eta,q,\,q\pm 1)}\kern-25pt(-1)^q\eta^n-\frac{C}{\log k}\bigg).\kern-20pt$}\end{block}
\end{frame}

\begin{frame}
\frametitle{Induced directed structure on a subvariety}
Let $Z$ be an irreducible algebraic subset of some Semple $k$-jet bundle
$X_k$ over~$X$ ($k$ arbitrary).\vskip3pt\pause
We define an induced directed structure $(Z,W)\hookrightarrow(X_k,V_k)$
by taking the linear subspace $W\subset T_Z\subset T_{X_k|Z}$ to 
be the closure of 
$T_{Z'}\cap V_k$ taken on a suitable Zariski open set 
$Z'\subset Z_{\rm reg}$ where the intersection has constant rank and 
is a subbundle of $T_{Z'}$.
\vskip3pt\pause
Alternatively, one could also take 
$W$ to be the closure of $T_{Z'}\cap V_k$ in the $k$-th stage
$(\cX_k,\cA_k)$ of the ``absolute Semple tower'' associated
with $(\cX_0,\cA_0)=(X,T_X)$ (so as to deal only with nonsingular
ambient Semple bundles).\\
\vskip3pt\pause
This produces an \alert{induced directed subvariety}\vskip-10pt
$$\alert{(Z,W)\subset(X_k,V_k)}.$$\vskip-2pt
It is easy to show that \alert{$\pi_{k,k-1}(Z)=X_{k-1}\Rightarrow
\rank W<\rank V_k=\rank V$}.
\end{frame}

\begin{frame}
\frametitle{Sufficient criterion for the GGL conjecture}
\vskip-5pt
\begin{block}{Definition} Let $(X,V)$ be a directed pair where
$X$ is projective algebraic. We say that \alert{$(X,V)$ is ``strongly of
general type''} if it is of general type and 
for every irreducible alg.\ subvariety $Z\subsetneq X_k$ that 
projects onto~$X$, $X_k\not\subset D_k:=P(T_{X_{k-1}/X_{k-2}})$, 
the induced directed structure $(Z,W)\subset(X_k,V_k)$ is 
of \alert{general type modulo $X_k\to X$},
i.e.\ \alert{${}^b\cK_W\otimes \cO_{X_k}(m)_{|Z}$ is big} for some $m\in\bQ_+$,
after a suitable \hbox{blow-up.\kern-15pt}
\end{block}\vskip-4pt
\pause
\begin{block}{Theorem (D-, 2014)} If \alert{$(X,V)$ is 
strongly of general type},
\alert{the Green-Griffiths-Lang conjecture holds true} for $(X,V)$, namely
there \alert{$\exists Y\subsetneq X$} such that
every non constant holomorphic curve $f:(\bC,T_{\bC})\to (X,V)$
satisfies \alert{$f(\bC)\subset Y$}.\end{block}
\pause
{\bf Proof:} Induction on rank$\,V$, using existence of jet differentials.
\end{frame}

\begin{frame}
\frametitle{Related stability property}
\vskip-5pt
\begin{block}{Definition} 
Fix an ample divisor $A$ on~$X$. For every 
irreducible subvariety $Z\subset X_k$ that projects onto $X_{k-1}$ for
$k\ge 1$, $Z\not\subset D_k$, and $Z=X=X_0$ for $k=0$, we define 
the \alert{slope} of the corresponding directed variety $(Z,W)$ to be
\alert{$\mu_A(Z,W)={}$}\vskip-12pt
\alert{$$
{\inf\big\{\lambda\in\bQ\,;\;\exists m\in\bQ_+,\;
{}^b\cK_W{\otimes}\big(\cO_{X_k}(m){\otimes}\pi_{k,0}^*\cO(\lambda A)
\big)_{|Z}~\hbox{big on $Z$}\big\}\over\rank W}.
$$}\vskip-10pt\pause%%
Notice that \alert{$(X,V)$ is 
of general type iff $\mu_A(X,V)<0$}.
\vskip3pt\pause
We say that $(X,V)$ is \alert{$A$-jet-stable} (resp.\ 
\alert{$A$-jet-semi-stable})
if \alert{$\mu_A(Z,W)<\mu_A(X,V)$} (resp.\ 
\alert{$\mu_A(Z,W)\le\mu_A(X,V)$}) for all $Z\subsetneq X_k$ 
as above.\end{block}
\pause
\claim{{\bf Observation.}} If $(X,V)$ is of general type and 
$A$-jet-semi-stable, then $(X,V)$ is strongly of general type. 
\end{frame}

\begin{frame}
\frametitle{Criterion for the generalized Kobayashi conjecture}
\vskip-5pt
\begin{block}{Definition} Let $(X,V)$ be a directed pair where
$X$ is projective algebraic. We say that \alert{$(X,V)$ is ``algebraically
jet-hyperbolic''} if for every irreducible 
alg.\ subvariety $Z\subsetneq X_k$ s.t.\ $X_k\not\subset D_k$,
the induced directed structure $(Z,W)\subset(X_k,V_k)$ either has $W=0$ or
is of \alert{general type modulo $X_k\to X$}.
\end{block}\vskip-4pt
\pause
\begin{block}{Theorem (D-, 2014)} If $(X,V)$ is \alert{algebraically
jet-hyperbolic}, then $(X,V)$ is \alert{Kobayashi (or Brody) hyperbolic}, 
i.e.\ there are no entire curves $f:(\bC,T_{\bC})\to (X,V)$.
\end{block}
\pause
Now, the hope is that a (very) generic complete intersection 
$X=H_1\cap\ldots\cap H_c\subset \bP^{n+c}$
of codimension $c$ and degrees~$(d_1,...,d_c)$ s.t.\ 
\alert{$\sum d_j\ge 2n+c$} yields $(X,T_X)$ algebraically jet-hyperbolic.
\end{frame}

\begin{frame}
\frametitle{Invariance of ``directed plurigenera'' ?}
One way to check the above property, at least with non optimal bounds,
would be to show some sort of Zariski openness of the properties
\alert{``strongly of general type''} or 
\alert{``algebraically jet-hyperbolic''}.\pause\
One would need e.g.\ to know the answer to
\begin{block}{Question} Let $(\cX,\cV)\to S$ be a proper family of directed
varieties over a base~$S$, such that $\pi:\cX\to S$ is a nonsingular 
deformation and the directed structure 
on $X_t=\pi^{-1}(t)$ is $V_t\subset T_{X_t}$, possibly singular.
Under which conditions is\vskip-8pt
$$\alert{t\mapsto h^0(X_t,\cK_{V_t}^{[m]})}$$\vskip-4pt
locally constant over $S$~?
\end{block}
This would be very useful since one can easily produce jet sections for
hypersurfaces $X\subset\bP^{n+1}$ admitting meromorphic connections with
low pole order (Siu, Nadel).
\end{frame}

\begin{frame}
\frametitle{The end}
\strut\vskip3mm
\pgfdeclareimage[height=6.5cm]{CalabiYau}{CalabiYau}
\strut\kern2cm\pgfuseimage{CalabiYau}
\end{frame}

\end{document}