% 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-190pt\rlap{\blank{J.-P.\ Demailly (Grenoble), Math.\ Institute of CAS, 14 Dec 2017}}\kern181pt\rlap{\blank{
Kobayashi conjecture on generic hyperbolicity}}\kern181pt
\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Kobayashi conjecture on the\\
generic hyperbolicity of algebraic\\
hypersurfaces in projective space}

%% \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 Seminar Talk at the Academy of Mathematics\\
and Systems Science, Chinese Academy of Sciences\\
\vskip7pt Beijing, December 14, 2017}

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% mathematical operators
\renewcommand{\Re}{\mathop{\rm Re}\nolimits}
\renewcommand{\Im}{\mathop{\rm Im}\nolimits}
\newcommand{\Pic}{\mathop{\rm Pic}\nolimits}
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\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}
  \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}{Definition} A complex space $X$ is said to be
  \alert{Kobayashi hyperbolic} if the Kobayashi pseudodistance 
  $d_{\rm Kob}:X\times X\to\bR_+$ is a distance\\
  (i.e.\ everywhere non degenerate).\pause
 \end{block}
  By an \alert{entire curve} we mean a non constant holomorphic map 
  $f:\bC\to X$ into a complex $n$-dimensional manifold.\vskip-3pt\pause

 \begin{block}{Theorem {\rm(Brody, 1978)}} For a \alert{compact} complex 
 manifold $X$, $dim_{\bC}X=n$, TFAE:\\
 (i) $X$ is \alert{Kobayashi hyperbolic}\\
 (ii) $X$ is \alert{Brody hyperbolic}, i.e. $\not\!\exists$ 
    entire curves $f:\bC\to X$\\
 (iii) The Kobayashi \alert{infinitesimal pseudometric} $\bfk_x$ is everywhere 
  non ${}\kern7mm$degenerate
 \end{block}\vskip-3pt
 \pause
 Our interest is the study of hyperbolicity for \hbox{\alert{projective
varieties}.\kern-15pt}\\ In dim $n=1$, $X$ is hyperbolic iff genus $g\ge 2$.
\end{frame}

\begin{frame}
\frametitle{Main conjectures}\vskip-9pt
\begin{block}{Conjecture of General Type (CGT)} 
$\bullet$ A compact complex  variety $X$ is \alert{volume hyperbolic} 
$\Longleftrightarrow$ $X$ is of 
\alert{\phantom{$\bullet$~}general type}, i.e.\ $K_X$ is big~
[implication $\Longleftarrow$ is well known].\pause\\
$\bullet$ $X$ \alert{Kobayashi (or Brody) 
hyperbolic} should imply \alert{$K_X$ ample}.
\end{block}\vskip-7pt
\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-7pt
\pause
\begin{block}{Arithmetic counterpart (Lang 1987) -- very optimistic ?} 
If $X$ is projective and defined over a number field $\bK_0$, the smallest 
locus $Y={\rm GGL}(X)$ in GGL's conjecture is also the \alert{smallest $Y$} 
such that \alert{$X(\bK)\smallsetminus Y$ is finite} $\forall\bK$
number field${}\supset\bK_0$.
\end{block}\vskip-7pt
\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 a result of M.\ Zaidenberg, the optimal bound must satisfy
\alert{$d_n\ge 2n+1$}, and one expects \alert{$d_n=2n+1$}.
\end{block}\pause
Using ``jet technology'' and \alert{deep results of McQuillan} for 
curve foliations on surfaces, the following has been proved:
\begin{block}{Theorem (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}
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}
By a combination of an algebraic existence theorem for jet differentials
and of Y.T.~Siu's technique of ``slanted vector fields'' (itself~derived from
ideas of H.~Clemens, L.~Ein and C.~Voisin), the following was proved:

\begin{block}{Theorem (S.~Diverio, J.~Merker, 
E.~Rousseau, 2009)} 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 $d_n=O(\exp(n^{1+\varepsilon}))\,:$
\centerline{%
$\alert{d_n=\left\lfloor{n^4\over 3}\big(n\log(n\log(24n))\big)^n\right\rfloor}
\quad\hbox{\claim{(D-, 2012)},}$}
\centerline{%
$\alert{d_n=(5n)^2\,n^n}\kern1.9cm\hbox{\claim{(Darondeau, 2015)}.}$}
\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}
In 2016, Brotbek gave a shorter and more geometric proof of Y.T. Siu's
result on the Kobayashi conjecture, using again jet \hbox{techniques.\kern-15pt}
\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{$d\gg 1$ large},
the hypersurface $X_\sigma=\sigma^{-1}(0)$ is \alert{hyperbolic}.
\end{block}
\pause%
The initial proof of Brotbek did not provide effective bounds. Through
various improvements, Deng Ya showed in his PhD thesis:
\begin{block}{Theorem (Y. Deng, May 2016)}
In the above setting, a generic hypersurface $X_\sigma=\sigma^{-1}(0)$
is hyperbolic as soon as\vskip3pt
\centerline{$\alert{d\geq
d_n=(n+1)^{n+2}(n+2)^{2n+7}=O(n^{3n+9})}.$}
\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
$T^*_{X_\sigma}$.\\
In particular, such a generic complete intersection is hyperbolic.
\end{block}
\pause%
Xie got the sufficient lower bound $d_j\ge d_{n,c}=N^{N^2}$, $N=n+c$.\pause
\vskip5pt

In his PhD thesis, Y.\ 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{Arrows} $\psi:(X,V)\to(Y,W)$ 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~$T_X$. One can show that ${}^b\cK_V$ is
equal to the integral closure of the image of the natural morphism
$$\cO(\Lambda^rT_X^*)\to \cO(\Lambda^r V^*)\to \cL_V:={\rm invert.~sheaf}~
\cO(\Lambda^r V^*)^{**}$$
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-12pt\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)~~[cont.]}
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.

\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 \claim{Green-Griffiths bundle $E^{\rm GG}_{k,m}V^*$}
  of polynomials of weighted degree $m$ written locally in coordinate charts
  as
  $$P(x\,;\,\xi_1,\ldots,\xi_k)=\sum
    a_{\alpha_1\alpha_2\ldots\alpha_k}(x)\xi_1^{\alpha_1}\ldots\xi_k^{\alpha_k},~~~
    \xi_s\in V,$$
  also viewed as \alert{algebraic differential operators}
  \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*}
\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
$$\lambda\cdot(\xi_1,\xi_1,\ldots,\xi_k)=(\lambda\xi_1,
  \lambda^2\xi_2,\ldots,\lambda^k\xi_k).$$
\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}
  \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}\pause

  \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 \hbox{$(V^*,h^*)$ big.\kern-15pt}\pause
  \begin{block}{Strategy : 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{$P(f_{[k]})\equiv 0.\kern-20pt$}\end{block}\vskip-4pt
\pause
\begin{block}{Theorem on existence of 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{And now ... the Semple jet bundles}
\vskip-3pt  
\wider[2em]{%
  \begin{itemize}
\item
\claim{{\bf Fonctor ``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*}\pause\vskip-36pt\strut
\item
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 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 $f$.}
\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 \rk 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 $r=\rk V$,
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{$\dim X_k=n+k(r-1)$, $\rk V_k=r$},\\ 
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{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{$\pi_{k,k-1}(Z)=X_{k-1}\Rightarrow
\rk W<\rk V_k=\rk V$}.
\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{Proof of the non optimal Kobayashi conjecture (Brotbek)}
\vskip-7pt
Let $A\to Z$ be a very ample line bundle, and $X_\sigma$ the hypersurface
associated with $\sigma\,{\in}\, H^0(Z,dA)$, $d\gg 1$. One looks at
special \hbox{sections\kern-15pt}
\alert{$$
\sigma=\smash{\textstyle\sum_{|I|=\delta}}\;a_I\tau^{(p+k)I},~~
a_I\in H^0(Z,\eta A),~\tau_j\in H^0(Z,A),~1\le j\le N
$$}%%
where the $\tau_j$ are generic and $d=\ell+(p+k)\delta$, $p\gg 1$.
Let $\cX\to S$ be the corresponding family of hypersurfaces,
$\sigma\in S$, and let
$\cX_k\to S$ be the Semple construction relative to $\cX\to S$.\pause\\
A construction similar to Nadel's meromorphic connections with low pole orders then produces certain Wronskian operators, and one shows that for generic $\sigma$, a certain fonctorial blow-up $\mu_k:\widehat\cX_k\to\cX_k$ of the $k$-th stage carries an ample invertible sheaf
\alert{$$
L_k=\mu_k^*\big(\cO_{\cX_k}(a_1,a_2,\ldots,a_k)\otimes\cJ_k\otimes\pi_{k,0}A^{-1}\big)~~\hbox{over $X_\sigma$},
$$}%%
where $\cJ_k$ is the ideal sheaf associated with a suitable family of
\alert{Wronkian operators}. This is enough to prove the conjecture.\qed

\end{frame}

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

\end{document}

    \alert{$E_{k,m}V^*\subset E^{\rm GG}_{k,m}V^*$ is the bundle of
    $\bG_k$-``invariant'' operators}, i.e.\ such that
    $$P((f\circ\varphi)_{[k]})=\varphi^{\prime m}P(f_{[k]})\circ\varphi,~~~
      \forall\varphi\in\bG_k.$$