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% - Style is ornate.
% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
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% In principle, this file can be redistributed and/or modified under
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%
% However, this file is supposed to be a template to be modified
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\title[\ \kern-190pt\rlap{\blank{J.-P.\ Demailly (Grenoble), 16th Takagi Lectures, Tokyo}}\kern181pt\rlap{\blank{On the Kobayashi and Green-Griffiths-Lang conjectures}}\kern181pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Recent progress towards\\
the Kobayashi and\\
Green-Griffiths-Lang conjectures}

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

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

\institute[]{Institut Fourier, Université de Grenoble Alpes\ \ \&\ \ Académie des Sciences de Paris}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[]% (optional)
{November 28-29, 2015\\
16th Takagi Lectures, University of Tokyo}

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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}
%%}


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  \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 pseudodistance and infinitesimal metric}
Let $X$ be a complex space. Given two points $p,q\in X$, 
consider a {\em chain  of analytic disks} from $p$ to $q$, i.e.\
holomorphic maps 
$$f_j:\Delta:=D(0,1)\to X~~\hbox{and points}~~
a_j,b_j\in\Delta,~~0\le j\le k~~\hbox{with}
$$\vskip-33pt
$$
p=f_0(a_0),~q=f_k(b_k),\quad f_j(b_j)=f_{j+1}(a_{j+1}),\quad 
0\le j\le k-1.
$$\pause
One defines the \alert{\em Kobayashi pseudodistance} $d_{\rm Kob}$ on $X$ to be
$$
d_{\rm Kob}(p,q)=\inf_{\{f_j,a_j,b_j\}}d_\Poin(a_1,b_1)+\cdots+d_\Poin(a_k,b_k).
$$
\pause
The \alert{\em Kobayashi-Royden infinitesimal pseudometric} on $X$ is the
Finsler pseudometric\vskip-19pt
$$
\bfk_x(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\Delta\to X,\,f(0)=x,\,
\lambda f'(0)=\xi\big\},~\xi\in T_{X,x}.
$$\vskip-5pt
The integrated pseudometric is precisely $d_{\rm Kob}$.
\end{frame}

\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 $1$, $X$ is hyperbolic iff genus $g\ge 2$.
\end{frame}

\begin{frame}
\frametitle{Kobayashi-Eisenman measures}
In a similar way, one can introduce the \alert{$p$-dimensional 
Kobayashi-Eisenman} infinitesimal metric on decomposable tensors
$\xi=\xi_1\wedge\ldots\wedge\xi_p$ of $\Lambda^pT_{X,x}$ (i.e.\ on 
the tautological line bundle over the Grassmann bundle Gr$(T_X,p)$) by
$$
\bfe^p_x(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\bB^p\to X,\,f(0)=x,\,
\lambda f'(0)\cdot\tau=\xi\big\},
$$
where $\bB^p\subset\bC^p$ is the unit ball and $\tau={\partial\over \partial
t_1}\wedge\ldots\wedge{\partial\over\partial t_p}$.\pause

 \begin{block}{Definition} A complex space $X$ is said to be
  \alert{$p$-measure hyperbolic} in the sense of Kobayashi-Eisenman
   if $\bfe^p$ is non degenerate on a dense Zariski open set.
 \end{block}

\alert{Volume hyperbolicity} refers to the case $p=n=\dim X$.
\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{Solution of the Bloch conjecture}\vskip-4pt
The following has been proved by Ochiai 77, Noguchi 77, 81, 84, Kawamata 80
in the algebraic situation.

\begin{block}{Theorem (Ochiai 77, Noguchi 77,81,84, Kawamata 80)}
Let $Z=\bC^n/\Lambda$ be an abelian variety (resp.\ a complex torus). Then the
(analytic) Zariski closure \alert{$\overline{f(\bC)}^{\rm Zar}$} of the image of 
every entire curve $f:\bC\to Z$ is the \alert{translate of a subtorus}.
\end{block}
\pause\vskip-6pt

\begin{block}{Corollary 1}
Let $X$ be a complex analytic subvariety of a complex torus $Z$ .
Assume that $X$ is of \alert{general type}. Then every 
entire curve drawn in $X$ is \alert{analytically degenerate}.
\end{block}
\pause\vskip-6pt

\begin{block}{Corollary 2}
Let $X$ be a complex analytic subvariety of a complex torus $Z$ .
Assume that $X$ \alert{does not contain any translate of a positive dimensional 
subtorus}. Then $X$ is \alert{Kobayashi hyperbolic}.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Results on the Kobayashi conjecture}
\begin{block}{Kobayashi conjecture (1970)}
$\bullet$ Let $X\subset\bP^{n+1}$ be a (very)
generic hypersurface of degree $d\ge d_n$ 
\phantom{$\bullet$~}large enough. Then $X$
is Kobayashi hyperbolic.\pause\\
$\bullet$ By a result of M.\ Zaidenberg (1987), the optimal bound must
\phantom{$\bullet$~}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 $X{\subset}\bP^3$ of 
\alert{degree $d\ge 21$} is \hbox{hyperbolic}.\\
Independently McQuillan got $d\ge 35$.
\end{block}
This was more recently 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 very large $d_n$} (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 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 $X\subset \bP^{n+1}$ of degree 
  \alert{$d\ge d_n:=2^{n^5}$} satisfies the GGL conjecture.\pause\\ 
  The bound was improved by \claim{\rm(D-, 2012)} to\vskip2pt \centerline{%
$\alert{d_n=\left\lfloor{n^4\over 3}\big(n\log(n\log(24n))\big)^n\right\rfloor}~~=O(\exp(n^{1+\varepsilon})),~~\forall\varepsilon>0$.}
\end{block}
  \pause
\begin{block}{Theorem (S.~Diverio, S.~Trapani, 2009)} Additionally,
  a generic hypersurface $X\subset\bP^4$ of degree \alert{$d\ge 593$}
  is hyperbolic.
\end{block}

\end{frame}

\begin{frame}
  \frametitle{Category of directed varieties}
\wider[2em]{%
  $~$\vskip-13pt
\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 of~$T_X$
    or, more generally, a (possibly singular) \alert{linear subspace}, i.e.\
    a closed irreducible analytic subspace of the total space $T_X$ 
    such that $\forall x\in X$, 
    $V_x:=V\cap T_{X,x}$ is \hbox{linear.\kern-20pt}\vskip4pt
    \pause
    \item
    \claim{{\bf Definition.}} {\it Category of directed varieties :}
    \vskip2pt
    -- \alert{Objects} : pairs $(X,V)$, $X$ variety/$\bC$ 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 $[V,V]\subset V$ (foliations)
    \pause
    \vskip6pt
    \item
    \claim{{\bf Fonctor ``1-jet'' :}} $(X,V)\mapsto (\tilde X,\tilde V)$ 
    where :\vskip-21pt
       \begin{eqnarray*}
       &&\strut\kern-60pt\tilde X=P(V)={}\hbox{bundle of projective spaces of lines in $V$}\\
       &&\strut\kern-60pt\pi:\tilde X=P(V)\to X,~~~(x,[v])\mapsto x,~~v\in V_x\\
       &&\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*}
  \end{itemize}}
\end{frame}

\begin{frame}
  \frametitle{Semple jet bundles (non singular case)}
\wider[2em]{%
  \begin{itemize} 
    \item
    For every entire curve $f:(\bC,T_\bC)\to(X,V)$ tangent to $V$\vskip-20pt
    \begin{eqnarray*}
    &&\strut\kern-60pt f_{[1]}(t):=(f(t),[f'(t)])\in P(V_{f(t)})\subset \tilde X\\
    &&\strut\kern-60pt f_{[1]}:(\bC,T_\bC)\to(\tilde X,\tilde V)~~
    \hbox{\alert{(projectivized 1st-jet)}}
    \end{eqnarray*}\ \vskip-34pt\
    \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}}\vskip-18pt
    \begin{eqnarray*}
    &&\strut\kern-60pt 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}\\
    &&\strut\kern-60pt 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
    &&\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}\\
    &&\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{$k$-jets of curves}
  $~$\vskip-16pt
   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 $X_k=P(V_{k-1})$}.
\vskip5pt
\pause
We define the bundle $J^kV$ of \alert{$k$-jets of curves tangent to $V$} by 
taking $J^kV_x$ to be the set of equivalence classes of germs 
$f:(\bC,0)\to(X,V)$ such that in some coordinates 
$f(t)=(f_1(t),\ldots,f_n(t))$ has a Taylor expansion
\alert{$$
f(t)=x+t\xi_1+\ldots +t^k\xi_k+O(t^{k+1}).
$$}
\pause
Here we take $\xi_s={1\over s!}\nabla^sf(0)$ with respect to some local
holomorphic connection on $V$ (obtained e.g.\ from a trivialization). Thus
$\xi_s\in V_x$ and
$$
J^kV_x\simeq V_x^{\oplus k}\simeq \bC^{kr}~~\hbox{(non intrinsically)}.
$$
\end{frame}

\begin{frame}
\frametitle{Semple bundles and reparametrization of curves}
Consider the group $\bG_k$ of $k$-jets of germs of 
biholomorphisms $\varphi:(\bC,0)\to(\bC,0)$, i.e.
$$\varphi(t)=\alpha_1t+\alpha_2t^2+\ldots+\alpha_kt^k+O(t^{k+1})$$
and the natural $\bG_k$ action on the right: 
$$J^kV\times\bG_k\to J^kV,~~~(f,\varphi)\mapsto f\circ\varphi.$$
The action is free on germs $J^kV^{\reg}$ of {\em regular curves} with
$\xi_1=f'(0)\ne 0$.\vskip-2pt\pause

\begin{block}{Theorem} $X_k$ is a smooth compactification of 
\alert{$J_kV^{\reg}/\bG_k$}.
\end{block}\pause
Now we want to deal with \alert{possibly singular} directed varieties
$(X,V)$, i.e.\ $X$ and $V$ both possibly singular.
\end{frame}

\begin{frame}
\frametitle{Singular directed varieties}\vskip-9pt
\begin{block}{Definition}
A singular directed variety is a pair $(X,V)$ where $X$
is a reduced complex space, and $V\subset T_X$ is a \alert{closed linear
subspace} of $T_X$.
\end{block}\vskip-3pt\pause
This means that we have an irreducible component $V_j$ lying over each
irreducible component $X_j$ of $X$.\vskip3pt\pause
Assume $X$ to be irreducible, $\dim X=n$. Every point $x\in X$ has a
neighborhood $U$ with an embedding $U\hookrightarrow\Omega$ as a closed
analytic subset in a smooth 
open set $\Omega\subset\bC^N$. Then $T_{X|U}$ is taken
to be the closure of $T_{U_{\reg}}$ in $T_\Omega$, and $V$ is always assumed
to be the \alert{closure of $V_{\rm reg|U}$} (part of $V$ that is a subbundle of
$T_{X_{\rm reg}|U}$).\vskip3pt\pause
If $X$ is {\em non singular} and $V\subset T_X$ is {\em singular}, $V$ is a
subbundle of $T_X$ over a Zariski open set $X'=X\smallsetminus Y$, and we 
have at least
an \alert{absolute Semple tower} $(X^a_k,V^a_k)$ associated with $(X,T_X)$.
\vskip3pt\pause
We then define $(X_k,V_k)$ to be the \alert{closure of $(X'_k,V'_k)$} 
[associated with $(X',V')$,~ $V'=V_{|X'}$] in $(X^a_k,V^a_k)$.
\end{frame}

\begin{frame}
\frametitle{Base resolution of singularities}
Let $(X,V)$ be singular pair. By Hironaka, there exists a \alert{modification}
$\mu:\widetilde X\to X$ (in the form of a composition of
blow-ups with smooth centers), such that $\widetilde X$ is non singular.
\vskip5pt\pause
Let $\mu_*:T_{\widetilde X}\to\mu^*T_X$ be the differential $d\mu$.
We define $\widetilde V=\mu^{-1}V\subset T_{\widetilde X}$ to be the \alert{closure of $(\mu_*)^{-1}(V_{|X'})$}, where $X'\subset X_{\rm reg}$ is a Zariski open set
over which $V_{|X'}$ is a subbundle of $T_{X_{\rm reg}}$ and 
$\mu:\mu^{-1}(X')\to X'$ is a biregular.
\vskip5pt\pause
We can then construct a Semple tower $(\widetilde X_k,\widetilde V_k)$
by taking the \alert{closure over regular points} of $(X'_k,V'_k)$ in the 
(regular) absolute tower $(\widetilde X^a_k,\widetilde V^a_k)$, where
$\widetilde V^a_0=T_{\widetilde X}$. 
\begin{block}{Big caution !}
In general, for $\dim X\ge 2$, one can never make $\widetilde V$ non 
singular, even by blowing up further~!
\end{block}
\end{frame}

\begin{frame}
\frametitle{Algebraic differential operators}
 Let $t\mapsto z=f(t)$ be a germ of curve, $f_{[k]}=(f',f'',\ldots,f^{(k)})$ 
\alert{its~$k$-jet} at any point $t=0$. 
We first look at the $\bC^*$-action induced
by {\em dilations} $\varphi(t)=\eta_\lambda(t)=\lambda t$.\vskip4pt\pause
Putting $\xi_s=\Delta^sf(0)$,
the $\bC^*$ action is obtained by computing the derivatives of $f(\lambda t)$,
hence it is given on $J^kV_x\simeq V_x^{\oplus k}$ by\vskip6pt
$(*)\kern1.5cm\lambda\cdot(\xi_1,\xi_2,\ldots,\xi_k)=(\lambda\xi_1,
\lambda^2\xi_2,\ldots,\lambda^k\xi_k).$\vskip8pt\pause
We consider the \claim{Green-Griffiths bundle $E^{\rm GG}_{k,m}V^*$}
of polynomials of weighted degree $m$ on $J^kV_x$ written locally 
in coordinate charts as\vskip6pt
$\strut\kern0.5cm P(x\,;\,\xi_1,\ldots,\xi_k)=\sum
   a_{\alpha_1\alpha_2\ldots\alpha_k}(x)\,\xi_1^{\alpha_1}\ldots\xi_k^{\alpha_k},~~~
    \xi_s\in V_x.$\vskip8pt\pause
Take $P$ to be a holomorphic section in $x$. It can then 
be viewed as an \alert{algebraic differential operator} \hbox{%
$P(f_{[k]})=P(f\,;\, f',f'',\ldots,f^{(k)})$,\kern-55pt}\vskip6pt
$\strut\kern0.5cm P(f_{[k]})(t)=\sum
    a_{\alpha_1\alpha_2\ldots\alpha_k}(f(t))~f'(t)^{\alpha_1}
    f''(t)^{\alpha_2}\ldots f^{(k)}(t)^{\alpha_k}.$
\end{frame}

\begin{frame}
 \frametitle{Direct image formula for Green-Griffiths bundles}\vskip-5pt
The homogeneity expressed by the $\bC^*$ action $(*)$ means that
$P((f\circ\eta_\lambda)_{[k]})=
\lambda^mP(f_{[k]})\circ\eta_\lambda$ for $\eta_\lambda(t)=\lambda t$, and our
polynomials are taken over multi-indices $(\alpha_1,\ldots,\alpha_k)$ such that
\vskip2pt
\alert{$\strut\kern1.5cm
|\alpha_1|+2|\alpha_2|+\ldots+k|\alpha_k|=m$}.\vskip-3pt
\begin{block}{Green Griffiths bundles} Consider $X_k^\GG:=
J^kV^{\ne{\rm const}}/\bC^*$. This defines a bundle
$\pi_k:X_k^\GG\to X$ of weighted projective spaces and by definition\vskip5pt
\alert{$\strut\kern1.5cm
\cO(E_{k,m}^\GG V^*)=(\pi_k)_*\cO_{X_k^\GG}(m)$}\vskip5pt
is the direct image of the $m$-th power of the tautological bundle (or~sheaf)
$\cO_{X_k^\GG}(1)$ on $X_k^\GG$.
\end{block}\vskip-3pt
In case $V$ is singular, we take {\em by definition} $\cO_{X_k^\GG}(m)$ to be
the sheaf of germs of polynomials $P(x;\xi_1,\ldots,\xi_k)$ that are 
\alert{locally bounded} with respect to a smooth ambient hermitian 
metric $h$ on $T_X$ (and the induced metric on $V_k$).
\end{frame}

\begin{frame}
\frametitle{Direct image formula for Semple bundles}
Now, look instead at the direct image of $\cO_{X_k}(m)$ on the Semple bundle 
$X_k=\overline{J^kV^{\reg}}/\bG_k$, by the 
projection $\pi_{k,0}:X_k\to X_0$ from the Semple tower\vskip6pt
$\strut\kern1.5cm X_k\to X_{k-1}\to \ldots X_1\to X_0=X$~
($X$ non singular).\vskip6pt\pause
\begin{block}{Semple direct image formula} 
The direct image sheaf\vskip6pt
\alert{$\strut\kern 2cm (\pi_{k,0})_*\cO_{X_k}(m)
  =\cO(E_{k,m}V^*)$}\vskip6pt
is the sheaf of sections of the bundle
$E_{k,m}V^*\subset E_{k,m}^\GG V^*$ of $\bG_k$-invariant 
algebraic differential operators $f\mapsto P(f_{[k]})$ such that\vskip6pt
\alert{$\strut\kern1.5cm
P((f\circ\varphi)_{[k]})=\varphi^{\prime m}P(f_{[k]})\circ\varphi,~~~
\forall\varphi\in\bG_k$.}\vskip6pt
(by definition, the sections are taken to be locally bounded with respect to 
an ambient smooth hermitian metric $h$ on $T_X$).
 \end{block}
\end{frame}
 
\begin{frame}
  \frametitle{Canonical sheaf of a singular pair (X,V)}
When $(X,V)$ is nonsingular, we simply set \alert{$K_V=\det(V^*)$}.\pause\\
When $X$ is non singular and $V$ singular, we first introduce the 
\hbox{rank~$1$\kern-10pt} sheaf \alert{$\bddK_V$}
of sections of $\det V^*$ that are \alert{locally bounded} with 
\hbox{respect\kern-10pt}
to a smooth ambient metric on~$T_X$. One can show that $\bddK_V$ is
equal to the integral closure of the image of the natural morphism
$$\Lambda^rT_X^*\to \Lambda^r V^*\to \cL_V:={\rm invert.~sheaf}~
(\Lambda^r V^*)^{**}$$
that is, if the image is $\cL_V\otimes\cJ_V$,~~ $\cJ_V\subset\cO_X$,
\alert{$$\bddK_V=\cL_V\otimes\overline{\cJ}_V,~~~~
\overline{\cJ}_V=\hbox{integral closure of}~\cJ_V.$$}\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
$$\alert{\bddK_V\subset\mu_*(\bddK_{\widetilde V})\subset \cL_V}$$\vskip-6pt
and $\mu_*(\bddK_{\widetilde V})$ 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
$$K^{[m]}_V=\lim_{\mu}\uparrow \mu_*(\bddK_{\widetilde V})^{\otimes m},~~~
(\bddK_V)^{\otimes m}\subset K^{[m]}_V\subset \cL_V^{\otimes m}$$\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 only for surfaces.

\begin{block}{Definition} We say that $(X,V)$ is of \alert{general type} if
\alert{the pluricanonical sheaf sequence is big}, i.e.\ 
$H^0(X,K^{[m]}_V)$ provides
a generic embedding of $X$ for a suitable $m\gg 1$.
\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{$K_V^\bullet$ 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 if $r=\rk V=1$, and more generally if 
  $V^*$ itself is big, i.e.\ $\exists A$ ample such that $S^mV^*\otimes\cO(-A)$
  generated by sections on a Zariski open set $X\smallsetminus Y$.\kern-15pt
  \pause\\
  \begin{block}{Ahlfors-Schwarz lemma} Let 
$\gamma=i\sum\gamma_{jk}dt_j\wedge d\overline t_k\ge 0$ be an a.e.\
positive hermitian form on the ball $B(0,R)\subset\bC^p$,
such that
\alert{$-{\rm Ricci}(\gamma):=i\,\ddbar\log\det\gamma\ge C\gamma$}
in the sense of currents, for some constant $C>0$. Then the 
$\gamma$-volume form is controlled by the Poincar\'e volume form~:\\
\alert{$\strut\kern1cm\displaystyle
\det(\gamma)\le\Big({p+1\over CR^2}\Big)^p{1\over(1-|t|^2/R^2)^{p+1}}.$}\\
In particular one has a bound 
\alert{$R\le \big({p+1\over C}\big)^{1/2}(\det(\gamma(0))^{-1/2p}$}.
\end{block}
\end{frame}

\begin{frame}
  \frametitle{Fundamental vanishing theorem}
  \claim{\bf Proof.} Construct a (singular) Finsler metric on $V$ by\\
$\Vert \xi\Vert_{V,h}^2:=\Big(\sum_j|\sigma_j(x)\cdot\xi^m|^2_{h_A^*}\Big)^{1/m}$
with $\xi\in V_x$, $\sigma_j\in H^0(X,S^mV^*\otimes\cO(-A)$.
Set $\gamma(t)=i\Vert f'(t)\Vert_{V,h}^2dt\wedge d\overline t$
on the disk $D(0,R)$. Then if $\gamma\not\equiv 0$, we have
\alert{$$
-{\rm Ricci}(\gamma)=i\partial\overline\partial\Vert f'(t)\Vert_{V,h}^2\ge
f^*\Theta_{V^*,h^*}\ge {1\over m}f^*\Theta_{A,h_A}\ge C\gamma,$$}
thus $R$ is bounded and one cannot have $R=+\infty$.\vskip4pt
\pause
  \begin{block}{Fundamental vanishing theorem for jet differentials}
   \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}
\end{frame}

\begin{frame}
  \frametitle{Proof of fundamental vanishing theorem}
  \claim{\bf Simple case}.
  First assume that $f$ is a Brody curve, i.e.\ $\Vert f'\Vert_\omega$ bounded
  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 relatively compact
  coordinate chart. Hence $u_A(t)=P(f_{[k]})(t)$ is bounded, and must thus be
  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 $u_A\equiv 0$.
  \vskip5pt
  \claim{\bf Case of an invariant jet differential}.
  Assume $P\in H^0(X,E_{k,m}V^*\otimes\cO(-A))$ is $\bG_k$-invariant. This is
the same as a section $\sigma\in H^0(X_k,\cO_{X_k}(m)\otimes\pi_{k,0}^*\cO(-A))$.
\end{frame}

\begin{frame}
  \frametitle{Proof of fundamental vanishing theorem (cont.)}
From the existence of $\sigma$ and the fact that $\cO_{X_k}(1)$ is 
relatively ample over $X_{k-1}$,
we infer the existence of a singular hermitian metric $h_\sigma$ 
on $\cO_{X_k}(-1)$
(essentially given by $|\xi^m\cdot\sigma|^{2/m}$ corrected with relatively ample
terms), such that \alert{$i\partial\overline\partial \log h_\sigma$} 
is bounded below by
a positive definite K\"ahler form $\omega$ on $X_k$, 
and the \alert{zeroes of $h_\sigma$} coincide
with the \alert{zero divisor $Z_\sigma$}.\vskip5pt\pause
Now $f_{[k-1]}:\bC\to X_{k-1}$ has a derivative $f'_{[k-1]}$ that can be viewed
as a section of the pull-back line bundle \alert{$f_{[k]}^*\cO_{X_k}(-1)$}.
\vskip5pt\pause
If we put
$\gamma(t)=i\,\Vert f'_{[k-1]}\Vert^2_{h_\sigma}dt\wedge d\overline t$, then
assuming $f(\bC)\not\subset Z_\sigma$, we get $\gamma\not\equiv 0$ on $\bC$
and
\alert{$$
-{\rm Ricci}(\gamma)=i\partial\overline\partial\log\gamma\ge
f_{[k]}^*\Theta_{\cO_{X_k}(1),h_\sigma^\star}\ge C
f_{[k]}^*\omega\ge C'\gamma.
$$}
This is a contradiction, hence $f(\bC)\subset Z_\sigma$, as desired.
\end{frame}

\begin{frame}
\frametitle{Existence theorem for jet differentials}\vskip-5pt
\claim{\bf Relation between invariant and non invariant jet differentials.}~
On a non invariant polynomial $P$ one can define in a natural way a
$\bG_k$-action by putting $(\varphi^*P)(f_{[k]}):=P((f\circ\varphi)_{[k]})(0)$.
\vskip3pt\pause
By expanding the derivatives, one finds
$$
(\varphi^*P)(f_{[k]})=\sum_{\alpha\in\bN^k,\,|\alpha|_w=m}\varphi^{(\alpha)}(0)\,
P_\alpha(f_{[k]})
$$\vskip-5pt
where $\alpha=(\alpha_1,\ldots,\alpha_k)\in\bN^k$,
$\varphi^{(\alpha)}=(\varphi')^{\alpha_1}(\varphi'')^{\alpha_2}\ldots
(\varphi^{(k)})^{\alpha_k}$,
$|\alpha|_w=\alpha_1+2\alpha_2+\ldots+k\alpha_k$ is the weighted degree
of~$\alpha$, and if one puts $\deg P=m$, $P_\alpha$ is a homogeneous polynomial 
of degree\vskip4pt
$\deg P_\alpha=m-(\alpha_2+2\alpha_3+\ldots+(k-1)\alpha_k)=
\alpha_1+\alpha_2+\ldots+\alpha_k.$\vskip-1pt\pause
\begin{block}{Fundamental existence theorem (D-, 2010)}
Let $(X,V)$ be of general type, such that
  $\bddK_V$ is a \alert{big} rank $1$ sheaf. Then
  \alert{$\exists$ many $P\in H^0(X,E_{k,m}V^*\otimes\cO(-A))$, 
$m{\gg}k{\gg}1$} $\Rightarrow$
  \alert{$\exists$~algebr.\ hypersurface $Z\subsetneq X_k$} such that
  \hbox{\alert{$f_{[k]}(\bC)\subset Z$},
  $\forall f:(\bC,T_\bC)\to(X,V)$\kern-20pt}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Holomorphic Morse inequalities}\vskip-4pt
\begin{block}{Theorem {\rm (D, 1985, L.\ Bonavero 1996)}} Let $L\to X$ be a 
holomorphic line bundle on a compact complex manifold. 
Assume $L$ equipped with a {\em singular hermitian metric} $h=e^{-\varphi}$
with analytic singularities in $\Sigma\subset X$, and 
$\theta={i\over 2\pi}\Theta_{L,h}$.~\pause Let\vskip5pt
\alert{$\strut\kern 2mm
X(\theta,q):=\big\{x\in X\smallsetminus\Sigma\,;\,\theta(x)~\hbox{has signature
$(n-q,q)$}\big\}$}\vskip5pt
be the $q$-index set of the $(1,1)$-form $\theta$, and\vskip5pt
\alert{$\strut\kern 1cm
X(\theta,E)=\bigcup_{j\in E}X(\theta,j),\quad E\subset\{0,\ldots,n\}.$}
\vskip5pt\pause
Then\vskip4pt
{\rm (i)}~~ \alert{$h^q(X,L^{\otimes m}\otimes\cI(m\varphi))\le {m^n\over n!}
\int_{X(\theta,q)}~(-1)^q\theta^n + o(m^n)$},
\vskip4pt
{\rm (ii)}~ \alert{$h^q(X,L^{\otimes m}\otimes\cI(m\varphi))\ge {m^n\over n!}
\int_{X(\theta,\{q-1,q,q+1\})}~(-1)^q\theta^n-o(m^n)$},\vskip7pt
where $\cI(m\varphi)\subset\cO_X$ denotes the \alert{multiplier ideal sheaf}
\vskip5pt
$\cI(m\varphi)_x=\big\{f\in\cO_{X,x}\,;\;\exists U\ni x~\hbox{s.t.}~
\int_U|f|^2e^{-m\varphi}dV<+\infty\big\}.$
\end{block}

\end{frame}

\begin{frame}
\frametitle{Finsler metric on the $k$-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{Probabilistic interpretation of the curvature}
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{Main cohomological estimate}
$\Rightarrow$ 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 (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 $\forall q\ge 0$ [$q=0$ most useful!], $\forall m\gg k\gg 1$ with
$m$ suffici-\\ ently divisible, the sheaf
$\cG_{k,m}=\cO(L_k^{\otimes m})\otimes\cI(h_k^m)$ satisfies bounds
\vskip3pt
\alert{$\displaystyle
h^q(X_k^\GG,\cG_{k,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)
$\vskip3pt\pause
$\displaystyle
h^q(X_k^\GG,\cG_{k,m})\ge{m^{n+kr-1}\over (n{+}kr{-}1)!}
{(\log k)^n\over n!\,(k!)^r}\bigg(
\int_{X(\eta,\{q,\,q\pm 1\})}\kern-27pt(-1)^q\eta^n-\frac{C'}{\log k}\bigg).\kern-20pt$}\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{$\pi_{k,k-1}(Z)=X_{k-1}\Rightarrow
\rk W<\rk V_k=\rk V$}.
\end{frame}

\begin{frame}
\frametitle{Partial solution of 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{$\bddK_W\otimes \cO_{X_k}(m)_{|Z}$ 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{$$
{\inf\big\{\lambda\in\bQ\,;\;\exists m\in\bQ_+,\;
\bddK_W{\otimes}\big(\cO_{X_k}(m){\otimes}\pi_{k,0}^*\cO(\lambda A)
\big)_{|Z}~\hbox{big on $Z$}\big\}\over\rank W}.
$$}\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{Approach of the 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
$$\alert{t\mapsto h^0(X_t,K_{V_t}^{[m]})}$$\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{Related work}

In 1993, Masuda and Noguchi gave examples of hyperbolic hypersurfaces 
in $\bP^n$ for arbitrary $n\ge 2$, of degree $d\ge d_n$ large enough. 
\vskip5pt\pause
In 2012, Y.T. Siu announced the generic hyperbolicity of of hyperbolic 
hypersurfaces of $\bP^n$ of degree $d\ge d_n$ large enough. This has
been confirmed in 2016 by Damian Brotbek (arXiv:1604.00311)
\vskip5pt\pause
In 2015, Dinh Tuan Huynh, showed that the complement of a small deformation
of the union of $2n+2$ hyperplanes in general position in $\bP^n$ is hyperbolic:
the resulting degree \alert{$d_n=2n+2$} is extremely close to optimality 
(if not optimal).
\vskip5pt\pause
Very recently, Gergely Berczi stated a positivity conjecture 
for Thom polynomials of Morin singularities, and showed that it would
imply a polynomial bound \alert{$d_n=2\,n^{10}$} for the 
generic hyperbolicity of hypersurfaces.
\end{frame}


\end{document}