%beamer -slides pdfmode
% 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), Johns Hopkins Univ, March 2, 2019}}\kern181pt\rlap{\blank{
Hyperbolicity of general algebraic hypersurfaces}}\kern181pt
\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Hyperbolicity of general algebraic\\
hypersurfaces of high degree}

%% \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 JAMI 2019 Conference: Geometric Analysis\\
in honor of Bernie Shiffman\\
\vskip7pt Johns Hopkins University, March 1--3, 2019}

%%\subject{Talks}
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\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{Bernie at the Skoda Conference in Paris, 2005}
\pgfdeclareimage[height=8cm]{dscn1780}{dscn1780}
\pgfuseimage{dscn1780}
\end{frame}


\begin{frame}
\frametitle{Bernie at the Skoda Conference in Paris, 2005}
\pgfdeclareimage[height=8cm]{dscn1871}{dscn1871}
\pgfuseimage{dscn1871}
\end{frame}

\begin{frame}
\frametitle{Kobayashi pseudodistance}
\vskip-7pt
\begin{block}{Kobayashi pseudodistance}
Let $X$ be a complex space, $\dim_\bC X=n$.\\
The \alert{Kobayashi pseudodistance $d^K_X$} is defined to be\\
\alert{$d^K_X(p,q)={}$infimum of sums $\sum_{i=0}^{N-1}
d_{\rm Poinc}(p_i,p_{i+1})$}\\
for all chains of points $p=p_0,p_1,\ldots,p_N=q\in X$ such that
$p_i=g_i(t_i),p_{i+1}=g_i(u_i)$ lie in the image of an analytic disc
$g_i:\bD\to X$, and $d_{\rm Poinc}(p_i,p_{i+1})=d_{\rm Poinc}(t_i,u_i)$
on that disc.
\end{block}
\pause\vskip-4pt
For $X$ algebraic, the following result shows that $d^K_X$
retains an algebraic flavor in spite of the analytic definition.
\pause\vskip-4pt
\begin{block}{Theorem (D, Lempert, Shiffman, Duke Math.\ J.\ 1994)}
If $X$ is projective algebraic, one obtains the same pseudodistance
by \alert{taking algebraic curves $C_i$ containing $p_i,p_{i+1}$} and computing
$d_{\rm Poinc}(p_i,p_{i+1})$ as the Poincaré distance on the universal cover
$\widehat C_i$ of the normalization of $C_i$ (with
$d_{\rm Poinc}(p_i,p_{i+1})=0$ if $g(C_i)\leq 1$).
\end{block}
\end{frame}
  
\begin{frame}
\frametitle{Kobayashi conjecture}
\vskip-7pt
\begin{block}{Definition}
 A complex space $X$ is said to be \alert{Kobayashi hyperbolic} if the
 Kobayashi pseudodistance $d^K_X$ is non degenerate, i.e.\ is a
 genuine distance. 
\end{block}
\pause\vskip-4pt
\begin{block}{Kobayashi conjecture (1970)}
Let $X^n\subset\bP^{n+1}$ be a (very)
\alert{general hypersurface of degree $d\ge d_n$ large enough}. Then $X$
is Kobayashi hyperbolic.
\end{block}\vskip-3pt\pause
Moreover, a very precise optimal degree $d_n$ should hold \hbox{(see below).\kern-10pt}\\
Of course, a smooth curve $X\subset \bP^2$ is hyperbolic iff genus $g\ge 2$,
i.e.\ \alert{$d\geq d_2=4$}.
\vskip3pt\pause
In 1999, McQuillan and
D.-El Goul proved the case $n=2$ (with $d_2=35$, \alert{$d_2=21$})
respectively.\pause\
Then \alert{Y.T. Siu} (Abel conf.\ 2002, final paper in Invent.\ Math.\ 2015)
detailed a strategy for the proof of the Kobayashi conjecture,
using Nevanlinna theory.
\end{frame}

\begin{frame}
\frametitle{Green-Griffiths-Lang conjecture}
\vskip-6pt
\begin{block}{Useful characterization of Kobayashi hyperbolicity (Brody, 1978)\kern-5pt}
A \alert{compact} complex space $X$ is Kobayashi hyperbolic iff
it is \alert{Brody~hyperbolic}, namely if $\not\!\exists$
entire \hbox{curves $f:\bC\to X$, $f\neq{}$const.}
\end{block}
\pause\vskip-3pt
\begin{block}{Green-Griffiths-Lang conjecture}
A projective variety $X$ is of \alert{general type} iff \alert{$\exists$ an
algebraic subvariety $Y\subsetneq X$ containing all images of
entire curves $f:\bC\to X$}.
\end{block}
\pause\vskip-3pt
\begin{block}{Conjectural corollary of GGL}
A projective variety $X$ is \alert{(Kobayashi/Brody) hyperbolic} iff
\alert{all its subvarieties $Y$ (possibly singular, including $X$) are of
general type}.
\end{block}
\vskip-3pt\pause
The latter property has been characterized by work of Zaidenberg,
Clemens, Ein, Voisin, Pacienza. In this way, 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{frame}


\begin{frame}
\frametitle{Geometric approaches of the Kobayashi conjecture}
\vskip-4pt
The main idea is to use the geometry of jet bundles, and more specifically
Semple bundles introduced in this context by D.\ in 1995.

\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{$d_n=(n+1)n^{2n+3}=O(n^{2n+4})$.}
\begin{block}{Theorem (D-, 2018, with a substantially 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{Use of algebraic differential operators}
  Let~ \alert{$\bC\to X,~~~t\mapsto f(t)=(f_1(t),\ldots,f_n(t))$}
  be a curve written in some local holomorphic coordinates 
  $(z_1,\ldots,z_n)$ on~$X$. It has a local Taylor expansion
  $$f(t)=x+t\xi_1+\ldots+t^k\xi_k+O(t^{k+1}),~~~
    \xi_s=\frac{1}{s!}\nabla^sf(0)$$
  where $\nabla$ is the trivial connection on $T_X$.\vskip3pt
  \pause
  One considers polynomials on the $k$-jet bundle $J^kX$ defined
  locally in coordinate charts by
  $$P(x\,;\,\xi_1,\ldots,\xi_k)=\sum
    a_{\alpha_1\alpha_2\ldots\alpha_k}(x)\xi_1^{\alpha_1}\ldots\xi_k^{\alpha_k},~~~
    \xi_s\in V.$$
  where the $a_\alpha$'s are holomorphic on $X$.\pause\
  One can view them as \alert{algebraic differential operators}
  on $k$-jets of curves $f:(\bC,0)_k\to X$ or entire curves
  $f:\bC\to X$
  $$
    P(f;f',\ldots,f^{(k)})(t)=\sum
    a_{\alpha_1\alpha_2\ldots\alpha_k}(f(t))~f'(t)^{\alpha_1}
    f''(t)^{\alpha_2}\ldots f^{(k)}(t)^{\alpha_k}.
  $$
\end{frame}

\begin{frame}
\frametitle{Invariant differential operators}
\vskip-4pt
Some operators $f\mapsto P(f)$, such as Wronskians
\alert{$f'\wedge \ldots\wedge f^{(k)}$}, or more generally
polynomial operators
of the form $Q(f',f'\wedge f'',\ldots, f'\wedge \ldots\wedge f^{(k)})$
satisfy the additional property that for any reparametrization
$\varphi:\bC\to\bC$, one has
$$
\alert{
P(f\circ \varphi)=\varphi^{\prime m}P(f)\circ \varphi}
$$
where $m$ is the weighted degree
$$
\alert{m=|\alpha_1|+2|\alpha_2|+\cdots+k|\alpha_k|}.
$$
They form a bundle denoted $E_{k,m}T^* _X$ (order $k$, weighted
degree $m$).
\pause\vskip-3pt
\begin{block}{Fundamental vanishing theorem}
\claim{\rm [Green-Griffiths 1979], [D.\ 1995],
 \hbox{[Siu-Yeung 1996]\kern-30pt}}\\
$\forall X$ projective manifold, $\forall A$ ample line bundle on $X$,
$\forall P\in H^0(X,E_{k,m}T^*_X\otimes\cO(-A))$ : global diff.\ operator
on~$X$, $\forall f:\bC\to X$ entire curve, one has 
\alert{$P(f)\equiv 0$}.
\end{block}
\end{frame}

\begin{frame}
  \frametitle{Proof of the fundamental vanishing theorem}
  \vskip-4pt
  \claim{\bf Proof in a simple case}. If $X$ has an entire curve $g:\bC\to X$,
  then the Brody reparametrization technique produces
  a \alert{Brody curve} $f:\bC\to X$, such 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{$u_A(t):=P(f;f',\ldots,f^{(k)})(t)$ 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 
  $f(\bC)\subset X$. But then $u_A$ vanishes somewhere and so
  \alert{$u_A\equiv 0$}.
  \pause\vskip4pt
  \claim{\bf General case.}
On can use the logarithmic derivative lemma (Siu-Yeung, 1996), or
the Alhfors lemma in case $P$ is invariant (D.~1995), combined with
an induction on $m=\deg P$ when $P$ is non invariant.
\end{frame}

\begin{frame}
\frametitle{Semple bundles and direct image formula}
\vskip-4pt
\begin{block}{Semple bundles}
One can construct of tower of $\bP^{n-1}$-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{$\dim X_k=n+k(n-1)$},\\
$X_k$ being equipped with a certain rank $n$
subbundle \alert{$V_k\subset T_{X_k}$},\\ 
and inductively $X_k=P(V_{k-1})$ over $X_{k-1}$,\\
such that $X_k$ is a desingularization of \alert{$J^kX\,/\!/\,\bG_k$},
where $\bG_k$ is the group
of $k$-jets of biholomorphisms $\varphi:(\bC,0)\to(\bC,0)$.
\end{block}

One takes in particular \alert{$V_0=T_X$} and \alert{$X_1=P(T_X)$}.
\pause
\begin{block}{Direct image formula}
The
\alert{tautological line bundles $\cO_{X_k}(m)$} on \alert{$X_k=P(V_{k-1})$}
have a direct image
\alert{$(\pi_{k,0})_*\cO_{X_k}(m)\simeq E_{k,m}T^*_X$}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Reinterpretation of the vanishing theorem}
\vskip-7pt
Any global differential operator $P\in H^0(X,E_{k,m}T^*_X\otimes\cO(-A))$ corresponds
to a section
$\alert{\sigma_P\in H^0(X_k,\cO_{X_k}(m)\otimes\pi_{k,0}^*\cO(-A))}$.
\pause\vskip5pt
Saying that a curve $f:\bC\to X$ satisfies the differential equation
$P(f;f',\ldots,f^{(k)})=0$ means that the $k$-jet $f_{[k]}:\bC\to X_k$
has its image $f_{[k]}(\bC)$ contained in the hypersurface
$\sigma_P^{-1}(0)\subset X_k$.\pause\vskip-4pt
\begin{block}{Consequence}
All entire curves lie in the so called Green-Griffiths locus\vskip4pt
\alert{\centerline{$\displaystyle
{\rm GG}(X)=\bigcap_k\pi_{k,0}({\rm GG}(X_k),~~~
{\rm GG}(X_k)=\bigcap_P\sigma_P^{-1}(0)\subset X_k,
$}}\vskip0pt
hence $(*)$: ${\rm GG}(X)\subsetneq X$ would imply the GGL conjecture.\\
\alert{Unfortunately $(*)$ not always true!}$\,$(Green,$\,$Lang,$\,$Diverio-Rousseau).
\end{block}
\pause\vskip-5pt
\begin{block}{Hope: existence theorem for jet differentials (D-, 2010)}
Let $X$ be of general type, i.e.\ $K_X$ big. Then for $m\gg k\gg 1$\vskip4pt
\alert{\centerline{$\displaystyle  
  h^0(X_k,\cO_{X_k}(m)\otimes \pi_{k,0}^*\cO(-A))\geq c_km^{\dim X_k},~~
  c_k>0.
$}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Sketch of proof of the existence theorem}
\vskip-7pt
One starts with a hermitian metric $h_X$ on $T_X$ such that its determinant
yields a singular hermitian metric $\det(h_X^*)$ on $K_X$ with
strictly positive curvature current. One can then equip $J^kX$
with a natural Finsler metric that induces a singular hermitian metric
on $\cO_{X_k}(1)$. One finally applies holomorphic Morse inequalities to
\vskip3pt
$\displaystyle~\kern5mm
L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big(-{\varepsilon\over kn}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)A\Big),~~\varepsilon\in\bQ_+^*,
$\vskip3pt
$\displaystyle~\kern5mm
\eta=\Theta_{K_X,\det h^*_X}-\varepsilon\Theta_{A,h_A}.$\vskip3pt
Then for all $q\ge 0$ and all $m\gg k\gg 1$ such that 
$m$ is sufficiently divisible, one gets upper and lower bounds\
[$q=0$ most useful!]\vskip3pt
\alert{$\displaystyle
h^q(X_k^\GG,\cO(L_k^{\otimes m}))\leq c_{k,n}m^{n+k(n-1)}
\bigg(\int_{X(\eta,q)}(-1)^q\eta^n+\frac{C}{\log k}\bigg)\kern-20pt
$\vskip3pt\pause
$\displaystyle
h^q(X_k^\GG,\cO(L_k^{\otimes m}))\geq c_{k,n}m^{n+k(n-1)}
\bigg(\int_{X(\eta,q,\,q\pm 1)}(-1)^q\eta^n-\frac{C}{\log k}\bigg).\kern-20pt$}
\end{frame}

\begin{frame}
\frametitle{Wronskian operators are easier to deal with ...}
\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 simplified~!}
\vskip-3pt
Take e.g.\ $X=\bP^N$, $A=\cO(1)$ very ample, $k\leq N$, $d\geq k$ and
$$s_j(z)=z_j^d~~\Longrightarrow~~
s_j\in H^0(X,A^d).
$$
{\pause}Then derivatives $D^\ell(s_j\circ f)$ are divisible by $z_j^{d-k}$
for $\ell\leq k$,\pause\\
and (taking $L=A^d$) we find
\alert{$$\eqalign{
\prod_{0\leq j\leq k}z_j^{-(d-k)}W(s_0,\ldots s_k)
&\in H^0(X,E_{k,k'}T^*_X\otimes A^{d(k+1)-(d-k)(k+1)})\cr
\noalign{\vskip-11pt}
&=H^0(X,E_{k,k'}T^*_X\otimes A^{k(k+1)}).\cr}
$$}\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
$k=N-1$, 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{$$P:=
\prod_{0\leq i\leq k+1}z_i^{-(d-k)}W(s_0,\ldots s_k)
\in H^0(X,E_{k,k'}T^*_X\otimes A^{k(k+2)-d}).
$$}
\end{frame}

\begin{frame}
\frametitle{A result of Shiffman-Zaidenberg}
\vskip-6pt  
By the vanishing theorem, if $d\geq k(k+2)+1=N^2$, then
\alert{$W(s_0,\ldots,s_k)(f)=0$},
which means that $f$ satisfies an extra Fermat type
equation \alert{$\sum_{j=0}^{N-1}c_jz_j^d=0$}, and one
can then use induction on $N$ to show that the
\alert{$f_j$'s are pairwise proportional} (Toda~'71, Fujimoto~'74, Green~'75).
\vskip4pt\pause
Using this result, inspired J.~Noguchi's construction of explicit
$n$-dim hyperbolic hypersurfaces (1996),
Shiffman-Zaidenberg \hbox{proved\kern-10pt}\vskip-3pt

\begin{block}{Theorem (Shiffman-Zaidenberg 2001)} For $N=2n$
and $d\geq N^2=4n^2$, the intersection of the Fermat hypersurface
$H=\{\sum z_j^d=0\}\subset\bP^{2n}$ with a sufficiently general
projective linear subspace
$\Lambda\simeq\bP^{n+1}\subset\bP^{2n}$ yields a hyperbolic hypersurface
\alert{$X:=H\cap\Lambda\subset\bP^{n+1}$}.
\end{block}
\pause\vskip-3pt
However, one gets here only one Wronskian, so that
the genericity of hyperbolicity in such low degrees \alert{$d=O(n^2)$}
is hard to establish.
\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{Strategy of proof of the Kobayashi conjecture
(Brotbek 2016, simplified by D. in 2018)}
\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{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}
\pause
The proof is achieved by induction on $\dim X$. By taking
$X'=\tau_J^{-1}(0)$ one can define further simplified Wronskian sections that
generate the universal line bundle $\cG$ everywhere on
$\widehat X_{t_0,k}$.~$\square$
\vskip5pt\pause
In order to improve the bounds (and eventually to prove the GGL conjecture),
one would need to achieve a betting understanding of the geometry of
Semple jet bundles and of base loci of jet differentials.
\end{frame}

\begin{frame}
\frametitle{Concept of algebraic jet hyberbolicity}
\vskip-5pt
Fix $X$ projective. An irreducible algebraic subset $Z\subset X_k$ of the
Semple $k$-jet bundle is said \alert{admissible} if it does not project
into the intermediate
``vertical divisors'' $P(T_{X_{\ell-1}/X_{\ell-2}})\subset X_\ell$,
$2\leq\ell\leq k$.
\vskip4pt\pause
One then defines an \alert{induced directed structure}
$(Z,W)\hookrightarrow(X_k,V_k)$
by putting $W=\overline{T_{Z'}\cap V_k}$, where the intersection is
taken on a suitable Zariski open set 
$Z'\subset Z_{\rm reg}$ where the intersection has constant rank and 
is a subbundle of $T_{Z'}$.\pause\
Then there is a naturally defined canonical sheaf
$\cK_W=\det W^*\otimes\cI_W$ where $\cI_W$ is a certain ideal sheaf
depending on the singularities of $W$.
\vskip-3pt\pause
\begin{block}{Definition} Let $X$ be of general type. We say that $X$ is
\alert{``algebraically jet hyperbolic''}, [resp.
\alert{``strongly of general type''}],
if for every admissible alg.\ subvariety $Z\subsetneq X_k$
[resp.\ such that $\pi_{k,0}(Z)=X$)], 
the induced directed structure $(Z,W)\,{\subset}\,(X_k,V_k)$ is 
of \alert{general type modulo $X_k\to X$}, i.e.\
\alert{$\cK_W\otimes \cO_{X_k}(m)_{|Z}$ is big} for some $m\in\bQ_+$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Relation between the GGL conjecture and hyperbolicity}
\vskip-5pt
\begin{block}{Theorem (D., 2015)} Let $X$ be a projective variety.
\vskip3pt
(i) If $X$ is strongly of general type, then $X$ satisfies the GGL
conjecture.
\vskip3pt
(ii) If $X$ is algebraically jet hyperbolic, then $X$ is Kobayashi
hyperbolic.
\end{block}
\pause
Another important result was obtained in 2018 by Eric Riedl and David Yang,
thanks to a general \alert{Grassmannian construction}.
\begin{block}{Theorem (Riedl, Yang 2018 - rough statement)}
Assume that a general hypersurface $X\subset \bP^n$ satisfies the
GGL conjecture for $d\geq d_{\rm GGL}(n)$. Then a general hypersurface $X\subset \bP^{n+1}$ satisfies the Kobayashi conjecture for $d\geq d_{\rm GGL}(2n)$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Recent result of J. Merker}
\vskip-5pt
Using the Riedl-Yang approach, J.~Merker recently proved

\begin{block}{Theorem (J.~Merker, January 2019)}
  Let $X\subset\bP^n$ be a general hypersurface of degree $d$.
\vskip3pt
(i) If $d\geq (\sqrt{n} \log n)^n$, then $X$ satisfies GGL.
\vskip3pt
(ii) If $d\geq (n \log n)^n$, then $X$ is Kobayashi hyperbolic.
\end{block}
\pause
Part (i) is obtained by producing jet differentials via holomorphic morse inequalities, and applying Siu's technique of slanted holomorphic vector fields, along with careful estimates.\vskip5pt\pause
Part (ii) is now a consequence of the above theorem of Riedl-Yang.
\vskip5pt\pause
(At this date -- February 2019 -- this is the best known estimate).
\vskip5pt\pause
In 2010, G.~B\'erczi has formulated a combinatorial
conjecture for Thom polynomials that would imply polynomial bounds.
\end{frame}

\begin{frame}
\frametitle{The end}
\strut\vskip-4.5mm
\centerline{\huge\bf Best wishes Bernie!}
\vskip3mm\strut\kern-7mm
\pgfdeclareimage[height=6.5cm]{bernie}{bernie}
\strut\kern3.7cm\pgfuseimage{bernie}
\end{frame}

\end{document}

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