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% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
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% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
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% 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
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\title[\ \kern-190pt\rlap{\blank{J.-P.\ Demailly (Grenoble), Institut Fourier, July 1, 2019}}\kern181pt\rlap{\blank{Existence of logarithmic and orbifold jet differentials}}\kern181pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Existence of logarithmic and\\
orbifold jet differentials}

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

\author[] % (optional, use only with lots of authors)
{\strut\vskip-24pt Jean-Pierre Demailly}

\institute[]{\strut\vskip-28pt
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
Joint work with F. Campana, L. Darondeau and E. Rousseau\vskip9pt
Summer School in Mathematics 2019\\
Foliations and Algebraic Geometry\\
Institut Fourier, Université Grenoble Alpes\\
June 17 -- July 5, 2019}

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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{Aim of the lecture}
\vskip-8pt  
\begin{itemize}
\item Our goal is to study (nonconstant) entire curves $f:\bC\to X$ drawn
in a projective variety/$\bC$. The variety $X$ is said to be \alert{Brody
($\Leftrightarrow$ Kobayashi) hyperbolic} if there are no such curves.\pause
\item More generally, if $\Delta=\sum\Delta_j$ is a reduced
\alert{normal crossing divisor} in $X$, we want to study entire
curves $f:\bC\to X\ssm\Delta$ drawn in
the complement of $\Delta$.
\vskip4pt
\pgfdeclareimage[height=3.5cm]{drawing1}{drawing1}
\strut\kern1.2cm\pgfuseimage{drawing1}
\pause\vskip2pt
If there are no such curves,
we say that the \alert{log pair $(X,\Delta)$} is Brody hyperbolic.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Aim of the lecture (continued)}
\vskip-8pt  
\begin{itemize}
\item Even more generally, if $\Delta=\sum(1-{1\over\rho_j})\Delta_j\subset X$
is a
\alert{normal crossing divisor}, we want to study entire
curves $f:\bC\to X$ meeting each component $\Delta_j$ of
$\Delta$ with multiplicity${}\geq \rho_j$.\vskip2pt
\pgfdeclareimage[height=3.5cm]{drawing2}{drawing2}
\strut\kern1.5cm\pgfuseimage{drawing2}
\pause\vskip0pt
The pair $(X,\Delta)$ is called an \alert{orbifold} (in the sense
\hbox{of Campana).\kern-15pt}\break
Here $\rho_j\in{}]1,\infty]$, where $\rho_j=\infty$ corresponds
to the \hbox{logarithmic\kern-10pt}\break
case. Usually $\rho_j\in\{2,3,...,\infty\}$, but $\rho_j\in
\bR_{>1}$ will be allowed.\vskip3pt
\pause\vskip-18pt\strut
\item The strategy is to show that under suitable conditions,
\hbox{orbifold\kern-15pt}\break
entire  curves must satisfy \alert{algebraic differential equations}.
\end{itemize}  
\end{frame}  
  
\begin{frame}
\frametitle{$k$-jets of curves and $k$-jet bundles}
\vskip-4pt  
Let \alert{$X$ be a nonsingular $n$-dimensional projective variety}
over~$\bC$.\pause

\begin{block}{Definition of k-jets}
For $k\in\bN^*$, a $k$-jet of curve $f_{[k]}:(\bC,0)_k\to X$ is an
equivalence class of germs of holomorphic curves
$f:(\bC,0)\to X$, written $f=(f_1,\ldots,f_n)$
in local coordinates $(z_1,\ldots,z_n)$ on an open subset $U\subset X$,
where two germs are declared to be
equivalent if~they have the same Taylor expansion of order $k$ at $0$~:
\vskip6pt
\alert{\centerline{$\displaystyle
f(t)=x+t\xi_1+t^2\xi_2+\cdots+t^k\xi_k+O(t^{k+1}),\quad t\in D(0,\varepsilon)\subset\bC,$}}\vskip6pt
and $x=f(0)\in U$, $\xi_s\in\bC^n$, $1\leq s\leq k$.
\end{block}\pause
\vskip-4pt 
\begin{block}{Notation}
Let \alert{$J^kX$} be the bundle of $k$-jets of curves, and
$\pi_k:J^kX\to X$ the natural projection, where the fiber
$(J^kX)_x=\pi_k^{-1}(x)$ consists of $k$-jets of curves $f_{[k]}$ such
that $f(0)=x$.
\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 $t=0$. Look at the $\bC^*$-action induced
by {\em dilations} \alert{$\lambda\cdot f(t):=f(\lambda t)$,
$\lambda\in\bC^*$}, for  $f_{[k]}\in J^kX$.\vskip4pt\pause
Taking$\,$a$\,$(local)$\,$connection$\,\nabla\,$on$\,T_X$$\,$and$\,$putting
\hbox{$\xi_s\,{=}\,f^{(s)}(0)\,{=}\,\nabla^sf(0)$,\kern-20pt}\break
we get a trivialization $J^kX\simeq(T_X)^{\oplus k}$ and
the $\bC^*$ action is given by\vskip6pt
$(*)\kern1.5cm\alert{\lambda\cdot(\xi_1,\xi_2,\ldots,\xi_k)=(\lambda\xi_1,
\lambda^2\xi_2,\ldots,\lambda^k\xi_k)}.$\vskip8pt\pause
We consider the \claim{Green-Griffiths sheaf $E_{k,m}(X)$}
of \alert{homogeneous polynomials of weighted degree $m$} on $J^kX$ defined
by\vskip6pt
\centerline{\alert{$P(x\,;\,\xi_1,\ldots,\xi_k)=\sum
a_{\alpha_1\alpha_2\ldots\alpha_k}(x)\,\xi_1^{\alpha_1}\ldots\xi_k^{\alpha_k},~~~
\sum_{s=1}^ks|\alpha_s|=m.$}}\vskip8pt\pause
Here, we assume the coefficients $a_{\alpha_1\alpha_2\ldots\alpha_k}(x)$ to be
holomorphic \hbox{in $x$,\kern-10pt}\break and view $P$
as a \alert{differential operator \hbox{%
$P(f)=P(f\,;\, f',f'',\ldots,f^{(k)})$},\kern-55pt}\vskip6pt
\centerline{\alert{$P(f)(t)=\sum
    a_{\alpha_1\alpha_2\ldots\alpha_k}(f(t))~f'(t)^{\alpha_1}
    f''(t)^{\alpha_2}\ldots f^{(k)}(t)^{\alpha_k}.$}}
\end{frame}

\begin{frame}
\frametitle{Graded algebra of algebraic differential operators}\vskip-3pt
In this way, we get a graded algebra \alert{$\bigoplus_m E_{k,m}(X)$}
of differential operators. As sheaf of rings, in each coordinate chart
$U\subset X$, it is a pure polynomial algebra isomorphic to
$$\alert{
\cO_X[f_j^{(s)}]_{1\leq j\leq n,\,1\leq s\leq k}}~~\hbox{where}~~
\alert{\deg f_j^{(s)}=s}.
$$\pause
If a change of coordinates $z\mapsto w=\psi(z)$ is performed on $U$,
the curve $t\mapsto f(t)$ becomes $t\mapsto\psi\circ f(t)$ and we
have inductively
\alert{$$
(\psi\circ f)^{(s)}=
(\psi'\circ f)\cdot f^{(s)}+Q_{\psi,s}(f',\ldots,f^{(s-1)})
$$}
where $Q_{\psi,s}$ is a polynomial of weighted degree $s$.\vskip4pt\pause
By filtering by the partial degree of $P(x;\xi_1,...,\xi_k)$
successively in $\xi_k$, $\xi_{k-1},...,\xi_1$, one gets a multi-filtration
on $E_{k,m}(X)$ such that the graded pieces are
\alert{$$
G^\bullet E_{k,m}(X)=\bigoplus_{\ell_1+2\ell_2+\cdots+k\ell_k=m}
S^{\ell_1}T^*_X\otimes \cdots\otimes S^{\ell_k}T^*_X.
$$}
\end{frame}

\begin{frame}
\frametitle{Logarithmic jet differentials}\vskip-3pt
Take a \alert{logarithmic pair $(X,\Delta)$},
$\Delta=\sum\Delta_j$ normal crossing divisor.\vskip4pt\pause
Fix a point $x\in X$ which belongs exactly to $p$ components, say
$\Delta_1,...,\Delta_p$, and take coordinates $(z_1,...,z_n)$
so that \alert{$\Delta_j=\{z_j=0\}$}.\vskip4pt\pause
$\Longrightarrow$ log differential operators : polynomials in the
derivatives
\alert{$$(\log f_j)^{(s)},~~1\leq j\leq p~~~\hbox{and}~~~
f_j^{(s)},~~p+1\leq j\leq n.$$}\pause
Alternatively, one gets an algebra of logarithmic jet differentials, denoted
\alert{$\bigoplus_m E_{k,m}(X,\Delta)$}, that can be expressed locally as
\alert{$$
\cO_X\big[(f_1)^{-1}f_1^{(s)},...,(f_p)^{-1}f_p^{(s)},
f_{p+1}^{(s)}, ...,f_n^{(s)}\big]_{1\leq s\leq k}.
$$}\pause
One gets a multi-filtration on $E_{k,m}(X,\Delta)$ with graded pieces\vskip-13pt
\alert{$$
G^\bullet E_{k,m}(X,\Delta)=\bigoplus_{\ell_1+2\ell_2+\cdots+k\ell_k=m}
S^{\ell_1}T^*_X\langle\Delta\rangle\otimes \cdots\otimes S^{\ell_k}
T^*_X\langle\Delta\rangle
$$}\vskip-8pt
where $T^*_X\langle\Delta\rangle$ is the logarithmic tangent bundle,
i.e., the locally \hbox{free\kern-15pt}\break sheaf generated
by \alert{${dz_1\over z_1},...,{dz_p\over z_p},dz_{p+1},...,dz_n$}.
\end{frame}

\begin{frame}
\frametitle{Orbifold jet differentials}\vskip-4pt
Consider an \alert{orbifold $(X,\Delta)$, 
$\Delta=\sum(1-{1\over\rho_j})\Delta_j$}~ a SNC divisor.\vskip4pt\pause
Assuming $\Delta_1=\{z_1=0\}$ and $f$ having multiplicity $q\geq \rho_1>1$
along~$\Delta_1$, then $f_1^{(s)}$ still vanishes at order${}\geq(q-s)_+$,
thus $(f_1)^{-\beta}f_1^{(s)}$ is bounded as soon as $\beta q\leq(q-s)_+$,
i.e.\ $\beta\leq(1-{s\over q})_+$. Thus, it is sufficient to ask that
\alert{$\beta\leq(1-{s\over \rho_1})_+$}.\pause\ At a point
$x\in |\Delta_1|\cap...\cap|\Delta_p|$, the condition for a monomial of the
form
\alert{$$
f_1^{-\beta_1}...\,f_p^{-\beta_p}\prod\nolimits_{s=1}^k(f^{(s)})^{\alpha_s},~~~
(f^{(s)})^{\alpha_s}=(f_1^{(s)})^{\alpha_{s,1}}...(f_n^{(s)})^{\alpha_{s,n}},
\leqno(*)
$$}
$\alpha_s\in\bN^n$, $\beta_1,...,\beta_p\in\bN$, to be bounded,
is to require that
\alert{
$$
\beta_j\leq\sum\nolimits_{s=1}^k\alpha_{s,j}\Big(1-{s\over \rho_j}\Big)_+,\quad
1\leq j\leq p.
\leqno(**)
$$}\pause\vskip-10pt
\begin{block}{Definition}
\alert{$E_{k,m}(X,\Delta)$} is taken to be the algebra generated by monomials
\alert{$(*)$} of degree $\sum s|\alpha_s|=m$, satisfying partial degree
inequalities \alert{$(**)$}.  
\end{block}
\end{frame}

\begin{frame}
\frametitle{Orbifold jet differentials [continued]}\vskip-4pt
It is important to notice that if we consider the log pair
$(X,\lceil\Delta\rceil)$ with $\lceil\Delta\rceil=\sum\Delta_j$,
then\vskip5pt
\alert{\centerline{$\displaystyle
\bigoplus_mE_{k,m}(X,\Delta)~~\hbox{is a graded subalgebra of}~~
\bigoplus_mE_{k,m}(X,\lceil\Delta\rceil).
$}}\pause\vskip5pt
The subalgebra $E_{k,m}(X,\Delta)$ still has a multi-filtration induced
by the one on $E_{k,m}(X,\lceil\Delta\rceil)$, and, at least for
$\rho_j\in\bQ$, we formally have\vskip5pt
\alert{\centerline{$\displaystyle
G^\bullet E_{k,m}(X,\Delta)\subset
\bigoplus_{\ell_1+2\ell_2+\cdots+k\ell_k=m}
S^{\ell_1}T^*_X\langle\Delta^{(1)}\rangle\otimes \cdots\otimes S^{\ell_k}
T^*_X\langle\Delta^{(k)}\rangle,
$}}\pause\vskip5pt
where \alert{$T^*_X\langle\Delta^{(s)}\rangle$} is the
\alert{``$s$-th orbifold cotangent sheaf''} \hbox{generated by\kern-15pt}
\vskip5pt
\alert{\centerline{$\displaystyle
z_j^{-(1-s/\rho_j)_+}d^{(s)}z_j,~~1\leq j\leq p,~~~d^{(s)}z_j,~~p+1\leq j\leq n
$}}\vskip5pt
(which makes sense only after taking some Galois cover of $X$ ramifying at
sufficiently large order along $\Delta_j$).
\end{frame}

\begin{frame}
\frametitle{Projectivized jets and direct image formula}\vskip-5pt
\begin{block}{Green Griffiths bundles} Consider $X_k:=
J^kX/\bC^*={\rm Proj}\bigoplus_mE_{k,m}(X)$. This defines a bundle
$\pi_k:X_k\to X$ of weighted projective spaces whose fibers
are the quotients of $(\bC^n)^k\ssm\{0\}$ by the $\bC^*$ action\vskip5pt
\centerline{$
\lambda\cdot(\xi_1,\ldots,\xi_k)=
(\lambda\xi_1,\lambda^2\xi_2,\ldots,\lambda^k\xi_k).$}\vskip5pt\pause
Correspondingly, there is a tautological rank $1$ sheaf $\cO_{X_k}(m)$
[only invertible when ${\rm lcm}(1,...,k)\mid m$], and a
direct image formula\vskip5pt
\alert{\centerline{$
E_{k,m}(X)=(\pi_k)_*\cO_{X_k}(m)$}}
\end{block}\pause
In the \alert{logarithmic case}, we define similarly\vskip5pt
\alert{\centerline{$
X_k\langle\Delta\rangle:={\rm Proj}\bigoplus_mE_{k,m}(X,\Delta)$}}
\pause\vskip5pt
and let $\cO_{X_k\langle\Delta\rangle}(1)$ be the corresponding tautological
sheaf, so that\vskip5pt
\alert{\centerline{$
E_{k,m}(X,\Delta)=(\pi_k)_*\cO_{X_k\langle\Delta\rangle}(m)$}}
\end{frame}

\begin{frame}
\frametitle{Generalized Green-Griffiths-Lang conjecture}
\vskip-5pt
\begin{block}{Generalized GGL conjecture (very optimistic ?)}
If $(X,\Delta)$ is an orbifold of general type, in the sense that
$K_X+\Delta$ is a big $\bR$-divisor, then there is a \alert{proper
algebraic subvariety $Y\subsetneq X$} containing all
\alert{orbifold entire curves $f:\bC\to(X,\Delta)$}
(not contained in $\Delta$ and having multiplicity${}\geq\rho_j$
along $\Delta_j$).
\end{block}\pause\vskip-2pt
One possible strategy is to show that such orbifold entire curves $f$
must satisfy a lot of algebraic differential equations of the form
\alert{$P(f;f',...,f^{(k)})=0$ for $k\gg 1$}. This is based on:\pause\vskip-2pt
\begin{block}{Fundamental vanishing theorem}
\claim{\rm [Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996], ...}\\
Let $A$ be an ample divisor on $X$. Then, for all
global jet differential operators on $(X,\Delta)$ with coefficients
vanishing on $A$,
i.e.\ \alert{$P\in H^0(X,E_{k,m}(X,\Delta)\otimes\cO(-A))$},
and for all orbifold entire curves \alert{$f:\bC\to (X,\Delta)$}, 
one has \alert{$P(f_{[k]})\equiv 0.$}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Proof of the fundamental vanishing theorem}
\vskip-5pt  
\claim{\bf Simple case}.
First consider the compact case $(\Delta=0)$, and 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 \alert{$u_A(t)=P(f_{[k]})(t)$ is bounded}, and
must thus 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 $u_A\equiv 0$.
\vskip4pt\pause
\claim{\bf Logarithmic and orbifold cases}. In the orbifold case, one
must use instead an \alert{``orbifold metric'' $\omega$}. Removing
the hypothesis $f'$~bounded is more tricky. One possible way is to
use the Ahlfors lemma and some representation theory.
\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{\centerline{$
X(\theta,\leq q)=\bigcup_{j\leq q}X(\theta,j).$}}
\vskip5pt\pause
Then\vskip4pt
\alert{$\displaystyle\sum_{j=0}^q(-1)^{q-j}
h^j(X,L^{\otimes m}\otimes\cI(m\varphi))\le {m^n\over n!}
\int_{X(\theta,\leq q)}~(-1)^q\theta^n + o(m^n)$},
\vskip4pt
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{Holomorphic Morse inequalities [continued]}\vskip-4pt
\begin{block}{Consequence of the holomorphic Morse inequalities}
For $q\,{=}\,1$, with the same notation as above, we get a
\alert{\hbox{lower bound\kern-10pt}}\vskip5pt
\alert{$\strut\kern17pt{}
h^0(X,L^{\otimes m})\geq h^0(x,L^{\otimes m}\otimes\cI(m\varphi))$\vskip5pt
$\strut\kern72pt{}
\geq h^0(x,L^{\otimes m}\otimes\cI(m\varphi))-h^1(x,L^{\otimes m}\otimes
\cI(m\varphi))$\vskip5pt
$\displaystyle\strut\kern72pt{}\geq{m^n\over n!}
\int_{X(\theta,\leq 1)}\theta^n - o(m^n).$}
\end{block}\pause
here $\theta$ is a real $(1,1)$ form of arbitrary signature
on $x$.\vskip5pt\pause
when $\theta=\alpha-\beta$ for some explicit (1,1)-forms
$\alpha,\beta\geq 0$ (not necessarily closed), an easy lemma yields
\vskip5pt
\alert{\centerline{
${\bf 1}_{X(\alpha-\beta,\leq 1)}~(\alpha-\beta)^n\geq\alpha^n-n\alpha^{n-1}\wedge
\beta$}}\vskip2pt
hence\vskip2pt
\alert{\centerline{$\displaystyle
h^0(X,L^{\otimes m})\geq{m^n\over n!}
\int_X(\alpha^n-n\alpha^{n-1}\wedge\beta) - o(m^n).$}}
\end{frame}

\begin{frame}
\frametitle{Finsler metric on the $k$-jet bundles}
\vskip-5pt
Assume that $T_X$ is equipped with a $C^\infty$ connection $\nabla$ and
a hermitian metric $h$.\pause\
One then defines a \alert{''weighted Finsler metric''} on $J^kX$ by 
taking~ $b={\rm lcm}(1,2,...,k)$~ and, at each point
\hbox{$x\,{=}f(0)$,\kern-15pt}\vskip-21pt%
$$\alert{
\Psi_{h_k}(f_{[k]}):=\Big(\sum_{1\le s\le k}\varepsilon_s\Vert\nabla^sf(0)
\Vert_{h(x)}^{2b/s}\Big)^{1/b},~~1=\varepsilon_1\gg
\varepsilon_2\gg\cdots\gg\varepsilon_k.}
$$\vskip-7pt\pause%
Letting $\xi_s\,{=}\,\nabla^sf(0)$,
this$\,$can$\,$be$\,$viewed$\,$as$\,$a$\,$metric
$h_k\,$\hbox{on$\,L_k\,{:=}\,\cO_{X_k}(1)$,\kern-25pt}\break
and the curvature form of $L_k$ is obtained by computing
${i\over 2\pi}\ddbar\log\Psi_{h_k}(f_{[k]})$ as a function
of $(x,\xi_1,\ldots,\xi_k)$.\vskip5pt\pause
Modulo negligible error terms of the form $O(\varepsilon_{s+1}/\varepsilon_s)$,
this gives\vskip-18pt
\alert{$$
\Theta_{L_k,h_k}=\omega_{{\rm FS},k}(\xi)+{i\over 2\pi}
\sum_{1\le s\le k}{1\over s}{|\xi_s|^{2b/s}\over \sum_t|\xi_t|^{2b/t}}
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}
{\xi_{s\alpha}\overline\xi_{s\beta}\over|\xi_s|^2}\,dz_i\wedge d\overline z_j
$$}\vskip-13pt%
where \alert{$(c_{ij\alpha\beta})$} are the coefficients of the curvature tensor
\hbox{\alert{$\Theta_{T_X^*,h^*}$}~and\kern-10pt} 
\alert{$\omega_{{\rm FS},k}$ is the weighted Fubini-Study metric}
on the fibers of \hbox{$X_k\to X$.\kern-10pt}
\end{frame}

\begin{frame}
\frametitle{Evaluation of Morse integrals}
\vskip-6pt
The above expression is simplified by using polar coordinates\vskip4pt
\centerline{\alert{$x_s=\vert\xi_s\vert_h^{2b/s}$,~~~
$u_s=\xi_s/\vert\xi_s\vert_h=\nabla^sf(0)/\vert\nabla^sf(0)\vert$.}}\pause
\vskip4pt
In such polar coordinates, one gets the formula\vskip-18pt
\alert{$$
\Theta_{L_k,h_k}=\omega_{{\rm FS},k}(\xi)+
{i\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j
$$}\vskip-13pt
where $\omega_{{\rm FS},k}(\xi)$ is positive definite in $\xi$.\vskip5pt\pause
By holomorphic Morse inequalities, we need to evaluate an integral\vskip-11pt
\alert{$$
\int_{X_k(\Theta_{L_h,h_k},\,\leq 1)}\Theta_{L_k,h_k}^{N_k},~~~
N_k=\dim X_k=n+(kn-1),
$$}\vskip-10pt
and we have to integrate over the parameters $z\in X$, $x_s\in\bR_+$ and
$u_s$ in the unit sphere bundle $\bS(T_X,1)\subset T_X$.\vskip3pt\pause
Since the weighted projective space can be viewed
as a circle quotient of the pseudosphere $\sum|\xi_s|^{2b/s}=1$, 
we can take here $\sum x_s=1$, i.e.\ $(x_s)$ in
the $(k-1)$-dimensional simplex $\bDelta^{k-1}$.\vskip-10pt\strut
\end{frame}

\begin{frame}
\frametitle{Probabilistic interpretation of the curvature}
Now, the signature of $\Theta_{L_k,h_k}$ depends only on the
vertical terms, \hbox{i.e.\kern-15pt}\vskip-11pt
\alert{$$
\sum_{1\le s\le k}{1\over s}x_sq(u_s),~~q(u_s):=
{i\over 2\pi}\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j.
$$}\vskip-8pt\pause
After averaging over $(x_s)\in\bDelta^{k-1}$ and computing
the rational number $\int\omega_{{\rm FS},k}(\xi)^{nk-1}={1\over (k!)^n}$,
what is left is to evaluate Morse integrals with respect to
$(u_s)$ of ``horizontal'' $(1,1)$-forms
given by sums $\sum\frac{1}{s}q(u_s)$, where $u_s$ 
are ``random points'' on the unit sphere. \vskip4pt\pause
As $k\,{\to}\,{+}\infty$, this sum yields asymptotically a
\hbox{\alert{``Monte-Carlo'' integral\kern-15pt}}\vskip-8pt
$$
\Big(1+\frac{1}{2}+\cdots+\frac{1}{k}\Big)\int_{u\,\in\,\bS(T_X,1)}q(u)\,du.
$$\vskip-1pt\pause
Since $q$ is quadratic in $u$, we have
\hbox{\alert{$\displaystyle\int_{u\,\in\,\bS(T_X,1)}q(u)\,du=\frac{1}{n}\Tr(q)$}
and\kern-10pt}\vskip0pt
\alert{$$
\Tr(q)=\Tr(\Theta_{T_X^*,h^*})=\Theta_{\det T_X^*,\det h^*}=
\Theta_{K_X,\det h^*}.$$}
\end{frame}

\begin{frame}
\frametitle{Probabilistic cohomology estimate}\vskip-6pt
\vskip-5pt
\begin{block}{Theorem 1 (D-, Pure and Applied Math.\ Quarterly 2011)}
Fix $A$ ample line bundle on $X$,
$(T_X,h)$, $(A,h_A)$ hermitian structures on $T_X$, $A$, and
\alert{$\omega_A=\Theta_{A,h_A}>0$}.\pause\ \ Let
\alert{$\eta_\varepsilon=\Theta_{K_X,\det h^*}-\varepsilon\omega_A$} and\vskip3pt
\alert{$\displaystyle~\kern5mm
L_k=\cO_{X_k}(1)\otimes
\pi_k^*\cO_X\Big(-{1\over kn}\Big(1+{1\over 2}+\cdots+{1\over k}\Big)
\varepsilon A\Big),~~\varepsilon\in\bQ_+.
$}\pause\vskip3pt
Then for $m$ sufficiently divisible, we have a lower
bound\vskip5pt
\alert{$\displaystyle
h^0(X_k,L_k^{\otimes m})=h^0\Big(X,E_{k,m}(X)\otimes
\cO_X\Big(-{m\varepsilon\over kn}
\Big(1\,{+}{1\over 2}\,{+}\,\ldots\,{+}\,{1\over k}\Big)A\Big)\Big)$}
\alert{$\displaystyle
\strut\kern58pt{}\geq{m^{n+kn-1}\over (n+kr-1)!}
{(\log k)^n\over n!\,(k!)^n}\bigg(
\int_{X(\eta,\,\leq 1)}\eta_\varepsilon^n-\frac{C}{\log k}\bigg).$}\end{block}
\vskip-8pt\pause
\begin{block}{Corollary}
If $K_X$ is big and $\varepsilon>0$ is small, then $\eta_\varepsilon$
can be taken${}>0$, so
\alert{$h^0(X_k,L_k^{\otimes m})\geq C_{n,k,\eta,\varepsilon}\,m^{n+kn-1}$}
with $C_{n,k,\eta,\varepsilon}>0$, for $m\gg k\gg 1$.
\end{block}
\vskip-5pt\pause
There are in fact similar upper/lower bounds for all
\alert{$h^q(X_k,L_k^{\otimes m})$}.
\end{frame}

\begin{frame}
\frametitle{Non probabilistic cohomology estimate}\vskip-6pt
The Monte-Carlo estimate can be replaced by a non probabilistic one,
if one assumes an explicit lower bound for the curvature tensor\vskip5pt
\alert{\centerline{$
\Theta_{T^*_X,h^*}\geq -\gamma\otimes \Id,$}}\vskip5pt
where $\gamma\geq 0$ is a smooth $(1,1)$-form on $X$.\pause\vskip4pt
In case $X\subset \bP^N$ and $A=\cO(1)$, one can always take
\alert{$\gamma=2\omega_A$} where $\omega_A=\Theta_{A,h_A}>0$.\pause\vskip5pt
By Morse inequalities for differences 
${\bf 1}_{X(\alpha-\beta,\leq 1)}~(\alpha-\beta)^n$, one gets\vskip-2pt
\begin{block}{Theorem 2 (D-, Acta Math.\ Vietnamica 2012)}
Assume $k\geq n$ and $m\gg 1$. With the same notation as in Theorem~1,
the dimensions
$h^0(X_k,L_k^{\otimes m})$ are bounded below by\vskip-18pt
\alert{$$
{m^{n+kn-1}\over n!k!^n(n+kn-1)!}\int_X \big(\Theta_{K_X}+n\gamma\big)^n-c_{n,k}
\big(\Theta_{K_X}+n\gamma\big)^{n-1}\wedge (\varepsilon\omega_A+n\gamma),
$$}\vskip-14pt\pause
with $c_{n,k}\in\bQ_{>0}$ explicit,
$c_{n,k}\leq 4^{n-1}n!\big(1+\frac{1}{2}+\cdots+\frac{1}{k}\big)^n$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Logarithmic situation}\vskip-6pt
In the case of a log pair $(X,\Delta)$, one reproduce essentially
the same calculations, by replacing the cotangent bundle $T^*_X$
with the logarithmic cotangent bundle \alert{$T^*_X\langle\Delta\rangle$}.
\pause\ This gives
\begin{block}{Theorem 3 (probabilistic estimate)}
Put \alert{$\eta_\varepsilon=\Theta_{K_X+\Delta,\det h^*}-\varepsilon\omega_A$}.
\pause  For
$m\gg k\gg 1$, the dimensions\vskip4pt
\alert{\centerline{$h^0\big(X,E_{k,m}(X,\Delta)\otimes
\cO_X\big(-{m\varepsilon\over kn}
\big(1+{1\over 2}+\cdots+{1\over k}\big)A\big)\big)$}}
\vskip4pt
are bounded below by\vskip4pt
\alert{\centerline{$\displaystyle
{m^{n+kn-1}\over (n+kr-1)!}
{(\log k)^n\over n!\,(k!)^n}\bigg(
\int_{X(\eta,\,\leq 1)}\eta_\varepsilon^n-\frac{C}{\log k}\bigg),\quad
C>0.$}}
\end{block}\pause\vskip-5pt

\begin{block}{Theorem 4 (non probabilistic estimate)}
Assume \alert{$\Theta_{T^*_X\langle\Delta\rangle}\geq-\gamma\otimes\Id$}.
\pause
For $k\geq n, m\gg 1$, there are bounds\vskip-18pt
\alert{$$\strut\kern-5pt
{m^{n+kn-1}\over n!k!^n(n+kn-1)!}\kern-3pt\int_X\kern-6pt
\big(\Theta_{K_X+\Delta}+n\gamma\kern-1pt\big)^{\kern-2pt n}-c_{n,k}
\big(\Theta_{K_X+\Delta}+n\gamma\big)^{\kern-2pt n-1}\kern-3pt
\wedge(\varepsilon\omega_A+n\gamma\kern-1pt)\kern-1pt.
$$}\vskip-35pt\strut
\end{block}
\end{frame}

\begin{frame}
\frametitle{Orbifold situation}\vskip-6pt
Consider now the orbifold case $(X,\Delta)$,
$\Delta=\sum(1-{1\over\rho_j})\Delta_j$.\vskip1pt\pause
In this case, the solution is to work on the logarithmic projectivized
jet bundle $X_k\langle\lceil\Delta\rceil\rangle$, with Finsler metrics
$\Psi_{h_k}(f_{[k]})$ of the form\vskip-18pt
$$\alert{
\Bigg(\sum_{1\le s\le k}\varepsilon_s
\Bigg(\sum_{j=1}^p|f_j|^{-2(1-{s\over\rho_j})_+}|f_j^{(s)}(0)|^2
+\sum_{j=p+1}^n|f_j^{(s)}(0)|^2\Bigg)_{\kern-4pt h_s(f(0))\kern6pt}^{b/s}\Bigg)^{1/b}},
$$\vskip-5pt
where $h_s$ is a hermitian metric on the $s$-th orbifold bundle
\hbox{$T^*_X\langle\Delta^{(s)}\rangle$.\kern-5pt}\pause\vskip-3pt

\begin{block}{Theorem 5 (non probabilistic estimate [probabilistic
doesn't work])}
Assume
\alert{$\Theta_{T^*_X\langle\Delta^{(s)}\rangle}\geq-\gamma_s\omega\otimes\Id$}
in the sense of Griffiths, with $\omega=\Theta_A$ ($A$ ample),
$\gamma_s\geq 0$, and let \alert{$\Theta_s=\Theta_{K_X+\Delta^{(s)}}$} for
$s\,{=}\,1,...,k$.\pause\ Then, for $k\geq n$ and $m\gg 1$,
\alert{$h^0\big(X,E_{k,m}(X,\Delta)\otimes\cO_X(-m\varepsilon A)
\big)\geq$}\vskip3pt
\alert{$\displaystyle {m^{n+kn-1}\over n!(k!)^n(n+kn-1)!}
\bigg[\int_X\bigwedge\nolimits_{s=1}^n\big(\Theta_s+n\gamma_s\omega\big)
-{(2n-1)!\over(n-1)!^2}\times{}$}\vskip1pt
\alert{$
\displaystyle\kern2cm
\Big(\sum\nolimits_{s=1}^k{\gamma_s\over s}\Big)\Big(
\sum\nolimits_{s=1}^k{1\over s}(\Theta_s+n\gamma_s\omega)\Big)^{n-1}
\wedge\omega-O(\varepsilon)\bigg]$}.\vskip-18pt\strut
\end{block}
\end{frame}

\begin{frame}
\frametitle{Application to projective space}
\vskip-3pt  
Consider \alert{$\bP^n$} equipped with an \alert{orbifold divisor
\hbox{$\Delta=\sum_{j=1}^N(1-{1\over \rho_j})\Delta_j$}.\kern-5pt}\pause
\vskip-2pt

\begin{block}{Lemma: lower bound on the curvature of the cotangent bundle}
Put $A=\cO_{\bP^n}(1)$, \alert{$d_j=\deg\Delta_j$} and
\alert{$\gamma_0=\max\big({d_j\over\rho_j},2\big)$}.\pause\ Then 
$\forall\gamma>\gamma_0$, there exists a suitable hermitian metric
on $T^*_{\bP^n}\langle\Delta\rangle$ such that
\vskip5pt
\centerline{\alert{$\Theta_{T^*_{\bP^n}\langle\Delta\rangle}+
\gamma\,\omega_A\otimes\Id>0$}~~~(in the sense of Griffiths).}
\end{block}\pause\vskip-6pt
  
\begin{block}{Corollary: sufficient condition of existence of orbifold
differentials}
A sufficient condition for the existence of
negatively twisted orbifold order $k=n$ jet differentials on 
$\bP^n\langle\Delta\rangle$ is\vskip5pt
\centerline{$\displaystyle\alert{
\rho_j\ge \rho>n,\quad
\sum\nolimits_{j=1}^N d_j\ge c_n\,\max\bigg({d_j\over\rho_j},2\bigg)
\prod_{s=1}^n\Big(1-{s\over\rho}\Big)^{-1}.}
$}\vskip5pt
with $c_n=O((2n\log n)^n)$ an explicit constant.
\end{block}\pause\vskip-3pt

Example: \alert{$N=1$, $\rho_1\geq 2c_n$, $d_1\ge 4c_n$}.
\end{frame}

\begin{frame}
\frametitle{Generalization: case of orbifold directed varieties}
\vskip-5pt  
One can also consider a smooth \alert{directed variety $(X,V)$} with
a subbundle or subsheaf $V\subset T_X$ (e.g.\ a foliation), equipped
\alert{with an orbifold divisor $\Delta$ transverse to $V$}.\vskip7pt
\pgfdeclareimage[height=3.5cm]{drawing3}{drawing3}
\pgfdeclareimage[height=3.5cm]{drawing4}{drawing4}
$\rlap{\smash{\hbox{\pgfuseimage{drawing3}}}}$\pause\pgfuseimage{drawing4}
\vskip-2.8cm
\strut\kern4cm One then looks at entire curves
\RGBColor{0 0.7 0}{$f:\bC\to X$}\\
\strut\kern4cm that are \alert{tangent to $V$} and satisfy\\
\strut\kern4cm the \alert{ramification conditions specified by $\Delta$}.
\pause\vskip1.2cm
It is possible to define orbifold directed structures
\hbox{\alert{$V\langle\Delta^{(s)}\rangle\,{\subset}\,
T_X\langle\Delta^{(s)}\rangle$}\kern-15pt}\break
and corresponding jet differential bundles \alert{$E_{k,m}(X,V,\Delta)$}.\pause
\begin{block}{Theorem 6}
An \alert{existence criterion}
for sections of $E_{k,m}(X,V,\Delta)$ holds as well.
\end{block}  
\end{frame}

\begin{frame}
\frametitle{The end}
\strut\vskip1mm
\centerline{\huge\bf Thank you for your attention!}
\pgfdeclareimage[height=6.5cm]{CalabiYau}{CalabiYau}
\strut\kern2cm\pgfuseimage{CalabiYau}
\end{frame}

\end{document}

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