%\documentclass[12pt]{beamer} %\usepackage{pgf,pgfnodes,pgfautomata,pgfheaps} % For printing instead a handout \documentclass[handout]{beamer} \usepackage{pgf,pgfnodes,pgfautomata,pgfheaps} \usepackage{pgfpages} \pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm] % 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 . % % In principle, this file can be redistributed and/or modified under % the terms of the GNU Public License, version 2. % % However, this file is supposed to be a template to be modified % for your own needs. For this reason, if you use this file as a % template and not specifically distribute it as part of a another % package/program, I grant the extra permission to freely copy and % modify this file as you see fit and even to delete this copyright % notice. \mode { % \setbeamertemplate{background canvas}[vertical shading][bottom=red!10, % top=blue!10] \usetheme{Warsaw} \usefonttheme[onlysmall]{structurebold} } % or whatever \usepackage{amsmath,amssymb} \usepackage[latin1]{inputenc} \usepackage{colortbl} \usepackage[english]{babel} % Or whatever. Note that the encoding and the font should match. If T1 % does not look nice, try deleting the line with the fontenc. \title[ \ \kern-190pt Jean-Pierre Demailly (Grenoble I), 21/12/2007\kern53pt Jet differential operators and hyperbolic varieties / Hangzhou] % (optional, use only with long paper titles) {Algebraic structure of the ring of jet differential operators and hyperbolic varieties} %% \subtitle{Presentation Subtitle} % (optional) \author[] % (optional, use only with lots of authors) {Jean-Pierre Demailly} \institute[]{Institut Fourier, Universit\'e de Grenoble I, France} % - Use the \inst command only if there are several affiliations. % - Keep it simple, no one is interested in your street address. \date[]% (optional) {December 21 / ICCM 2007, Hangzhou} %%\subject{Talks} % This is only inserted into the PDF information catalog. Can be left % out. % If you have a file called "university-logo-filename.xxx", where xxx % is a graphic format that can be processed by latex or pdflatex, % resp., then you can add a logo as follows: \definecolor{ColClaim}{rgb}{0,0,0.8} \def\claim#1{{\color{ColClaim}#1}} \definecolor{Alert}{rgb}{0.8,0,0} \def\alert#1{{\color{Alert}#1}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Ker}{\operatorname{Ker}} \newcommand{\tors}{\operatorname{torsion}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\reg}{\operatorname{reg}} \newcommand{\bB}{{\mathbb B}} \newcommand{\bC}{{\mathbb C}} \newcommand{\bD}{{\mathbb D}} \newcommand{\bN}{{\mathbb N}} \newcommand{\bP}{{\mathbb P}} \newcommand{\bQ}{{\mathbb Q}} \newcommand{\bR}{{\mathbb R}} \newcommand{\bZ}{{\mathbb Z}} \newcommand{\cA}{{\mathcal A}} \newcommand{\cC}{{\mathcal C}} \newcommand{\cD}{{\mathcal D}} \newcommand{\cE}{{\mathcal E}} \newcommand{\cF}{{\mathcal F}} \newcommand{\cH}{{\mathcal H}} \newcommand{\cI}{{\mathcal I}} \newcommand{\cK}{{\mathcal K}} \newcommand{\cM}{{\mathcal M}} \newcommand{\cN}{{\mathcal N}} \newcommand{\cO}{{\mathcal O}} \newcommand{\cP}{{\mathcal P}} \newcommand{\cX}{{\mathcal X}} \newcommand{\dbar}{\overline\partial} \newcommand{\ddbar}{\partial\overline\partial} \newcommand{\ovl}{\overline} \newcommand{\wt}{\widetilde} \newcommand{\lra}{\longrightarrow} \newcommand{\bul}{{\scriptscriptstyle\bullet}} % mathematical operators \renewcommand{\Re}{\mathop{\rm Re}\nolimits} \renewcommand{\Im}{\mathop{\rm Im}\nolimits} \newcommand{\Pic}{\mathop{\rm Pic}\nolimits} \newcommand{\codim}{\mathop{\rm codim}\nolimits} \newcommand{\Id}{\mathop{\rm Id}\nolimits} \newcommand{\Sing}{\mathop{\rm Sing}\nolimits} \newcommand{\Supp}{\mathop{\rm Supp}\nolimits} \newcommand{\Vol}{\mathop{\rm Vol}\nolimits} \newcommand{\rank}{\mathop{\rm rank}\nolimits} \newcommand{\pr}{\mathop{\rm pr}\nolimits} \newcommand{\NS}{\mathop{\rm NS}\nolimits} \newcommand{\GG}{{\mathop{\rm GG}\nolimits}} \newcommand{\NE}{\mathop{\rm NE}\nolimits} \newcommand{\ME}{\mathop{\rm ME}\nolimits} \newcommand{\SME}{\mathop{\rm SME}\nolimits} \newcommand{\alg}{{\rm alg}} \newcommand{\nef}{{\rm nef}} \newcommand{\num}{\nu} \newcommand{\ssm}{\mathop{\Bbb r}} \newcommand{\smallvee}{{\scriptscriptstyle\vee}} % figures inserted as PostScript files \special{header=/home/demailly/psinputs/mathdraw/grlib.ps} \long\def\InsertFig#1 #2 #3 #4\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" #3}}#4$}} \long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}} \def\ovl{\overline} \def\build#1^#2_#3{\mathrel{\mathop{\null#1}\limits^{#2}_{#3}}} \def\bibitem[#1]#2#3{\medskip{\bf[#1]} #3} \begin{document} % Delete this, if you do not want the table of contents to pop up at % the beginning of each subsection: %%\AtBeginSubsection[] %%{ %% \begin{frame} %% \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=1cm]{icm-logo}{logo_iccm} \pgfuseimage{icm-logo} \pgfdeclareimage[height=1cm]{ujf-logo}{logo_ujf} \pgfuseimage{ujf-logo} \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} %%\def\pause{} \begin{frame} \frametitle{Kobayashi metric / hyperbolic manifolds} \begin{itemize} \item Let $X$ be a complex manifold, $n=\dim_\bC X$. \item $X$ is said to be \alert{hyperbolic in the sense of Brody} if there are no non-constant entire holomorphic curves $f:\bC\to X$. \pause \item Brody has shown that for $X$ compact, hyperbolicity is equivalent to the \alert{non degeneracy of the Kobayashi pseudo-metric} : $x\in X$, $\xi\in T_X$ $$k_x(\xi)=\inf\{\lambda>0\,;\;\exists f:\bD\to X,\;f(0)=x,\; \lambda f_*(0)=\xi\}$$\ \vskip-30pt\ \pause \item Hyperbolic varieties are especially interesting for their expected diophantine properties :\vskip1pt \claim{{\bf Conjecture} {\rm (S.\ Lang)}} {\it If a projective variety $X$ defined over~$\bQ$ is hyperbolic, then $X(\bQ)$ is finite.} \end{itemize} \end{frame} \begin{frame} \frametitle{Hyperbolicity and curvature} \begin{itemize} \item Case $n=1$ (compact Riemann surfaces):\\ $$ \begin{matrix} &X=\bP^1\kern50pt &(g=0,~~T_X>0)\cr &X=\bC/(\bZ+\bZ\tau)\hfill &(g=1,~~T_X=0)\cr \end{matrix} $$ obviously non hyperbolic : $\exists f:\bC\to X$. \pause \item If $g\ge 2,~~X\simeq \bD/\Gamma~~(T_X<0)$, then $X$ hyperbolic. \pause \item \claim{{\bf The $n$-dimensional case} {\rm (Kobayashi)}}\\ {\it If $T_X$ is negatively curved $(T^*_X>0$, i.e.\ ample$)$, then $X$ is hyperbolic.} \item \claim{{\bf Examples :}} $X=\Omega/\Gamma$, $\Omega$ bounded symmetric domain. \pause \item \claim{{\bf Conjecture GT.}} {\it Conversely, if a compact manifold $X$ is hyperbolic, then it should be of general type, i.e.\ $K_X=\Lambda^nT^*_X$ should be big and nef $($Ricci${}<0$, possibly with some degeneration$)$.} \end{itemize} \end{frame} \begin{frame} \frametitle{Conjectural characterizations of hyperbolicity} \begin{itemize} \item \claim{{\bf Theorem.}} {\it Let $X$ be projective algebraic. Consider the following properties :\\ {\rm (P1)} $X$ is \alert{hyperbolic}\\ {\rm (P2)} Every subvariety $Y$ of $X$ is of \alert{general type}.\\ {\rm (P3)} $\exists\varepsilon>0$, $\forall C\subset X$ algebraic curve $$ 2g(\bar C)-2\ge \varepsilon\deg(C).$$ \ \kern27pt$(X$ \alert{``algebraically hyperbolic''}$)$\\ {\rm (P4)} $X$ possesses a \alert{jet-metric with negative curvature} on its $k$-jet bundle $X_k$ $[$to be defined later$]$, for $k\ge k_0\gg 1$.\vskip3pt Then $\hbox{\rm(P4)}\Rightarrow \hbox{\rm(P1)},~\hbox{\rm(P2)},~ \hbox{\rm(P3)}$,\\ \ \kern26pt$\hbox{\rm(P1)}\Rightarrow \hbox{\rm(P3)}$,\\ and if Conjecture GT holds, $\hbox{\rm(P1)}\Rightarrow \hbox{\rm(P2)}$.} \pause \item It is expected that all 4 properties (P1), (P2), (P3), (P4) are equivalent for projective varieties. \end{itemize} \end{frame} \begin{frame} \frametitle{Green-Griffiths-Lang conjecture} \begin{itemize} \item \claim{{\bf Conjecture}} {\rm (Green-Griffith-Lang = GGL)} {\it Let $X$ be a projective variety of general type. Then there exists an algebraic variety $Y\subsetneq X$ such that for all non-constant holomorphic $f:\bC\to X$ one has $f(\bC)\subset Y$.} \pause \item Combining the above conjectures, we get :\vskip1pt \claim{{\bf Expected consequence}} {\rm (of GT + GGL)}\\ {\it {\rm (P1)} $X$ is \alert{hyperbolic}\\ {\rm (P2)} Every subvariety $Y$ of $X$ is of \alert{general type}\\ are equivalent.} \pause \item The main idea in order to attack GGL is to use differential equations. Let $$ \bC\to X,~~~t\mapsto f(t)=(f_1(t),\ldots,f_n(t)) $$ be a curve written in some local holomorphic coordinates $(z_1,\ldots,z_n)$ on $X$. \end{itemize} \end{frame} \begin{frame} \frametitle{Definition of algebraic differential operators} \begin{itemize} \item Consider \alert{algebraic differential operators} which can be written locally in multi-index notation \begin{eqnarray*} P(f_{[k]})&=&P(f',f'',\ldots,f^{(k)})\\ &=&\sum a_{\alpha_1\alpha_2\ldots\alpha_k}(f(t))~f'(t)^{\alpha_1} f''(t)^{\alpha_2}\ldots f^{(k)}(t)^{\alpha_k} \end{eqnarray*} where $a_{\alpha_1\alpha_2\ldots\alpha_k}(z)$ are holomorphic coefficients on $X$ and $t\mapsto z=f(t)$ is a curve, $f_{[k]}= (f',f'',\ldots,f^{(k)})$ \alert{its $k$-jet}. \pause Obvious $\bC^*$-action : $$\lambda \cdot f(t)=f(\lambda t),~~~ (\lambda \cdot f) ^{(k)}(t)=\lambda^kf^{(k)}(\lambda t)$$ $\Rightarrow$ \alert{weighted degree} $m=|\alpha_1|+2|\alpha_2|+\ldots+k|\alpha_k|$.\vskip 3pt \pause \item \claim{{\bf Definition.}} {\it $E^\GG_{k,m}$ is the sheaf $($bundle$)$ of algebraic differential operators of order $k$ and weighted degree $m$.} \end{itemize} \end{frame} \begin{frame} \frametitle{Vanishing theorem for differential operators} \begin{itemize} \item \claim{{\bf Fundamental vanishing theorem}\\ {\rm (Green-Griffiths $'$78, Demailly $'$95, Siu $'$96)}}\\ {\it Let $P\in H^0(X,E^\GG_{k,m}\otimes\cO(-A))$ be a global algebraic differential operator whose coefficients vanish on some ample divisor $A$. Then for any $f:\bC\to X$, \alert{$P(f_{[k]})\equiv 0$}.} \pause \item \claim{{\it Proof}}. One can assume that $A$ is very ample and intersects $f(\bC)$. Also assume $f'$ bounded (this is not so restrictive by Brody !). Then all $f^{(k)}$ are bounded by Cauchy inequality. Hence $$\bC\ni t\mapsto P(f',f'',\ldots, f^{(k)})(t)$$ is a bounded holomorphic function on $\bC$ which vanishes at some point. Apply Liouville's theorem !\qed \end{itemize} \end{frame} \begin{frame} \frametitle{Geometric interpretation of vanishing theorem} \begin{itemize} \item Let $X_k^{\GG}=J_k(X)^*/\bC^*$ be the \alert{projectivized $k$-jet bundle} of $X$ $=$ quotient of non constant $k$-jets by $\bC^*$-action.\\ Fibers are weighted projective spaces.\vskip1pt \claim{{\bf Observation.}} {\it If $\pi_k:X_k^\GG\to X$ is canonical projection and $\cO_{X_k^\GG}(1)$ is the \alert{tautological line bundle}, then $$ E^\GG_{k,m}=(\pi_k)_*\cO_{X_k^\GG}(m) $$}\vskip-15pt\ \pause \item Saying that $f:\bC\to X$ satisfies the differential equation $P(f_{[k]})=0$ means that $$f_{[k]}(\bC)\subset Z_P$$ where $Z_P$ is the zero divisor of the section $$\sigma_P\in H^0(X_k^\GG,\cO_{X_k^\GG}(m)\otimes\pi_k^*\cO(-A))$$ associated with $P$. \end{itemize} \end{frame} \begin{frame} \frametitle{Consequence of fundamental vanishing theorem} \begin{itemize} \item \claim{{\bf Consequence of fundamental vanishing theorem.}}{\it \\ If $P_j\in H^0(X,E^\GG_{k,m}\otimes\cO(-A))$ is a basis of sections then the image $f(\bC)$ lies in $Y=\pi_k(\bigcap Z_{P_j})$, hence property asserted by the GGL conjecture holds true if there are ``enough independent differential equations'' so that $$Y=\pi_k(\bigcap_j Z_{P_j})\subsetneq X.$$} \pause \item However, \alert{some differential equations are useless}. On a surface with coordinates $(z_1,z_2)$, a Wronskian equation $f'_1f''_2-f'_2f''_1=0$ tells us that $f(\bC)$ sits on a line, but $f''_2(t)=0$ says that the second component is linear affine in time, an essentially \alert{meaningless information} which is lost by a change of parameter $t\mapsto\varphi(t)$. \end{itemize} \end{frame} \begin{frame} \frametitle{Invariant differential operators} \begin{itemize} \item The $k$-th order Wronskian operator $$W_k(f)=f'\wedge f''\wedge \ldots\wedge f^{(k)}$$ (locally defined in coordinates) has degree $m=\frac{k(k+1)}{2}$ and \alert{$$W_k(f\circ\varphi)=\varphi^{\prime m}W_k(f)\circ \varphi.$$} \ \vskip-20pt\ \pause \item \claim{{\bf Definition.}} {\it A differential operator $P$ of order $k$ and degree $m$ is said to be invariant by reparametrization if $$P(f\circ\varphi)=\varphi^{\prime m}P(f)\circ \varphi$$ for any parameter change $t\mapsto\varphi(t)$. Consider their set $$E_{k,m}\subset E^\GG_{k,m}~~~\hbox{$($a subbundle$)$}$$} \ \vskip-12pt (Any polynomial $Q(W_1,W_2,\ldots W_k)$ is invariant, but for $k\ge 3$ there are other invariant operators.) \end{itemize} \end{frame} \begin{frame} \frametitle{Category of directed manifolds} \begin{itemize} \item \claim{{\bf Definition.}} {\it Category of directed manifolds :} \vskip3pt -- \alert{Objects} are pairs $(X,V)$ where $X$ is a complex manifold \phantom {-- }and $V\subset T_X$ (subbundle or subsheaf)\\ -- \alert{Arrows} $\psi:(X,V)\to(Y,W)$ are holomorphic maps \phantom {-- }with $\psi_*V\subset W$\\ \pause -- \alert{``Absolute case''} $(X,T_X)$\\ -- \alert{``Relative case''} $(X,T_{X/S})$ where $X\to S$\\ -- \alert{``Integrable case''} when $[V,V]\subset V$ (foliations) \pause \vskip7pt \item \claim{{\bf Fonctor ``1-jet'' :}} $(X,V)\mapsto (\tilde X,\tilde V)$ where : \begin{eqnarray*} &&\tilde X=P(V)={}\hbox{bundle of projective spaces of lines in $V$}\\ &&\pi:\tilde X=P(V)\to X,~~~(x,[v])\mapsto x,~~v\in V_x\\ &&\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} \begin{itemize} \item For every entire curve $f:(\bC,T_\bC)\to(X,V)$ tangent to $V$ \begin{eqnarray*} &&f_{[1]}(t):=(f(t),[f'(t)])\in P(V_{f(t)})\subset \tilde X\\ &&f_{[1]}:(\bC,T_\bC)\to(\tilde X,\tilde V)~~ \hbox{\alert{(projectivized 1st-jet)}} \end{eqnarray*}\ \vskip-28pt\ \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}} \begin{eqnarray*} &&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}\\ &&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 &&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}\\ &&0\to\cO_{X_k}\to \pi_k^\star V_{k-1}\otimes\cO_{X_k}(1) \to T_{X_k/X_{k-1}}\to 0~~\hbox{\alert{(Euler)}} \end{eqnarray*} \end{itemize} \end{frame} \begin{frame} \frametitle{Direct image formula} \begin{itemize} \item For $n=\dim X$ and $r=\rk V$, get a \alert{tower of $\bP^{r-1}$-bundles} $$\pi_{k,0}:X_k\build\to^{\pi_k}_{}X_{k-1}\to\cdots\to X_1 \build\to^{\pi_1}_{}X_0=X$$ with \alert{$\dim X_k=n+k(r-1)$, $\rk V_k=r$},\\ and \alert{tautological line bundles $\cO_{X_k}(1)$ on $X_k=P(V_{k-1})$}. \pause \item \claim{{\bf Theorem.}} {\it $X_k$ is a smooth compactification of $$X_k^{\GG,\reg}/G_k=J_k^{\GG,\reg}/G_k$$ where $G_k$ is the group of $k$-jets of germs of biholomorphisms of $(\bC,0)$, acting on the right by reparametrization: $(f,\varphi)\mapsto f\circ\varphi$, and $J_k^{\reg}$ is the space of $k$-jets of regular curves.} \pause \item \claim{{\bf Direct image formula.}} {\it \alert{$(\pi_{k,0})_*\cO_{X_k}(m) =E_{k,m}V^*={}$} invariant algebraic differential operators $f\mapsto P(f_{[k]})$\\ acting on germs of curves $f:(\bC,T_\bC)\to (X,V)$.} \end{itemize} \end{frame} \begin{frame} \frametitle{Results obtained so far} \begin{itemize} \item Using this technology and \alert{deep results of McQuillan} for curve foliations on surfaces, D.\ -- El Goul proved in 1998\\ \claim{{\bf Theorem.} {\rm (solution of Kobayashi conjecture)}}\\ {\it A very generic surface $X{\subset}\bP^3$ of \alert{degree${}\ge 21$} is hyperbolic.} (McQuillan got independently degree${}\ge 35$). \vskip3pt \pause \item The result was improved in 2004 by M.~P\v{a}un, \alert{degree${}\ge 18$} is enough, with ``generic'' instead of ``very generic''. Paun's technique exploits a new idea of Y.T.~Siu based on C.~Voisin's work, which consists of studying vector fields on the the \alert{universal jet space of the universal family of hypersurfaces of $\bP^{n+1}$} (with $n=2$ here). \pause \item \claim{{\bf Dimension 3 case.} {\rm (Erwan Rousseau, 2006--2007)}}\\ If $X\subset \bP^4$ is a \alert{generic $3$-fold} of degree $d$, then\\ -- for $d\ge 97$, every $f{:}\,\bC\to X$ satisfies a diff.\ equation.\\ -- for $d\ge 593$, every $f{:}\,\bC\to X$ is algebraically degenerate. \pause \end{itemize} \end{frame} \begin{frame} \frametitle{Algebraic structure of differential rings} \begin{itemize} \item Although very interesting, results are currently limited by \alert{lack of knowledge on jet bundles and differential operators} \item \claim{{\bf Unknown !}} {\it Is the ring of germs of invariant differential operators on $(\bC^n,T_{\bC^n})$ at the origin\\ \ \kern28pt $\displaystyle\cA_{k,n}=\bigoplus_m E_{k,m}T^*_{\bC^n}~~~ \hbox{finitely generated ?}$} \pause \item At least this is OK for $\forall n$,~$k\le 2$ and $n=2$, $k\le 4$: \alert{ \begin{eqnarray*} &&\kern-10pt\cA_{1,n}=\cO[f'_1,\ldots,f'_n]\\ &&\kern-10pt\cA_{2,n}=\cO[f'_1,\ldots,f'_n,W^{[ij]}],~~~ W^{[ij]}=f'_if''_j-f'_jf''_i\\ &&\kern-10pt \cA_{3,2}=\cO[f'_1,f'_2,W_1,W_2][W]^2,~~~W_i=f'_iDW-3f''_iW\\ &&\kern-10pt \cA_{4,2}=\cO[f'_1,f'_2,W_{11},W_{22},S][W]^6,~~~W_{ii}=f'_iDW_i-5f''_iW_i \end{eqnarray*}} where $W=f'_1f''_2-f'_2f''_1$~~~ is $2$-dim Wronskian and \phantom{where~~}$S=(W_1DW_2-W_2DW_1)/W$.~~~ Also known:\vskip2pt \ \kern6pt \alert{$\cA_{3,3}$ (E.~Rousseau, 2004), $\cA_{5,2}$ (J.\ Merker, 2007)} \end{itemize} \end{frame} \begin{frame} \frametitle{Strategy : evaluate growth of differential operators} \begin{itemize} \item The strategy of the proofs is that the algebraic structure of $\cA_{k,n}$ allows to compute the Euler characteristic $\chi(X,E_{k,m}\otimes A^{-1})$, e.g.\ on surfaces $$\chi(X,E_{k,m}\otimes A^{-1})=\frac{m^4}{648}(13c_1^2 - 9c_2) +O(m^3).$$ \pause \item Hence for $13c_1^2 - 9c_2>0$, using \alert{Bogomolov's vanishing theorem for $H^2(X,(T^*_X)^{\otimes m}\otimes A^{-1})$ for $m\gg 0$}, one gets $$h^0(X,E_{k,m}\otimes A^{-1})\ge\chi=h^0-h^1=\frac{m^4}{648}(13c_1^2 - 9c_2) +O(m^3)$$ \pause \item Therefore many global differential operators exist for surfaces with $13c_1^2 - 9c_2>0$, e.g.\ surfaces of degree large enough in $\bP^3$, $d\ge 15$ (\alert{end of proof uses stability}) \end{itemize} \end{frame} \begin{frame} \frametitle{Trouble / more general perspectives} \begin{itemize} \item \alert{Trouble is}, in higher dimensions $n$, intermediate coho- mology groups $H^q(X,E_{k,m}T^*_X)$, $00.$$}}\ \vskip-29pt\ \pause \item My PhD student S.\ Diverio has recently worked out this strategy for hypersurfaces $X\subset\bP^{n+1}$, with \begin{eqnarray*} &&L=\bigotimes\nolimits_{1\le j6$). One can check that \alert{$$d_2=15,~~~d_3=82,~~~d_4=329,~~~d_5=1222,~~~d_6~\hbox{exists}.$$} \end{itemize} \end{frame} \bibitem[Demailly85]{Demailly85} Demailly, J.-P.: \emph{Champs Magnétiques et Inégalités de Morse pour la $d"$-cohomologie}. Ann.\ Inst.\ Fourier (Grenoble) {\bf 35} (1985), no.\ 4, 189--229. \bibitem[Demailly95]{Demailly95} Demailly, J.-P.: \emph{Algebraic Criteria for Kobayashi Hyperbolic Projective Varieties and Jet Differentials}. Algebraic geometry -- Santa Cruz 1995, 285--360, Proc.\ Sympos.\ Pure Math., 62, Part 2, Amer.\ Math.\ Soc., Providence, RI, 1997. \bibitem[D-EG00]{D-EG00} Demailly, J.-P., El Goul, J.: \emph{Hyperbolicity of Generic Surfaces of High Degree in Projective $3$-Space}. Amer.\ J.\ Math.\ {\bf 122} (2000), no.\ 3, 515--546. \bibitem[F-H91]{F-H91} Fulton, W., Harris, J.: \emph{Representation Theory: A First Course}. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991, xvi+551~pp. \bibitem[G-G79]{G-G79} Green, M., Griffiths, P.: \emph{Two Applications of Algebraic Geometry to Entire Holomorphic Mappings}. The Chern Symposium 1979 (Proc.\ Internat.\ Sympos., Berkeley, Calif., 1979), pp.\ 41--74, Springer, New York-Berlin, 1980. \bibitem[Kobayashi70]{Kobayashi70} Kobayashi S.: \emph{Hyperbolic Manifolds and Holomorphic Mappings}. Marcel Dekker, Inc., New York 1970 ix+148 pp. \bibitem[Lang86]{Lang86} Lang S.: \emph{Hyperbolic and Diophantine analysis}, Bull.\ Amer.\ Math.\ Soc.\ {\bf 14} (1986), no.\ 2, 159--205. \bibitem[Rousseau05]{Rousseau05} Rousseau, E: \emph{Weak Analytic Hyperbolicity of Generic Hypersurfaces of High Degree in the Complex Projective Space of Dimension $4$}. arXiv:math/0510285v1 [math.AG]. \bibitem[Rousseau06a]{Rousseau06a} Rousseau, E.: \emph{\'Etude des Jets de Demailly-Semple en Dimension $3$}. Ann.\ Inst.\ Fourier (Grenoble) {\bf 56} (2006), no.\ 2, 397--421. \bibitem[Rousseau06b]{Rousseau06b} Rousseau, E: \emph{\'Equations Différentielles sur les Hypersurfaces de $\Bbb P\sp 4$}. J.\ Math.\ Pures Appl.\ (9) {\bf 86} (2006), no.\ 4, 322--341. \bibitem[Siu04]{Siu04} Siu, Y.-T.: \emph{Hyperbolicity in Complex Geometry}. The legacy of Niel Henrik Abel, 543--566, Springer, Berlin, 2004. \bibitem[Trapani95]{Trapani95} Trapani, S.: \emph{Numerical criteria for the positivity of the difference of ample divisors}, Math.\ Z.\ {\bf 219} (1995), no.\ 3, 387--401. \bibitem[Vojta87]{Vojta87} Vojta, P.: \emph{Diophantine Approximations and Value Distribution Theory}, Springer-Verlag, Lecture Notes in Mathematics no.\ 1239, 1987. \end{document}