% Hyperbolic algebraic varieties and holomorphic differential equations % % expanded version of VIASM Lecture (Hanoi, August 25-26, 2012) % Jean-Pierre Demailly % % Universit\'e de Grenoble I, Institut Fourier % Plain-TeX file \def\ref#1{??#1??} % page setting \magnification=1200 \pretolerance=500 \tolerance=1000 \brokenpenalty=5000 \hsize=14cm \hoffset=-3mm \vsize=21cm \voffset=-3mm \parskip3.5pt plus 1.5pt minus 1pt \parindent=0.6cm % new fonts definitions \let\em=\it \font\seventeenbf=ec-lmbx10 at 17.28pt \font\fourteenbf=ec-lmbx10 at 14.4pt \font\twelvebf=ec-lmbx10 at 12pt \font\eightbf=ec-lmbx10 \font\sixbf=ec-lmbx10 \font\twelvei=cmmi10 at 12pt \font\eighti=cmmi8 \font\sixi=cmmi6 \font\twelverm=ec-lmr10 at 12pt \font\eightrm=ec-lmr10 at 8pt \font\sixrm=ec-lmr10 at 6pt \font\eightsy=cmsy8 \font\sixsy=cmsy6 \font\eightit=ec-lmri10 at 8pt \font\eightsl=ec-lmro10 at 8pt \font\eighttt=ec-lmtt10 at 8pt \font\seventeenbsy=cmbsy10 at 17.28pt \font\fourteenbsy=cmbsy10 at 14.4pt \font\twelvebsy=cmbsy10 at 12pt \font\tenbsy=cmbsy10 \font\eightbsy=cmbsy8 \font\sevenbsy=cmbsy7 \font\sixbsy=cmbsy6 \font\fivebsy=cmbsy5 \font\tenmsa=msam10 \font\eightmsa=msam8 \font\sevenmsa=msam7 \font\fivemsa=msam5 \newfam\msafam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \def\msa{\fam\msafam\tenmsa} \font\tenmsb=msbm10 \font\eightmsb=msbm8 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\Bbb{\fam\msbfam\tenmsb} \font\tenCal=eusm10 \font\sevenCal=eusm7 \font\fiveCal=eusm5 \newfam\Calfam \textfont\Calfam=\tenCal \scriptfont\Calfam=\sevenCal \scriptscriptfont\Calfam=\fiveCal \def\Cal{\fam\Calfam\tenCal} \font\teneuf=eusm10 \font\teneuf=eufm10 \font\seveneuf=eufm7 \font\fiveeuf=eufm5 \newfam\euffam \textfont\euffam=\teneuf \scriptfont\euffam=\seveneuf \scriptscriptfont\euffam=\fiveeuf \def\euf{\fam\euffam\teneuf} \let\goth=\euf 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\def\Bbb{\fam\msbfam\eightmsb}% \abovedisplayskip=9pt plus 2pt minus 6pt \abovedisplayshortskip=0pt plus 2pt \belowdisplayskip=9pt plus 2pt minus 6pt \belowdisplayshortskip=5pt plus 2pt minus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@ntpc=\eightrm \let\smallf@ntpc=\sixrm \normalbaselines\rm} \catcode`\@=12 \def\eightpointbf{% \textfont0=\eightbf \scriptfont0=\sixbf \scriptscriptfont0=\fivebf \textfont1=\eightbfit \scriptfont1=\sixbfit \scriptscriptfont1=\fivebfit \textfont2=\eightbsy \scriptfont2=\sixbsy \scriptscriptfont2=\fivebsy \eightbf \baselineskip=10pt} \def\tenpointbf{% \textfont0=\tenbf \scriptfont0=\sevenbf \scriptscriptfont0=\fivebf \textfont1=\tenbfit \scriptfont1=\sevenbfit \scriptscriptfont1=\fivebfit \textfont2=\tenbsy \scriptfont2=\sevenbsy \scriptscriptfont2=\fivebsy \tenbf} \def\twelvepointbf{% \textfont0=\twelvebf \scriptfont0=\eightbf \scriptscriptfont0=\sixbf \textfont1=\twelvebfit \scriptfont1=\eightbfit \scriptscriptfont1=\sixbfit \textfont2=\twelvebsy \scriptfont2=\eightbsy \scriptscriptfont2=\sixbsy \twelvebf \baselineskip=14.4pt} \def\fourteenpointbf{% \textfont0=\fourteenbf \scriptfont0=\tenbf \scriptscriptfont0=\sevenbf \textfont1=\fourteenbfit \scriptfont1=\tenbfit \scriptscriptfont1=\sevenbfit \textfont2=\fourteenbsy \scriptfont2=\tenbsy \scriptscriptfont2=\sevenbsy \fourteenbf \baselineskip=17.28pt} \def\seventeenpointbf{% \textfont0=\seventeenbf \scriptfont0=\twelvebf \scriptscriptfont0=\eightbf \textfont1=\seventeenbfit\scriptfont1=\twelvebfit\scriptscriptfont1=\eightbfit \textfont2=\seventeenbsy \scriptfont2=\twelvebsy \scriptscriptfont2=\eightbsy \seventeenbf \baselineskip=20.736pt} % main item macros \newdimen\srdim \srdim=\hsize \newdimen\irdim \irdim=\hsize \def\NOSECTREF#1{\noindent\hbox to \srdim{\null\dotfill ???(#1)}} \def\SECTREF#1{\noindent\hbox to \srdim{\csname REF\romannumeral#1\endcsname}} \def\INDREF#1{\noindent\hbox to \irdim{\csname IND\romannumeral#1\endcsname}} \newlinechar=`\^^J \def\openauxfile{ \immediate\openin1\jobname.aux \ifeof1 \message{^^JCAUTION\string: you MUST run TeX a second time^^J} \let\sectref=\NOSECTREF \let\indref=\NOSECTREF \else \input \jobname.aux \message{^^JCAUTION\string: if the file has just been modified you may have to run TeX twice^^J} \let\sectref=\SECTREF \let\indref=\INDREF \fi \message{to get correct page numbers displayed in Contents or Index Tables^^J} \immediate\openout1=\jobname.aux \let\END=\end \def\end{\immediate\closeout1\END}} \newbox\titlebox \setbox\titlebox\hbox{\hfil} \newbox\sectionbox \setbox\sectionbox\hbox{\hfil} \def\folio{\ifnum\pageno=1 \hfil \else \ifodd\pageno \hfil {\eightpoint\copy\sectionbox\kern8mm\number\pageno}\else {\eightpoint\number\pageno\kern8mm\copy\titlebox}\hfil \fi\fi} \footline={\hfil} \headline={\folio} \def\blankline{\phantom{}\hfil\vskip0pt} \def\titlerunning#1{\setbox\titlebox\hbox{\eightpoint #1}} \def\title#1{\noindent\hfil$\smash{\hbox{\seventeenpointbf #1}}$\hfil \titlerunning{#1}\medskip} \def\titleleft#1{\noindent$\smash{\hbox{\seventeenpointbf #1}}$\hfil \titlerunning{#1}\medskip} \def\supersection#1{% \par\vskip0.5cm\penalty -100 \vbox{\baselineskip=17.28pt\noindent{{\fourteenpointbf #1}}} \vskip3pt \penalty 500 \titlerunning{#1}} \newcount\numbersection \numbersection=-1 \def\sectionrunning#1{\setbox\sectionbox\hbox{\eightpoint #1} \immediate\write1{\string\def \string\REF \romannumeral\numbersection \string{% \noexpand#1 \string\dotfill \space \number\pageno \string}}} \def\section#1{% \par\vskip0.5cm\penalty -100 \vbox{\baselineskip=14.4pt\noindent{{\twelvepointbf #1}}} \vskip2pt \penalty 500 \advance\numbersection by 1 \sectionrunning{#1}} \def\subsection#1|{% \par\vskip0.25cm\penalty -100 \vbox{\noindent{{\tenpointbf #1}}} \vskip1pt \penalty 500} \newcount\numberindex \numberindex=0 \def\startindex#1\par{\message{#1}} \def\index#1{#1 \advance\numberindex by 1 \immediate\write1{\string\def \string\IND \romannumeral\numberindex \string{% \ifnum\pageno=\pageno \fi \noexpand#1 \string\dotfill \space \string\S \number\numbersection, p.\string\ \space\number\pageno \string}}} \newdimen\itemindent \itemindent=\parindent \def\setitemindent#1{\setbox0=\hbox{#1~}\itemindent=\wd0} \def\item#1{\par\noindent\hangindent\itemindent% \rlap{#1}\kern\itemindent\ignorespaces} \def\itemitem#1{\par\noindent\hangindent2\itemindent% \kern\itemindent\rlap{#1}\kern\itemindent\ignorespaces} \def\itemitemitem#1{\par\noindent\hangindent3\itemindent% \kern2\itemindent\rlap{#1}\kern\itemindent\ignorespaces} \long\def\claim#1|#2\endclaim{\par\vskip 5pt\noindent {\tenpointbf #1.}\ {\em #2}\par\vskip 5pt} \def\proof{\noindent{\em Proof}} \def\bottomnote#1#2{\footnote{\hbox{#1}}{\eightpoint #2\vskip-2\parskip \vskip-\baselineskip}} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \catcode`\@=11 \newcount\@tempcnta \newcount\@tempcntb \def\timeofday{{% \@tempcnta=\time \divide\@tempcnta by 60 \@tempcntb=\@tempcnta \multiply\@tempcntb by -60 \advance\@tempcntb by \time \ifnum\@tempcntb > 9 \number\@tempcnta:\number\@tempcntb \else\number\@tempcnta:0\number\@tempcntb\fi}} \catcode`\@=12 \def\bibitem#1&% {\hangindent=1.66cm\hangafter=1 \noindent\rlap{\hbox{\eightpointbf #1}}\kern1.66cm{\rm #2}{\it #3}{\rm #4.}} % blackboard symbols \def\bB{{\Bbb B}} \def\bC{{\Bbb C}} \def\bD{{\Bbb D}} \def\bG{{\Bbb G}} \def\bH{{\Bbb H}} \def\bN{{\Bbb N}} \def\bP{{\Bbb P}} \def\bQ{{\Bbb Q}} \def\bR{{\Bbb R}} \def\bT{{\Bbb T}} \def\bU{{\Bbb U}} \def\bZ{{\Bbb Z}} \def\bOne{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} % calligraphic symbols \def\cA{{\Cal A}} \def\cB{{\Cal B}} \def\cC{{\Cal C}} \def\cD{{\Cal D}} \def\cE{{\Cal E}} \def\cF{{\Cal F}} \def\cI{{\Cal I}} \def\cJ{{\Cal J}} \def\cK{{\Cal K}} \def\cL{{\Cal L}} \def\cM{{\Cal M}} \def\cO{{\Cal O}} \def\cR{{\Cal R}} \def\cS{{\Cal S}} \def\cU{{\Cal U}} \def\cV{{\Cal V}} \def\cW{{\Cal W}} \def\cX{{\Cal X}} \def\cY{{\Cal Y}} % gothic symbols \def\ggl{{\goth gl}} \def\gpgl{{\goth pgl}} \def\gsl{{\goth sl}} \def\gm{{\goth m}} % bf or bfit mathematical characters \def\bfk{{\bf k}} \def\bfe{{\bf e}} \def\bfO{{\bf 0}} \def\bfa{{\bfit a}} \def\bfb{{\bfit b}} % special symbols %%\def\ii{{\rm i}\,} \def\ii{{\rm i}} \def\ld{,\ldots,} \def\bu{{\scriptstyle\bullet}} \def\ort{\mathop{\hbox{\kern1pt\vrule width4pt height0.4pt depth0pt \vrule width0.4pt height7pt depth0pt\kern3pt}}} \def\bigglp#1({\raise-#1\hbox{$\bigg($}} \def\biggrp#1){\raise-#1\hbox{$\bigg)$}} %\def\QED{{\hfill$\quad$QED}\medskip} \def\square{\hbox{% \vrule height 1.5ex width 0.1ex depth 0ex\kern-0.1ex \vrule height 1.5ex width 1.5ex depth -1.4ex\kern-1.5ex \vrule height 0.1ex width 1.5ex depth 0ex\kern-0.1ex \vrule height 1.5ex width 0.1ex depth 0ex}} \def\qed{\strut~\hfill\square\vskip6pt plus2pt minus1pt} %\def\semidirect{\mathop{\kern2pt\vrule depth-0.3pt height4.3pt %\kern-2pt\times}\nolimits} \def\hexnbr#1{\ifnum#1<10 \number#1\else \ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else \ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi} \def\msatype{\hexnbr\msafam} \def\msbtype{\hexnbr\msbfam} \mathchardef\restriction="3\msatype16 \let\restr\restriction \mathchardef\compact="3\msatype62 \mathchardef\complement="0\msatype7B \mathchardef\smallsetminus="2\msbtype72 \let\ssm\smallsetminus \mathchardef\subsetneq="3\msbtype28 \mathchardef\supsetneq="3\msbtype29 \mathchardef\leqslant="3\msatype36 \let\le\leqslant \mathchardef\geqslant="3\msatype3E \let\ge\geqslant \mathchardef\ltimes="2\msbtype6E \mathchardef\rtimes="2\msbtype6F % hats and tildes and over/underlines \let\ol=\overline \let\ul=\underline \let\wt=\widetilde \let\wh=\widehat \def\swt#1|{\smash{\widetilde#1}} \def\swh#1|{\smash{\widehat#1}} \def\build#1|#2|#3|{\mathrel{\mathop{\null#1}\limits^{#2}_{#3}}} \def\buildo#1^#2{\mathrel{\mathop{\null#1}\limits^{#2}}} \def\buildu#1_#2{\mathrel{\mathop{\null#1}\limits_{#2}}} % arrows \let\lra=\longrightarrow \def\lhra{\lhook\joinrel\longrightarrow} \mathchardef\rsa"3\msatype20 \def\vlra#1|{\hbox to#1mm{\rightarrowfill}} \def\vlhra#1|{\lhook\joinrel\hbox to#1mm{\rightarrowfill}} \let\Llra=\Longleftrightarrow \def\lraww{\mathrel{\rlap{$\longrightarrow$}\kern-1pt\longrightarrow}} \def\hdashpiece{\hbox{\vrule height2.45pt depth-2.15pt width2.3pt\kern1.2pt}} \def\dashto{\mathrel{\hdashpiece\hdashpiece\kern-0.5pt\hbox{\tenmsa K}}} \def\dasharrow{\mathrel{\hdashpiece\hdashpiece\hdashpiece \kern-0.3pt\hbox{\tenmsa K}}} \def\vdashpiece{\smash{\hbox{\vrule height2.3pt depth0pt width0.3pt}}} \def\vdashto{\mathrel{\smash{\hbox{\vbox{ \bgroup\baselineskip=3.5pt \vdashpiece\vdashpiece\smash{\raise2pt\hbox{\kern-3.1pt\tenex y}} \egroup}}}}} \def\vdasharrow{\mathrel{\smash{\hbox{\vbox{ \bgroup\baselineskip=3.5pt \vdashpiece\vdashpiece\vdashpiece \smash{\raise2pt\hbox{\kern-3.1pt\tenex y}} \egroup}}}}} \catcode`\@=11 \newdimen\@rrowlength \@rrowlength=6ex \def\ssrelbar{\vrule width\@rrowlength height0.64ex depth-0.56ex\kern-4pt} \def\llra#1{\@rrowlength=#1\ssrelbar\rightarrow} \catcode`\@=12 % mathematical operators \def\lcm{\mathop{\rm lcm}\nolimits} \def\card{\mathop{\rm card}\nolimits} \def\Re{\mathop{\rm Re}\nolimits} \def\Im{\mathop{\rm Im}\nolimits} \def\Id{\mathop{\rm Id}\nolimits} \def\Ker{\mathop{\rm Ker}\nolimits} \def\End{\mathop{\rm End}\nolimits} \def\Sol{\mathop{\rm Sol}\nolimits} \def\Hom{\mathop{\rm Hom}\nolimits} \def\Sym{\mathop{\rm Sym}\nolimits} \def\SM{\mathop{\rm SM}\nolimits} \def\Herm{\mathop{\rm Herm}\nolimits} \def\Aut{\mathop{\rm Aut}\nolimits} \def\Tr{\mathop{\rm Tr}\nolimits} \def\GL{\mathop{\rm GL}\nolimits} \def\PGL{\mathop{\rm PGL}\nolimits} \def\Alb{\mathop{\rm Alb}\nolimits} \def\Pic{\mathop{\rm Pic}\nolimits} \def\Psh{\mathop{\rm Psh}\nolimits} \def\Proj{\mathop{\rm Proj}\nolimits} \def\Supp{\mathop{\rm Supp}\nolimits} \def\Vol{\mathop{\rm Vol}\nolimits} \def\Ricci{\mathop{\rm Ricci}\nolimits} \def\Vect{\mathop{\rm Vect}\nolimits} \def\Span{\mathop{\rm Span}\nolimits} \def\codim{\mathop{\rm codim}\nolimits} \def\rank{\mathop{\rm rank}\nolimits} \def\div{\mathop{\rm div}\nolimits} \def\ord{\mathop{\rm ord}\nolimits} \def\mod{\mathop{\rm mod}\nolimits} \def\pr{\mathop{\rm pr}\nolimits} \def\Gr{\mathop{\rm Gr}\nolimits} \def\Ch{\mathop{\rm Ch}\nolimits} \def\Bs{\mathop{\rm Bs}\nolimits} \def\dbar{{\overline\partial}} \def\ddbar{{\partial\overline\partial}} % subscript and superscript operands \def\reg{{\rm reg}} \def\sing{{\rm sing}} \def\Sing{{\rm Sing}} \def\FS{{\rm FS}} \def\GG{{\rm GG}} \def\DR{{\rm DR}} \def\loc{{\rm loc}} \def\dev{{\rm dev}} \def\Zar{{\rm Zar}} \def\Const{{\rm Const}} % figures inserted as PostScript files \special{header=/home/demailly/psinputs/mathdraw/mdrlib.ps} \long\def\InsertFig#1 #2 #3 #4\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" /rotmem where not {(/home/demailly/psinputs/mathdraw/mdrlib.ps) run} if #3}}#4$}} \long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}} % Main text \openauxfile \title{On the existence of logarithmic} \title{and orbifold jet differentials} \titlerunning{F.\ Campana, L.\ Darondeau, J.-P.~Demailly, E.\ Rousseau, orbifold jet differentials} \bigskip \centerline{\twelvebf Frédéric Campana, Lionel Darondeau,}\medskip \centerline{\twelvebf Jean-Pierre Demailly, Erwan Rousseau} \vskip50pt \section{0. Introduction and main definitions} The present research is mainly concerned with the existence of logarithmic and orbifold jet differentials on projective varieties. For the sake of generality, and in view of potential applications to the case of foliations, we work throughout this paper in the category of directed varieties, and generalize them by introducing the concept of directed orbifold. \claim 0.1. Definition|Let $X$ be a complex manifold or variety. A directed structure $(X,V)$ on $X$ is defined to be a subsheaf $V\subset\cO(T_X)$ such that $\cO(T_X)/V$ is torsion free. A~morphism of directed varieties $\Psi:(X,V)\to (Y,W)$ is a holomorphic map $\Psi:X\to Y$ such that $d\Psi(V)\subset\Psi^*W$. We say that $(X,V)$ is non singular if $X$ is non singular and $V$ is locally free, i.e., is a holomorphic subbundle of $T_X$. \endclaim We refer to the {\it absolute case} as being the situation when $V=T_X$, the {\it relative case} when $V=T_{X/S}$ for some fibration $X\to S$, and the {\it foliated case} when $V$ is integrable, i.e.\ $[V,V]\subset V$, that is, $V$ is the tangent sheaf to a holomorphic foliation. We now combine these concepts with orbifold structures in the sense of Campana [Cam04]. \claim 0.2. Definition|A directed orbifold is a triple $(X,V,\Delta)$ where $(X,V)$ is a directed variety and $\Delta=\sum(1-{1\over\rho_j})\Delta_j$ an effective divisor where the $\Delta_j$ are irreducible hypersurfaces and $\rho_j\in{}]1,\infty]$. We denote by $\lceil\Delta\rceil=\sum\Delta_j$ the corresponding reduced divisor, and by $|\Delta|=\bigcup \Delta_j$ its support. \vskip2pt \item{\rm(a)} We will say that $(X,V,\Delta)$ is non singular if $(X,V)$ is non singular and $\Delta$ is a simple normal crossing divisor such that $\Delta$ is transverse to $V$. If $r=\rank(V)$, we mean by this that there are at most $r$ components $\Delta_j$ meeting at any point $x\in X$, and that for any $p$-tuple $(j_1,\ldots,j_p)$ of indices, $1\leq p\leq r$, we have $\dim V_x\cap \bigcap_{j=1}^pT_{\Delta_{j_\ell},x}=r-p$ at any point $x\in\bigcap_{j=1}^p\Delta_{j_\ell}$. \vskip2pt \item{\rm(b)} If $(X,V,\Delta)$ is non singular, the canonical divisor of $(X,V,\Delta)$ is defined to be $$ K_{V,\Delta}=K_V+\Delta $$ $($in additive notation$)$, where $K_V=\det V^*$. \vskip2pt \item{\rm(c)} The so called logarithmic case corresponds to all multiplicities $\rho_j=\infty$ being taken infi\-nite, so that $\Delta= \sum\Delta_j=\lceil\Delta\rceil$.\vskip0pt \endclaim In case $V=T_X$, we recover the concept of orbifold introduced in [Cam04], except possibly for the fact that we allow here $\rho_j\in\bR$, $\rho_j>1$ (even though the case $\rho_j\in\bN^*$ is of greater interest). It would certainly be interesting to investigate the case when $(X,V,\Delta)$ is singular, by allowing singularities in $V$ and tangencies between $V$ and $\Delta$, and to study whether the results discussed in this paper can be extended in some way, e.g.\ by introducing suitable multiplier ideal sheaves taking care of singularities, as was done in [Dem15] for the study of directed varieties $(X,V)$. For the sake of technical simplicity, we will refrain to do so here, and will therefore leave for future work the study of singular directed orbifolds. \claim 0.3. Definition|Let $(X,V,\Delta)$ be a singular directed orbifold. We say that $f:\bC\to X$ is an orbifold entire curve if $f$ is a non constant holomorphic map such that$\;:$\vskip2pt \item{\rm(a)} $f$ is tangent to $V$ $($i.e.\ $f'(t)\in V_{f(t)}$ at every point, or equivalently $f:(\bC,T_\bC)\to (X,V)$ is a morphism of directed varieties$\,;$ \vskip2pt \item{\rm(b)} $f(\bC)$ is not identically contained in $|\Delta|\,;$ \vskip2pt \item{\rm(c)} at every point $t_0\in\bC$ such that $f(t_0)\in\Delta_j$, $f$ meets $\Delta_j$ with ramification number${}\geq\rho_j$, i.e., if $\Delta_j=\{z_j=0\}$ near $f(t_0)$, then $z_j\circ f(t)$ vanishes with multiplicity${}\geq\rho_j$ at $t_0$. \vskip2pt\noindent In the case of a logarithmic component $\Delta_j$ $(\rho_j=\infty)$, condition {\rm(c)} is to be replaced by the assumption \vskip2pt \item{$({\rm c}')$} $f(\bC)$ does not meet~$\Delta_j$.\vskip0pt \endclaim \noindent One can now consider a category of directed orbifolds as follows. \claim 0.4. Definition|Consider directed orbifolds $(X,V,\Delta)$, $(Y,W,\Lambda)$ with $$ \Delta=\sum\Big(1-{1\over\rho_{\Delta,i}}\Big)\Delta_i,\qquad \Lambda=\sum\Big(1-{1\over\rho_{\Lambda,j}}\Big)\Lambda_j. $$ A morphism $\Psi:(X,V,\Delta)\to(Y,W,\Lambda)$ is a morphism $\Psi:(X,V)\to(Y,W)$ of directed varieties satisfying the additional following properties {\rm(a,b,c)}. \vskip2pt \item{\rm(a)} for every component $\Lambda_j$, $\Psi^{-1}(\Lambda_j)$ consists of a union of components $\Delta_i$, $i\in I(j)$, eventually after adding a number of extra components $\Delta_i$ with $\rho_{\Delta,i}=1\;;$ \vskip2pt \item{\rm(b)} in case $\rho_{\Lambda,j}<\infty$, for every $i\in I(j)$ and $z\in\Delta_i$, the derivatives $D^\alpha\Psi(z)$ of $\Psi$ at $z$, computed in suitable local coordinates on $X$ and $Y$, vanish for all multi-indices $\alpha\in\bN^n$ with $0<|\alpha|<\rho_{\Lambda,j}/\rho_{\Delta,i}\;;$ \vskip2pt \item{\rm(c)} if $\Lambda_j$ is a logarithmic component $(\rho_{\Lambda,j}=\infty)$, then $\Phi^{-1}(\Lambda_j)= \bigcup_{i\in I(j)}\Delta_i$ where the $(\Delta_i)_{i\in I(j)}$ consist of logarithmic components $(\rho_{\Delta,i}=\infty)$.\vskip0pt \vskip0pt \endclaim \noindent It is easy to check that the composite of directed orbifold morphisms is actually a directed orbifold morphism, and that the composition of an orbifold entire curve $f:\bC\to(X,V,\Delta)$ with a directed orbifold morphism $\Psi:(X,V,\Delta)\to(Y,W,\Lambda)$ produces an orbifold entire curve $\Psi\circ f:\bC\to(Y,W,\Lambda)$. One of our main goals is to investigate the following generalized Green-Griffiths conjecture \claim 0.5. Conjecture|Let $(X,V,\Delta)$ be a non singular directed orbifold of generated type, in the sense that the canonical divisor $K_V+\Delta$ is big. Then then should exist an algebraic subvariety $Y\subsetneq X$ containing all orbifold entire curves $f:\bC\to(X,V,\Delta)$. \endclaim \noindent As in the absolute case ($V=T_X$, $\Delta=0$), the idea is to show, at least as a first step towards the conjecture, that orbifold entire curves must satisfy suitable algebraic differential equations. In section~1, we introduce graded algebras $$ \bigoplus_{m\in\bN}E_{k,m}V^*\langle\Delta\rangle\leqno(0.6) $$ of sheaves of ``orbifold jet differentials''. These sheaves correspond to algebraic differential operators $P(f;f',f'',\ldots,f^{(k)})$ acting on germs of $k$-jets of curves that are tangent to $V$ and satisfy the ramification conditions prescribed by~$\Delta$. The strategy relies on the following standard vanishing theorem. \claim 0.7. Proposition|Let $(X,V,\Delta)$ be a projective non singular directed orbifold, and $A$ an ample divisor on $X$. Then, for every orbifold entire curve $f:\bC\to(X,V,\Delta)$ and every global jet differential operator $P\in H^0(X,E_{k,m}V^*\langle\Delta\rangle \otimes\cO_X(-A))$, we have $P(f;f',f'',\ldots,f^{(k)})=0$. \endclaim \noindent The next step consists precisely of finding sufficient conditions that ensure the existence of many global sections $P\in H^0(X,E_{k,m}V^*\langle\Delta\rangle\otimes\cO_X(-A))$. In this direction, among other more general results, we prove \claim 0.8. Theorem|Let $(X,V,\Delta)$ be a $n$-dimensional projective non singular directed orbifold. We assume that $\Delta=(1-{1\over\rho_1})\Delta_1$ has ramification index $\rho_1\geq n+1$, with a single component $\Delta_1\in|d_1A|$ of degree $d_1$ with respect to a {\rm very ample} divisor $A$ on~$X$. Then, for $\rho_1\geq n+1$, $\varepsilon\in\bQ_{>0}$ small and $$ n-1+d_1> n\,2^{2n-1}\Big(1+{1\over 2}+\ldots+{1\over n}\Big)^n {\rho_1^n\over{\rho_1-1\choose n}}, $$ there exist many $($i.e.\ at least $c\,m^{n+n^2-1}$, $c>0)$ orbifold jet differentials of order~$n$~in $$ H^0(X,E_{n,m}T^*_X\langle\Delta\rangle\otimes\cO_X(-m\varepsilon A)) $$ for $m\gg 1$ sufficiently divisible. \endclaim \section{1. Logarithmic and orbifold jet differentials} \subsection 1.A. Directed varieties and associated jet differentials| Let $(X,V)$ be a non singular directed variety. We set $n=\dim_\bC X$, $r=\rank_\bC V$, and following the exposition of [Dem97], we denote by $\pi_k:J^kV\to X$ the bundle of $k$-jets of holomorphic curves tangent to $V$ at each point. The canonical bundle of $V$ is defined to be $$ K_V=\det(V^*)=\Lambda^rV^*.\leqno(1.1) $$ If $f:(\bC,0)\to X$, $t\mapsto f(t)$ is a germ of holomorphic curve tangent to $V$, we denote by $f_{[k]}(0)$ its $k$-jet at ~$t=0$. For $x_0\in X$ given, we take a coordinate system $(z_1,\ldots,z_n)$ centered at $x_0$ such that $V_{x_0}=\Span({\partial\over\partial z_\mu})_{1\leq \mu\leq r}$. Then there exists a neighborhood $U$ of $x_0$ such that $V_{|U}$ admits a holomorphic frame $(e_\mu)_{1\leq\mu\leq r}$ of the form $$ e_\mu(z)={\partial\over\partial z_\mu}+\sum_{r+1\leq \lambda\leq n} a_{\lambda\mu}(z){\partial\over\partial z_\lambda},\quad 1\leq\mu\leq r,\leqno(1.2) $$ with $a_{\lambda\mu}(0)=0$. Germs of curves $f:(\bC,0)\to X$ tangent to $V_{|U}$ are obtained by integrating the system of ordinary differential equations $$ f'_\lambda(t)=\sum_{1\leq\mu\leq r}a_{\lambda\mu}(f(t))\,f'_\mu(t),\quad r+1\leq \lambda\leq n,\leqno(1.3) $$ when we write $f=(f_1,\ldots,f_n)$ in coordinates. Therefore any such germ of curve $f$ is uniquely determined by its initial point $z=f(0)$ and its projection $\tilde f=(f_1,\ldots,f_r)$ on the first $r$ coordinates. By definition, every $k$-jet $f_{[k]}\in J^kV_z=\pi_k^{-1}(z)$ is uniquely determined by its initial point $f(0)=z\simeq(z_1,\ldots,z_n)$ and the Taylor expansion of order $k$ $$ \tilde f(t)-\tilde f(0)=t\xi_1+{1\over 2!}t^2\xi_2+\cdots+{1\over k!} t^k\xi_k+O(t^{k+1}),\quad t\in D(0,\varepsilon),~\xi_s\in\bC^r,~1\leq s\leq k. \leqno(1.4) $$ Alternatively, we can pick an arbitrary local holomorphic connection $\nabla$ on $V_{|U}$ and represent the $k$-jet $f_{[k]}(0)$ by $(\xi_1,\ldots,\xi_k)$, where $\xi_s=\nabla^sf(0)\in V_z$ is defined inductively by $\nabla^1 f=f'$ and $\nabla^sf=\nabla_{f'}(\nabla^{s-1}f)$. This gives a local biholomorphic trivialization of $J^kV_{|U}$ of the form $$ J_kV_{|U}\to V_{|U}^{\oplus k},\qquad f_{[k]}(0)\mapsto(\xi_1,\ldots,\xi_k)=(\nabla f(0),\ldots,\nabla f^k(0))\,; \leqno(1.5) $$ the particular choice of the ``trivial connection'' $\nabla_0$ of $V_{|U}$ that turns $(e_\mu)_{1\leq\mu\leq r}$ into a parallel frame precisely yields the components $\xi_s\in V_{|U}\simeq\bC^r$ appearing in (1.4). We could of course also use a $C^\infty$ connection $\nabla=\nabla_0+\Gamma$ where $\Gamma\in C^\infty(U,T^*_X\otimes\Hom(V,V))$, and in this case, the corresponding trivialization (1.5) is just a $C^\infty$ diffeomorphism; the advantage, though, is that we can always produce such a global $C^\infty$ connection $\nabla$ by using a partition of unity on~$X$, and then (1.5) becomes a global $C^\infty$ diffeomorphism. Now, there is a global holomorphic $\bC^*$ action on $J^kV$ given at the level of germs by $f\mapsto\alpha\cdot f$ where $\alpha\cdot f(t):=f(\alpha t)$, $\alpha\in\bC^*$. With respect to our trivializations (1.5), this is the weighted $\bC^*$ action defined by $$ \alpha\cdot(\xi_1,\xi_2,\ldots,\xi_k)=(\alpha\xi_1, \alpha^2\xi_2,\ldots,\alpha^k\xi_k),\quad \xi_s\in V.\leqno(1.6) $$ We see that $J^kV\to X$ is an algebraic fiber bundle with typical fiber $\bC^{rk}$, and that the projectivized $k$-jet bundle $$ X_k(V):=(J^kV\ssm\{0\})/\bC^*,\qquad \pi_k:\smash{X_k(V)}\to X \leqno(1.7) $$ is a $P(1^{[r]},2^{[r]},\ldots,k^{[r]})$ weighted projective bundle over $X$, of total dimension $$ \dim X_k(V)=n+kr-1. \leqno(1.8) $$ \claim 1.9. Definition|We define $\cO_X(E_{k,m} V^*)$ to be the sheaf over $X$ of holomorphic functions $P(z\,;\,\xi_1,\ldots,\xi_k)$ on $J^kV$ that are weighted polynomials of degree $m$ in~$(\xi_1,\ldots,\xi_m)$. \endclaim \noindent In coordinates and in multi-index notation, we can write $$ P(z\,;\,\xi_1,\ldots,\xi_k)= \sum_{\scriptstyle\alpha_1,\ldots,\alpha_k\in\bN^r\atop \scriptstyle|\alpha_1|+2|\alpha_2|+\cdots+k|\alpha_k|=m} a_{\alpha_1\ldots\alpha_k}(z)\,\xi_1^{\alpha_1}\ldots\xi_k^{\alpha_k} $$ where the $a_{\alpha_1\ldots\alpha_k}(z)$ are holomorphic functions in $z=(z_1,\ldots,z_n)$ and $\xi_s^{\alpha_s}$ actually means $$ \xi_s^{\alpha_s}=\xi_{s,1}^{\alpha_{s,1}}\ldots\,\xi_{s,r}^{\alpha_{s,r}}\quad \hbox{for}~~ \xi_s=(\xi_{s,1},\ldots,\xi_{s,r})\in\bC^r,~~ \alpha_s=(\alpha_{s,1},\ldots,\alpha_{s,r})\in\bN^r, $$ and $|\alpha_s|=\sum_{j=1}^r\alpha_{s,j}$. Such sections can be interpreted as algebraic differential operators acting on holomorphic curves $f:D(0,R)\to X$ tangent to $V$, by putting $P(f):=u$ where $$ u(t)=\sum_{\scriptstyle\alpha_1,\ldots,\alpha_k\in\bN^r\atop \scriptstyle|\alpha_1|+2|\alpha_2|+\cdots+k|\alpha_k|=m} a_{\alpha_1\ldots\alpha_k}(f(t))\;f'(t)^{\alpha_1}\ldots\,f^{(k)}(t)^{\alpha_k}. \leqno(1.10) $$ Here $f^{(s)}(t)^{\alpha_s}$ is actually to be expanded as $$ f^{(s)}(t)^{\alpha_s}=f_1^{(s)}(t)^{\alpha_{s,1}}\ldots\,f_r^{(s)}(t)^{\alpha_{s,r}} $$ with respect to the components $f_j^{(s)}$ defined in (1.4). We also set $u=P(f\,;\,f',f'',\ldots,f^{(k)})$ when we want to make more explicit the dependence of the expression in terms of the derivatives of~$f$. We thus get a sheaf of graded algebras $$ \bigoplus_{m\in\bN}\cO_X(E_{k,m}V^*).\leqno(1.11) $$ Locally in coordinates, the algebra is isomorphic to the weighted polynomial ring $$ \cO_X\big[f_j^{(s)}\big]_{1\leq j\leq r,\,1\leq s\leq k},\quad \deg f_j^{(s)}=s\leqno(1.12) $$ over $\cO_X$. An immediate consequence of these definitions is~: \claim 1.13. proposition|The projectivized bundle $\pi_k:X_k(V)\to X$ can be identified with $$ \Proj\Bigg(\bigoplus_{m\in\bN}\cO_X(E_{k,m}V^*)\Bigg)\to X, \leqno{\rm(a)} $$ and, if $\cO_{X_k(V)}(m)$ denote the associated tautological sheaves, we have the direct image formula $$ (\pi_k)_*\smash{\cO_{X_k(V)}(m)}=\cO_X(E_{k,m} V^*). \leqno{\rm(b)} $$ \endclaim \claim 1.14. Remark|{\rm These objects where denoted $X_k^\GG$ and $E_{k,m}^\GG V^*$ in our previous paper [Dem97], as a reference to the work of Green-Griffiths [GG79], but we will avoid here the superscript GG to simplify the notation.} \endclaim \noindent Thanks to the Fa\`a di Bruno formula, a change of coordinates $w=\psi(z)$ on $X$ leads to a transformation rule $$ (\psi\circ f)^{(k)}=\psi'\circ f\cdot f^{(k)}+Q_\psi(f',\ldots,f^{(k-1}) $$ where $Q_\psi$ is a polynomial of weighted degree $k$ in the lower order derivatives. This shows that the transformation rule of the top derivative is linear and, as a consequence, the partial degree in $f^{(k)}$ of the polynomial $P(f\,;\,f',\ldots,f^{k)})$ is intrinsically defined. By taking the corresponding filtration and factorizing the monomials $(f^{(k)})^{\alpha_k}$ with polynomials in $f',f'',\ldots,f^{(k-1)}$, we get graded pieces $$ G^\bullet(E_{k,m}V^*)=\bigoplus_{\ell_k\in\bN} E_{k-1,m-k\ell_k}V^*\otimes S^{\ell_k}V^*. $$ By considering successively the partial degrees with respect to $f^{(k)}$, $f^{(k-1)}$, $\ldots\,$, $f'',f'$ and merging inductively the resulting filtrations, we get a multi-filtration such that $$ G^\bullet(E_{k,m}V^*)=\bigoplus_{\ell_1,\ldots,\ell_k\in\bN,\, \ell_1+2\ell_2+\cdots+k\ell_k=m}S^{\ell_1}V^*\otimes S^{\ell_2}V^*\otimes\cdots \otimes S^{\ell_k}V^*.\leqno(1.15) $$ \subsection 1.B. Logarithmic directed varieties| We now turn ourselves to the logarithmic case. Let $(X,V,\Delta)$ be a non singular logarithmic variety, where $\Delta=\sum\Delta_j$ is a simple normal crossing divisor. Fix a point $x_0\in X$. By the assumption that $\Delta$ is transverse to $V$, we can then select holomorphic coordinates $(z_1,\ldots,z_n)$ centered at $x_0$ such that $V_{x_0}=\Span({\partial\over\partial z_j})_{1\leq j\leq r}$ and $\Delta_j=\{z_j=0\}$, $1\leq j\leq p$, are the components of $\Delta$ that contain $x_0$ (here $p\leq r$ and we can have $p=0$ if $x_0\notin|\Delta|$). What we want is to introduce an algebra of differential operators, defined locally near $x_0$ as the weighted polynomial ring $$ \cO_X\big[(\log f_j)^{(s)}_{1\leq j\leq p}\,,(f_j^{(s)})_{p+1\leq j\leq r} \big]_{1\leq s\leq k},\quad \deg f_j^{(s)}=\deg(\log f_j)^{(s)}=s,\leqno(1.16) $$ or equivalently $$ \cO_X\big[(f_j^{-1}f_j^{(s)})_{1\leq j\leq p}\,,(f_j^{(s)})_{p+1\leq j\leq r} \big]_{1\leq s\leq k},\quad \deg f_j^{(s)}=s,~\deg f_j^{-1}=0.\leqno(1.16') $$ For this we notice that $$ \eqalign{ (\log f_1)''&=(f_1^{-1}f_1')'=f_1^{-1}f_1''-(f_1^{-1}f_1')^2,\cr \noalign{\vskip4pt} (\log f_1)'''&=f_1^{-1}f_1'''-3(f_1^{-1}f_1')(f_1^{-1}f_1'')+2 (f_1^{-1}f_1')^3,\ldots\,.\cr} $$ A similar argument easily shows that the above graded rings do not depend on the particular choice of coordinates made, as soon as they satisty $\Delta_j=\{z_j=0\}$. Now (as is well known in the absolute case $V=T_X$), we have a corresponding logarithmic directed structure $V\langle\Delta\rangle$ and its dual $V^*\langle\Delta\rangle$. If the coordinates $(z_1,\ldots,z_n)$ are chosen so that $V_{x_0}=\{dz_{r+1}=\ldots=dz_n=0\}$, then the fiber $V\langle\Delta\rangle_{x_0}$ is spanned by the derivations $$ z_1{\partial\over\partial z_1},\ldots,z_p{\partial\over\partial z_p},~ {\partial\over\partial z_{p+1}},\ldots,{\partial\over\partial z_r}. $$ The dual sheaf $\cO_X(V^*\langle\Delta\rangle)$ is the locally free sheaf generated by $$ {dz_1\over z_1},\ldots,{dz_p\over z_p},~dz_{p+1},\ldots,dz_r $$ [where the $1$-forms are considered in restriction to $\cO_X(V\langle\Delta\rangle)\subset\cO_X(V)\,$]. It follows from this that $\cO_X(V\langle\Delta\rangle)$ and $\cO_X(V^*\langle\Delta\rangle)$ are locally free sheaves of rank~$r$. By taking $\det(V^*\langle\Delta\rangle)$ and using the above generators, we find $$ \det(V^*\langle\Delta\rangle)=\det(V^*)\otimes\cO_X(\Delta)=K_V+\Delta \leqno(1.17) $$ in additive notation. Quite similarly to 1.13 and 1.15, we have~: \claim 1.18. Proposition|Let $\bigoplus_{m\in\bN}\cO_X(E_{k,m}V^* \langle\Delta\rangle)$ be the graded algebra defined in coordinates by $(1.16)$ or $(1.16')$. We define the logarithmic $k$-jet bundle to be $$ X_k(V\langle\Delta\rangle):= \Proj\Bigg(\bigoplus_{m\in\bN}\cO_X(E_{k,m}V^*\langle\Delta\rangle)\Bigg)\to X. \leqno{\rm(a)} $$ If $\cO_{X_k(V\langle\Delta\rangle)}(m)$ denote the associated tautological sheaves, we get the direct image formula $$ (\pi_k)_*\smash{\cO_{X_k(V\langle\Delta\rangle)}(m)}=\cO_X(E_{k,m} V^* \langle\Delta\rangle).\leqno{\rm(b)} $$ Moreover, the mult-filtration by the partial degrees in the derivatives $f_j^{(s)}$ has graded pieces $$ G^\bullet\big(E_{k,m}V^*\langle\Delta\rangle\big)= \bigoplus_{\ell_1,\ldots,\ell_k\in\bN,\, \ell_1+2\ell_2+\cdots+k\ell_k=m} S^{\ell_1}V^*\langle\Delta\rangle\otimes S^{\ell_2}V^*\langle\Delta\rangle\otimes\cdots\otimes S^{\ell_k}V^*\langle\Delta\rangle. \leqno{\rm(c)} $$ \endclaim \subsection 1.C. Orbifold directed varieties| We finally consider a non singular directed orbifold $(X,V,\Delta)$, where $\Delta=\sum(1-{1\over\rho_j})\Delta_j$ is a simple normal crossing divisor transverse to~$V$. Let $\lceil\Delta\rceil=\sum\Delta_j$ be the corresponding reduced divisor. By \S$\,$1.B, we have associated logarithmic sheaves $\cO_X(E_{k,m}V^*\langle\lceil\Delta\rceil\rangle)$. We want to introduce a graded subalgebra $$ \bigoplus_{m\in\bN}\cO_X(E_{k,m}V^*\langle\Delta\rangle)~~\subset~~ \bigoplus_{m\in\bN}\cO_X(E_{k,m}V^*\langle\lceil\Delta\rceil\rangle) \leqno(1.19) $$ in such a way that for every germ $P\in \cO_X(E_{k,m}V^*\langle\Delta\rangle)$ and every germ of orbifold curve $f:(\bC,0)\to(X,V,\Delta)$ the germ of meromorphic function $P(f)(t)$ is bounded at $t=0$ (hence holomorphic). Assume that $\Delta_1=\{z_1=0\}$ and that $f$ has multiplicity $q\geq \rho_1>1$ along~$\Delta_1$ at $t=0$. 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 $\beta\leq(1-{s\over \rho_1})_+$. At a point $x_0\in |\Delta_1|\cap\ldots\cap|\Delta_p|$, a sufficient condition for a monomial of the form $$ f_1^{-\beta_1}\ldots\,f_p^{-\beta_p}\prod_{s=1}^k\prod_{j=1}^r (f_j^{(s)})^{\alpha_{s,j}}, \quad \alpha_s=(\alpha_{s,j})\in\bN^r,~\beta_1,\ldots,\beta_p\in\bN \leqno(1.20) $$ to be bounded is to require that the multiplicities of poles satisfy $$ \beta_j\leq\sum_{s=1}^k\alpha_{s,j}\Big(1-{s\over \rho_j}\Big)_+,\quad 1\leq j\leq p. \leqno(1.20') $$ \claim 1.21. Definition|The subalgebra $\bigoplus_{m\in\bN}\cO_X(E_{k,m}V^*\langle\Delta\rangle)$ is taken to be the graded ring generated by monomials $(1.20)$ of degree $\sum s|\alpha_s|=m$, satisfying the pole multiplicity conditions $(1.20')$. These conditions do not depend on the choice of coordinates, hence we get a globally and intrinsically defined sheaf of algebras on~$X$. \endclaim \proof. We only have to prove the last assertion. Consider a change of variables $w=\psi(z)$ such that $\Delta_j$ can still be expressed as $\Delta_j=\{w_j=0\}$. Then, for $j=1,\ldots,p$, we can write $w_j=z_ju_j(z)$ with an invertible holomorphic factor~$u_j$. We need to check that the monomials~(1.20) computed with $g=\psi\circ f$ are holomorphic combinations of those associated with $f$. However, we have $g_j=f_ju_j(f)$, hence $g_j^{(s)}=\sum_{0\leq\ell\leq s}{s\choose\ell} f_j^{(\ell)}(u_j(f))^{(s-\ell)}$ by the Leibniz formula, and we see that $$ g_1^{-\beta_1}\ldots\,g_p^{-\beta_p}\prod_{s=1}^k\prod_{j=1}^r (g_j^{(s)})^{\alpha_{s,j}} $$ expands as a linear combination of monomials $$ f_1^{-\beta_1}\ldots\,f_p^{-\beta_p}\prod_{s=1}^k\prod_{j=1}^r \prod_{m=1}^{\alpha_{s,j}}f_j^{(\ell_{s,j,m})},\quad \ell_{s,j,m}\leq s, $$ multiplied by holomorphic factors of the form $$ \prod_{j=1}^pu_j(f)^{-\beta_j}\times \prod_{s=1}^k\prod_{j=1}^r\prod_{m=1}^{\alpha_{s,j}}(u_j(f))^{(s−\ell_{j,s,m})}. $$ However, we have $$ \beta_j\leq\sum_{s=1}^k\alpha_{s,j}\Big(1-{s\over \rho_j}\Big)_+ \leq\sum_{s=1}^k\sum_{m=1}^{\alpha_{s,j}} \Big(1-{\ell_{s,j,m}\over \rho_j}\Big)_+,\quad $$ so the $f$-monomials satisfy again the required multiplicity conditions for the poles~$f_j^{-\beta_j}$.\qed \noindent The above conditions $(1.20')$ suggest to introduce a sequence of ``differentiated'' orbifold divisors $$ \Delta^{(s)}=\sum_j\bigg(1-{s\over\rho_j}\bigg)_+\Delta_j. \leqno(1.22) $$ We say that $\Delta^{(s)}$ is the order $s$ orbifold divisor associated to~$\Delta$. By definition, the logarithmic components ($\rho_j=\infty$) of $\Delta$ remain logarithmic in $\Delta^{(s)}$, while all others eventually disappear when $s$ is large. Now, we introduce (in a purely formal way) a sheaf of rings $\smash{\wt\cO}_X=\cO_X[z_j^\bullet]$ by adjoining all positive real powers of coordinates $z_j$ such that \hbox{$\Delta_j=\{z_j=0\}$} is locally a component of~$\Delta$. Locally over~$X$, this can be done by taking the universal cover $Y$ of a punctured polydisk $$ D^*(0,r):=\prod_{1\leq j\leq p}D^*(0,r_j)\times \prod_{p+1\leq j\leq n}D(0,r_j)~~\subset~~ D(0,r):=\prod_{1\leq j\leq n}D(0,r_j) $$ in the local coordinates $z_j$ on $X$. If $\gamma:Y\to D^*(0,r) \hookrightarrow X$ is the covering map and $U\subset D(0,r)$ is an open subset, we can then consider the functions of~$\smash{\wt\cO}_X(U)$ as being defined on $\gamma^{-1}(U\cap D^*(0,r))$. In case $X$ is projective, one can even achieve such a construction globally by taking $Y$ to be the universal cover of a complement $X\ssm(|\Delta|\cup|\Delta'|)$, where $\Delta'=\sum\Delta'_\ell$ is a sum of very ample divisors such that $\Delta+\Delta'$ has simple normal crossings, and $\Delta_j\sim \Delta'_{\ell_1(j,m)}-\Delta'_{\ell_2(j,m)}$ with $\bigcup_mX\ssm (\Delta'_{\ell_1(j,m)}\cup\Delta'_{\ell_2(j,m)})=X$ for each $j$. In this setting, the subalgebra $\bigoplus_m\cO_X(E_{k,m}V^*\langle\Delta\rangle)$ still has a multi-filtration induced by the one on $\bigoplus_m\cO_X(E_{k,m}V^*\langle\lceil\Delta\rceil\rangle)$, and by extending the structure sheaf $\cO_X$ into $\smash{\wt\cO}_X$, we get an inclusion $$ \wt\cO_X(G^\bullet E_{k,m}V^*\langle\Delta\rangle)\subset \bigoplus_{\ell_1+2\ell_2+\cdots+k\ell_k=m} \wt\cO_X(S^{\ell_1}V^*\langle\Delta^{(1)})\rangle\otimes \cdots \otimes\wt\cO_X(S^{\ell_k}V^*\langle\Delta^{(k)}\rangle), \leqno(1.23) $$ $\wt\cO_X(V^*\langle\Delta^{(s)}\rangle)$ is the ``$s$-th orbifold (dual) directed structure'', generated by the order $s$ differentials $$ z_j^{-(1-s/\rho_j)_+}d^{(s)}z_j,~~1\leq j\leq p,~~~d^{(s)}z_j,~~p+1\leq j\leq r. \leqno(1.24) $$ By construction, we have $$ \det(\wt\cO_X(V^*\langle\Delta^{(s)}\rangle))=\wt\cO_X(K_V+\Delta^{(s)}). \leqno(1.25) $$ \claim 1.26. Remark|{\rm When $\rho_j=a_j/b_j\in\bQ_+$, one can find a finite ramified Galois cover $\gamma:Y\to X$ from a smooth projective variety $Y$ onto~$X$, such that the compositions $(z_j\circ\gamma)^{1/a_j}$ become single-valued functions $w_j$ on $Y$. In this way, the pull-back $\cO_Y(\gamma^*V^*\langle\Delta^{(s)}\rangle)$ is actually a locally free $\cO_Y$-module. However, we will adopt here an alternative viewpoint that suits much better our approach, namely that such an object is to be considered as a logarithmic directed structure equipped with an hoc class of hermitian singular metrics, see \S$\,$???. It then becomes possible to deal without trouble with real coefficients $\rho_j\in{}]1,\infty]$, and without having to worry about the existence of Galois covers (which a priori might not hold if $X$ is not projective). } \endclaim \section{2. Preliminaries on holomorphic Morse inequalities} \subsection 2.A. Basic results| We first recall the basic results concerning holomorphic Morse inequalities for smooth hermitian line bundles, first proved in [Dem85]. \claim 2.1. Theorem|Let $X$ be a compact complex manifolds, $E\to X$ a holomorphic vector bundle of rank $r$, and $(L,h)$ a hermitian line bundle. We denote by $\Theta_{L,h}={\ii\over 2\pi}D_h^2=-{\ii\over2\pi}\ddbar\log h$ the curvature form of $(L,h)$ and introduce the open subsets of $X$ $$ \cases{ X(L,h,q)=\big\{x\in X\,;\;\Theta_{L,h}(x)~\hbox{has signature $(n-q,q)$}\big\}, \cr \noalign{\vskip5pt} \displaystyle X(L,h,S)=\bigcup_{q\in S} X(L,h,q),\quad \forall S\subset\{0,1,\ldots,n\}.\cr} \leqno(*) $$ Then, for all $q=0,1,\ldots,n$, the dimensions $h^q(X,E\otimes L^m)$ of cohomology groups of the tensor powers $E\otimes L^m$ satisfy the following ``Strong Morse inequalities'' as $m\to +\infty\,:$ $$\sum_{0\le j\le q} (-1)^{q-j}h^j(X,E\otimes L^m) \le r {m^n\over n!} \int_{X(L,h,\le q)}(-1)^q\Theta_{L,h}^n+o(m^n), \leqno\SM(q):$$ with equality $\chi(X,E\otimes L^m)= r{m^n\over n!}\int_X \Theta_{L,h}^n + o(m^n)$ for the Euler characteristic $(q=n)$. \endclaim \noindent As a consequence, one gets upper and lower bounds for all cohomology groups, and especially a very useful criterion for the existence of sections of large multiples of $L$. \vskip2mm \claim 2.2. Corollary|Under the above hypotheses, we have \vskip2pt \item{\rm(a)} Upper bound for $h^q$ $($Weak Morse inequalities$)\,:$ $$h^q(X,E\otimes L^m)\le r {m^n\over n!}\int_{X(L,h,q)} (-1)^q \Theta_{L,h}^n + o(m^n)~.$$ \vskip2pt \item{\rm(b)} Lower bound for $h^0\,:$ $$ h^0(X,E\otimes L^m)\geq h^0-h^1\ge r{m^n\over n!}\int_{X(L,h,\le 1)}\Theta_{L,h}^n -o(m^n)~.$$ Especially $L$ is big as soon as $\int_{X(L,h,\le 1)}\Theta_{L,h}^n>0$ for some hermitian metric $h$ on~$L$. \vskip2pt \item{\rm(c)} Lower bound for $h^q\,:$ $$ h^q(X,E\otimes L^m)\geq h^q-h^{q-1}-h^{q+1}\geq r{m^n\over n!}\int_{X(L,h,\{q,q\pm 1\})} (-1)^q \Theta_{L,h}^n + o(m^n)~.$$ \endclaim \proof. (a) is obtained by taking $\SM(q)+\SM(q\,{-}\,1)$, (b) is equivalent to $-\SM(1)$ and (c) is equivalent to $-(\SM(q\,{+}\,1)+\SM(q\,{-}\,2))$.\qed \noindent The following simple lemma is the key to derive algebraic Morse inequalities from their analytic form (cf.\ [Dem94], Theorem~12.3). \claim 2.3.~Lemma|Let $\eta=\alpha-\beta$ be a difference of semipositive $(1,1)$-forms on an $n$-dimensional complex manifold~$X$, and let $\bOne_{\eta,\le q}$ be the characteristic function of the open set where $\eta$ is non degenerate with a number of negative eigenvalues at most equal to~$q$. Then $$ (-1)^q\bOne_{\eta,\le q}~\eta^n\le \sum_{0\le j\le q}(-1)^{q-j} {n\choose j}\alpha^{n-j}\wedge\beta^j, $$ in particular $$ \bOne_{\eta,\le 1}~\eta^n\ge \alpha^n-n\alpha^{n-1}\wedge \beta\qquad\hbox{for $q=1$.} $$ \endclaim \proof. Without loss of generality, we can assume $\alpha>0$ positive definite, so that $\alpha$ can be taken as the base hermitian metric on~$X$. Let us denote by $$ \lambda_1\ge\lambda_2\ge\ldots\ge\lambda_n\ge 0 $$ the eigenvalues of $\beta$ with respect to $\alpha$. The eigenvalues of $\eta=\alpha-\beta$ are then given by $$ 1-\lambda_1\le\ldots\le 1-\lambda_q\le 1-\lambda_{q+1}\le\ldots\le 1-\lambda_n, $$ hence the open set $\{\lambda_{q+1}<1\}$ coincides with the support of $\bOne_{\eta,\le q}$, except that it may also contain a part of the degeneration set $\eta^n=0$. On the other hand we have $${n\choose j}\alpha^{n-j}\wedge\beta^j=\sigma_n^j(\lambda)\,\alpha^n,$$ where $\sigma_n^j(\lambda)$ is the $j$-th elementary symmetric function in the $\lambda_j$'s. Thus, to prove the lemma, we only have to check that $$\sum_{0\le j\le q}(-1)^{q-j}\sigma_n^j(\lambda)- \bOne_{\{\lambda_{q+1}<1\}}(-1)^q\prod_{1\le j\le n}(1-\lambda_j)\ge 0.$$ This is easily done by induction on~$n$ (just split apart the parameter $\lambda_n$ and write $\sigma_n^j(\lambda)= \sigma_{n-1}^j(\lambda)+\sigma_{n-1}^{j-1}(\lambda)\,\lambda_n$).\qed \claim 2.4.~Corollary|Assume that $\eta=\Theta_{L,h}$ can be expressed as a difference $\eta=\alpha-\beta$ of smooth $(1,1)$-forms $\alpha,\beta\geq 0$. Then we have $$\sum_{0\le j\le q} (-1)^{q-j}h^j(X,E\otimes L^m) \leq r {m^n\over n!} \int_X\sum_{0\le j\le q}(-1)^{q-j}{n\choose j}\alpha^{n-j}\wedge\beta^j+o(m^n), \leqno\SM(q):$$ and in particular, for $q=1$, $$h^0(X,E\otimes L^m)\geq h^0-h^1\geq r{m^n\over n!}\int_X\alpha^n-n\alpha^{n-1}\wedge\beta+o(m^n). $$ \endclaim \claim 2.5. Remark|{\rm These estimates are consequences of Theorem~2.1 and Lemma 2.3, by taking the integral over $X$. The estimate for $h^0$ was stated and studied by Trapani [Tra93]. In the special case $\alpha=\Theta_{A,h_A}>0$, $\beta=\Theta_{B,h_B}>0$ where $A,B$ are ample line bundles, a direct proof can be obtained by purely algebraic means, via the Riemann-Roch formula. However, we will later have to use Corollary 2.4 in case $\alpha$ and $\beta$ are not closed, a situation in which no algebraic proof seems to exist.} \endclaim \subsection 2.B. Singular holomorphic Morse inequalities| The case of singular hermitian metrics has been considered in Bonavero's PhD thesis {\rm [Bon93]} and will be important for~us. We assume that $L$ is equipped with a singular hermitian metric $h=e^{-\varphi}$ with analytic singularities, i.e., on an neighborhood $V\ni x_0$ of an arbirary point $x_0\in X$, the weight $\varphi$ is of the form $$ \varphi(z)=c\log\sum_{1\leq j\leq N}|g_j|^2+u(z) $$ where $g_j\in\cO_X(V)$ and $u\in C^\infty(V)$. In such a situation, one introduces the multiplier ideal sheaves $$ \cI(h^m)=\cI(k\varphi)=\big\{f\in\cO_{X,x},\;\;\exists V\ni x,~ \int_V|f(z)|^2e^{-m\varphi(z)}d\lambda(z)<+\infty\big\}\leqno(2.6) $$ and the $q$-index sets $$ X(L,h,q)=\big\{x\in X\ssm\varphi^{-1}(-\infty)\,;\; \Theta_{L,h}(x)~\hbox{has signature $(n-q,q)$}\big\}. \leqno(2.7) $$ \claim 2.8. Theorem {\rm([Bon93])}|Morse inequalities still hold in the context of singular hermitian metric with analytic singularities, provided the cohomology groups under consideration are twisted by the appropriate multiplier ideal sheaves, i.e.\ replaced by $H^q(X,E\otimes L^m\otimes\cI(h^m))$. \endclaim \subsection 2.C. Morse inequalities and semi-continuity| Let $\cX\to S$ be a proper and flat morphism of reduced complex spaces, and let $(X_t)_{t\in S}$ be the fibers. Given a sheaf $\cE$ over $\cX$ of locally free $\cO_\cX$-modules of rank $r$, inducing on the fibres a family of sheaves $(E_t\to X_t)_{t\in S}$, the following semicontinuity property holds ([CRAS]): \claim 2.9. Proposition|For every $q\geq0$, the alternate sum $$ t\mapsto h^q (X_t,E_t)-h^{ q-1} (X_t,E_t)+. . .+(-1)^q h^0 (X_t,E_t) $$ is upper semicontinuous with respect to the (analytic) Zariski topology on~$S$. \endclaim Now, if $\cL\to\cX$ is an invertible sheaf equipped with a smooth hermitian metric $h$, and if $(h_t)$ are the fiberwise metrics on the family $(L_t\to X_t)_{t\in S}$, we get $$ \sum_{j=0}^q(-1)^{q-j}h^j(X_t,E_t\otimes L_t^{\otimes m}) \leq r{m^n\over n!}\int_{X(L_0,h_0,\leq q)} (-1)^q\Theta_{L_0,h_0}^n + \delta(t)m^n, \leqno(2.10) $$ where $\delta(t)\to 0$ as $t\to 0$. In fact, the proof of holomorphic Morse inequalities shows that the inequality holds uniformly on every relatively compact $S'\compact S$, with $$ I(t)=\int_{X(L_t,h_t,\leq q)}(-1)^q\Theta_{L_t,h_t}^n= \int_X (-1)^q\bOne_{X(L_t,h_t,\leq q)}\Theta_{L_t,h_t}^n $$ in the right hand side, and $t\mapsto I(t)$ is clearly continuous with respect to the ordinary topology. In other words, the Morse integral computed on the central fibers provides uniform upper bounds for cohomology groups of $E_t\otimes L_t^{\otimes m}$ when $t$ is close to $0$ in ordinary topology (and also, as a consequence, for $t$ in a complement $S\ssm \bigcup S_m$ of at most countably many analytic strata $S_m\subsetneq S$). \claim 2.11. Remark|{\rm Similar results would hold when $h$ is a singular hermitian metric with analytic singularities on $\cL\to\cX$, under the restriction that the families of multiplier ideal sheaves $(\cI(h_t^m))_{t\in S}$ ``never jump''.} \endclaim \subsection 2.D. Case of filtered bundles| Let $E\to X$ be a vector bundle over a variety, equipped with a filtration (or multifiltration) $F^p(E)$, and let $G=\bigoplus G^p(E)\to X$ be the graded bundle associated to this filtration. \claim 2.12. Lemma|In the above setting, one has for every $q\geq 0$ $$ \sum_{j=0}^q(-1)^{q-j}h^j(X,E)\leq\sum_{j=0}^q(-1)^{q-j}h^j(X,G). $$ \endclaim \proof. One possible argument is to use the well known fact that there is a family of filtered bundles $(E_t\to X)_{t\in \bC}$ (with the same graded pieces $G^p(E_t)=G^p(E)$), such that $E_t\simeq E$ for all $t\neq 0$ and $E_0\simeq G$. The result is then an immediate consequence of the semi-continuity result~2.9. A more direct very elementary argument can be given as follows: by transitivity of inequalities, it is sufficient to prove the result for simple filtrations; then, by induction on the length of filtrations, it is sufficient to prove the result for exact sequences $0\to S\to E\to Q\to 0$ of vector bundles on $X$. Consider the associated (truncated) long exact sequence in cohomology: $$ \eqalign{ 0\to H^0(S)\to H^0(E)\to H^0(Q)&\build\to|\delta_1||\cdots\cr &\build\to|\delta_{q-1}|| H^q(S)\to H^q(E)\to H^q(Q)\build\to|\delta_q|| \Im(\delta_q)\to 0.\cr} $$ By the rank theorem of linear algebra, $$ 0\leq\rank(\delta_q) = (-1)^q\sum_{j=0}^q(-1)^j(h^j(X,Q)- h^j(X,E)+ h^j(X,S)). $$ The result follows, since here $h^j(X,G)=h^j(X,Q)+h^j(X,S)$. \qed \subsection 2.E. Rees deformation construction (after Cadorel)| In this short paragraph, we outline a nice algebraic interpretation by Beno\^it Cadorel of certain semi-continuity arguments for cohomology group dimensions that underline the analytic approach of [Dem11, Lemma~2.12 and Prop.~2.13] and [Dem12, Prop.~9.28]; we will anyway explain again the essential points in \S3, since we have to deal here with a more general situation. Recall after [Cad, Prop.~4.2, Prop.~4.5], that Rees deformation construction allows one to construct natural deformations of Green-Griffiths jets spaces to weighted projectivized bundles. Let $(X,\Delta)$ be a smooth orbifold pair, and $\pi\colon Y\to(X,\Delta)$ be an adapted covering. For $k \in \bZ_{+}$, recall that the Green-Griffiths jet bundle of graded algebras $E_{k,\bullet }\Omega_{(\pi,\Delta)}\to Y$ admits a natural filtration, the {\it Green-Griffiths filtration} ([GrGr,CDR]) , with associated graded bundle (of graded algebras): $$ E_{k,\bullet}^{\rm lin}\Omega_{(\pi,\Delta)} = \bigoplus_{N\geq1} \bigoplus_{\ell_1+2\ell_2+\cdots+k\ell_k=N} \Sym^{\ell_1}\big(\Omega_{\pi,\Delta^{(1)}}\big) \otimes \cdots \otimes \Sym^{\ell_1}\big(\Omega_{\pi,\Delta^{(k)}}\big). $$ Applying the Proj functor, one gets a weighted projective bundle: $$ \bP_{(1,\cdots,k)}\left( \Omega_{\pi,\Delta^{(1)}} \oplus \cdots \oplus \Omega_{\pi,\Delta^{(k)}} \right) = \Proj\Big( E_{k,\bullet}^{\rm lin}\Omega_{(\pi,\Delta)} \Big) \build\to|\rho_k|| Y, $$ Then, following mutadis mutandus the arguments of Cadorel, one constructs a family $Y\build\leftarrow|p_k||\cY_k\to \bC$ parametrized by $\bC$, with a canonical line bundle $\cO_{\cY_k}(1)$ such that: \item{$\bu$} the central fiber $\cY_{k,0}$ is $ \bP_{(1,\cdots,k)}\left( \Omega_{\pi,\Delta^{(1)}} \oplus \cdots \oplus \Omega_{\pi,\Delta^{(k)}} \right) $ and the restriction of $\cO_{\cY_k}(1)$ conincide with the canonical line bundle of this weighted projective bundle. Hence ${\rho_k}_\ast O_{\cY_{k,0}}(N)= E_{k,N}^{\rm lin}\Omega_{(\pi,\Delta)}$. \item{$\bu$} other fibers $\cY_{k,\lambda}$ are isomorphic to the singular variety $J_k(\pi,\Delta)???diagup???\bC^*$, for the natural $\bC^*$-action by homotheties, and ${\rho_k}_\ast\cO_{\cY_{k,\lambda}}(N)\simeq E_{k,N}\Omega_{(\pi,\Delta)}$. \medskip \noindent Applying the semicontinuity result of [CRAS], and working with holomorphic inequalities, we obtain a control about dimensions of cohomology spaces of $E_{k,N}\Omega$ in terms of dimensions of cohomology spaces of the much simpler $E_{k,N}^{\rm lin}\Omega$. In particular, one can work directly on the weighted projective space with symmetric differentials, and it is not necessary to define higher order jet metrics. \claim Lemma|For any $j\in\bZ_+$: $$ (-1)^{j} \sum_{i=0}^{j} (-1)^{i} h^i(Y,E_{k,N}^{\rm lin}\Omega_{(\pi,\Delta)}) \geq (-1)^{j} \sum_{i=0}^{j} (-1)^{i} h^i(Y,E_{k,N}\Omega_{(\pi,\Delta)}). $$ In particular: $$ h^0(Y,E_{k,N}\Omega_{(\pi,\Delta)}) \geq h^0(Y,E_{k,N}^{\rm lin}\Omega_{(\pi,\Delta)}) - h^1(Y,E_{k,N}^{\rm lin}\Omega_{(\pi,\Delta)}). $$ \endclaim \section{3. On the curvature of jet metrics} Near any given point $z_0\in X$, we can choose a local holomorphic coordinate frame $(e_\lambda(z))_{1\le\lambda\leq r}$ of $V$ on an open set $U\ni z_0$, such that $$ \langle e_\lambda(z),e_\mu(z)\rangle =\delta_{\lambda\mu}+ \sum_{1\le i,j\le n,\,1\le\lambda,\mu\le r}c_{ij\lambda\mu}z_i\overline z_j+ O(|z|^3)\leqno(1.22) $$ for suitable complex coefficients $(c_{ij\lambda\mu})$. It is a standard fact that such a normalized coordinate system always exists, and that the Chern curvature tensor ${\ii\over 2\pi}D^2_{V,h}$ of $(V,h)$ at $z_0$ is given by $$ \Theta_{V,h}(z_0)=-{\ii\over 2\pi} \sum_{i,j,\lambda,\mu} c_{ij\lambda\mu}\,dz_i\wedge d\overline z_j\otimes e_\lambda^*\otimes e_\mu. \leqno(1.23) $$ Therefore, $(c_{ij\lambda\mu})$ are the components of $-\Theta_{V,h}$. Up to taking the transposed tensor with respect to $\lambda,\mu$, these coefficients are also the components of the curvature tensor $\Theta_{V^*,h^*}$ of the dual bundle $(V^*,h^*)$. We can also use instead a $C^\infty$ connection $\nabla$ on $V$, with the advantage that such a connection always exists globally on $X$ by a well known partition of unity argument. The drawback is that the identification $J_kV\to V^{\oplus k}$ is now just $C^\infty$, but this will not create any difficulty in what follows. Fix an integer $p\in\bN^*$ that is a multiple of $\lcm(1,2,\ldots,k)$, and positive numbers $1=\varepsilon_1\gg\varepsilon_2\gg\cdots\gg \varepsilon_k>0$. We define a global weighted Finsler metric on $J^kV$ by putting for any $k$-jet $f\in J^kV_z$ $$ \Psi_{h,p,\varepsilon}(f):=\Bigg( \sum_{1\le s\le k}\varepsilon_s^{2p}\Vert\nabla^s f(0) \Vert_{h(z)}^{2p/s}\Bigg)^{1/p}, \leqno(1.24) $$ where $\Vert~~\Vert_{h(z)}$ is the Hermitian metric $h$ of $V$ evaluated on the fiber $V_z$, $z=f(0)$. The function $\Psi_{h,p,\varepsilon}$ satisfies the fundamental homogeneity property $$ \Psi_{h,p,\varepsilon}(\alpha\cdot f)=|\alpha|^2\,\Psi_{h,p,\varepsilon}(f) \leqno(1.25) $$ with respect to the $\bC^*$ action on $J^kV$, in other words, it induces a Hermitian metric on the dual $L_k^*$ of the tautological $\bQ$-line bundle $L_k=\cO_{X_k}(1)$ over $X_k$. The curvature of $L_k$ is given by $$ \pi_k^*\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}=dd^c\log\Psi_{h,p,\varepsilon} \leqno(1.26) $$ Our next goal is to compute precisely the curvature and to apply holomorphic Morse inequalities to $L\to X_k$ with the above metric. It might look a priori like an untractable problem, since the definition of $\Psi_{h,p,\varepsilon}$ is a rather unnatural one. However, the ``miracle'' is that the asymptotic behavior of $\Psi_{h,p,\varepsilon}$ as $\varepsilon_s/\varepsilon_{s-1}\to 0$ is in some sense uniquely defined and very natural. It will lead to a computable asymptotic formula, which is moreover simple enough to produce useful results. \claim 1.27. Lemma|Let us consider the global $C^\infty$ bundle isomorphism $J^kV\to V^{\oplus k}$ associated with an arbitrary global $C^\infty$ connection $\nabla$ on $V\to X$, and introduce the rescaling transformation $$\rho_{\nabla,\varepsilon}(\xi_1,\xi_2,\ldots,\xi_k)= (\varepsilon_1^1\xi_1,\varepsilon_2^2\xi_2,\ldots, \varepsilon_k^k\xi_k)\quad \hbox{on fibers $J^kV_z$, $z\in X$}. $$ Such a rescaling commutes with the $\bC^*$-action. Moreover, if $p$ is a multiple of $\lcm(1,2,\ldots,k)$ and the ratios $\varepsilon_s/\varepsilon_{s-1}$ tend to~$0$ for all $s=2,\ldots,k$, the rescaled Finsler metric \hbox{$\Psi_{h,p,\varepsilon}\circ\rho_{\nabla,\varepsilon}^{-1} (\xi_1,\ldots,\xi_k)$} converges towards the limit $$ \bigg(\sum_{1\le s\le k}\Vert \xi_s\Vert^{2p/s}_h\bigg)^{1/p} $$ on every compact subset of $V^{\oplus k}\ssm\{0\}$, uniformly in $C^\infty$ topology, and the limit is independent of the connection~$\nabla$. \endclaim \proof. Let us pick another $C^\infty$ connection $\wt\nabla= \nabla+\Gamma$ where $\Gamma\in C^\infty(U,T^*_X\otimes \Hom(V,V))$. Then $\wt\nabla^2f=\nabla^2f+\Gamma(f)(f')\cdot f'$, and inductively we get $$ \wt\nabla^sf=\nabla^sf+P_s(f\,;\,\nabla^1f,\ldots,\nabla^{s-1}f) $$ where $P(z\,;\,\xi_1,\ldots,\xi_{s-1})$ is a polynomial with $C^\infty$ coefficients in $z\in U$, which is of weighted homogeneous degree $s$ in $(\xi_1,\ldots,\xi_{s-1})$. In other words, the corresponding isomorphisms $J^kV\simeq V^{\oplus k}$ correspond to each other by a $\bC^*$-homogeneous transformation $(\xi_1,\ldots,\xi_k)\mapsto (\wt\xi_1,\ldots,\wt\xi_k)$ such that $$ \wt\xi_s=\xi_s+P_s(z\,;\,\xi_1,\ldots,\xi_{s-1}). $$ Let us introduce the corresponding rescaled components $$ (\xi_{1,\varepsilon},\ldots,\xi_{k,\varepsilon})= (\varepsilon_1^1\xi_1,\ldots,\varepsilon_k^k\xi_k),\qquad (\wt\xi_{1,\varepsilon},\ldots,\wt\xi_{k,\varepsilon})= (\varepsilon_1^1\wt\xi_1,\ldots,\varepsilon_k^k\wt\xi_k). $$ Then $$ \eqalign{ \wt\xi_{s,\varepsilon} &=\xi_{s,\varepsilon}+ \varepsilon_s^s\,P_s(x\,;\,\varepsilon_1^{-1}\xi_{1,\varepsilon},\ldots, \varepsilon_{s-1}^{-(s-1)}\xi_{s-1,\varepsilon})\cr &=\xi_{s,\varepsilon}+O(\varepsilon_s/\varepsilon_{s-1})^s\, O(\Vert\xi_{1,\varepsilon}\Vert+\cdots+\Vert\xi_{s-1,\varepsilon} \Vert^{1/(s-1)})^s\cr} $$ and it is easily seen, as a simple consequence of the mean value inequality $|\Vert x\Vert^\gamma-\Vert y\Vert^\gamma|\leq\gamma\sup_{z\in[x,y]} \Vert z\Vert^{\gamma-1}\Vert x-y\Vert$, that the ``error term'' in the difference $\Vert\wt\xi_{s,\varepsilon}\Vert^{2p/s}-\Vert\xi_{s,\varepsilon}\Vert^{2p/s}$ is bounded by $$ (\varepsilon_s/\varepsilon_{s-1})^s\, \big(\Vert\xi_{1,\varepsilon}\Vert+\cdots+ \Vert\xi_{s-1,\varepsilon}\Vert^{1/(s-1)}+ \Vert\xi_{s,\varepsilon}\Vert^{1/s}\big)^{2p}. $$ When $p/s$ is an integer, similar bounds hold for all derivatives $D_{z,\xi}^\beta(\Vert\wt\xi_{s,\varepsilon}\Vert^{2p/s}- \Vert\xi_{s,\varepsilon}\Vert^{2p/s})$ and the lemma follows.\qed Now, we fix a point $z_0\in X$, a local holomorphic frame $(e_\lambda(z))_{1\le\lambda\leq r}$ satisfying (1.22) on a neighborhood $U$ of~$z_0$, and the {\it holomorphic} connection $\nabla$ on $V_{|U}$ such that $\nabla e_\lambda=0$. Since the uniform estimates of Lemma~1.27 also apply locally (provided they are applied on a relatively compact open subset $U'\compact U$), we can use the corresponding holomorphic trivialization $J^kV_{|U}\simeq V_{|U}^{\oplus k}\simeq U\times(\bC^r)^{\oplus k}$ to make our calculations. We do this in terms of the rescaled components $\xi_s=\varepsilon_s^s\nabla^sf(0)$. Then, uniformly on compact subsets of $J^kV_{|U}\ssm\{0\}$, we have $$ \Psi_{h,p,\varepsilon}\circ\rho_{\nabla,\varepsilon}^{-1}(z\,;\,\xi_1,\ldots,\xi_k) =\bigg(\sum_{1\le s\le k}\Vert\xi_s\Vert^{2p/s}_{h(z)}\bigg)^{1/p} +O(\max((\varepsilon_s/\varepsilon_{s-1})^{1/p}), $$ and the error term remains of the same magnitude when we take any derivative $D_{z,\xi}^\beta$. By (1.22) we find $$ \Vert \xi_s\Vert_{h(z)}^2= \sum_\lambda|\xi_{s\lambda}|^2+ \sum_{i,j,\lambda,\mu}c_{ij\lambda\mu}\,z_i\overline z_j \,\xi_{s\lambda}\overline \xi_{s\mu}+O(|z|^3|\xi|^2). $$ The question is thus reduced to evaluating the curvature of the weighted Finsler metric on $V^{\oplus k}$ defined by $$ \eqalign{ \Psi(z\,;\,\xi_1,\ldots,\xi_k) &=\bigg(\sum_{1\le s\le k}\Vert\xi_s\Vert^{2p/s}_{h(z)}\bigg)^{1/p}\cr &=\bigg(\sum_{1\le s\le k}\Big(\sum_\lambda|\xi_{s\lambda}|^2+ \sum_{i,j,\lambda,\mu}c_{ij\lambda\mu}\,z_i\overline z_j\, \xi_{s\lambda}\overline\xi_{s\mu}\Big)^{p/s}\bigg)^{1/p}+O(|z|^3).\cr} $$ We set $|\xi_s|^2=\sum_\lambda|\xi_{s\lambda}|^2$. A straightforward calculation yields $$ \eqalign{ &\log\Psi(z\,;\,\xi_1,\ldots,\xi_k)=\cr &~~{}={1\over p}\log\sum_{1\le s\le k}|\xi_s|^{2p/s}+ \sum_{1\le s\le k}{1\over s}\,{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}} \sum_{i,j,\lambda,\mu}c_{ij\lambda\mu}z_i\overline z_j {\xi_{s\lambda}\overline\xi_{s\mu}\over|\xi_s|^2}+O(|z|^3).\cr} $$ By (1.26), the curvature form of $L_k=\cO_{X_k}(1)$ is given at the central point $z_0$ by the following formula. \claim 1.28. Proposition| With the above choice of coordinates and with respect to the rescaled components $\xi_s=\varepsilon_s^s\nabla^sf(0)$ at $z_0\in X$, we have the approximate expression $$ \Theta_{L_k,\Psi^*_{h,p,\varepsilon}}(z_0,[\xi])\simeq \omega_{a,r,p}(\xi)+{\ii\over 2\pi} \sum_{1\le s\le k}{1\over s}\,{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}} \sum_{i,j,\lambda,\mu}c_{ij\lambda\mu} {\xi_{s\lambda}\overline\xi_{s\mu}\over|\xi_s|^2}\,dz_i\wedge d\overline z_j $$ where the error terms are $O(\max_{2\le s\le k}(\varepsilon_s/\varepsilon_{s-1})^s)$ uniformly on the compact variety $X_k$. Here $\omega_{a,r,p}$ is the $($degenerate$)$ K\"ahler metric associated with the weight $a=(1^{[r]},2^{[r]},\ldots,k^{[r]})$ of the canonical $\bC^*$ action on $J^kV$. \endclaim \subsection 1.A. Probabilistic estimate of the curvature of $k$-jet bundles| Thanks to the uniform approximation, we can (and will) neglect the error terms in the calculations below. Since $\omega_{a,r,p}$ is positive definite on the fibers of $X_k\to X$ (at least outside of the axes $\xi_s=0$), the index of the $(1,1)$ curvature form $\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}(z,[\xi])$ is equal to the index of the $(1,1)$-form $$ \gamma_k(z,\xi):={\ii\over 2\pi} \sum_{1\le s\le k}{1\over s}\,{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}} \sum_{i,j,\lambda,\mu}c_{ij\lambda\mu}(z) {\xi_{s\lambda}\overline\xi_{s\mu}\over|\xi_s|^2}\,dz_i\wedge d\overline z_j \leqno(1.29) $$ depending only on the differentials $(dz_j)_{1\le j\le n}$ on~$X$. The $q$-index integral of $(L_k,\Psi^*_{h,p,\varepsilon})$ on $X_k$ is therefore equal to $$ \eqalign{ &\int_{X_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}\cr &\qquad{}={(n+kr-1)!\over n!(kr-1)!} \int_{z\in X}\int_{\xi\in P(1^{[r]},\ldots,k^{[r]})} \omega_{a,r,p}^{kr-1}(\xi)\bOne_{\gamma_k,q}(z,\xi)\gamma_k(z,\xi)^n\cr} $$ where $\bOne_{\gamma_k,q}(z,\xi)$ is the characteristic function of the open set of points where $\gamma_k(z,\xi)$ has signature $(n-q,q)$ in terms of the $dz_j$'s. Notice that since $\gamma_k(z,\xi)^n$ is~a determinant, the product $\bOne_{\gamma_k,q}(z,\xi)\gamma_k(z,\xi)^n$ gives rise to a continuous function on~$X_k$. Formula 1.20 with $r_1=\ldots=r_k=r$ and $a_s=s$ yields the slightly more explicit integral $$ \eqalign{ &\int_{X_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}= {(n+kr-1)!\over n!(k!)^r}~~\times\cr &\qquad\int_{z\in X}\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k} \bOne_{g_k,q}(z,x,u)g_k(z,x,u)^n\, {(x_1\ldots x_k)^{r-1}\over (r-1)!^k}\,dx\,d\mu(u),\cr} $$ where $g_k(z,x,u)=\gamma_k(z,x_1^{1/2p}u_1,\ldots,x_k^{k/2p}u_k)$ is given by $$ g_k(z,x,u)={\ii\over 2\pi}\sum_{1\le s\le k}{1\over s}\,x_s \sum_{i,j,\lambda,\mu}c_{ij\lambda\mu}(z)\, u_{s\lambda}\overline u_{s\mu}\,dz_i\wedge d\overline z_j \leqno(1.30) $$ and $\bOne_{g_k,q}(z,x,u)$ is the characteristic function of its $q$-index set. Here $$ d\nu_{k,r}(x)=(kr-1)!\,{(x_1\ldots x_k)^{r-1}\over (r-1)!^k}\,dx \leqno(1.31) $$ is a probability measure on $\Delta_{k-1}$, and we can rewrite $$ \leqalignno{ &\int_{X_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1} ={(n+kr-1)!\over n!(k!)^r(kr-1)!}~~\times\cr &\qquad\int_{z\in X} \int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k} \bOne_{g_k,q}(z,x,u)g_k(z,x,u)^n\,d\nu_{k,r}(x)\,d\mu(u).&(1.32)\cr} $$ Now, formula (1.30) shows that $g_k(z,x,u)$ is a ``Monte Carlo'' evaluation of the curvature tensor, obtained by averaging the curvature at random points $u_s\in S^{2r-1}$ with certain positive weights $x_s/s\,$; we should then think of the \hbox{$k$-jet} $f$ as some sort of random variable such that the derivatives $\nabla^kf(0)$ are uniformly distributed in all directions. Let us compute the expected value of $(x,u)\mapsto g_k(z,x,u)$ with respect to the probability measure $d\nu_{k,r}(x)\,d\mu(u)$. Since $\int_{S^{2r-1}}u_{s\lambda}\overline u_{s\mu}d\mu(u_s)={1\over r} \delta_{\lambda\mu}$ and $\int_{\Delta_{k-1}}x_s\,d\nu_{k,r}(x)={1\over k}$, we find $$ {\bf E}(g_k(z,\bu,\bu))={1\over kr} \sum_{1\le s\le k}{1\over s}\cdot{\ii\over 2\pi}\sum_{i,j,\lambda} c_{ij\lambda\lambda}(z)\,dz_i\wedge d\overline z_j. $$ In other words, we get the normalized trace of the curvature, i.e. $$ {\bf E}(g_k(z,\bu,\bu))={1\over kr} \Big(1+{1\over 2}+\ldots+{1\over k}\Big)\Theta_{\det(V^*),\det h^*}, \leqno(1.33) $$ where $\Theta_{\det(V^*),\det h^*}$ is the $(1,1)$-curvature form of $\det(V^*)$ with the metric induced by~$h$. It is natural to guess that $g_k(z,x,u)$ behaves asymptotically as its expected value ${\bf E}(g_k(z,\bu,\bu))$ when $k$ tends to infinity. If we replace brutally $g_k$ by its expected value in (1.32), we get the integral $$ {(n+kr-1)!\over n!(k!)^r(kr-1)!}{1\over (kr)^n} \Big(1+{1\over 2}+\ldots+{1\over k}\Big)^n\int_X\bOne_{\eta,q}\eta^n, $$ where $\eta:=\Theta_{\det(V^*),\det h^*}$ and $\bOne_{\eta,q}$ is the characteristic function of its $q$-index set in~$X$. The leading constant is equivalent to $(\log k)^n/n!(k!)^r$ modulo a multiplicative factor $1+O(1/\log k)$. By working out a more precise analysis of the deviation, the following result has been proved in [Dem11] and [Dem12]. \claim 1.34. Probabilistic estimate| Fix smooth Hermitian metrics $h$ on $V$ and $\omega={\ii\over 2\pi} \sum\omega_{ij}dz_i\wedge d\overline z_j$ on $X$. Denote by $\Theta_{V,h}=-{\ii\over 2\pi}\sum c_{ij\lambda\mu}dz_i\wedge d\overline z_j\otimes e_\lambda^*\otimes e_\mu$ the curvature tensor of $V$ with respect to an $h$-orthonormal frame $(e_\lambda)$, and put $$ \eta(z)=\Theta_{\det(V^*),\det h^*}={\ii\over 2\pi}\sum_{1\le i,j\le n}\eta_{ij} dz_i\wedge d\overline z_j,\qquad \eta_{ij}=\sum_{1\leq\lambda\leq r}c_{ij\lambda\lambda}. $$ Finally consider the $k$-jet line bundle $L_k=\smash{\cO_{X_k}(1)}\to X_k$ equipped with the induced metric $\Psi^*_{h,p,\varepsilon}$ $($as defined above, with $1=\varepsilon_1\gg\varepsilon_2\gg\ldots\gg \varepsilon_k>0)$. When $k$ tends to infinity, the integral of the top power of the curvature of $L_k$ on its $q$-index set $X_k(L_k,q)$ is given by $$ \int_{X_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}= {(\log k)^n\over n!\,(k!)^r}\bigg( \int_X\bOne_{\eta,q}\eta^n+O((\log k)^{-1})\bigg) $$ for all $q=0,1,\ldots,n$, and the error term $O((\log k)^{-1})$ can be bounded explicitly in terms of $\Theta_V$, $\eta$ and $\omega$. Moreover, the left hand side is identically zero for $q>n$. \endclaim The final statement follows from the observation that the curvature of $L_k$ is positive along the fibers of $X_k\to X$, by the plurisubharmonicity of the weight (this is true even when the partition of unity terms are taken into account, since they depend only on the base); therefore the $q$-index sets are empty for $q>n$. It will be useful to extend the above estimates to the case of sections of $$ L_{F,k}=\cO_{X_k}(1)\otimes \pi_k^*\cO\Big(-{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big) \leqno(1.35) $$ where $F\in\Pic_\bQ(X)$ is an arbitrary $\bQ$-line bundle on~$X$ and $\pi_k:X_k\to X$ is the natural projection. We assume here that $F$ is also equipped with a smooth Hermitian metric $h_F$. In formulas (1.32--1.34), the curvature $g_{F,k}(z,x,u)$ of $L_{F,k}$ takes the form $$ g_{F,k}(z,x,u)=g_k(z,x,u)- {1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)\Theta_{F,h_F}(z), \leqno(1.36) $$ and by the same calculations its normalized expected value is $$ \eta_F(z):={1\over{1\over kr}(1+{1\over 2}+\ldots+{1\over k})} {\bf E}(g^F_k(z,\bu,\bu))=\Theta_{\det V^*,\det h^*}(z)-\Theta_{F,h_F}(z). \leqno(1.37) $$ Then the variance estimate for $g_{F,k}$ is the same as the variance estimate for $g_k$, and the recentered $L^p$ bounds are still valid, since our forms are just shifted by subtracting the constant smooth term $\Theta_{F,h_F}(z)$. The probabilistic estimate 1.34 is therefore still true in exactly the same form for $L_{F,k}$, provided we use $g_{F,k}$ and $\eta_F$ instead of $g_k$ and $\eta$. An application of holomorphic Morse inequalities gives the desired cohomology estimates for $$ \eqalign{ h^q\Big(X,E_{k,m} V^*&{}\otimes \cO\Big(-{m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big)\cr &{}=h^q(X_k,\cO_{X_k}(m)\otimes \pi_k^*\cO\Big(-{m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big), \cr} $$ provided $m$ is sufficiently divisible to give a multiple of $F$ which is a $\bZ$-line bundle. \claim 1.38. Theorem| Let $(X,V)$ be a directed manifold, $F\to X$ a $\bQ$-line bundle, $(V,h)$ and $(F,h_F)$ smooth Hermitian structure on $V$ and $F$ respectively. We define $$ \eqalign{ L_{F,k}&=\cO_{X_k}(1)\otimes \pi_k^*\cO\Big(-{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big),\cr \eta_F&=\Theta_{\det V^*,\det h^*}-\Theta_{F,h_F} =\Theta_{\det V^*\otimes F^*,\det h^*}.\cr} $$ Then for all $q\ge 0$ and all $m\gg k\gg 1$ such that m is sufficiently divisible, we have $$\leqalignno{\kern20pt h^q(X_k,\cO(L_{F,k}^{\otimes m}))&\le {m^{n+kr-1}\over (n+kr-1)!} {(\log k)^n\over n!\,(k!)^r}\bigg( \int_{X(\eta_F,q)}(-1)^q\eta_F^n+O((\log k)^{-1})\bigg),&\hbox{\rm(a)}\cr h^0(X_k,\cO(L_{F,k}^{\otimes m}))&\ge {m^{n+kr-1}\over (n+kr-1)!} {(\log k)^n\over n!\,(k!)^r}\bigg( \int_{X(\eta_F,\le 1)}\eta_F^n-O((\log k)^{-1})\bigg),&\hbox{\rm(b)}\cr \cr \chi(X_k,\cO(L_{F,k}^{\otimes m}))&={m^{n+kr-1}\over (n+kr-1)!} {(\log k)^n\over n!\,(k!)^r}\big( c_1(V^*\otimes F^*)^n+O((\log k)^{-1})\big).&\hbox{\rm(c)}\cr \cr} $$ \vskip-4pt \endclaim Green and Griffiths [GrGr79] already checked the Riemann-Roch calculation (1.38$\,$c) in the special case $V=T_X^*$ and $F=\cO_X$. Their proof is much simpler since it relies only on Chern class calculations, but it cannot provide any information on the individual cohomology groups, except in very special cases where vanishing theorems can be applied; in fact in dimension 2, the Euler characteristic satisfies $\chi=h^0-h^1+h^2\le h^0+h^2$, hence it is enough to get the vanishing of the top cohomology group $H^2$ to infer $h^0\ge\chi\,$; this works for surfaces by means of a well-known vanishing theorem of Bogomolov which implies in general $$H^n\bigg(X,E_{k,m} T_X^*\otimes \cO\Big(-{m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big)\bigg)=0 $$ as soon as $K_X\otimes F^*$ is big and $m\gg 1$. In fact, thanks to Bonavero's singular holomorphic Morse inequalities [Bon93], everything works almost unchanged in the case where $V\subset T_X$ has singularities and $h$ is an admissible metric on $V$ (see Definition~1.7). We only have to find a blow-up $\mu:\smash{\wt X}_k\to X_k$ so that the resulting pull-backs $\mu^*L_k$ and $\mu^*V$ are locally free, and $\mu^*\det h^*$, $\mu^*\Psi_{h,p,\varepsilon}$ only have divisorial singularities. Then $\eta$ is a $(1,1)$-current with logarithmic poles, and we have to deal with smooth metrics on $ \mu^*L_{F,k}^{\otimes m}\otimes\cO(-mE_k)$ where $E_k$ is a certain effective divisor on $X_k$ (which, by our assumption in 1.7, does not project onto $X$). The cohomology groups involved are then the twisted cohomology groups $$ H^q(X_k,\cO(L_{F,k}^{\otimes m})\otimes\cJ_{k,m}) $$ where $\cJ_{k,m}=\mu_*(\cO(-mE_k))$ is the corresponding multiplier ideal sheaf, and the Morse integrals need only be evaluated in the complement of the poles, that is on $X(\eta,q)\ssm S$ where $S=\Sing(V)\cup\Sing(h)$. Since $$ (\pi_k)_*\big(\cO(L_{F,k}^{\otimes m})\otimes\cJ_{k,m}\big)\subset E_{k,m} V^*\otimes \cO\Big(-{m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big) $$ we still get a lower bound for the $H^0$ of the latter sheaf (or for the $H^0$ of the un-twisted line bundle $\cO(L_k^{\otimes m})$ on $\smash{X_k}$). If we assume that $K_V\otimes F^*$ is big, these considerations also allow us to obtain a strong estimate in terms of the volume, by using an approximate Zariski decomposition on a suitable blow-up of~$(X,V)$. The following corollary implies in particular Theorem~1.3. \claim 1.31. Corollary| If $F$ is an arbitrary $\bQ$-line bundle over~$X$, one has $$ \eqalign{ h^0\bigg(&X_k,\cO_{X_k}(m)\otimes\pi_k^*\cO \Big(-{m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\bigg)\cr &\ge {m^{n+kr-1}\over (n+kr-1)!} {(\log k)^n\over n!\,(k!)^r}\Big( \Vol(K_V\otimes F^*)-O((\log k)^{-1})\Big)-o(m^{n+kr-1}),\cr} $$ when $m\gg k\gg 1$, in particular there are many sections of the $k$-jet differentials of degree $m$ twisted by the appropriate power of $F$ if $K_V\otimes F^*$ is big. \endclaim \proof. The volume is computed here as usual, i.e.\ after performing a suitable modifi\-cation $\mu:\smash{\wt X}\to X$ which converts $K_V$ into an invertible sheaf. There is of course nothing to prove if $K_V\otimes F^*$ is not big, so we can assume $\Vol(K_V\otimes F^*)>0$. Let us fix smooth Hermitian metrics $h_0$ on $T_X$ and $h_F$ on $F$. They induce a metric $\mu^*(\det h_0^{-1}\otimes h_F^{-1})$ on $\mu^*(K_V\otimes F)$ which, by our definition of $K_V$, is a smooth metric. By the result of Fujita [Fuj94] on approximate Zariski decomposition, for every $\delta>0$, one can find a modification $\mu_\delta:\smash{\wt X_\delta}\to X$ dominating $\mu$ such that $$ \mu_\delta^*(K_V\otimes F^*) =\cO_{\wt X_\delta}(A+E) $$ where $A$ and $E$ are $\bQ$-divisors, $A$ ample and $E$ effective, with $$\Vol(A)=A^n\ge \Vol(K_V\otimes F^*)-\delta.$$ If we take a smooth metric $h_A$ with positive definite curvature form $\Theta_{A,h_A}$, then we get a singular Hermitian metric $h_Ah_E$ on $\mu_\delta^*(K_V\otimes F^*)$ with poles along $E$, i.e.\ the quotient $h_Ah_E/\mu^*(\det h_0^{-1}\otimes h_F^{-1})$ is of the form $e^{-\varphi}$ where $\varphi$ is quasi-psh with log poles $\log|\sigma_E|^2$ (mod $C^\infty(\smash{\wt X_\delta}))$ precisely given by the divisor~$E$. We then only need to take the singular metric $h$ on $T_X$ defined by $$ h=h_0e^{{1\over r}(\mu_\delta)^*\varphi} $$ (the choice of the factor ${1\over r}$ is there to correct adequately the metric on $\det V$). By construction $h$ induces an admissible metric on $V$ and the resulting curvature current $\eta_F=\Theta_{K_V,\det h^*}-\Theta_{F,h_F}$ is such that $$ \mu_\delta^*\eta_F = \Theta_{A,h_A} +[E],\qquad \hbox{$[E]={}$current of integration on $E$.} $$ Then the $0$-index Morse integral in the complement of the poles is given by $$ \int_{X(\eta,0)\ssm S}\eta_F^n=\int_{\wt X_\delta}\Theta_{A,h_A}^n=A^n\ge \Vol(K_V\otimes F)-\delta $$ and (1.39) follows from the fact that $\delta$ can be taken arbitrary small.\qed \subsection \S1.B. Logarithmic and orbifold curvature estimates| Let again $(X,V)$ be a non singular directed variety, where $V\subset T_X$ is a subbundle. We consider an arbifold directed structure $(X,V,\Delta)$ where $\Delta=\sum(1-{1\over \rho_j})\Delta_j$ is a normal crossing divisor that is assumed to intersect $V$ transversally everywhere; this implies that at most $r$ components $\Delta_j$ can meet, if we denote $r=\rank(V)$. One then performs very similar calculations, but with adapted Finsler metrics. Fix a point $z_0$ at which $p$ components $\Delta_j$ meet, and use coordinates $(z_1,\ldots,z_n)$ such that $V_{z_0}$ is spanned by $({\partial\over\partial z_1},\ldots,{\partial\over\partial z_r})$ and $\Delta_j$ is defined by $z_j=0$, $1\leq j\leq p\leq r$. In the logarithmic case $\rho_j=\infty$, the logarithmic dual bundle $\cO(V^*\langle\Delta\rangle)$ is spanned by $$ {dz_1\over z_1},\ldots,{dz_p\over z_p},~dz_{p+1},\ldots,dz_n. $$ The logarithmic jet differentials are just polynomials in $$ {d^sz_1\over z_1},\ldots,{d^sz_p\over z_p},~d^sz_{p+1},\ldots,d^sz_n,\quad 1\leq s\leq k, $$ and the corresponding $(\varepsilon_1,\ldots,\varepsilon_k)$-rescaled Finsler metric is $$ \Bigg(\sum_{s=1}^k\varepsilon_s^{2p}\bigg(\sum_{j=1}^p |f_j|^{-2}|f_j^{(s)}|^2+\sum_{j=p+1}^r|f_j^{(s)}|^2\bigg)^{2p/s}\Bigg)^{1/p}. \leqno(1.40) $$ Alternatively, we could replace $|f_j|^{-2}|f_j^{(s)}|^2$ by $|(\log f_j)^{(s)}|^2$ which has the same leading term and differs by a weighted degree $s$ polynomial in the $f_j^{-1}f_j^{(\ell)}$, $\ell0$. We assume here that the $s$-th directed (dual) orbifold bundle $V^*\langle\Delta^{(s)}\rangle$ (cf.\ \S$\,$1.B) possesses a hermitian metric $\wt h_s^*$ such that its curvature tensor satisfies an inequality $$ \Theta_{V^*\langle\Delta^{(s)}\rangle,\wt h_s^*}+\gamma\otimes \Id_{V^*\langle\Delta^{(s)}\rangle} \ge 0\leqno(1.41') $$ in the sense of Griffiths, for some smooth $(1,1)$-form $\gamma=\ii\sum\gamma_{ij}dz_i\wedge d\overline z_j\geq 0$ on $X$. Now, instead of replacing $\Theta_V$ with its trace free part $\wt\Theta_V$ and exploiting a Monte Carlo convergence process, we replace $\Theta_{V^*\langle\Delta^{(s)}\rangle}$ with $\Theta_{V^*\langle\Delta^{(s)}\rangle}^\gamma= \Theta_{V^*\langle\Delta^{(s)}\rangle}+\gamma\otimes\Id\geq 0$, i.e.\ the curvature coefficients $c^{(s)}_{ij\lambda\mu}$ by $c_{ij\lambda\mu}^{(s,\gamma)}=c^{(s)}_{ij\lambda\mu}+\gamma_{ij}\delta_{\lambda\mu}$. This has the effect of replacing $\Theta_{\det V^*\langle\Delta^{(s)}\rangle}= \Tr\Theta_{V^*\langle\Delta^{(s)}\rangle}$ by $\Theta_{\det V^*\langle\Delta^{(s)}\rangle}+r\gamma$. Also, we take a line bundle $F=\varepsilon A$, $\varepsilon\in\bQ_{>0}$. Then our earlier formulas (1.28), (1.35), (1.36) become $$ \leqalignno{ &g_k^{\gamma}(z,x,u)={\ii\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s \sum_{i,j,\lambda,\mu}c^{(s,\gamma)}_{ij\lambda\mu}(z)\, u_{s\lambda}\overline u_{s\mu}\,dz_i\wedge d\overline z_j\ge 0, &(1.42)\cr &L_{\varepsilon,k}:=\cO_{X_k}(1)\otimes \pi_k^*\cO\Big(-{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big) \varepsilon A\Big), &(1.43)\cr &\Theta_{L_{\varepsilon,k}}=\omega_{a,r,p}(\xi)+g_{\varepsilon,k}(z,x,u)\quad \hbox{where}&(1.44)\cr &g_{\varepsilon,k}(z,x,u)=g_k^\gamma(z,x,u)- {1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big) (\varepsilon\omega_A+r\gamma).\cr} $$ The gain that we have is that $\Theta_{L_{\varepsilon,k}}=g_{A,k}$ is now expressed as a difference of semipositive $(1,1)$-forms, and we can exploit \medskip We apply here Lemma 1.45 with $\lambda,\mu$ replaced by $$ \alpha_k=g_k^\gamma(z,x,u),\qquad\beta_k=\beta^{(s),\gamma}_k= {1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big) \big(\varepsilon\omega_A+r\gamma), $$ which are both semipositive by our assumptions. Then (1.32) leads to $$ \leqalignno{\qquad &\int_{X_k(L_k,\le 1)}\Theta_{L_{\varepsilon,k},\Psi^*_{h,p,\varepsilon}}^{n+kr-1}\cr &\quad{}={(n+kr-1)!\over n!(k!)^r(kr-1)!}\int_{z\in X} \int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k} \bOne_{\alpha_k-\beta_k,\le 1}\;(\alpha_k-\beta_k)^n\,d\nu_{k,r}(x)\,d\mu(u)\cr &\quad{}\ge{(n+kr-1)!\over n!(k!)^r(kr-1)!}\int_{z\in X} \int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k} \big(\alpha_k^n-n\alpha_k^{n-1}\wedge\beta_k\big)\,d\nu_{k,r}(x)\,d\mu(u). &(1.46)\cr} $$ The resulting integral now produces a ``closed formula'' which can be expressed solely in terms of Chern classes (at least if we assume that $\gamma$ is the Chern form of some semipositive line bundle). It is just a matter of routine to find a sufficient condition for the positivity of the integral. One can first observe that $\alpha_k$ is bounded from above by taking the trace of $(c^{(s,\gamma)}_{ij\lambda\mu})$, in this way we get $$ 0\le\alpha_k\le\sum_{1\le s\le k}{x_s\over s}\big( \Theta_s+r\gamma\big)\quad\hbox{where}~~ \Theta_s=\Theta_{\det V^*\langle\Delta^{(s)}\rangle} \leqno(1.47) $$ and where the right hand side no longer depends on $u\in (S^{2r-1})^k$. Also, $\alpha_k=g_k^\gamma$ can be written as a sum of semipositive $(1,1)$-forms $$ g_k^\gamma =\sum_{1\le s\le k}{x_s\over s}\theta^{s,\gamma}(u_s),\qquad \theta^{s,\gamma}(u)=\sum_{i,j,\lambda,\mu}c_{ij\lambda\mu}^{(s,\gamma)} u_\lambda\ol u_\mu\,dz_i\wedge d\ol z_j, $$ hence for $k\ge n$ we have $$ \alpha_k^n=(g_k^\gamma)^n\ge n!\sum_{1\le s_1<\ldots0. \leqno(1.54) $$ \claim 1.55.~Lemma|Let $X\subset\bP^N$ be a projective variety and $(X,V,\Delta)$ an orbifold directed structure where $\Delta$ is a normal crossing divisor in $X$ transverse to~$V$. Then the orbifold vector bundle $V\langle D\rangle$ possesses a smooth hermitian metric such that the induced curvature tensor of $V^*\langle D\rangle \otimes\cO_X(2)$ is Griffiths semi-positive. \endclaim \proof. Very straightforward calculation, for the obvious metric induced by the Fubini-Study metric on $\bP^N$~!! \noindent An interesting special orbifold example is the case when $X=\bP^n$, $V=T_X$, $A=\cO(1)$ and $\Delta=\sum (1-{1\over\rho_j})\Delta_j$ is a normal crossing divisor, with components $\Delta_j$ of degree $d_j$. Then $$ \det V^*\langle\Delta^{(s)}\rangle =\cO_{\bP^n}(-n-1+\sum_j d_j(1-s/\rho_j)_+), $$ and we can take $\gamma=2\omega_A$ by lemma 1.53. For $k=n$ and $\varepsilon\in\bQ_{>0}$ small, the existence of orbifold jet differentials is guaranteed under the condition $$ {1\over n!} \prod_{s=1}^n\Big(n-1+\sum_jd_j(1-s/\rho_j)_+\Big) -2n\,c_{n,n,n}\bigglp2pt(\sum_{1\leq s\leq n} {n-1+\sum_jd_j(1-s/\rho_j)_+\over s}\biggrp2pt)^{n-1}>0. $$ If $\Delta$ contains a component $(1-{1\over \rho_1})\Delta_1$ of ramification index $\rho_1\in\bN$, $\rho_1\geq n+1$, the inequality is satisfied as soon as $d_1$ is large enough (we can simply drop the other components and argue with $\Delta=(1-{1\over \rho_1})\Delta_1$). Indeed, the inequalities $$ \eqalign{ &n-1+d_1(1-\ell/\rho_1)_+\geq (1-{\ell/\rho_1})(n-1+d_1) ={\rho_1-\ell\over \rho_1}(n-1+d_1),\cr \noalign{\vskip4pt} &\sum_{1\leq s\leq n}{n-1+d_1(1-s/\rho_1)_+\over s}\leq \Big(1+{1\over 2}+\cdots+{1\over n}\Big)(n-1+d_1)\cr} $$ show that is enough to have $$ {\rho_1-1\choose n}{1\over\rho_1^n} \big(n-1+d_1\big)^n-2n\,c_{n,n,n} \Big(1+{1\over 2}+\cdots+{1\over n}\Big)^{n-1}\big(n-1+d_1\big)^{n-1}>0. $$ Finally, (1.53) implies $$ 2n\,c_{n,n,n}\leq n\,2^{2n-1}\Big(1+{1\over 2}+\cdots+{1\over n}\Big) $$ and we get \claim 1.56.~Corollary|Consider on $\bP^n$ a smooth irreducible orbifold divisor $\Delta$ of ramification index $\rho_1=n+1$, namely $\Delta=(1-{1\over \rho_1})\Delta_1$, with a single component $\Delta_1$ of degree $\deg\Delta_1=d_1$. Then, for $\rho_1\geq n+1$, $\varepsilon\in\bQ_{>0}$ small and $$ n-1+d_1> n\,2^{2n-1}\Big(1+{1\over 2}+\ldots+{1\over n}\Big)^n {\rho_1^n\over{\rho_1-1\choose n}}, $$ there exist many $($i.e.\ at least $c\,m^{n+n^2-1}$, $c>0)$ orbifold jet differentials of order~$n$~in $$ H^0(X,E_{n,m}T^*_{\bP^n}\langle\Delta\rangle\otimes\cO_{\bP^n}(-m\varepsilon)). $$ \endclaim \bigskip [Cad] "Jet differentials on toroidal compactifications of ball quotients" Beno\^it Cadorel. [CRAS] JP Demailly "Propriétés de semi-continuité de la cohomologie et de la dimension de Kodaira-Iitaka" C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 341-346 \parindent=0cm (version of June 28, 2012, printed on \today, \timeofday) \vskip5pt \end % Local Variables: % TeX-command-default: "TeX" % End: By elementary integrations by parts and induction on $k,\,r_1,\ldots,r_k$, it can be checked that $$ \int_{x\in\Delta_{k-1}} \prod_{1\le s\le k}x_s^{r_s-1}dx_1\ldots dx_{k-1} ={1\over (|r|-1)!}\prod_{1\le s\le k}(r_s-1)!~. \leqno(1.20) $$ This implies that $(|r|-1)!\prod_{1\le s\le k} {x_s^{r_s-1}\over(r_s-1)!}\,dx$ is a probability measure on $\Delta_{k-1}$.