% Hyperbolicity book, S. Diverio et al % % Jean-Pierre Demailly % Universit\'e Grenoble Alpes, Institut Fourier % % LaTeX file \documentclass[11pt,leqno]{amsart} \usepackage{amsmath, amssymb} \usepackage{graphicx,color,hyperref} \usepackage{verbatim,enumitem} \textheight=23.5cm \voffset=-1.75cm \textwidth=16.5cm \hoffset=-1.75cm % main item macros \def\smallskip{\vskip3pt plus1pt minus1pt} \def\medskip{\vskip5pt plus2pt minus1pt} \def\plainsubsection#1|{% \par\vskip0.25cm\penalty -100 \centerline{{\sc #1}} \vskip3pt plus 1pt minus 0pt \penalty 500} \long\def\claim#1|#2\endclaim{\par\vskip 6pt\noindent {\bf #1.}\ {\em #2}\par\vskip 5pt} \def\plainproof{\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 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\def\displ@y{\global\dt@ptrue\openup\jot\m@th \everycr{\noalign{\ifdt@p \global\dt@pfalse \ifdim\prevdepth>-1000\p@ \vskip-\lineskiplimit \vskip\normallineskiplimit \fi \else \penalty\interdisplaylinepenalty \fi}}} \def\@lign{\tabskip\z@skip\everycr{}} % restore inside \displ@y \def\displaylines#1{\displ@y \tabskip\z@skip \halign{\hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}\crcr #1\crcr}} \def\eqalignno#1{\displ@y \tabskip\plaincentering \halign to\displaywidth{\hfil$\@lign\displaystyle{##}$\tabskip\z@skip &$\@lign\displaystyle{{}##}$\hfil\tabskip\plaincentering &\llap{$\@lign##$}\tabskip\z@skip\crcr #1\crcr}} \def\leqalignno#1{\displ@y \tabskip\plaincentering \halign to\displaywidth{\hfil$\@lign\displaystyle{##}$\tabskip\z@skip &$\@lign\displaystyle{{}##}$\hfil\tabskip\plaincentering &\kern-\displaywidth\rlap{$\@lign##$}\tabskip\displaywidth\crcr #1\crcr}} \def\plaincases#1{\left\{\,\vcenter{\normalbaselines\m@th \ialign{$##\hfil$&\quad##\hfil\crcr#1\crcr}}\right.} \def\plainmatrix#1{\null\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \def\plainpmatrix#1{\left(\plainmatrix{#1}\right)} \newcommand\myleaders{\leavevmode\leaders\hbox{\kern1pt.}\hfill} \renewcommand{\tocsection}[3]{% \indentlabel{\@ifnotempty{#2}{\ignorespaces#1 \rlap{\kern0.1em #2.}\hphantom{00.\ \ }}}#3\myleaders} \catcode`\@=12 % Main text \title[A simple proof of the Kobayashi conjecture] {A simple proof of the Kobayashi conjecture on the hyperbolicity of general algebraic hypersurfaces} \author{Jean-Pierre Demailly} \date{} \begin{document} \maketitle \tableofcontents \setcounter{section}{-1} \section{Introduction} For a compact complex space~$X$, a well known result of Brody [Bro78] asserts that the hyperbolicity property introduced by Kobayashi [Kob67a, Kob67b] is equivalent to the nonexistence of noncons\-tant entire holomorphic curves $f:\bC\to X$. The aim of this chapter is to describe some geometric techniques that are useful to investigate the existence or nonexistence of such curves. A central conjecture due to Green-Griffiths [GrGr80] and Lang [Lang86] stipulates that for every projective variety $X$ of general type over~$\bC$, there exists a proper algebraic subvariety $Y$ of $X$ containing all nonconstant entire curves. According to Green-Griffiths [GrGr80], jet bundles can be used to give sufficient conditions for Kobayashi hyperbolicity. As in [Dem95], we introduce the formalism of directed varieties and Semple towers [Sem54] to express these conditions in terms of intrinsic algebraic differential equations that entire curves must satisfy; see the ``fundamental vanishing theorem'' 3.23 below. An important application is a confirmation of an old-standing conjecture of Kobayashi (cf.\ [Kob70]): a general hypersurface $X$ of complex projective space $\bP^{n+1}$ of degree $d\geq d_n$ large enough is Kobayashi hyperbolic. The main arguments are based on techniques introduced in 2016 by Damian Brotbek [Brot17]; they make use of Wronskian differential operators and their associated multiplier ideals. Shortly afterwards, Ya Deng [Deng16] found how to make the method effective, and produced in this way an explicit value of~$d_n$. We describe here a proof based on a simplification of their ideas, producing a very similar bound, namely $d_n=\lfloor\frac{1}{3}(en)^{2n+2}\rfloor$ (cf.\ [Dem18]). This extends in particular earlier results of Demailly-El Goul [DeEG97], McQuillan [McQ99], P\u{a}un [Pau08], Diverio-Merker-Rousseau [DMR10], Diverio-Trapani [DT10] and [Siu15]. According to work of Clemens[86], Zaidenberg [Zai87], Ein [Ein88, Ein91], Voisin [Voi96] and Pacienza [Pac04], every subvariety of a general algebraic hypersurface hypersurface $X$ of $\bP^{n+1}$ is of general type for degrees $d\geq \delta_n$, with an optimal lower bound given by $\delta_n=2n+1$ for $2\leq n\leq 4$ and $\delta_n=2n$ for $n\geq 5$ -- that the same bound $d_n=\delta_n$ holds for Kobayashi hyperbolicity would then be a consequence of the Green-Griffiths-Lang conjecture. In the same vein, we present a construction of hyperbolic hypersurfaces of $\bP^{n+1}$ for all degrees $d\geq 4n^2$. The main idea is inspired from the method of Shiffman-Zaidenberg [ShZa02]; by using again Wronskians, it is possible to give a direct and self-contained argument. I wish to thank Damian Brotbek, Ya Deng, Simone Diverio, Gianluca Pacienza, Erwan Rousseau, Mihai P\u{a}un and Mikhail Zaidenberg for very stimulating discussions on these questions. These notes owe a lot to their work. \section{Hyperbolicity concepts} \plainsubsection 1.A. Kobayashi pseudodistance and pseudometric| We first recall a few basic facts concerning the concept of hyperbolicity, according to S.~Kobayashi [Kob67a, Kob67b, Kob70, Kob76]. Let $X$ be a complex space. Given two points $p,q\in X$, let us consider a {\em chain of analytic disks} from $p$ to $q$, that is a sequence of holomorphic maps $f_0,f_1,\dots,f_k:\bD\to X$ from the unit disk $\bD=D(0,1)\subset\bC$ to $X$, together with pairs of points $a_0,b_0,\dots,a_k,b_k$ of $\bD$ such that $$ p=f_0(a_0),\quad q=f_k(b_k),\quad f_i(b_i)=f_{i+1}(a_{i+1}),\qquad i=0,\dots,k-1. $$ Denoting this chain by $\alpha$, we define its length $\ell(\alpha)$ to be $$ \ell(\alpha)=d_P(a_1,b_1)+\cdots+d_P(a_k,b_k),\leqno(1.1') $$ where $d_P$ is the Poincar\'e distance on $\bD$, and the {\em Kobayashi pseudodistance} $d^K_X$ on $X$ to be $$ d^K_X(p,q)=\inf_{\alpha}\ell(\alpha).\leqno(1.1'') $$ A {\em Finsler metric} (resp.\ {\em pseudometric}\/) on a vector bundle $E$ is a homogeneous positive (resp.\ nonnegative) function $N$ on the total space $E$, that is, $$ N(\lambda\xi)=|\lambda|\,N(\xi)\qquad \hbox{for all $\lambda\in\bC$ and $\xi\in E$,} $$ but in general $N$ is not assumed to be subbadditive (i.e.\ convex) on the fibers of~$E$. A~Finsler (pseudo-)metric on $E$ is thus nothing but a hermitian (semi-)norm on the tautological line bundle $\cO_{P(E)}(-1)$ of lines of~$E$ over the projectivized bundle $Y=P(E)$. The {\em Kobayashi-Royden infinitesimal pseudometric} on $X$ is the Finsler pseudometric on the tangent bundle $T_X$ defined by $$ \bfk_X(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\bD\to X,\,f(0)=x,\, \lambda f'(0)=\xi\big\},\qquad x\in X,~\xi\in T_{X,x}.\leqno(1.2) $$ If $\Phi:X\to Y$ is a morphism of complex spaces, by considering the compositions $\Phi\circ f:\bD\to Y$, this definition immediately implies the monotonicity property $\Phi^*\bfk_Y\leq \bfk_X$, i.e.\ $$ \bfk_Y(\Phi_*\xi)\leq \bfk_X(\xi)\quad\hbox{for all $x\in X$ and $\xi\in T_{X,x}$.} \leqno(1.3) $$ When $X$ is a manifold, it follows from the work of H.L.~Royden ([Roy71], [Roy74]) that $d^K_X$ is the integrated pseudodistance associated with the pseudometric, i.e. $$ d^K_X(p,q)=\inf_\gamma\int_\gamma\bfk_X(\gamma'(t))\,dt, \leqno(1.4) $$ where the infimum is taken over all piecewise smooth curves joining $p$ to $q\,$; in the case of complex spaces, a similar formula holds, involving jets of analytic curves of arbitrary order, cf.\ S.~Venturini [Ven96]. When $X$ is a non-singular projective variety, it has been shown in [DeLS94] that the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal pseudometric can be computed by looking only at analytic disks that are contained in algebraic curves. \claim 1.5.~Definition|A complex space $X$ is said to be {\em hyperbolic} $($in the sense of Kobayashi$)$ if $d^K_X$ is actually a distance, namely if $d^K_X(p,q)>0$ for all pairs of distinct points $(p,q)$ in~$X$. \endclaim \plainsubsection 1.B. Brody criterion| In the above context, we have the following well-known result of Brody [Bro78]. Its main interest is to relate hyperbolicity to the non-existence of entire curves. \claim 1.6.~Brody reparametrization lemma|Let $\omega$ be a hermitian metric on~$X$ and let $f:\bD\to X$ be a holomorphic map. For every $\varepsilon>0$, there exists a radius $R\ge(1-\varepsilon)\|f'(0)\|_\omega$ and a homographic transformation $\psi$ of the disk $D(0,R)$ onto $(1-\varepsilon)\bD$ such that $$ \|(f\circ\psi)'(0)\|_\omega=1,\qquad \|(f\circ\psi)'(t)\|_\omega \le{1\over 1-|t|^2/R^2}\quad\hbox{for every $t\in D(0,R)$.} $$ \endclaim \plainproof. Select $t_0\in\bD$ such that $(1-|t|^2)\|f'((1-\varepsilon)t)\|_\omega$ reaches its maximum for $t=t_0$. The reason for this choice is that $(1-|t|^2)\|f'((1-\varepsilon)t)\|_\omega$ is the norm of the differential $f'((1-\varepsilon)t):T_\bD\to T_X$ with respect to the Poincar\'e metric $|dt|^2/(1-|t|^2)^2$ on $T_\bD$, which is conformally invariant under $\Aut(\bD)$. One then adjusts $R$ and $\psi$ so that $\psi(0)=(1-\varepsilon)t_0$ and $|\psi'(0)|\,\|f'(\psi(0))\|_\omega=1$. As $|\psi'(0)|={1-\varepsilon\over R}(1-|t_0|^2)$, the only possible choice for $R$ is $$ R=(1-\varepsilon)(1-|t_0|^2)\|f'(\psi(0))\|_\omega\ge (1-\varepsilon) \|f'(0)\|_\omega. $$ The inequality for $(f\circ\psi)'$ follows from the fact that the Poincar\'e norm is maximum at the origin, where it is equal to $1$ by the choice of~$R$. Using the Ascoli-Arzel\`a theorem we obtain immediately: \claim 1.7.~Corollary {\rm(Brody)}|Let $(X,\omega)$ be a compact complex hermitian manifold. Given a sequence of holomorphic mappings \hbox{$f_\nu:\bD\to X$} such that $\lim\|f_\nu'(0)\|_\omega=+\infty$, one can find a sequence of homographic transformations $\psi_\nu:D(0,R_\nu)\to(1-1/\nu)\bD$ with $\lim R_\nu=+\infty$, such that, after passing possibly to a subsequence, $(f_\nu\circ\psi_\nu)$ converges uniformly on every compact subset of $\bC$ towards a nonconstant holomorphic map $g:\bC\to X$ with $\|g'(0)\|_\omega=1$ and $\sup_{t\in\bC}\|g'(t)\|_\omega\le 1$. \endclaim An entire curve $g:\bC\to X$ such that $\sup_\bC\|g'\|_\omega=M<+\infty$ is called a {\em Brody curve}; this concept does not depend on the choice of $\omega$ when $X$ is compact, and one can always assume $M=1$ by rescaling the parameter~$t$. \claim 1.8.~Brody criterion|Let $X$ be a {\rm compact} complex manifold. The following properties are equivalent. \plainitem{\rm(a)} $X$ is hyperbolic. \plainitem{\rm(b)} $X$ does not possess any entire curve $f:\bC\to X$. \plainitem{\rm(c)} $X$ does not possess any Brody curve $g:\bC\to X$. \plainitem{\rm(d)} The Kobayashi infinitesimal metric $\bfk_X$ is uniformly bounded below, namely $$ \bfk_X(\xi)\ge c\|\xi\|_\omega,\qquad c>0, $$ for any hermitian metric $\omega$ on $X$.\vskip5pt \noindent When property {\rm (b)} holds, $X$ is said to be {\rm Brody hyperbolic}. \endclaim \plainproof. (a)${}\Rightarrow{}$(b) If $X$ possesses an entire curve $f:\bC\to X$, then by looking at arbitrary large analytic disks $f:D(t_0,R)\subset\bC$ and rescaling them on $\bD$ as $t\mapsto f(t_0+Rt)$, it is easy to see that the Kobayashi distance of any two points in $f(\bC)$ is zero, so $X$ is not hyperbolic.\smallskip \noindent (b)${}\Rightarrow{}$(c) is trivial. \smallskip \noindent (c)${}\Rightarrow{}$(d) If (d) does not hold, there exists a sequence of tangent vectors $\xi_\nu\in T_{X,x_\nu}$ with $\Vert\xi_\nu\|_\omega=1$ and $\bfk_X(\xi_\nu)\to 0$. By definition, this means that there exists an analytic curve $f_\nu:\bD\to X$ with $f(0)=x_\nu$ and $\Vert f'_\nu(0)\Vert_\omega\ge (1-{1\over \nu})/\bfk_X(\xi_\nu)\to+\infty$. One can then produce a Brody curve $g=\bC\to X$ by Corollary~1.7, contradicting~(c).\smallskip \noindent (d)${}\Rightarrow{}$(a). In fact (d) implies after integrating that $d^K_X(p,q)\ge c\,d_\omega(p,q)$ where $d_\omega$ is the geodesic distance associated with~$\omega$, so $d^K_X$ must be non degenerate.\qed As a consequence, any projective variety containing a rational curve $C$ (i.e.\ a curve normalized by $\overline C\simeq\bP^1_\bC\simeq\bC\cup\{\infty\}$ or an elliptic curve (i.e.\ a curve normalized by a nonsingular elliptic curve~$\bC/(\bZ\oplus\bZ\tau)$) is non-hyperbolic. An immediate consequence of the Brody criterion is the openness property of hyperbolicity for the metric topology: \claim 1.9. Proposition|Let $\pi:\cX\to S$ be a holomorphic family of compact complex manifolds. Then the set of $s\in S$ such that the fiber $X_s=\pi^{-1}(s)$ is hyperbolic is open in the metric topology. \endclaim \plainproof. Let $\omega$ be an arbitrary hermitian metric on $\cX$, $(X_{s_\nu})_{s_\nu\in S}$ a sequence of non-hyperbolic fibers, and $s=\lim s_\nu$. By the Brody criterion, one obtains a sequence of entire maps $f_\nu:\bC\to X_{s_\nu}$ such that $\|f'_\nu(0)\|_\omega=1$ and $\|f'_\nu\|_\omega\le 1$. Ascoli's theorem shows that there is a subsequence of $f_\nu$ converging uniformly to a limit $f:\bC\to X_s$, with $\|f'(0)\|_\omega=1$. Hence $X_s$ is not hyperbolic and the collection of non-hyperbolic fibers is closed in $S$.\qed \medskip \plainsubsection 1.C. Relationship of hyperbolicity with algebraic properties| In the case of projective algebraic varieties, Kobayashi hyperbolicity is expected to be an algebraic property. In fact, the following classical conjectures would give a necessary and sufficient algebraic characterization. Recall that a projective variety $X$ of dimension $n=\dim_\bC X$ is said to be of \emph{general type} if the canonical bundle $K_{\widetilde X}=\Lambda^nT^*_{\widetilde X}$ of some desingularization $\widetilde X$ of $X$ is big. When $n=\dim_\bC X= 1$, this is equivalent to say that $X$ is not rational or elliptic. \claim 1.10. Some classical conjectures|Let $X$ be a projective variety. {\plainitemindent=7mm \plainitem{\rm(i)} {\rm(Green-Griffiths-Lang conjecture)} If $X$ is of general type, there should exist a proper algebraic variety $Y\subsetneq X$ $($possibly empty$\,)$ containing all nonconstant entire curves $f:\bC\to X$. \plainitem{\rm(ii)} Conversely, if $X$ is Kobayashi hyperbolic and nonsingular, it is expected that $K_X$ should be ample. More generally, if $X$ is singular, any desingularization $\widetilde X$ should be of general type. \plainitem{\rm(iii)} {\rm(Conjectural algebraic characterization of Kobayashi hyperbolicity)}. A projective variety $X$ is Kobayashi hyperbolic if and only if every positive dimensional algebraic subvariety $Y\subset X$ $($including $X$ itself$\,)$ is of general type.\vskip0pt} \endclaim \medskip In fact, since every analytic subspace of Kobayashi hyperbolic space is again hyperbolic by definition, it is not difficult to see by induction on dimension that 1.10~(iii) would follow formally from 1.10~(i) and~(ii) [the ``if'' part is a consequence of 1.10~(i), and the ``only if'' part follows from 1.10~(ii)]. Thanks to fundamental work of Clemens [Cle86], Ein [Ein88, Ein91] and Voisin [Voi96], it is known that every subvariety $Y$ of a generic algebraic hypersurface $X\subset \bP^{n+1}$ of degree $d\ge 2n+1$ is of general type for $n\geq 2\,$; Pacienza [Pac04] has even shown that this holds for $d\ge 2n$ when $n\geq 5$. The Green-Griffiths-Lang conjecture would then imply that these hypersurfaces are Kobayashi hyperbolic. \claim 1.11. Definition|Let $X$ be a projective algebraic manifold, and $A$ a very ample line bundle on $X$. We say that $X$ is {\rm algebraically hyperbolic} if there exists $\varepsilon>0$ such that every closed irreducible curve $C\subset X$ has a normalization $\overline C$ such that its Euler characteristic satisfies $$ -\chi(\ol C)=2g(\ol C)-2\ge\varepsilon\,\deg_A(C), $$ where $g(\ol C)$ is the genus and $\deg_A(C)=C\cdot A=\int_C c_1(A)$. \endclaim \claim 1.12. Theorem|Every Kobayashi hyperbolic projective variety is algebraically hyperbolic. More generally, if $X$ is a hyperbolic compact complex manifold equipped with a hermitian metric $\omega$, there exists $\varepsilon>0$ such that every closed irreducible curve $C\subset X$ satisfies $$ 2g(\ol C)-2\ge\varepsilon\,\deg_\omega(C)\quad \hbox{where $\deg_\omega(C)=\int_C\omega$.} $$ \endclaim \plainproof\ ([Dem95]). When $\Gamma$ is a nonsingular compact curve of genus at least~$2$, the uniformization theorem implies that the universal cover $\rho:\widehat \Gamma\to \Gamma$ is isomorphic to the unit disk $\bD$, and one then sees that the Kobayashi metric $\bfk_\Gamma$ is induced by he Kobayashi metric of the disk, i.e. $$ \bfk_\bD^2={\ii dz\wedge d\overline z\over (1-|z|^2)^2}. $$ These metrics have constant negative curvature $-{\ii\over 2\pi}\ddbar\log\bfk_\Gamma^2=-{1\over \pi}\bfk_\Gamma^2$, hence $$ {1\over \pi}\int_\Gamma\bfk_\Gamma^2=-\chi(\Gamma)=2g(\Gamma)-2 $$ by the Gauss-Bonnet formula. Now, if $X$ is hyperbolic and $C\subset X$ is a closed analytic curve, the monotonicity formula (1.3) applied to the normalization map $\nu:\overline C\to X$ implies $\bfk_{\overline C}\geq\nu^*\bfk_X$, and we also have $\bfk_X^2\geq c^2\omega$ for some $c>0$ by 1.8~(d). Therefore $$ 2g(\overline C)-2={1\over \pi}\int_{\overline C}\bfk_{\overline C}^2 \geq{1\over \pi}\int_{\overline C}\nu^*\bfk_X^2= {1\over \pi}\int_{C}\bfk_X^2 \geq{c^2\over \pi}\int_{C}\omega={c^2\over \pi}\deg_{\omega}(C). $$ It is not very difficult to check that the proof can be extended to the case of singular hyperbolic compact complex spaces (a smooth hermitian metric on $X$ being a metric that has extensions with respect to local embeddings of $X$ in open sets $U\subset\bC^N$).\qed \claim 1.13.~Proposition|Let $\cX\to S$ be an algebraic family of projective algebraic manifolds, given by a projective morphism $\cX\to S$. Then the set of $t\in S$ such that the fiber $X_t$ is algebraically hyperbolic is open with respect to the countable Zariski topology \endclaim \plainproof. After replacing $S$ by a Zariski open subset, we may assume that the total space $\cX$ itself is quasi-projective. Let $\omega$ be the K\"ahler metric on~$\cX$ obtained by pulling back the Fubini-Study metric via an embedding in a projective space. If integers $d>0$, $g\ge 0$ are fixed, the set $A_{d,g}$ of $t\in S$ such that $X_t$ contains an algebraic $1$-cycle $C=\sum m_jC_j$ with $\deg_\omega(C)=d$ and $g(\ol C)=\sum m_j\,g(\ol C_j)\le g$ is a closed algebraic subset of $S$ (this follows from the existence of a relative cycle space of curves of given degree, and from the fact that the geometric genus is Zariski lower semicontinuous). Now, the set of non algebraically hyperbolic fibers is by definition $$ \bigcap_{k>0}~~\bigcup_{2g-20\,;\,\exists f:(\bD,T_{\bD})\to (X,V),\,f(0)=x,\,\lambda f'(0)=\xi\big\}. $$ \endclaim We say that $(X,V)$ is {\em infinitesimally hyperbolic} if $\bfk_{(X,V)}$ is positive definite on every fiber~$V_x$ and satisfies a uniform lower bound $\bfk_{(X,V)}(\xi)\ge\varepsilon\|\xi\|_\omega$ in terms of any smooth hermitian metric $\omega$ on $X$, when $x$ runs over a compact subset of~$X$. When $X$ is compact, the Brody criterion shows that this is equivalent to the nonexistence of nonconstant entire curves $f:(\bC,T_\bC)\to (X,V)$, or even to the nonexistence of entire curves $g:(\bC,T_\bC)\to (X,V)$ with $\sup \Vert g'(t)\Vert_\omega=\Vert g'(0)\Vert_\omega=1$. In this context we have the \claim 2.3. Generalized Green-Griffiths-Lang conjecture| Let $(X,V)$ be a projective directed manifold where $V\subset T_X$ is nonsingular $($i.e.\ a subbundle of $T_X)$. Assume that $(X,V)$ is of ``general type'' in the sense that $K_V:=\det V^*$ is a big line bundle. Then there should exist a proper algebraic subvariety $Y\subsetneq X$ containing the images $f(\bC)$ of all entire curves $f:\bC\to X$ tangent to~$V$. \endclaim A similar statement can be made when $V$ is singular, but then $K_V$ has to be replaced by a certain (nonnecessarily invertible) rank 1 sheaf of ``locally bounded'' forms of $\cO(\det V^*)$, with respect to a smooth hermitian form $\omega$ on $T_X$. The reader will find a more precise definition in [Dem18]. \plainsubsection 2.B. The $1$-jet functor| The basic idea is to introduce a functorial process which produces a new complex directed manifold $(\swt{X},\swt{V})$ from a given one~$(X,V)$. The new structure $(\swt{X},\swt{V})$ plays the role of a space of $1$-jets over~$X$. First assume that $V$ is {\em non-singular}. We let $$ \swt{X}=P(V),\qquad \swt{V}\subset T_{\swt{X}} \leqno(2.4) $$ be the projectivized bundle of lines of $V$, together with a subbundle $\swt{V}$ of $T_{\swt{X}}$ defined as follows: for every point $(x,[v])\in \swt{X}$ associated with a vector $v\in V_x\ssm\{0\}$, $$ \swt{V}_{(x,[v])}=\big\{\xi\in T_{\swt{X},\,(x,[v])}\,;\,\pi_*\xi\in \bC v\big\},\qquad\bC v\subset V_x\subset T_{X,x},\leqno(2.4') $$ where $\pi:\swt{X}=P(V)\to X$ is the natural projection and $\pi_*: T_{\swt{X}}\to\pi^* T_X$ is its differential. On $\swt{X}=P(V)$ we have the tautological line bundle $\cO_{\swt{X}}(-1)\subset\pi^* V$ such that $\cO_{\swt{X}}(-1)_{(x,[v])}=\bC v$. The bundle $\swt{V}$ is characterized by the two exact sequences $$ \leqalignno{ &0\lra T_{\swt{X}/X}\lra\swt{V}\build\lra^{\pi_*}_{}\cO_{\swt{X}}(-1) \lra 0,&(2.5)\cr &0\lra\cO_{\swt{X}}\lra \pi^* V\otimes\cO_{\swt{X}}(1) \lra T_{\swt{X}/X}\lra 0,&(2.5')\cr} $$ where $T_{\swt{X}/X}$ denotes the relative tangent bundle of the fibration $\pi:\swt{X}\to X$. The first sequence is a direct consequence of the definition of $\swt{V}$, whereas the second is a relative version of the Euler exact sequence describing the tangent bundle of the fibers $P(V_x)$. From these exact sequences we infer $$ \dim\swt{X}=n+r-1,\qquad \rank\swt{V}=\rank V=r,\leqno(2.6) $$ and by taking determinants we find $\det(T_{\swt{X}/X})= \pi^*\det V\otimes\cO_{\swt{X}}(r)$, thus $$ \det\swt{V}=\pi^*\det V\otimes\cO_{\swt{X}}(r-1).\leqno(2.7) $$ By definition, $\pi:(\swt{X},\swt{V})\to(X,V)$ is a morphism of complex directed manifolds. Clearly, our construction is functorial, i.e., for every morphism of directed manifolds $\Phi:(X,V)\to(Y,W)$, there is a commutative diagram $$ \plainmatrix{(\swt{X},\swt{V})&\build\lra^{\textstyle\pi}_{}&(X,V)\cr \wt\Phi~\smash{\raise 1.2em\hbox{$\vdasharrow$}}&&\big\downarrow\Phi\cr (\swt{Y},\swt{W})&\build\lra^{\textstyle\pi}_{}&\,(Y,W),\cr} $$ where the left vertical arrow is the meromorphic map $P(V)\merto P(W)$ induced by the differential $\Phi_*:V\to\Phi^* W$ ($\swt{\Phi}$ is actually holomorphic if $\Phi_*:V\to\Phi^* W$ is injective). \plainsubsection 2.C. Lifting of curves to the $1$-jet bundle| Suppose that we are given a holomorphic curve $f:D_R\to X$ parametrized by the disk $D_R$ of centre $0$ and radius $R$ in the complex plane, and that $f$ is a tangent curve of the directed manifold, i.e., $f'(t)\in V_{f(t)}$ for every $t\in D_R$. If $f$ is nonconstant, there is a well defined and unique tangent line $[f'(t)]$ for every~$t$, even at stationary points, and the map $$ \swt{f}:D_R\to\swt{X},\qquad t\mapsto\swt{f}(t):=(f(t),[f'(t)])\leqno(2.8) $$ is holomorphic (at a stationary point $t_0$, we just write $f'(t)=(t-t_0)^su(t)$ with $s\in\bN^*$ and $u(t_0)\ne 0$, and we define the tangent line at $t_0$ to be $[u(t_0)]$, hence $\swt{f}(t)=(f(t),[u(t)])$ near $t_0\,$; even for $t=t_0$, we still denote $[f'(t_0)]=[u(t_0)]$ for simplicity of notation). By definition $f'(t)\in\cO_{\swt{X}}(-1)_{\swt{f}(t)}=\bC\,u(t)$, hence the derivative $f'$ defines a section $$ f':T_{D_R}\to\swt{f}^*\cO_{\swt{X}}(-1).\leqno(2.9) $$ Moreover $\pi\circ\swt{f}=f$, therefore $$ \pi_*\swt{f}'(t)=f'(t)\in\bC u(t)\Longrightarrow \swt{f}'(t)\in\swt{V}_{(f(t),u(t))}=\swt{V}_{\swt{f}(t)} $$ and we see that $\swt{f}$ is a tangent trajectory of $(\swt{X},\swt{V})$. We say that $\swt{f}$ is the {\em canonical lifting} of $f$ to~$\swt{X}$. Conversely, if $g:D_R\to\swt{X}$ is a tangent trajectory of $(\swt{X},\swt{V})$, then by definition of $\swt{V}$ we see that $f=\pi\circ g$ is a tangent trajectory of $(X,V)$ and that $g=\swt{f}$ (unless $g$ is contained in a vertical fiber $P(V_x)$, in which case $f$ is constant). For any point $x_0\in X$, there are local coordinates $(z_1\ld z_n)$ on a neighborhood $\Omega$ of $x_0$ such that the fibers $(V_z)_{z\in\Omega}$ can be defined by linear equations $$ V_z=\Big\{\xi=\sum_{1\le j\le n}\xi_j{\partial\over\partial z_j}\,;\, \xi_j= \sum_{1\le k\le r}a_{jk}(z)\xi_k~\hbox{\rm for $j=r+1\ld n$}\Big\}, \leqno(2.10) $$ where $(a_{jk})$ is a holomorphic $(n-r)\times r$ matrix. It follows that a vector $\xi\in V_z$ is completely determined by its first $r$ components $(\xi_1\ld\xi_r)$, and the affine chart $\xi_j\ne 0$ of $P(V)_{\restriction \Omega}$ can be described by the coordinate system $$ \Big(z_1\ld z_n;{\xi_1\over\xi_j}\ld{\xi_{j-1}\over\xi_j}, {\xi_{j+1}\over\xi_j}\ld{\xi_r\over\xi_j}\Big).\leqno(2.11) $$ Let $f\simeq(f_1\ld f_n)$ be the components of $f$ in the coordinates $(z_1\ld z_n)$ (we suppose here $R$ so small that $f(D_R)\subset\Omega$). It should be observed that $f$ is uniquely determined by its initial value $x$ and by the first $r$ components $(f_1\ld f_r)$. Indeed, as $f'(t)\in V_{f(t)}\,$, we can recover the other components by integrating the system of ordinary differential equations $$ f_j'(t)=\sum_{1\le k\le r}a_{jk}(f(t))f_k'(t),\qquad j>r,\leqno(2.12) $$ on a neighborhood of~$0$, with initial data $f(0)=x$. We denote by $m=m(f,t_0)$ the {\em multiplicity} of $f$ at any point $t_0\in D_R$, that is, $m(f,t_0)$ is the smallest integer $m\in\bN^*$ such that $f_j^{(m)}(t_0)\ne 0$ for some~$j$. By (2.12), we can always suppose $j\in\{1\ld r\}$, for example $f_r^{(m)}(t_0)\ne 0$. Then $f'(t)=(t-t_0)^{m-1}u(t)$ with $u_r(t_0)\ne 0$, and the lifting $\swt{f}$ is described in the coordinates of the affine chart $\xi_r\ne 0$ of $P(V)_{\restriction\Omega}$ by $$ \swt{f}\simeq\Big(f_1\ld f_n;{f_1'\over f_r'}\ld{f_{r-1}'\over f_r'}\Big). \leqno(2.13) $$ \plainsubsection 2.D. The Semple tower| Let $X$ be a complex $n$-dimensional manifold. Following ideas of \hbox{Green-Griffiths} [GrGr80], we let $J_kX\to X$ be the bundle of $k$-jets of germs of parametrized curves in~$X$, that is, the set of equivalence classes of holomorphic maps $f:(\bC,0)\to(X,x)$, with the equivalence relation $f\sim g$ if and only if all derivatives $f^{(j)}(0)=g^{(j)}(0)$ coincide for $0\le j\le k$, when computed in some local coordinate system of $X$ near~$x$. The projection map $J_kX\to X$ is simply $f\mapsto f(0)$. If $(z_1\ld z_n)$ are local holomorphic coordinates on an open set $\Omega\subset X$, the elements $f$ of any fiber $J_kX_x$, $x\in\Omega$, can be seen as $\bC^n$-valued maps $$f=(f_1\ld f_n):(\bC,0)\to\Omega\subset\bC^n,$$ and they are completely determined by their Taylor expansion of order $k$ at~$t=0$ $$ f(t)=x+t\,f'(0)+{t^2\over 2!}f''(0)+\cdots+{t^k\over k!}f^{(k)}(0)+ O(t^{k+1}). $$ In these coordinates, the fiber $J_kX_x$ can thus be identified with the set of $k$-tuples of vectors $(\xi_1,\ldots,\xi_k)=(f'(0)\ld f^{(k)}(0))\in(\bC^n)^k$. It follows that $J_kX$ is a holomorphic fiber bundle with typical fiber $(\bC^n)^k$ over $X$ (however, $J_kX$ is not a vector bundle for $k\ge 2$, because of the nonlinearity of coordinate changes. According to the philosophy of directed structures, one can also introduce the concept of jet bundle in the general situation of complex directed manifolds. If $X$ is equipped with a holomorphic subbundle $V\subset T_X$, one associates to $V$ a $k$-jet bundle $J_kV$ as follows. \claim 2.14.~Definition|Let $(X,V)$ be a complex directed manifold. We define $J_kV\to X$ to be the bundle of $k$-jets of curves \hbox{$f:(\bC,0)\to X$} which are tangent to $V$, i.e., such that $f'(t)\in V_{f(t)}$ for all $t$ in a neighborhood of~$0$, together with the projection map $f\mapsto f(0)$ onto~$X$. \endclaim It is easy to check that $J_kV$ is actually a subbundle of~$J_kX$. In fact, by using (2.10) and (2.12), we see that the fibers $J_kV_x$ are parametrized by $$ \big((f_1'(0)\ld f_r'(0));(f_1''(0)\ld f_r''(0));\ldots; (f_1^{(k)}(0)\ld f_r^{(k)}(0))\big)\in(\bC^r)^k $$ for all $x\in\Omega$, hence $J_kV$ is a locally trivial $(\bC^r)^k$-subbundle of~$J_kX$. Alternatively, we can pick a local holomorphic connection $\nabla$ on $V$ such that for any germs $w=\sum_{1\leq j\leq n}w_j{\partial\over \partial z_j} \in\cO(T_{X,x})$ and $v=\sum_{1\leq\lambda\leq r}v_\lambda e_\lambda\in\cO(V)_x$ in a local trivializing frame $(e_1,\ldots,e_r)$ of~$V_{\restriction \Omega}$ we have $$ \nabla_wv(x)=\sum_{1\leq j\leq n,\,1\leq\lambda\leq r} w_j{\partial v_\lambda\over\partial z_j}e_\lambda(x)+ \sum_{1\leq j\leq n,\,1\leq\lambda,\mu\leq r} \Gamma_{j\lambda}^\mu(x)w_jv_\lambda\,e_\mu(x). \leqno(2.15) $$ We can of course take the frame obtained from (2.10) by lifting the vector fields $\partial/\partial z_1,\ldots,\partial/\partial z_r$, and the ``trivial connection'' given by the zero Christoffel symbols $\Gamma=0$. One then obtains a trivialization $J^kV_{\restriction \Omega}\simeq V^{\oplus k}_{\restriction \Omega}$ by considering $$ J_kV_x\ni f\mapsto(\xi_1,\xi_2,\ldots,\xi_k)=(\nabla f(0),\nabla^2f(0),\ldots,\nabla^k f(0))\in V_x^{\oplus k} $$ and computing inductively the successive derivatives $\nabla f(t)=f'(t)$ and $\nabla^sf(t)$ via $$ \nabla^sf=(f^*\nabla)_{d/dt}(\nabla^{s-1}f) =\sum_{1\leq\lambda\leq r} {d\over dt}\Big(\nabla^{s-1}f\Big)_\lambda e_\lambda(f)+ \sum_{1\leq j\leq n,\,1\leq\lambda,\mu\leq r} \Gamma_{j\lambda}^\mu(f)f'_j\Big(\nabla^{s-1}f\Big)_\lambda e_\mu(f). $$ This identification depends of course on the choice of $\nabla$ and cannot be defined globally in general (unless we are in the rare situation where $V$ has a global holomorphic connection.\qed \medskip We now describe a convenient process for constructing ``projectivized jet bundles'', which will later appear as natural quotients of our jet bundles~$J_kV$ (or~rather, as suitable desingularized compactifications of the quotients). Such spaces have already been considered since a long time, at least in the special case $X=\bP^2$, $V=T_{\bP^2}$ (see Gherardelli [Ghe41], Semple [Sem54]), and they have been mostly used as a tool for establishing enumerative formulas dealing with the order of contact of plane curves (see [Coll88], [CoKe94]); the article [ASS97] is also concerned with such generalizations of jet bundles, as well as [LaTh96] by Laksov and Thorup. One defines inductively the {\em projectivized $k$-jet bundle $X_k$} (or {\em Semple $k$-jet bundle}) and the associated subbundle $V_k\subset T_{X_k}$ by $$ (X_0,V_0)=(X,V),\qquad (X_k,V_k)=(\swt{X}_{k-1},\swt{V}_{k-1}). \leqno(2.16) $$ In other words, $(X_k,V_k)$ is obtained from $(X,V)$ by iterating $k$-times the lifting construction $(X,V)\mapsto(\swt{X},\swt{V})$ described in~\S$\,$2.B. By (2.4--2.9), we find $$ \dim X_k=n+k(r-1),\qquad\rank V_k=r,\leqno(2.17) $$ together with exact sequences $$ \leqalignno{ &0\lra T_{X_k/X_{k-1}}\lra V_k\build{\vlra{11}}^{(\pi_k)_*}_{} \cO_{X_k}(-1)\lra 0,&(2.18)\cr &0\lra\cO_{X_k}\lra\pi_k^* V_{k-1}\otimes\cO_{X_k}(1) \lra T_{X_k/X_{k-1}}\lra 0, &(2.18')\cr} $$ where $\pi_k$ is the natural projection $\pi_k:X_k\to X_{k-1}$ and $(\pi_k)_*$ its differential. Formula (5.4) yields $$ \det V_k=\pi_k^*\det V_{k-1}\otimes\cO_{X_k}(r-1).\leqno(2.19) $$ Every nonconstant tangent trajectory \hbox{$f:D_R\to X$} of $(X,V)$ lifts to a well defined and unique tangent trajectory $f_{[k]}:D_R\to X_k$ of $(X_k,V_k)$. Moreover, the derivative $f_{[k-1]}'$ gives rise to a section $$ f_{[k-1]}':T_{D_R}\to f_{[k]}^*\cO_{X_k}(-1).\leqno(2.20) $$ In coordinates, one can compute $f_{[k]}$ in terms of its components in the various affine charts (5.9) occurring at each step: we get inductively $$ f_{[k]}=(F_1\ld F_N),\qquad f_{[k+1]}=\Big(F_1\ld F_N, {F_{s_1}'\over F_{s_r}'}\ld{F_{s_{r-1}}'\over F_{s_r}'}\Big),\leqno(2.21) $$ where $N=n+k(r-1)$ and $\{s_1\ld s_r\}\subset\{1\ld N\}$. If $k\ge 1$, $\{s_1\ld s_r\}$ contains the last $r-1$ indices of $\{1\ld N\}$ corresponding to the ``vertical'' components of the projection $X_k\to X_{k-1}$, and in general, $s_r$ is an index such that $m(F_{s_r},0)=m(f_{[k]},0)$, that is, $F_{s_r}$ has the smallest vanishing order among all components $F_s$ ($s_r$ may be vertical or not, and the choice of $\{s_1\ld s_r\}$ need not be unique). By definition, there is a canonical injection $\cO_{X_k}(-1) \hookrightarrow\pi_k^* V_{k-1}$, and a composition with the projection $(\pi_{k-1})_*$ (analogue for order $k-1$ of the arrow~$(\pi_k)_*$ in the sequence (2.18)) yields for all $k\ge 2$ a canonical line bundle morphism $$ \cO_{X_k}(-1)\lhra\pi_k^* V_{k-1}\build{\vlra{16}}^ {(\pi_k)^*(\pi_{k-1})_*}_{}\pi_k^*\cO_{X_{k-1}}(-1),\leqno(2.22) $$ which admits precisely \hbox{$D_k=P(T_{X_{k-1}/X_{k-2}}) \subset P(V_{k-1})=X_k$} as its zero divisor (clearly, $D_k$ is a hyperplane subbundle of~$X_k$). Hence we find $$ \cO_{X_k}(1)=\pi_k^*\cO_{X_{k-1}}(1)\otimes\cO(D_k).\leqno(2.23) $$ Now, we consider the composition of projections $$ \pi_{j,k}=\pi_{j+1}\circ\cdots\circ\pi_{k-1}\circ\pi_k:X_k\lra X_j. \leqno(2.24) $$ Then $\pi_{0,k}:X_k\to X_0=X$ is a locally trivial holomorphic fiber bundle over~$X$, and the fibers $X_{k,x}=\pi_{0,k}^{-1}(x)$ are $k$-stage towers of $\bP^{r-1}$-bundles. Since we have (in both directions) morphisms $(\bC^r,T_{\bC^r})\leftrightarrow(X,V)$ of directed manifolds which are bijective on the level of bundle morphisms, the fibers are all isomorphic to a ``universal'' non-singular projective algebraic variety of dimension $k(r-1)$ which we will denote by~$\cR_{r,k}\,$; it is not hard to see that $\cR_{r,k}$ is rational (as will indeed follow from the proof of Theorem~3.11 below). \claim 2.25. Remark|{\rm When $(X,V)$ is singular, one can easily extend the construction of the Semple tower by functoriality. In fact, assume that $X$ is a closed analytic subset of some open set $Z\subset\bC^N$, and that $X'\subset X$ is a Zariski open subset on which $V_{\restriction X'}$ is a subbundle of $T_{X'}$. Then we consider the injection of the nonsingular directed manifold $(X',V')$ into the absolute structure $(Z,W)$, $W=T_Z$. This yields an injection $(X'_k,V'_k)\hookrightarrow (Z_k,W_k)$, and we simply define $(X_k,V_k)$ to be the closure of $(X'_k,V'_k)$ into $(Z_k,W_k)$. It is not hard to see that this is indeed a closed analytic subset of the same dimension $n+k(r-1)$, where $r=\rank V'$.} \endclaim \section{Jet differentials and Green-Griffiths bundles} \plainsubsection 3.A. Green-Griffiths jet differentials| We first introduce the concept of jet differentials in the sense of Green-Griffiths [GrGr80]. The goal is to provide an intrinsic geometric description of holomorphic differential equations that a germ of curve $f:(\bC,0)\to X$ may satisfy. In the sequel, we fix a directed manifold $(X,V)$ and suppose implicitly that all germs of curves $f$ are tangent to~$V$. Let $\bG_k$ be the group of germs of $k$-jets of biholomorphisms of $(\bC,0)$, that is, the group of germs of biholomorphic maps $$ t\mapsto\varphi(t)=a_1t+a_2t^2+\cdots+a_kt^k,\qquad a_1\in\bC^*,~a_j\in\bC,~j\ge 2, $$ in which the composition law is taken modulo terms $t^j$ of degree $j>k$. Then $\bG_k$ is a $k$-dimensional nilpotent complex Lie group, which admits a natural fiberwise right action on $J_kV$. The action consists of reparametrizing $k$-jets of maps $f:(\bC,0)\to X$ by a biholomorphic change of parameter $\varphi:(\bC,0)\to(\bC,0)$, that is, $(f,\varphi)\mapsto f\circ\varphi$. There is an exact sequence of groups $$ 1\to \bG'_k\to \bG_k\to\bC^*\to 1, $$ where $\bG_k\to\bC^*$ is the obvious morphism $\varphi\mapsto\varphi'(0)$, and $\bG'_k=[\bG_k,\bG_k]$ is the group of $k$-jets of biholomorphisms tangent to the identity. Moreover, the subgroup $\bH\simeq\bC^*$ of homotheties $\varphi(t)=\lambda t$ is a (non-normal) subgroup of $\bG_k$, and we have a semidirect decomposition $\bG_k=\bG'_k\ltimes\bH$. The corresponding action on $k$-jets is described in coordinates by $$ \lambda\cdot(f',f'',\ldots,f^{(k)})= (\lambda f',\lambda^2f'',\ldots,\lambda^kf^{(k)}). $$ Following [GrGr80], we introduce the vector bundle $E^\GG_{k,m}V^*\to X$ whose fibers are complex valued polynomials $Q(f',f'',\ldots,f^{(k)})$ on the fibers of $J_kV$, of weighted degree $m$ with respect to the $\bC^*$ action defined by $H$, that is, such that $$ Q(\lambda f',\lambda^2 f'',\ldots,\lambda^k f^{(k)})=\lambda^m Q(f',f'',\ldots,f^{(k)})\leqno(3.1) $$ for all $\lambda\in\bC^*$ and $(f',f'',\ldots,f^{(k)})\in J_kV$. Here we view $(f',f'',\ldots,f^{(k)})$ as indeterminates with components $$ \big((f_1'\ld f_r');(f_1''\ld f_r'');\ldots;(f_1^{(k)}\ld f_r^{(k)})\big) \in(\bC^r)^k. $$ Notice that the concept of polynomial on the fibers of $J_kV$ makes sense, for all coordinate changes $z\mapsto w=\Psi(z)$ on $X$ induce polynomial transition automorphisms on the fibers of $J_kV$, given by a formula $$ (\Psi\circ f)^{(j)}=\Psi'(f)\cdot f^{(j)}+\sum_{s=2}^{s=j}{~} \sum_{j_1+j_2+\cdots+j_s=j}c_{j_1\ldots j_s}\Psi^{(s)}(f)\cdot (f^{(j_1)}\ld f^{(j_s)})\leqno(3.2) $$ with suitable integer constants $c_{j_1\ldots j_s}$ (this is easily checked by induction on~$s$). In the case $V=T_X$, we get the bundle of ``absolute'' jet differentials $E^\GG_{k,m}T^*_X$. If $Q\in E^\GG_{k,m}V^*$ is decomposed into multihomogeneous components of multidegree $(\ell_1,\ell_2\ld\ell_k)$ in $f',f''\ld f^{(k)}$ (the decomposition is of course coordinate dependent), these multidegrees must satisfy the relation $$ \ell_1+2\ell_2+\cdots+k\ell_k=m. $$ The bundle $E^\GG_{k,m}V^*$ will be called the {\em bundle of jet differentials of order $k$ and weighted degree~$m$}. It is clear from (3.2) that a coordinate change $f\mapsto\Psi\circ f$ transforms every monomial $(f^{(\bullet)})^\ell=(f')^{\ell_1}(f'')^{\ell_2}\cdots(f^{(k)})^{\ell_k}$ of partial weighted degree $|\ell|_s:=\ell_1+2\ell_2+\cdots+s\ell_s$, $1\le s\le k$, into a polynomial $((\Psi\circ f)^{(\bullet)})^\ell$ in $(f',f''\ld f^{(k)})$ which has the same partial weighted degree of order $s$ if $\ell_{s+1}=\cdots=\ell_k=0$, and a larger or equal partial degree of order $s$ otherwise. Hence, for each $s=1\ld k$, we get a well defined (i.e., coordinate invariant) decreasing filtration $F_s^\bullet$ on $E^\GG_{k,m}V^*$ as follows: $$ F^p_s(E^\GG_{k,m}V^*)=\left\{ {\displaystyle \hbox{\rm $Q(f',f''\ld f^{(k)})\in E^\GG_{k,m}V^*$ involving} \atop\hbox{\rm only monomials $(f^{(\bullet)})^\ell$ with $|\ell|_s\ge p$} \hfill}\right\},\qquad \forall p\in\bN.\leqno(3.3) $$ The graded terms $\Gr^p_{k-1}(E^\GG_{k,m}V^*)$ associated with the filtration $F^p_{k-1}(E^\GG_{k,m}V^*)$ are precisely the homogeneous polynomials $Q(f'\ld f^{(k)})$ whose monomials $(f^{\bullet})^\ell$ all have partial weighted degree $|\ell|_{k-1}=p$ (hence their degree $\ell_k$ in~$f^{(k)}$ is such that $m-p=k\ell_k$, and $\Gr^p_{k-1}(E^\GG_{k,m}V^*)=0$ unless $k$ divides $m-p$). The transition automorphisms of the graded bundle are induced by coordinate changes $f\mapsto\Psi\circ f$, and they are described by substituting the arguments of~$Q(f'\ld f^{(k)})$ according to formula (3.2), namely $f^{(j)}\mapsto(\Psi\circ f)^{(j)}$ for $j1$, but they only have quotient singularities, see [Dol81]$\,$; here $J_kV\ssm\{0\}$ is the set of nonconstant jets of order~$k\,$; we refer e.g.\ to Hartshorne's book [Har77] for a definition of the $\Proj$ functor). As such, it possesses a canonical sheaf $\cO_{X^\GG_k}(1)$ such that $\cO_{X^\GG_k}(m)$ is invertible when $m$ is a multiple of $\lcm(1,2,\ldots,k)$. Under the natural projection $\pi_k:X^\GG_k\to X$, the direct image $(\pi_k)_*\cO_{X^\GG_k}(m)$ coincides with polynomials $$ P(z\,;\,\xi_1,\ldots,\xi_k)=\sum_{\alpha_\ell\in\bN^r,\,1\le\ell\le k} a_{\alpha_1\ldots\alpha_k}(z)\,\xi_1^{\alpha_1}\ldots\xi_k^{\alpha_k} \leqno(3.8) $$ of weighted degree $|\alpha_1|+2|\alpha_2|+\ldots+k|\alpha_k|=m$ on $J^kV$ with holomorphic coefficients; in other words, we obtain precisely the sheaf of sections of the bundle $E_{k,m}^\GG V^*$ of jet differentials of order $k$ and degree $m$. \claim 3.9. Proposition|By construction, if $\pi_k:X_k^\GG\to X$ is the natural projection, we have the direct image formula $$ (\pi_k)_*\cO_{X^\GG_k}(m)=\cO(E_{k,m}^\GG V^*) $$ for all $k$ and $m$. \endclaim \plainsubsection 3.B. Invariant jet differentials| In the geometric context, we are not really interested in the bundles $(J_kV\ssm\{0\})/\bC^*$ themselves, but rather on their quotients $(J_kV\ssm\{0\})/\bG_k$ (would such nice complex space quotients exist!). We will see that the Semple bundle $X_k$ constructed in \S$\,$2.D plays the role of such a quotient. First we introduce a canonical bundle subalgebra of~$E^\GG_{k,\bu}V^*$. \claim 3.10.~Definition|We introduce a subbundle $E_{k,m}V^*\subset E^\GG_{k,m}V^*$, called the bundle of invariant jet differentials of order~$k$ and degree $m$, defined as follows: $E_{k,m}V^*$ is the set of polynomial differential operators $Q(f',f'',\ldots,f^{(k)})$ which are invariant under arbitrary changes of parametrization, i.e., for every $\varphi\in \bG_k$ $$ Q\big((f\circ\varphi)',(f\circ\varphi)'',\ldots,(f\circ\varphi)^{(k)})= \varphi'(0)^m Q(f',f'',\ldots,f^{(k)}). $$ \endclaim Alternatively, $E_{k,m}V^*=(E^\GG_{k,m}V^*)^{\bG'_k}$ is the set of invariants of $E^\GG_{k,m}V^*$ under the action of~$\bG'_k$. Clearly, $E_{\infty,\bu}V^*=\bigcup_{k\ge 0}\bigoplus_{m\ge0}E_{k,m}V^*$ is a subalgebra of $E^\GG_{k,m}V^*$ (observe however that this algebra is not invariant under the derivation~$D^\GG$, since e.g.\ $f_j''=D^\GG f_j$ is not an invariant polynomial). \claim 3.11.~Theorem|Suppose that $V$ has rank $r\ge 2$. Let $\pi_{0,k}: X_k\lra X$ be the Semple jet bundles constructed in section~{\rm 2.B}, and let $J_kV^\reg$ be the bundle of regular $k$-jets of maps $f:(\bC,0)\to X$, that is, jets $f$ such that $f'(0)\ne 0$. \plainitem{\rm(i)} The quotient $J_kV^\reg/\bG_k$ has the structure of a locally trivial bundle over~$X$, and there is a holomorphic embedding $J_kV^\reg/\bG_k\hookrightarrow X_k$ over $X$, which identifies $J_kV^\reg/\bG_k$ with $X_k^\reg$ $($thus $X_k$ is a relative compactification of $J_kV^\reg/\bG_k$ over~$X)$. \plainitem{\rm(ii)} The direct image sheaf $$ (\pi_{0,k})_*\cO_{X_k}(m)\simeq\cO(E_{k,m}V^*) $$ can be identified with the sheaf of holomorphic sections of $E_{k,m}V^*$. \plainitem{\rm(iii)} For every $m>0$, the relative base locus of the linear system $|\cO_{X_k}(m)|$ is equal to the set $X_k^\sing$ of singular $k$-jets. Moreover, $\cO_{X_k}(1)$ is relatively big over~$X$. \vskip0pt \endclaim \plainproof. (i) For $f\in J_kV^\reg$, the lifting $\swt{f}$ is obtained by taking the derivative $(f,[f'])$ without any cancellation of zeroes in~$f'$, hence we get a uniquely defined $(k-1)$-jet $\swt{f}:(\bC,0)\to\swt{X}$. Inductively, we get a well defined $(k-j)$-jet $f_{[j]}$ in~$X_j$, and the value $f_{[k]}(0)$ is independent of the choice of the representative $f$ for the $k$-jet. As the lifting process commutes with reparametrization, i.e., $(f\circ\varphi)^\sim=\swt{f}\circ\varphi$ and more generally $(f\circ\varphi)_{[k]}=f_{[k]}\circ\varphi$, we conclude that there is a well defined set-theoretic map $$ J_kV^\reg/\bG_k\to X_k^\reg,\qquad f~\mod~\bG_k\mapsto f_{[k]}(0). $$ This map is better understood in coordinates as follows. Fix coordinates $(z_1\ld z_n)$ near a point $x_0\in X$, such that $V_{x_0}=\Vect(\partial/\partial z_1\ld\partial/\partial z_r)$. Let $f=(f_1\ld f_n)$ be a regular $k$-jet tangent to~$V$. Then there exists $i\in\{1,2\ld r\}$ such that $f_i'(0)\ne 0$, and there is a unique reparametrization $t=\varphi(\tau)$ such that $f\circ\varphi=g =(g_1,g_2\ld g_n)$ with $g_i(\tau)=\tau$ (we just express the curve as a graph over the $z_i$-axis, by means of a change of parameter $\tau=f_i(t)$, i.e.\ $t=\varphi(\tau)=f_i^{-1}(\tau)$). Suppose $i=r$ for the simplicity of notation. The space $X_k$ is a $k$-stage tower of $\bP^{r-1}$-bundles. In~the corresponding inhomogeneous coordinates on these $\bP^{r-1}$'s, the point $f_{[k]}(0)$ is given by the collection of derivatives $$ \big((g_1'(0)\ld g_{r-1}'(0));(g_1''(0)\ld g_{r-1}''(0));\ldots; (g_1^{(k)}(0)\ld g_{r-1}^{(k)}(0))\big). $$ [Recall that the other components $(g_{r+1}\ld g_n)$ can be recovered from $(g_1\ld g_r)$ by integrating the differential system (5.10)]. Thus the map $J_kV^\reg/\bG_k\to X_k$ is a bijection onto $X_k^\reg$, and the fibers of these isomorphic bundles can be seen as unions of $r$ affine charts ${}\simeq(\bC^{r-1})^k$, associated with each choice of the axis $z_i$ used to describe the curve as a graph. The change of parameter formula ${d\over d\tau}={1\over f_r'(t)}{d\over dt}$ expresses all derivatives $g_i^{(j)}(\tau)=d^jg_i/d\tau^j$ in terms of the derivatives $f_i^{(j)}(t)=d^jf_i/dt^j$ $$ \leqalignno{ (g_1'\ld g_{r-1}')&=\Big({f'_1\over f'_r}\ld{f'_{r-1}\over f'_r}\Big);\cr (g_1''\ld g_{r-1}'')&=\Big({f''_1f'_r-f''_rf'_1\over f^{\prime 3}_r}\ld {f''_{r-1}f'_r-f''_rf'_{r-1}\over f^{\prime 3}_r}\Big);~\ldots~;&(3.12)\cr \qquad(g_1^{(k)}\ld g_{r-1}^{(k)})&=\Big({f^{(k)}_1f'_r-f^{(k)}_rf'_1\over f^{\prime k+1}_r}\ld{f^{(k)}_{r-1}f'_r-f^{(k)}_rf'_{r-1}\over f^{\prime k+1}_r}\Big)+(\hbox{order}0$. We see that $\cO_{X_k}(1)$ is big by considering the sections of $\cO_{X_k}(2k-1)$ associated with the polynomials $Q(f'\ld f^{(k)})=f_r^{\prime 2k-1}g_i^{(j)}$, $1\le i\le r-1$, $1\le j\le k$; indeed, these sections separate all points in the open chart $f_r'\ne 0$ of~$X_{k,x}^\reg$. Now, we check that every section $\sigma$ of $\cO_{X_k}(m)$ over $X_{k,x}$ must vanish on $X_{k,x}^\sing$. Pick an arbitrary element $w\in X_k^\sing$ and a germ of curve \hbox{$f:(\bC,0)\to X$} such that $f_{[k]}(0)=w$, $f_{[k-1]}'(0)\ne 0$ and $s=m(f,0)\gg 0$ (such an $f$ exists by Corollary~6.14). There are local coordinates $(z_1\ld z_n)$ on $X$ such that $f(t)=(f_1(t)\ld f_n(t))$ where $f_r(t)=t^s$. Let $Q$, $\swh{Q}$ be the polynomials associated with $\sigma$ in these coordinates and let $(f')^{\alpha_1}(f'')^{\alpha_2}\cdots (f^{(k)})^{\alpha_k}$ be a monomial occurring in $Q$, with $\alpha_j\in\bN^r$, $|\alpha_j|=\ell_j$, \hbox{$\ell_1+2\ell_2+\cdots+k\ell_k=m$}. Putting $\tau=t^s$, the curve $t\mapsto f(t)$ becomes a Puiseux expansion $\tau\mapsto g(\tau)=(g_1(\tau)\ld g_{r-1}(\tau),\tau)$ in which $g_i$ is a power series in~$\tau^{1/s}$, starting with exponents of $\tau$ at least equal to~$1$. The derivative $g^{(j)}(\tau)$ may involve negative powers of $\tau$, but the exponent is always${}\ge 1+{1\over s}-j$. Hence the Puiseux expansion of $\swh{Q}(g',g''\ld g^{(k)})$ can only involve powers of $\tau$ of exponent \hbox{$\ge~{}-\max_\ell((1-{1\over s})\ell_2+\cdots+(k-1-{1\over s})\ell_k)$}. Finally $f_r'(t)=st^{s-1}=s\tau^{1-1/s}$, thus the lowest exponent of $\tau$ in $Q(f'\ld f^{(k)})$ is at least equal to $$ \eqalign{ \Big(1-{1\over s}\Big)m-\max_\ell\Big(&\Big(1-{1\over s}\Big)\ell_2+\cdots+ \Big(k-1-{1\over s}\Big)\ell_k\Big)\cr &\ge\min_\ell\Big(1-{1\over s}\Big)\ell_1+\Big(1-{1\over s}\Big)\ell_2 +\cdots+\Big(1-{k-1\over s}\Big)\ell_k,\cr} $$ where the minimum is taken over all monomials $(f')^{\alpha_1}(f'')^{\alpha_2}\cdots (f^{(k)})^{\alpha_k}$, $|\alpha_j|=\ell_j$, occurring in $Q$. Choosing $s\ge k$, we already find that the minimal exponent is positive, hence $Q(f'\ld f^{(k)})(0)=0$ and $\sigma(w)=0$ by (3.14).\qed Theorem 3.11~(iii) shows that $\cO_{X_k}(1)$ is never relatively ample over $X$ for $k\ge 2$. In order to overcome this difficulty, we define for every \hbox{$\abu=(a_1\ld a_k)\in\bZ^k$} a line bundle $\cO_{X_k}(\abu)$ on $X_k$ such that $$ \cO_{X_k}(\abu)=\pi_{1,k}^*\cO_{X_1}(a_1)\otimes \pi_{2,k}^*\cO_{X_2}(a_2)\otimes\cdots\otimes\cO_{X_k}(a_k). \leqno(3.16) $$ By (6.9), we have $\pi_{j,k}^*\cO_{X_j}(1)= \cO_{X_k}(1)\otimes\cO_{X_k} (-\pi_{j+1,k}^* D_{j+1}-\cdots-D_k)$, thus by putting $D^*_j= \pi_{j+1,k}^* D_{j+1}$ for $1\le j\le k-1$ and $D^*_k=0$, we find an identity $$ \leqalignno{ &\cO_{X_k}(\abu)=\cO_{X_k}(b_k)\otimes\cO_{X_k}(-\bbu\cdot D^*), \qquad\hbox{\rm where}&(3.17)\cr &\bbu=(b_1\ld b_k)\in\bZ^k,\quad b_j=a_1+\cdots+a_j,&\cr &\bbu\cdot D^*=\sum_{1\le j\le k-1}b_j\,\pi_{j+1,k}^* D_{j+1}.&\cr} $$ In particular, if $\bbu\in\bN^k$, i.e., $a_1+\cdots+a_j\ge 0$, we get a morphism $$ \cO_{X_k}(\abu)=\cO_{X_k}(b_k)\otimes\cO_{X_k}(-\bbu\cdot D^*) \to\cO_{X_k}(b_k).\leqno(3.18) $$ The following result gives a sufficient condition for the relative nefness or ampleness of weighted jet bundles. \claim 3.19.~Proposition|Take a very ample line bundle $A$ on $X$, and consider on $X_k$ the line bundle $$ L_k=\cO_{X_k}(3^{k-1},3^{k-2},\ldots,3,1)\otimes \pi_{k,0}^*A^{\otimes 3^k} $$ defined inductively by $L_0=A$ and $L_k=\cO_{X_k}(1)\otimes\pi_{k,k-1}^*L_{k-1}^{\otimes 3}$. Then $V_k^*\otimes L_k^{\otimes 2}$ is a nef vector bundle on $X_k$, which is in fact generated by its global sections, for all $k\geq 0$. Equivalently $$ L'_k=\cO_{X_k}(1)\otimes\pi_{k,k-1}^*L_{k-1}^{\otimes 2}= \cO_{X_k}(2\cdot 3^{k-2},2\cdot 3^{k-3},\ldots,6,2,1)\otimes \pi_{k,0}^*A^{\otimes 2\cdot 3^{k-1}} $$ is nef over~$X_k$ $($and generated by sections$)$ for all $k\geq 1$. \endclaim Let us recall that a line bundle $L\to X$ on a projective variety $X$ is said to nef if \hbox{$L\cdot C\geq 0$} for~all~irreducible algebraic curves $C\subset X$, and that a vector bundle $E\to X$ is said to be nef if $\cO_{\bP(E)}(1)$ is nef on $\bP(E):=P(E^*)\,$; any vector bundle generated by global sections is nef (cf.~[DePS94] for more details). The statement concerning $L'_k$ is obtained by projectivizing the vector bundle $E=V_{k-1}^*\otimes L_{k-1}^{\otimes 2}$ on $X_{k-1}$, whose associated tautological line bundle is $\cO_{\bP(E)}(1)=L'_k$ on \hbox{$\bP(E)=P(V_{k-1})=X_k$}. Also one gets inductively that $$ L_k=\cO_{\bP(V_{k-1}\otimes L_{k-1}^{\otimes 2})}(1)\otimes\pi_{k,k-1}^*L_{k-1} \quad\hbox{\it is very ample on $X_k$}.\leqno(3.20) $$ \smallskip \plainproof. Let $X\subset \bP^N$ be the embedding provided by $A$, so that $A=\cO_{\bP^N}(1)_{\restriction X}$. As is well known, if $Q$ is the tautological quotient vector bundle on $\bP^N$, the twisted cotangent bundle $$ T^*_{\bP^N}\otimes\cO_{\bP^N}(2)=\Lambda^{N-1}Q $$ is nef; hence its quotients $T_X^*\otimes A^{\otimes 2}$ and $V_0^*\otimes L_0^{\otimes 2}=V^*\otimes A^{\otimes 2}$ are nef (any tensor power of nef vector bundles is nef, and so is any quotient). We now proceed by induction, assuming $V_{k-1}^*\otimes L_{k-1}^{\otimes 2}$ to be nef, $k\geq 1$. By taking the second wedge power of the central term in $(6.4')$, we get an injection $$ 0\lra T_{X_k/X_{k-1}}\lra \Lambda^2\big(\pi_k^\star V_{k-1}\otimes \cO_{X_k}(1)\big). $$ By dualizing and twisting with $\cO_{X_{k-1}}(2)\otimes\pi_k^\star L_{k-1}^{\otimes 2}$, we find a surjection $$ \pi_k^\star\Lambda^2(V_{k-1}^\star\otimes L_{k-1})\lra T_{X_k/X_{k-1} }^\star\otimes\cO_{X_k}(2)\otimes\pi_k^\star L_{k-1}^{\otimes 2}\lra 0. $$ By the induction hypothesis, we see that $T_{X_k/X_{k-1}}^\star\otimes \cO_{X_k}(2)\otimes\pi_k^\star L_{k-1}^{\otimes 2}$ is nef. Next, the dual of (6.4) yields an exact sequence $$ 0\lra\cO_{X_k}(1)\lra V_k^\star\lra T_{X_k/X_{k-1}}^\star\lra 0. $$ As an extension of nef vector bundles is nef, the nefness of $V_k^*\otimes L_k^{\otimes 2}$ will follow if we check that $\cO_{X_k}(1)\otimes L_k^{\otimes 2}$ and $T^\star_{X_k/X_{k-1}}\otimes L_k^{\otimes 2}$ are both nef. However, this follows again from the induction hypothesis if we observe that the latter implies $$ L_k\geq \pi_{k,k-1}^* L_{k-1}\quad\hbox{and}\quad L_k\geq \cO_{X_k}(1)\otimes\pi_{k,k-1}^* L_{k-1} $$ in the sense that $L''\geq L'$ if the ``difference'' $L''\otimes (L')^{-1}$ is nef. All statements remain valid if we replace ``nef'' with ``generated by sections'' in the above arguments.\qed \claim 3.21. Corollary|A $\bQ$-line bundle $\cO_{X_k}(\abu)\otimes \pi_{k,0}^*A^{\otimes p}$, $\abu\in\bQ^k$, $p\in\bQ$, is nef $($resp.\ ample$)$ on $X_k$ as soon as $$ \hbox{$a_j\geq 3a_{j+1}$ for $j=1,2,\ldots,k-2$ and $a_{k-1}\geq 2a_k\geq 0$, $p\geq 2\sum a_j$}, $$ resp.\ $$ \hbox{$a_j\geq 3a_{j+1}$ for $j=1,2,\ldots,k-2$ and $a_{k-1}>2a_k>0$, $p> 2\sum a_j$}. $$ \endclaim \plainproof. This follows easily by taking convex combinations of the $L_j$ and $L'_j$ and applying Proposition~3.19 and our observation (3.20).\qed \claim 3.22.~Remark|{\rm As $\bG_k$ is a non-reductive group, it is a priori unclear whether the graded ring $\cA_{n,k,r}=\bigoplus_{m\in\bZ} E_{k,m}V^\star$ (taken pointwise over $X$) is finitely generated. This can be checked manually ([Dem07a], [Dem07b]) for $n=2$ and $k\le 4$. Rousseau [Rou06] also checked the case $n=3$, $k=3$, and then Merker [Mer08, Mer10] proved the finiteness for $n=2,3,4$, $k\leq 4$ and $n=2$, $k=5$. Recently, B\'erczi and Kirwan [BeKi12] made an attempt to prove the finiteness in full generality, but it appears that the general case is still unsettled.} \endclaim \plainsubsection 3.C. Fundamental vanishing theorem| We prove here a fundamental vanishing theorem due to Siu and Yeung ([SiYe96, SiYe97], [Siu97]). Their original proof makes use of Nevanlinna theory, especially of the logarithmic derivative lemma, see also [Dem97] for a more detailed account (in French). An alternative simpler proof based on the Ahlfors lemma and on algebraic properties of jet differentials can be found in [Dem18] (cf.~also [Dem95]). \claim 3.23. Fundamental vanishing theorem| Let $(X,V)$ be a projective directed manifold and $A$ an ample divisor on~$X$. Then $P(f\,;\,f',f'',\ldots,f^{(k)})=0$ for every entire curve $f:(\bC,T_\bC)\to (X,V)$ and every global section $P\in H^0(X,E^\GG_{k,m}V^*\otimes \cO(-A))$. \endclaim \plainproof. We first give a proof of~3.23 in the special case where $f$ is a Brody curve, i.e.\ $\sup_{t\in\bC}\Vert f'(t)\Vert_\omega<+\infty$ with respect to a given Hermitian metric $\omega$ on $X$. In fact, the proof is much simpler in that case, and thanks to the Brody criterion~1.8, this is sufficient to establish the hyperbolicity of $(X,V)$. After raising $P$ to a power $P^s$ and replacing $\cO(-A)$ with $\cO(-sA)$, one can always assume that $A$ is a very ample divisor. We interpret $\smash{E^\GG_{k,m}V^*} \otimes \cO(-A)$ as the bundle of complex valued differential operators whose coefficients $a_\alpha(z)$ vanish along~$A$. Fix a finite open covering of $X$ by coordinate balls $B(p_j,R_j)$ such that the balls $B_j(p_j,R_j/4)$ still cover $X$. As $f'$ is bounded, there exists $\delta>0$ such that for $f(t_0)\in B(p_j,R_j/4)$ we have $f(t)\in B(p_j,R_j/2)$ whenever $|t-t_0|<\delta$, uniformly for every $t_0\in\bC$. The Cauchy inequalities applied to the components of $f$ in each of the balls imply that the derivatives $f^{(j)}(t)$ are bounded on~$\bC$, and therefore, since the coefficients $a_\alpha(z)$ of $P$ are also uniformly bounded on each of the balls $B(p_j,R_j/2)$ we conclude that $g:=P(f\,;\,f',f'',\ldots,f^{(k)})$ is a bounded holomorphic function on $\bC$. After moving $A$ in the linear system $|A|$, we may further assume that $\Supp A$ intersects $f(\bC)$. Then $g$ vanishes somewhere, hence $g\equiv 0$ by Liouville's theorem, as expected. Next we consider the case where $P\in H^0(X,E_{k,m}V^*\otimes \cO(-A))$ is an invariant differential operator. We may of course assume $P\neq 0$. Then we get an associated non-zero section $\sigma\in H^0(X_k,\cO_{X_k}(m)\otimes \pi_{k,0}^*\cO(-A))$. Thanks to Corollary 3.21, the line bundle $$ L=\cO_{X_k}(a_\bu)\otimes \pi_{k,0}^*\cO(pA)= \cO_{X_k}(m')\otimes\cO_{X_k}(-b_\bu\cdot D^*)\otimes\pi_{k,0}^*\cO(pA) $$ is ample on $X_k$ for suitable $b_\bu\geq 0$ and $m',p>0$. Let $h_L$ be a smooth metric on $L$ such that $\omega_k=\Theta_{L,h_L}$ is a K\"ahler metric on~$X_k$. Then we can produce a singular hermitian metric $h$ on $\cO_{X_k}(-1)$ by putting $$ \Vert\xi\Vert_h =(\Vert\sigma^p\cdot\xi^{pm+m'}\Vert_{h_L^{-1}})^{1/(pm+m')}, \xi\in\cO_{X_k}(-1), $$ and viewing $\sigma^p\cdot\xi^{pm+m'}$ as an element in $\cO_{X_k}(-m')\otimes\pi_{k,0}^*\cO(-pA)\subset \cO(L^{-1})$. The metric $h$ has a weight $e^\varphi$ that is continuous, with zeroes contained in the union of $\{\sigma=0\}$ and of the vertical divisor~$D^*$. Moreover the curvature tensor $\Theta_{\cO_{X_k}(1),h^{-1}}={i\over 2\pi}\ddbar\log h$ satisfies by construction $\Theta_{\cO_{X_k}(1),h^{-1}}\geq (pm+m')^{-1}\omega_k$. On the other hand, the continuity of the weight of $h$ and the compactness of $X_k$ imply that there exists a constant $C>0$ such that $\Vert d\pi_{k,k-1}(\eta)\Vert_h\leq C\Vert\eta\Vert_{\omega_k}$ for all vectors $\eta\in V_k$ (notice that $\xi=d\pi_{k,k-1}(\eta)\in\cO_{X_k}(-1)$). Now, the derivative $f_{[k-1]}'$ can be seen as a section of $f_{[k]}^*\cO_{X_k}(-1)$, and we use this to define a singular hermitian metric $\gamma(t)\,i\,dt\wedge d\overline t$ on $\bC$ by taking $$ \gamma(t)=\Vert f_{[k-1]}'(t)\Vert_{h(f_{[k]}(t))}^2. $$ If $f_{[k]}(\bC)$ is not contained in the divisor $\{\sigma=0\}$, then $\gamma$ is not identically zero and, in the sense of distributions, we find $$ {i\over 2\pi}\ddbar\log\gamma \geq f_{[k]}^*\Theta_{\cO_{X_k}(1),h^{-1}} \geq (pm+m')^{-1}f_{[k]}^*\omega_k \geq C^{-1}(pm+m')^{-1}\gamma. $$ The final inequality comes from the inequality relating $h$ and $\omega_k$ when we take $\eta=f_{[k]}'(t)$ and $\xi=f_{[k-1]}'(t)$. However, the Ahlfors lemma shows that a hermitian metric on $\bC$ with negative curvature bounded away from~$0$ cannot exist, thus we must have $f_{[k]}(\bC)\subset\{\sigma=0\}$. This proves our vanishing theorem in the case where $P$ is invariant. The general case of a nonnecessarily invariant operator $P$ will not be used here; a proof can be obtained by decomposing $P$ into invariant parts and using an induction on $m$ (cf.\ [Dem18] for details), or alternatively by means of Nevanlinna theory arguments ([SiYe97], [Siu97], see also [Dem97]).\qed Especially, we can apply the above vanishing theorem for any global invariant jet differential $P\in H^0(X,E_{k,m}V^*\otimes \cO(-A))$. In that case, $P$ corresponds bijectively to a section $$ \sigma\in H^0(X_k,\cO_{X_k}(m)\otimes\pi_{k,0}^*\cO(-A)), \leqno(3.24) $$ and assuming $P\neq 0$, the vanishing theorem can be reinterpreted by stating that $f_{[k]}(\bC)$ is contained in the zero divisor $Z_\sigma\subset X_k$. Let $\Delta_k=\bigcup_{2\le \ell\le k} \pi_{k,\ell}^{-1}(D_\ell$) be the union of the vertical divisors (see (2.22) and (2.23)). Then $f_{[k]}(\bC)$ cannot be contained in $\Delta_k$ (as otherwise we would have $f'(t)=0$ identically). We define the $k$-stage Green-Griffiths locus of $(X,V)$ to be the Zariski closure $$ \GG_k(X,V)=\overline{(X_k\smallsetminus\Delta_k)\cap \bigcap_{m\in\bN}\left(\hbox{base locus of }\cO_{X_k}(m)\otimes \pi_{k,0}^*\cO(-A)\right)} \leqno(3.25) $$ (trivially independent of the choice of~$A$), and $$ \GG(X,V)=\bigcap_{k\in\bN^*}\pi_{k,0}\big(\GG_k(X,V)). \leqno(3.26) $$ Then Theorem~3.23 implies that $f_{[k]}(\bC)$ must be contained in $\GG_k(X,V)$ for every entire curve $f:(\bC,T_\bC)\to (X,V)$, and also that $f(\bC)\subset\GG(X,V)$. \claim 3.27. Corollary|If $\GG(X,V)=\emptyset$, then $(X,V)$ is hyperbolic. In particular, if there exists $k\geq 1$ and a weight $a_\bu\in\bN^k$ such that $\cO_{X_k}(a_\bu)$ is ample on $X_k$, then $(X,V)$ is hyperbolic. \endclaim It should be observed that Corollary 3.27 yields a sufficient condition for hyperbolicity, but this is not a necessary condition. In fact, if we take $X=C_1\times C_2$ to be a product of curves of genus${}\geq 2$ and $V=T_X$, it is easily checked that $\GG(X)=\GG(X,T_X)=X$. More general examples have been found by Diverio and Rousseau [DR15]. In a similar way, the Green-Griffiths-Lang conjecture holds for $(X,V)$ if $Y:=\GG(X,V)\subsetneq X$, but this is only a sufficient condition. The following fundamental existence theorem, however, has been proved in [Dem11], using holomorphic Morse inequalities of [Dem85] as an essential tool. We only state the main result, as it will not be used here. \claim 3.28. Theorem|Let $(X,V)$ be a projective directed manifold of general type, in the sense that the sheaf $K_V$ of locally bounded sections of $\cO(\det V^*)$ is big. Let $A$ be an ample $\bQ$-divisor on $X$ such that $\cO(\det V^*)\otimes\cO(-A)$ is still ample. Then $$ H^0\bigg(X_k,\cO_{X_k}(m)\otimes\pi_{k,0}^* \cO\Big(-{m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)A\Big)\bigg)\neq 0 $$ for $m\gg k\gg 1$ and $m$ sufficiently divisible $($so that the multiple of $A$ is an integral divisor$)$. In particular $\GG_k(X,V)\subsetneq X_k$ for $k\gg 1$. \endclaim \section{Existence of hyperbolic hypersurfaces of low degree} We give here a self-contained proof of the existence of hyperbolic surfaces of low degree in $\bP^{n+1}$, using various techniques borrowed from the work of Toda [Toda71], Fujimoto [Fuj74], Green [Gre75], Nadel [Nad89], Siu-Yeung [SiYe96], Masuda-Noguchi [MaNo96] and Shiff\-man-Zaidenberg [ShZa02]. The main idea is to produce ad hoc differential equations for entire curves by means of Wronskian operators. This can be seen as a variation of Nadel's approach, that was actually based on Wronkians associated with meromorphic connections -- Wronskian operators have the advantage of being much easier to handle than general jet differentials, thanks to their straightforward relationship with linear degeneracy. \plainsubsection 4.A. General Wronskian operators| This section follows closely the work of D.~Brotbek [Brot17]. Let $U$ be an open set of a complex manifold $X$, $\dim X=n$, and $s_0,\ldots,s_k\in\cO_X(U)$ be holomorphic functions. To these functions, we can associate a Wronskian operator of order $k$ defined by $$ W_k(s_0,\ldots,s_k)(f)=\left| \plainmatrix{ s_0(f) & s_1(f) & \ldots &s_k(f)\cr D(s_0(f)) & D(s_1(f)) & \ldots &D(s_k(f))\cr \noalign{\vskip4pt} \vdots & \vdots & &\vdots\cr \noalign{\vskip4pt} D^k(s_0(f)) & D^k(s_1(f)) & \ldots &D^k(s_k(f))\cr}\right|,\leqno(4.1) $$ where $f:t\mapsto f(t)\in U\subset X$ is a germ of holomorphic curve (or a $k$-jet of curve), and $D=\frac{d}{dt}$. For a biholomorphic change of variable $\varphi$ of $(\bC,0)$, we find by induction on $\ell$ polynomial differential operators $Q_{\ell,i}$ of order${}\leq\ell$ acting on $\varphi$ satisfying $$ D^\ell(s_j(f\circ \varphi))=\varphi^{\prime\ell}D^\ell(s_j(f))\circ\varphi +\sum_{i<\ell}Q_{\ell,i}(\varphi',\ldots,\varphi^{(\ell)}) \, D^i(s_j(f))\circ\varphi. $$ It follows easily from this that $$ W_k(s_0,\ldots,s_k)(f\circ\varphi)=(\varphi')^{1+2+\cdots+k} W_k(s_0,\ldots,s_k)(f)\circ\varphi, $$ hence $W_k(s_0,\ldots,s_k)(f)$ is an invariant differential operator of degree $k'=\frac{1}{2}k(k+1)$. Especially, we get in this way a section that we denote $$ W_k(s_0,\ldots,s_k)= \left| \plainmatrix{ s_0 & s_1 & \ldots &s_k\cr D(s_0) & D(s_1) & \ldots &D(s_k)\cr \noalign{\vskip4pt} \vdots & \vdots & &\vdots\cr \noalign{\vskip4pt} D^k(s_0) & D^k(s_1) & \ldots &D^k(s_k)\cr}\right|\in H^0(U,E_{k,k'}T^*_X).\leqno(4.2) $$ \claim 4.3. Proposition|These Wronskian operators satisfy the following properties. \vskip2pt \plainitem{\rm(a)} $W_k(s_0,\ldots,s_k)$ is $\bC$-multilinear and alternate in $(s_0,\ldots,s_k)$. \vskip2pt\plainitem{\rm(b)} For any $g\in\cO_X(U)$, we have $$W_k(gs_0,\ldots,gs_k)=g^{k+1}W_k(s_0,\ldots,s_k).$$ \endclaim \noindent Property 4.3~(b) is an easy consequence of the Leibniz formula $$ D^\ell(g(f)s_j(f))=\sum_{k=0}^{\ell}{\ell\choose k}D^k(g(f)) D^{\ell-k}(s_j(f)), $$ by performing linear combinations of rows in the determinants. This property implies in its turn that one can define more generally an operator $$ W_k(s_0,\ldots,s_k)\in H^0(U,E_{k,k'}T^*_X\otimes L^{k+1}) \leqno(4.4) $$ for any $(k+1)$-tuple of sections $s_0,\ldots,s_k\in H^0(U,L)$ of a holomorphic line bundle $L\to X$. In fact, when we compute the Wronskian in a local trivialization of $L_{\restriction U}$, Property 4.3~(b) shows that the determinant is independent of the trivialization. Moreover, if $g\in H^0(U,G)$ for some line bundle $G\to X$, we have $$ W_k(gs_0,\ldots,gs_k)=g^{k+1}W_k(s_0,\ldots,s_k)\in H^0(U,E_{k,k'}T^*_X \otimes L^{k+1}\otimes G^{k+1}).\leqno(4.5) $$ For global sections $\sigma_0,\ldots,\sigma_k\in H^0(X,L)$, we thus get a Wronskian operator $$ W_k(s_0,\ldots,s_k)\in H^0(X_k,\cO_{X_k}(k')\otimes\pi_{k,0}^*L^{k+1}) \leqno(4.6) $$ on the $k$-stage $X_k$ of the Semple tower. Very roughly, the idea for the construction of hyperbolic hypersurfaces is apply the fundamental vanishing theorem 3.23 to show that all entire curves have to satisfy certain Wronskian equations, leading in fine to exclude their existence. However, the vanishing theorem only holds for jet differentials in $H^0(X_k,\cO_{X_k}(k')\otimes\pi_{k,0}^*A^{-1})$ with $A>0$, while the existence of suitable sections $s_j\in H^0(X,L)$ can be achieved only when $L$ is ample, so the strategy seems a priori unapplicable. It turns out that one can sometimes arrange the Wronkian operator coefficients to be divisible by a section $\sigma_\Delta\in H^0(X,\cO_X(\Delta))$ possessing a large zero divisor~$\Delta$, so that $$ \sigma_\Delta^{-1}W_k(s_0,\ldots,s_k)\in H^0\big(X_k,\cO_{X_k}(k')\otimes\pi_{k,0}^*(L^{k+1}\otimes\cO_X(-\Delta))\big), \leqno(4.7) $$ and we can then hope that $L^{k+1}\otimes\cO_X(-\Delta))<0$. The strategy is to find a variety $X$ and sections $\sigma_0,\ldots,\sigma_k\in H^0(X,L)$ for which the associated Wronskian $W_k(s_0,\ldots,s_k)$ is highly divisible. \plainsubsection 4.B. Hyperbolicity of certain Fermat-Waring hypersurfaces| Let $Z$ be a non-singular $(n+1)$-dimensional projective variety, and let $A$ be a very ample divisor on~$Z\,$; the fundamental example is of course $Z=\bP^{n+1}$ and $A=\cO_{\bP^{n+1}}(1)$. Our goal is to show that a well chosen ($n$-dimensional) hypersurface $X=\{x\in Z\,;\;\sigma(x)=0\}$ defined by a section $\sigma\in H^0(Z,A^d)$, $d\gg 1$, is Kobayashi hyperbolic. The construction explained below follows closely the ideas of Shiffman-Zaidenberg [ShZa02] and is based similarly on a use of Fermat-Waring type hypersurfaces. Our proof is however completely self-contained. The reader can consult Brody-Green [BrGr77], Nadel [Nad89] and Masuda-Noguchi [MaNo96] for constructions based on other techniques. \claim 4.8. Theorem|Let $Z$ be a non-singular $(n+1)$-dimensional projective variety, $A$ a very ample divisor on~$Z$, and $\tau_j\in H^0(Z,A)$, $0\leq j\leq N$, sufficiently general sections. Then for $N\geq 2n$ and $d\geq N^2$, the hypersurface $X=\sigma^{-1}(0)$ associated with $\sigma=\sum_{0\leq j\leq N}\tau_j^d\in H^0(Z,A^d)$ is Kobayashi hyperbolic. \endclaim In particular, Theorem~4.8 provides examples of hyperbolic hypersurfaces of $\bP^{n+1}$ for all $n\geq 1$ and all degrees $d\geq 4n^2$. A~substantially improved bound $d\geq \lceil(n+3)^2/4\rceil$ has been obtained recently by [DTH16] via a deformation argument for certain unions of hyperplanes, but the methods are quite different from the techniques used here. As in [ShZa02], the main step of our proof is the following proposition due to Toda [Toda71], Fujimoto [Fuj74] and Green [Gre75]. \claim 4.9. Proposition|Let $g_j:\bC\to\bC$, $0\leq j\leq N$, be non-zero entire functions such that the curve $g=[g_0:\ldots:g_N]:\bC\to\bP^N$ satisfies \hbox{$\sum_{0\leq j\leq N}g_j^d=0$}. If $d\geq N^2$, there exists a partition $J_1,\ldots,J_q$ of $\{0,1,\ldots,N\}$ such that $|J_s|\geq 2$, $g_j/g_i$ is constant for all $i,j\in J_s$, and $\sum_{j\in J_s}g_j^d=0$ for all $s=1,2,\ldots,q$. If $g$ is nonconstant, we must have $q\geq 2$. \endclaim \plainproof. The result is true for $N=1$ (with a single $J_1=\{0,1\}$), and for higher values $N\geq 2$ we apply induction and use vanishing arguments for Wronskians. The map $g=[g_0:\ldots:g_N]:\bC\to\bP^N$ can be seen as an entire curve drawn in the (smooth, irreducible) Fermat hypersurface $Y=\sum_{0\leq j\leq N}z_j^d$ of $\bP^N$. We set $k=N-1$ and consider on $Y$ the Wronskian operator $$ W_k(s_0,\ldots,s_k)\quad\hbox{where $s_j(z)=z_j^d$, $\quad s_j\in H^0(Y,\cO(d))$}. $$ Then $$ W_{N-1}(s_0,\ldots,s_{N-1})\in H^0(Y,E_{k,k'}T^*_Y\otimes\cO(Nd)). $$ Since $D^\ell(s_j)$ is divisible by $z_j^{d-k}$ for $\ell\leq k$, we conclude that $W_{N-1}(s_0,\ldots,s_{N-1})$ is divisible by $\prod_{jk$ for $f$ to reach all points of the fiber $\pi_{k,0}^{-1}(x_0)$, the order $2k-1$ being sufficient by [Dem95, Proposition~5.11], but this fact does not play any role here). A consequence of universality is that $\cJ^X_k$ does not depend on coordinates nor on the geometry of~$X$.\qed \noindent The above discussion combined with Lemma 5.3 leads to the following statement. \claim 5.4. Proposition|Assume that $L$ generates all $k$-jets of sections $($e.g.\, take $L=A^p$ with $A$ very ample and $p\geq k)$, and let $\Sigma\subset H^0(X,L)$ be a linear system that also generates $k$-jets of sections at any point of $X$. Then we have a universal isomorphism $$ \mu_k^*\big(\cO_{X_k}(k')\otimes\pi_{k,0}^*L^{k+1})\otimes \cO_{\widehat X_{k,\Sigma}}(-F_k) \simeq(\Phi\circ\mu_k)^*\cO_{\Gr_{k+1}(\Sigma)}(1), $$ where $\mu_k:\widehat X_k\to X_k$ is the normalized blow-up of the $($maximal$\,)$ ideal sheaf $\cJ^X_k\subset\cO_{X_k}$ associated with order~$k$ Wronskians, and $F_k$ the universal divisor of $\widehat X_k$ resolving $\cJ^X_k$. \endclaim \plainsubsection 5.B. Specialization to suitable hypersurfaces| As in \S4.B, let $Z$ be a non-singular $(n+1)$-dimensional projective variety polarized with a very ample divisor~$A$. We are going to show that a sufficiently general algebraic hypersurface \hbox{$X=\{x\in Z\,;\;\sigma(x)=0\}$} defined by $\sigma\in H^0(Z,A^d)$ is Kobayashi hyperbolic when $d$ is large. Brotbek's main idea developed in [Brot17] is that a carefully selected hypersurface (of a more complicated type than the Fermat-Waring hypersurfaces considered in \S4) may have enough Wronskian sections to directly imply the ampleness of some tautological jet line bundle -- a Zariski open property. Here, we take $\sigma$ be a sum of terms $$ \sigma=\sum_{0\leq j\leq N}a_jm_j^\delta,\quad a_j\in H^0(Z,A^\rho),~~m_j\in H^0(Z,A^b),~~n0$ for $\delta$ large enough $($all other parameters being fixed or bounded$\,)$, and under this assumption, the fundamental vanishing theorem implies that all entire curves $f:\bC\to X$ are annihilated by these Wronskian operators. \endclaim \proof\ In fact, $$ (k+1)d=(k+c-N)d+(N+1-c)d=(k+c-N)d+(N+1-c)(\delta b+\rho) $$ and we get (5.12) by subtraction.\qed \plainsubsection 5.D. Control of the base locus for sufficiently general coefficients $a_j$ in $\sigma$| The next step is to control more precisely the base locus of these Wronskians and to find conditions on $N$, $k$, $c$, $d=b\delta+\rho$ ensuring that the base locus is empty for a generic choice of the sections $a_j$ in $\sigma=\sum a_jm_j$. Although we will not formally use it, the next lemma is useful to realize that the base locus is related to a natural rank condition. \claim 5.13. Lemma|Set $u_j:=a_jm_j^\delta$. The base locus in $X_k^\reg$ of the above Wronskians $W_{k,s,\hat\tau,a,J\restriction X}$, when $s,\,\hat\tau$ vary, consists of jets $f_{[k]}(0)\in X_k^\reg$ such that the matrix $(D^\ell(u_j\circ f)(0))_{0\leq\ell\leq k,\,j\in\complement J}$ is not of maximal rank $($i.e., of rank${}<\card\complement J=N+1-c)\,;$ if $\delta>k$, this includes all jets $f_{[k]}(0)$ such that $f(0)\in \bigcup_{I \neq J}\tau_I^{-1}(0)$. When $J$ also varies, the base locus of all $W_{k,s,\hat\tau,a,J\restriction X}$ in the Zariski open set $X'_k:=X_k^\reg\ssm\bigcup_{|I|=c}\tau_I^{-1}(0)$ consists of all $k$-jets such that $\rank(D^\ell(u_j\circ f)(0))_{0\leq\ell\leq k,\,0\leq j\leq N}\leq N-c$. \endclaim \plainproof. If $\delta>k$ and $m_j\circ f(0)=0$ for some $j\in J$, we have in fact $D^\ell(u_j\circ f)(0)=0$ for all derivatives $\ell\leq k$, because the exponents involved in all factors of the differentiated monomial $a_jm_j^\delta$ are at least equal to $\delta-k>0$. Hence the rank of the matrix cannot be maximal. Now, assume that $m_j\circ f(0)\neq 0$ for all $j\in\complement J$, i.e.\ $$ x_0:=f(0)\in X\ssm\bigcup_{j\in\complement J}m_j^{-1}(0)= X\ssm \bigcup_{I \neq J}\tau_I^{-1}(0).\leqno(5.14) $$ We take sections $\hat\tau_j$ so that $\hat\tau_j(x_0)\neq 0$, and then adjust the $k$-jet of the sections $s_1,\ldots,s_r$ in order to generate any matrix of derivatives $(D^\ell(s_j(f)\hat\tau_j(f)^{d-k})(0))_{0\leq\ell\leq k,\, j\in\complement J}$ (the fact that $f'(0)\neq 0$ is used for this!). Therefore, by expanding the determinant according to the last $N+1-c$ columns, we see that the base locus is defined by the equations $$\det(D^\ell(u_j(f))(0))_{\ell\in L,\,j\in\complement J}=0,\qquad \forall L\subset\{0,1,\ldots,k\},~~|L|=N+1-c,\leqno(5.15) $$ equivalent to the non-maximality of the rank. The last assertion follows by a simple linear algebra argument.\qed For a finer control of the base locus, we adjust the family of coefficients $$ a=(a_j)_{0\leq j\leq N}\in S:=H^0(Z,A^\rho)^{\oplus(N+1)}\leqno(5.16) $$ in our section $\sigma=\sum a_jm_j^\delta\in H^0(Z,A^d)$, and denote by $X_a=\sigma^{-1}(0)\subset Z$ the corresponding hypersurface. By Lemma~5.7, we know that there is a Zariski open set $U\subset S$ such that $X_a$ is smooth and $\prod \tau_I=0$ is a simple normal crossing divisor in $X_a$ for all $a\in U$. We consider the Semple tower $X_{a,k}:=(X_a)_k$ of $X_a$, the ``universal blow-up'' $\mu_{a,k}:\widehat X_{a,k}\to X_{a,k}$ of the Wronskian ideal sheaf $\cJ_{a,k}$ such that $\mu_{a,k}^*\cJ_{a,k}=\cO_{\widehat X_{a,k}}(-F_{a,k})$ for some ``Wronskian divisor'' $F_{a,k}$ in~$\widehat X_{a,k}$. By the universality of this construction, we can also embed $X_{a,k}$ in the Semple tower $Z_k$ of $Z$, blow up the Wronskian ideal sheaf $\cJ^Z_k$ of $Z_k$ to get a Wronskian divisor $F_k$ in $\widehat Z_k$ where $\mu_k:\widehat Z_k\to Z_k$ is the blow-up map. Then $F_{a,k}$ is the restriction of $F_k$ to $\widehat X_{a,k}\subset\widehat Z_k$. Our section $\widetilde W_{k,a,\hat\tau,s,J\restriction X_a}$ is the restriction of a \emph{meromorphic} section defined on $Z$, namely $$ (\hat\tau^{\alpha_J}\tau^{\beta_J})^{-1}W_{k,s,\hat\tau,a,J}= (\hat\tau^{\alpha_J}\tau^{\beta_J})^{-1}W_k\big(s_1\hat\tau_1^{d-k},...\,,s_r\hat\tau_r^{d-k},(a_jm_j^\delta)_{j\in\complement J}\big). \leqno(5.17) $$ It induces over the Zariski open set $Z'=Z\ssm\bigcup_I\tau_I^{-1}(0)$ a holomorphic section $$ \sigma_{k,s,\hat\tau,a,J} \in H^0\big(\widehat Z'_k, \mu_k^*(\cO_{Z_k}(k')\otimes\pi_{k,0}^*A^{-p})\otimes \cO_{\widehat Z_k}(-F_k)\big) \leqno(5.18) $$ (notice that the relevant factors $\hat\tau_j$ remain divisible on the whole variety~$Z$). By construction, thanks to the divisibility property explained in Lemma~5.10, the restriction of this section to $\widehat X'_{a,k}= \widehat X_{a,k}\cap\widehat Z'_k$ extends holomorphically to~$\widehat X_{a,k}$, i.e.\ $$ \sigma_{k,s,\hat\tau,a,J\restriction \widehat X_{a,k}} \in H^0\big(\widehat X_{a,k},\mu_{a,k}^*(\cO_{X_{a,k}}(k')\otimes\pi_{k,0}^*A^{-p}) \otimes\cO_{\widehat X_{a,k}}(-F_{a,k})\big). \leqno(5.19) $$ (Here the fact that we took $\widehat X_{k,a}$ to be normal avoids any potential issue in the division process, as $\widehat X_{k,a}\cap \mu_k^{-1}\big(\pi_{k,0}^{-1}\bigcap_{I\in\cG}\tau_I^{-1}(0)\big)$ has the expected codimension${}=\card\cG$ for any family $\cG$). \claim 5.20. Lemma|Let $V$ be a finite dimensional vector space over $\bC$, $\Psi:V^p\to\bC$ a non-zero alternating multilinear form, and let $m,c\in\bN$, $cn+1+kn$, there exists $a\in U\subset S$ such that the base locus of the family of sections $\sigma_{k,s,\hat\tau,a,J}$ in $\widehat X_{a,k}$ lies over $\bigcup_IX_a\cap\tau_I^{-1}(0)$. \endclaim \plainproof. The idea is similar to [Brot17, Lemma 3.8], but somewhat simpler in the present context. Let us consider a point $\xi\in\widehat Z'_k$ and the $k$-jet $f_{[k]}=\mu_k(\xi)\in Z'_k$, so that $x=f(0)\in Z'=Z\ssm\bigcup_I\tau_I^{-1}(0)$. Let us take the $\hat\tau_j$ such that $\hat\tau_j(x)\neq 0$. Then, we do not have to pay attention to the non-vanishing factors $\hat\tau^{\alpha_J}\tau^{\beta_J}$, and the $k$-jets of sections $m_j$ and $\hat\tau_j^{d-k}$ are invertible near~$x$. Let $e_A$ be a local generator of $A$ near $x$ and $e_\cL$ a local generator of the invertible sheaf $$ \cL=\mu_k^*\cO_{Z_k}(k')\otimes\cO_{\widehat Z_k}(-F_k) $$ near $\xi\in\widehat Z'_k$. Let $J^k\cO_{Z,x}=\cO_{Z,x}/\gm_{Z,x}^{k+1}$ be the vector space of $k$-jets of functions on $Z$ at $x$. By definition of the Wronskian ideal and of the associated divisor~$F_k$, we have a \emph{non-zero} alternating multilinear form $$ \Psi:(J^k\cO_{Z,x})^{k+1}\to\bC,\qquad (g_0,\ldots,g_k)\mapsto \mu_k^*W_k(g_0,\ldots,g_k)(\xi)/e_\cL(\xi). $$ The simultaneous vanishing of our sections at $\xi$ is equivalent to the vanishing of $$ \Psi\big( s_1\hat\tau_1^{d-k}e_A^{-d},\ldots,s_r\hat\tau_r^{d-k}e_A^{-d}, (a_jm_j^\delta e_A^{-d})_{j\in\complement J}\big) \leqno(5.22) $$ for all $(s_1,\ldots,s_r)$. Since $A$ is very ample and $\rho\geq k$, the power $A^\rho$ generates $k$-jets at every point $x\in Z$, hence the morphisms $$ H^0(Z,A^\rho)\to J^k\cO_{Z,x},\quad a\mapsto am_j^\delta e_A^{-d}\quad\hbox{and}\quad H^0(Z,A^k)\to J^k\cO_{Z,x},\quad s\mapsto s\hat\tau_j^{d-k}e_A^{-d} $$ are surjective. Lemma~5.20 applied with $r=k+c-N$ and $(p,m)$ replaced by $(k+1,N+1)$ implies that the codimension of families $a=(a_0,\ldots,a_N)\in S=H^0(Z,A^\rho)^{\oplus(N+1)}$ for which $\sigma_{k,s,\hat\tau,a,J}(\xi)=0$ for all choices of $s$, $\hat\tau$ and $J$ is at least $(c+1)(k+c-N+1)$, i.e.\ the dimension is at most $\dim S-(c+1)(k+c-N+1)$. When we let $\xi$ vary over $\widehat Z_k'$ which has dimension $(n+1)+kn$ and take into account the fibration $(a,\xi)\mapsto\xi$, the dimension estimate of Proposition~5.21 follows. Under the assumption $$ (c+1)(k+c-N+1)>n+1+kn\leqno(5.23) $$ we have $\dim \Gamma<\dim S$, hence the image of the projection $\Gamma\to S$, $(a,\xi)\mapsto a$ is a constructible algebraic subset distinct from~$S$. This concludes the proof.\qed Our final goal is to completely eliminate the base locus. Proposition 5.21 indicates that we have to pay attention to the intersections $X_a\cap\tau_I^{-1}(0)$. For $x\in Z$, we let $\cG$ be the family of hyperplane sections $\tau_I=0$ that contain $x$. We introduce the set $P=\{0,1,\ldots,N\}\ssm\bigcup_{I\in\cG}I$ and the smooth intersection $$ Z_\cG=Z\cap\bigcap_{I\in\cG}\tau_I^{-1}(0), $$ so that $N'+1:=\card P\geq N+1-c\card\cG$ and $\dim Z_\cG=n+1-\card\cG$. If $a\in U$ is such that $x\in X_a$, we also look at the intersection $$ X_{\cG,a}=X_a\cap\bigcap_{I\in\cG}\tau_I^{-1}(0), $$ which is a smooth hypersurface of $Z_\cG$. In that situation, we consider Wronskians $W_{k,s,\hat\tau,a,J}$ as defined above, but we now take $J\subset P$, $\card J=c$, $\complement J=P\ssm J$, $r'=k+c-N'$. \claim 5.24. Lemma|In the above setting, if we assume $\delta>k$, the restriction $W_{k,s,\hat\tau,a,J\restriction X_{\cG,a}}$ is still divisible by a monomial $\hat\tau^{\alpha_J}\tau^{\beta_J}$ such that $$ |\alpha_J|+|\beta_J|=(k+c-N')(d-2k)+(N'+1-c)(\delta-k)b+(\delta-k). $$ Therefore, if $$ p'=|\alpha_J|+|\beta_J|-(k+1)d=(\delta-k)-(k+c-N')2k-(N'+1-c)(kb+\rho) $$ as in $(5.12)$, we obtain again holomorphic sections \begin{align*} &\widetilde W_{k,s,\hat\tau,a,J\restriction X_{\cG,a}}:= (\hat\tau^{\alpha_J}\tau^{\beta_J})^{-1} W_{k,s,\hat\tau,a,J\restriction X_{\cG,a}} \in H^0(X_{\cG,a},E_{k,k'}T^*_X\otimes A^{-p'}),\\ \noalign{\vskip4pt} &\sigma_{k,s,\hat\tau,a,J\restriction \pi_{k,0}^{-1}(X_{\cG,a})} \in H^0\big(\pi_{k,0}^{-1}(X_{\cG,a}), \mu_{a,k}^*(\cO_{X_{a,k}}(k')\otimes\pi_{k,0}^*A^{-p'}) \otimes\cO_{\widehat X_{a,k}}(-F_{a,k})\big). \end{align*} \endclaim \plainproof. The arguments are similar to those employed in the proof of Lemma~5.10. Let $f_{[k]}\in X_{a,k}$ be a $k$-jet such that $f(0)\in X_{\cG,a}$ (the $k$-jet need not be entirely contained in $X_{\cG,a}$). Putting $j_0=\min(\complement J)$, we observe that we have on $X_{\cG,a}$ an identity $$ a_{j_0}m_{j_0}^\delta=-\sum_{i\in P\ssm\{j_0\}}a_im_i^\delta =-\sum_{i\in J}a_im_i^\delta-\sum_{P\ssm (J\cup\{j_0\})}a_im_i^\delta $$ because $m_i=\prod_{I\ni i}\tau_I=0$ on $X_{\cG,a}$ when $i\in\complement P=\bigcup_{I\in\cG}I$ (one of the factors $\tau_I$ is such that $I\in\cG$, hence $\tau_I=0$). If we compose with a germ $t\mapsto f(t)$ such that $f(0)\in X_{\cG,a}$ (even though $f$ does not necessarily lie entirely in $X_{\cG,a}$), we get $$ a_{j_0}m_{j_0}^\delta(f(t))= -\sum_{i\in J}a_im_i^\delta(f(t))-\sum_{P\ssm (J\cup\{j_0\})}a_im_i^\delta(f(t)) +O(t^{k+1}) $$ as soon as $\delta>k$. Hence we have an equality for all derivatives $D^\ell({\scriptstyle\bullet})$, $\ell\leq k$ at $t=0$, and $$ W_{k,s,\hat\tau,a,J\restriction X_{\cG,a}}(f_{[k]})=-\sum_{i\in J} W_k\big(s_1\hat\tau_1^{d-k}, \ldots,s_{r'}\hat\tau_{r'}^{d-k}, a_im_i^\delta,(a_jm_j^\delta)_{j\in P\ssm(J\cup\{j_0\})} \big)_{\restriction X_{\cG,a}}(f_{[k]}). $$ Then, again, $\tau_J^{\delta-k}$ is a new additional common factor of all terms in the sum, and we conclude as in Lemma 5.10 and Corollary 5.11.\qed Now, we analyze the base locus of these new sections on $$ \bigcup_{a\in U}\mu_{a,k}^{-1}\pi_{k,0}^{-1}(X_{\cG,a})\subset \mu_{k}^{-1}\pi_{k,0}^{-1}(Z_\cG)\subset\widehat Z_k. $$ As $x$ runs in $Z_\cG$ and $N'n+1-\card\cG+kn= \dim \mu_{k}^{-1}\pi_{k,0}^{-1}(Z_\cG).\leqno(5.23') $$ A proof entirely similar to that of Proposition 5.21 shows that for a generic choice of $a\in U$, the base locus of these sections on $\widehat X_{\cG,a,k}$ projects onto $\bigcup_{I\in\complement\cG}X_{\cG,a}\cap\tau_I^{-1}(0)$. Arguing inductively on $\card\cG$, the base locus can be shrinked step by step down to empty set (but it is in fact sufficient to stop when $X_{\cG,a}\cap\tau_I^{-1}(0)$ reaches dimension $0$). \plainsubsection 5.E. Nefness and ampleness of appropriate tautological line bundles| At this point, we have produced a smooth family $\cX_S\to U\subset S$ of particular hypersurfaces in $Z$, namely $X_a=\hbox{$\{\sigma_a(z)=0\}$}$, $a\in U$, for which a certain ``tautological'' line bundle has an empty base locus for sufficiently general coefficients: \claim 5.25. Corollary|Under condition $(5.23)$ and the hypothesis $p>0$ in $(5.12)$, the following properties hold. \plainitem{\rm(a)} The line bundle $$ \cL_a:=\mu_{a,k}^*(\cO_{X_{a,k}}(k')\otimes\pi_{k,0}^*A^{-1}) \otimes\cO_{\widehat X_{a,k}}(-F_{a,k}) $$ is nef on $\widehat X_{a,k}$ for general $a\in U'$, where $U'\subset U$ is a dense Zariski open set. \plainitem{\rm(b)} Let $\Delta_a=\sum_{2\leq\ell\leq k}\lambda_\ell D_{a,\ell}$ be a positive rational combination of vertical divisors of the Semple tower and $q\in\bN$, $q\gg 1$, an integer such that $$ \cL_a':=\cO_{X_{a,k}}(1)\otimes\cO_{a,k}(-\Delta_a)\otimes\pi_{k,0}^*A^q $$ is ample on $X_{a,k}$. Then the $\bQ$-line bundle $$ \cL_{a,\varepsilon,\eta}:= \mu_{a,k}^*(\cO_{X_{a,k}}(k')\otimes\cO_{X_{a,k}}(-\varepsilon\Delta_a) \otimes\pi_{k,0}^*A^{-1+q\varepsilon}) \otimes\cO_{\widehat X_{a,k}}(-(1+\varepsilon\eta)F_{a,k}) $$ is ample on $\widehat X_{a,k}$ for $a\in U'$, for some $q\in\bN$ and $\varepsilon,\eta\in\bQ_{>0}$ arbitrarily small.\vskip0pt \endclaim \plainproof. (a) This would be obvious if we had global sections generating $\cL_a$ on the whole of $\widehat X_{a,k}$, but our sections are only defined on a stratification of~$\widehat X_{a,k}$. In any case, if $C\subset\widehat X_{a,k}$ is an irreducible curve, we take a maximal family $\cG$ such that $C\subset X_{\cG,a,k}$. Then, by what we have seen, for $a\in U$ general enough, we can find global sections of $\cL_a$ on $\widehat X_{\cG,a,k}$ such that $C$ is not contained in their base locus. Hence $\cL_a\cdot C\geq 0$ and $\cL_a$ is nef for $a$ in a dense Zariski open set $U'\subset U$. \medskip \noindent (b) The existence of $\Delta_a$ and $q$ follows from Proposition~3.19 and Corollary~3.21, which even provide universal values for $\lambda_\ell$ and~$q$. After taking the blow up $\mu_{a,k}:\widehat X_{a,k}\to X_{a,k}$ (cf.\ (4.8)), we infer that $$ \cL'_{a,\eta}:=\mu_{a,k}^*\cL_a'\otimes\cO_{\widehat X_{a,k}} (-\eta F_{a,k})=\mu_{a,k}^*\big(\cO_{X_{a,k}}(1)\otimes\cO_{X_{a,k}}(-\Delta_a) \otimes\pi_{k,0}^*A^q\big)\otimes\cO_{\widehat X_{a,k}}(-\eta F_{a,k}) $$ is ample for $\eta>0$ small. The result now follows by taking a combination $$ \cL_{a,\varepsilon,\eta}=\cL_a^{1-\varepsilon/k'}\otimes (\cL'_{a,\eta})^\varepsilon.\eqno\square $$ \claim 5.26. Corollary|Let $\cX\to\Omega$ be the universal family of hypersurfaces $X_\sigma=\{\sigma(z)=0\}$, $\sigma\in \Omega$, where $\Omega\subset P(H^0(Z,A^d))$ is the dense Zariski open set over which the family is smooth. On the ``Wronskian blow-up'' $\widehat X_{\sigma,k}$ of $X_{\sigma,k}$, let us consider the line bundle $$ \cL_{\sigma,\varepsilon,\eta}:= \mu_{\sigma,k}^*(\cO_{X_{\sigma,k}}(k')\otimes\cO_{X_{\sigma,k}} (-\varepsilon\Delta_\sigma)\otimes\pi_{k,0}^*A^{-1+q\varepsilon}) \otimes\cO_{\widehat X_{\sigma,k}}(-(1+\varepsilon\eta)F_{\sigma,k}) $$ associated with the same choice of constants as in Cor.~$5.25$. Then $\cL'_{\sigma,\varepsilon,\eta}$ is ample on $\widehat X_{\sigma,k}$ for $\sigma$ in a dense Zariski open set $\Omega'\subset \Omega$. \endclaim \proof By 5.25~(b), we can find $\sigma_0\in H^0(Z,A^d)$ such that $X_{\sigma_0}=\sigma_0^{-1}(0)$ is smooth and $\cL_{\sigma_0,\varepsilon,\eta}^m$ is an ample line bundle on $\widehat X_{\sigma_0,k}$ ($m\in \bN^*$). As ampleness is a Zariski open condition, we infer that $\cL_{\sigma,\varepsilon,\eta}^m$ remains ample for a general section~$\sigma\in H^0(Z,A^d)$, i.e.\ for $[\sigma]$ in some Zariski open set $\Omega'\subset\Omega$. Since $\mu_{\sigma,k}(F_{\sigma,k})$ is contained in the vertical divisor of $X_{\sigma,k}$, we conclude by Corollary~3.27 that $X_\sigma$ is Kobayashi hyperbolic for $[\sigma]\in\Omega$.\qed \plainsubsection 5.F. Final conclusion and computation of degree bounds| At this point, we fix our integer parameters to meet all conditions that have been found. We must have $N\geq c(n+1)$ by Lemma 5.7, and for such a large value of $N$, condition (5.23) can hold only when $c\geq n$, so we take $c=n$ and $N=n(n+1)$. Inequality (5.23) then requires $k$ large enough, $k=n^3+n^2+1$ being the smallest possible value. We find $$ b={N\choose c-1}={n^2+n\choose n-1}=n\frac{(n^2+n)\ldots(n^2+2)}{n!}. $$ We have $n^2+k=n^2(1+k/n^2)\sqrt{2\pi n}\,(n/e)^n$, hence $$ b<\frac{n^{2n-1}\exp((2+\cdots+n)/n^2)}{\sqrt{2\pi n}\,(n/e)^n}< \frac{e^{n+\frac{1}{2}+\frac{1}{2n}}}{\sqrt{2\pi}}\,n^{n-\frac{3}{2}}. $$ Finally, we divide $d-k$ by $b$, get in this way $d-k=b\delta+\lambda$, $0\leq \lambda0$ is achieved as soon as $$ d\geq d_n=k+b\big(1+(2n^3+3)k+(n^2+2n+1)(kb+k+b-1)\big), $$ where $$ k=n^3+n^2+1,\quad b={n^2+n\choose n-1}. $$ The dominant term in $d_n$ is $k(n^2+2n+1)b^2\sim e^{2n+1}n^{2n+2}/2\pi$. By means of more precise numerical calculations and of Stirling's asymptotic expansion for $n!$, one can check in fact that \hbox{$d_n\leq \lfloor(n+4)\,(en)^{2n+1}/2\pi\rfloor$} for $n\geq 4$ (which is also an equivalent and a close approximation as~$n\to+\infty$), while $d_1=61$, $d_2=6685$, $d_3=2825761$. We can now state the main result of this section. \claim 5.27. Theorem|Let $Z$ be a projective $(n+1)$-dimensional manifold and $A$ a very ample line bundle on $Z$. Then, for a general section $\sigma\in H^0(Z,A^d)$ and $d\geq d_n$, the hypersurface $X_\sigma=\sigma^{-1}(0)$ is Kobayashi hyperbolic. The bound $d_n$ for the degree can be taken to be $$ d_n=\lfloor(n+4)\,(en)^{2n+1}/2\pi\rfloor \quad\hbox{for $n\ge 4$}, $$ and for $n\leq 3$, one can take $d_1=4$, $d_2=6685$, $d_3=2825761$. \endclaim For $n=1$, we have already seen in \S$\,$4.B that $d_1=4$ works (rather than the insane value~$d_1=61$). A simpler (and less refined) choice is $\tilde d_n=\lfloor \frac{1}{3}(en)^{2n+2}\rfloor$, which is valid for all~$n$. These bounds are only slightly weaker than the ones found by Ya Deng in his PhD thesis [Deng16, Deng17], namely $\tilde d_n=O(n^{2n+6})$. \plainsubsection 5.G. Further comments| \noindent {\bf 5.28.} Our bound $d_n$ is rather large, but just as in Ya Deng's effective approach of Brotbek's theorem [Deng17], the bound holds for a property that looks substantially stronger than hyperbo\-licity, namely the ampleness of the pull-back of some (twisted) jet bundle \hbox{$\mu_k^*\cO_{\widehat X_k}(\abu)\otimes\cO_{\widehat X_k}(-F'_k)$}. It is certainly desirable to look for more general jet differentials than Wronskians, and to relax the positivity demands on tautological line bundles to ensure hyperbolicity (see e.g.\ [Dem14]). However, the required calculations appear to be much more involved. \medskip \noindent {\bf 5.29.} After this chapter was written, Riedl and Yang [RiYa18] proved the important and somewhat surprising result that the lower bound estimates $d_\GG(n)$ and $d_\Kob(n)$, respectively for the Green-Griffiths-Lang and Kobayashi conjectures for general hypersurfaces in $\bP^{n+1}$, can be related by $d_\Kob(n):=d_\GG(2n-2)$. This should be understood in the sense that a solution of the generic $(2n-2)$-dimensional Green-Griffiths conjecture for $d\geq d_\GG(2n-2)$ implies a solution of the $n$-dimensional Kobayashi conjecture for the same lower bound. We refer to [RiYa18] for the precise statement, which requires an extra assumption on the algebraic dependence of the Green-Griffiths locus with respect to a variation of coefficients in the defining polynomials. In combination with [DMR10], this gives a completely new proof of the Kobayashi conjecture, and the order~$1$ bound $d_\GG(n)=O(\exp(n^{1+\varepsilon}))$ of [Dem12] implies a similar bound $d_\Kob(n)=O(\exp(n^{1+\varepsilon}))$ for the Kobayashi conjecture -- just a little bit weaker than what our direct proof gave (Theorem 5.26). In [MeTa19], Merker and Ta were able to improve the Green-Griffiths bound to $d_\GG(n)=o(\sqrt{n}\log n)^n$, using a strengthening of Darondeau's estimates [Dar16a, Dar16b], along with very delicate calculations. The Riedl-Yang result then implies $d_\Kob(n)=O((n\log n)^{n+1})$, which was the best bound known when [MeTa19] appeared. \medskip \noindent {\bf 5.30.} In the unpublished preprint [Dem15], we introduced an alternative strategy for the proof of the Kobayashi conjecture which appears to be still incomplete at this point. We nevertheless hope that a refined version could one day lead to linear bounds such as $d_\Kob(n)=2n+1$. The rough idea was to establish a $k$-jet analogue of Claire Voisin's proof [Voi96] of the Clemens conjecture. Unfortunately, Lemma 5.1.18 as stated in [Dem15] is incorrect -- the assertion concerning the $\Delta$ divisor introduced there simply does not hold. It is however conceivable that a weaker statement holds, in the form of a control of the degree of the divisor $\Delta$, and in a way that would still be sufficient to imply similar consequences for the generic positivity of tautological jet bundles.\medskip \noindent {\bf 5.31.} In [Ber19], G.~B\'erczi stated a positivity conjecture for Thom polynomials of Morin singularities (see also [BeSz12]), and showed that it would imply a polynomial bound $d_n=2\,n^9+1$ for the generic hyperbolicity of hypersurfaces. \medskip \noindent {\bf 5.32.} In September 2019, B\'erczi and Kirwan [BeKi19] introduced new deep ideas in non-reductive geometric invariant theory that actually lead to polynomials bounds for the Kobayashi conjecture. 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