% Hyperbolic algebraic varieties and holomorphic differential equations
%
% expanded version of VIASM Lecture (Hanoi, August 25-26, 2012)
% Jean-Pierre Demailly 
%
% Universit\'e de Grenoble I, Institut Fourier
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% Main text

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\title{Hyperbolic algebraic varieties}
\title{and holomorphic differential equations}

\titlerunning{J.-P.~Demailly, Hyperbolic algebraic varieties
and holomorphic differential equations}
\medskip

\centerline{\twelvebf Jean-Pierre Demailly}\medskip
\centerline{\twelverm Universit\'e de Grenoble I, Institut Fourier}
\bigskip
\centerline{\twelverm VIASM Annual Meeting 2012}
\smallskip
\centerline{\twelverm Hanoi -- August 25-26, 2012}
\vskip50pt

\long\def\donotprint{
\noindent
{\bf Key words:} Kobayashi hyperbolic variety, directed manifold,
genus of a curve, jet bundle, jet differential, jet metric, Chern connection
and curvature, negativity of jet curvature, variety of general type, 
Green-Griffiths conjecture, Lang conjecture

\vskip5pt
\noindent
{\bf A.M.S.~Classification (1991):} 32H20, 32L10, 53C55, 14J40
\vskip10pt}

\noindent{\bf Contents}\vskip5pt
{\eightpoint
\sectref{0} \sectref{1} \sectref{2} \sectref{3} \sectref{4} \sectref{5}
\sectref{6} \sectref{7} \sectref{8} \sectref{9} \sectref{10} \sectref{11}
}
\vskip15pt

\section{\S0. Introduction}

The goal of these notes is to explain recent results in the theory of
complex varieties, mainly projective algebraic ones, through a few geometric 
questions pertaining to hyperbolicity in the sense of Kobayashi. A complex space
$X$ is said to be hyperbolic if analytic disks $f:\bD\to X$ 
through a given point form a normal family. If $X$ is not
hyperbolic, a basic question is to analyze entire holomorphic
curves $f:\bC\to X$, and especially to understand the Zariski closure $Y\subset X$
of the union $\bigcup f(\bC)$ of all those curves. A tantalizing conjecture by 
Green-Griffiths and Lang says that $Y$ is a proper algebraic subvariety of $X$
whenever $X$ is a projective variety of general type. It is also 
expected that very generic algebraic hypersurfaces $X$ of high degree
in complex projective space $\bP^{n+1}$ are Kobayashi hyperbolic, i.e.\
without any entire holomorphic curves $f:\bC\to X$.
A convenient framework for this study is the category of
``directed manifolds'', that is, the category of pairs $(X,V)$ where $X$ is a
complex manifold and $V$ a holomorphic subbundle of~$T_X$, possibly with 
singularities -- this includes for instance the case of holomorphic 
foliations. If $X$ is compact,
the pair $(X,V)$ is hyperbolic if and only if there are no nonconstant entire
holomorphic curves $f:\bC\to X$ tangent to~$V$, as a consequence of
the Brody criterion. We describe here the construction of certain
jet bundles $J_kX$, $J_k(X,V)$, and corresponding
projectivized $k$-jet bundles $P_kV$. These bundles, which were introduced
in various contexts (Semple in 1954, Green-Griffiths in 1978)
allow to analyze hyperbolicity in terms of certain negativity 
properties of the curvature. For instance, $\pi_k:P_kV \to X$ is a 
tower of projective bundles over $X$ and carries a canonical line
bundle $\cO_{P_kV}(1)\,$; the hyperbolicity of $X$ is then conjecturally
equivalent to the existence of suitable singular hermitian metrics of negative
curvature on $\cO_{P_kV}(-1)$ for $k$ large enough. The direct images 
$(\pi_k)_*\cO_{P_kV}(m)$ can be viewed as bundles of algebraic differential
operators of order $k$ and degree~$m$, acting on germs of curves and invariant
under reparametrization. 

Following an approach initiated by Green and Griffiths, one can use
the Ahlfors-Schwarz lemma in the situation where the jet bundle carries
a (possibly singular) metric of negative curvature, to infer that
every nonconstant entire curve $f:\bC\to V$ tangent to $V$ must be
contained in the base locus of the metric.  A related result is the
fundamental vanishing theorem asserting that entire curves must be solutions
of the algebraic differential equations provided by global sections of
jet bundles, whenever their coefficients vanish on a given ample
divisor; this result was obtained in the mid 1990's as the conclusion of 
contributions by Bloch, Green-Griffiths, Siu-Yeung and the author. It can
in its turn be used to prove various important geometric statements. One of them
is the Bloch theorem, which was confirmed at the end of the 1970's by Ochiai
and Kawamata, asserting that the Zariski closure of an entire curve in a 
complex torus is a translate of a subtorus. 

Since then many developments occurred, for a large part via the technique 
of constructing jet differentials --  either by direct calculations or by various indirect
methods: Riemann-Roch calculations, vanishing theorems~... In 1997, McQuillan introduced
his ``diophantine approximation'' method, which was soon recognized to be an important tool
in the study of holomorphic foliations,  in parallel with Nevanlinna theory and the
construction of Ahlfors currents. Around 2000, Siu showed that generic hyperbolicity
results in the 
direction of the Kobayashi conjecture could be investigated by combining the algebraic 
techniques of Clemens, Ein and Voisin with the existence of certain ``vertical'' meromorphic 
vector  fields on the jet space of the universal hypersurface of high degree;
these vector fields are actually used to differentiate the global sections of the jet 
bundles involved, so as to produce new sections with a better control on the base locus.
Also, in 2007, Demailly pioneered the use of holomorphic Morse inequalities to construct
jet differentials; in 2010, Diverio, Merker and Rousseau were able in that way
to prove the Green-Griffiths conjecture for generic hypersurfaces of high degree in 
projective space -- their proof also makes an essential use of Siu's differentiation 
technique via meromorphic vector fields, as improved by P\u{a}un and
Merker in 2008. The last sections of the notes are devoted to explaining the holomorphic 
Morse inequality technique; as an application,
one obtains a partial answer to the Green-Griffiths conjecture in a very wide 
context~: in particular, for every projective variety of general type~$X$, there exists
a global algebraic differential operator $P$ on $X$ (in fact many such operators $P_j$)
such that every entire curve $f:\bC\to X$ must satisfy
the differential equations $P_j(f;f',\ldots,f^{(k)})=0$. We also recover from there
the result of Diverio-Merker-Rousseau on the generic Green-Griffiths conjecture
(with an even better bound asymptotically as the dimension tends to infinity), as well as
a recent recent of Diverio-Trapani (2010) on the hyperbolicity of generic 3-dimensional
hypersurfaces in $\bP^4$.

\section{\S1. Basic hyperbolicity concepts}

\subsection \S1.A. Kobayashi hyperbolicity|
We first recall a few basic facts concerning the concept of
hyperbolicity, according to S.~Kobayashi [Kob70, Kob76]. 
Let $X$ be a complex space. An {\em analytic disk} in $X$ a holomorphic map 
from the unit disk $\Delta=D(0,1)$ to $X$. Given two points $p,q\in X$, 
consider a {\em chain  of analytic disks} from $p$ to $q$, that is a chain of points 
$p=p_0,p_1,\dots,p_k=q$ of $X$, pairs of points $a_1,b_1,\dots,a_k,b_k$ 
of $\Delta$ and holomorphic maps $f_1,\dots,f_k:\Delta\to X$ such that
$$
f_i(a_i)=p_{i-1},\quad f_i(b_i)=p_i,\qquad i=1,\dots,k.
$$
Denoting this chain by $\alpha$, define its length $\ell(\alpha)$ by
$$
\ell(\alpha)=d_P(a_1,b_1)+\cdots+d_P(a_k,b_k)\leqno(1.1')
$$ 
and a pseudodistance $d^K_X$ on $X$ by 
$$
d^K_X(p,q)=\inf_{\alpha}\ell(\alpha).\leqno(1.1'')
$$
This is by definition the {\em Kobayashi pseudodistance} of $X$.
In the terminology of Kobayashi [Kob75],
a {\em Finsler metric} (resp.\ {\em pseudometric}\/) on a vector bundle $E$
is a homogeneous positive (resp.\ nonnegative) positive 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
$$
\bfk_X(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\Delta\to X,\,f(0)=x,\,
\lambda f'(0)=\xi\big\},\qquad x\in X,~\xi\in T_{X,x}.\leqno(1.2)
$$
Here, if $X$ is not smooth at $x$, we take $T_{X,x}=(\gm_{X,x}/\gm^2_{X,x})^*$ 
to be the Zariski tangent space, i.e.\ the tangent space of a minimal smooth
ambient vector space containing the germ $(X,x)$; all tangent vectors
may not be reached by analytic disks and in those cases we put 
$\bfk_X(\xi)=+\infty$. When $X$ is a smooth 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.
$$
d^K_X(p,q)=\inf_\gamma\int_\gamma\bfk_X(\gamma'(t))\,dt,
$$
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].

\claim 1.3.~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

When $X$ is hyperbolic, it is interesting to investigate when the Kobayashi
metric is complete: one then says that $X$ is a {\em complete hyperbolic} space.
However, we will be mostly concerned with compact spaces here, so completeness
is irrelevant in that case.

Another important property is the {\em monotonicity} of the Kobayashi
metric with respect to holomorphic mappings. In fact, if $\Phi:X\to Y$
is a holomorphic map, it is easy to see from the definition that
$$
d^K_Y(\Phi(p),\Phi(q))\le d^K_X(p,q),\qquad\hbox{for all $p,q\in X$.}\leqno(1.4)
$$
The proof merely consists of taking the composition $\Phi\circ f_i$ for all
clains of analytic disks connecting $p$ and $q$ in~$X$. Clearly the Kobayashi 
pseudodistance $d^K_\bC$ on $X=\bC$ is identically zero, as one can see by looking at
arbitrarily large analytic disks $\Delta\to\bC$, $t\mapsto\lambda t$. Therefore,
if there is any (non constant) {\em entire curve} $\Phi:\bC\to X$, namely a non constant
holomorphic map defined on the whole complex plane~$\bC$, then by monotonicity
$d^K_X$ is identically zero on the image $\Phi(\bC)$ of the curve, and therefore 
$X$ cannot be hyperbolic. When $X$ is hyperbolic, it follows that $X$ cannot contain
rational curves $C\simeq\bP^1$, or elliptic curves $\bC/\Lambda$, or more generally
any non trivial image $\Phi:W=\bC^p/\Lambda\to X$ of a $p$-dimensional complex torus
(quotient of $\bC^p$ by a lattice).

\subsection \S1.B. The case of complex curves (i.e.\ Riemann surfaces)|

The only case where hyperbolicity is easy to assess
is the case of curves $\smash{(\dim_\bC X=1)}$. In fact, as the disk is simply connected,
every holomorphic map $f:\Delta\to X$ lifts to the universal cover 
$\widehat f:\Delta\to\widehat X$, so that $f=\rho\circ\widehat f$ where
$\rho:\widehat X\to X$ is the projection map. 

Now, by the Poincar\'e-Koebe uniformization theorem, every simply
connected Riemann surface is biholomorphic to $\bC$, the unit disk
$\Delta$ or the complex projective line $\bP^1$. The complex projective line
$\bP^1$ has no smooth \'etale quotient since every automorphism of $\bP^1$ 
has a fixed point; therefore the only case where $\widehat X\simeq\bP^1$ is when
$X\simeq\bP^1$ already. Assume now that $\widehat X\simeq\bC$. Then $\pi_1(X)$ operates
by translation on $\bC$ (all other automorphisms are affine nad have fixed points), and
the discrete subgroups of $(\bC,+)$ are isomorphic to $\bZ^r$, $r=0,1,2$. We then obtain
respectively $X\simeq\bC$, $X\simeq\bC/2\pi i\bZ\simeq\bC^*=\bC\ssm\smash{\{0\}}$ and
$X\simeq\bC/\Lambda$ where $\Lambda$ is a lattice, i.e.\ $X$ is an elliptic curve.
In all those cases, any entire function $\widehat f:\bC\to\bC$ gives rise to an
entire curve $f:\bC\to X$, and the same is true when $X\simeq\bP^1=\bC\cup\{\infty\}$.

Finally, assume that $\widehat X\simeq\Delta$; by what we have just seen, this must occur as
soon as $X\not\simeq\bP^1,\bC,\bC^*,\bC/\Lambda$. Let us take on $X$ the infinitesimal 
metric $\omega_P$
which is the quotient of the Poincar\'e metric on $\Delta$. The Schwarz-Pick lemma shows that
$d^K_\Delta=d_P$ coincides with the Poincar\'e metric on $\Delta$, and it follows easily by
the lifting argument that we have $\bfk_X=\omega_P$. In particular, $d^K_X$ is non
degenerate and is just the quotient of the Poincar\'e metric on $\Delta$, i.e.\
$$d^K_X(p,q)=\inf_{p'\in\rho^{-1}(p),\,q'\in\rho^{-1}(q)}d_P(p',q').$$
We can summarize this discussion as follows.

\claim 1.5.~Theorem|Up to bihomorphism, any smooth Riemann surface $X$ belongs to one
$($and only one$)$ of the following three types.
\vskip-2pt
\item{\rm(a)} $($rational curve$)$ $X\simeq\bP^1$.
\vskip-2pt
\item{\rm(b)} $($parabolic type$)$ $\widehat X\simeq\bC$, $X\simeq \bC,\,\bC^*$ or $X\simeq
\bC/\Lambda$ $($elliptic curve$)$
\vskip-2pt
\item{\rm(c)} $($hyperbolic type$)$ $\widehat X\simeq\Delta$. All compact curves
$X$ of genus $g\ge 2$ enter in this category, as well as $X=\bP^1\ssm\{a,b,c\}\simeq
\bC\ssm\{0,1\}$, or $X=\bC/\Lambda\ssm\{a\}$ $($elliptic curve minus one
point$)$.\vskip0pt
\endclaim

In some rare cases, the one-dimensional case can be used to study the case of higher 
dimensions. For instance, it is easy to see by looking at projections
that the Kobayashi pseudodistance on a product 
$X\times Y$ of complex spaces is given by
$$
\leqalignno{
&d^K_{X\times Y}((x,y),(x',y'))=\max\big(d^K_X(x,x'),d^K_Y(y,y')\big),&(1.6)\cr
&\bfk_{X\times Y}(\xi,\xi')=\max\big(\bfk_X(\xi),\bfk_Y(\xi')\big),&(1.6')\cr}
$$
and from there it follows that a product of hyperbolic spaces is hyperbolic. As a
consequence $(\bC\ssm\{0,1\})^2$, which is also a complement of five lines in $\bP^2$,
is hyperbolic. 

\subsection \S1.C. Brody criterion for hyperbolicity|
Throughout this subsection, we assume that $X$ is a complex
manifold. In this 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 1.7.~Brody reparametrization lemma|Let $\omega$ be a hermitian
metric on~$X$ and let $f:\Delta\to X$ 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
$(1-\varepsilon)\Delta$ 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

\proof. Select $t_0\in\Delta$ 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 $f'((1-\varepsilon)t):T_\Delta\to T_X$
with respect to the Poincar\'e metric $|dt|^2/(1-|t|^2)^2$ on $T_\Delta$,
which is conformally invariant under $\Aut(\Delta)$. 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 1.8.~Corollary {\rm(Brody)}|Let $(X,\omega)$ be a compact complex hermitian manifold.
Given a sequence of holomorphic
mappings \hbox{$f_\nu:\Delta\to X$} such that
$\lim\|f_\nu'(0)\|_\omega=+\infty$, one can find a sequence of homographic
transformations $\psi_\nu:D(0,R_\nu)\to(1-1/\nu)\Delta$ 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 non constant 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 1.9.~Brody criterion|Let $X$ be a compact complex manifold. The following properties
are equivalent.\smallskip
\item{\rm(a)} $X$ is hyperbolic.
\vskip0pt
\item{\rm(b)} $X$ does not possess any entire curve $f:\bC\to X$.
\vskip0pt
\item{\rm(c)} $X$ does not possess any Brody curve $g:\bC\to X$.
\vskip0pt
\item{\rm(d)} The Kobayashi infinitesimal metric $\bfk_X$ is uniformly bouded below, namely
$$
\bfk_X(\xi)\ge c\|\xi\|_\omega,\qquad c>0,
$$
for any hermitian metric $\omega$ on $X$.\vskip0pt
\endclaim

\proof. (a)$\Rightarrow$(b) If $X$ possesses an entire curve $f:\bC\to X$, then by 
looking at arbitrary large disks $D(0,R)\subset\bC$, it is easy to see that the 
Kobayashi distance of any two points in $f(\bC)$ is zero, so $X$ is not hyperbolic. 

\noindent
(b)$\Rightarrow$(c) is trivial.

\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 $f_\nu:\Delta\to X$ 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~1.8, contradicting~(c).

\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.\square


Notice also that if $f:\bC\to X$ is an entire curve such that $\|f'\|_\omega$ is 
unbounded, one can apply the Corollary~1.8 to $f_\nu(t):=f(t+a_\nu)$ 
where the sequence $(a_\nu)$ is chosen such that $\|f'_\nu(0)\|_\omega=
\|f(a_\nu)\|_\omega\to+\infty$. Brody's result then produces repametrizations
$\psi_\nu:D(0,R_\nu)\to D(a_\nu,1-1/\nu)$ and a Brody curve
$g=\lim f\circ\psi_\nu:\bC\to X$ such that $\sup\|g'\|_\omega=1$ 
and $g(\bC)\subset\overline{f(\bC)}$. It may happen that the image $g(\bC)$ of such a
limiting curve is disjoint from $f(\bC)$. In fact Winkelmann [Win07] has
given a striking example, actually a projective $3$-fold $X$ obtained by blowing-up a
3-dimensional abelian variety~$Y$, such that every Brody curve $g:\bC\to X$ lies in the
exceptional divisor $E\subset X\,$; however, entire curves $f:\bC\to X$ can be dense, as
one can see by taking $f$ to be the lifting of a generic complex line embedded in the
abelian variety~$Y$. For further precise information on the localization of 
Brody curves, we  refer the reader to the remarkable results of [Duv08].

The absence of entire holomorphic curves in a given complex manifold
is often referred to as {\em Brody hyperbolicity}. Thus, in the
compact case, Brody hyperbolicity and Kobayashi hyperbolicity
coincide (but Brody hyeperbolicity is in general a strictly weaker
property when $X$ is non compact).

\subsection \S1.D. Geometric applications|

We give here two immediate consequences of the Brody criterion: the openness property of 
hyperbolicity and a hyperbolicity criterion for subvarieties of complex tori.
By definition, a {\em holomorphic family} of compact complex manifolds is a 
holomorphic proper submersion $\cX\to S$ between two complex manifolds. 

\claim 1.10. 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 Euclidean
topology.
\endclaim

\proof. 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 $S$.\square
\medskip

Consider now an $n$-dimensional complex torus~$W$, i.e.\ an additive quotient 
$W=\bC^n/\Lambda$,  where $\Lambda\subset\bC^n$ is a (cocompact) lattice.
By taking a composition of entire curves $\bC\to\bC^n$ with
the projection $\bC^n\to W$ we obtain an infinite dimensional space 
of entire curves in~$W$.

\claim 1.11.~Theorem|Let $X\subset W$ be a compact complex submanifold of a complex torus. Then $X$ is hyperbolic if and only if it does not contain any translate of a subtorus.
\endclaim

\proof. If $X$ contains some translate of a subtorus, then it contains lots of 
entire curves and so $X$ is not hyperbolic.

Conversely, suppose that $X$ is not hyperbolic. Then by the Brody
criterion there exists an entire curve $f:\bC\to X$ such that
$\|f'\|_\omega\le \|f'(0)\|_\omega=1$, where $\omega$ is the flat
metric on $W$ inherited from $\bC^n$. This means that any lifting
$\wt f=(\wt f,\dots,\wt f_\nu):\bC\to\bC^n$ is
such that
$$
\sum_{j=1}^n|f_j'|^2\le 1.
$$
Then, by Liouville's theorem, $\wt f'$ is constant and therefore $\wt f$ is affine. But then the closure of the image of $f$ is a translate $a+H$ of a connected 
(possibly real) subgroup $H$ of $W$. We conclude that $X$ contains the analytic 
Zariski closure of $a+H$, namely $a+H^\bC$ where $H^\bC\subset W$ is the smallest 
closed complex subgroup of $W$ containing $H$.\square
\medskip

\section{\S2. Directed manifolds}

\subsection \S2.A. Basic definitions concerning directed manifolds|
Let us consider a pair $(X,V)$ consisting of a $n$-dimensional complex manifold $X$
equipped with a {\em linear subspace $V\subset T_X$}: assuming
$X$ connected, this is by definition an irreducible closed analytic subspace of 
the total space of $T_X$ such that each fiber
$V_x=V\cap T_{X,x}$ is a vector subspace of~$T_{X,x}$; the rank $x\mapsto \dim_\bC V_x$
is Zariski lower semicontinuous, and it may a priori jump. We will refer to 
such a pair as being a (complex) {\em directed manifold}. A morphism
$\Phi:(X,V)\to (Y,W)$ in the category of (complex) directed manifolds is
a holomorphic map such that $\Phi_*(V)\subset W$.

The rank $r\in\{0,1,\ldots,n\}$ of $V$ is by definition the dimension of 
$V_x$ at a generic point. The dimension may be larger at non generic points;
this happens e.g.\ on $X=\bC^n$ for the rank~$1$ linear space $V$
generated by the Euler vector field: $V_z=\bC\sum_{1\le j\le n} 
z_j\smash{\partial\over\partial z_j}$ for $z\ne 0$, and $V_0=\bC^n$. 
Our philosophy is that directed manifolds are also useful to study the
``absolute case'', i.e.\ the case $V=T_X$, because there are certain fonctorial
constructions which are quite natural in the category of directed manifolds
(see e.g.\ \S$\,$5,~6,~7). We think of directed manifolds as a kind
of ``relative situation'', covering e.g.\ the case when $V$ is the
relative tangent space to a holomorphic map $X\to S$. In general, we can 
associate to $V$ a sheaf $\cV=\cO(V)\subset\cO(T_X)$ of holomorphic sections.
These sections need not generate the fibers of $V$ at singular points, as one 
sees already in the case of the Euler vector field when $n\ge 2$. However, $\cV$
is a {\em saturated} subsheaf of $\cO(T_X)$, i.e.\ $\cO(T_X)/\cV$ has no torsion:
in fact, if the components of a section have a common divisorial component, one can
always simplify this divisor and produce a new section without any such common divisorial
component. Instead of defining directed manifolds by picking a linear space $V$, one 
could equivalently define them by considering saturated coherent subsheaves
$\cV\subset\cO(T_X)$. One could also take the dual viewpoint, looking at arbitrary quotient
morphisms $\Omega^1_X\to\cW=\cV^*$ (and recovering $\cV=\cW^*=\Hom_\cO(\cW,\cO)$,
as $\cV=\cV^{**}$ is reflexive). We want to stress here that no assumption need be made 
on the Lie bracket tensor $[~,~]:\cV\times \cV\to \cO(T_X)/\cV$, i.e.\ we do not 
assume any kind of integrability for~$\cV$ or $\cW$.

The singular set $\Sing(V)$ is by definition the set of points where
$\cV$ is not locally free, it can also be defined as the indeterminacy set of 
the (meromorphic) classifying map $\alpha:X\dasharrow G_r(T_X)$, $z\mapsto
V_z$ to the Grasmannian of $r$ dimensional subspaces of~$T_X$. We thus have 
$V_{|X\ssm \Sing(V)}=\alpha^*S$ where $S\to G_r(T_X)$ is the tautological 
subbundle of $G_r(T_X)$. The singular set $\Sing(V)$ is an
analytic subset of $X$ of codim${}\ge 2$, hence $V$ is always a
holomorphic subbundle outside of codimension~$2$. Thanks to this remark,
one can most often treat linear spaces as vector bundles (possibly modulo passing
to the Zariski closure along $\Sing(V)$).

\subsection \S2.B. Hyperbolicity properties of directed manifolds|
\noindent
Most of what we have done in \S1 can be extended to the category of directed manifolds.

\claim 2.1.~Definition|Let $(X,V)$ be a complex directed manifold.
\smallskip
\item{\rm i)} The Kobayashi-Royden infinitesimal metric
of $(X,V)$ is the Finsler metric on $V$ defined for any $x\in X$ and
$\xi\in V_x$ by
$$
\bfk_{(X,V)}(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\Delta\to X,\,f(0)=x,\,
\lambda f'(0)=\xi,\,f'(\Delta)\subset V\big\}.
$$
Here $\Delta\subset\bC$ is the unit disk and the map $f$ is an arbitrary
holomorphic map which is tangent to~$V$, i.e., such that $f'(t)\in V_{f(t)}$
for all $t\in\Delta$. We say that $(X,V)$ is 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 $x$
describes a compact subset of~$X$.
\smallskip
\item{\rm ii)} More generally, the Kobayashi-Eisenman infinitesimal
pseudometric of $(X,V)$ is the pseudometric defined on all decomposable
$p$-vectors $\xi=\xi_1\wedge\cdots\wedge\xi_p\in\Lambda^pV_x$, $1\le p\le
r=\rank V$, by
$$
\bfe^p_{(X,V)}(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\bB_p\to X,\,f(0)=x,\,
\lambda f_*(\tau_0)=\xi,\,f_*(T_{\bB_p})\subset V\big\}
$$
where $\bB_p$ is the unit ball in $\bC^p$ and $\tau_0=\partial/\partial t_1
\wedge\cdots\wedge\partial/\partial t_p$ is the unit $p$-vector of $\bC^p$
at the origin. We say that $(X,V)$ is infinitesimally $p$-measure hyperbolic
if $\bfe^p_{(X,V)}$ is positive definite on every fiber~$\Lambda^pV_x$
and satisfies a locally uniform lower bound in terms of any smooth metric.
\vskip0pt
\endclaim

If $\Phi:(X,V)\to (Y,W)$ is a morphism of directed manifolds, it is
immediate to check that we have the monotonicity property
$$
\leqalignno{
&\bfk_{(Y,W)}(\Phi_*\xi)\le \bfk_{(X,V)}(\xi),
\qquad\forall\xi\in V,&(2.2)\cr
&\bfe^p_{(Y,W)}(\Phi_*\xi)\le \bfe^p_{(X,V)}(\xi),
\qquad\forall\xi=\xi_1\wedge\cdots\wedge\xi_p\in\Lambda^pV.&(2.2^p)\cr}
$$
The following proposition shows that virtually all reasonable definitions
of the hyperbolicity property are equivalent if $X$ is compact (in
particular, the additional assumption that there is locally uniform
lower bound for $\bfk_{(X,V)}$ is not needed). We merely say in that case
that $(X,V)$ is {\em hyperbolic}.

\claim 2.3.~Proposition|For an arbitrary directed manifold $(X,V)$,
the Kobayashi-Royden infinitesimal metric $\bfk_{(X,V)}$ is upper
semicontinuous on the total space of~$V$. If $X$ is compact, $(X,V)$ is
infinitesimally hyperbolic if and only if there are no non constant entire
curves $g:\bC\to X$ tangent to~$V$. In that case, $\bfk_{(X,V)}$ is a
continuous $($and positive definite$)$ Finsler metric on~$V$.
\endclaim

\proof. The proof is almost identical to the standard proof for $\bfk_X$,
for which we refer to Royden [Roy71, Roy74]. One of the main ingredients is that
one can find a Stein neighborhood of the graph of any analytic disk (thanks to
a result of [Siu76], cf.\ also [Dem90a] for more general results). This allows
to obtain ``free'' small deformations of any given analytic disk, as there are many 
holomorphic vector fields on a Stein manifold.\square

Another easy observation is that the concept of $p$-measure hyperbolicity gets 
weaker and weaker as $p$ increases (we leave it as an exercise to the reader, this
is mostly just linear algebra).

\claim 2.4. Proposition|If $(X,V)$ is $p$-measure hyperbolic, then it is
$(p+1)$-measure hyperbolic for all $p\in\{1,\ldots,r-1\}$.
\endclaim

Again, an argument extremely similar to the proof of 1.10 shows that relative hyperbo\-licity 
is an open property.

\claim 2.5. Proposition|Let $(\cX,\cV)\to S$ be a holomorphic family
of compact directed manifolds $($by this, we mean a proper holomorphic
map $\cX\to S$ together with an analytic linear subspace $\cV\subset T_{\cX/S}\subset
T_{\cX}$ of the relative tangent bundle, defining a deformation $(X_s,V_s)_{s\in S}$
of the fibers$)$. Then the set of $s\in S$ such that the fiber $(X_s,V_s)$
is hyperbolic is open in $S$ with respect to the Euclidean topology.
\endclaim

Let us mention here an impressive result proved by Marco Brunella
[Bru03, Bru05, Bru06] concerning the behavior of the Kobayashi metric
on foliated varieties.

\claim 2.6. Theorem {\rm (Brunella)}|Let $X$ be a compact K\"ahler manifold
equipped with a $($possibly singular$)$ rank $1$ holomorphic foliation 
which is not a foliation by rational curves. Then the canonical bundle 
$K_{\cF}=\cF^*$ of the foliation is pseudoeffective $($i.e.\ the curvature
of $K_{\cF}$ is${}\ge 0$ in the sense of currents$)$.
\endclaim

The proof is obtained by putting on $K_{\cF}$ precisely the metric induced
by the Kobayashi metric on the leaves whenever they are generically 
hyperbolic (i.e.\ covered by the unit disk). The case of parabolic leaves
(covered by $\bC$) has to be treated separately.

\section{\S3. Algebraic hyperbolicity}

In the case of projective algebraic varieties, hyperbolicity is
expected to be related to other properties of a more algebraic
nature. Theorem~3.1 below is a first step in this direction.

\claim 3.1.~Theorem|Let $(X,V)$ be a compact complex directed manifold
and let $\sum\omega_{jk}dz_j\otimes d\ol z_k$ be a hermitian metric on~$X$,
with associated positive $(1,1)$-form
$\omega={i\over 2}\sum\omega_{jk}dz_j\wedge d\ol z_k$. Consider the following
three properties, which may or not be satisfied by~$(X,V)\,:$
\smallskip
\item{\rm i)} $(X,V)$ is hyperbolic.
\smallskip
\item{\rm ii)} There exists $\varepsilon>0$ such that every compact
irreducible curve $C\subset X$ tangent to $V$ satisfies
$$
-\chi(\ol C)=2g(\ol C)-2\ge\varepsilon\,\deg_\omega(C)
$$
where $g(\ol C)$ is the genus of the normalization $\ol C$ of $C$,
$\chi(\ol C)$ its Euler charac\-teristic and $\deg_\omega(C)=\int_C\omega$.
$($This property is of course independent of~$\omega.)$
\smallskip
\item{\rm iii)} There does not exist any non constant holomorphic map
$\Phi:Z\to X$ from an abelian variety $Z$ to $X$ such that
$\Phi_*(T_Z)\subset V$.
\smallskip\noindent
Then {\rm i)$\,\Rightarrow\,$ii)$\,\Rightarrow\,$iii)}.
\endclaim

\proof. i)$\,\Rightarrow\,$ii). If $(X,V)$ is hyperbolic, there is a
constant $\varepsilon_0>0$ such that $\bfk_{(X,V)}(\xi)\ge\varepsilon_0
\|\xi\|_\omega$ for all $\xi\in V$. Now, let $C\subset X$ be a compact
irreducible curve tangent to $V$ and let $\nu:\ol C\to C$ be its
normalization. As $(X,V)$ is hyperbolic, $\ol C$ cannot be a rational
or elliptic curve, hence $\ol C$ admits the disk as its universal
covering $\rho:\Delta\to\ol C$.

The Kobayashi-Royden metric $\bfk_\Delta$ is the Finsler metric
$|dz|/(1-|z|^2)$ associated with the Poincar\'e metric
$|dz|^2/(1-|z|^2)^2$ on~$\Delta$, and $\bfk_{\ol C}$ is such that
$\rho^* \bfk_{\ol C}=\bfk_\Delta$. In other words, the metric
$\bfk_{\ol C}$ is induced by the unique hermitian metric on $\ol C$
of constant Gaussian curvature~$-4$. If $\sigma_\Delta=
{i\over 2}dz\wedge d\ol z/(1-|z|^2)^2$ and $\sigma_{\ol C}$ are
the corresponding area measures, the Gauss-Bonnet formula
(integral of the curvature${}=2\pi\,\chi(\ol C)$) yields
$$
\int_{\ol C}d\sigma_{\ol C}=-{1\over 4}\int_{\ol C}{\rm curv}
(\bfk_{\ol C})=-{\pi\over 2}\chi(\ol C)
$$
On the other hand, if $j:C\to X$ is the inclusion, the monotonicity
property (2.2) applied to the holomorphic map $j\circ\nu:\ol C\to X$
shows that
$$
\bfk_{\ol C}(t)\ge \bfk_{(X,V)}\big((j\circ\nu)_* t\big)
\ge \varepsilon_0\big\|(j\circ\nu)_* t\big\|_\omega,
\qquad\forall t\in T_{\ol C}.
$$
From this, we infer $d\sigma_{\ol C}\ge\varepsilon_0^2(j\circ\nu)^*
\omega$, thus
$$
-{\pi\over 2}\chi(\ol C)=\int_{\ol C}d\sigma_{\ol C}\ge
\varepsilon_0^2\int_{\ol C}(j\circ\nu)^*\omega=
\varepsilon_0^2\int_C\omega.
$$
Property ii) follows with $\varepsilon=2\varepsilon_0^2/\pi$.
\smallskip

\noindent ii)$\,\Rightarrow\,$iii). First observe that ii) excludes
the existence of elliptic and rational curves tangent to~$V$. Assume that
there is a non constant holomorphic map $\Phi:Z\to X$ from an abelian
variety $Z$ to $X$ such that $\Phi_*(T_Z)\subset V$. We must have
$\dim\Phi(Z)\ge 2$, otherwise $\Phi(Z)$ would be a
curve covered by images of holomorphic maps $\bC\to\Phi(Z)$, and so
$\Phi(Z)$ would be elliptic or rational, contradiction. Select a
sufficiently general curve $\Gamma$ in $Z$ (e.g., a curve obtained as an
intersection of very generic divisors in a given very ample linear system 
$|L|$ in~$Z$). Then all isogenies $u_m:Z\to Z$, $s\mapsto ms$ map $\Gamma$ 
in a $1:1$ way to curves $u_m(\Gamma)\subset Z$, except maybe for
finitely many double points of $u_m(\Gamma)$ (if $\dim Z=2$). It follows
that the normalization of $u_m(\Gamma)$ is isomorphic to $\Gamma$. If
$\Gamma$ is general enough, similar arguments show that the images
$$
C_m:=\Phi(u_m(\Gamma))\subset X
$$
are also generically $1:1$ images of $\Gamma$, thus $\ol C_m\simeq\Gamma$
and $g(\ol C_m)=g(\Gamma)$. We would like to show that $C_m$ has 
degree${}\ge\Const\,m^2$. This is indeed rather easy to check if
$\omega$ is K\"ahler, but the general case is slightly more involved.
We write
$$
\int_{C_m}\omega=\int_\Gamma (\Phi\circ u_m)^*\omega=
\int_Z[\Gamma]\wedge u_m^*(\Phi^*\omega),
$$
where $\Gamma$ denotes the current of integration over $\Gamma$. Let us
replace $\Gamma$ by an arbitrary translate $\Gamma+s$, $s\in Z$, and
accordingly, replace $C_m$ by $C_{m,s}=\Phi\circ u_m(\Gamma+s)$. For
$s\in Z$ in a Zariski open set, $C_{m,s}$ is again a generically $1:1$ image
of $\Gamma+s$. Let us take the average of the last integral identity with
respect to the unitary Haar measure $d\mu$ on~$Z$. We find
$$
\int_{s\in Z}\left(\int_{C_{m,s}}\omega\right)d\mu(s)=\int_Z\left(\int_{s
\in Z}[\Gamma+s]\,d\mu(s)\right)\wedge u_m^*(\Phi^*\omega).
$$
Now, $\gamma:=\int_{s\in Z}[\Gamma+s]\,d\mu(s)$ is a translation
invariant positive definite form of type $(p-1,p-1)$ on $Z$, where
$p=\dim Z$, and $\gamma$ represents the same cohomology class as
$[\Gamma]$, i.e.\ $\gamma\equiv c_1(L)^{p-1}$. Because of the
invariance by translation, $\gamma$ has constant coefficients and so
$(u_m)_*\gamma=m^2\gamma$. Therefore we get
$$
\int_{s\in Z}d\mu(s)\int_{C_{m,s}}\omega=m^2\int_Z \gamma\wedge
\Phi^*\omega.
$$
In the integral, we can exclude the algebraic set of values $z$ such that
$C_{m,s}$ is not a generically $1:1$ image of $\Gamma+s$, since this set
has measure zero. For each $m$, our integral identity implies that there
exists an element $s_m\in Z$ such that $g(\ol C_{m,s_m})=g(\Gamma)$ and
$$
\deg_\omega(C_{m,s_m})=\int_{C_{m,s_m}}\omega\ge m^2
\int_Z \gamma\wedge\Phi^*\omega.
$$
As $\int_Z \gamma\wedge\Phi^*\omega>0$, the curves $C_{m,s_m}$ have
bounded genus and their degree is growing quadratically with $m$,
contradiction to property ii).\square

\claim 3.2.~Definition|We say that a projective directed manifold
$(X,V)$ is ``algebraically hyperbolic'' if it satisfies property
$3.1$~{\rm ii)}, namely, if there exists $\varepsilon>0$ such that
every algebraic curve $C\subset X$ tangent to $V$ satisfies
$$
2g(\ol C)-2\ge\varepsilon\,\deg_\omega(C).
$$
\endclaim

A nice feature of algebraic hyperbolicity is that it satisfies an algebraic
analogue of the openness property.

\claim 3.3.~Proposition|Let $(\cX,\cV)\to S$ be an algebraic family
of projective algebraic directed manifolds $($given by a projective
morphism $\cX\to S)$. Then the set of $t\in S$ such that the fiber
$(X_t,V_t)$ is algebraically hyperbolic is open with respect to
the ``countable Zariski topology'' of $S$ $($by definition, this is
the topology for which closed sets are countable unions of algebraic
sets$)$.
\endclaim

\proof. 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 $C=\sum m_jC_j$ tangent to $V_t$ 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
$$
\bigcap_{k>0}~~\bigcup_{2g-2<d/k}~A_{d,g}.
$$
This concludes the proof (of course, one has to know that the countable
Zariski topology is actually a topology, namely that the class
of countable unions of algebraic sets is stable under arbitrary
intersections; this can be easily checked by an induction on
dimension).\square

\claim 3.4.~Remark|{\rm More explicit versions of the openness
property have been dealt with in the literature. H.~Clemens
([Cle86] and [CKL88]) has shown that on a very generic surface of degree
$d\ge 5$ in $\bP^3$, the curves of type $(d,k)$ are of genus $g>kd(d-5)/2$
(recall that a very generic surface $X\subset\bP^3$ of degree${}\ge 4$ has
Picard group generated by $\cO_X(1)$ thanks to the Noether-Lefschetz theorem,
thus any curve on the surface is a complete intersection with another
hypersurface of degree $k\,$; such a curve is said to be of type $(d,k)\,$;
genericity is taken here in the sense of the countable Zariski topology).
Improving on this result of Clemens, Geng Xu [Xu94] has shown that
every curve contained in a very generic surface of degree $d\ge 5$ satisfies
the sharp bound $g\ge d(d-3)/2-2$. This actually shows that a very generic
surface of degree $d\ge 6$ is algebraically hyperbolic. Although a very 
generic quintic surface has no rational or elliptic curves, it seems to be
unknown whether a (very) generic quintic surface is algebraically hyperbolic
in the sense of Definition~3.2.

In higher dimension, L.~Ein ([Ein88], [Ein91]) proved that every subvariety
of a very generic hypersurface $X\subset\bP^{n+1}$ of degree $d\ge 2n+1$ $(n\ge 2)$,
is of general type. This was reproved by a simple efficient technique by
C.~Voisin in [Voi96].}
\endclaim

\claim 3.5.~Remark|{\rm It would be interesting to know whether
algebraic hyperbolicity is open with respect to the Euclidean topology$\,$;
still more interesting would be to know whether Kobayashi hyperbolicity
is open for the countable Zariski topology (of course, both properties
would follow immediately if one knew that algebraic hyperbolicity and Kobayashi
hyperbolicity  coincide, but they seem otherwise highly non trivial to
establish). The latter openness property has raised an important amount
of work around the following more particular question: is a (very) generic
hypersurface $X\subset\bP^{n+1}$ of degree $d$ large enough
(say $d\ge 2n+1$) Kobayashi hyperbolic$\,$? Again, ``very generic'' is to
be taken here in the sense of the countable Zariski topology. Brody-Green
[BrGr77] and Nadel [Nad89] produced examples of hyperbolic
surfaces in $\bP^3$ for all degrees $d\ge 50$, and Masuda-Noguchi
[MaNo93] gave examples of such hypersurfaces in $\bP^n$ for
arbitrary $n\ge 2$, of degree $d\ge d_0(n)$ large enough. The question
of studying the hyperbolicity of complements $\bP^n\ssm D$ of generic
divisors is in principle closely related to this; in fact if
$D=\{P(z_0\ld z_n)=0\}$ is a smooth generic divisor of degree $d$,
one may look at the hypersurface
$$
X=\big\{z_{n+1}^d=P(z_0\ld z_n)\big\}\subset\bP^{n+1}
$$
which is a cyclic $d\,{:}\,1$ covering of $\bP^n$. Since any holomorphic
map $f:\bC\to\bP^n\ssm D$ can be lifted to $X$, it is clear that the
hyperbolicity of $X$ would imply the hyperbolicity of $\bP^n\ssm D$.
The hyperbolicity of complements of divisors in $\bP^n$ has been
investigated by many authors.\square}
\endclaim 

In the ``absolute case''~$V=T_X$, it seems reasonable to expect that
properties 3.1~i), ii) are equivalent, i.e.\ that
Kobayashi and algebraic hyperbolicity coincide. However, it was observed
by Serge Cantat [Can00] that property 3.1~(iii) is not sufficient to imply
the hyperbolicity of $X$, at least when $X$ is a general complex
surface: a general (non algebraic) K3 surface is known to have no elliptic
 curves and does not admit either any surjective map from an abelian
variety; however such a surface is not Kobayashi hyperbolic. We are uncertain
about the sufficiency of 3.1~(iii) when $X$ is assumed to be projective.

\section{\S4. The Ahlfors-Schwarz lemma for metrics of negative curvature}

One of the most basic ideas is that hyperbolicity should somehow be
related with suitable negativity properties of the curvature. For
instance, it is a standard fact already observed in Kobayashi [Kob70]
that the negativity of $T_X$ (or the ampleness of $T^*_X$) implies
the hyperbolicity of~$X$. There are many ways of improving or
generalizing this result. We present here a few simple examples of such
generalizations. 

\subsection \S4.A. Exploiting curvature via potential theory|
If $(V,h)$ is a holomorphic vector bundle equipped with
a smooth hermitian metric, we denote by $\nabla_h=\nabla'_h+\nabla''_h$
the associated Chern connection and by $\Theta_{V,h}={i\over 2\pi}
\nabla_h^2$ its Chern curvature tensor.

\claim 4.1.~Proposition|Let $(X,V)$ be a compact directed
manifold. Assume that $V$ is non singular and that
$V^*$ is ample. Then $(X,V)$ is hyperbolic.
\endclaim

\proof\ (from an original idea of [Kob75]). Recall that a vector bundle
$E$ is said to be ample if $S^mE$ has enough global sections
$\sigma_1\ld\sigma_N$ so as to generate $1$-jets of sections at any
point, when $m$ is large. One obtains a Finsler metric $N$ on $E^*$
by putting
$$
N(\xi)=\Big(\sum_{1\le j\le N}|\sigma_j(x)\cdot\xi^m|^2\Big)^{1/2m},\qquad
\xi\in E^*_x,
$$
and $N$ is then a strictly plurisubharmonic function on the total space
of $E^*$ minus the zero section (in other words, the line
bundle $\cO_{P(E^*)}(1)$ has a metric of positive curvature). By
the ampleness assumption on $V^*$, we thus have a Finsler
metric $N$ on $V$ which is strictly plurisubharmonic outside the zero
section. By the Brody lemma, if $(X,V)$ is not hyperbolic, there is a
non constant entire curve $g:\bC\to X$ tangent to $V$ such that
$\sup_\bC\|g'\|_\omega\le 1$ for some given hermitian metric $\omega$
on~$X$. Then $N(g')$ is a bounded subharmonic function on $\bC$ which
is strictly subharmonic on $\{g'\ne 0\}$. This is a contradiction, for
any bounded subharmonic function on $\bC$ must be constant.\square
\medskip

\subsection \S4.B. Ahlfors-Schwarz lemma|

Proposition 4.1 can be generalized a little bit further by means of
the Ahlfors-Schwarz lemma (see e.g.\ [Lang87]; we refer to [Dem85]
for the generalized version presented here; the proof is merely an 
application of the maximum principle plus a regularization argument).

\claim 4.2.~Ahlfors-Schwarz lemma|Let $\gamma(t)=\gamma_0(t)\,i\,dt
\wedge d\ol t$ be a hermitian metric on $\Delta_R$ where $\log\gamma_0$ is
a subharmonic function such that $i\,\ddbar\log\gamma_0(t)\ge A\,\gamma(t)$
in the sense of currents, for some positive constant $A$. Then $\gamma$ can
be compared with the Poincar\'e metric of $\Delta_R$ as follows:
$$
\gamma(t)\le{2\over A}{R^{-2}|dt|^2\over(1-|t|^2/R^2)^2}.
$$
More generally, let $\gamma=i\sum\gamma_{jk}dt_j\wedge d\ol t_k$ be an
almost everywhere positive hermitian form on the ball $B(0,R)\subset\bC^p$,
such that $-\Ricci(\gamma):=i\,\ddbar\log\det\gamma\ge A\gamma$ in the
sense of currents, for some constant $A>0$ $($this means in particular
that $\det\gamma=\det(\gamma_{jk})$ is such that $\log\det\gamma$ is
plurisubharmonic$)$. Then the $\gamma$-volume form is controlled by
the Poincar\'e volume form~:
$$
\det(\gamma)\le\Big({p+1\over AR^2}\Big)^p{1\over(1-|t|^2/R^2)^{p+1}}.
$$
\endclaim


\subsection 4.C. Applications of the Ahlfors-Schwarz lemma to hyperbolicity|

Let $(X,V)$ be a {\em compact} directed manifold. We assume throughout this subsection
that $V$ is {\em non singular}.

\claim 4.3.~Proposition|Assume $V^*$ is ``very big'' in the following sense: there
exists an ample line bundle $L$ and a sufficiently large integer $m$
such that the global sections in $H^0(X,S^mV^*\otimes L^{-1})$
generate all fibers over $X\ssm Y$, for some analytic subset $Y\subsetneq X$.
Then all entire curves $f:\bC\to X$ tangent to $V$ satisfy
$f(\bC)\subset Y$ $[$under our assumptions, $X$ is a projective algebraic
manifold and $Y$ is an algebraic subvariety, thus it is legitimate to say
that the entire curves are ``algebraically degenerate''$]$.
\endclaim

\proof. Let $\sigma_1\ld\sigma_N\in H^0(X,S^mV^*\otimes L^{-1})$ be
a basis of sections generating $S^mV^*\otimes L^{-1}$ over $X\ssm Y$.
If $f:\bC\to X$ is tangent to~$V$, we define a semipositive hermitian form
$\gamma(t)=\gamma_0(t)\,|dt|^2$ on $\bC$ by putting
$$
\gamma_0(t)=\sum\|\sigma_j(f(t))\cdot f'(t)^m\|_{L^{-1}}^{2/m}
$$
where $\|~~\|_L$ denotes a hermitian metric with positive
curvature on~$L$. If $f(\bC)\not\subset Y$, the form $\gamma$ is not
identically $0$ and we then find 
$$
i\,\ddbar\log\gamma_0\ge {2\pi\over m}f^*\Theta_L
$$
where $\Theta_L$ is the curvature form. The positivity assumption combined
with an obvious homogeneity argument yield
$$
{2\pi\over m}f^*\Theta_L\ge\varepsilon\|f'(t)\|_\omega^2\,|dt|^2\ge
\varepsilon'\,\gamma(t)
$$
for any given hermitian metric $\omega$ on~$X$. Now, for any $t_0$ with
$\gamma_0(t_0)>0$, the Ahlfors-Schwarz lemma shows that $f$ can only exist 
on a disk $D(t_0,R)$ such that $\gamma_0(t_0)\le {2\over\varepsilon'}R^{-2}$,
contradiction.\square

There are similar results for $p$-measure hyperbolicity, e.g.

\claim 4.4.~Proposition|Assume that $\Lambda^pV^*$ is ample. 
Then $(X,V)$ is infinitesimally
$p$-measure hyperbolic. More generally, assume that $\Lambda^pV^*$
is very big with base locus contained in $Y\subsetneq X$ $($see $3.3)$.
Then $\bfe^p$ is non degenerate over $X\ssm Y$.
\endclaim

\proof. By the ampleness assumption, there is a smooth Finsler metric $N$ on
$\Lambda^pV$ which is strictly plurisubharmonic outside the zero section.
We select also a hermitian metric $\omega$ on~$X$. For any holomorphic
map $f:\bB_p\to X$ we define a semipositive hermitian metric
$\swt\gamma|$ on $\bB_p$ by putting $\swt\gamma|=f^*\omega$. Since
$\omega$ need not have any good curvature estimate, we introduce
the function $\delta(t)=N_{f(t)}(\Lambda^pf'(t)\cdot\tau_0)$, where
$\tau_0=\partial/\partial t_1\wedge\cdots\wedge\partial/\partial t_p$,
and select a metric $\gamma=\lambda\swt\gamma|$ conformal to
$\swt\gamma|$ such that $\det\gamma=\delta$. Then $\lambda^p$ is equal
to the ratio $N/\Lambda^p\omega$ on the element $\Lambda^pf'(t)\cdot\tau_0
\in\Lambda^pV_{f(t)}$. Since $X$ is compact, it is clear that the
conformal factor $\lambda$ is bounded by an absolute constant
independent of~$f$. From the curvature assumption we then get
$$
i\,\ddbar\log\det\gamma=i\,\ddbar\log\delta\ge (f,\Lambda^pf')^*
(i\,\ddbar\log N)\ge\varepsilon f^*\omega\ge\varepsilon'\,\gamma.
$$
By the Ahlfors-Schwarz lemma we infer that $\det\gamma(0)\le C$ for some
constant $C$, i.e., $N_{f(0)}(\Lambda^pf'(0)\cdot\tau_0)\le C'$. This means
that the Kobayashi-Eisenman pseudometric $\bfe^p_{(X,V)}$ is positive
definite everywhere and uniformly bounded from below. In the
case $\Lambda^pV^*$ is very big with base locus $Y$, we use essentially
the same arguments, but we then only have $N$ being positive definite
on $X\ssm Y$.\square

\claim 4.5.~Corollary {\rm([Gri71], KobO71])}|If $X$ is a projective
variety of general type, the Kobayashi-Eisenmann volume form $\bfe^n$,
$n=\dim X$, can degenerate only along a proper algebraic set
$Y\subsetneq X$.
\endclaim

\subsection \S4.C. Main conjectures concerning hyperbolicity|
One of the earliest conjectures in hyperbolicity theory is the following statement
due to Kobayashi ([Kob70], [Kob76]). 

\claim 4.6.~Conjecture {\rm(Kobayashi)}|\vskip0pt
\item{\rm(a)}A $($very$)$ generic hypersurface
$X\subset\bP^{n+1}$ of degree $d\ge d_n$ large enough is hyperbolic.
\vskip0pt
\item{\rm(b)}The complement $\bP^n\ssm H$ of a $($very$)$ generic hypersurface
$H\subset\bP^n$ of degree $d\ge d'_n$ large enough is hyperbolic.
\vskip0pt
\endclaim

In its original form, Kobayashi conjecture did not give the lower bounds
$d_n$ and $d'_n$. Zaidenberg proposed the bounds $d_n=2n+1$ (for $n\ge 2$) and
$d'_n=2n+1$ (for $n\ge 1$), based on the results of Clemens, Xu, Ein and Voisin
already mentioned, and the following observation (cf.\ [Zai87], [Zai93]).

\claim 4.7.~Theorem {\rm(Zaidenberg)}|The complement of a general 
hypersurface of degree $2n$ in $\bP^n$ is not hyperbolic.
\endclaim 

The converse of Corollary 4.5 is also expected to be true, namely, the generic
non degeneracy of $\bfe^n$ should imply that $X$ is of general type, but
this is only known for surfaces (see [GrGr80] and [MoMu82]):

\claim 4.8.~Conjecture {\rm(Green-Griffiths [GrGr80])}|A projective
algebraic variety $X$ is measure hyperbolic $($i.e.\ $\bfe^n$
degenerates only along a proper algebraic subvariety$)$ if and only if
$X$ is of general type.
\endclaim

An essential step in the proof of the necessity of having general type
subvarieties would be to show that manifolds of Kodaira
dimension $0$ (say, Calabi-Yau manifolds and holomorphic symplectic
manifolds, all of which have $c_1(X)=0$) are not measure hyperbolic, 
e.g.\ by exhibiting enough families of curves $C_{s,\ell}$ covering $X$
such that $(2g(\ol C_{s,\ell})-2)/\deg(C_{s,\ell})\to 0$. 
Another (even stronger) conjecture which we will investigate at the end
of these notes is

\claim 4.9.~Conjecture {\rm(Green-Griffiths [GrGr80])}|If $X$ is a
variety of general type, there exists a proper algebraic set $Y\subsetneq
X$ such that every entire holomorphic curve $f:\bC\to X$ is contained
in~$Y$.
\endclaim

One of the early important result in the direction of Conjecture 4.9 is the
proof of the Bloch theorem, as proposed by Bloch [Blo26a] and Ochiai
[Och77]. The Bloch theorem is the special case of 4.9 when the
irregularity of $X$ satisfies $q=h^0(X,\Omega^1_X)>\dim X$. Various
solutions have then been obtained in fundamental papers of Noguchi
[Nog77,~81,~84], Kawamata [Kaw80] and Green-Griffiths [GrGr80], by
means of different techniques. See section \S$\,$10 for a proof based
on jet bundle techniques. A much more recent result is the striking
statement due to Diverio, Merker and Rousseau [DMR10], confirming 4.9
when  $X\subset\bP^{n+1}$ is a generic non singular hypersurface of 
sufficiently large degree~$d\ge\smash{2^{n^5}}$ (cf.~\S16).
Conjecture 4.9 was also considered by S.~Lang [Lang86, Lang87] in view
of arithmetic counterparts of the above geometric statements.

\claim 4.10.~Conjecture {\rm(Lang)}|A projective algebraic
variety $X$ is hyperbolic if and only if all its algebraic subvarieties
$($including $X$ itself\/$)$ are of general type.
\endclaim

\claim 4.11.~Conjecture {\rm(Lang)}|Let $X$ be a projective variety
defined over a number field~$K$.\vskip0pt
\item{\rm(a)} If $X$ is hyperbolic, then the set of $K$-rational points
is finite.
\vskip0pt
\item{\rm(a${}'$)} Conversely, if the set of $K'$-rational points is finite for every finite
extension $K'\supset K$, then $X$ is hyperbolic.
\vskip0pt
\item{\rm(b)} If $X$ is of general type, then the set of $K$-rational points is 
not Zariski dense.
\vskip0pt
\item{\rm(b${}'$)} Conversely, if the set of $K'$-rational points is not Zariski dense
for any extension $K'\supset K$, then $X$ is of general type.
\vskip0pt
\endclaim

In fact, in 4.11~(b), if $Y\subsetneq X$ is the ``Green-Griffiths locus'' 
of $X$, it is expected
that $X\ssm Y$ contains only finitely many rational $K$-points. Even when dealing
only with the geometric statements, there are several interesting
connections between these conjectures.

\claim 4.12.~Proposition|Conjecture $4.9$ implies the ``if'' part of 
conjecture $4.10$, and Conjecture~$4.8$ implies the ``only if'' part
of Conjecture $4.10$, hence $(4.8$ and $4.9)$ $\Rightarrow$ $(4.10)$.
\endclaim

\proof. In fact if Conjecture 4.9 holds and every subariety $Y$ of $X$ is
of general type, then it is easy to infer that every entire curve
$f:\bC\to X$ has to be constant by induction on $\dim X$, because
in fact $f$ maps $\bC$ to a certain subvariety~$Y\subsetneq X$. Therefore
$X$ is hyperbolic.

Conversely, if Conjecture 4.8 holds and $X$ has a certain subvariety
$Y$ which is not of general type, then $Y$ is not measure
hyperbolic. However Proposition 2.4 shows that hyperbolicity implies
measure hyperbolicity. Therefore $Y$ is not hyperbolic
and so $X$ itself is not hyperbolic either.\square

\claim 4.13.~Proposition|Assume that the Green-Griffiths conjecture $4.9$ holds.
Then the Kobayashi conjecture $4.6$~{\rm(a)} holds with $d_n=2n+1$.
\endclaim

\proof. We know by Ein [Ein88, Ein91] and Voisin [Voi96] that a very generic
hypersurface $X\subset\bP^{n+1}$ of degree $d\ge 2n+1$, $n\ge 2$, has all
its subvarieties that are of general type. We have seen that the Green-Griffiths 
conjecture $4.9$ implies the hyperbolicity of $X$ in this circumstance.\square

\section{\S5. Projectivization of a directed manifold}

\subsection \S5.A. The $1$-jet fonctor|
The basic idea is to introduce a fonctorial 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~$X$. We let
$$
\swt X|=P(V),\qquad \swt V|\subset T_{\swt X|}
$$
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 $v\in V_x\ssm\{0\}$,
$$
\swt V|_{(x,[v])}=\big\{\xi\in T_{\swt X|,\,(x,[v])}\,;\,\pi_*\xi\in
\bC v\big\},\qquad\bC v\subset V_x\subset T_{X,x},\leqno(5.1)
$$
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 a 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
$$
\leqalignno{
&0\lra T_{\swt X|/X}\lra\swt V|\build\lra|{\pi_*}||\cO_{\swt X|}(-1)
\lra 0,&(5.2)\cr
&0\lra\cO_{\swt X|}\lra \pi^* V\otimes\cO_{\swt X|}(1)
\lra T_{\swt X|/X}\lra 0,&(5.2')\cr}
$$
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
$$
\dim\swt X|=n+r-1,\qquad \rank\swt V|=\rank V=r,\leqno(5.3)
$$
and by taking determinants we find $\det(T_{\swt X|/X})=
\pi^*\det V\otimes\cO_{\swt X|}(r)$, thus
$$
\det\swt V|=\pi^*\det V\otimes\cO_{\swt X|}(r-1).\leqno(5.4)
$$
By definition, $\pi:(\swt X|,\swt V|)\to(X,V)$ is a morphism of
complex directed manifolds. Clearly, our construction is fonctorial, i.e.,
for every morphism of directed manifolds $\Phi:(X,V)\to(Y,W)$, there
is a commutative diagram
$$
\matrix{(\swt X|,\swt V|)&\build\lra|{\textstyle\pi}||&(X,V)\cr
\wt\Phi\vdasharrow&&\Big\downarrow\Phi\cr
(\swt Y|,\swt W|)&\build\lra|{\textstyle\pi}||&(Y,W)\cr}\leqno(5.5)
$$
where the left vertical arrow is the meromorphic map $P(V)\dasharrow P(W)$
induced by the differential $\Phi_*:V\to\Phi^* W$ ($\swt\Phi|$ is
actually holomorphic if $\Phi_*:V\to\Phi^* W$ is injective).

\subsection \S5.B. Lifting of curves to the $1$-jet bundle|

Suppose that we are given a holomorphic curve $f:\Delta_R\to X$
parametrized by the disk $\Delta_R$ 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 $t\in \Delta_R$.
If $f$ is non constant, there is a well defined and unique tangent line
$[f'(t)]$ for every~$t$, even at stationary points, and the map
$$
\swt f|:\Delta_R\to\swt X|,\qquad
t\mapsto\swt f|(t):=(f(t),[f'(t)])\leqno(5.6)
$$
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
$$
f':T_{\Delta_R}\to\swt f|^*\cO_{\swt X|}(-1).\leqno(5.7)
$$
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 $g:\Delta_R\to\swt X|$ 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
$$
V_z=\Big\{\xi=\sum_{1\le j\le n}\xi_j{\partial\over\partial z_j}\,;\,
\xi_j= \sum_{1\le k\le r}a_{jk}(z)\xi_k~\hbox{\rm for $j=r+1\ld n$}\Big\},
\leqno(5.8)
$$
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 $P(V)_{\restr
\Omega}$ can be described by the coordinate system
$$
\Big(z_1\ld z_n;{\xi_1\over\xi_j}\ld{\xi_{j-1}\over\xi_j},
{\xi_{j+1}\over\xi_j}\ld{\xi_r\over\xi_j}\Big).\leqno(5.9)
$$
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 $f(\Delta_R)\subset\Omega$).
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
$$
f_j'(t)=\sum_{1\le k\le r}a_{jk}(f(t))f_k'(t),\qquad j>r,\leqno(5.10)
$$
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
$t_0\in\Delta_R$, 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 (5.10), we can always
suppose $j\in\{1\ld r\}$, for example $\smash{f_r^{(m)}}(t_0)\ne 0$. 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
$P(V)_{\restr\Omega}$ by
$$
\swt f|\simeq\Big(f_1\ld f_n;{f_1'\over f_r'}\ld{f_{r-1}'\over f_r'}\Big).
\leqno(5.11)
$$

\subsection \S5.C. Curvature properties of the $1$-jet bundle|

We end this section with a few curvature computations.
Assume that $V$ is equipped with a smooth hermitian metric~$h$. Denote by
$\nabla_h=\nabla'_h+\nabla''_h$ the associated Chern connection and
by $\Theta_{V,h}={i\over 2\pi}\nabla_h^2$ its Chern curvature tensor.
For every point $x_0\in X$, there exists a ``normalized'' holomorphic frame
$(e_\lambda)_{1\le\lambda\le r}$ on a neighborhood of~$x_0$, such that
$$
\langle e_\lambda,e_\mu\rangle_h=\delta_{\lambda\mu}-
\sum_{1\le j,k\le n}c_{jk\lambda\mu}z_j\ol z_k+O(|z|^3),\leqno(5.12)
$$
with respect to any holomorphic coordinate system $(z_1\ld z_n)$ centered
at~$x_0$. A~computation of $d'\langle e_\lambda,e_\mu\rangle_h=
\langle\nabla'_h e_\lambda,e_\mu\rangle_h$ and
$\nabla^2_h e_\lambda=d''\nabla'_h e_\lambda$ then gives
$$
\leqalignno{\nabla'_h e_\lambda
&=-\sum_{j,k,\mu}c_{jk\lambda\mu}\ol z_k\,dz_j\otimes e_\mu+O(|z|^2),\cr
\Theta_{V,h}(x_0)
&={i\over 2\pi}\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}dz_j\wedge
d\ol z_k\otimes e_\lambda^*\otimes e_\mu.&(5.13)\cr}
$$
The above curvature tensor can also be viewed as a hermitian form
on $T_X\otimes V$. In fact, one associates with $\Theta_{V,h}$ the hermitian
form $\langle\Theta_{V,h}\rangle$ on $T_X\otimes V$ defined for all
$(\zeta,v)\in T_X\times_X V$ by
$$
\langle\Theta_{V,h}\rangle(\zeta\otimes v)=
\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\zeta_j\ol\zeta_kv_\lambda\ol v_\mu.\leqno(5.14)
$$
Let $h_1$ be the hermitian metric on the tautological line bundle
$\cO_{P(V)}(-1)\subset\pi^* V$ induced by the metric $h$ of~$V$. We
compute the curvature $(1,1)$-form $\Theta_{h_1}(\cO_{P(V)}(-1))$
at an arbitrary point $(x_0,[v_0])\in P(V)$, in terms of $\Theta_{V,h}$. For
simplicity, we suppose that the frame $(e_\lambda)_{1\le\lambda\le r}$ has
been chosen in such a way that $[e_r(x_0)]=[v_0]\in P(V)$ and $|v_0|_h=1$.
We get holomorphic local coordinates $(z_1\ld z_n\,;\,\xi_1\ld\xi_{r-1})$
on a neighborhood of $(x_0,[v_0])$ in $P(V)$ by assigning
$$
(z_1\ld z_n\,;\,\xi_1\ld\xi_{r-1})\longmapsto(z,[\xi_1e_1(z)+\cdots
+\xi_{r-1}e_{r-1}(z)+e_r(z)])\in P(V).
$$
Then the function
$$
\eta(z,\xi)=\xi_1e_1(z)+\cdots+\xi_{r-1}e_{r-1}(z)+e_r(z)
$$
defines a holomorphic section of $\cO_{P(V)}(-1)$ in a neighborhood
of~$(x_0,[v_0])$. By using the expansion (5.12) for $h$, we find
$$
\leqalignno{
|\eta|_{h_1}^2=|\eta|_h^2=1+|\xi|^2&{}-\sum_{1\le j,k\le n}c_{jkrr}
z_j\ol z_k+O((|z|+|\xi|)^3),\cr
\Theta_{h_1}(\cO_{P(V)}(-1))_{(x_0,[v_0])}
&=-{i\over 2\pi}\ddbar\log|\eta|_{h_1}^2\cr
&={i\over 2\pi}\Big(\sum_{1\le j,k\le n}c_{jkrr}dz_j\wedge d\ol z_k
-\sum_{1\le\lambda\le r-1}d\xi_\lambda\wedge d\ol\xi_\lambda\Big).
&(5.15)}
$$

\section{\S6. Jets of curves and Semple jet bundles}

Let $X$ be a complex $n$-dimensional manifold. Following ideas of 
\hbox{Green-Griffiths} [GrGr80], we let $J_k\to X$ 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 $J_k\to X$ 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 $J_{k,x}$,
$x\in\Omega$, can be seen as $\bC^n$-valued maps $$f=(f_1\ld
f_n):(\bC,0)\to\Omega\subset\bC^n,$$ and they are completetely
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 $J_{k,x}$ 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 $J_k$ is a holomorphic fiber bundle with typical fiber
$(\bC^n)^k$ over $X$ (however, $J_k$ is not a vector bundle for $k\ge 2$,
because of the nonlinearity of coordinate changes; see formula (7.2)
in~\S$\,$7).

According to the philosophy developed throughout this paper, we describe
the concept of jet bundle in the general situation of complex directed
manifolds. If $X$ is equipped with a holomorphic subbundle $V\subset T_X$,
we associate to $V$ a $k$-jet bundle $J_kV$ as follows.

\claim 6.1.~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~$J_k$. In
fact, by using (5.8) and (5.10), 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~$J_k$. Alternatively, we can pick a
local holomorphic connection $\nabla$ on $V$, defined on some open
set $\Omega\subset X$, and compute inductively the successive derivatives
$$\nabla f=f',\qquad \nabla^jf=\nabla_{f'}(\nabla^{j-1}f)$$
with respect to $\nabla$ along the cure $t\mapsto f(t)$. Then
$$
(\xi_1,\xi_2,\ldots,\xi_k)=(\nabla f(0),\nabla^2f(0),\ldots,\nabla^k f(0))\in
V_x^{\oplus k}
$$
provides a ``trivialization'' $J^kV_{|\Omega}\simeq\smash{V^{\oplus k}_{|\Omega}}$. 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).\square
\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 [ASS92] is also
concerned with such generalizations of jet bundles, as well as  [LaTh96] 
by Laksov and Thorup.

We define inductively the {\em projectivized $k$-jet bundle $P_kV=X_k$}
(or {\em Semple $k$-jet bundle}) and the associated subbundle 
$V_k\subset T_{X_k}$ by
$$
(X_0,V_0)=(X,V),\qquad (X_k,V_k)=(\swt X|_{k-1},\swt V|_{k-1}).
\leqno(6.2)
$$
In other words, $(P_kV,V_k)=(X_k,V_k)$ is obtained from $(X,V)$ by iterating
$k$-times the lifting construction $(X,V)\mapsto(\swt X|,\swt V|)$ described
in~\S$\,$5. By (5.2--5.7), we find
$$
\dim P_kV=n+k(r-1),\qquad\rank V_k=r,\leqno(6.3)
$$
together with exact sequences
$$
\leqalignno{
&0\lra T_{P_kV/P_{k-1}V}\lra V_k\build{\vlra11|}|{(\pi_k)_*}||
\cO_{P_kV}(-1)\lra 0,&(6.4)\cr
&0\lra\cO_{P_kV}\lra\pi_k^* V_{k-1}\otimes\cO_{P_kV}(1)
\lra T_{P_kV/P_{k-1}V}\lra 0.
&(6.4')\cr}
$$
where $\pi_k$ is the natural projection $\pi_k:P_kV\to P_{k-1}V$ and
$(\pi_k)_*$ its differential. Formula (5.4) yields
$$
\det V_k=\pi_k^*\det V_{k-1}\otimes\cO_{P_kV}(r-1).\leqno(6.5)
$$
Every non constant tangent trajectory
\hbox{$f:\Delta_R\to X$} of $(X,V)$ lifts to a well defined and
unique tangent trajectory $f_{[k]}:\Delta_R\to P_kV$ of $(P_kV,V_k)$.
Moreover, the derivative $f_{[k-1]}'$ gives rise to a section
$$
f_{[k-1]}':T_{\Delta_R}\to f_{[k]}^*\cO_{P_kV}(-1).\leqno(6.6)
$$
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
$$
f_{[k]}=(F_1\ld F_N),\qquad f_{[k+1]}=\Big(F_1\ld F_N,
{F_{s_1}'\over F_{s_r}'}\ld{F_{s_{r-1}}'\over F_{s_r}'}\Big)\leqno(6.7)
$$
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
$P_kV\to P_{k-1}V$, 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 $\cO_{P_kV}(-1)
\hookrightarrow\pi_k^* V_{k-1}$, and a composition with the
projection $(\pi_{k-1})_*$ (analogue for order $k-1$ of the
arrow~$(\pi_k)_*$ in sequence (6.4)) yields for all $k\ge 2$ a
canonical line bundle morphism
$$
\cO_{P_kV}(-1)\lhra\pi_k^* V_{k-1}\build{\vlra16|}|
{(\pi_k)^*(\pi_{k-1})_*}||\pi_k^*\cO_{P_{k-1}V}(-1),\leqno(6.8)
$$
which admits precisely \hbox{$D_k=P(T_{P_{k-1}V/P_{k-2}V})
\subset P(V_{k-1})=P_kV$} as its zero divisor (clearly, $D_k$ is a
hyperplane subbundle of~$P_kV$). Hence we find
$$
\cO_{P_kV}(1)=\pi_k^*\cO_{P_{k-1}V}(1)\otimes\cO(D_k).\leqno(6.9)
$$
Now, we consider the composition of projections
$$
\pi_{j,k}=\pi_{j+1}\circ\cdots\circ\pi_{k-1}\circ\pi_k:P_kV\lra P_jV.
\leqno(6.10)
$$
Then $\pi_{0,k}:P_kV\to X=P_0V$ is a locally trivial holomorphic fiber bundle
over~$X$, and the fibers $P_kV_x=\smash{\pi_{0,k}^{-1}(x)}$ 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'' nonsingular projective algebraic 
variety of dimension $k(r-1)$ which we will denote by~$\bR_{r,k}\,$; it is
not hard to see that $\bR_{r,k}$ is rational (as will indeed follow from
the proof of Theorem~7.11 below). The following Proposition will help us
to understand a little bit more about the geometric structure of~$P_kV$.
As usual, we define the {\em multiplicity} $m(f,t_0)$ of a curve 
$f:\Delta_R\to X$ at a point $t\in\Delta_R$ to be the smallest integer
$s\in\bN^*$ such that $f^{(s)}(t_0)\ne 0$, i.e., the largest $s$ such
that $\delta(f(t),f(t_0))= O(|t-t_0|^s)$ for any hermitian or riemannian
geodesic distance $\delta$ on~$X$. As $f_{[k-1]}=\pi_k\circ f_{[k]}$,
it is clear that the sequence $m(f_{[k]},t)$ is non increasing with~$k$.

\claim 6.11.~Proposition|Let $f:(\bC,0)\to X$ be a non constant germ of
curve tangent to~$V$. Then for all $j\ge 2$ we have $m(f_{[j-2]},0)\ge
m(f_{[j-1]},0)$ and the inequality is strict if and only if $f_{[j]}(0)\in
D_j$. Conversely, if $w\in P_kV$ is an arbitrary element and 
$m_0\ge m_1\ge\cdots\ge m_{k-1}\ge 1$ is a sequence of integers
with the property that
$$
\forall j\in\{2\ld k\},\qquad m_{j-2}>m_{j-1}\quad\hbox{if and only if
$\pi_{j,k}(w)\in D_j$,}
$$
there exists a germ of curve $f:(\bC,0)\to X$ tangent to $V$ such that
$f_{[k]}(0)=w$ and $m(f_{[j]},0)=m_j$ for all~$j\in\{0\ld k-1\}$.
\endclaim

\proof. i) Suppose first that $f$ is given and put $m_j=m(f_{[j]},0)$. By
definition, we have $f_{[j]}=(f_{[j-1]},[u_{j-1}])$ where
$f_{[j-1]}'(t)=t^{m_{j-1}-1}u_{j-1}(t)\in V_{j-1}$, $u_{j-1}(0)\ne 0$.
By composing with the differential of the projection
$\pi_{j-1}:P_{j-1}V\to P_{j-2}V$, we find $f_{[j-2]}'(t)=t^{m_{j-1}-1}
(\pi_{j-1})_* u_{j-1}(t)$. Therefore
$$
m_{j-2}=m_{j-1}+\ord_{t=0}(\pi_{j-1})_* u_{j-1}(t),
$$
and so $m_{j-2}>m_{j-1}$ if and only if $(\pi_{j-1})_* u_{j-1}(0)=0$,
that is, if and only if $u_{j-1}(0)\in T_{P_{j-1}V/P_{j-2}V}$, or equivalently
$f_{[j]}(0)=(f_{[j-1]}(0),[u_{j-1}(0)])\in D_j$.
\medskip

\noindent ii) Suppose now that $w\in P_kV$ and $m_0\ld m_{k-1}$ are given.
We denote by $w_{j+1}=(w_j,[\eta_j])$, $w_j\in P_jV$, $\eta_j\in V_j$, the
projection of $w$ to~$P_{j+1}V$. Fix coordinates $(z_1\ld z_n)$ on~$X$
centered at $w_0$ such that the $r$-th component $\eta_{0,r}$ of $\eta_0$
is non zero. We prove the existence of the germ $f$ by induction on~$k$,
in the form of a Taylor expansion
$$
f(t)=a_0+t\,a_1+\cdots+t^{d_k}a_{d_k}+O(t^{d_k+1}),\qquad
d_k=m_0+m_1+\cdots+m_{k-1}.
$$
If $k=1$ and $w=(w_0,[\eta_0])\in P_1V_x$, we simply take
$f(t)=w_0+t^{m_0}\eta_0+O(t^{m_0+1})$. In general, the induction
hypothesis applied to $P_kV=P_{k-1}(V_1)$ over $X_1=P_1V$ yields
a curve $g:(\bC,0)\to X_1$ such that $g_{[k-1]}=w$ and $m(g_{[j]},0)=m_{j+1}$
for $0\le j\le k-2$. If $w_2\notin D_2$, then $[g_{[1]}'(0)]=[\eta_1]$ is
not vertical, thus $f=\pi_1\circ g$ satisfies $m(f,0)=m(g,0)=m_1=m_0$ and
we are done.

If $w_2\in D_2$, we express $g=(G_1\ld G_n;G_{n+1}\ld G_{n+r-1})$ as
a Taylor expansion of order $m_1+\cdots+m_{k-1}$ in the coordinates (5.9)
of the affine chart $\xi_r\ne 0$. As $\eta_1=\lim_{t\to 0}g'(t)/t^{m_1-1}$
is vertical, we must have $m(G_s,0)>m_1$ for $1\le j\le n$. It follows from
(6.7) that $G_1\ld G_n$ are never involved in the calculation of the
liftings $g_{[j]}$. We can therefore replace $g$ by $f\simeq(f_1\ld f_n)$
where $f_r(t)=t^{m_0}$ and $f_1\ld f_{r-1}$ are obtained by integrating
the equations $f_j'(t)/f_r'(t)=G_{n+j}(t)$, i.e., $f_j'(t)=m_0t^{m_0-1}
G_{n+j}(t)$, while $f_{r+1}\ld f_n$ are obtained by integrating (5.10).
We then get the desired Taylor expansion of order $d_k$ for~$f$.\square

Since we can always take $m_{k-1}=1$ without restriction, we get in
particular: 

\claim 6.12.~Corollary|Let $w\in P_kV$ be an arbitrary element. Then
there is a germ of curve $f:(\bC,0)\to X$ such that $f_{[k]}(0)=w$ and
$\smash{f_{[k-1]}'(0)}\ne 0$ $($thus the liftings $f_{[k-1]}$ and $f_{[k]}$
are regular germs of curve$)$. Moreover, if $w_0\in P_kV$ and $w$ is
taken in a sufficiently small neighborhood of $w_0$, then the germ $f=f_w$
can be taken to depend holomorphically on~$w$.
\endclaim

\proof. Only the holomorphic dependence of $f_w$ with respect to
$w$ has to be guaranteed. If $f_{w_0}$ is a solution for $w=w_0$, we
observe that $(f_{w_0})_{[k]}'$ is a non vanishing section of $V_k$ along
the regular curve defined by $(f_{w_0})_{[k]}$ in $P_kV$. We can thus
find a non vanishing section $\xi$ of $V_k$ on a neighborhood of $w_0$ in
$P_kV$ such that $\xi=(f_{w_0})_{[k]}'$ along that curve. We define
$t\mapsto F_w(t)$ to be the trajectory of $\xi$ with initial point $w$,
and we put $f_w=\pi_{0,k}\circ F_w$. Then $f_w$ is the required
family of germs.\square

Now, we can take $f:(\bC,0)\to X$ to be regular at the origin (by this,
we mean $f'(0)\ne 0$) if and only if $m_0=m_1=\cdots=m_{k-1}=1$, which is
possible by Proposition~6.11 if and only if $w\in P_kV$ is such that
$\pi_{j,k}(w)\notin D_j$ for all $j\in\{2\ld k\}$. For this reason,
we define
$$
\eqalign{
P_kV^\reg&=\bigcap_{2\le j\le k}\pi_{j,k}^{-1}(P_jV\ssm D_j),\cr
P_kV^\sing&=\bigcup_{2\le j\le k}\pi_{j,k}^{-1}(D_j)=P_kV\ssm P_kV^\reg,
\cr}
\leqno(6.13)
$$
in other words, $P_kV^\reg$ is the set of values $f_{[k]}(0)$ reached
by all regular germs of curves~$f$. One should take care however that
there are singular germs which reach the same points $f_{[k]}(0)\in P_kV^\reg$,
e.g., any $s$-sheeted covering $t\mapsto f(t^s)$. On the other hand,
if $w\in P_kV^\sing$, we can reach $w$ by a germ $f$ with $m_0=m(f,0)$
as large as we want.

\claim 6.14.~Corollary|Let $w\in P_kV^\sing$ be given, and let
$m_0\in\bN$ be an arbitrary integer larger than the number of components
$D_j$ such that $\pi_{j,k}(w)\in D_j$. Then there is a germ of curve
$f:(\bC,0)\to X$ with multiplicity $m(f,0)=m_0$ at the origin, such that
$f_{[k]}(0)=w$ and $\smash{f_{[k-1]}'(0)}\ne 0$.
\endclaim

\section{\S7. Jet differentials}

\subsection \S7.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
$$
Q(\lambda f',\lambda^2 f'',\ldots,\lambda^k f^{(k)})=\lambda^m
Q(f',f'',\ldots,f^{(k)})\leqno(7.1)
$$
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
$$
(\Psi\circ f)^{(j)}=\Psi'(f)\cdot f^{(j)}+\sum_{s=2}^{s=j}{~}
\sum_{j_1+j_2+\cdots+j_s=j}c_{j_1\ldots j_s}\Psi^{(s)}(f)\cdot
(f^{(j_1)}\ld f^{(j_s)})\leqno(7.2)
$$
with suitable integer constants $c_{j_1\ldots j_s}$ (this is easily
checked by induction on~$s$). In the ``absolute case'' $V=T_X$, we simply
write $E^\GG_{k,m}T^*_X=E^\GG_{k,m}$. If \hbox{$V\subset W\subset T_X$} 
are holomorphic subbundles, there are natural inclusions
$$
J_kV\subset J_kW\subset J_k,\qquad P_kV\subset P_kW\subset P_k.
$$
The restriction morphisms induce surjective arrows
$$
E^\GG_{k,m}\to E^\GG_{k,m}W^*\to E^\GG_{k,m}V^*,
$$
in particular $E^\GG_{k,m}V^*$ can be seen as a quotient of $E^\GG_{k,m}$.
(The notation $V^*$ is used here to make the contravariance property
implicit from the notation). Another useful consequence of these inclusions
is that one can extend the
definition of $J_kV$ and $P_kV$ to the case where $V$ is an arbitrary
linear space, simply by taking the closure of $\smash{J_kV_{X\ssm\Sing(V)}}$
and $\smash{P_kV_{X\ssm\Sing(V)}}$ in the smooth bundles 
$J_k$ and $P_k$, respectively.

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 (7.2)
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:
$$
F^p_s(E^\GG_{k,m}V^*)=\left\{
{\displaystyle
\hbox{\rm $Q(f',f''\ld f^{(k)})\in E^\GG_{k,m}V^*$ involving}
\atop\hbox{\rm only monomials $(f^{(\bullet)})^\ell$ with $|\ell|_s\ge p$}
\hfill}\right\},\qquad
\forall p\in\bN.\leqno(7.3)
$$
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 $k|m-p$).  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 (7.2), namely
$f^{(j)}\mapsto(\Psi\circ f)^{(j)}$ for $j<k$, and
$f^{(k)}\mapsto\Psi'(f)\circ f^{(k)}$ for $j=k$ (when $j=k$, the other
terms fall in the next stage $F^{p+1}_{k-1}$ of the filtration).
Therefore $f^{(k)}$ behaves as an element of $V\subset T_X$ under
coordinate changes.  We thus find
$$
G_{k-1}^{m-k\ell_k}(E^\GG_{k,m}V^*)=E^\GG_{k-1,m-k\ell_k}V^*\otimes
S^{\ell_k}V^*.
\leqno(7.4)
$$
Combining all filtrations $F_s^\bullet$ together, we find inductively a
filtration $F^\bullet$ on $E^\GG_{k,m}V^*$ such that the graded terms are
$$
\Gr^\ell(E^\GG_{k,m}V^*)=S^{\ell_1}V^*\otimes S^{\ell_2}V^*\otimes
\cdots\otimes S^{\ell_k}V^*,\qquad\ell\in\bN^k,\quad
|\ell|_k=m.\leqno(7.5)
$$

The bundles $E^\GG_{k,m}V^*$ have other interesting properties. In fact,
$$
E^\GG_{k,\bu}V^*:=\bigoplus_{m\ge 0}E^\GG_{k,m}V^*
$$
is in a natural way a bundle of graded algebras (the product is
obtained simply by taking the product of polynomials). There are
natural inclusions \hbox{$E^\GG_{k,\bu}V^*\subset E^\GG_{k+1,\bu}
V^*$} of algebras, hence $E^\GG_{\infty,\bu}V^*=\bigcup_{k\ge 0}
E^\GG_{k,\bu}V^*$ is also an algebra. Moreover, the sheaf of
holomorphic sections $\cO(E^\GG_{\infty,\bu} V^*)$ admits a
canonical derivation $\nabla^\GG$ given by a collection of $\bC$-linear maps
$$
\nabla^\GG:\cO(E^\GG_{k,m}V^*)\to\cO(E^\GG_{k+1,m+1}V^*),
$$
constructed in the following way. A holomorphic section of
$E^\GG_{k,m}V^*$ on a coordinate open set $\Omega\subset X$ can be
seen as a differential operator on the space of germs
$f:(\bC,0)\to\Omega$ of the form
$$
Q(f)=\sum_{|\alpha_1|+2|\alpha_2|+\cdots+k|\alpha_k|=m}
a_{\alpha_1\ldots\alpha_k}(f)\,(f')^{\alpha_1}(f'')^{\alpha_2}\cdots
(f^{(k)})^{\alpha_k}\leqno(7.6)
$$
in which the coefficients $a_{\alpha_1\ldots\alpha_k}$ are holomorphic
functions on $\Omega$. Then $\nabla Q$ is given by the formal derivative
$(\nabla Q)(f)(t)=d(Q(f))/dt$ with respect to the
$1$-dimensional parameter $t$ in~$f(t)$. For example, in dimension 2,
if $Q\in H^0(\Omega,\cO(E^\GG_{2,4}))$ is the section of weighted
degree $4$
$$
Q(f)=a(f_1,f_2)\,f_1^{\prime 3}f_2'+b(f_1,f_2)\,f_1^{\prime\prime 2},
$$
we find that $\nabla Q\in H^0(\Omega,\cO(E^\GG_{3,5}))$ is given by
$$
\eqalign{(\nabla Q)&(f)=
{\partial a\over\partial z_1}(f_1,f_2)\,f_1^{\prime 4}f_2'+
{\partial a\over\partial z_2}(f_1,f_2)\,f_1^{\prime 3}f_2^{\prime 2}+
{\partial b\over\partial z_1}(f_1,f_2)\,f_1'f_1^{\prime\prime 2}\cr
&{}+{\partial b\over\partial z_2}(f_1,f_2)\,f_2'f_1^{\prime\prime 2}
+a(f_1,f_2)\,\big(3f_1^{\prime 2}f_1''f_2'+f_1^{\prime 3}f_2'')+
b(f_1,f_2)\,\,2f_1''f_1'''.\cr}
$$
Associated with the graded algebra bundle $E^\GG_{k,\bu}V^*$, we 
have an analytic fiber bundle
$$X_k^\GG:=\Proj(E^\GG_{k,\bu}V^*)=(J_kV\ssm\{0\})/\bC^*\leqno(7.7)$$
over $X$, which has weighted projective spaces
$\bP(1^{[r]},2^{[r]}\ld k^{[r]})$ as fibers (these
weighted projective spaces are singular for $k>1$, but they only have
quotient singularities, see [Dol81]$\,$; here $J_kV\ssm\{0\}$ is the set
of non constant jets of order~$k\,$; we refer e.g.\ to Hartshorne's book
[Har77] for a definition of the $\Proj$ fonctor). 
As such, it possesses a canonical sheaf $\smash{\cO_{X^\GG_k}}(1)$ such that
$\smash{\cO_{X^\GG_k}}(m)$ is invertible when $m$ is a multiple of
$\lcm(1,2,\ldots,k)$. Under the natural projection 
$\pi_k:\smash{X^\GG_k}\to X$, the direct image 
$(\pi_k)_*\smash{\cO_{X^\GG_k}(m)}$ coincides with polynomials
$$
P(z\,;\,\xi_1,\ldots,\xi_k)=\sum_{\alpha_\ell\in\bN^r,\,1\le\ell\le k} 
a_{\alpha_1\ldots\alpha_k}(z)\,\xi_1^{\alpha_1}\ldots\xi_k^{\alpha_k}
\leqno(7.8)
$$
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 7.9. 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

\subsection \S7.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 $P_kV$ 
constructed in \S$\,$6 plays the role of such a quotient. First we 
introduce a canonical bundle subalgebra of~$E^\GG_{k,\bu}V^*$.

\claim 7.10.~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~$\nabla^\GG$, since e.g.\ 
$f_j''=\nabla^\GG f_j$ is not an invariant polynomial). 
In addition to this, there are natural induced filtrations
$F^p_s(E_{k,m}V^*)=E_{k,m}V^*\cap F^p_s(E^\GG_{k,m}V^*)$
(all locally trivial over~$X$). These induced filtrations will play an
important role later on.

\claim 7.11.~Theorem|Suppose that $V$ has rank $r\ge 2$. Let $\pi_{0,k}:
P_kV\lra X$ be the Semple jet bundles constructed in section~$6$,
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 $f'(0)\ne 0$. 
\smallskip
\item{\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
$J_kV^\reg/\bG_k\hookrightarrow P_kV$ over $X$, which identifies
$J_kV^\reg/\bG_k$ with $P_kV^\reg$ $($thus $P_kV$ is a relative
compactification of $J_kV^\reg/\bG_k$ over~$X)$.
\smallskip
\item{\rm ii)} The direct image sheaf
$$
(\pi_{0,k})_*\cO_{P_kV}(m)\simeq\cO(E_{k,m}V^*)
$$
can be identified with the sheaf of holomorphic sections of
$E_{k,m}V^*$.
\smallskip
\item{\rm iii)} For every $m>0$, the relative base locus of the linear
system $|\cO_{P_kV}(m)|$ is equal to the set $\smash{P_kV^\sing}$
of singular $k$-jets. Moreover, $\cO_{P_kV}(1)$ is relatively big
over~$X$.
\vskip0pt
\endclaim

\proof. 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~$P_jV$, 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
$$
J_kV^\reg/\bG_k\to P_kV^\reg,\qquad f~\mod~\bG_k\mapsto f_{[k]}(0).
$$
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 $P_kV$ 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 $J_kV^\reg/\bG_k\to P_kV$ is a bijection onto $P_kV^\reg$, 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
$f_i^{(j)}(t)=d^jf_i/dt^j$
$$
\leqalignno{
(g_1'\ld g_{r-1}')&=\Big({f'_1\over f'_r}\ld{f'_{r-1}\over f'_r}\Big);\cr
(g_1''\ld g_{r-1}'')&=\Big({f''_1f'_r-f''_rf'_1\over f^{\prime 3}_r}\ld
{f''_{r-1}f'_r-f''_rf'_{r-1}\over f^{\prime 3}_r}\Big);~\ldots~;&(7.12)\cr
\qquad(g_1^{(k)}\ld g_{r-1}^{(k)})&=\Big({f^{(k)}_1f'_r-f^{(k)}_rf'_1\over
f^{\prime k+1}_r}\ld{f^{(k)}_{r-1}f'_r-f^{(k)}_rf'_{r-1}\over
f^{\prime k+1}_r}\Big)+(\hbox{order}<k).\cr}
$$
Also, it is easy to check that $f_r^{\prime 2k-1}g_i^{(k)}$ is an
invariant polynomial in $f'$, $f''\ld f^{(k)}$ of total degree $2k-1$, i.e.,
a section of $E_{k,2k-1}$. 

\noindent ii) Since the bundles $P_kV$ and $E_{k,m}V^*$ are both locally
trivial over $X$, it is sufficient to identify sections $\sigma$ of
$\cO_{P_kV}(m)$ over a fiber $P_kV_x=\pi_{0,k}^{-1}(x)$ with the fiber
$E_{k,m}V^*_x$, at any point  $x\in X$. Let $f\in J_kV_x^\reg$
be a regular $k$-jet at~$x$. By (6.6), the derivative $f_{[k-1]}'(0)$
defines an element of the fiber of $\cO_{P_kV}(-1)$ at $f_{[k]}(0)\in P_kV$.
Hence we get a well defined complex valued operator
$$
Q(f',f''\ld f^{(k)})=\sigma(f_{[k]}(0))\cdot(f_{[k-1]}'(0))^m.
\leqno(7.13)
$$
Clearly, $Q$ is holomorphic on $J_kV_x^\reg$ (by the holomorphicity
of~$\sigma$), and the $\bG_k$-invariance condition of Def.~7.10 is satisfied
since $f_{[k]}(0)$ does not depend on reparametrization and
\hbox{$(f\circ\varphi)_{[k-1]}'(0)=f_{[k-1]}'(0)\varphi'(0)$}.
Now, $J_kV_x^\reg$ is the complement of a
linear subspace of codimension $n$ in $J_kV_x$, hence $Q$ extends
holomorphically to all of $J_kV_x\simeq(\bC^r)^k$ by Riemann's
extension theorem (here we use the hypothesis $r\ge 2\,$; if $r=1$, the
situation is anyway not interesting since $P_kV=X$ for all~$k$). Thus
$Q$ admits an everywhere convergent power series
$$
Q(f',f''\ld f^{(k)})=\sum_{\alpha_1,\alpha_2\ld\alpha_k\in\bN^r}
a_{\alpha_1\ldots\alpha_k}\,(f')^{\alpha_1}(f'')^{\alpha_2}\cdots
(f^{(k)})^{\alpha_k}.
$$
The $\bG_k$-invariance (7.10) implies in particular that $Q$ must be
multihomogeneous in the sense of (7.1), and thus $Q$ must be a
polynomial. We conclude that $Q\in E_{k,m}V^*_x$, as desired.

Conversely, Corollary~6.12 implies that there is a holomorphic family
of germs $f_w:(\bC,0)\to X$ such that $(f_w)_{[k]}(0)=w$ and
$(f_w)_{[k-1]}'(0)\ne 0$, for all $w$ in a neighborhood of any given
point $w_0\in P_kV_x$. Then every $Q\in E_{k,m}V^*_x$ yields a
holomorphic section $\sigma$ of $\cO_{P_kV}(m)$ over the fiber $P_kV_x$
by putting
$$
\sigma(w)=Q(f_w',f_w''\ld f_w^{(k)})(0)\,\big((f_w)_{[k-1]}'(0)\big)^{-m}.
\leqno(7.14)
$$

\noindent iii) By what we saw in i-ii), every section $\sigma$ of
$\cO_{P_kV}(m)$ over the fiber $P_kV_x$ is given by a polynomial
$Q\in E_{k,m}V^*_x$, and this polynomial can be expressed
on the Zariski open chart $f'_r\ne 0$ of $P_kV^\reg_x$ as
$$
Q(f',f''\ld f^{(k)})=f_r^{\prime m}\swh Q|(g',g''\ld g^{(k)}),
\leqno(7.15)
$$
where $\swh Q|$ is a polynomial and $g$ is the reparametrization of $f$ such
that $g_r(\tau)=\tau$. In fact $\swh Q|$ is obtained from $Q$ by substituting
$f'_r=1$ and $f^{(j)}_r=0$ for~$j\ge 2$, and conversely $Q$ can be recovered
easily from $\swh Q|$ by using the substitutions (7.12).

In this context, the jet differentials $f\mapsto f'_1\ld f\mapsto f'_r$ can
be viewed as sections of $\cO_{P_kV}(1)$ on a neighborhood
of the fiber $P_kV_x$. Since these sections vanish exactly
on $P_kV^\sing$, the relative base locus of $\cO_{P_kV}(m)$ is
contained in $P_kV^\sing$ for every~$m>0$. We see that $\cO_{P_kV}(1)$ is 
big by considering the sections of $\cO_{P_kV}(2k-1)$ 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~$P_kV^\reg_x$. 

Now, we check that every section $\sigma$ of $\cO_{P_kV}(m)$ over $P_kV_x$
must vanish on $P_kV^\sing_x$. Pick an arbitrary element $w\in P_kV^\sing$
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 (7.14).\square

Theorem (7.11~iii) shows that $\cO_{P_kV}(1)$ is never relatively ample
over $X$ for $k\ge 2$. In order to overcome this difficulty, we define for
every \hbox{$\bfa=(a_1\ld a_k)\in\bZ^k$} a line bundle
$\cO_{P_kV}(\bfa)$ on $P_kV$ such that
$$
\cO_{P_kV}(\bfa)=\pi_{1,k}^*\cO_{P_1V}(a_1)\otimes
\pi_{2,k}^*\cO_{P_2V}(a_2)\otimes\cdots\otimes\cO_{P_kV}(a_k).
\leqno(7.16)
$$
By (6.9), we have $\pi_{j,k}^*\cO_{P_jV}(1)=
\cO_{P_kV}(1)\otimes\cO_{P_kV}
(-\pi_{j+1,k}^* D_{j+1}-\cdots-D_k)$, 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
$$
\leqalignno{
&\cO_{P_kV}(\bfa)=\cO_{P_kV}(b_k)\otimes\cO_{P_kV}(-\bfb\cdot D^*),
\qquad\hbox{\rm where}&(7.17)\cr
&\bfb=(b_1\ld b_k)\in\bZ^k,\quad b_j=a_1+\cdots+a_j,&\cr
&\bfb\cdot D^*=\sum_{1\le j\le k-1}b_j\,\pi_{j+1,k}^* D_{j+1}.&\cr}
$$
In particular, if $\bfb\in\bN^k$, i.e., $a_1+\cdots+a_j\ge 0$, we get a
morphism
$$
\cO_{P_kV}(\bfa)=\cO_{P_kV}(b_k)\otimes\cO_{P_kV}(-\bfb\cdot D^*)
\to\cO_{P_kV}(b_k).\leqno(7.18)
$$

\claim 7.19.~Remark|{\rm As in Green-Griffiths [GrGr80], Riemann's
extension theorem shows that for every meromorphic map $\Phi:X\dasharrow Y$
there are well-defined pullback morphisms
$$
\Phi^*:H^0(Y,E^\GG_{k,m})\to H^0(X,E^\GG_{k,m}),\qquad
\Phi^*:H^0(Y,E_{k,m})\to H^0(X,E_{k,m}).
$$
In particular the dimensions $h^0(X,E^\GG_{k,m})$ and $h^0(X,E^\GG_{k,m})$
are bimeromorphic invariants of~$X$. The same is true for spaces of
sections of any subbundle of $E^\GG_{k,m}$ or $E_{k,m}$ constructed by
means of the canonical filtrations $F_s^\bullet$.}
\endclaim

\claim 7.20.~Remark|{\rm As $\bG_k$ is a non reductive group, it is not
a priori clear that the graded ring $\cA_{n,k,r}=\bigoplus_{m\in\bZ}
E_{k,m}V^\star$ is finitely generated (pointwise). This can be checked by hand
([Dem07a], [Dem07b]) for $n=2$ and $k\ge 4$. Rousseau [Rou06b] also checked
the case $n=3$, $k=3$, and then Merker [Mer08] proved the finiteness
for $n=2$, $k=5$. Recently, B\'erczi and Kirwan [BeKi10] found a nice
geometric argument proving the finiteness in full generality.}
\endclaim

\section{\S8. $k$-jet metrics with negative curvature}

The goal of this section is to show that hyperbolicity is closely
related to the existence of $k$-jet metrics with suitable negativity
properties of the curvature. The connection between these properties is
in fact a simple consequence of the Ahlfors-Schwarz lemma. Such ideas have
been already developed long ago by Grauert-Reckziegel [GRec65],
Kobayashi [Kob75] for $1$-jet metrics (i.e., Finsler metrics on~$T_X$)
and by Cowen-Griffiths [CoGr76], Green-Griffiths [GrGr80] and Grauert
[Gra89] for higher order jet metrics. 

\subsection \S8.A. Definition of $k$-jet metrics|
Even in the standard case $V=T_X$, the definition given below differs from 
that of [GrGr80], in which the $k$-jet metrics are not supposed to be 
$\bG'_k$-invariant. We prefer to deal here with $\bG'_k$-invariant objects, because
they reflect better the intrinsic geometry. Grauert [Gra89] actually deals
with $\bG'_k$-invariant metrics, but he apparently does not take care of
the way the quotient space $J_k^\reg V/\bG_k$ can be compactified;
also, his metrics are always induced by the Poincar\'e metric, and it is
not at all clear whether these metrics have the expected curvature
properties (see 8.14 below). In the present situation, it is
important to allow also hermitian metrics possessing some singularities
(``singular hermitian metrics'' in the sense of [Dem90b]).

\claim 8.1.~Definition|Let $L\to X$ be a holomorphic line bundle over
a complex manifold~$X$. We say that $h$ is a singular metric on~$L$ if
for any trivialization $L_{\restriction U}\simeq U\times\bC$ of~$L$,
the metric is given by $|\xi|_h^2=|\xi|^2e^{-\varphi}$ for some real
valued weight function $\varphi\in L^1_\loc(U)$. The curvature current
of $L$ is then defined to be the closed $(1,1)$-current
$\Theta_{L,h}={i\over 2\pi}\ddbar\varphi$, computed in the sense of
distributions. We say that $h$ admits a closed subset $\Sigma\subset
X$ as its degeneration set if $\varphi$ is locally bounded on
$X\ssm\Sigma$ and is unbounded on a neighborhood of any point
of~$\Sigma$.
\endclaim

An especially useful situation is the case when the curvature of $h$
is positive definite. By this, we mean that there exists a smooth
positive definite hermitian metric $\omega$ and a continuous positive
function $\varepsilon$ on $X$ such that $\Theta_{L,h}\ge\varepsilon
\omega$ in the sense of currents, and we write in this case
$\Theta_{L,h}\gg0$. We need the following basic fact (quite standard
when $X$ is projective algebraic; however we want to avoid
any algebraicity assumption here, so as to be able to cover the case 
of general complex tori in \S$\,$10).

\claim 8.2.~Proposition|Let $L$ be a holomorphic line bundle on a compact
complex manifold~$X$.
\smallskip
\item{\rm i)} $L$ admits a singular hermitian metric $h$ with positive
  definite curvature current $\Theta_{L,h}\gg0$ if and only if $L$ is~big.
\smallskip\noindent
Now, define $B_m$ to be the base locus of the linear system 
$|H^0(X,L^{\otimes m})|$ and let 
$$
\Phi_m:X\ssm B_m\to\bP^N
$$ 
be the corresponding meromorphic map. Let $\Sigma_m$ be the closed 
analytic set equal to the union of $B_m$ and of the set of points 
$x\in X\ssm B_m$ such that the fiber $\Phi_m^{-1}(\Phi_m(x))$ is 
positive dimensional.
\smallskip
\item{\rm ii)} If $\Sigma_m\ne X$ and $G$ is any line bundle,
  the base locus of $L^{\otimes k}\otimes G^{-1}$ is contained in~$\Sigma_m$
  for~$k$ large. As a consequence, $L$ admits a singular hermitian
  metric $h$ with degeneration set~$\Sigma_m$ and with $\Theta_{L,h}$
  positive definite on~$X$. 
\smallskip
\item{\rm iii)} Conversely, if $L$ admits a hermitian metric $h$ with
  degeneration set~$\Sigma$ and positive definite curvature
  current~$\Theta_{L,h}$, there exists an integer~$m>0$ such that the
  base locus $B_m$ is contained in~$\Sigma$ and
  $\Phi_m:X\ssm\Sigma\to\bP_m$ is an embedding.
\vskip0pt 
\endclaim

\proof. i) is proved e.g.\ in [Dem90b,~92], and ii) and iii) are
well-known results in the basic theory of linear systems.\square
\medskip

We now come to the main definitions. By (6.6), every regular $k$-jet
$f\in J_kV$ gives rise to an element $f_{[k-1]}'(0)\in \cO_{P_kV}(-1)$.
Thus, measuring the ``norm of $k$-jets'' is the same as taking a
hermitian metric on $\cO_{P_kV}(-1)$. 

\claim 8.3.~Definition|A smooth, $($resp.\ continuous, resp.\ singular$)$
$k$-jet metric on a complex directed manifold $(X,V)$ is a
hermitian metric $h_k$ on the line bundle $\cO_{P_kV}(-1)$ over~$P_kV$
$($i.e.\ a Finsler metric on the vector bundle $V_{k-1}$ over $P_{k-1}V)$,
such that the weight functions $\varphi$ representing the metric are smooth
$($resp.\ conti\-nuous, $L^1_\loc)$. We let $\Sigma_{h_k}\subset P_kV$ be
the singularity set of the metric, i.e., the closed subset of points in a 
neighborhood of which the weight $\varphi$ is not locally bounded.
\endclaim

We will always assume here that the weight function $\varphi$ is quasi psh.
Recall that a function $\varphi$ is said to be quasi psh if $\varphi$
is locally the sum of a plurisubharmonic function and of a smooth
function (so that in particular $\varphi\in L^1_\loc$). Then the curvature 
current
$$
\Theta_{h_k^{-1}}(\cO_{P_kV}(1))={i\over 2\pi}\ddbar\varphi.
$$
is well defined as a current and is locally bounded from below by a 
negative $(1,1)$-form with constant coefficients.

\claim 8.4.~Definition|Let $h_k$ be a $k$-jet metric on~$(X,V)$. We say that
$h_k$ has negative jet curvature $($resp.\ negative total jet curvature$)$
if $\Theta_{h_k}(\cO_{P_kV}(-1))$ is negative definite along the
subbundle $V_k\subset T_{P_kV}$ $($resp.\ on all of $T_{P_kV})$, i.e.,
if there is $\varepsilon>0$ and a smooth hermitian metric $\omega_k$ on
$T_{P_kV}$ such that
$$
\langle\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\rangle(\xi)
\ge\varepsilon|\xi|^2_{\omega_k},\qquad\forall\xi\in V_k\subset T_{P_kV}
\quad(\hbox{resp.}\quad\forall\xi\in T_{P_kV}).
$$
$($If the metric $h_k$ is not smooth, we suppose that its weights $\varphi$
are quasi psh, and the curvature inequality is taken in the sense of
distributions.$)$
\endclaim

It is important to observe that for $k\ge 2$ there cannot exist any smooth
hermitian metric $h_k$ on $\cO_{P_kV}(1)$ with positive definite curvature
along $T_{X_k/X}$, since $\cO_{P_kV}(1)$ is not relatively ample over~$X$.
However, it is relatively big, and Prop.~8.2~i) shows that $\cO_{P_kV}(-1)$
admits a singular hermitian metric with negative total jet curvature 
(whatever the singularities of the metric are) if and only if $\cO_{P_kV}(1)$
is big over $P_kV$. It is therefore crucial to allow singularities in the
metrics in Def.~8.4.

\subsection \S8.B. Special case of $1$-jet metrics|

A $1$-jet metric $h_1$ 
on $\cO_{P_1V}(-1)$ is the same as a Finsler metric $N=\sqrt{h_1}$
on~$V\subset T_X$. Assume until the end of this paragraph that $h_1$ is
smooth. By the well known Kodaira embedding theorem, the
existence of a smooth metric $h_1$ such that
$\Theta_{h_1^{-1}}(\cO_{P_1V}(1))$ is positive on all of $T_{P_1V}$ is
equivalent to $\cO_{P_1V}(1)$ being ample, that is, $V^*$~ample.

\claim 8.5 Remark|{\rm In the absolute case $V=T_X$, there are only few examples of varieties
$X$ such that $T^*_X$ is ample, mainly quotients of the ball
$\bB_n\subset\bC^n$ by a discrete cocompact group of automorphisms.}
\endclaim

The $1$-jet negativity condition considered in Definition~8.4 is much
weaker. For example, if the hermitian metric $h_1$ comes from a 
(smooth) hermitian metric $h$ on $V$, then formula (5.15) implies that
$h_1$ has negative total jet curvature (i.e.\ $\Theta_{h_1^{-1}}
(\cO_{P_1V}(1))$ is positive) if and only if $\langle\Theta_{V,h}\rangle
(\zeta\otimes v)<0$ for all $\zeta\in T_X\ssm\{0\}$, $v\in V\ssm\{0\}$, 
that is, if $(V,h)$ is {\em negative in the sense of Griffiths}. 
On the other hand, $V_1\subset T_{P_1V}$ consists by definition of 
tangent vectors $\tau\in T_{P_1V,(x,[v])}$ whose horizontal projection
${}^H\!\tau$ is proportional to~$v$, thus $\Theta_{h_1}(\cO_{P_1V}(-1))$
is negative definite on $V_1$ if and only if $\Theta_{V,h}$ satisfies
the much weaker condition that the {\em holomorphic sectional curvature}
$\langle\Theta_{V,h}\rangle(v\otimes v)$ is negative on every complex
line.\square

\subsection \S8.C. Vanishing theorem for invariant jet differentials|

We now come back to the general situation of jets of arbitrary order~$k$.
Our first observation is the fact that the $k$-jet negativity property
of the curvature becomes actually weaker and weaker as $k$ increases.

\claim 8.6.~Lemma|Let $(X,V)$ be a compact complex directed
manifold. If $(X,V)$ has a $(k-1)$-jet metric $h_{k-1}$ with negative 
jet curvature, then there is a $k$-jet metric $h_k$ with negative jet
curvature such that $\Sigma_{h_k}\subset\pi_k^{-1}(\Sigma_{h_{k-1}})\cup D_k$.
$($The same holds true for negative total jet curvature$)$.
\endclaim

\proof. Let $\omega_{k-1}$, $\omega_k$ be given smooth hermitian metrics on
$T_{P_{k-1}V}$ and $T_{P_kV}$. The hypothesis implies
$$
\langle\Theta_{h_{k-1}^{-1}}(\cO_{P_{k-1}V}(1))\rangle(\xi)\ge
\varepsilon|\xi|_{\omega_{k-1}}^2,\qquad\forall\xi\in V_{k-1}
$$
for some constant~$\varepsilon>0$. On the other hand, as
$\cO_{P_kV}(D_k)$ is relatively ample over $P_{k-1}V$ ($D_k$ is a hyperplane
section bundle), there exists a smooth metric $\wt h$ on $\cO_{P_kV}(D_k)$
such that
$$
\langle\Theta_{\wt h}(\cO_{P_kV}(D_k))\rangle(\xi)\ge
\delta|\xi|_{\omega_k}^2-C|(\pi_k)_*\xi|_{\omega_{k-1}}^2,\qquad
\forall\xi\in T_{P_kV}
$$
for some constants $\delta,C>0$. Combining both inequalities (the second one
being applied to $\xi\in V_k$ and the first one to $(\pi_k)_*\xi\in
V_{k-1}$), we get
$$
\eqalign{
\langle\Theta_{(\pi_k^* h_{k-1})^{-p}\wt h\,}(\pi_k^*\cO_{P_{k-1}V}(p)
\otimes{}&\cO_{P_kV}(D_k))\rangle(\xi)\ge\cr
&\ge\delta|\xi|_{\omega_k}^2+(p\varepsilon-C)
|(\pi_k)_*\xi|_{\omega_{k-1}}^2,\qquad\forall\xi\in V_k.\cr}
$$
Hence, for $p$ large enough, $(\pi_k^* h_{k-1})^{-p}\wt h$ has positive
definite curvature along~$V_k$. Now, by (6.9), there is a sheaf injection
$$
\cO_{P_kV}(-p)=\pi_k^*\cO_{P_{k-1}V}(-p)\otimes\cO_{P_kV}(-pD_k)
\hookrightarrow\big(\pi_k^*\cO_{P_{k-1}V}(p)\otimes\cO_{P_kV}(D_k)
\big)^{-1}
$$
obtained by twisting with $\cO_{P_kV}((p-1)D_k)$.  Therefore
$h_k:=((\pi_k^* h_{k-1})^{-p}\wt h)^{-1/p}=(\pi_k^*
h_{k-1})\wt h^{-1/p}$ induces a singular metric on $\cO_{P_kV}(1)$ in
which an additional degeneration divisor $p^{-1}(p-1)D_k$ appears.
Hence we get $\Sigma_{h_k}=\pi_k^{-1}\Sigma_{h_{k-1}}\cup D_k$ and
$$
\Theta_{h_k^{-1}}(\cO_{P_kV}(1))={1\over p}\Theta_{(\pi_k^* 
h_{k-1})^{-p}\wt h}+{p-1\over p}[D_k]
$$
is positive definite along~$V_k$. The same proof works in the case of
negative total jet curvature.\square

One of the main motivations for the introduction of $k$-jets metrics is the
following list of algebraic sufficient conditions.

\claim 8.7.~Algebraic sufficient conditions|{\rm We suppose here that $X$
is projective algebraic, and we make one of the additional assumptions i),
ii) or iii) below.

\noindent i) Assume that there exist integers $k,m>0$ and $\bfb\in\bN^k$
such that the line bundle $\cO_{P_kV}(m)\otimes\cO_{P_kV}(-\bfb\cdot D^*)$
is ample over~$P_kV$. Set $A=\cO_{P_kV}(m)\otimes\cO_{P_kV}(-\bfb\cdot
D^*)$. Then there is a smooth hermitian metric $h_A$ on $A$ with
positive definite curvature on~$P_kV$. By means of the morphism
$\mu:\cO_{P_kV}(-m)\to A^{-1}$, we get an induced metric $h_k=(\mu^*
h_A^{-1})^{1/m}$ on $\cO_{P_kV}(-1)$ which is degenerate on the support of
the zero divisor $\div(\mu)=\bfb\cdot D^*$. Hence
$\Sigma_{h_k}=\Supp(\bfb\cdot D^*)\subset P_kV^\sing$ and
$$
\Theta_{h_k^{-1}}(\cO_{P_kV}(1))={1\over m}\Theta_{h_A}(A)+{1\over m}
[\bfb\cdot D^*]\ge{1\over m}\Theta_{h_A}(A)>0.
$$
In particular $h_k$ has negative total jet curvature.

\noindent ii) Assume more generally that there exist integers $k,m>0$ and
an ample line bundle $L$ on $X$ such that
$H^0(P_kV,\cO_{P_kV}(m)\otimes\pi_{0,k}^* L^{-1})$ has non zero
sections $\sigma_1\ld\sigma_N$. Let $Z\subset P_kV$ be the base locus of
these sections; necessarily $Z\supset P_kV^\sing$ by 7.11~iii). By taking a
smooth metric $h_L$ with positive curvature on $L$, we get a singular
metric $h_k'$ on $\cO_{P_kV}(-1)$ such that
$$
h_k'(\xi)=\Big(\sum_{1\le j\le N}|\sigma_j(w)\cdot\xi^m|_{h_L^{-1}}^2
\Big)^{1/m},\qquad w\in P_kV,\quad\xi\in\cO_{P_kV}(-1)_w.
$$
Then $\Sigma_{h_k'}=Z$, and by computing ${i\over 2\pi}\ddbar\log h_k'(\xi)$
we obtain
$$
\Theta_{h_k^{\prime\,-1}}(\cO_{P_kV}(1))\ge{1\over m}\pi_{0,k}^*\Theta_L.
$$
By (7.17) and an induction on $k$, there exists $\bfb\in\bQ^k_+$ such
that $\cO_{P_kV}(1)\otimes\cO_{P_kV}(-\bfb\cdot D^*)$ is relatively ample
over~$X$. Hence $A=\cO_{P_kV}(1)\otimes\cO_{P_kV}(-\bfb\cdot D^*)\otimes
\pi_{0,k}^* L^{\otimes p}$ is ample on $X$ for $p\gg 0$. The
arguments used in i) show that there is a $k$-jet metric $h_k''$ on
$\cO_{P_kV}(-1)$ with $\Sigma_{h_k''}=\Supp(\bfb\cdot D^*)=P_kV^\sing$ and
$$
\Theta_{h_k^{\prime\prime\,-1}}(\cO_{P_kV}(1))=\Theta_A+[\bfb\cdot D^*]-
p\,\pi_{0,k}^*\Theta_L,
$$
where $\Theta_A$ is positive definite on $P_kV$. The metric
$h_k=(h_k^{\prime\,mp}h_k'')^{1/(mp+1)}$ then satisfies 
$\Sigma_{h_k}=\Sigma_{h_k'}=Z$ and
$$
\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\ge{1\over mp+1}\Theta_A>0.
$$

\noindent iii) If $E_{k,m}V^*$ is ample, there is an ample
line bundle $L$ and a sufficiently high symmetric power such that
$S^p(E_{k,m}V^*)\otimes L^{-1}$ is generated by sections. These
sections can be viewed as sections of $\cO_{P_kV}(mp)\otimes\pi_{0,k}^*
L^{-1}$ over $P_kV$, and their base locus is exactly $Z=P_kV^\sing$ by
7.11~iii). Hence the $k$-jet metric $h_k$ constructed in ii) has negative
total jet curvature and satisfies $\Sigma_{h_k}=P_kV^\sing$.\square}
\endclaim

An important fact, first observed by [GRe65] for $1$-jet metrics and by
[GrGr80] in the higher order case, is that $k$-jet negativity implies
hyperbolicity. In particular, the existence of enough global jet
differentials implies hyperbolicity.

\claim 8.8.~Theorem|Let $(X,V)$ be a compact complex directed
manifold. If $(X,V)$ has a $k$-jet metric $h_k$ with negative jet
curvature, then every entire curve $f:\bC\to X$ tangent to $V$ is such
that $f_{[k]}(\bC)\subset\Sigma_{h_k}$. In particular, if $\Sigma_{h_k}
\subset P_kV^\sing$, then $(X,V)$ is hyperbolic.
\endclaim

\proof. The main idea is to use the Ahlfors-Schwarz lemma, following the
approach of [GrGr80]. However we will give here all necessary details
because our setting is slightly different. Assume that there is a $k$-jet
metric $h_k$ as in the hypotheses of Theorem~8.8. Let 
$\omega_k$ be a smooth hermitian metric on $T_{P_kV}$. By 
hypothesis, there exists $\varepsilon>0$ such that
$$
\langle\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\rangle(\xi)\ge\varepsilon
|\xi|_{\omega_k}^2\qquad\forall\xi\in V_k.
$$
Moreover, by (6.4), $(\pi_k)_*$ maps $V_k$ continuously to
$\cO_{P_kV}(-1)$ and the weight $e^\varphi$ of $h_k$ is locally bounded from
above. Hence there is a constant $C>0$ such that
$$
|(\pi_k)_*\xi|_{h_k}^2\le C|\xi|_{\omega_k}^2,\qquad\forall\xi\in V_k.
$$
Combining these inequalities, we find
$$
\langle\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\rangle(\xi)\ge
{\varepsilon\over C}|(\pi_k)_*\xi|_{h_k}^2,\qquad\forall\xi\in V_k.
$$
Now, let $f:\Delta_R\to X$ be a non constant holomorphic map tangent to~$V$
on the disk~$\Delta_R$. We use the line bundle morphism (6.6)
$$
F=f_{[k-1]}':T_{\Delta_R}\to f_{[k]}^*\cO_{P_kV}(-1)
$$
to obtain a pullback metric
$$
\gamma=\gamma_0(t)\,dt\otimes d\ol t=F^* h_k\qquad
\hbox{on $T_{\Delta_R}$}.
$$
If $f_{[k]}(\Delta_R)\subset\Sigma_{h_k}$ then $\gamma\equiv 0$. Otherwise,
$F(t)$ has isolated zeroes at all singular points of $f_{[k-1]}$ and so
$\gamma(t)$ vanishes only at these points and at points of the degeneration
set $(f_{[k]})^{-1}(\Sigma_{h_k})$ which is a polar set in~$\Delta_R$.
At other points, the Gaussian curvature of $\gamma$ satisfies
$$
{i\,\ddbar\log\gamma_0(t)\over\gamma(t)}
={-2\pi\,(f_{[k]})^*\Theta_{h_k}(\cO_{P_kV}(-1))\over F^* h_k}
={\langle\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\rangle(f_{[k]}'(t))\over
|f_{[k-1]}'(t)|_{h_k}^2}\ge {\varepsilon\over C},
$$
since $f_{[k-1]}'(t)=(\pi_k)_* f_{[k]}'(t)$. The Ahlfors-Schwarz lemma
4.2 implies that $\gamma$ can be compared with the Poincar\'e
metric as follows:
$$
\gamma(t)\le{2C\over\varepsilon}{R^{-2}|dt|^2\over(1-|t|^2/R^2)^2}\quad
\Longrightarrow\quad|f_{[k-1]}'(t)|_{h_k}^2\le
{2C\over\varepsilon}{R^{-2}\over(1-|t|^2/R^2)^2}.
$$
If $f:\bC\to X$ is an entire curve tangent to~$V$ such that
$f_{[k]}(\bC)\not\subset\Sigma_{h_k}$, the above estimate implies
as $R\to+\infty$ that $f_{[k-1]}$ must be a constant, hence also~$f$. Now,
if $\Sigma_{h_k}\subset P_kV^\sing$, the inclusion $f_{[k]}(\bC)\subset
\Sigma_{h_k}$ implies $f'(t)=0$ at every point, hence $f$ is a constant
and $(X,V)$ is hyperbolic.\square

Combining Theorem~8.8 with 8.7~ii) and iii), we get the following
consequences.

\claim 8.9.~Corollary|Assume that there exist integers $k,m>0$ and an
ample line bundle $L$ on $X$ such that $H^0(P_kV,\cO_{P_kV}(m)\otimes
\pi_{0,k}^* L^{-1})\simeq H^0(X,E_{k,m}(V^*)\otimes L^{-1})$
has non zero sections $\sigma_1\ld\sigma_N$. Let $Z\subset P_kV$ be the
base locus of these sections. Then every entire curve $f:\bC\to X$
tangent to $V$ is such that $f_{[k]}(\bC)\subset Z$. In other words,
for every global $\bG_k$-invariant polynomial differential operator
$P$ with values in $L^{-1}$, every entire curve $f$ must satisfy
the algebraic differential equation $P(f)=0$.\square
\endclaim

\claim 8.10.~Corollary|Let $(X,V)$ be a compact complex directed
manifold. If $E_{k,m}V^*$ is ample for some positive integers
$k,m$, then $(X,V)$ is hyperbolic.\square
\endclaim

\claim 8.11.~Remark|{\rm Green and Griffiths [GrGr80] stated that Corollary
8.9 is even true with sections $\sigma_j\in H^0(X,E^\GG_{k,m}(V^*)
\otimes L^{-1})$, in the special case $V=T_X$ they consider. We refer
to [SiYe97] by Siu and Yeung for a detailed proof of 
this fact, based on a use of the well-known logarithmic derivative lemma in
Nevanlinna theory (the original proof given in [GrGr80] does not seem
to be complete, as it relies on an unsettled pointwise version of 
the Ahlfors-Schwarz lemma for general jet differentials); other proofs
seem to have been circulating in the literature in the last years.
We give here a very short proof for the case when $f$ is supposed to have a
bounded derivative (thanks to the Brody criterion, 
this is enough if one is merely interested in proving
hyperbolicity, thus Corollary 8.10 will be valid with $E^\GG_{k,m}V^*$
in place of $E_{k,m}V^*$). In fact, if $f'$ is bounded, one can apply
the Cauchy inequalities to all components $f_j$ of $f$ with respect to
a finite collection of coordinate patches covering~$X$. As $f'$ is bounded,
we can do this on sufficiently small discs $D(t,\delta)\subset\bC$ 
of constant radius $\delta>0$. Therefore
all derivatives $f'$, $f''$, $\ldots\,f^{(k)}$ are bounded. From this
we conclude that $\sigma_j(f)$ is a bounded section of $f^*
L^{-1}$. Its norm $|\sigma_j(f)|_{L^{-1}}$ (with respect to any
positively curved metric $|~~|_L$ on $L$) is a bounded subharmonic
function, which is moreover strictly subharmonic at all points where
$f'\ne 0$ and $\sigma_j(f)\ne 0$. This is a contradiction unless $f$ is
constant or $\sigma_j(f)\equiv 0$.\square}
\endclaim

The above results justify the following definition and problems.

\claim 8.12.~Definition|We say that $X$, resp.\ $(X,V)$, has non
degenerate negative $k$-jet curvature if there exists a $k$-jet metric
$h_k$ on $\cO_{P_kV}(-1)$ with negative jet curvature such that
$\Sigma_{h_k}\subset P_kV^\sing$.
\endclaim

\claim 8.13.~Conjecture|Let $(X,V)$ be a compact directed manifold.
Then $(X,V)$ is hyperbolic if and only if $(X,V)$ has nondegenerate 
negative $k$-jet curvature for $k$ large enough.
\endclaim

This is probably a hard problem. In fact, we will see in the next
section that the smallest admissible integer $k$ must depend on the
geometry of $X$ and need not be uniformly bounded as soon as $\dim X\ge
2$ (even in the absolute case~$V=T_X$). On the other hand, if $(X,V)$ is
hyperbolic, we get for each integer $k\ge 1$ a generalized
Kobayashi-Royden metric $\bfk_{(P_{k-1}V,V_{k-1})}$ on $V_{k-1}$ (see
Definition~2.1), which can be also viewed as a $k$-jet metric $h_k$ on
$\cO_{P_kV}(-1)\,$; we will call it the {\em Grauert $k$-jet metric}
of $(X,V)$, although it formally differs from the jet metric considered 
in [Gra89] (see also [DGr91]). By looking at the projection
$\pi_k:(P_kV,V_k)\to(P_{k-1}V,V_{k-1})$, we see that the sequence $h_k$
is monotonic, namely $\pi_k^*h_k\le h_{k+1}$ for every~$k$. If
$(X,V)$ is hyperbolic, then $h_1$ is nondegenerate and therefore by
monotonicity $\Sigma_{h_k}\subset P_kV^\sing$ for $k\ge 1$. Conversely,
if the Grauert metric satisfies $\Sigma_{h_k} \subset P_kV^\sing$, it 
is easy to see that $(X,V)$ is hyperbolic. The following problem is
thus especially meaningful.

\claim 8.14.~Problem|Estimate the $k$-jet curvature
$\Theta_{h_k^{-1}}(\cO_{P_kV}(1))$ of the Grauert metric $h_k$
on $(P_kV,V_k)$ as $k$ tends to $+\infty$.
\endclaim

\subsection \S8.D. Vanishing theorem for non invariant $k$-jet differentials|

We prove here a more general vanishing theorem which strengthens 
Theorem~8.8 and Corollary~8.9. In this form, the result is due to
Siu and Yeung ([SiYe96a, SiYe97], [Siu97], cf.\ also  [Dem97] for a more
detailed account (in French)).

\claim 8.15. Fundamental vanishing theorem|
Let $(X,V)$ be a directed projective
variety and $f:(\bC,T_\bC)\to (X,V)$ an entire curve tangent to $V$.
Then for every global section $P\in H^0(X,\smash{E^\GG_{k,m}V^*}
\otimes \cO(-A))$ where $A$ is an ample divisor of $X$, one has 
\hbox{$P(f\,;\,f',f'',\ldots,f^{(k)})=0$}.
\endclaim

\proof. After raising $P$ to a power $P^s$ and replacing $\cO(-A)$ with $\cO(-sA)$, 
one can always assume that $A$ is 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~$A$.

Let us first give the proof of~(8.15) 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 $X$. 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.

The proof for the general case where $f'$ is unbounded is slightly more subtle (cf.\ [Siu87]),
and  makes use of Nevanlinna theory, especially the logarithmic derivative lemma. Assume
that $g=P(f'\ld f^{(k)})$ does not vanish identically. Fix a hermitian metric $h$
on $\cO(-A)$ such that $\omega:=\Theta_{\cO(A),h^{-1}}>0$ is a K\"ahler metric.  
The starting point is the inequality
$$
{i\over 2\pi}\ddbar\log\Vert g\Vert_h^2=
{i\over 2\pi}\ddbar\log\Vert P(f'\ld f^{(k)})\Vert_h^2\ge f^*\omega.
$$
In fact, as we are on $\bC$, the Lelong-Poincar\'e equation shows that the 
left hand side is equal to the right
hand side plus a certain linear combination of Dirac measures at points
where $P(f'\ld f^{(k)})$ vanishes. Let us consider the growth and
proximity functions
$$
\leqalignno{
T_{f,\omega}(r)&:=\int_{r_0}^r{d\rho\over\rho}\int_{D(0,\rho)}f^*\omega,&(8.16)\cr
m_{g}(r)&:={1\over 2\pi}\int_0^{2\pi}\log_+\Vert g(r\,e^{i \theta})\Vert_h^2\,d\theta.&(8.17)\cr}
$$
We get
$$
T_{f,\omega}(r)\le\int_{r_0}^r{d\rho\over\rho}\int_{D(0,\rho)}
{i\over 2\pi}\ddbar\log\Vert g\Vert_h^2=m_g(r)+\hbox{Const}\leqno(8.18)
$$
thanks to the Jensen formula. Now, consider a (finite) family of rational
functions $(u_j)$ on $X$ such that one can extract local coordinates
from local determinations of the logarithms $\log u_j$ at any point of $X$ 
(if $X$ is embedded in some projective space, it is sufficient to take 
rational functions of the form $u_j(z)=\ell_j(z)/\ell'_j(z)$ where $\ell_j,\ell'_j$
are linear forms; we also view the $u_j$'s as rational maps
$u_j:X\dasharrow\bP^1$). One can then express locally $\smash{P(f'\ld f^{(k)})}$ 
as a polynomial 
$Q$ in the logarithmic derivatives $D^p(\log u_j\circ f)$, with
holomorphic coefficients in~$f$, i.e.,
$$
g=P(f'\ld f^{(k)})=Q\big(f,D^p(\log u_j\circ f)_{p,j}\big),\qquad
Q(z,v_{p,j})=\sum a_\alpha(z)v^\alpha.
$$
By compactness of $X$, we infer
$$
m_g(r)={1\over 2\pi}\int_0^{2\pi}\log_+\Vert g(r\,e^{i\theta})\Vert_h^2\,d\theta
\le C_1\sum_{j,\,1\le p\le k} m_{D^p(\log u_j\circ f)}(r)+C_2\leqno(8.19)
$$
with suitable constants $C_1$, $C_2$. The logarithmic derivative lemma states that
for every meromorphic function $h:\bC\to\bP^1$ we have
$$
m_{D^p\log h}(r)\le \log r+(1+\varepsilon)\log_+T_{h,\omega_\FS}(r)+
O(1)\quad/\!/,
$$
where the notation $/\!/$ indicates as usual that the inequality holds true outside a
set of finite Lebesgue measure in $[0,+\infty[$. We apply this to $h=u_j\circ f$ and
use the standard fact that $T_{u_j\circ f,\omega_\FS}(r)\le C_j T_{f,\omega}(r)$.
We find in this way
$$
m_{D^p(\log u_j\circ f)}(r)\le C_3\big(\log r+\log_+ 
T_{f,\omega}(r)\big)\quad/\!/.\leqno(8.20)
$$ 
By putting (8.18--8.20) together, one obtains
$$
T_{f,\omega}(r)\le C\big(\log r+\log_+ T_{f,\omega}(r)\big)\quad/\!/.
$$
We infer from here that $T_{f,\omega}(r)=O(\log r)$, hence $f(\bC)$ has a finite
total area. By well known facts of Nevanlinna theory, we conclude that 
$C=\ol{f(\bC)}$ is a rational curve and that $f$ extends as
a rational map $\bP^1\to X$. In particular the derivative $f'$ is bounded, but
then the first case of the proof can be applied to conclude that
$g=P(f'\ld f^{(k)})\equiv 0$.\square

\subsection \S8.E. Bloch theorem|

The core of the result can be expressed as a characterization of the
Zariski closure of an entire curve drawn on a complex torus. The proof
is a simple consequence of the Ahlfors-Schwarz lemma (more specifically 
Theorem~8.8), combined with a jet bundle argument. We refer to
[Och], [GrG80] (also [Dem95]) for a detailed proof.

\claim 8.21.~Theorem|Let $Z$ be a complex torus and let $f:\bC\to Z$ be
a holomorphic map. Then the $($analytic$)$ Zariski closure
$\ol{f(\bC)}{}^\Zar$ is a translate of a subtorus, i.e.\ of the form
$a+Z'$, $a\in Z$, where $Z'\subset Z$ is a subtorus.
\endclaim

The converse is of course also true: for any subtorus $Z'\subset Z$, we 
can choose a dense line $L\subset Z'$, and the corresponding map
$f:\bC\simeq a+L\hookrightarrow Z$ has Zariski closure
$\ol{f(\bC)}{}^\Zar=a+Z'$.

\section{\S9. Morse inequalities and
the Green-Griffiths-Lang conjecture}

The goal of this section is to study the existence and properties of
entire curves $f:\bC\to X$ drawn in a complex irreducible
$n$-dimensional variety~$X$, and more specifically to show that they 
must satisfy certain global algebraic or differential equations 
as soon as $X$ is projective of general type. By means of
holomorphic Morse inequalities and a probabilistic analysis of
the cohomology of jet spaces, we are able to prove a significant step 
of a generalized version of the Green-Griffiths-Lang conjecture on the
algebraic degeneracy of entire curves. The use of holomorphic Morse inequalities 
was first suggested in [Dem07a], and then carried out in an algebraic context
by S.~Diverio in his PhD work ([Div08, Div09]). The general more analytic
and more powerful results presented here first appeared in [Dem11].
We refer to [Dem12] for a more detailed exposition.

\subsection \S9.A. Introduction|
Our main target is the following deep conjecture concerning the algebraic 
degeneracy of entire curves, which generalizes the similar absolute
statements given in \S$\,$4 (see also [GrGr79], [Lang86, Lang87]).

\claim 9.1. Generalized Green-Griffiths-Lang conjecture| Let $(X,V)$ be a 
projective directed manifold such that the canonical sheaf $K_V$ is big 
$($in the absolute case $V=T_X$, this means that $X$ is a variety of
general type, and in the relative case we will say that $(X,V)$ is of
general type$)$. Then there should exist an algebraic subvariety 
$Y\subsetneq X$ such that every non constant entire curve $f:\bC\to X$ 
tangent to $V$ is contained in~$Y$.
\endclaim

The precise meaning of $K_V$ and of its bigness will be explained
below \hbox{--}~our definition {\em does not coincide} with other
frequently used definitions and is in our view better suited to
the study of entire curves of $(X,V)$.  One says that $(X,V)$ is
Brody-hyperbolic when there are no entire curves tangent to $V$.
According to (generalized versions of) conjectures of Kobayashi
[Kob70, Kob76] the hyperbolicity of $(X,V)$ should imply that $K_V$ is
big, and even possibly ample, in a suitable sense. It~would then
follow from conjecture (9.1) that $(X,V)$ is hyperbolic if and only if
for every irreducible variety $Y\subset X$, the linear subspace
$$
V_{\wt Y}=\overline{\smash{T_{\wt Y\ssm E}}
\cap\mu_*^{-1}V}\subset T_{\wt Y}\leqno(9.2)
$$
has a big canonical sheaf whenever $\mu:\smash{\wt Y}\to Y$ is a desingularization and $E$
is the exceptional locus.

By defi\-nition, proving the algebraic degeneracy means finding
a non zero polynomial $P$ on $X$ such that all entire curves $f:\bC\to X$
satisfy $P(f)=0$. As already explained in \S$\,$8, all known 
methods of proof are based on establishing 
first the existence of certain algebraic differential equations 
$P(f\,;\,f',f'',\ldots,f^{(k)})=0$ of some order $k$, and then trying to find
enough such equations so that they cut out a proper algebraic
locus $Y\subsetneq X$. We use for this global sections of
$H^0(X,E^\GG_{k,m}V^*\otimes \cO(-A))$ where $A$ is ample, and
apply the fundamental vanishing theorem~8.15.
It is expected that the global sections of
$H^0(X,E^\GG_{k,m}V^*\otimes \cO(-A))$ are precisely those which
ultimately define the algebraic locus $Y\subsetneq X$ where the curve
$f$ should lie.  The problem is then reduced to (i) showing that there are 
many non zero sections of $H^0(X,E^\GG_{k,m}V^*\otimes \cO(-A))$ and
(ii) understanding what is their joint base locus. 
The first part of this program is the main result of this section.

\claim 9.3. Theorem| Let $(X,V)$ be a directed projective variety
such that $K_V$ is big and let $A$ be an ample divisor. Then 
for $k\gg 1$ and $\delta\in\bQ_+$ small enough, $\delta\le c(\log k)/k$,
the number of sections
$h^0(X,\smash{E^\GG_{k,m}V^*}\otimes\cO(-m\delta A))$ has maximal growth, 
i.e.\ is larger that $c_km^{n+kr-1}$ for some~$m\ge m_k$, 
where $c,\,c_k>0$, $n=\dim X$ and $r=\rank V$. In particular, entire curves
$f:(\bC,T_\bC)\to(X,V)$ satisfy $($many$)$ algebraic differential 
equations.
\endclaim

The statement is very elementary to check when $r=\rank V=1$, and 
therefore when $n=\dim X=1$. In higher dimensions  $n\ge 2$, only 
very partial results were known at this point, 
concerning merely the absolute case $V=T_X$. In dimension~$2$,
Theorem~9.3 is a consequence of the Riemann-Roch calculation of 
Green-Griffiths [GrGr79], 
combined with a vanishing theorem due to Bogomolov [Bog79] -- the latter 
actually only applies to the top cohomology group $H^n$, and things 
become much more delicate when extimates of intermediate cohomology
groups are needed. In higher dimensions, Diverio [Div08, Div09] proved the 
existence of sections of $H^0(X,\smash{E^\GG_{k,m}V^*}\otimes\cO(-1))$
whenever $X$ is a hypersurface of $\bP^{n+1}_\bC$ of high degree $d\ge
d_n$, assuming $k\ge n$ and $m\ge m_n$. More recently, Merker [Mer10]
was able to treat the case of arbitrary hypersurfaces of general type,
i.e.\ $d\ge n+3$, assuming this time $k$ to be very large.  The latter
result is obtained through explicit algebraic calculations of the
spaces of sections, and the proof is computationally very
intensive. B\'erczi [Ber10] also obtained related results with a
different approach based on residue formulas, assuming $d\ge 2^{7n\log n}$.

All these approaches are algebraic in nature. Here,
however, our techniques are based on more elaborate curvature estimates in 
the spirit of Cowen-Griffiths [CoGr76]. They require holomorphic Morse 
inequalities (see 9.10 below) -- and we
do not know how to translate our method in an algebraic setting.
Notice that holomorphic Morse inequalities are essentially
insensitive to singularities, as we can pass to non singular models
and blow-up $X$ as much as we want: if $\mu:\smash{\wt X}\to X$ is a
modification then $\mu_*\cO_{\wt X}=\cO_X$ and $R^q\mu_*\cO_{\wt X}$
is supported on a codimension $1$ analytic subset (even codimension $2$ if
$X$ is smooth). It~follows from the
Leray spectral sequence that the cohomology estimates for $L$ on $X$ or for
$\smash{\wt L}=\mu^*L$ on $\smash{\wt X}$ differ by negligible terms, i.e.
$$
h^q(\wt X,\wt L^{\otimes m})-h^q(X,L^{\otimes m})=O(m^{n-1}).\leqno(9.4)
$$
Finally, singular holomorphic Morse inequalities (in the form obatined by L.~Bonavero 
[Bon93]) allow us to work with singular Hermitian metrics $h$; this is the
reason why we will only require to have big line bundles rather than
ample line bundles. In the case of linear subspaces $V\subset T_X$, we
introduce singular Hermitian metrics as follows.

\claim 9.5. Definition| A singular Hermitian metric on a linear
subspace $V\subset T_X$ is a metric $h$ on the fibers of $V$ such that
the function $\log h:\xi\mapsto\log|\xi|_h^2$ is locally integrable 
on the total space of $V$.
\endclaim

Such a metric can also be viewed as a singular Hermitian metric on
the tauto\-logical line bundle $\cO_{P(V)}(-1)$ on the projectivized bundle
$P(V)=V\ssm\{0\}/\bC^*$, and therefore its dual metric $h^*$ defines a 
curvature current $\Theta_{\cO_{P(V)}(1),h^*}$ of type $(1,1)$ on
$P(V)\subset P(T_X)$, such that
$$
p^*\Theta_{\cO_{P(V)}(1),h^*} ={\ii\over 2\pi}\ddbar\log h,\qquad
\hbox{where $p:V\ssm\{0\}\to P(V)$}.
$$
If $\log h$ is quasi-plurisubharmonic $($or quasi-psh, which means psh 
modulo addition of a smooth function$)$ on $V$, then $\log h$ is indeed
locally integrable, and we have moreover
$$\Theta_{\cO_{P(V)}(1),h^*}\ge -C\omega\leqno(9.6)$$
for some smooth positive $(1,1)$-form on $P(V)$ and some constant $C>0\;$;
conversely, if (9.6) holds, then $\log h$ is quasi-psh.

\claim 9.7. Definition| We will say that a singular Hermitian metric $h$ on
$V$ is {\rm admissible} if $h$ can be written as $h=e^\varphi h_{0|V}$ where
$h_0$ is a smooth positive definite Hermitian on $T_X$ and $\varphi$
is a quasi-psh weight with analytic singularities on $X$, as in 
Definition~$9.5$. 
Then $h$ can be seen as a singular Hermitian metric on $\cO_{P(V)}(1)$,
with the property that it induces a smooth positive definite metric
on a Zariski open set $X'\subset X\ssm \Sing(V)\,;$ we will denote by
$\Sing(h)\supset\Sing(V)$ the complement of the largest such Zariski
open set $X'$.
\endclaim

If $h$ is an admissible metric, we define 
$\cO_h(V^*)$ to be the sheaf of germs of holomorphic sections 
sections of $\smash{V^*_{|X\ssm\Sing(h)}}$ which are $h^*$-bounded 
near $\Sing(h)$; 
by the assumption on the analytic singularities, 
this is a coherent sheaf (as the direct image of some coherent sheaf on
$P(V)$), and actually, since $h^*=e^{-\varphi}h_0^*$, it is 
a subsheaf of the
sheaf $\cO(V^*):=\cO_{h_0}(V^*)$ associated with a smooth positive
definite metric $h_0$  on $T_X$. If $r$ is the generic rank of $V$ and
$m$ a positive integer, we define similarly $K_{V,h}^m$ to be sheaf of 
germs of holomorphic sections of 
$(\det V^*_{|X'})^{\otimes m}=(\Lambda^rV^*_{|X'})^{\otimes m}$ which 
are $\det h^*$-bounded, and $K_V^m:=K_{V,h_0}^m$.

If $V$ is defined by $\alpha:X\dasharrow G_r(T_X)$, 
there always exists a modification $\mu:\smash{\wt X}\to X$ such that
the composition $\alpha\circ\mu:\smash{\wt X}\to G_r(\mu^* T_X)$
becomes holomorphic, and then $\mu^*V_{|\mu^{-1}(X\ssm \Sing(V))}$ 
extends as a locally trivial subbundle of $\mu^* T_X$ which we will
simply denote by $\mu^*V$. If $h$ is an admissible metric on $V$,
then $\mu^*V$ can be equipped with the
metric $\mu^*h=e^{\varphi\circ\mu}\mu^*h_0$ where $\mu^*h_0$
is smooth and positive definite. We may assume that $\varphi\circ\mu$
has divisorial singularities (otherwise just perform further blow-ups
of $\smash{\wt X}$ to achieve this). We then see that there is an integer 
$m_0$ such that for all multiples $m=pm_0$ the pull-back $\mu^*K_{V,h}^m$ 
is an invertible sheaf on $\smash{\wt X}$, and $\det h^*$ induces 
a smooth non singular metric on~it (when $h=h_0$, we can even take $m_0=1$).
By definition we always 
have $K_{V,h}^m=\mu_*(\mu^*K_{V,h}^m)$ for any $m\ge 0$.
In the sequel, however, we think of $K_{V,h}$ not really as a coherent 
sheaf, but rather as the ``virtual'' 
$\bQ$-line bundle $\mu_*\smash{(\mu^*K_{V,h}^{m_0})^{1/m_0}}$, and we say 
that $K_{V,h}$ is big
if $h^0(X, K_{V,h}^m)\ge cm^n$ for $m\ge m_1$, with $c>0\,$, i.e.\
if the invertible sheaf $\mu^*K_{V,h}^{m_0}$ is big in the usual sense.

At this point, it is important to observe that ``our'' canonical sheaf
$K_V$ differs from the sheaf $\cK_V:=i_*\cO(K_V)$ associated with
the injection $i:X\ssm \Sing(V)\hookrightarrow X$,
which is usually referred to as being the ``canonical sheaf'',
at least when $V$ is the space of tangents to a foliation. 
In fact, $\cK_V$ is always an invertible sheaf and there is
an obvious inclusion $K_V\subset\cK_V$. More precisely, the image of 
$\cO(\Lambda^rT_X^*)\to\cK_V$ is equal to $\cK_V\otimes_{\cO_X}\cJ$
for a certain coherent ideal $\cJ\subset\cO_X$, and the condition to 
have $h_0$-bounded sections on $X\ssm\Sing(V)$
precisely means that our sections are bounded by Const$\sum|g_j|$
in terms of the generators $(g_j)$ of $\cK_V\otimes_{\cO_X}\cJ$, i.e.\ 
$K_V=\cK_V\otimes_{\cO_X}\ol\cJ$ where $\ol\cJ$ is the integral
closure of~$\cJ$. More generally, 
$$
K_{V,h}^m=\cK_V^m\otimes_{\cO_X}\ol\cJ_{h,m_0}^{m/m_0}\leqno(9.8)
$$
where $\ol\cJ_{h,m_0}^{m/m_0}\subset\cO_X$ is the $(m/m_0)$-integral closure 
of a certain ideal sheaf \hbox{$\cJ_{h,m_0}\subset\cO_X$},
which can itself be assumed to be integrally closed; in our \hbox{previous}
discussion, $\mu$ is chosen so that $\mu^*\cJ_{h,m_0}$ is invertible
on $\smash{\wt X}$.

The discrepancy already occurs
e.g.\ with the rank~$1$ linear space $V\subset T_{\bP^n_\bC}$ consisting at each
point $z\ne 0$ of the tangent to the line $(0z)$ (so that
necessarily $V_0=T_{\bP^n_\bC,0}$). As a sheaf (and not as a linear space), 
$i_*\cO(V)$ is the invertible sheaf generated by the vector field 
$\xi=\sum z_j{\partial/\partial z_j}$ on the affine open set
$\bC^n\subset\bP^n_\bC$, and therefore $\cK_V:=i_*\cO(V^*)$ is generated
over $\bC^n$ by the unique $1$-form $u$ such that $u(\xi)=1$. 
Since $\xi$ vanishes at $0$, the generator $u$ is {\em unbounded} 
with respect to a smooth metric $h_0$
on  $T_{\bP^n_\bC}$, and it is easily seen that $K_V$ is the non 
invertible sheaf $K_V=\cK_V\otimes \gm_{\bP^n_\bC,0}$. We can make it invertible
by considering the blow-up $\mu:\smash{\wt X}\to X$ of $X=\bP^n_\bC$
at $0$, so that $\mu^*K_V$ is isomorphic to $\mu^*\cK_V\otimes\cO_{\wt X}(-E)$ 
where $E$ is the exceptional divisor. The integral curves $C$ of $V$ are of
course lines through~$0$, and when a standard parametrization is used,
their derivatives do not vanish at~$0$, while the sections of 
$i_*\cO(V)$ do -- another sign
that $i_*\cO(V)$ and $i_*\cO(V^*)$ are the {\em wrong objects} to consider. 
Another standard example is obtained by taking a generic pencil of
elliptic curves $\lambda P(z)+\mu Q(z)=0$ of degree $3$ in $\bP_\bC^2$,
and the linear space $V$ consisting of the tangents to the fibers
of the rational map $\bP_\bC^2\dasharrow\bP_\bC^1$
defined by $z\mapsto Q(z)/P(z)$. Then $V$ is given by
$$
0\lra i_*\cO(V)\longrightarrow\cO(T_{\bP_\bC^2})
\build\llra{8ex}|{PdQ-QdP}||\cO_{\bP^2_\bC}(6)\otimes\cJ_S\lra 0
$$
where $S=\Sing(V)$ consists of the 9 points $\{P(z)=0\}\cap\{Q(z)=0\}$, and
$\cJ_S$ is the corresponding ideal sheaf of~$S$. Since 
$\det\cO(T_{\bP^2})=\cO(3)$, we see that $\cK_V=\cO(3)$ is ample, which seems
to contradict 9.1 since all leaves are elliptic curves. There is however
no such contradiction, because $K_V=\cK_V\otimes\cJ_S$ is not big in our sense
(it has degree $0$ on all members of the elliptic pencil). A similar example
is obtained with a generic pencil of conics, in which case $\cK_V=\cO(1)$
and $\card S=4$.

For a given admissible Hermitian structure $(V,h)$, we define similarly
the sheaf $E^\GG_{k, m}V^*_h$ to be the sheaf of 
polynomials defined over $X\ssm\Sing(h)$ which are \hbox{``$h$-bounded''}.
This means that when they are viewed as polynomials $P(z\,;\,\xi_1,\ldots,
\xi_k)$ in terms of $\xi_j=(\nabla_{h_0}^{1,0})^jf(0)$ where
$\nabla_{h_0}^{1,0}$ is the $(1,0)$-component of the induced Chern 
connection on $(V,h_0)$, there is a uniform bound 
$$
\big|P(z\,;\,\xi_1,\ldots,\xi_k)\big|\le 
C\Big(\sum\Vert \xi_j\Vert_h^{1/j}\Big)^m\leqno(9.9)
$$
near points of $X\ssm X'$ (see section~2 for more details on
this). Again, by a direct image argument, one sees that
$\smash{E^\GG_{k, m}V^*_h}$ is always a coherent sheaf. The sheaf
$\smash{E^\GG_{k, m}V^*}$ is defined to be $\smash{E^\GG_{k, m}V^*_h}$ 
when $h=h_0$ (it is actually independent of the choice of $h_0$, as
follows from arguments similar to those given in section~2). Notice
that this is exactly what is needed to extend the proof of the vanishing
theorem~8.15 to the case of a singular linear space $V\,$; the value 
distribution theory argument can only work when the functions 
$P(f\,;\;f',\ldots,f^{(k)})(t)$ do not exhibit poles, and this is guaranteed
here by the boundedness assumption.

Our strategy can be described as follows. We consider the
Green-Griffiths bundle of $k$-jets $X_k^\GG=J^kV\ssm\{0\}/\bC^*$,
which by (9.3) consists of a fibration in {\em weighted projective
spaces}, and its associated tautological sheaf 
$$L=\cO_{X_k^\GG}(1),$$
viewed rather as a virtual $\bQ$-line bundle $\smash{\cO_{X_k^\GG}}
(m_0)^{1/m_0}$ with $m_0=\lcm(1,2,\,...\,,k)$.
Then, if $\pi_k:\smash{X_k^\GG}\to X$ is the natural projection, we have
$$
E^\GG_{k,m}=(\pi_k)_*\cO_{X_k^\GG}(m)\quad\hbox{and}\quad
R^q(\pi_k)_*\cO_{X_k^\GG}(m)=0~\hbox{for $q\ge 1$}.
$$
Hence, by the Leray spectral sequence we get for every invertible sheaf
$F$ on $X$ the isomorphism
$$
H^q(X,E^\GG_{k,m}V^*\otimes F)\simeq 
H^q(X_k^\GG,\cO_{X_k^\GG}(m)\otimes\pi_k^*F).
$$
The latter group can be evaluated thanks to holomorphic Morse inequalities.
Let us recall the main statement.

\claim 9.10. Holomorphic Morse inequalities {\rm ([Dem85])}|Let $X$
be a compact complex  manifolds, $E\to X$ a holomorphic vector bundle of
rank $r$, and $(L,h)$ a hermitian line bundle. The dimensions
$h^q(X,E\otimes L^k)$ of cohomology groups of the tensor powers 
$E\otimes L^k$ satisfy the following asymptotic estimates as $k\to +\infty\,:$

\noindent {\rm(WM)} Weak Morse inequalities$\,:$
$$h^q(X,E\otimes L^k)\le r {k^n\over n!}\int_{X(L,h,q)} (-1)^q \Theta_{L,h}^n + o(k^n)~.$$
\noindent {\rm(SM)} Strong Morse inequalities$\,:$
$$\sum_{0\le j\le q} (-1)^{q-j}h^j(X,E\otimes L^k) \le r {k^n\over n!}
\int_{X(L,h,\le q)}(-1)^q\Theta_{L,h}^n+o(k^n)~.$$
\noindent {\rm(RR)} Asymptotic Riemann-Roch formula$\,:$
$$\chi(X,E\otimes L^k) := \sum_{0\le j\le n} (-1)^jh^j(X,E\otimes L^k)
= r{k^n\over n!}\int_X \Theta_{L,h}^n + o(k^n)~.$$
\vskip-10pt
\endclaim

\noindent Moreover $($cf.\ Bonavero's PhD thesis {\rm [Bon93]}$)$, 
if $h=e^{-\varphi}$ is a singular hermitian metric with analytic 
singularities, the estimates are still true provided all cohomology groups
are replaced by cohomology groups 
$H^q(X,E\otimes L^k\otimes\cI(h^k))$ twisted with the multiplier
ideal sheaves 
$$
\cI(h^k)=\cI(k\varphi)=\big\{f\in\cO_{X,x},\;\;\exists V\ni x,~
\int_V|f(z)|^2e^{-k\varphi(z)}d\lambda(z)<+\infty\big\}.
$$
The special case of 9.10~(SM) when $q=1$ yields a very useful
criterion for the existence of sections of large multiples of $L$.
\vskip2mm

\vbox{\claim 9.11. Corollary|Under the above hypotheses, we have
$$
h^0(X,E\otimes L^k)\ge h^0(X,E\otimes L^k)-h^1(X,E\otimes L^k)\ge
 r{k^n\over n!}\int_{X(L,h,\le 1)}\Theta_{L,h}^n -o(k^n)~.$$
Especially $L$ is big as soon as $\int_{X(L,h,\le 1)}\Theta_{L,h}^n>0$
for some hermitian metric $h$ on~$L$.
\endclaim}

Now, given a directed manifold $(X,V)$, we can associate with any 
admissible metric $h$ on $V$ a metric
(or rather a natural family) of metrics on $L=\smash{\cO_{X_k^\GG}}(1)$. 
The space $\smash{X_k^\GG}$ always possesses quotient singularities if $k\ge 2$ 
(and even some more if $V$ is singular), but we do not really care since 
Morse inequalities still work in this setting thanks to Bonavero's
generalization. As we will see, it is then
possible to get nice asymptotic formulas as $k\to +\infty$. They appear to
be of a {\em probabilistic nature} if we take the components of the $k$-jet
(i.e.\ the successive derivatives $\xi_j=f^{(j)}(0)$, $1\le j\le k$) as 
random variables. This probabilistic behaviour was somehow  already
visible in the Riemann-Roch calculation of~[GrGr79].
In this way, assuming $K_V$ big, we produce a lot of sections
$\sigma_j=\smash{H^0(X_k^\GG,\cO_{X_k^\GG}}(m)\otimes\pi_k^*F)$, corresponding
to certain divisors $Z_j\subset \smash{X_k^\GG}$. The hard problem which
is left in order to complete a proof of the generalized
Green-Griffiths-Lang conjecture is to compute the base locus
$Z=\bigcap Z_j$ and to show that $Y=\pi_k(Z)\subset X$ must be a
proper algebraic variety. 

\subsection \S9.B. Hermitian geometry of weighted projective spaces|

The goal of this section is to introduce natural K\"ahler metrics on
weighted projective spaces, and to evaluate the corresponding volume forms.
Here we put $d^c={\ii\over 4\pi}(\dbar-\partial)$ so that
$dd^c={\ii\over 2\pi}\ddbar$. The normalization of the $d^c$ operator is
chosen such that we have precisely $(dd^c\log|z|^2)^n=\delta_0$ for the
Monge-Amp\`ere operator in $\bC^n$. Given a $k$-tuple
of ``weights'' $a=(a_1,\ldots,a_k)$, i.e.\ of integers $a_s>0$ with
\hbox{${\rm gcd}(a_1,\ldots,a_k)=1$}, we introduce the weighted projective
space $P(a_1,\ldots,a_k)$ to be the quotient of \hbox{$\bC^k\ssm\{0\}$} by
the corresponding weighted $\bC^*$ action:
$$
P(a_1,\ldots,a_k)=\bC^k\ssm\{0\}/\bC^*,\qquad
\lambda\cdot z =(\lambda^{a_1}z_1,\ldots,\lambda^{a_k}z_k).
\leqno(9.12)
$$
As is well known, this defines a toric $(k-1)$-dimensional algebraic variety 
with quotient singularities. On this variety, we introduce the possibly
singular (but almost everywhere smooth and non degenerate) K\"ahler form 
$\omega_{a,p}$ defined by
$$
\pi_a^*\omega_{a,p}=dd^c\varphi_{a,p},\qquad
\varphi_{a,p}(z)={1\over p}\log\sum_{1\le s\le k}|z_s|^{2p/a_s},
\leqno(9.13)
$$
where $\pi_a:\bC^k\ssm\{0\}\to P(a_1,\ldots,a_k)$ is the canonical projection
and $p>0$ is a positive constant. It is clear that $\varphi_{p,a}$ is real
analytic on $\bC^k\ssm\{0\}$ if $p$ is an integer and a common multiple of 
all weights $a_s$, and we will implicitly pick such a $p$ later on 
to avoid any difficulty. Elementary calculations  give the following
well-known formula for the volume
$$
\int_{P(a_1,\ldots,a_k)}\omega_{a,p}^{k-1}={1\over a_1\ldots a_k}\leqno(9.14)
$$
(notice that this is independent of $p$, as it is obvious 
by Stokes theorem, since the cohomology class of $\omega_{a,p}$ does 
not depend on $p$).

Our later calculations will require a slightly more general setting.
Instead of looking at $\bC^k$, we consider the weighted $\bC^*$ action
defined by
$$
\bC^{|r|}=\bC^{r_1}\times\ldots\times\bC^{r_k},\qquad
\lambda\cdot z =(\lambda^{a_1}z_1,\ldots,\lambda^{a_k}z_k).
\leqno(9.15)
$$
Here $z_s\in\bC^{r_s}$ for some $k$-tuple $r=(r_1,\ldots,r_k)$ and
$|r|=r_1+\ldots+r_k$. This gives rise to a weighted projective space 
$$
\leqalignno{
&P(a_1^{[r_1]},\ldots,a_k^{[r_k]})=P(a_1,\ldots,a_1,\ldots,a_k,\ldots,a_k),\cr
&\pi_{a,r}:\bC^{r_1}\times\ldots\times\bC^{r_k}
\ssm\{0\}\longrightarrow P(a_1^{[r_1]},\ldots,a_k^{[r_k]})&(9.16)\cr}
$$
obtained by repeating $r_s$ times each weight $a_s$. On this space, we
introduce the degenerate K\"ahler metric $\omega_{a,r,p}$ such that
$$
\pi_{a,r}^*\omega_{a,r,p}=dd^c\varphi_{a,r,p},\qquad
\varphi_{a,r,p}(z)={1\over p}\log\sum_{1\le s\le k}|z_s|^{2p/a_s}
\leqno(9.17)
$$
where $|z_s|$ stands now for the standard Hermitian norm 
$(\sum_{1\le j\le r_s}|z_{s,j}|^2)^{1/2}$ on~$\bC^{r_s}$. This metric
is cohomologous to the corresponding ``polydisc-like'' 
metric $\omega_{a,p}$ already defined, and therefore Stokes theorem implies
$$
\int_{P(a_1^{[r_1]},\ldots,a_k^{[r_k]})}
\omega_{a,r,p}^{|r|-1}={1\over a_1^{r_1}\ldots a_k^{r_k}}.
\leqno(9.18)
$$
Using standard results of integration theory (Fubini, change of variable 
formula...), one obtains:

\claim 9.19. Proposition| Let $f(z)$ be a bounded function on
$P(a_1^{[r_1]},\ldots,a_k^{[r_k]})$ which is continuous outside of
the hyperplane sections $z_s=0$. We also view $f$ as a $\bC^*$-invariant
continuous function on $\prod(\bC^{r_s}\ssm\{0\})$. Then
$$
\eqalign{
&\int_{P(a_1^{[r_1]},\ldots,a_k^{[r_k]})}f(z)\,\omega_{a,r,p}^{|r|-1}\cr
&={(|r|-1)!\over \prod_s a_s^{r_s}}\kern-2pt
\int_{(x,u)\in\Delta_{k-1}\times\prod S^{2r_s-1}}\kern-3pt
f(x_1^{a_1/2p}u_1,\ldots,x_k^{a_k/2p}u_k)\kern-4.5pt
\prod_{1\le s\le k}\kern-2.5pt
{x_s^{r_s-1}\over(r_s-1)!}\,dx\,d\mu(u)\kern-3pt
\cr}
$$
where $\Delta_{k-1}$ is the $(k-1)$-simplex $\{x_s\ge 0$, $\sum x_s=1\}$,
$dx=dx_1\wedge\ldots\wedge dx_{k-1}$ its standard measure, and where
$d\mu(u)=d\mu_1(u_1)\ldots d\mu_k(u_k)$ is the rotation invariant 
probability measure on the product $\prod_sS^{2r_s-1}$ of unit 
spheres in $\bC^{r_1}\times\ldots\times\bC^{r_k}$. As a consequence
$$
\lim_{p\to+\infty}
\int_{P(a_1^{[r_1]},\ldots,a_k^{[r_k]})}f(z)\,\omega_{a,r,p}^{|r|-1}=
{1\over \prod_s a_s^{r_s}}\int_{\prod S^{2r_s-1}}f(u)\,d\mu(u).
$$
\endclaim

Also, by elementary integrations by parts and induction on 
$k,\,r_1,\ldots,r_k$, it can be checked that
$$
\int_{x\in\Delta_{k-1}}
\prod_{1\le s\le k}x_s^{r_s-1}dx_1\ldots dx_{k-1}
={1\over (|r|-1)!}\prod_{1\le s\le k}(r_s-1)!~.
\leqno(9.20)
$$
This implies that $(|r|-1)!\prod_{1\le s\le k}
{x_s^{r_s-1}\over(r_s-1)!}\,dx$ is a probability measure on $\Delta_{k-1}$.

\subsection \S9.C. Probabilistic estimate of the curvature of $k$-jet bundles|

Let $(X,V)$ be a compact complex directed non singular variety. To
avoid any technical difficulty at this point, we first assume that $V$
is a holomorphic vector subbundle of $T_X$, equipped with a smooth
Hermitian metric $h$.

According to the notation already specified in \S$\,$7, we
denote by $J^kV$ the bundle of $k$-jets of holomorphic curves
$f:(\bC,0)\to X$ tangent to $V$ at each point. Let us set $n=\dim_\bC
X$ and $r=\rank_\bC V$. Then $J^kV\to X$ is an algebraic fiber bundle
with typical fiber $\bC^{rk}$, and we get a projectivized $k$-jet bundle 
$$
X^\GG_k:=(J^kV\ssm\{0\})/\bC^*,\qquad \pi_k:\smash{X^\GG_k}\to X
\leqno(9.21)
$$
which is a $P(1^{[r]},2^{[r]},\ldots,k^{[r]})$ weighted projective 
bundle over $X$, and we have the direct image formula
$(\pi_k)_*\smash{\cO_{X^\GG_k}(m)}=\cO(E_{k,m}^\GG V^*)$
(cf.\ Proposition~7.9). In the sequel, we do not make a direct 
use of coordinates, 
because they need not be related in any way to the Hermitian 
metric $h$ of $V$.  Instead, we choose a local holomorphic coordinate frame
$(e_\alpha(z))_{1\le\alpha\le r}$ of $V$ on a neighborhood $U$ of~$x_0$, 
such that
$$
\langle e_\alpha(z),e_\beta(z)\rangle =\delta_{\alpha\beta}+
\sum_{1\le i,j\le n,\,1\le\alpha,\beta\le r}c_{ij\alpha\beta}z_i\overline z_j+
O(|z|^3)\leqno(9.22)
$$
for suitable complex coefficients $(c_{ij\alpha\beta})$. It is a standard fact
that such a normalized coordinate system always exists, and that the 
Chern curvature tensor ${\ii\over 2\pi}D^2_{V,h}$ of $(V,h)$ at $x_0$ 
is then given by
$$
\Theta_{V,h}(x_0)=-{\ii\over 2\pi}
\sum_{i,j,\alpha,\beta}
c_{ij\alpha\beta}\,dz_i\wedge d\overline z_j\otimes e_\alpha^*\otimes e_\beta.
\leqno(9.23)
$$
Consider a local holomorphic connection $\nabla$ on $V_{|U}$ (e.g.\ the 
one which turns $(e_\alpha)$ into a parallel frame), and take 
$\xi_k=\nabla^kf(0)\in V_x$ defined inductively 
by $\nabla^1 f=f'$ and $\nabla^sf=\nabla_{f'}(\nabla^{s-1}f)$. This
gives a local identification
$$
J_kV_{|U}\to V_{|U}^{\oplus k},\qquad
f\mapsto(\xi_1,\ldots,\xi_k)=(\nabla f(0),\ldots,\nabla f^k(0)),
$$
and the weighted $\bC^*$ action on $J_kV$ is expressed in this setting by
$$
\lambda\cdot(\xi_1,\xi_2,\ldots,\xi_k)=(\lambda\xi_1,
\lambda^2\xi_2,\ldots,\lambda^k\xi_k).
$$
Now, we fix a finite open covering 
$(U_\alpha)_{\alpha\in I}$ of~$X$ by open coordinate charts such that
$V_{|U_\alpha}$ is trivial, along with holomorphic connections 
$\nabla_\alpha$ on $V_{|U_\alpha}$. Let $\theta_\alpha$ be a partition of
unity of $X$ subordinate to the covering $(U_\alpha)$. Let us fix 
$p>0$ and small parameters $1=\varepsilon_1\gg\varepsilon_2\gg\ldots\gg
\varepsilon_k>0$. Then we define a global 
weighted Finsler metric on $J^kV$ by putting for any $k$-jet $f\in J^k_xV$
$$
\Psi_{h,p,\varepsilon}(f):=\Big(\sum_{\alpha\in I}
\theta_\alpha(x)\sum_{1\le s\le k}\varepsilon_s^{2p}\Vert\nabla^s_\alpha f(0)
\Vert_{h(x)}^{2p/s}\Big)^{1/p}
\leqno(9.24)
$$
where $\Vert~~\Vert_{h(x)}$ is the Hermitian metric $h$ of $V$ evaluated
on the fiber $V_x$, $x=f(0)$. The function $\Psi_{h,p,\varepsilon}$ satisfies
the fundamental homogeneity property 
$$
\Psi_{h,p,\varepsilon}(\lambda\cdot f)=\Psi_{h,p,\varepsilon}(f)\,|\lambda|^2
\leqno(9.25)
$$
with respect to the $\bC^*$ action on $J^kV$, in other words, it induces
a Hermitian metric on the dual $L^*$ of the tautological $\bQ$-line bundle
$L_k=\cO_{X_k^\GG}(1)$ over $X_k^\GG$. The curvature of $L_k$ is given by
$$
\pi_k^*\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}=dd^c\log\Psi_{h,p,\varepsilon}
\leqno(9.26)
$$
Our next goal is to compute precisely the curvature and to apply
holomorphic Morse inequalities to $L\to X_k^\GG$ with the above metric.
It might look a priori like an untractable problem, since the definition of
$\Psi_{h,p,\varepsilon}$ is a rather unnatural one. However, the ``miracle''
is that the asymptotic behavior of $\Psi_{h,p,\varepsilon}$ as
$\varepsilon_s/\varepsilon_{s-1}\to 0$ is in some sense uniquely defined 
and very natural.
It will lead to a computable asymptotic formula, which is moreover
simple enough to produce useful results.

\claim 9.27. Lemma| On each coordinate chart $U$ equipped with
a holomorphic connection $\nabla$ of $V_{|U}$, let us define 
the components of a $k$-jet $f\in J^kV$ by $\xi_s=\nabla^sf(0)$,
and consider the rescaling transformation 
$$\rho_{\nabla,\varepsilon}(\xi_1,\xi_2,\ldots,\xi_k)=
(\varepsilon_1^1\xi_1,\varepsilon_2^2\xi_2,\ldots,
\varepsilon_k^k\xi_k)\quad
\hbox{on $J^k_xV$, $x\in U$}
$$
$($it commutes with the $\bC^*$-action but is otherwise unrelated and 
not canonically defined over $X$ as it depends on the choice of 
$\nabla)$. Then, if $p$ is a multiple of $\lcm(1,2,\ldots,k)$ and
$\varepsilon_s/\varepsilon_{s-1}\to 0$ for all $s=2,\ldots,k$, the
rescaled function $\Psi_{h,p,\varepsilon}\circ\rho_{\nabla,\varepsilon}^{-1}
(\xi_1,\ldots,\xi_k)$ converges towards
$$
\bigg(\sum_{1\le s\le k}\Vert \xi_s\Vert^{2p/s}_h\bigg)^{1/p}
$$
on every compact subset of $J^kV_{|U}\ssm\{0\}$,
uniformly in $C^\infty$ topology.
\endclaim

\proof. Let $U\subset X$ be an open set on which $V_{|U}$ is trivial
and equipped with some holomorphic connection $\nabla$. Let us pick
another holomorphic connection $\wt\nabla=
\nabla+\Gamma$ where $\Gamma\in H^0(U,\Omega^1_X\otimes
\Hom(V,V)$. Then $\wt\nabla^2f=\nabla^2f+\Gamma(f)(f')\cdot f'$, and
inductively we get
$$
\wt\nabla^sf=\nabla^sf+P_s(f\,;\,\nabla^1f,\ldots,\nabla^{s-1}f)
$$
where $P(x\,;\,\xi_1,\ldots,\xi_{s-1})$ is a polynomial with holomorphic 
coefficients in $x\in U$ which is of weighted homogeneous degree
$s$ in $(\xi_1,\ldots,\xi_{s-1})$. In other words, the corresponding change
in the parametrization of $J^kV_{|U}$ is given by a $\bC^*$-homogeneous
transformation
$$
\wt\xi_s=\xi_s+P_s(x\,;\,\xi_1,\ldots,\xi_{s-1}).
$$
Let us introduce the corresponding rescaled components
$$
(\xi_{1,\varepsilon},\ldots,\xi_{k,\varepsilon})=
(\varepsilon_1^1\xi_1,\ldots,\varepsilon_k^k\xi_k),\qquad
(\wt\xi_{1,\varepsilon},\ldots,\wt\xi_{k,\varepsilon})=
(\varepsilon_1^1\wt\xi_1,\ldots,\varepsilon_k^k\wt\xi_k).
$$
Then
$$
\eqalign{
\wt\xi_{s,\varepsilon}
&=\xi_{s,\varepsilon}+
\varepsilon_s^s\,P_s(x\,;\,\varepsilon_1^{-1}\xi_{1,\varepsilon},\ldots,
\varepsilon_{s-1}^{-(s-1)}\xi_{s-1,\varepsilon})\cr
&=\xi_{s,\varepsilon}+O(\varepsilon_s/\varepsilon_{s-1})^s\,
O(\Vert\xi_{1,\varepsilon}\Vert+\ldots+\Vert\xi_{s-1,\varepsilon}
\Vert^{1/(s-1)})^s\cr}
$$
and the error terms are thus polynomials of fixed degree with arbitrarily
small coefficients as $\varepsilon_s/\varepsilon_{s-1}\to 0$. Now, the 
definition of $\Psi_{h,p,\varepsilon}$ consists of glueing the sums
$$
\sum_{1\le s\le k}\varepsilon_s^{2p}\Vert\xi_k\Vert_h^{2p/s}=
\sum_{1\le s\le k}\Vert\xi_{k,\varepsilon}\Vert_h^{2p/s}
$$
corresponding to $\xi_k=\nabla_\alpha^sf(0)$ by means of the partition
of unity $\sum\theta_\alpha(x)=1$. We see that by using the rescaled
variables $\xi_{s,\varepsilon}$ the changes occurring when replacing a
connection $\nabla_\alpha$ by an alternative one $\nabla_\beta$ are
arbitrary small in $C^\infty$ topology, with error terms uniformly
controlled in terms of the ratios $\varepsilon_s/\varepsilon_{s-1}$ on
all compact subsets of $V^k\ssm\{0\}$. This shows that in $C^\infty$
topology,
$\Psi_{h,p,\varepsilon}\circ\rho_{\nabla,\varepsilon}^{-1}(\xi_1,\ldots,\xi_k)$
converges uniformly towards $\smash{(\sum_{1\le s\le
    k}\Vert\xi_k\Vert_h^{2p/s})^{1/p}}$, whatever the trivializing
open set $U$ and the holomorphic connection $\nabla$ used to evaluate
the components and perform the rescaling are.\square

Now, we fix a point $x_0\in X$ and a local holomorphic frame 
$(e_\alpha(z))_{1\le\alpha\le r}$ satisfying (9.22) on a neighborhood $U$ 
of~$x_0$. We introduce the rescaled components 
$\xi_s=\varepsilon_s^s\nabla^sf(0)$ on $J^kV_{|U}$ and compute
the curvature of
$$
\Psi_{h,p,\varepsilon}\circ\rho_{\nabla,\varepsilon}^{-1}(z\,;\,\xi_1,\ldots,\xi_k)
\simeq\bigg(\sum_{1\le s\le k}\Vert\xi_s\Vert^{2p/s}_h\bigg)^{1/p}
$$
(by Lemma 9.27, the errors can be taken arbitrary small in 
$C^\infty$ topology). We write $\xi_s=\sum_{1\le\alpha\le r}\xi_{s\alpha}
e_\alpha$. By (9.22) we have
$$
\Vert \xi_s\Vert_h^2=
\sum_\alpha|\xi_{s\alpha}|^2+
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}z_i\overline z_j\xi_{s\alpha}
\overline \xi_{s\beta}+O(|z|^3|\xi|^2).
$$
The question is to evaluate the curvature of the weighted metric defined by
$$
\eqalign{
\Psi(z\,;\,\xi_1,\ldots,\xi_k)
&=\bigg(\sum_{1\le s\le k}\Vert\xi_s\Vert^{2p/s}_h\bigg)^{1/p}\cr
&=\bigg(\sum_{1\le s\le k}\Big(\sum_\alpha|\xi_{s\alpha}|^2+\!\!
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}z_i\overline z_j\xi_{s\alpha}
\overline\xi_{s\beta}
\Big)^{p/s}\bigg)^{1/p}\kern-4pt{}+O(|z|^3).\cr}
$$
We set $|\xi_s|^2=\sum_\alpha|\xi_{s\alpha}|^2$. A straightforward 
calculation yields
$$
\eqalign{
&\log\Psi(z\,;\,\xi_1,\ldots,\xi_k)=\cr
&~~{}={1\over p}\log\sum_{1\le s\le k}|\xi_s|^{2p/s}+
\sum_{1\le s\le k}{1\over s}{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}}
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}z_i\overline z_j
{\xi_{s\alpha}\overline\xi_{s\beta}\over|\xi_s|^2}+O(|z|^3).\cr}
$$
By (9.26), the curvature form of $L_k=\cO_{X_k^\GG}(1)$ 
is given at the central point $x_0$ by the following formula.

\claim 9.28. Proposition| With the above choice of coordinates and with
respect to the rescaled components $\xi_s=\varepsilon_s^s\nabla^sf(0)$ at 
$x_0\in X$, we have the approximate expression
$$
\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}(x_0,[\xi])\simeq
\omega_{a,r,p}(\xi)+{\ii\over 2\pi}
\sum_{1\le s\le k}{1\over s}{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}}
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}
{\xi_{s\alpha}\overline\xi_{s\beta}\over|\xi_s|^2}\,dz_i\wedge d\overline z_j
$$
where the error terms  are
$O(\max_{2\le s\le k}(\varepsilon_s/\varepsilon_{s-1})^s)$ uniformly on
the compact variety $X_k^\GG$. Here $\omega_{a,r,p}$ is the $($degenerate$)$
K\"ahler metric associated with the weight $a=(1^{[r]},2^{[r]},\ldots,k^{[r]})$ 
of the canonical $\bC^*$ action on $J^kV$.
\endclaim

Thanks to the uniform approximation, we can (and will) neglect the error 
terms in the calculations below. Since $\omega_{a,r,p}$ is positive definite
on the fibers of $X_k^\GG\to X$ (at least outside of the axes $\xi_s=0$), 
the index of the $(1,1)$
curvature form $\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}(z,[\xi])$ is equal
to the index of the $(1,1)$-form
$$
\gamma_k(z,\xi):={\ii\over 2\pi}
\sum_{1\le s\le k}{1\over s}{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}}
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}(z)
{\xi_{s\alpha}\overline\xi_{s\beta}\over|\xi_s|^2}\,dz_i\wedge d\overline z_j
\leqno(9.29)
$$
depending only on the differentials $(dz_j)_{1\le j\le n}$ on~$X$. The 
$q$-index integral of $(L_k,\Psi^*_{h,p,\varepsilon})$ on $X^\GG_k$ is 
therefore equal to
$$
\eqalign{
&\int_{X^\GG_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}=\cr
&\qquad{}={(n+kr-1)!\over n!(kr-1)!}
\int_{z\in X}\int_{\xi\in P(1^{[r]},\ldots,k^{[r]})}
\omega_{a,r,p}^{kr-1}(\xi)\bOne_{\gamma_k,q}(z,\xi)\gamma_k(z,\xi)^n\cr}
$$
where $\bOne_{\gamma_k,q}(z,\xi)$ is the characteristic function of the open
set of points where $\gamma_k(z,\xi)$ has signature $(n-q,q)$ in terms of
the $dz_j$'s. Notice that since $\gamma_k(z,\xi)^n$ is~a determinant, the
product $\bOne_{\gamma_k,q}(z,\xi)\gamma_k(z,\xi)^n$ gives rise to a continuous
function on~$X^\GG_k$. Formula 9.20 with $r_1=\ldots=r_k=r$ and
$a_s=s$ yields the slightly more explicit
integral
$$
\eqalign{
&\int_{X^\GG_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}=
{(n+kr-1)!\over n!(k!)^r}~~\times\cr
&\qquad\int_{z\in X}\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
\bOne_{g_k,q}(z,x,u)g_k(z,x,u)^n\,
{(x_1\ldots x_k)^{r-1}\over (r-1)!^k}\,dx\,d\mu(u),\cr}
$$
where $g_k(z,x,u)=\gamma_k(z,x_1^{1/2p}u_1,\ldots,x_k^{k/2p}u_k)$ is given by
$$
g_k(z,x,u)={\ii\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j
\leqno(9.30)
$$
and $\bOne_{g_k,q}(z,x,u)$ is the characteristic function of its $q$-index 
set. Here 
$$
d\nu_{k,r}(x)=(kr-1)!\,{(x_1\ldots x_k)^{r-1}\over (r-1)!^k}\,dx
\leqno(9.31)
$$
is a probability measure on $\Delta_{k-1}$, and we can rewrite
$$
\leqalignno{
&\int_{X^\GG_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}
={(n+kr-1)!\over n!(k!)^r(kr-1)!}~~\times\cr
&\qquad\int_{z\in X}
\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
\bOne_{g_k,q}(z,x,u)g_k(z,x,u)^n\,d\nu_{k,r}(x)\,d\mu(u).&(9.32)\cr}
$$
Now, formula (9.30) shows that $g_k(z,x,u)$ is a ``Monte Carlo''
evaluation of the curvature tensor, obtained by averaging the curvature
at random points $u_s\in S^{2r-1}$ with certain positive weights $x_s/s\,$; 
we should then think of the \hbox{$k$-jet}
$f$ as some sort of random variable such that the derivatives 
$\nabla^kf(0)$ are uniformly distributed in all directions. Let us compute
the expected value of
$(x,u)\mapsto g_k(z,x,u)$ with respect to the probability measure
$d\nu_{k,r}(x)\,d\mu(u)$. Since 
$\int_{S^{2r-1}}u_{s\alpha}\overline u_{s\beta}d\mu(u_s)={1\over r}
\delta_{\alpha\beta}$ and $\int_{\Delta_{k-1}}x_s\,d\nu_{k,r}(x)={1\over k}$,
we find
$$
{\bf E}(g_k(z,\bu,\bu))={1\over kr}
\sum_{1\le s\le k}{1\over s}\cdot{\ii\over 2\pi}\sum_{i,j,\alpha}
c_{ij\alpha\alpha}(z)\,dz_i\wedge d\overline z_j.
$$
In other words, we get the normalized trace of the curvature, i.e.
$$
{\bf E}(g_k(z,\bu,\bu))={1\over kr}
\Big(1+{1\over 2}+\ldots+{1\over k}\Big)\Theta_{\det(V^*),\det h^*},
\leqno(9.33)
$$
where $\Theta_{\det(V^*),\det h^*}$ is the $(1,1)$-curvature form of
$\det(V^*)$ with the metric induced by~$h$. It is natural to guess that 
$g_k(z,x,u)$ behaves asymptotically as its expected value
${\bf E}(g_k(z,\bu,\bu))$ when $k$ tends to infinity. If we replace brutally 
$g_k$ by its expected value in (9.32), we get the integral
$$
{(n+kr-1)!\over n!(k!)^r(kr-1)!}{1\over (kr)^n}
\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^n\int_X\bOne_{\eta,q}\eta^n,
$$
where $\eta:=\Theta_{\det(V^*),\det h^*}$ and $\bOne_{\eta,q}$ is the
characteristic function of its $q$-index set in~$X$. The leading constant is
equivalent to $(\log k)^n/n!(k!)^r$ modulo 
a multiplicative factor $1+O(1/\log k)$. By working out a more precise analysis
of the deviation, the following result has been proved in [Dem11] and 
[Dem12].

\claim 9.34. Probabilistic estimate|
Fix smooth Hermitian metrics $h$ on $V$ and
$\omega={\ii\over 2\pi} \sum\omega_{ij}dz_i\wedge d\overline z_j$ on $X$. 
Denote by $\Theta_{V,h}=-{\ii\over 2\pi}\sum
c_{ij\alpha\beta}dz_i\wedge d\overline z_j\otimes e_\alpha^*\otimes
e_\beta$ the curvature tensor of $V$ with respect to an $h$-orthonormal frame
$(e_\alpha)$, and put
$$
\eta(z)=\Theta_{\det(V^*),\det h^*}={\ii\over 2\pi}\sum_{1\le i,j\le n}\eta_{ij}
dz_i\wedge d\overline z_j,\qquad
\eta_{ij}=\sum_{1\le\alpha\le r}c_{ij\alpha\alpha}.
$$
Finally consider the $k$-jet line bundle $L_k=\smash{\cO_{X_k^\GG}(1)}\to
X_k^\GG$ equipped with the induced metric $\Psi^*_{h,p,\varepsilon}$
$($as defined above, with $1=\varepsilon_1\gg\varepsilon_2\gg\ldots\gg
\varepsilon_k>0)$. When $k$ tends 
to infinity, the integral of the top power of the curvature of $L_k$ on its
$q$-index set $X^\GG_k(L_k,q)$ is given by
$$
\int_{X^\GG_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}=
{(\log k)^n\over n!\,(k!)^r}\bigg(
\int_X\bOne_{\eta,q}\eta^n+O((\log k)^{-1})\bigg)
$$
for all $q=0,1,\ldots,n$, and the error term $O((\log k)^{-1})$ can be 
bounded explicitly in terms of $\Theta_V$, $\eta$ and $\omega$. Moreover, the 
left hand side is identically zero for $q>n$.
\endclaim

The final statement follows from the observation that the curvature of
$L_k$ is positive along the fibers of $X_k^\GG\to X$, by the 
plurisubharmonicity of the weight (this is true even 
when the partition of unity terms are taken into account, since they
depend only on the base); therefore the $q$-index sets are empty for
$q>n$. It will be useful to extend the above estimates to the 
case of sections of
$$
L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big({1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)
\leqno(9.35)
$$
where $F\in\Pic_\bQ(X)$ is an arbitrary $\bQ$-line bundle on~$X$ and 
$\pi_k:X_k^\GG\to X$ is the natural projection. We assume here
that $F$ is also equipped with a smooth Hermitian metric $h_F$. In formulas
(9.32--9.34), the renormalized curvature $\eta_k(z,x,u)$ of $L_k$ takes 
the form
$$
\eta_k(z,x,u)={1\over {1\over kr}(1+{1\over 2}+\ldots+{1\over k})}g_k(z,x,u)+
\Theta_{F,h_F}(z),
\leqno(9.36)
$$
and by the same calculations its  expected value is
$$
\eta(z):={\bf E}(\eta_k(z,\bu,\bu))=\Theta_{\det V^*,\det h^*}(z)+
\Theta_{F,h_F}(z).
\leqno(9.37)
$$
Then the variance estimate for $\eta_k-\eta$ is unchanged, and the
$L^p$ bounds for $\eta_k$ are still valid, since our forms are just shifted
by adding the constant smooth term $\Theta_{F,h_F}(z)$. The probabilistic
estimate 9.34 is therefore still true in exactly the same form, provided
we use (9.35 -- 9.37) instead of the previously defined $L_k$, $\eta_k$
and~$\eta$. An application of holomorphic Morse inequalities gives the 
desired cohomology estimates for 
$$
\eqalign{
h^q\Big(X,E_{k,m}^\GG V^*&{}\otimes
\cO\Big({m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big)\cr
&{}=h^q(X_k^\GG,\cO_{X_k^\GG}(m)\otimes
\pi_k^*\cO\Big({m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big),
\cr}
$$
provided $m$ is sufficiently divisible to give a multiple of $F$ which
is a $\bZ$-line bundle.

\claim 9.38. Theorem| Let $(X,V)$ be a directed manifold, $F\to X$ a
$\bQ$-line bundle, $(V,h)$ and $(F,h_F)$ smooth Hermitian structure on $V$ 
and $F$ respectively. We define
$$
\eqalign{
L_k&=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big({1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big),\cr
\eta&=\Theta_{\det V^*,\det h^*}+\Theta_{F,h_F}.\cr}
$$
Then for all $q\ge 0$ and all $m\gg k\gg 1$ such that 
m is sufficiently divisible, we have
$$\leqalignno{\kern20pt
h^q(X_k^\GG,\cO(L_k^{\otimes m}))&\le {m^{n+kr-1}\over (n+kr-1)!}
{(\log k)^n\over n!\,(k!)^r}\bigg(
\int_{X(\eta,q)}(-1)^q\eta^n+O((\log k)^{-1})\bigg),&\hbox{\rm(a)}\cr
h^0(X_k^\GG,\cO(L_k^{\otimes m}))&\ge {m^{n+kr-1}\over (n+kr-1)!}
{(\log k)^n\over n!\,(k!)^r}\bigg(
\int_{X(\eta,\le 1)}\eta^n-O((\log k)^{-1})\bigg),&\hbox{\rm(b)}\cr
\cr
\chi(X_k^\GG,\cO(L_k^{\otimes m}))&={m^{n+kr-1}\over (n+kr-1)!}
{(\log k)^n\over n!\,(k!)^r}\big(
c_1(V^*\otimes F)^n+O((\log k)^{-1})\big).&\hbox{\rm(c)}\cr
\cr}
$$
\vskip-4pt
\endclaim

Green and Griffiths [GrGr79] already checked the Riemann-Roch
calculation (9.38$\,$c) in the special case
$V=T_X^*$ and $F=\cO_X$. Their proof is much simpler since it relies only
on Chern class calculations, but it cannot provide any information on
the individual cohomology groups, except in very special cases where
vanishing theorems can be applied; in fact in dimension 2, the
Euler characteristic satisfies $\chi=h^0-h^1+h^2\le h^0+h^2$, hence
it is enough to get the vanishing of the top cohomology group $H^2$
to infer $h^0\ge\chi\,$; this works for surfaces by means of a well-known
vanishing theorem of Bogomolov which implies in general
$$H^n\bigg(X,E_{k,m}^\GG T_X^*\otimes
\cO\Big({m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big)\bigg)=0
$$
as soon as $K_X\otimes F$ is big and $m\gg 1$.

In fact, thanks to Bonavero's singular holomorphic Morse inequalities 
[Bon93], everything works almost unchanged 
in the case where $V\subset T_X$
has singularities and $h$ is an admissible metric on $V$ (see Definition~9.7).
We only have to find a blow-up $\mu:\smash{\wt X}_k\to X_k$ so that
the resulting pull-backs $\mu^*L_k$ and $\mu^*V$ are locally free,
and $\mu^*\det h^*$, $\mu^*\Psi_{h,p,\varepsilon}$ only have divisorial
singularities. Then $\eta$ is a $(1,1)$-current with logarithmic poles,
and we have to deal with smooth metrics on $
\mu^*L_k^{\otimes m}\otimes\cO(-mE_k)$ where $E_k$ is a certain effective 
divisor on $X_k$ (which, by our assumption in 9.7, does not project onto
$X$). The cohomology groups involved are then the twisted
cohomology groups
$$
H^q(X_k^\GG,\cO(L_k^{\otimes m})\otimes\cJ_{k,m})
$$
where $\cJ_{k,m}=\mu_*(\cO(-mE_k))$ is the corresponding multiplier ideal sheaf,
and the Morse integrals need only be evaluated in the complement of the 
poles, that is on $X(\eta,q)\ssm S$ where $S=\Sing(V)\cup\Sing(h)$. Since
$$
(\pi_k)_*\big(\cO(L_k^{\otimes m})\otimes\cJ_{k,m}\big)\subset
E_{k,m}^\GG V^*\otimes
\cO\Big({m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big)
$$
we still get a lower bound for the $H^0$ of the latter sheaf (or for the $H^0$
of the un-twisted line bundle $\cO(L_k^{\otimes m})$ on $\smash{X_k^\GG}$).
If we assume that $K_V\otimes F$ is big, these considerations
also allow us to obtain a strong estimate in terms of the volume, by
using an approximate Zariski decomposition on a suitable blow-up of~$(X,V)$.
The following corollary implies in particular Theorem~9.3.

\claim 9.39. Corollary|
If $F$ is an arbitrary $\bQ$-line bundle over~$X$, one has
$$
\eqalign{
h^0\bigg(&X_k^\GG,\cO_{X_k^\GG}(m)\otimes\pi_k^*\cO
\Big({m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\bigg)\cr
&\ge {m^{n+kr-1}\over (n+kr-1)!}
{(\log k)^n\over n!\,(k!)^r}\Big(
\Vol(K_V\otimes F)-O((\log k)^{-1})\Big)-o(m^{n+kr-1}),\cr}
$$
when $m\gg k\gg 1$, in particular there are many sections of the
$k$-jet differentials of degree $m$ twisted by the appropriate
power of $F$ if $K_V\otimes F$ is big.
\endclaim

\proof. The volume is computed here as usual, i.e.\ after performing a
suitable modifi\-cation $\mu:\smash{\wt X}\to X$ which converts $K_V$ into 
an invertible sheaf. There is of course nothing to prove if
$K_V\otimes F$ is not big, so we can assume $\Vol(K_V\otimes F)>0$.
Let us fix smooth Hermitian metrics $h_0$ on $T_X$
and $h_F$ on $F$. They induce a metric $\mu^*(\det h_0^{-1}\otimes
h_F)$ on $\mu^*(K_V\otimes F)$ which, by our definition of $K_V$, is
a smooth metric. By the result of Fujita [Fuj94] on
approximate Zariski decomposition, for every $\delta>0$, one can find
a modification $\mu_\delta:\smash{\wt X_\delta}\to X$ dominating
$\mu$ such that
$$
\mu_\delta^*(K_V\otimes F) =\cO_{\wt X_\delta}(A+E)
$$
where $A$ and $E$ are $\bQ$-divisors, $A$ ample and $E$ effective,
with 
$$\Vol(A)=A^n\ge \Vol(K_V\otimes F)-\delta.$$
If we take a smooth metric $h_A$ with positive definite curvature form
$\Theta_{A,h_A}$, then we get a singular Hermitian metric $h_Ah_E$ on
$\mu_\delta^*(K_V\otimes F)$ with poles along $E$, i.e.\ the quotient
$h_Ah_E/\mu^*(\det h_0^{-1}\otimes h_F)$ is of the form $e^{-\varphi}$ where
$\varphi$ is quasi-psh with log poles $\log|\sigma_E|^2$ 
(mod $C^\infty(\smash{\wt X_\delta}))$ precisely given
by the divisor~$E$. We then only need to take the singular metric $h$
on $T_X$ defined by
$$
h=h_0e^{{1\over r}(\mu_\delta)^*\varphi}
$$
(the choice of the factor ${1\over r}$ is there to correct adequately 
the metric on $\det V$). By construction $h$ induces an 
admissible metric on $V$ and the resulting 
curvature current $\eta=\Theta_{K_V,\det h^*}+\Theta_{F,h_F}$ is such that
$$
\mu_\delta^*\eta = \Theta_{A,h_A} +[E],\qquad
\hbox{$[E]={}$current of integration on $E$.}
$$
Then the $0$-index Morse integral in the complement of the poles 
is given by
$$
\int_{X(\eta,0)\ssm S}\eta^n=\int_{\wt X_\delta}\Theta_{A,h_A}^n=A^n\ge
\Vol(K_V\otimes F)-\delta
$$
and (9.39) follows from the fact that $\delta$ can be taken arbitrary 
small.\square

\claim 9.40. Example| {\rm In some simple cases, the above estimates can 
lead to very explicit results. Take for instance $X$ to be a smooth
complete intersection of multidegree $(d_1,d_2,\ldots,d_s)$ in $\bP^{n+s}_\bC$
and consider the absolute case $V=T_X$. Then 
$K_X=\cO_X(d_1+\ldots+d_s-n-s-1)$ and one can check via
explicit bounds of the error terms (cf.\ [Dem11], [Dem12])
that a sufficient condition for the existence of
sections is
$$
k\ge\exp\Big(7.38\,n^{n+1/2}\Big({\sum d_j+1\over\sum d_j-n-s-a-1}\Big)^n\Big).
$$
This is good in view of the fact that we can cover arbitrary smooth 
complete intersections of general type. On the other hand, even when the
degrees $d_j$ tend to $+\infty$, we still get a large lower bound
$k\sim \exp(7.38\,n^{n+1/2})$ on the order of jets, and this is far 
from being optimal$\,$: Diverio [Div08, Div09] has shown e.g.\ that one can take
$k=n$ for smooth hypersurfaces of high degree, using the 
algebraic Morse inequalities of Trapani [Tra95].
The next paragraph uses essentially the same idea, in our more analytic
setting.\square}
\endclaim

\subsection \S9.D. Non probabilistic estimate of the Morse integrals|

We assume here that the curvature tensor $(c_{ij\alpha\beta})$ satisfies a lower
bound
$$
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}\xi_i\ol\xi_ju_\alpha\ol u_\beta\ge -\sum\gamma_{ij}\xi_i
\ol\xi_j\;|u|^2,
\qquad\forall\xi\in T_X,~u\in V\leqno(9.41)
$$
for some semipositive $(1,1)$-form $\gamma={i\over 2\pi}\sum\gamma_{ij}(z)\,
dz_i\wedge d\ol z_j$ on~$X$. This is the
same as assuming that the curvature tensor of $(V^*,h^*)$ satisfies
the semipositivity condition
$$
\Theta_{V^*,h^*}+\gamma\otimes\Id_{V^*}\ge 0\leqno(9.41')
$$
in the sense of Griffiths, or equivalently $\Theta_{V,h}-\gamma\otimes\Id_V\le 0$.
Thanks to the compactness of~$X$, such a form $\gamma$ always
exists if $h$ is an admissible metric on~$V$. Now, instead of replacing $\Theta_V$
with its trace free part $\wt\Theta_V$ and exploiting a Monte Carlo convergence
process, we replace $\Theta_V$ with $\Theta_V^\gamma=\Theta_V-\gamma\otimes\Id_V\le 0$, 
i.e.\ $c_{ij\alpha\beta}$ by 
$c_{ij\alpha\beta}^\gamma=c_{ij\alpha\beta}+\gamma_{ij}\delta_{\alpha\beta}$. Also, we take
a line bundle $F=A^{-1}$ with $\Theta_{A,h_A}\ge 0$, i.e.\ $F$ seminegative.
Then our earlier formulas (9.28), (9.35), (9.36) become instead
$$
\leqalignno{
&g_k^\gamma(z,x,u)={\ii\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}^\gamma(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j\ge 0,
&(9.42)\cr
&L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big(-{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)A\Big),
&(9.43)\cr
&\Theta_{L_k}=\eta_k(z,x,u)={1\over {1\over kr}(1+{1\over 2}+\ldots+{1\over k})}
g_k^\gamma(z,x,u)-(\Theta_{A,h_A}(z)+r\gamma(z)).&(9.44)\cr}
$$
In fact, replacing $\Theta_V$ by $\Theta_V-\gamma\otimes\Id_V$ has the effect of
replacing $\Theta_{\det V^*}=\Tr\Theta_{V^*}$ by $\Theta_{\det V^*}+r\gamma$. The major
gain that we have is that $\eta_k=\Theta_{L_k}$ is now expressed as a difference
of semipositive $(1,1)$-forms, and we can exploit the following simple lemma, which is
the key to derive algebraic Morse inequalities from their analytic form
(cf.\ [Dem94], Theorem~12.3).

\claim 9.45.~Lemma|Let $\eta=\alpha-\beta$ be a difference of semipositive 
$(1,1)$-forms on an $n$-dimensional complex manifold~$X$, 
and let $\bOne_{\eta,\le q}$ be the characteristic function of the
open set where $\eta$ is non degenerate with a number of negative eigenvalues 
at most equal to~$q$.
Then
$$
(-1)^q\bOne_{\eta,\le q}~\eta^n\le \sum_{0\le j\le q}(-1)^{q-j}\alpha^{n-j}\beta^j,
$$
in particular
$$
\bOne_{\eta,\le 1}~\eta^n\ge \alpha^n-n\alpha^{n-1}\wedge \beta\qquad\hbox{for $q=1$.}
$$
\endclaim

\proof. Without loss of generality, we can assume $\alpha>0$ positive definite, so that
$\alpha$ can be taken as the base hermitian metric on~$X$. Let us denote by
$$
\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_n\ge 0
$$
the eigenvalues of $\beta$ with respect to $\alpha$. The eigenvalues of $\eta=\alpha-\beta$
are then given by 
$$
1-\lambda_1\le\ldots\le 1-\lambda_q\le 1-\lambda_{q+1}\le\ldots\le 1-\lambda_n,
$$
hence the open set $\{\lambda_{q+1}<1\}$ coincides with the support of 
$\bOne_{\eta,\le q}$, except that it may also contain a part of 
the degeneration set $\eta^n=0$. On the other hand we have
$${n\choose j}\alpha^{n-j}\wedge\beta^j=\sigma_n^j(\lambda)\,\alpha^n,$$
where $\sigma_n^j(\lambda)$ is the $j$-th elementary symmetric function in the $\lambda_j$'s.
Thus, to prove the lemma, we only have to check that
$$\sum_{0\le j\le q}(-1)^{q-j}\sigma_n^j(\lambda)-
\bOne_{\{\lambda_{q+1}<1\}}(-1)^q\prod_{1\le j\le n}(1-\lambda_j)\ge 0.$$
This is easily done by induction on~$n$ (just split apart the parameter
$\lambda_n$ and write $\sigma_n^j(\lambda)=
\sigma_{n-1}^j(\lambda)+\sigma_{n-1}^{j-1}(\lambda)\,\lambda_n$).\square
\medskip

We apply here Lemma 9.45 with
$$
\alpha=g_k^\gamma(z,x,u),\qquad\beta=\beta_k=
{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)(\Theta_{A,h_A}+r\gamma),
$$
which are both semipositive by our assumption. The analogue of (9.32) leads to
$$
\eqalign{
&\int_{X^\GG_k(L_k,\le 1)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}\cr
&\quad{}={(n+kr-1)!\over n!(k!)^r(kr-1)!}\int_{z\in X}
\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
\bOne_{g_k^\gamma-\beta_k,\le 1}\;(g_k^\gamma-\beta_k)^n\,d\nu_{k,r}(x)\,d\mu(u)\cr
&\quad{}\ge{(n+kr-1)!\over n!(k!)^r(kr-1)!}\int_{z\in X}
\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
((g_k^\gamma)^n-n(g_k^\gamma)^{n-1}\wedge\beta_k)\,d\nu_{k,r}(x)\,d\mu(u).\cr}
$$
The resulting integral now produces a ``closed formula'' which can be
expressed solely in terms of Chern classes (at least if we assume that
$\gamma$ is the Chern form of some semipositive line bundle). It is
just a matter of routine to find a sufficient condition for the
positivity of the integral. One can first observe that $g_k^\gamma$ is
bounded from above by taking the trace of $(c_{ij\alpha\beta})$, in
this way we get
$$
0\le g_k^\gamma\le\bigg(\sum_{1\le s\le k}{x_s\over s}\bigg)\big(\Theta_{\det V^*}+r\gamma\big)
$$
where the right hand side no longer depends on $u\in (S^{2r-1})^k$. 
Also, $g_k^\gamma$ can be written as a sum of semipositive $(1,1)$-forms
$$
g_k^\gamma =\sum_{1\le s\le k}{x_s\over s}\theta^\gamma(u_s),\qquad
\theta^\gamma(u)=\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}^\gamma
u_\alpha\ol u_\beta\,dz_i\wedge d\ol z_j,
$$
hence for $k\ge n$ we have
$$
(g_k^\gamma)^n\ge n!\sum_{1\le s_1<\ldots<s_n\le k}
{x_{s_1}\ldots x_{s_n}\over s_1\ldots s_n}\,
\theta^\gamma(u_{s_1})\wedge \theta^\gamma(u_{s_2})\wedge\ldots\wedge \theta^\gamma(u_{s_n}).
$$
Since $\int_{S^{2r-1}}\theta^\gamma(u)\,d\mu(u)={1\over r}\Tr(\Theta_{V^*}+\gamma)=
{1\over r}\Theta_{\det V^*}+\gamma$, we infer from this
$$
\eqalign{
&\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
(g_k^\gamma)^n\,d\nu_{k,r}(x)\,d\mu(u)\cr
&\qquad\ge 
n!\sum_{1\le s_1<\ldots<s_n\le k}
{1\over s_1\ldots s_n}\Big(\int_{\Delta_{k-1}}x_1\ldots x_n\,d\nu_{k,r}(x)\Big)
\Big({1\over r}\Theta_{\det V^*}+\gamma\Big)^n.\cr}
$$
By putting everything together, we conclude:

\claim 9.46.~Theorem|Assume that $\Theta_{V^*}\ge-\gamma\otimes\Id_{V^*}$ with a
semipositive $(1,1)$-form $\gamma$ on $X$. Then the Morse integral of the line bundle
$$
L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big(-{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)A\Big),\qquad
A\ge 0
$$
satisfies for $k\ge n$ the inequality
$$
\leqalignno{
&{1\over (n+kr-1)!}\int_{X^\GG_k(L_k,\le 1)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}\cr
&\qquad{}\ge{1\over n!(k!)^r(kr-1)!}
\int_Xc_{n,r,k}\big(\Theta_{\det V^*}+r\gamma\big)^n-c'_{n,r,k}
\big(\Theta_{\det V^*}+r\gamma\big)^{n-1}\wedge
\big(\Theta_{A,h_A}+r\gamma\big)&(*)\cr}
$$
where
$$
\eqalign{
c_{n,r,k}&={n!\over r^n}\Big(\sum_{1\le s_1<\ldots<s_n\le k}
{1\over s_1\ldots s_n}\Big)\int_{\Delta_{k-1}}x_1\ldots x_n\,d\nu_{k,r}(x),\cr
c'_{n,r,k}&={n\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)
\int_{\Delta_{k-1}}\Big(\sum_{1\le s\le k}{x_s\over s}\Big)^{n-1}\,d\nu_{k,r}(x).
\cr}
$$
Especially we have a lot of sections in $H^0(X_k^\GG,mL_k)$, $m\gg 1$, as soon
as the difference occurring in $(*)$ is positive.
\endclaim

The statement is also true for $k<n$, but then $c_{n,r,k}=0$ and the lower bound $(*)$ cannot
be positive. By Corollary 9.11, it still provides a non trivial lower bound for
$h^0(X_k^\GG,mL_k)-h^1(X_k^\GG,mL_k)$, though. For $k\ge n$ we have $c_{n,r,k}>0$ and
$(*)$ will be positive if $\Theta_{\det V^*}$ is large enough. By Formula~9.20 we have
$$
c_{n,r,k}={n!\,(kr-1)!\over (n+kr-1)!}
\smash{\sum_{1\le s_1<\ldots<s_n\le k}{1\over s_1\ldots s_n}\ge
{(kr-1)!\over (n+kr-1)!}},\leqno(9.47)
$$
(with equality for $k=n$), and by ([Dem11], Lemma 2.20~(b)) we get
the upper bound 
$$
{c'_{n,k,r}\over c_{n,k,r}}\le {(kr+n-1)r^{n-2}\over k/n}
\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^n\smash{\bigg[
1+{1\over 3}\sum_{m=2}^{n-1}{2^m(n-1)!\over(n-1-m)!}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^{-m}\bigg]}.
$$
The case $k=n$ is especially interesting. For $k=n\ge 2$ one can show
(with $r\le n$ and $H_n$ denoting the harmonic sequence) that
$$
{c'_{n,k,r}\over c_{n,k,r}}\le {n^2+n-1\over 3}n^{n-2}\exp\Big({2(n-1)\over H_n}+n\log H_n\Big)
\le {1\over 3} \big(n\log(n\log 24n)\big)^n.
\leqno(9.48)
$$
We will later need the particular values that can be obtained by
direct calculations (cf.\ Formula (9.20 and [Dem11, Lemma 2.20]).
$$
\leqalignno{
\noalign{\vskip-3pt}
&c_{2,2,2}={1\over 20},\kern4.8pt\qquad c'_{2,2,2}={9\over 16},\kern9.8pt\qquad
{c'_{2,2,2}\over c_{2,2,2}}={45\over 4},&(9.49_2)\cr
&c_{3,3,3}={1\over 990},\qquad c'_{3,3,3}={451\over 4860},\qquad
{c'_{3,3,3}\over c_{3,3,3}}={4961\over 54}.&(9.49_3)\cr
\noalign{\vskip-5pt}}
$$

\section{\S10. Hyperbolicity properties of  hypersurfaces of high degree}

\subsection \S10.A. Global generation of the twisted tangent space of the universal family|

In [Siu02, Siu04], Y.T.~ Siu developed a new stategy to produce jet
differentials, involving meromorphic vector fields on the total space of
jet bundles -- these vector fields are used to differentiate the
sections of $E_{k,m}^\GG$ so as to produce new ones with less zeroes. The approach
works especially well on universal families of hypersurfaces in
projective space, thanks to the good positivity properties of the
relative tangent bundle, as shown by L.~Ein [Ein88, Ein91] and C.~Voisin [Voi96].
This allows at least to prove the hyperbolicity of generic surfaces
and generic 3-dimensional hypersurfaces of sufficiently high
degree. We reproduce here the improved approach given by [Pau08] for the
twisted global generation of the tangent space of the space of
vertical two jets. The situation of $k$-jets in arbitrary dimension $n$ is 
substantially more involved, details can be found in [Mer09].

Consider the universal hypersurface $\cX\subset\bP^{n+1}\times\bP^{N_d}$ of degree $d$ 
given by the equation
$$
\sum_{|\alpha|=d}A_{\alpha}\,Z^\alpha=0,
$$
where $[Z]\in\bP^{n+1}$, $[A]\in\bP^{N_d}$, $\alpha=(\alpha_0,\dots,\alpha_{n+1})\in\bN^{n+2}$
and
$$
N_d={n+d+1\choose d}-1.
$$
Finally, we denote by $\cV\subset\cX$ the vertical tangent space, i.e.\ the kernel of the
projection
$$
\pi:\cX\to U\subset\bP^{N_d}
$$
where $U$ is the Zariski open set parametrizing smooth hypersurfaces, and by $J_k\cV$ the 
bundle of $k$-jets of curves tangent to $\cV$, i.e.\ curves contained in the fibers
$X_s=\pi^{-1}(s)$. The goal is to describe certain meromorphic vector fields on the total space
of $J_k\cV$. In the special case $n=2$, $k=2$ considered by P\u{a}un [Pau08], 
one fixes the affine open set 
$$\cU_0=\{Z_0\ne 0\}\times\{A_{0d00}\ne 0\}\simeq\bC^{3}\times\bC^{N_d}$$
in $\bP^{3}\times\bP^{N_d}$ with the corresponding inhomogeneous coordinates $(z_j=Z_j/Z_0)_{j=1,2,3}$ and $(a_{\alpha}=A_\alpha/A_{0d00})_{|\alpha|=d,\alpha_1<d}$. Since $\alpha_0$ is determined by $\alpha_0=d-(\alpha_1+\alpha_2+\alpha_3)$, with a slight abuse of notation in the sequel, $\alpha$ will be seen as a multiindex $(\alpha_1,\alpha_2,\alpha_3)$ in $\bN^3$, with moreover the convention that $a_{d00}=1$. On this affine open set we have
$$
\cX_0:=\cX\cap\cU_0=\bigg\{z_1^d+\sum_{|\alpha|\le d,\alpha_1<d}a_{\alpha}\,z^\alpha=0\bigg\}.
$$
We now write down equations for the open variety $J_2\cV_0$, where we indicated with $\cV_0$ the restriction of $\cV\subset T_{\cX}$, the kernel of the differential of the second projection, to $\cX_0$: elements in $J_2\cV_0$ are therefore $2$-jets of germs of  \lq\lq vertical\rq\rq{} holomorphic curves in $\cX_0$, that is curves tangent to vertical fibers. The equations, which live in a natural way in $\bC^3_{z_j}\times\bC^{N_d}_{a_\alpha}\times\bC^3_{z_j'}\times\bC^3_{z_j''}$,
stand as follows.
$$
\eqalign{
&\sum_{|\alpha|\le d}a_{\alpha}\,z^{\alpha}=0,\cr
&\sum_{1\le j\le 3}~\sum_{|\alpha|\le d}a_{\alpha}\,{\partial z^\alpha\over\partial z_j}\,z_j'=0,\cr
&\sum_{1\le j\le 3}~\sum_{|\alpha|\le d}a_{\alpha}\,{\partial z^\alpha\over\partial z_j}\,z_j''
+\sum_{1\le j,k\le 3}~\sum_{|\alpha|\le d}a_{\alpha}\,{\partial^2 z^\alpha\over\partial z_j\partial z_k}\,z_j'z_k'=0.\cr}
$$
Let $\cW_0$ to be the closed algebraic subvariety of $J_2\cV_0$ defined by
$$
\cW_0=\{(z,a,z',z'')\in J_2\cV_0\mid z'\wedge z''=0\}
$$
and let $\cW$ be the Zariski closure of $\cW_0$ in $J_2\cV$: we call this set the {\em Wronskian locus} of $J_2\cV$. Explicit calculations (cf.\ [Pau08]) then
produce the following vector fields:

\noindent
{\em First family of tangent vector fields.} For any multiindex $\alpha$ such that $\alpha_1\ge 3$, consider the vector field
$$
\theta_{\alpha}^{300}={\partial\over\partial a_{\alpha}}-3z_1\,{\partial\over\partial a_{\alpha-\delta_1}}+3z_1^2\,{\partial\over\partial a_{\alpha-2\delta_1}}-z_1^3\,{\partial\over\partial a_{\alpha-3\delta_1}},
$$
where $\delta_j\in\bN^4$ is the multiindex whose $j$-th component is equal to $1$ and the others are zero. For the multiindexes $\alpha$ which verify $\alpha_1\ge 2$ and $\alpha_2\ge 1$, define
$$
\eqalign{
\theta_{\alpha}^{210}&={\partial\over\partial a_{\alpha}}-2z_1\,{\partial\over\partial a_{\alpha-\delta_1}}-z_2\,{\partial\over\partial a_{\alpha-\delta_2}}+z_1^2\,{\partial\over\partial a_{\alpha-2\delta_1}} \cr
&\qquad+2z_1z_2\,{\partial\over\partial a_{\alpha-\delta_1-\delta_2}}-z_1^2z_2\,{\partial\over\partial a_{\alpha-2\delta_1-\delta_2}}.\cr}
$$
Finally, for those $\alpha$ for which $\alpha_1,\alpha_2,\alpha_3\ge 1$, set
$$
\eqalign{
\theta_{\alpha}^{111}&={\partial\over\partial a_{\alpha}}-z_1\,{\partial\over\partial a_{\alpha-\delta_1}}-z_2\,{\partial\over\partial a_{\alpha-\delta_2}}-z_3\,{\partial\over\partial a_{\alpha-\delta_3}} \cr
&\qquad+z_1z_2\,{\partial\over\partial a_{\alpha-\delta_1-\delta_2}}+z_1z_3\,{\partial\over\partial a_{\alpha-\delta_1-\delta_3}}+z_2z_3\,{\partial\over\partial a_{\alpha-\delta_2-\delta_3}} \cr
&\qquad\quad -z_1z_2z_3{\partial\over\partial a_{\alpha-\delta_1-\delta_2-\delta_3}}.\cr}
$$
\noindent{\em Second family of tangent vector fields.}
We construct here the holomorphic vector fields in order to span the $\partial/\partial z_j$-directions. For $j=1,2,3$, consider the vector field
$$
{\partial\over\partial z_j}-\sum_{|\alpha+\delta_j|\le d}(\alpha_j+1)a_{\alpha+\delta_j}\,{\partial\over\partial a_{\alpha}}.
$$
\noindent{\em Third family of tangent vector fields.}
In order to span the jet directions, consider a vector field of the following form:
$$
\theta_B=\sum_{|\alpha|\le d,\alpha_1<d}p_{\alpha}(z,a,b)\,{\partial\over\partial a_{\alpha}}+\sum_{1\le j\le 3}~\sum_{k=1}^2\xi_j^{(k)}\,{\partial\over\partial z_j^{(k)}},
$$
where $\xi^{(k)}=B\cdot z^{(k)}$, $k=1,2$, and $B=(b_{jk})$ varies among $3\times 3$ invertible matrices with complex entries. By studying more carefully these three families of vector fields, one obtains:

\claim 10.1.~Theorem|The twisted tangent space 
$T_{J_2\cV}\otimes\cO_{\bP^3}(7)\otimes\cO_{\bP^{N_d}}(1)$
is generated over by its global sections over the complement $J_2\cV\ssm\cW$ of the Wronskian locus~$\cW$. Moreover, one can choose generating global sections that are invariant with respect to the action of $\bG_2$ on~$J_2\cV$.
\endclaim

By similar, but more computationally intensive arguments [Mer09], one
can investigate the higher dimensional case. The following result
strengthens the initial announcement of~[Siu04].

\claim 10.2.~Theorem|Let $J_k^{\rm vert}(\cX)$ be the space of vertical $k$-jets of
the universal hypersurface
$$
\cX\subset\bP^{n+1}\times\bP^{N_d}
$$
parametrizing all projective hypersurfaces $X\subset\bP^{n+1}$ of degree $d$.
Then for $k=n$, there exist constants $c_n$ and $c'_n$ such that the twisted tangent bundle
$$
T_{J^{\rm vert}_k(\cX)}\otimes\cO_{\bP^{n+1}}(c_n)\otimes\cO_{\bP^{N_d}}(c'_n)
$$
is generated by its global $\bG_k$-invariant sections outside a certain exceptional 
algebraic subset
$\Sigma\subset J^{\rm vert}_k(\cX)$. One can take either $c_n={1\over 2}(n^2+5n)$, $c'_n=1$ and
$\Sigma$ defined by the vanishing of certain Wronskians, or $c_n=n^2+2n$ and a
smaller set $\wt\Sigma\subset\Sigma$ defined by the vanishing of the $1$-jet part.
\endclaim

\subsection 10.B. General strategy of proof|

Let again $\cX\subset\bP^{n+1}\times\bP^{N_d}$ be the universal hypersurface of degree $d$
in $\bP^{n+1}$.

\noindent
(10.3) {\em Assume that we can prove the existence of a non zero polynomial differential operator
$$
P\in H^0(\cX,E_{k,m}^\GG T^*_\cX\otimes\cO(-A)),
$$
where $A$ is an ample divisor on $\cX$, at least over some Zariski open set $U$
in the base of the projection $\pi:\cX\to U\subset\bP^{N_d}$.}

Observe that we now have a lot of techniques to do this; the existence of $P$ over 
the family follows from lower semicontinuity in the Zariski topology, once we know 
that such a section $P$ exists on a generic fiber $X_s=\pi^{-1}(s)$. 
Let $\cY\subset\cX$ be the set of points $x\in\cX$ where $P(x)=0$, as an element in
the fiber of the vector bundle $E_{k,m}^\GG T^*_\cX\otimes\cO(-A))$ at $x$. Then
$\cY$ is a proper algebraic subset of $\cX$, and after shrinking $U$ we may assume
that $Y_s=\cY\cap X_s$ is a proper algebraic subset of $X_s$ for every~$s\in U$.

\noindent
(10.4) {\em Assume also, according to Theorems $10.1$ and $10.2$, that we have enough global 
holomorphic $\bG_k$-invariant vector fields $\theta_i$ on 
$J_k\cV$ with values in the pull-back of some ample divisor $B$ on $\cX$,
in such a way that they generate  $T_{J_k\cV}\otimes p^*_kB$ over 
the dense open set $(J_k\cV)^\reg$ of regular $k$-jets, i.e.\ 
$k$-jets with non zero first derivative $($here $p_k:J_k\cV\to \cX$ 
is the natural projection\/$)$.}

Considering jet differentials $P$ as functions on $J_k\cV$, the idea is 
to produce new ones by taking differentiations
$$
Q_j:=\theta_{j_1}\ldots\theta_{j_\ell}P,\qquad 0\le \ell\le m,~
j=(j_1,\ldots,j_\ell).
$$
Since the $\theta_j$'s are $\bG_k$-invariant, they are in particular
$\bC^*$-invariant, thus
$$
Q_j\in H^0(\cX,E_{k,m}^\GG T^*_\cX\otimes\cO(-A+\ell B))
$$
(and $Q$ is in fact $\bG_k'$ invariant as soon as $P$ is). In order to
be able to apply the vanishing theorems of \S$\,$8, we need $A-mB$ to
be ample, so $A$ has to be large compared to $B$. If $f:\bC\to X_s$ is
an entire curve contained in some fiber $X_s\subset\cX$, its lifting
$j_k(f):\bC\to J_k\cV$ has to lie in the zero divisors of all sections
$Q_j$.  However, every non zero polynomial of degree $m$ has at any
point some non zero derivative of order $\ell\le m$. Therefore, at any
point where the $\theta_i$ generate the tangent space to $J_k\cV$, we
can find some non vanishing section~$Q_j$.  By the assumptions on the
$\theta_i$, the base locus of the $Q_j$'s is contained in the union of
$p_k^{-1}(\cY)\cup (J_k\cV)^\sing$; there is of course no way of
getting a non zero polynomial at points of $\cY$ where $P$
vanishes. Finally, we observe that $j_k(f)(\bC)\not\subset
(J_k\cV^\sing$ (otherwise $f$ is constant).  Therefore
$j_k(f)(\bC)\subset p_k^{-1}(\cY)$ and thus $f(\bC)\subset\cY$, i.e.
$f(\bC)\subset Y_s=\cY\cap X_s$.

\claim 10.5.~Corollary|Let $\cX\subset\bP^{n+1}\times\bP^{N_d}$ be the
universal hypersurface of degree $d$ in $\bP^{n+1}$. If $d\ge d_n$ is
taken so large that conditions $(10.3)$ and $(10.4)$ are met with
$A-mB$ ample, then the generic fiber $X_s$ of the universal family
$\cX\to U$ satisfies the Green-Griffiths conjecture, namely all entire
curves $f:\bC\to X_s$ are contained in a proper algebraic subvariety
$Y_s\subset X_s$, and the $Y_s$ can be taken to form an algebraic
subset $\cY\subset\cX$.
\endclaim

This is unfortunately not enough to get the hyperbolicity of $X_s$,
because we would have to know that $Y_s$ itself is
hyperbolic. However, one can use the following simple observation due
to Diverio and Trapani [DT10]. The starting point is the following
general, straightforward remark. Let $\cE\to \cX$ be a holomorphic
vector bundle let $\sigma\in H^0(\cX,\cE)\ne 0$; then, up to
factorizing by an effective divisor $D$ contained in the common zeroes
of the components of $\sigma$, one can view $\sigma$ as a section
$$
\sigma\in H^0(\cX,\cE\otimes\cO_\cX(-D)),
$$
and this section now has a zero locus without divisorial
components. Here, when $n\ge 2$, the very generic fiber $X_s$ has
Picard number one by the Noether-Lefschetz theorem, and so, after
shrinking $U$ if necessary, we can assume that $\cO_\cX(-D)$ is the
restriction of $\cO_{\bP^{n+1}}(-p)$, $p\ge 0$ by the effectivity of
$D$. Hence $D$ can be assumed to be nef. After performing this
simplification, $A-mB$ is replaced by $A-mB+D$, which is still ample
if $A-mB$ is ample.  As a consequence, we may assume $\codim\cY\ge 2$,
and after shrinking $U$ again, that all $Y_s$ have $\codim Y_s\ge 2$.

\claim 10.6.~Additional statement|In corollary $10.5$, under the same
hypotheses $(10.3)$ and $(10.4)$, one can take all fibers $Y_s$ to
have $\codim Y_s\ge 2$.
\endclaim

This is enough to conclude that $X_s$ is hyperbolic if $n=\dim X_s\le
3$. In fact, this is clear if $n=2$ since the $Y_s$ are then reduced
to points. If $n=3$, the $Y_s$ are at most curves, but we know by Ein
and Voisin that a generic hypersurface $X_s\subset\bP^4$ of degree
$d\ge 7$ does not possess any rational or elliptic curve.  Hence $Y_s$
is hyperbolic and so is $X_s$, for $s$ generic.\square

\claim 10.7. Corollary|Assume that $n=2$ or $n=3$, and that
$\cX\subset\bP^{n+1}\times\bP^{N_d}$ is the universal hypersurface of
degree $d\ge d_n\ge 2n+1$ so large that conditions $(10.3)$ and
$(10.4)$ are met with $A-mB$ ample. Then the very generic hypersurface
$X_s\subset\bP^{n+1}$ of degree $d$ is hyperbolic.
\endclaim

\subsection \S10.C. Proof of the Green-griffiths conjecture for
generic hypersurfaces in $\bP^{n+1}$|

The most striking progress made at this date on the Green-Griffiths conjecture
itself is a recent result of Diverio, Merker and Rousseau [DMR10],
confirming the statement when $X\subset\bP^{n+1}_\bC$ is a generic 
hypersurface of large degree~$d$, with a (non optimal) sufficient
lower bound $d\ge\smash{2^{n^5}}$. Their proof is based in an essential
way on Siu's strategy as developed in~\S$\,$10.B, combined with the
earlier techniques of [Dem95]. Using our improved bounds from~\S$\,$9.D,
we obtain here a better estimate (actually of exponential order 
one $O(\exp(n^{1+\varepsilon}))$ rather than order $5$).

\vbox{\claim 10.8.~Theorem|A generic hypersurface $X\subset\bP^{n+1}$ of 
degree $d\ge d_n$ with
$$
d_2=286,\quad d_3=7316,\quad
d_n= \left\lfloor{n^4\over 3}\big(n\log(n\log(24n))\big)^n\right\rfloor
\quad\hbox{for $n\ge 4$},
$$
satisfies the Green-Griffiths conjecture.
\endclaim}

\proof. Let us apply Theorem~9.46 with $V=T_X$, $r=n$ and $k=n$. 
The main starting point is the well
known fact that $T^*_{\bP^{n+1}}\otimes\cO_{\bP^{n+1}}(2)$ is 
semipositive (in fact, generated by its sections). Hence the exact
sequence
$$
0\to \cO_{\bP^{n+1}}(-d)\to T^*_{\bP^{n+1}|X}\to T^*_X\to 0
$$
implies that $T^*_X\otimes\cO_X(2)\ge 0$. We can therefore take 
$\gamma=\Theta_{\cO(2)}=2\omega$ where $\omega$ is the Fubini-Study metric.
Moreover $\det V^*=K_X=\cO_X(d-n-2)$ has curvature $(d-n-2)\omega$,
hence $\Theta_{\det V^*}+r\gamma=(d+n-2)\omega$.
The Morse integral to be computed when $A=\cO_X(p)$ is
$$
\int_X \Big(c_{n,n,n}(d+n-2)^n-c'_{n,n,n}(d+n-2)^{n-1}(p+2n)\Big)\omega^n,
$$
so the critical condition we need is
$$d+n-2>{c'_{n,n,n}\over c_{n,n,n}}(p+2n).$$
On the other hand, Siu's differentiation technique requires 
${m\over n^2}(1+{1\over 2}+\ldots+{1\over n})A-mB$ to be ample, where
$B=\cO_X(n^2+2n)$ by Merker's result 10.2. This ampleness condition
yields
$$
{1\over n^2}\Big(1+{1\over 2}+\ldots+{1\over n}\Big)p-(n^2+2n)>0,
$$
so one easily sees that it is enough to take $p=n^4-2n$ for $n\ge
3$. Our estimates (9.48) and (9.49) give the expected 
bound~$d_n$.\square
\medskip

Thanks to 10.6, one also obtains the generic hyperbolicity of $2$ and
$3$-dimensional hypersurfaces of large degree.

\claim 10.9.~Theorem|For $n=2$ or $n=3$, a generic hypersurface
$X\subset\bP^{n+1}$ of degree $d\ge d_n$ is Kobayashi hyperbolic.
\endclaim

By using more explicit calculations of Chern classes (and invariant jets
rather than Green-Griffiths jets) Diverio-Trapani [DT10] obtained the
better lower bound $d\ge d_3=593$ in dimension~$3$. In the case of
surfaces, Paun [Pau08] obtained $d\ge d_2=18$, using deep results of
McQuillan [McQ98].

One may wonder whether it is possible to use jets of order $k<n$ in
the proof of 10.8 and 10.9. Diverio [Div08] showed that the answer
is negative (his proof is based on elementary facts of representation
theory and a vanishing theorem of Br\"uckmann-Rackwitz [BR90]):

\claim 10.10.~Proposition {\rm([Div08])}|Let 
$X\subset\bP^{n+1}$ be a smooth hypersurface. Then 
$$
H^0(X,E_{k,m}^\GG T^*_X)=0
$$
for $m\ge 1$ and $1\le k<n$. More generally, if $X\subset\bP^{n+s}$ is
a smooth complete intersection of codimension~$s$, there are no global jet differentials
for $m\ge 1$ and $k<n/s$.
\endclaim
\vfill\eject

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}

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Universit\'e Joseph Fourier Grenoble I\hfil\break
Institut Fourier (Math\'ematiques)\hfil\break
UMR 5582 du C.N.R.S., BP 74\hfil\break 38402
Saint-Martin d'H\`eres, France\hfil\break
{\em e-mail:}\/ demailly@fourier.ujf-grenoble.fr

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