% 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}
\smallskip
\centerline{\twelverm (long expository version)}

\vskip10pt
\noindent{\bf 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 $\bP^4$.
\vskip5pt

%%\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, holomorphic Morse inequalities

\vskip5pt
\noindent
{\bf A.M.S.~Classification (1991):} 32H20, 32L10, 53C55, 14J40
\vskip15pt

\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}
\sectref{12} \sectref{13} \sectref{14} \sectref{15} \sectref{16} \sectref{17}
\sectref{18}
}\vskip8pt
\vfill\eject

\section{\S0. Preliminaries of complex differential geometry}

\subsection \S0.A. Dolbeault cohomology and sheaf cohomology|
Let $X$ be a $\bC$-analytic manifold of dimension $n$. We denote by
$\Lambda^{p,q}T^*_X$ the bundle of differential forms of bidegree
$(p,q)$ on $X$, i.e., differential forms which can be written as
$$u=\sum_{|I|=p,\,|J|=q}u_{I,J}dz_I\wedge d\ol z_J.$$
Here $(z_1\ld z_n)$ denote local holomorphic coordinates,
\hbox{$I=(i_1\ld i_p)$}, \hbox{$J=(j_1\ld j_q)$} are multiindices
(increasing sequences of integers in the range $[1\ld n]$, of lengths
$|I|=p$, $|J|=q$), and
$$dz_I:=dz_{i_1}\wedge\ldots\wedge dz_{i_p},\qquad
d\ol z_J:=d\ol z_{j_1}\wedge\ldots\wedge d\ol z_{j_q}.$$
Let $\cE^{p,q}$ be the sheaf of germs of complex valued differential
$(p,q)$-forms with $C^\infty$ coefficients. Recall that the exterior
derivative $d$ splits as $d=\partial+\dbar$ where
$$\eqalign{
\partial u&=\sum_{|I|=p,\,|J|=q,1\le k\le n}{\partial u_{I,J}
\over\partial z_k}dz_k\wedge dz_I\wedge d\ol z_J,\cr
\dbar u&=\sum_{|I|=p,\,|J|=q,1\le k\le n}{\partial u_{I,J}
\over\partial\ol z_k}d\ol z_k\wedge dz_I\wedge d\ol z_J\cr}$$
are of type $(p+1,q)$, $(p,q+1)$ respectively. (Another frequently used
alternative notation is $d=d'+d''$, where $d'=\partial$, $d''=\dbar$).
The well-known Dolbeault-Grothendieck lemma asserts that any $\dbar $-closed
form of type $(p,q)$ with $q>0$ is locally $\dbar$-exact (this is the
analogue for $\dbar $ of the usual Poincar\'e lemma for~$d$, see e.g.\
[H\"or66]). In other words, the complex of sheaves
$(\cE^{p,\bu},\dbar)$ is exact in degree $q>0\,$; in degree $q=0$, $\Ker \dbar$
is the sheaf $\Omega^p_X$ of germs of holomorphic forms of degree $p$
on~$X$.

More generally, if $F$ is a holomorphic vector bundle
of rank $r$ over~$X$, there is a natural $\dbar$ operator acting
on the space $C^\infty(X,\Lambda^{p,q}T^*_X\otimes F)$ of smooth
$(p,q)$-forms with values in~$F$; if $s=\sum_{1\le\lambda\le r}s_\lambda
e_\lambda$ is a $(p,q)$-form expressed in terms of a local holomorphic
frame of $F$, we simply define $\smash{\dbar s}:=\sum 
\smash{\dbar s_\lambda}\otimes e_\lambda$,
observing that the holomorphic transition matrices involved in changes
of holomorphic frames do not affect the computation of~$\dbar$.
It is then clear that the Dolbeault-Grothendieck lemma still
holds for $F$-valued forms. For every integer $p=0,1\ld n$, the
{\em Dolbeault Cohomology} groups $H^{p,q}(X,F)$ are defined to be the
cohomology groups of the complex of global $(p,q)$ forms (graded by $q$):
$$H^{p,q}(X,F)=H^q\big(C^\infty(X,\Lambda^{p,\bu}T^*_X\otimes F)\big).
\leqno(0.1)$$
Now, let us recall the following fundamental result from sheaf theory
(De Rham-Weil isomorphism theorem): let $(\cL^{\bu},d)$ be a resolution
of a sheaf $\cA$ by acyclic sheaves, i.e.\ a complex of sheaves
$(\cL^\bu,\delta)$ such that there is an exact sequence of sheaves
$$0\lra\cA\buildo\lra^j \cL^0\buildo \lra^{\delta^0}\cL^1\lra\cdots
\lra\cL^q\buildo \lra^{\delta^q}\cL^{q+1}\lra\cdots~,$$
and $H^s(X,\cL^q)=0$ for all $q\ge 0$ and $s\ge 1$. Then there
is a functorial isomorphism 
$$H^q\big(\Gamma(X,\cL^\bu)\big)\lra H^q(X,\cA).\leqno(0.2)$$
We apply this to the following situation: let $\cE(F)^{p,q}$ be the
sheaf of germs of $C^\infty$ sections of $\Lambda^{p,q}T^*_X\otimes
F$. Then $(\cE(F)^{p,\bu},\dbar)$ is a resolution of the locally free
$\cO_X$-module $\Omega^p_X\otimes\cO(F)$ (Dolbeault-Grothendieck
lemma), and the sheaves $\cE(F)^{p,q}$ are acyclic as modules over the
soft sheaf of rings~$\cC^\infty$. Hence by (0.2) we get

\claim 0.3. Dolbeault isomorphism theorem {\rm (1953)}|For every
holomorphic vector bundle $F$ on~$X$, there is a canonical isomorphism
$$H^{p,q}(X,F)\simeq H^q(X,\Omega^p_X\otimes\cO(F)).\eqno\square$$
\endclaim

If $X$ is projective algebraic and $F$ is an algebraic vector bundle,
Serre's GAGA theorem [Ser56] shows that the algebraic sheaf cohomology
group $H^q(X,\Omega^p_X\otimes\cO(F))$ computed with algebraic sections
over Zariski open sets is actually isomorphic to the analytic
cohomology group. These results are the most basic tools to attack
algebraic problems via analytic methods. Another important tool
is the theory of plurisubharmonic functions and positive
currents introduced by K.~Oka and P.~Lelong in the decades 1940-1960.

\subsection \S0.B. Plurisubharmonic functions|
Plurisubharmonic functions have been introduced independently
by Lelong and Oka in the study of holomorphic convexity. We refer to
[Lel67, 69] for more details.

\claim 0.4. Definition|A function $u:\Omega\lra[-\infty,+\infty[$ defined
on an open subset $\Omega\subset\bC^n$ is said to be plurisubharmonic
$($psh for short$)$ if
\smallskip
\item{\rm (a)} $u$ is upper semicontinuous~$;$
\smallskip
\item{\rm (b)} for every complex line $L\subset\bC^n$, 
$u_{\restriction\Omega\cap L}$ is subharmonic on $\Omega\cap L$, that is,
for all $a\in\Omega$ and $\xi\in\bC^n$ with $|\xi|<d(a,\complement\Omega)$,
the function $u$ satisfies the mean value inequality
$$u(a)\le{1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta.$$
\vskip0pt\noindent
The set of psh functions on $\Omega$ is denoted by
$\Psh(\Omega)$.
\endclaim

We list below the most basic properties of psh functions.
They all follow easily from the definition.

\claim 0.5. Basic properties|\rm
\smallskip
\item{\rm (a)} Every function $u\in\Psh(\Omega)$ is subharmonic, namely it
satisfies the mean value inequality on Euclidean balls or spheres:
$$u(a)\le{1\over\pi^nr^{2n}/n!}\int_{B(a,r)}u(z)\,d\lambda(z)$$
for every $a\in\Omega$ and $r<d(a,\complement\Omega)$. Either $u\equiv-\infty$
or $u\in L^1_\loc$ on every connected component of~$\Omega$.
\smallskip
\item{\rm (b)} For any decreasing sequence of psh
functions $u_k\in\Psh(\Omega)$, the limit $u=\lim u_k$ is psh
on $\Omega$.
\smallskip
\item{\rm (c)} Let $u\in\Psh(\Omega)$ be such that $u\not\equiv-\infty$ 
on every connected component of~$\Omega$. If $(\rho_\varepsilon)$ is a 
family of smoothing kernels, then $u*\rho_\varepsilon$ is $C^\infty$ and 
psh on
$$\Omega_\varepsilon=\big\{x\in\Omega\,;\,d(x,\complement\Omega)>\varepsilon
\big\},$$
the family $(u*\rho_\varepsilon)$ is increasing in $\varepsilon$ and 
$\lim_{\varepsilon\to 0}u*\rho_\varepsilon=u$.
\smallskip
\item{\rm (d)} Let $u_1\ld u_p\in\Psh(\Omega)$ and
$\chi:\bR^p\lra\bR$ be a convex function such that $\chi(t_1\ld t_p)$
is increasing in each $t_j$. Then $\chi(u_1\ld u_p)$ is psh
on $\Omega$. In particular~ $u_1+\cdots+u_p$, $\max\{u_1\ld u_p\}$, 
$\log(e^{u_1}+\cdots+e^{u_p})$ are psh on~$\Omega$.\square
\smallskip
\endclaim

\claim 0.6. Lemma|A function $u\in C^2(\Omega,\bR)$ is psh on
$\Omega$ if and only if the hermitian form $Hu(a)(\xi)=\sum_{1\le j,k\le n}
\partial^2 u/\partial z_j\partial\ol z_k(a)\,\xi_j\ol \xi_k$
is semipositive at every point $a\in\Omega$.
\endclaim

\proof. This is an easy consequence of the following standard formula
$${1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta - u(a)=
{2\over\pi}\int_0^1{dt\over t}\int_{|\zeta|<t}Hu(a+\zeta\xi)(\xi)\,
d\lambda(\zeta),$$
where $d\lambda$ is the Lebesgue measure on $\bC$.
Lemma~0.6 is a strong evidence that plurisubharmonicity is the natural
complex analogue of linear convexity.\square
\medskip

For non smooth functions, a similar characterization of plurisubharmonicity
can be obtained by means of a regularization process.

\claim 0.7. Theorem|If $u\in\Psh(\Omega)$, $u\not\equiv-\infty$ on every
connected component of $\Omega$, then for all $\xi\in\bC^n$
$$Hu(\xi)=\sum_{1\le j,k\le n}{\partial^2 u\over\partial z_j\partial\ol z_k}
\,\xi_j\ol\xi_k\in\cD'(\Omega)$$
is a positive measure. Conversely, if $v\in\cD'(\Omega)$ is such that 
$Hv(\xi)$ is a positive measure for every $\xi\in\bC^n$, there exists a 
unique function $u\in\Psh(\Omega)$ which is locally integrable on $\Omega$
and such that $v$ is the distribution associated to $u$.\square
\endclaim

In order to get a better geometric insight of this notion, we assume more
generally that $u$ is a function on a complex $n$-dimensional manifold 
$X$. If \hbox{$\Phi:X\to Y$} is a holomorphic mapping and if $v\in C^2(Y,\bR)$, we
have the commutation relation $\hbox{$\ddbar(v\circ \Phi)$}=
\Phi^*(\ddbar v)$, hence
$$H(v\circ \Phi)(a,\xi)=Hv\big(\Phi(a),\Phi'(a)\cdot\xi\big).$$
In particular $Hu$, viewed as a hermitian form on $T_X$, does not depend
on the choice of coordinates $(z_1\ld z_n)$. Therefore, the notion of
psh function makes sense on any complex manifold. More generally,
we have

\claim 0.8. Proposition|If $\Phi:X\lra Y$ is a holomorphic map and
$v\in\Psh(Y)$, then $v\circ \Phi\in\Psh(X)$.\square
\endclaim

\claim 0.9. Example|\rm It is a standard fact that $\log|z|$ is psh
(i.e.\ subharmonic) on~$\bC$. Thus $\log|f|\in\Psh(X)$ for every
holomorphic function $f\in H^0(X,\cO_X)$. More generally 
$$\log\big(|f_1|^{\alpha_1}+\cdots+|f_q|^{\alpha_q}\big)\in\Psh(X)$$
for every $f_j\in H^0(X,\cO_X)$ and $\alpha_j\ge 0$ (apply Property~0.5$\,$d
with $u_j=\alpha_j\,\log|f_j|$). We will be especially interested in the
singularities obtained at points of the zero variety $f_1=\ldots=f_q=0$,
when the $\alpha_j$ are rational numbers.\square
\endclaim

\claim 0.10. Definition|A psh function $u\in\Psh(X)$ will be said to
have analytic singularities if $u$ can be written locally as
$$u={\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)+v,$$
where $\alpha\in\bR_+$, $v$ is a locally bounded function and the $f_j$
are holomorphic functions. If $X$ is algebraic, we say that $u$ has
algebraic singularities if $u$ can be written as above on sufficiently
small Zariski open sets, with $\alpha\in\bQ_+$ and $f_j$ algebraic.
\endclaim

We then introduce the ideal $\cJ=\cJ(u/\alpha)$ of germs of holomorphic
functions $h$ such that $|h|\le Ce^{u/\alpha}$ for some constant~$C$, i.e.
$$|h|\le C\big(|f_1|+\cdots+|f_N|\big).$$
This is a globally defined ideal sheaf on $X$, locally equal to the
integral closure $\ol\cI$ of the ideal sheaf $\cI=(f_1\ld f_N)$, thus $\cJ$
is coherent on~$X$. If $(g_1\ld g_{N'})$ are local generators of $\cJ$,
we still have
$$u={\alpha\over 2}\log\big(|g_1|^2+\cdots+|g_{N'}|^2\big)+O(1).$$
If $X$ is projective algebraic and $u$ has analytic singularities with
$\alpha\in\bQ_+$, then $u$ automatically has algebraic singularities.
From an algebraic point of view, the singularities of $u$ are in 1:1
correspondence with the ``algebraic data'' $(\cJ,\alpha)$. 

\subsection \S0.C. Positive currents|
The reader can consult [Fed69] for a more thorough treatment of
current theory. Let us first recall a few basic definitions. A {\em
current} of degree $q$ on an oriented differentiable manifold $M$ is
simply a differential $q$-form $T$ with distribution coefficients.
The space of currents of degree $q$ over $M$ will be denoted by
$\cD^{\prime q}(M)$. Alternatively, a current of degree $q$ can be seen
as an element $T$ in the dual space $\cD'_p(M):=\big(\cD^p(M)\big)'$ of
the space $\cD^p(M)$ of smooth differential forms of degree $p=\dim M-q$ with
compact support; the duality pairing is given by
$$\langle T,\alpha\rangle=\int_M T\wedge\alpha,~~~~
\alpha\in\cD^p(M).\leqno(0.11)$$
A basic example is the {\em current of integration} $[S]$ over a compact
oriented submanifold $S$ of $M\,$:
$$\langle[S],\alpha\rangle=\int_S\alpha,~~~~{\rm deg}\alpha=p=\dim_\bR S.
\leqno(0.12)$$
Then $[S]$ is a current with measure coefficients, and
Stokes' formula shows that $d[S]=(-1)^{q-1}[\partial S]$, in particular
$d[S]=0$ if $S$ has no boundary. Because of this example, the integer $p$
is said to be the dimension of~$T$ when $T\in\cD_p'(M)$. The
current $T$ is said to be {\em closed} if $dT=0$.

On a complex manifold $X$, we have similar notions of bidegree and
bidimension; as in the real case, we denote by
$$\cD^{\prime p,q}(X)=\cD'_{n-p,n-q}(X),\qquad n=\dim X,$$
the space of currents of bidegree $(p,q)$ and bidimension $(n-p,n-q)$ on~$X$.
According to [Lel57], a current $T$
of bidimension $(p,p)$ is said to be {\em $($weakly$)$ positive} if
for every choice of smooth $(1,0)$-forms $\alpha_1\ld\alpha_p$ on~$X$
the distribution
$$T\wedge\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\ol\alpha_p~~~~\hbox{\rm is a positive measure.}
\leqno(0.13)$$

\claim 0.14. Exercise|\rm If $T$ is positive, show that the
coefficients $T_{I,J}$ of $T$ are complex measures, and that,
up to constants, they are dominated by the trace measure
$$\sigma_T=T\wedge{1\over p!}\beta^p=2^{-p}\sum T_{I,I},\qquad
\beta={\ii\over 2}\ddbar|z|^2={\ii\over 2}\sum_{1\le j\le n}dz_j\wedge
d\ol z_j,$$
which is a positive measure.\hfil\break

{\em Hint}. Observe that $\sum T_{I,I}$ is invariant by unitary changes
of coordinates and that the $(p,p)$-forms
$\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge\ii\alpha_p\wedge\ol\alpha_p$
generate $\Lambda^{p,p}T^*_{\bC^n}$ as a $\bC$-vector space.\square
\endclaim

A current $T=\ii\sum_{1\le j,k\le n}T_{jk}dz_j\wedge dz_k$
of bidegree $(1,1)$ is easily seen to be positive if and only if the
complex measure $\sum\lambda_j\ol\lambda_k T_{jk}$ is a positive
measure for every $n$-tuple $(\lambda_1\ld\lambda_n)\in\bC^n$.

\claim 0.15. Example|\rm If $u$ is a (not identically $-\infty$)
psh function on $X$, we can associate with $u$ a (closed) positive
current $T=\ii\partial\ol\partial u$ of bidegree~$(1,1)$. Conversely,
every closed positive current of bidegree $(1,1)$ can be written under
this form on any open subset $\Omega\subset X$ such that
$H^2_{DR}(\Omega,\bR)=H^1(\Omega,\cO)=0$, e.g.\ on small coordinate
balls (exercise to the reader).\square
\endclaim

It is not difficult to show that a product $T_1\wedge\ldots\wedge
T_q$ of positive currents of bidegree $(1,1)$ is positive whenever
the product is well defined (this is certainly the case if all
$T_j$ but one at most are smooth; much finer conditions will be
discussed in Section~2).

We now discuss another very important example of closed positive current.
In fact, with every closed analytic set $A\subset X$ of pure dimension $p$
is associated a current of integration
$$\langle [A],\alpha\rangle=\int_{A_\reg}\alpha,~~~~\alpha\in\cD^{p,p}(X),
\leqno(0.16)$$
obtained by integrating over the regular points of~$A$. In order to show
that (0.16) is a correct definition of a current on~$X$, one must show that
$A_\reg$ has locally finite area in a neighborhood of $A_\sing$.
This result, due to [Lel57] is shown as follows. Suppose that $0$
is a singular point of $A$. By the local parametrization theorem for
analytic sets, there is a linear change of coordinates on $\bC^n$
such that all projections
$$\pi_I:(z_1\ld z_n)\mapsto (z_{i_1}\ld z_{i_p})$$
define a finite ramified covering of the intersection $A\cap\Delta$ with
a small polydisk $\Delta$ in $\bC^n$ onto a small
polydisk $\Delta_I$ in~$\bC^p$. Let $n_I$ be the sheet number. Then the
\hbox{$p$-dimensional} area of $A\cap\Delta$ is bounded above by the sum
of the areas of its projections counted with multiplicities, i.e.
$$\hbox{\rm Area}(A\cap\Delta)\le\sum n_I\hbox{\rm Vol}(\Delta_I).$$
The fact that $[A]$ is positive is also easy. In fact
$$\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\ol\alpha_p=|\det(\alpha_{jk})|^2\,
\ii w_1\wedge\ol w_1\wedge\ldots\wedge
\ii w_p\wedge\ol w_p$$
if $\alpha_j=\sum\alpha_{jk}dw_k$ in terms of local coordinates
$(w_1\ld w_p)$ on $A_\reg$. This shows that all such forms are $\ge 0$
in the canonical orientation defined by
$\ii w_1\wedge\ol w_1\wedge\ldots\wedge\ii w_p\wedge\ol w_p$.
More importantly, Lelong [Lel57] has shown that $[A]$ is $d$-closed in~$X$,
even at points of $A_\sing$. This last result can be seen today as a
consequence of the Skoda-El Mir extension theorem. For this we need the
following definition: a {\em complete pluripolar} set is a set $E$ such
that there is an open covering $(\Omega_j)$ of $X$ and psh functions $u_j$ on
$\Omega_j$ with $E\cap\Omega_j=u_j^{-1}(-\infty)$. Any (closed) analytic
set is of course complete pluripolar (take $u_j$ as in Example~0.9). 

\claim 0.17. Theorem {\rm(Skoda [Sko82], El Mir [EM84], Sibony [Sib85])}|
Let $E$ be a closed complete pluripolar set in $X$,
and let $T$ be a closed positive current on $X\ssm E$ such that the
coefficients $T_{I,J}$ of $T$ are measures with locally finite
mass near~$E$. Then the trivial extension $\wt T$ obtained by extending
the measures $T_{I,J}$ by $0$ on $E$ is still closed on~$X$.\square
\endclaim

The proof proceeds by rather direct mass estimates and will be omitted here.
Lelong's result $d[A]=0$ is obtained by applying the Skoda-El Mir
theorem to $T=[A_\reg]$ on $X\ssm A_\sing$. 

\claim 0.18. Corollary|Let $T$ be a closed positive current on
$X$ and let $E$ be a complete pluripolar set. Then $\bOne_E T$
and $\bOne_{X\ssm E}T$ are closed positive currents. In fact,
\hbox{$\wt T=\bOne_{X\ssm E}T$} is the trivial extension of
$T_{\restriction X\ssm E}$ to~$X$, and
$\bOne_ET=T-\wt T$.\square
\endclaim

As mentioned above, any current $T=\ii \ddbar u$ associated with a psh
function $u$ is a closed positive $(1,1)$-current. In the special
case $u=\log|f|$ where $f\in H^0(X,\cO_X)$ is a non zero holomorphic
function, we have the important

\claim 0.19. Lelong-Poincar\'e equation|Let $f\in H^0(X,\cO_X)$ 
be a non zero holomorphic function,
$Z_f=\sum m_jZ_j$, $m_j\in\bN$, the zero divisor of~$f$ and
$[Z_f]=\sum m_j[Z_j]$ the associated current of integration. Then
$${\ii\over\pi}\partial\ol\partial\log|f|=[Z_f].$$
\endclaim

\proof\ {\em$($sketch$)$.} It is clear that $\ii \ddbar\log|f|=0$ in a neighborhood
of every point $x\notin\Supp(Z_f)=\bigcup Z_j$, so it is enough to check
the equation in a neighborhood of every point of $\Supp(Z_f)$. Let $A$ be
the set of singular points of $\Supp(Z_f)$, i.e.\ the union of the
pairwise intersections $Z_j\cap Z_k$ and of the singular loci
$Z_{j,\sing}$; we thus have $\dim A\le n-2$. In a neighborhood of any
point $x\in\Supp(Z_f)\ssm A$ there are local coordinates $(z_1\ld z_n)$
such that $f(z)=z_1^{m_j}$ where $m_j$ is the multiplicity of $f$ along
the component $Z_j$ which contains~$x$ and $z_1=0$ is an equation for
$Z_j$ near~$x$. Hence
$${\ii\over\pi}\ddbar\log|f|=m_j{\ii\over\pi}\ddbar\log|z_1|=m_j[Z_j]$$
in a neighborhood of $x$, as desired (the identity comes from the
standard formula ${\ii\over\pi}\ddbar\log|z|={}$ Dirac measure $\delta_0$
in $\bC$). This shows that the equation holds on $X\ssm A$. Hence
the difference ${\ii\over\pi}\ddbar\log|f|-[Z_f]$ is a closed current of
degree $2$ with measure coefficients, whose support is contained in~$A$.
By Exercise~0.20, this current must be $0$, because $A$ has too small
dimension to carry its support ($A$~is~stratified by submanifolds of real
codimension $\ge 4$).\square
\medskip

\claim 0.20. Exercise|\rm Let $T$ be a current of degree $q$
on a real manifold~$M$, such that both $T$ and $dT$ have
measure coefficients (``normal current''). Suppose that $\Supp T$
is contained in a real submanifold $A$ with $\codim_\bR A>q$. Show
that $T=0$.\hfil\break

\noindent
{\em Hint:} Let $m=\dim_\bR M$ and let $(x_1\ld x_m)$ be a coordinate
system in a neighborhood $\Omega$ of a point $a\in A$ such that
$A\cap\Omega=\{x_1=\ldots=x_k=0\}$, $k>q$. Observe that
$x_jT=x_jdT=0$ for $1\le j\le k$, thanks to the hypothesis on
supports and on the normality of~$T$, hence $dx_j\wedge
T=d(x_jT)-x_jdT=0$, $1\le j\le k$. Infer from this that
all coefficients in $T=\sum_{|I|=q}T_Idx_I$ vanish.\square
\endclaim

\subsection \S0.D. Hermitian vector bundles, connections and curvature|
The goal of this section is to recall the most basic definitions of
hemitian differential geometry related to the concepts of connection,
curvature and first Chern class of a line bundle.

Let $F$ be a complex vector bundle of rank $r$ over a smooth
differentiable manifold~$M$. A {\em connection} $D$ on $F$ is a
linear differential operator of order $1$
$$D:C^\infty(M,\Lambda^qT^*_M\otimes F) \to C^\infty(M,
\Lambda^{q+1}T^*_M\otimes F)$$
such that
$$D(f\wedge u) = df\wedge u + (-1)^{\deg\,f}f\wedge Du\leqno(0.21)$$
for all forms $f\in C^\infty(M,\Lambda^pT^*_M)$, $u\in C^\infty(X,
\Lambda^qT^*_M\otimes F)$. On an open set $\Omega \subset M$ where $F$
admits a trivialization $\theta:F_{|\Omega }\buildo\lra^\simeq \Omega
\times \bC^r$, a connection $D$ can be written
$$Du\simeq_\theta du+\Gamma \wedge u$$
where $\Gamma \in C^\infty(\Omega ,\Lambda^1T^*_M\otimes\Hom
(\bC^r,\bC^r))$ is an arbitrary matrix of $1$-forms and $d$ acts
componentwise. It is then easy to check that
$$D^2u\simeq_\theta (d\Gamma +\Gamma \wedge \Gamma )\wedge u\quad
\hbox{on~}\Omega.$$
Since $D^2$ is a globally defined operator, there is a global $2$-form
$$
R_D\in C^\infty(M,\Lambda^2T^*_M\otimes\Hom_\bC(F,F))
$$
such that $D^2u=R_D\wedge u$ for every form $u$ with values in~$F$. Locally,
$R_D$ is given~by 
$$
R_D\simeq_\theta d\Gamma+\Gamma\wedge\Gamma
\leqno(0.22)
$$
where $\Gamma$ is the connection matrix.

Assume now that $F$ is endowed with a $C^\infty$ hermitian metric $h$ along
the fibers and that the isomorphism $F_{|\Omega }\simeq \Omega \times
\bC^r$ is given by a $C^\infty$ frame $(e_\lambda)$. We then have
a canonical sesquilinear pairing $\{\bu,\bu\}=\{\bu,\bu\}_h$
$$\leqalignno{
C^\infty(M,\Lambda^pT^*_M\otimes F)\times
C^\infty(M,\Lambda^qT^*_M\otimes F)
&\longrightarrow C^\infty(M,\Lambda^{p+q}T^*_M\otimes\bC)&(0.23)\cr
(u,v) &\longmapsto \{u,v\}_h\cr}$$
given by
$$\{u,v\}_h=\sum_{\lambda ,\mu }u_\lambda \wedge \overline v_\mu \langle
e_\lambda,e_\mu \rangle_h,\qquad
u=\sum u_\lambda \otimes e_\lambda,\quad 
v=\sum v_\mu \otimes e_\mu.$$
We will frequently omit the subscript $h$ when no confusion can arise.
The connection $D$ is said to be {\em hermitian} (with respect to $h$) 
if it satisfies the additional property
$$d\{u,v\}=\{Du,v\}+(-1)^{{\rm deg~}u}\{u,Dv\}.$$
Assuming that $(e_\lambda )$ is orthonormal, one easily checks that $D$
is hermitian if and only if $\Gamma^*=-\Gamma$, i.e.\ $\Gamma$ is
hermitian skew symmetric. In this case
$R_D^*=-R_D$ [$\,$observe that $(\Gamma\wedge\Gamma)^*=-
\Gamma^*\wedge\Gamma^*$ and more generally 
$(A\wedge B)^*=-B^*\wedge A^*$ for products of matrices of $1$-forms,
since reversing the order of the product of $1$-forms changes the sign$\,$].
Therefore the $2$-form $\Theta_D:={\ii\over2\pi}R_D={\ii\over2\pi}D^2$ 
takes values in hermitian symmetric tensors $\Herm(F,F)$, i.e.
$$\Theta_D \in C^\infty(M,\Lambda^2 T^*_M\otimes_\bR\Herm(F,F))$$
where $\Herm(F,F)\subset \Hom(F,F)$ is the real subspace of hermitian
endomorphisms. (The reason for introducing the additional factor $2\pi$
will appear below).
\medskip

\claim 0.24. Special case|\rm For a bundle $F$ of rank 1, the
connection form $\Gamma$ of a hermitian connection $D$ can be seen as
a $1$-form with purely imaginary coefficients (i.e.\ $\Gamma=\ii A$, 
$A$ real).
Then we have $R_D=d\Gamma=\ii dA$, therefore $\Theta_F={\ii\over 2\pi}R_D=
-{1\over 2\pi}dA$ is a  {\em $d$-closed and real} 2-form. 
The {\em $($real$\,)$ first Chern class} of $F$ is defined to be 
the cohomology class
$$c_1(F)_\bR=\{\Theta_D\}\in H^2_\DR(M,\bR).$$
This cohomology class is actually independent of the connection $D$ taken on
$F\,$: any other connection $D_1$ differs by a global 1-form, i.e.\
$D_1u=Du+B\wedge u$, so that $\Theta_{D_1}=\Theta_D-{1\over 2\pi}dB$. 
It is well-known that $c_1(F)_\bR$
is the image in $H^2(M,\bR)$ of an integral class $c_1(F)\in
H^2(M,\bZ)\,$; by using the exponential exact sequence
$$0\to\bZ\to\cE\to\cE^*\to 0,$$
$c_1(F)$ can be defined in \v Cech cohomology theory as the image
by the coboundary map $H^1(M,\cE^*)\to H^2(M,\bZ)$ of the cocycle
$\{g_{jk}\}\in H^1(M,\cE^*)$ defining $F\,$; see e.g.\ [GrH78] for
details. This is the essential reason for the introduction of a factor
${\ii\over 2\pi}$ in the definition of $\Theta_D$.\square
\endclaim

We now concentrate ourselves on the complex analytic case. If $M=X$ is a
complex manifold $X$, every connection $D$ on a complex $C^\infty$ vector
bundle $F$ can be split in a unique way as a
sum of a $(1,0)$ and of a $(0,1)$-connection, $D=D'+D''$. In a local
trivialization $\theta$ given by a $C^\infty$ frame, one can write 
$$\leqalignno{
D'u &\simeq_\theta d'u + \Gamma '\wedge u,&(0.25')\cr
D''u &\simeq_\theta d''u + \Gamma ''\wedge u,&(0.25'')\cr}$$
with $\Gamma =\Gamma '+\Gamma ''$. The connection is hermitian if and
only if $\Gamma '=-(\Gamma '')^*$ in any orthonormal frame. Thus there
exists a unique hermitian connection $D$ corresponding to a prescribed
$(0,1)$ part $D''$. 

Assume now that the hermitian bundle $(F,h)$ itself has 
a {\em holomorphic} structure.
The unique hermitian connection $D_h$ for which $D''_h$ is the $\dbar$ operator
defined in \S0.A is called the {\em Chern connection} of $F$. In a local
holomorphic frame $(e_\lambda )$ of $E_{|\Omega }$, the metric $h$ is then 
given by a hermitian matrix $H=(h_{\lambda\mu})$, $h_{\lambda\mu}
=\langle e_\lambda,e_\mu \rangle$. We have
$$\{u,v\}=\sum_{\lambda,\mu}h_{\lambda\mu}u_\lambda \wedge
\ol v_\mu=u^\dagger\wedge H\ol v,$$
where $u^\dagger$ is the transposed matrix of $u$. Easy computations yield
$$\eqalign{d\{u,v\}
&=(du)^\dagger \wedge H\ol v+(-1)^{\deg u} u^\dagger
\wedge (dH\wedge \ol v + H\ol{dv})\cr
&=\big(du +\ol H{\,}^{-1}d'\ol H \wedge u\big)^\dagger \wedge H\ol v
+(-1)^{\deg u}u^\dagger\wedge(\ol{dv+ \ol H{\,}^{-1}d'\ol H\wedge v)}\cr}$$
using the fact that $dH = d'H+\ol{d'\ol H}$ and
$\ol H^\dagger=H$. Therefore the Chern connection $D_h$ coincides with the
hermitian connection defined by
$$
\left\{\eqalign{
D_hu&{}\simeq_\theta du+\ol H{\,}^{-1}d'\ol H \wedge u,\cr
D_h'&{}\simeq_\theta d'+\ol H{\,}^{-1} d'\ol H\wedge\bu=
\ol H{\,}^{-1} d'(\ol H\bu),~~~~D''_h = d''.\cr}\right.\leqno(0.26)
$$
It is clear from the above relations (0.26) that $D_h^{\prime 2}=
D_h^{\prime\prime 2}=0$.  Consequently $D_h^2$ is given by to
\hbox{$D_h^2=D_h'D_h''+D_h''D_h'$}, and the curvature tensor 
$R_{D_h}$ is of type~$(1,1)$. Since $d'd''+d''d'=0$, we get
$$
\eqalign{
(D_h'D_h''+D_h''D_h')u&\simeq_\theta\ol H{\,}^{-1}d'\ol H\wedge d''u 
+d''(\ol H{\,}^{-1}d'\ol H\wedge u)\cr
&=d''(\ol H{\,}^{-1}d'\ol H)\wedge u.\cr}
$$
By the above calculation $R_{D_h}$ is given by the  matrix of $(1,1)$-forms
$$
R_{D_h}\simeq_\theta d''(\ol H{\,}^{-1}d'\ol H)=\ol
H{\,}^{-1}d''d'\ol H -\ol H{\,}^{-1}d''\ol H\wedge \ol
H{\,}^{-1}d'\ol H
$$
Since $H=\ol H^\dagger$ is hermitian symmetric
and transposition reverses products, we find again in this setting 
that $R_{D_h}$ is hermitian skew symmetric
$$
R_{D_h}^* \simeq_\theta\ol H{\,}^{-1}
\ol{R}_F^\dagger \ol H=-R_{D_h}.
$$

\claim 0.27. Definition and proposition|The Chern curvature tensor of
$(F,h)$ is defined to be $\Theta_{F,h}:=\Theta_{D_h}={\ii\over 2\pi}R_{D_h}$ 
where $D_h$ is the Chern connection. It is such that
$$
\Theta_{F,h} \in 
C^\infty\big(X,\Lambda^{1,1}_\bR T^*_X\otimes_\bR \Herm(F,F)\big)\subset
C^\infty\big(X,\Lambda^{1,1} T^*_X\otimes_\bC \Hom(F,F)\big).
$$
If $\theta: F_{\restriction\Omega}\to\Omega\times\bC^r$ is a holomorphic
trivialization and if $H$ is the hermitian matrix representing the
metric along the fibers of $F_{\restriction\Omega}$, then
$$\Theta_{F,h}\simeq_\theta{\ii\over 2\pi}
d''(\ol H{\,}^{-1} d'\ol H)\quad\hbox{\rm on}~\Omega.\eqno\square$$
$[\,$We will frequently omit the subscript $h$ and write simply $D_h=D$,
$\Theta_{F,h}=\Theta_F$ when no confusion can arise$\,]$.
\endclaim

The next proposition shows that the Chern curvature tensor is the
obstruction to the existence of orthonormal holomorphic frames:
a holomorphic frame can be made ``almost orthonormal" only up to
curvature terms of order $2$ in a neighborhood of any point.

\claim 0.28. Proposition|For every point $x_0\in X$ and every holomorphic
coordinate system $(z_j)_{1\le j\le n}$ at $x_0$, there exists a
holomorphic frame $(e_\lambda)_{1\le\lambda\le r}$ of $F$ in a neighborhood 
of $x_0$ such that
$$\langle e_\lambda(z),e_\mu(z)\rangle=\delta_{\lambda\mu}- 
\sum_{1\le j,k\le n}c_{jk\lambda\mu}\,z_j\ol z_k+O(|z|^3)$$ 
where $(c_{jk\lambda\mu})$ are the coefficients of the Chern curvature
tensor $\Theta_F(x_0)$, namely
$$\Theta_F(x_0)={\ii\over 2\pi}
\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}dz_j\wedge d\ol z_k\otimes e_\lambda^*\otimes e_\mu.$$
Such a frame $(e_\lambda)$ is called a normal
coordinate frame at~$x_0$.
\endclaim

\proof. Let $(h_\lambda)$ be a holomorphic frame of $F$. After replacing
$(h_\lambda)$ by suitable linear combinations with constant
coefficients, we may assume that $\big(h_\lambda(x_0)\big)$ is an
orthonormal basis of $F_{x_0}$.  Then the inner products 
$\langle h_\lambda,h_\mu\rangle$ have an expansion 
$$\langle h_\lambda(z),h_\mu(z)\rangle=\delta_{\lambda\mu}+
\sum_j(a_{j\lambda\mu}\,z_j+a'_{j\lambda\mu}\,\ol z_j)+O(|z|^2)$$
for some complex coefficients $a_{j\lambda\mu}$, $a'_{j\lambda\mu}$ such
that $a'_{j\lambda\mu}=\ol a_{j\mu\lambda}$.  Set first
$$g_\lambda(z)=h_\lambda(z)-\sum_{j,\mu}a_{j\lambda\mu}\,z_j\,h_\mu(z).$$
Then there are coefficients $a_{jk\lambda\mu}$, $a'_{jk\lambda\mu}$,
$a''_{jk\lambda\mu}$ such that
$$\eqalign{
\langle g_\lambda(z),g_\mu(z)\rangle&=\delta_{\lambda\mu}+O(|z|^2)\cr
&=\delta_{\lambda\mu}+\sum_{j,k}\big(a_{jk\lambda\mu}\,z_j\ol z_k+
a'_{jk\lambda\mu}\,z_jz_k+a''_{jk\lambda\mu}\ol z_j\ol z_k\big)+
O(|z|^3).}$$ 
The holomorphic frame $(e_\lambda)$ we are looking for is 
$$e_\lambda(z)=
g_\lambda(z)-\sum_{j,k,\mu}a'_{jk\lambda\mu}\,z_jz_k\,g_\mu(z).$$
Since $a''_{jk\lambda\mu}=\ol a'_{jk\mu\lambda}$, we easily find
$$\eqalign{
\langle e_\lambda(z),e_\mu(z)\rangle&=\delta_{\lambda\mu}+
\sum_{j,k}a_{jk\lambda\mu}\,z_j\ol z_k+O(|z|^3),\cr
d'\langle e_\lambda,e_\mu\rangle&=\{D'e_\lambda,e_\mu\}=
\sum_{j,k}a_{jk\lambda\mu}\,\ol z_k\,dz_j+O(|z|^2),\cr
\Theta_F\cdot e_\lambda
&=D''(D'e_\lambda)=\sum_{j,k,\mu}a_{jk\lambda\mu}\,d\ol z_k\wedge dz_j
\otimes e_\mu+O(|z|),\cr}$$ 
therefore $c_{jk\lambda\mu}=-a_{jk\lambda\mu}$.\square
\medskip

According to (0.27), one can associate canonically with the curvature tensor of
$F$ a hermitian form on $T_X\otimes F$ defined by
$$\wt\Theta_F(\xi\otimes v)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\xi_j\ol\xi_kv_\lambda\ol v_\mu,\qquad
\xi\in T_X,~~v\in F.\leqno(0.29)$$
This leads in a natural way to positivity concepts,
following definitions introduced by Kodaira [Kod53], Nakano [Nak55] and
Griffiths [Gri69].

\claim 0.30. Definition|The hermitian vector bundle $F$ is said to be
\smallskip
\item{\rm (a)} positive in the sense of Nakano if
$\wt\Theta_F(\tau)>0$ for all non zero tensors $\tau=\sum
\tau_{j\lambda}\partial/\partial z_j\otimes e_\lambda\in T_X\otimes F$.
\smallskip
\item{\rm (b)} positive in the sense of Griffiths if
$\wt\Theta_F(\xi\otimes v)>0$ for all non zero decomposable tensors
$\xi\otimes v\in T_X\otimes F\,;$
\smallskip\noindent
Corresponding semipositivity concepts are defined by relaxing the strict
inequalities.
\endclaim

\claim 0.31. Special case of rank 1 bundles|\rm Assume that
$F$ is a line bundle. The hermitian matrix $H=(h_{11})$ associated
to a trivialization $\theta:F_{\restriction\Omega}\simeq\Omega\times\bC$
is simply a positive function which we find convenient to denote by
$e^{-\varphi}$, \hbox{$\varphi\in C^\infty(\Omega,\bR)$}. In this case
the curvature form $R_{F,h}$ can be identified to the $(1,1)$-form
$\ddbar\varphi$, and thus we get a real $(1,1)$-form
$$
\Theta_{F,h}={\ii\over2\pi}\ddbar\varphi.
$$
Hence $F$ is semipositive (in either the Nakano or
Griffiths sense) if and only if $\varphi$ is psh, resp.\ positive if
and only if $\varphi$ is {\em strictly psh}. In this setting, the
Lelong-Poincar\'e equation can be generalized as follows: let
$\sigma\in H^0(X,F)$ be a non zero holomorphic section. Then
$$
{\ii\over 2\pi}\ddbar\log\|\sigma\|_h=[Z_\sigma]-{\ii\over 2\pi}\Theta_{F,h}.\leqno(0.32)
$$
Formula (0.32) is immediate if we write $\|\sigma\|_h^2=|\theta(\sigma)|^2
e^{-\varphi}$ and if we apply (0.19) to the holomorphic function
$f=\theta(\sigma)$. As we shall see later, it is very important for
the applications to consider also singular hermitian metrics.
\endclaim

\claim 0.33. Definition|A singular $($hermitian$)$ metric $h$ on a line
bundle $F$ is a metric $h$ which is given in any trivialization
$\theta:F_{\restriction\Omega}
\buildo \lra^\simeq \Omega\times\bC$ by
$$\|\xi\|_h^2=|\theta(\xi)|^2\,e^{-\varphi(x)},~~~~x\in\Omega,~\xi\in F_x$$
where $\varphi$ is an arbitrary measurable function 
in $L^1_\loc(\Omega)$, called the
weight of the metric with respect to the trivialization $\theta$.
\endclaim

If $\theta':F_{\restriction\Omega'}\lra\Omega'\times\bC$ 
is another trivialization, $\varphi'$ the associated weight and
$g\in\cO^*(\Omega\cap\Omega')$ the transition function, then
$\theta'(\xi)=g(x)\,\theta(\xi)$ for $\xi\in F_x$, and so
$\varphi'=\varphi+\log|g|^2$ on $\Omega\cap\Omega'$. The curvature form 
of $F$ is then given formally by the closed $(1,1)$-current
$\Theta_{F,h}={\ii\over 2\pi}\ddbar\varphi$ on $\Omega\,$; our assumption
$\varphi\in L^1_\loc(\Omega)$ guarantees that $\Theta_{F,h}$ exists in
the sense of distribution theory. As in the smooth case, $\Theta_{F,h}$ 
is globally defined on $X$ and independent of the choice of
trivializations, and its De Rham cohomology class is the image of the first
Chern class $c_1(F)\in H^2(X,\bZ)$ in $H^2_{DR}(X,\bR)$.
Before going further, we discuss two basic examples. 

\claim 0.34. Example|\rm Let $D=\sum\alpha_jD_j$ be a divisor with 
coefficients $\alpha_j\in\bZ$ and let $F=\cO(D)$ be the associated
invertible sheaf of meromorphic functions $u$ such that
${\rm div}(u)+D\ge 0\,$; the corresponding line bundle can be
equipped with the singular metric defined by $\|u\|=|u|$. If $g_j$ is a
generator of the ideal of $D_j$ on an open set $\Omega\subset X$ then 
$\theta(u)=u\prod\smash{g_j^{\alpha_j}}$ defines a trivialization of
$\cO(D)$ over $\Omega$, thus our singular metric is associated to the
weight $\varphi=\sum\alpha_j\log|g_j|^2$. By the Lelong-Poincar\'e equation,
we find
$$\Theta_{\cO(D)}={\ii\over 2\pi}\ddbar\varphi=[D],$$
where $[D]=\sum\alpha_j[D_j]$ denotes the current of integration
over~$D$.\square
\endclaim

\claim 0.35. Example|\rm Assume that $\sigma_1\ld \sigma_N$ are non zero 
holomorphic sections of $F$. Then we can define a natural (possibly
singular) hermitian metric $h^*$
on $F^*$ by
$$\|\xi^*\|^2_{h^*}=\sum_{1\le j\le n}\big|\xi^*.\sigma_j(x)\big|^2
~~~~\hbox{\rm for}~~\xi^*\in F^*_x.$$
The dual metric $h$ on $F$ is given by
$$\|\xi\|^2_h={|\theta(\xi)|^2\over|\theta(\sigma_1(x))|^2+\ldots+
|\theta(\sigma_N(x))|^2}
\leqno(0.35\,{\rm a})
$$
with respect to any trivialization $\theta$. The associated weight function is
thus given by $\varphi(x)=\log\big(\sum_{1\le j\le N}|\theta(\sigma_j(x))|^2
\big)$. In this case $\varphi$ is a psh function, thus
$\Theta_{F,h}$ is a closed positive current, given explicity by
$$
\Theta_{F,h}={\ii\over 2\pi}\ddbar\varphi={\ii\over 2\pi}
\ddbar\log\Big(\sum_{1\le j\le N}|\theta(\sigma_j(x))|^2\Big).
\leqno(0.35\,{\rm b})
$$
Let us denote by $\Sigma$
the linear system defined by $\sigma_1\ld \sigma_N$ and by
$B_\Sigma=\bigcap\sigma_j^{-1}(0)$ its base locus. We have a meromorphic
map
$$\Phi_\Sigma:X\ssm B_\Sigma\to\bP^{N-1},\qquad
x\mapsto(\sigma_1(x):\sigma_2(x):\ldots:\sigma_N(x)).$$
Then $\Theta_F$ is equal to the pull-back 
by $\Phi_\Sigma$ over $X\ssm B_\Sigma$ of the so called 
{\em Fubini-Study metric} on~$\bP^{N-1}\,$:
$$
\omega_\FS={\ii\over 2\pi}
\ddbar\log(|z_1|^2+\ldots+ |z_N|^2)
\leqno\rlap{\kern\hsize\llap{\hbox{$\square$}}}(0.35\,{\rm c})
$$
\endclaim

\claim 0.36. Ample and very ample line bundles|A holomorphic line
bundle $F$ over a compact complex manifold $X$ is said to be
\vskip0pt
\item{\rm (a)} very ample if the map $\Phi_{|F|}:X\to\bP^{N-1}$ associated
to the complete linear system $|F|=P(H^0(X,F))$ is a regular embedding
$($by this we mean in particular that the base locus is empty, i.e.\
$B_{|F|}=\emptyset)$.
\vskip0pt
\item{\rm (b)} ample if some multiple $mF$, $m>0$, is very ample.
\endclaim

Here we use an additive notation for $\Pic(X)=H^1(X,\cO^*)$, hence
the symbol $mF$ denotes the line bundle $F^{\otimes m}$. By
Example~0.35, every ample line bundle $F$ has a smooth hermitian metric
with positive definite curvature form; indeed, if the linear system
$|mF|$ gives an embedding in projective space, then we get a smooth
hermitian metric on $F^{\otimes m}$, and the $m$-th root yields a
metric on $F$ such that $\Theta_F={1\over m}
\Phi_{|mF|}^*\omega_\FS$. Conversely, the Kodaira embedding theorem
[Kod54] tells us that every positive line bundle $F$ is ample.

\claim 0.37.~Big line bundles|Let $F$ be a holomorphic line
bundle over a projective manifold $X$. The following properties 
are equivalent.
\vskip0pt
\item{\rm (a)} multiples of $F$ have a maximal growth of sections$\,:$
$$
h^0(X,mF)\ge cm^n\qquad\hbox{where $c>0$ and $n=\dim_\bC X$.}
$$
\vskip0pt
\item{\rm (b)} For any $($resp.\ $($b${}')$ for some$)$ ample line bundle, there exists an
integer $m>0$ such that $\cO(mF)\simeq\cO(E+A)$ where $E$ is an effective divisor.
\vskip0pt
\item{\rm (c)} There exists a sinular hermitian metric $h$ on $F$ such that, in the sense
of currents, we have
$$
\Theta_{F,h}\ge\varepsilon\omega
$$
for some $\varepsilon>0$ and some smooth positive $(1,1)$-form $\omega$ on~$X$.
\smallskip\noindent
Under these conditions, the line bundle $F$ is said to be {\rm big }. 
\endclaim

\noindent
{\em Sketch of proof.} (a)${}\Rightarrow{}$(b) For $p\gg 1$, $pA$ and $(p-1)A$ are very ample.
Hence, after replacing $A$ by $pA$, we can suppose that $A$ is very ample, represented by
a smooth divisor $A\in|A|$. The exact sequence
$$
0\to\cO(mF-A)\to \cO(mF)\to \cO_A(mF_{|A})\to 0
$$
yields 
$$
0\to H^0(X,mF-A)\to H^0(X,mF)\to H^0(A,mF_{|A})
$$
where $h^0(X,mF)\ge c m^n$ and $h^0(A,mF_{|A})\le Cm^{n-1}$. Therefore $H^0(X,mF-A)$
has a section $\sigma$ for $m$ large. If $E$ is the zero divisor of $\sigma$ we find
$\cO(mF-A)\simeq\cO(E)$, hence $\cO(mF)\simeq\cO(E+A)$.

\noindent
(b)${}\Rightarrow{}$(c) Tahe a smooth hermitian metric $h_A$ on $\cO(A)$ with 
$\omega=\Theta_{A,h_A}>0$ and the singular metric $h_E$ on $\cO(E)$ given by the canonical 
section of divisor $E$, so that $\Theta_{E,h_E}=[E]\ge 0$. Then the resulting metric 
$h_F=(h_Ah_E)^{1/m}$ on $F$ satisfies
$$
\Theta_{F,h_F}={1\over m}\big(\Theta_{A,h_A}+\Theta_{E,h_E}\big)\ge {1\over m}\omega.
$$
(c)${}\Rightarrow{}$(a) is a standard consequence of $L^2$ estimates for 
the $\overline\partial$ operator acting on sections of $\cO(mF)$.\square
\medskip

\claim 0.38.~Definition| A projective manifold $X$ is said to be of general type if
its canonical bundle $K_X=\Omega^n_X=\det(T_X^*)$ is big.
\endclaim

if $X'$ is birational to $X$, then $H^0(X',mK_{X'})\simeq H^0(X,mK_X)$, so the concept
is birationally invariant by 0.37~(a). If $X\subset \bP^{n+1}$ is a smooth hypersurface
of degree~$d$, then $K_X\simeq\cO(d-n-2)_{|X}$, hence $X$ is of general type if and only
if $d\ge n=3$. More generally if $X=H_{d_1}\cap\ldots\cap H_{d_s}$ is a smooth
complete intersection of multidegree $(d_1,\ldots,d_s)$ in $\bP^{n+s}$, then
$K_X\simeq\cO(\sum d_j-n-s-1)_{|X}$, hence $X$ is of general type if and only if
$\sum d_j\ge n+s+2$.

\subsection \S0.F. Poincar\'e metric|

Let $D(0,R)\subset\bC$ be the disk of radius $R$ in the complex plane, with complex
coordinate~$t$. On $D(0,R)$, there exists a particular riemannian metric
$$
ds_P^2={R^{-2}dt\otimes d\overline t \over (1-|t|^2/R^2)^2}=
{R^2dt\otimes d\overline t \over (R^2-|t|^2)^2},\qquad t\in D(0,R)
$$
which enjoys several interesting properties: it is called the 
{\em Poincar\'e metric} (of course, it is most often enough consider the case 
of the unit disk $\Delta=D(0,1)$). One of its main properties is the invariance 
by the  group of holomorphic automorphisms $\Aut(D(0,R))$. Let us recall that 
these automorphisms are of the form
$$
t\mapsto \lambda{t-a\over 1-\overline a\,t/R^2},\qquad\lambda\in\bC,~
|\lambda|=1,~~a\in D(0,R).
$$
The corresponding geodesic distance obtained by integrating along rectifiable
paths and taking the infimum is given by
$$
d_P(a,b)=\tanh^{-1}\left|{R^{-1}(a-b)\over 1-a\overline b/R^2}\right|,\qquad a,b\in D(0,R)
\qquad\hbox{Poincar\'e distance)}.
$$
The distance $d_P$ is complete on $D(0,R)$ and invariant under $\Aut(D(0,R))$.
The geodesics are diameters of $D(0,R)$ and circle arcs that are orthogonal 
to $\partial D(0,R)$. It is also well known that all holomorphic maps
$g:D(0,R)\to D(0,R)$ are (weak) contractions with respect to~$ds_P^2$, i.e.
$$
{|g'(t)|/R\over 1-|g(t)|^2/R^2}\le {1 \over 1-|t|^2/R^2},\qquad t\in D(0,R)
\qquad\hbox{(Schwarz-Pick lemma)}.\leqno(0.39)
$$
As a consequence
$$
d_P(g(a),g(b))\le d_P(a,b)\qquad\hbox{on $D(0,R)$}.\leqno(0.40)
$$

\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. More generally:

\claim 1.7.~Proposition|The complement $X=\bP^2\ssm\bigcup_{i=1,...,5}\ell_i$ of five lines 
in generic position in $\bP^2$ $($any three of them non concurrent$)$ 
is hyperbolic.
\endclaim

\proof. By a linear change of coordinates, we can achieve that $\ell_1$ is the line at 
infinity, so that $\bP^2\ssm\ell_1\simeq\bC^2$, and that $\ell_2=\{x=0\}$, $\ell_3=\{y=0\}$,
$\ell_4=\{x+y=1\}$ in the coordinates $(x,y)$ of $\bC^2$. Then there is a holomorphic
map
$$
\Phi_{1,2,3,4}:\bP^2\ssm\bigcup_{i=1,...,4}\ell_i\to\Gamma=\bC\ssm\{0,1\},\qquad (x,y)\mapsto -y/x.
$$
In a similar way, for any $4$-tuple $j=(j_1,j_2,j_3,j_4)$ of distinct indices in
$\{1,2,3,4,5\}$, we get a holomorphic map
$$
\Phi_j:\bP^2\ssm\bigcup_{i=j_1,...,j_4,\,i\ne j}\ell_i\to\Gamma=\bC\ssm\{0,1\}.
$$
The monotonicity property implies $d^K_X(p,q)\ge\max_jd^K_\Gamma(\Phi_j(x),\Phi_j(y))$.
As $\Gamma=\bC\ssm\{0,1\}$ is hyperbolic and the fibers of the $\Phi_j$ are distinct
pencils of lines, one easily concludes that $d^K_X$ separates points, hence $X$ is 
hyperbolic.\square

\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 1.8.~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~$R$.\square

\claim 1.9.~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~$t$. 

\proof. The existence of $\psi_\nu$ follows from the Brody reparametrization lemma applied with $f=f_\nu$ and $\varepsilon=1/\nu$. Let us denote $g_\nu=f_\nu\circ\psi_\nu:D(0,R_\nu)\to X$. 
Lemma~1.8 implies
$$
\|g_\nu'(0)\|_\omega=1,\quad \|g_\nu'(t)\|_\omega\le{1\over 1-|t|^2/R_\nu^2}\quad\hbox{and}\quad R_\nu\ge(1-1/\nu)\|f'_\nu(0)\|_\omega\to+\infty.
$$
The conclusion follows from the Ascoli-Arzel\`a theorem: the family $\{g_\nu\}$ is with values in a compact space (hence pointwise bounded) and the estimate on its derivatives shows that it is equi-Lipschitz on any given compact subset $K$ of $\bC$ (for $\nu\ge\nu_0(K)$), hence equicontinuous there. Thus, by a diagonal process, one gets a subsequence that converges to an entire map $g:\bC\to X$; moreover, $\|g'(0)\|_\omega=\lim \|g_\nu'(0)\|_\omega=1$, so that $g$ is non constant and also $\|g'(t)\|_\omega=\lim \|g'_\nu(t)\|_\omega\le 1$.\square

\claim 1.10.~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~1.9, 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~1.9 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 $f(\bC)$; this is in fact extremely frequent in
dimension $n\ge 3$. For instance, if $a\in\bR\ssm\bQ$, one can
check that the curve $f(t)=(e^{2\pi i t^2},e^{2\pi i at^2},1)\in\bC^3\subset\bP^3$ 
reparametrized by $\smash{\psi_\nu(t)=\nu^{1/2}+{1\over 2 }\nu^{-1/2}t}$ produces
limiting curves of the form 
$$
g_\theta(t)=\lim f\circ\psi_\nu(t)=(e^{2\pi i t},e^{2\pi i(at+\theta)}, 1),\qquad
\theta\in\bR,
$$
and the angle
$\theta=\lim a\nu \mod\bZ$ can be taken to be arbitrary thanks to the irrationality of~$a$.
However, $g_\theta(\bC)$ is disjoint from $f(\bC)$ when $\theta\notin\bZ+a\bZ$. 
Winkelmann [Win07] has
given a more 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. The following example shows that in the non compact case one may have 
Brody hyperbolic domains which are not Kobayashi hyperbolic.

\claim 1.11. Example|{\rm 
Consider the domain in $\bC^2$ defined by
$$
D=\{(z,w)\in\bC^2\,;\;|z|<1,~|zw|<1\}\ssm\{(0,w)\,;\;|w|\ge 1\}.
$$
The mapping $\Phi:D\to\bC^2$ which sends $(z,w)\mapsto (z,zw)$ has as
image the unit bidisk and is one-to-one except on the set
$\{z=0\}$. If $f:\bC\to D$ is holomorphic, then $\Phi\circ f$
is constant by Liouville's theorem. Thus, either $f$ is constant or
$f$ maps $\bC$ into the set $\{(0,w)\in D\}$. But this set is
equivalent to the unit disk, hence $f$ is constant in any case.
Therefore $D$ is Brody hyperbolic.

Now, since $\Phi$ is holomorphic, it is distance decreasing with respect to
the Kobayashi pseudodistance, so we have that $d^K_D(p,q)>0$ for $p\ne q$
unless both $p$ and $q$ lie in the subset $\{(0,w)\in D\}$. Suppose
then that we are in this case, and consider the points $p=(0,0)$, $q=(0,a)$,
$p_\nu=(1/\nu,a)$, $q_\nu=(1/\nu,a)$ with $|a|<1$. The holomorphic
mapping $f_\nu:t\mapsto (1/\nu,\nu t)$
maps  the unit disk $\Delta$ into $D$, and we have $p_\nu=f_\nu(0)$, $q_\nu=f_\nu(a/\nu)$, 
hence $d^K_D(p_\nu,q_\nu)\le d_P(0,a/\nu)\to 0$ as $\nu\to+\infty$. However, it is
equally clear that $\lim d^K_D(p,p_\nu)=\lim d^K_D(q,q_\nu)=0$, therefore
$d^K_D(p,q)=0$ by the triangle inequality. This implies that $D$ is not
Kobayashi hyperbolic. A similar construction of a Hartogs domain 
$D'=\{|w|<e^{-u(z)},\;|z|<1/2\}$ where  
$$u(z)=u_{\alpha,\beta,\gamma}(z)=
1+\sum_{\nu\ge \nu_0} \beta^{-\nu}\max(\log|z-e^{-\gamma^\nu}|,-\alpha^\nu),
\qquad\alpha>\beta>\gamma>1,~\nu_0\gg 1
$$ 
is subharmonic, everywere finite and not locally bounded near $0$, produces
a Stein domain $D'\subset D$ that is also Brody hyperbolic but not Kobayashi
hyperbolic. In fact we have $u(0)=1-\sum_{\nu\ge\nu_0}(\gamma/\beta)^\nu\ge 0$ 
for $\nu_0$ large enough, and
$$\beta^{-\nu}\max(\log|z-e^{-\gamma^\nu}|,-\alpha^\nu)\ge {1\over\nu^2}\log|z|\leqno(*)$$
(so that $u(z)\ge\log|z|$ and $D'\subset D$) : this is clear
if $\nu^{-2}\log|z|\le -(\alpha/\beta)^\nu$, i.e.\ $|z|\le\exp(-\nu^2(\alpha/\beta)^\nu)\,$;
on the other hand, for $|z|\ge\exp(-\nu^2(\alpha/\beta)^\nu)$, we certainly have 
$|z|\ge 2\,e^{-\gamma^\nu}$ when we choose $\alpha/\beta<\gamma$ and $\nu_0\gg 1$, 
hence $\log|z-e^{-\gamma^\nu}|\ge \log|{1\over 2}z|\ge 2\log|z|$ and $(*)$ holds as well;
one can take for instance $\alpha=4$, $\beta=3$, $\gamma=2$ to achieve these conditions.
Now, for $\varepsilon_\nu=e^{-\gamma^\nu}\to 0$, we find
$u(\varepsilon_\nu)\le 1-(\alpha/\beta)^\nu\to -\infty$ and the sequence of analytic disks
$f_\nu(t)=(\varepsilon_\nu,e^{-u(\varepsilon_\nu)}t)$, $t\in \Delta$, contradicts
Kobayashi hyperbolicity.\square}
\endclaim

\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 1.12. 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 1.13.~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 $\bfk_X$,
so we only give a brief outline of the ideas. In order to prove the
upper semicontinuity, let $\xi_0\in V_{x_0}$ and $\varepsilon>0$ be
given. Then there is a curve $f:\Delta\to X$ tangent to $V$ such that
$f(0)=x_0$ and $\lambda\,f'(0)=\xi_0$ with
$0<\lambda<\bfk_X(\xi_0)+\varepsilon$. Take $\lambda=1$ for simplicity,
and replace $\xi_0$ by $\lambda^{-1}\xi_0$. We may assume that $f$ is a
proper embedding, otherwise we replace $(X,V)$ by $(X',V')=
(X\times\Delta,\pr_1^* V\oplus\pr_2^* T_\Delta)$, $f$ by
$f\times\Id_\Delta$, $\xi_0$ by $\xi_0\oplus 1$, and use a monotonicity
argument for the projection $\pr_1:X'\to X$. If $f$ is an embedding,
then $f(\Delta)$ is a Stein submanifold of~$X$, and thus $f(\Delta)$
has a Stein neighborhood~$\Omega$ by a well-known result due to [Siu76]
(cf.\ also [Dem90a] for more general results). As $\Omega$ is Stein, 
there exists a section $\theta\in H^0(\Omega,\cO(V))$ extending $f'\in
H^0(f(\Delta),\cO(V))$. The map $f$ can be viewed as the solution of
the differential equation $f'=\theta(f)$ with initial value $f(0)=x_0$.
Take a small perturbation $g'=\theta_\eta(g)$ with initial value
$g(0)=x$, where $\theta_\eta=\theta+\sum\eta_j s_j$ and $s_1\ld s_N$
are finitely many sections of $H^0(\Omega,\cO(V))$ which generate $V$
in a neighborhood of~$x_0$. We can achieve that $g'(0)=\theta_\eta(x)$
is equal to any prescribed vector $\xi\in V_x$ close to
$\xi_0=\theta(x_0)$, and the solution $g$ exists on
$(1-\varepsilon)\Delta$ if the perturbation is small enough. We
conclude that $\bfk_{(X,V)}$ is upper semicontinuous by considering
\hbox{$t\mapsto g((1-\varepsilon)t)$}.

If there exists a non constant entire curve $g:\bC\to X$ tangent
to~$V$, it is clear that $\bfk_{(X,V)}(g'(t))\equiv 0$, hence $(X,V)$
cannot be hyperbolic. Conversely, if $X$ is compact and if there are no
non constant entire curves \hbox{$g:\bC\to X$} tangent to $V$, the Brody
lemma implies that there is an absolute bound
\hbox{$\|f'(0)\|_\omega\le C$} for all holomorphic maps $f:\Delta\to X$
tangent to~$V\,$; hence $\bfk_{(X,V)}(\xi)\ge C^{-1}\|\xi\|_\omega$ and
$(X,V)$ is infinitesimally hyperbolic. By reparametrizing $f$ with an
arbitrary automorphism of~$\Delta$, we find $\|f'(t)\|_\omega\le
C/(1-|t|^2)$. The space of maps $f:\Delta\to X$ tangent to $V$ is
therefore compact for the topology of uniform convergence on compact
subsets of~$\Delta$, thanks to Ascoli's theorem. We easily infer from
this that $\bfk_{(X,V)}$ is lower semicontinuous on~$V$.\square

Another easy observation is that the concept of $p$-measure hyperbolicity gets 
weaker and weaker as $p$ increases~:

\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

\proof. Asserting that  $(X,V)$ is $p$-measure hyperbolic means that 
for all maps $f:\bB_p\to X$ tangent to~$V$ with $f(0)=x$, there is
a uniform upper bound $\Vert\Lambda^pf_*(0)\Vert_\omega\le A$ for
$\Lambda^pf_*(0):\Lambda^pT_{\bB_p}\to\Lambda^pV$ with respect to a given
hermitian metric $\omega$ on $X$. Consider $g:\bB_{p+1}\to X$ tangent 
to $V$ with $g(0)=x$ fixed. Let us restrict $g$ 
to all $p$-dimensional balls $\bB_{p+1}\cap H$ where $H$ is a hyperplane 
in~$\bB^{p+1}$. Applying this to 
$f=g_{\restriction\bB_{p+1}\cap H}$ and $H$ arbitrary, one gets 
a bound for $\Vert(\Lambda^pg_*(0))_{\restriction H}\Vert_\omega$ and
therefore a bound for $\Vert\Lambda^pg_*(0)\Vert_\omega$. However, there are
orthonormal bases of $\bC^{p+1}$ and $V\simeq\bC^r$ such that
$u:=g_*(0):\bC^{p+1}\to V$ has a diagonal matrix with diagonal entries 
$\lambda_j\in\bR_+$ (the $\lambda_j$'s are the square roots of the 
eigenvalues of the hermitian form $\tau\mapsto \Vert u(\tau)\Vert_\omega^2$). 
Then
$$\eqalign{
&\Vert\Lambda^ku\Vert_\omega^2=\sum_{i_1<\ldots<i_k}
(\lambda_{i_1}\ldots\lambda_{i_k})^2,\qquad\hbox{especially}\quad
\Vert\Lambda^{p+1}u\Vert_\omega^2=(\lambda_1\ldots\lambda_{p+1})^2,\quad
\hbox{hence}\cr
&\Vert\Lambda^{p+1}u\Vert_\omega^{2p}=
\prod_{j=1}^{p+1}(\lambda_1\ldots\widehat{\lambda_j}\ldots\lambda_{p+1})^2
\le\Vert\Lambda^pu\Vert_\omega^{2(p+1)},\qquad
\hbox{i.e.}\quad
\Vert\Lambda^{p+1}u\Vert_\omega\le \Vert\Lambda^pu\Vert_\omega^{1+1/p}.\cr}
$$
This implies our claim. In fact, the proof also shows that if 
$\smash{\bfe^p_{(X,V)}}(\xi)\ge A^{-1}
\Vert\xi\Vert_\omega$ for decomposable $\xi\in \Lambda^pV$, then
$\smash{\bfe^{p+1}_{(X,V)}}(\xi)\ge A^{-(1+1/p)}\Vert\xi\Vert_\omega$ for decomposable 
$\xi\in \Lambda^{p+1}V$.\square\medskip

We conclude this section by showing that (relative) hyperbolicity is also
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

\proof. Take a sequence of non hyperbolic fibers $(X_{s_\nu},V_{s_\nu})$
with $s_\nu\to s$ and fix a hermitian metric $\omega$ on $\cX$. By
the Brody lemma, there is a sequence of entire holomorphic maps
$g_\nu:\bC\to X_{s_\nu}$ tangent to $V_{s_\nu}$, such that
$\|g_\nu'(0)\|_\omega=1$ and $\|g'_\nu\|\le 1$. Ascoli's theorem
shows that there is a subsequence of $(g_\nu)$ converging uniformly
to a limit $g:\bC\to X_s$, tangent to~$V_s$, with $\|g'(0)\|_\omega=1$.
Hence $(X_s,V_s)$ is not hyperbolic, and the collection of non
hyperbolic fibers is closed in~$S$.\square

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. Statement and proof of The Ahlfors-Schwarz lemma|

Proposition 4.1 can be generalized a little bit further by means of
the Ahlfors-Schwarz lemma (see e.g.\ [Lang87]).

\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

\proof. It is of course sufficient to deal with the more general case of
a ball in $\bC^p$. First assume that $\gamma$ is smooth and
positive definite on $\ol B(0,R)$. Take a point $t_0\in B(0,R)$ at which
$(1-|t|^2/R^2)^{p+1}\det(\gamma(t))$ is maximum. The logarithmic
$i\,\ddbar$-derivative of this function at $t_0$ must be${}\le0$, hence
$$
i\,\ddbar\log\det\gamma(t)_{t=t_0}-
(p+1)\,i\,\ddbar\log(1-|t|^2/R^2)^{-1}_{t=t_0}\le 0.
$$
The hypothesis on the Ricci curvature implies
$$
A^p\,\gamma(t_0)^p\le\big(i\,\ddbar\log\det\gamma(t)_{t=t_0}\big)^p
\le(p+1)^p\,\big(i\,\ddbar\log(1-|t|^2/R^2)^{-1}_{t=t_0}\big)^p.
$$
An easy computation shows that the determinant of
$i\,\ddbar\log(1-|t|^2/R^2)^{-1}$ is equal to
$R^{-2p}(1-|t|^2/R^2)^{-p-1}$. From this, we conclude that
$$
(1-|t|^2/R^2)^{p+1}\det\gamma(t)\le(1-|t_0|^2/R^2)^{p+1}\det\gamma(t_0)
\le\Big({p+1\over AR^2}\Big)^p.
$$
If $\gamma$ is not smooth, we use a regularization argument. Namely, we
shrink $R$ a little bit and look at the maximum of the function
$$
u(t)=(1-|t|^2/R^2)^{p+1}\exp\big(\rho_\varepsilon*\log\det\gamma(t)
\big)
$$
where $(\rho_\varepsilon)$ is a family of regularizing kernels.
The argument goes through because
$$
i\,\ddbar(\rho_\varepsilon*\log\det\gamma)\ge A\,\rho_\varepsilon
*\gamma
$$
and $\log\det(\rho_\varepsilon*\gamma)\ge\rho_\varepsilon*
\log\det\gamma$ by concavity of the $\log\det$ function.\square

\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
$$
\leqalignno{
|\eta|_{h_1}^2=|\eta|_h^2=1+|\xi|^2&{}-\sum_{1\le j,k\le n}c_{jkrr}
z_j\ol z_k+O((|z|+|\xi|)^3),\cr
\Theta_{h_1}(\cO_{P(V)}(-1))_{(x_0,[v_0])}
&=-{i\over 2\pi}\ddbar\log|\eta|_{h_1}^2\cr
&={i\over 2\pi}\Big(\sum_{1\le j,k\le n}c_{jkrr}dz_j\wedge d\ol z_k
-\sum_{1\le\lambda\le r-1}d\xi_\lambda\wedge d\ol\xi_\lambda\Big).
&(5.15)}
$$
Now, the connection $\nabla_h$ on $V$ defines on $\swt X|=P(V)$ a
$C^\infty$ decomposition
$$
T_{\swt X|}={}^H\!T_{\swt X|}\oplus{}^V\!T_{\swt X|},\qquad
{}^H\!T_{\swt X|,(x,[v])}\simeq T_{X,x},\quad
{}^V\!T_{\swt X|,(x,[v])}\simeq T_{P(V_x),[v]},
$$
in horizontal and vertical components. With respect to this
decomposition, (5.15) can be rewritten as
$$
\langle\Theta_{h_1}(\cO_{P(V)}(-1))\rangle_{(x_0,[v_0])}(\tau)
=\langle\Theta_{V,h}\rangle_{x_0}({}^H\!\tau\otimes v_0)-|{}^V\!\tau|_\FS^2
\leqno(5.16)
$$
where $|~~|_\FS$ is the Fubini-Study metric along the fibers $T_{P(V_x)}$.
By definition of $\swt V|$, we have $\swt V|_{(x,[v])}\subset V_x\oplus
T_{P(V_x),[v]}$ with respect to the decomposition. By this observation, if
we equip $P(V)$ with the Fubini-Study metric rescaled by~$\rho^2>0$, the
metric $h$ on $V$ induces a canonical hermitian metric $\swt h|_\rho$
on~$\swt V|$ such that
$$
|w|^2_{\wt h_\rho}=|{}^H\!w|^2_h+\rho^2|{}^V\!w|^2_h\qquad
\hbox{for $w\in\swt V|_{(x_0,[v_0])}$},
$$
where ${}^H\!w\in\bC v_0\subset V_{x_0}$ and ${}^V\!w\in T_{P(V_{x_0}),[v_0]}$
is viewed as an element of $v_0^\perp\subset V_{x_0}$. A computation (left to
the reader) gives the formula
$$
\leqalignno{
\langle\Theta_{\swt h|_\rho}(\swt V|)\rangle_{(x_0,[v_0])}(\tau\otimes w)
&=\langle\Theta_{V,h}\rangle_{x_0}({}^H\!\tau\otimes v_0)~
(|{}^H\!w|_h^2-\rho^2|{}^V\!w|_h^2)\cr
&\phantom{{}={}}{+}\;
\rho^2\langle\Theta_{V,h}\rangle_{x_0}({}^H\!\tau\otimes{}^V\!w)\cr
&\phantom{{}={}}{+}\;
\rho^2\big(|\langle{}^V\!\tau,{}^V\!w\rangle_h|^2+|{}^V\!\tau|_h^2
|{}^V\!w|_h^2\big)-|{}^V\!\tau|_h^2|{}^H\!w|_h^2&(5.17)\cr
&\phantom{{}={}}{+}\;
O(\rho)|\tau|_\omega^2|w|^2_{\wt h_\rho},\qquad
\tau\in T_{\swt X|},~w\in\swt V|,\cr}
$$
where $|\tau|_\omega^2$ is computed from a fixed hermitian metric $\omega$
on~$T_X$.

\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%
\bottomnote{*}{{Also, the paper [LaTh96] by Laksov and Thorup deals in depth with 
certain algebraic-theoretic properties of 
jet differentials. The formalism of ``higher order'' differentials has
been part of the mathematical folklore during the 18th and 19th centuries
(without too much concern, in those times, on the existence of precise 
definitions$\,!$). During the 20th century, this formalism almost disappeared,
before getting revived in several ways. See e.g.\ the interested article by
P.A.~Meyer [Mey89], which was originally motivated by applications to
probability theory.}}.

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.~Proposition|Let $\bfa=(a_1\ld a_k)\in\bZ^k$ and
$m=a_1+\cdots+a_k$.
\smallskip
\item{\rm i)} We have the direct image formula
$$
(\pi_{0,k})_*\cO_{P_kV}(\bfa)\simeq\cO(\ol F^\bfa E_{k,m}V^*)
\subset\cO(E_{k,m}V^*)
$$
where $\ol F^\bfa E_{k,m}V^*$ is the subbundle of polynomials
\hbox{$Q(f',f''\ld f^{(k)})\in E_{k,m}V^*$\kern-1.8pt} involving only
monomials $(f^{(\bullet)})^\ell$ such that
$$
\ell_{s+1}+2\ell_{s+2}+\cdots+(k-s)\ell_k\le a_{s+1}+\cdots+a_k
$$
for all $s=0\ld k-1$.
\smallskip
\item{\rm ii)} If $a_1\ge 3a_2\ld a_{k-2}\ge 3a_{k-1}$ and $a_{k-1}\ge
2a_k\ge 0$, the line bundle $\cO_{P_kV}(\bfa)$ is relatively nef over~$X$.
\smallskip
\item{\rm iii)} If $a_1\ge 3a_2\ld a_{k-2}\ge 3a_{k-1}$ and $a_{k-1}>
2a_k>0$, the line bundle $\cO_{P_kV}(\bfa)$ is relatively ample
over~$X$.\vskip0pt
\endclaim

\proof. i) By (7.18), we find a sheaf injection
$$
(\pi_{0,k})_*\cO_{P_kV}(\bfa)\hookrightarrow
(\pi_{0,k})_*\cO_{P_kV}(m)=\cO(E_{k,m}V^*).
$$
Given a section $\sigma$ of $\cO_{P_kV}(\bfa)$ over a fiber $P_kV_x$,
the associated polynomial 
$$Q(f',f''\ld f^{(k)})\in E_{k,m}V^*_x$$
is given by the identity
$$
Q(f',f''\ld f^{(k)})=\sigma(f_{[k]}(0))\cdot(f'(0))^{a_1}
\cdot(f_{[1]}'(0))^{a_2}\cdots(f_{[k-1]}'(0))^{a_k}.
$$
Indeed, we see this from (7.13) and from the fact that $f_{[k-1]}'(0)$
is mapped to $f_{[j-1]}'(0)$ by the projection morphism
$$
(\pi_{j-1,k-1})_*:\cO_{P_kV}(-1)\to\pi_{j,k}^*\cO_{P_jV}(-1)
$$
(cf.\ (6.8)), which is dual to the corresponding morphism (7.18). Now,
we prove the inclusion $(\pi_{0,k})_*\cO_{P_kV}(\bfa)
\subset\cO(\ol F^\bfa E_{k,m}V^*)$ by induction on~$k$. For $s=0$,
the desired inequality comes from the weighted homogeneity condition, hence
we may assume~$s\ge 1$. Let $f$ run over all regular germs having their first
derivative $f'(0)$ fixed. This means that $\sigma$ is viewed as a section of
$\pi_{2,k}^*\cO_{P_2V}(a_2)\otimes\cdots \otimes\cO_{P_kV}(a_k)$
on the fibers of the projection $P_kV=P_{k-1}V_1\to X_1=P_1V$.
Then we get a polynomial $Q_1\in E_{k-1,m-a_1}V_1^*$ such that
$$
Q_1(f_{[1]}',f_{[1]}''\ld f_{[1]}^{(k-1)})=Q(f',f''\ld f^{(k)}).
$$
In the affine chart $f'_r\ne 0$, the map $f_{[1]}$ is defined in
coordinates by 
$$f_{[1]}\simeq(f_1\ld f_n\,;\,f_1'/f_r'\ld f_{r-1}'/f_r').$$
Its derivative $f_{[1]}'\in V_1$ can thus be described by
$f_{[1]}'\simeq\big((f_1'/f_r')'\ld(f_{r-1}'/f_r')',f_r'\big)$, by taking
$r-1$ vertical components and a horizontal one. All this becomes much
simpler if we replace $f$ by $g=f\circ f_r^{-1}$, since $g_r(t)=t$ and
$g_r'(t)=1$. Then we get
$$
\eqalign{
(g',g''\ld g^{(k)})&\simeq\big((g_1'\ld g_{r-1}',1),(g_1''\ld g_{r-1}'',0)
\ld(g_1^{(k)}\ld g_{r-1}^{(k)},0)\big),\cr
(g_{[1]}',g_{[1]}''\ld g_{[1]}^{(k)})&\simeq
\big((g_1''\ld g_{r-1}'',1),(g_1'''\ld g_{r-1}''',0)\ld
(g_1^{(k)}\ld g_{r-1}^{(k)},0)\big)\cr}
$$
in the corresponding charts of $J_kV$ and $J_{k-1}V_1$. The inequality
(7.19$\,$i) for the monomials $(g^{(\bu)})^\ell$ of $Q(g',g''\ld g^{(k)})$ 
follows clearly from the corresponding inequality on the monomials
$(g_{[1]}^{(\bu)})^\ell$ of $Q_1$, when $(k,s)$ is replaced by $(k-1,s-1)$.
Now, thanks to (7.12), we get $Q(f',f''\ld f^{(k)})=(f_r')^m
Q(g',g''\ld g^{(k)})$, and the desired inequality (7.19$\,$i)
for the monomials $(f^{(\bu)})^\ell$ follows easily. In the opposite
direction, if we are given a section $Q(f',f''\ld f^{(k)})\in\cO(\ol F^\bfa
E_{k,m}V^*)$, we see by induction on $k$ that $Q$ defines
a section of
$$
\cO_{P_1V}(a_1)\otimes(\pi_{1,k})_*
\big(\pi_{2,k}^*\cO_{P_2V}(a_2)\otimes\cdots\otimes\cO_{P_kV}(a_k)\big)
$$
on $P_1V$, and we conclude that we get a section of
$(\pi_{0,k})_*\cO_{P_kV}(\bfa)$ by taking the direct image by
$(\pi_1)_*$.
\medskip

\noindent ii-iii) By induction on $k$, we construct a relatively ample
line bundle $L_{k-1}$ on $P_{k-1}V$ such that
$\cO_{P_kV}(1)\otimes\pi_k^* L_{k-1}$ is relatively nef; by
definition, this is equivalent to saying that the vector bundle
$V_{k-1}^*\otimes L_{k-1}$ is relatively nef (for the notion of a
nef vector bundle, see e.g.\ [DPS94]). Since $\cO_{P_1V}(1)$ is relatively
ample, we can start with $L_0=\cO_X$. Suppose that $L_{k-1}$ has been
constructed. The dual of (6.4) yields an exact sequence
$$
0\lra\cO_{P_kV}(1)\lra V_k^*\lra T_{P_kV/P_{k-1}V}^*\lra 0.
$$
As an extension of nef vector bundles is nef, it is enough to select $L_k$
in such a way that
$$
\cO_{P_kV}(1)\otimes L_k\quad\hbox{and}\quad
T^*_{P_kV/P_{k-1}V}\otimes L_k\quad\hbox{are relatively nef.}\leqno(7.20)
$$
By taking the second wedge power of the central term in $(6.4')$, we get an
injection
$$
0\lra T_{P_kV/P_{k-1}V}\lra \Lambda^2\big(\pi_k^* V_{k-1}\otimes
\cO_{P_kV}(1)\big).
$$
By dualizing and twisting with $\cO_{P_{k-1}V}(2)\otimes\pi_k^*
L_{k-1}^{\otimes 2}$, we find a surjection
$$
\pi_k^*\Lambda^2(V_{k-1}^*\otimes L_{k-1})\lra T_{P_kV/P_{k-1}V
}^*\otimes\cO_{P_kV}(2)\otimes\pi_k^* L_{k-1}^{\otimes 2}\lra 0.
$$
As $V_{k-1}^*\otimes L_{k-1}$ is relatively nef by the induction
hypothesis, we obtain that its quotient $T_{P_kV/P_{k-1}V}^*\otimes
\cO_{P_kV}(2)\otimes\pi_k^* L_{k-1}^{\otimes 2}$
is also relatively nef. Hence Condition (7.17) is achieved if we take
$L_k\ge\pi_k^* L_{k-1}$ and
$L_k\ge\cO_{P_kV}(2)\otimes\pi_k^* L_{k-1}^{\otimes 2}$ (the ordering
relation $\ge$ is the one given by the cone of relatively nef line bundles).
We need only define $L_k$ inductively by
$$
L_k=\cO_{P_kV}(2)\otimes\pi_k^* L_{k-1}^{\otimes 3}.
$$
The relative ampleness of $L_k$ is then clear by induction, since
$\cO_{P_kV}(1)\otimes\pi_k^* L_{k-1}$ is relatively nef over $X$ and
relatively ample over $P_{k-1}V$. The resulting formula for $L_k$ is
$$
L_k=\cO_{P_kV}\big((2\cdot 3^{k-1},2\cdot 3^{k-2}\ld 6,2)\big).
$$
By definition, we then find
$$
\cO_{P_kV}(1)\otimes\pi_k^* L_{k-1}=
\cO_{P_kV}\big((2\cdot 3^{k-2},2\cdot 3^{k-3}\ld 6,2,1)\big)
\qquad\hbox{relatively nef}.
$$
These properties imply ii) and iii) by taking suitable convex
combinations.\square 

\claim 7.21.~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 7.22.~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], so we will only briefly
sketch the details. If $L$ is big, then $X$ is Moishezon and we can
even assume that $X$ is projective algebraic after taking a suitable
modification $\swt X|$ (apply Hironaka [Hir64]; observe moreover that
the direct image of a strictly positive current is strictly positive).
So, assume that $X$ is projective algebraic. Then it is well-known
that some large multiple of $L$ can be written as $L^{\otimes m}=
\cO_X(D+A)$ with divisors $D,\,A$ such that $D$ is effective and $A$
ample. The invertible sheaf $\cO_X(D)$ can be viewed as a subsheaf of
the sheaf of meromorphic functions. We get a singular metric $|s|^2$
on sections of $\cO_X(D)$ by just taking the square of the modulus of
$s$ viewed as a complex valued (meromorphic) function. By the
Lelong-Poincar\'e equation, the curvature current of that metric is
equal to the current of integration~$[D]\ge 0$ over the divisor~$D$.
We thus get $\Theta_L={1\over m}([D]+\Theta_A)\ge {1\over
  m}\Theta_A\gg 0$ for a suitable choice of the metric on~$\cO_X(A)$.
In the other direction, if $\Theta_{L,h}$ is positive, one can
construct a ``lot~of'' sections in $H^0(X,L^{\otimes m})$, $m\gg 0$,
by using H\"ormander's $L^2$~estimates; the H\"ormander-Bombieri-Skoda
technique implies that these sections can be taken to have arbitrary
jets at all points in a given finite subset of $X\ssm\Sigma$, if
$\Sigma$ is the degeneration set of~$h$. This also proves property~iii).

\noindent
ii) The assumption $\Sigma_m\ne X$ shows that there is a generically
finite meromorphic map from $X$ to an algebraic variety, and this implies
again that $X$ is Moishezon. By blowing-up the ideal 
$$
\cI_m=\Im\big(H^0(X,L^{\otimes m})\otimes\cO_X(L^{\otimes -m})\to\cO_X\big)
\subset\cO_X
$$
and resolving the singularities, we obtain a smooth modification
$\mu:\wt X\to X$ and a line bundle $\wt L=\mu^*(L^{\otimes m})
\otimes \cO_{\wt X}(-E)$ $($where $E$ is a $\mu$-exceptional divisor
with support in $\mu^{-1}(\Sigma_m)$, such that $\wt L$ is base point
free; after possibly blowing-up again, we may assume furthermore that 
$\swt X|$ is projective algebraic.  Clearly, it is enough to prove the
result for $\wt L$, and we are thus reduced to the case when $L$ is
base point free and $X$ is projective algebraic. We may finally assume
that $G$ is very ample (other we add a large ample divisor to $G$ to
make it very ample). In this situation, we have a holomorphic map
$\Phi_m:X\to\bP^N$ such that $L^{\otimes m}=\Phi_m^{-1}(\cO(1))$, and
$\Phi_m$ is finite-to-one outside~$\Sigma_m$. Hence, if $x\in
X\ssm\Sigma_m$, the set $\Phi_m^{-1}(\Phi_m(x))$ is finite, and we can
take a smooth divisor $D\in|G|$ such that
$D\cap\Phi_m^{-1}(\Phi_m(x))=\emptyset$. Thus
$\Phi_m(D)\not\ni\varphi_m(x)$ in~$\bP^N$. It follows that there
exists a hypersurface $H=\sigma^{-1}(0)\in|\cO_{\bP^N}(k)|$ of
sufficiently large degree~$k$, such that $H$ contains $\Phi_m(D)$ but
does not pass through~$\Phi_m(x)$. Then $\Phi_m^*\sigma$ can be
viewed as a section of $\Phi_m^*\cO_{\bP^N}(k)\otimes\cO_X(-D)=
L^{\otimes km}\otimes G^{-1}$, and $\Phi_m^*\sigma$ does not
vanish at~$x$. By the Noetherian property, there exists $k_0$ such
that the base locus of $L^{\otimes km}\otimes G^{-1}$ is contained in
$\Sigma_m$ for $k\ge k_0$ large. Claim~ii) follows.\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 (5.16) 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
Definitions 1.2 and 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

\section{\S9. Algebraic criterion for the negativity of jet curvature}

Our goal is to show that the negativity of $k$-jet curvature
implies strong restrictions of an algebraic nature, similar to
property 3.1~ii). Using this we give examples, for any prescribed
integer $k$, of hyperbolic projective surfaces which do not admit
any $k$-jet metric of negative jet curvature.

\claim 9.1.~Theorem|Let $(X,V)$ be a compact complex directed
manifold and let $\omega$ be a hermitian metric on~$X$. If $(X,V)$ has
negative $k$-jet curvature, there exists a constant $\varepsilon>0$
such that every closed irreducible curve $C\subset X$ tangent to $V$
satisfies
$$
-\chi(\ol C)=2g(\ol C)-2\ge\varepsilon\,\deg_\omega(C)+
\sum_{t\in\ol C}(m_{k-1}(t)-1)>0
$$
where $g(\ol C)$ is the genus of the normalization $\nu:\ol C\to
C\subset X$, and $m_k(t)$ is the multiplicity at point $t$ of the
$k$-th lifting $\nu_{[k]}:\ol C\to P_kV$ of $\nu$.
\endclaim

\proof. By (6.6), we get a lifting $\nu_{[k]}:\ol C\to P_kV$ of the
normalization map~$\nu$, and there is a canonical map
$$
\nu_{[k-1]}':T_{\ol C}\to\nu_{[k]}^*\cO_{P_kV}(-1).
$$
Let $t_j\in\ol C$ be the singular points of $\nu_{[k-1]}$, and let
$m_j=m_{k-1}(t_j)$ be the corresponding multiplicity. Then $\nu_{[k-1]}'$
vanishes at order $m_j-1$ at $t_j$ and thus we find
$$
T_{\ol C}\simeq\nu_{[k]}^*\cO_{P_kV}(-1)\otimes\cO_{\ol C}
\Big(-\sum(m_j-1)p_j\Big).
$$
Taking any $k$-jet metric $h_k$ with negative jet curvature on
$\cO_{P_kV}(-1)$, the Gauss-Bonnet formula yields
$$
2g(\ol C)-2=\int_{\ol C}\Theta_{T_{\ol C}^*}=
\sum(m_j-1)+\int_{\ol C}\nu_{[k]}^*\Theta_{h_k^{-1}}(\cO_{P_kV}(1)).
$$
Now, the curvature hypothesis implies
$$
\langle\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\rangle(\xi)\ge\varepsilon'
|\xi|_{\omega_k}^2\ge\varepsilon''|(\pi_{0,k})_*\xi|_\omega^2
\qquad\forall\xi\in V_k,
$$
for some $\varepsilon',\,\varepsilon''>0$ and some smooth hermitian metric
$\omega_k$ on~$P_kV$. As \hbox{$\pi_{0,k}\circ\nu_{[k]}=\nu$}, we infer 
from this $\nu_{[k]}^*\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\ge\nu^*\omega$,
hence
$$
\int_{\ol C}\nu_{[k]}^*\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\ge
{\varepsilon''\over 2\pi}\int_{\ol C}\nu^*\omega=
\varepsilon\,\deg_\omega(C)
$$
with $\varepsilon=\varepsilon''/2\pi$. Theorem~9.1 follows.\square

\claim 9.2.~Theorem|Let $k\ge 1$ be any positive integer. Then there
is a nonsingular algebraic surface $X$ $($depending on $k)$ which is
hyperbolic, but does not carry any nondegenerate $k$-jet metric with
negative jet curvature. In fact, given any two curves $\Gamma,\Gamma'$
of genus at least~$2$, the surface $X$ may be constructed as a fibration
$X\to\Gamma$ in which one of the fibers $C_\bfO$ is singular and has
$\Gamma'$ as its normalization.
\endclaim

\proof. The idea is to construct $X$ in such a way that the singular fiber
$C$ which is normalized by $\Gamma'$ violates the inequality obtained in
Theorem~9.1. For this we need only having a singular point $t_0$ such
that $m_{k-1}(t_0)-1>2g(\ol C)-2$, i.e., $m_{k-1}(t_0)\ge 2g(\Gamma')$.
Moreover, as $\Gamma$ is hyperbolic, $X$ will be hyperbolic if and only if
all fibers of $X\to\Gamma$ have geometric genus at least~$2$.

We first construct from $\Gamma'$ a singular curve $\Gamma''$ with
normalization $\ol\Gamma''=\Gamma'$, simply by modifying the structure sheaf
$\cO_{\Gamma'}$ at one given point $w_0\in\Gamma'$. Let $t$ be a holomorphic
coordinate on $\Gamma'$ at $w_0$. We replace $\cO_{\Gamma',w_0}=\bC\{t\}$ by
$\cO_{\Gamma'',w_0}=\bC\{t^a,t^b\}$, where $a<b$ are relatively prime
integers. The corresponding singularity is described by the germ of embedding
$t\mapsto f(t)=(t^a,t^b)$ in $(\bC^2,0)$. Now, $f'(t)=(at^{a-1},bt^{b-1})$,
thus $[f'(t)]\in\bP^1\simeq\bC\cup\{\infty\}$ is given by
$[f'(t)]={b\over a}t^{b-a}$. By induction, we see that the singularity
of the $j$-th lifting $f_{[j]}$ is described by the embedding
$$
t\mapsto(t^a,t^b,c_1t^{b-a}\ld c_jt^{b-ja})\in\bC^{j+2},\qquad
c_j=a^{-j}b(b-a)\cdots(b-(j-1)a)
$$
if $b>ja$. Then we have $m(f_{[j]},0)=\min(a,b-ja)$. If we take for
instance $a=2g(\Gamma')$ and $b=ka+1$, then $m(f_{[k-1]},0)=a$.
We embed $\Gamma''$ in some projective space $\bP^n$ and let $C=p(\Gamma'')$
to be a generic projection to a plane $\bP^2\subset\bP^n$ in such a way
that $C$ has only $x_0=p(w_0)$ and some nodes (ordinary double points) as its
singular points. By construction, the Zariski tangent space to $\Gamma''$
at $w_0$ is 2-dimensional, so we may assume that $p$ projects that
plane injectively into $T_{\bP^2}$. Then we get a curve $C\subset\bP^2$
with $\ol C=\Gamma'$, such that $m(\nu_{[k-1]},w_0)=a=2g(\ol C)$, if
$\nu:\ol C\to\bP^2$ is the normalization.
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\vskip6mm

Let $P_0(z_0,z_1,z_2)=0$ be an equation of $C$ in $\bP^2$. Since
$C$ has geometric genus at least $2$, we have $d=\deg P_0\ge 4$. We complete
$P_0$ into a basis $(P_0\ld P_N)$ of the space of homogeneous polynomials
of degree $d$, and consider the universal family
$$
\cF=\big\{([z_0:z_1:z_2],[\lambda_0,\lambda_1\ld\lambda_N])\in
\bP^2\times\bP^N\,;\,\sum\lambda_jP_j(z)=0\big\}\subset
\bP^2\times\bP^N
$$
of curves $C_\lambda=\{\sum\lambda_jP_j(z)=0\}$ of degree $d$ in
$\bP^2$. As is well known, the set $Z$ of points $\lambda\in\bP^N$ such
that $C_\lambda$ is a singular curve is an algebraic hypersurface, and
the set $Z'\subset Z$ of points $\lambda$ such that $C_\lambda$ has not
just a node in its singularity set satisfies $\codim Z'\ge 2$. The curve
$C=C_\bfO$ itself corresponds to the point $\bfO=[1:0:\cdots:0]\in Z'$.
Since $\codim Z'\ge 2$, we can embed $\Gamma$ in $\bP^N$ in such a way that
$\Gamma\cap Z'=\{\bfO\}$. We then take $X\to\Gamma$ to be the family of
curves $(C_\lambda)_{\lambda\in\Gamma}$. If $X$ is singular, we move
$\Gamma$ by a generic automorphism of $\bP^N$ leaving $\bfO$ fixed. Then,
since $\cF$ is smooth (it is a smooth $\bP^{N-1}$ subbundle of
$\bP^2\times\bP^N$ over $\bP^2$), Bertini's theorem implies that
$X\ssm C_\bfO$ will become nonsingular. That $X$ will be also nonsingular
near $C_\bfO$ depends only on the following first order condition: if
$[1:\alpha\lambda_1^0:\cdots:\alpha\lambda_N^0]$, $\alpha\in\bC$, is
the tangent line to $\Gamma$ at $\bfO$, then $\sum_{j\ge 1}
\lambda_j^0P_j(z)$ does not vanish at any of the singular points
of~$C_\bfO$. Now, all nonsingular fibers $C_\lambda$ of the fibration
$X\to\Gamma$ have genus $(d-1)(d-2)/2\ge 3$, and the singular ones
other than $C_\bfO$ only have one node, so their genus is
$(d-1)(d-2)/2-1\ge 2$.\square

If we make an assumption on the total jet curvature (as is the case
with the algebraic sufficient conditions~8.7), Theorem~9.1 can
be strengthened to curves which are not necessarily tangent to~$V$,
again by introducing the concept of deviation. We start with a general
purpose statement.

\claim 9.4.~Proposition|Let $(X,V)$ be a compact complex directed
manifold and let $L$ be a holomorphic line bundle over~$X$.
Assume that $L$ is equipped with a singular hermitian metric $h$ of
degeneration set $\Sigma_h$, such that the curvature $($computed in the
sense of distributions$)$ satisfies
$$
\Theta_{L,h}\ge\alpha,\qquad \alpha_{\restriction V} \ge 
\delta\omega_{\restriction V}
$$
where $\delta$ is a positive constant, $\omega$ a smooth hermitian
metric and $\alpha$ is a continuous real $(1,1)$-form on~$X$. Then for
every compact irreducible curve $C\subset X$ not contained in~$\Sigma_h$,
there exists a constant $\varepsilon>0$ such that the following a priori 
inequality holds
$$
\max\big(L\cdot C,\dev^2_\omega(C/V)\big)\ge \varepsilon\,\deg_\omega(C).
$$
\endclaim

\proof. By the continuity of $\alpha$ and the compactness of~$X$, our
assumption $\alpha_{\restriction V}\ge \delta\omega$ implies that there is
a constant $M>0$ such that 
$$
\alpha+ M\,\omega^{V^\perp}\ge{\delta\over 2}\omega
$$ 
(to get this, one merely needs to apply the Cauchy-Schwarz inequality to
mixed terms $V^*\otimes(V^\perp)^*$ in a hermitian form on~$V$).
In particular, we find
$$
\Theta_{L,h}+M\,\omega^{V^\perp} \ge {\delta\over 2}\omega
$$
This inequality gives rise to a corresponding numerical inequality on
every irreducible curve $C\not\subset\Sigma_h$, for the difference
has a well defined and nonnegative restriction to~$C$ (we use here the
fact that the weight of $h$ is quasi-psh and locally bounded at some
point of~$C$, hence locally integrable along~$C$). From this we infer
$$
L\cdot C+M\,\dev^2_\omega(C/V)\ge
{\delta\over 2}\deg_\omega(C),
$$
and the left hand side is at most equal to $(M+1) \max\big(L\cdot C,
\dev^2_\omega(C/V)\big)$.\square

\claim 9.4.~Proposition|Let $(X,V)$ be a compact complex directed 
manifold. Assume that there are integers $k,m>0$ and $\bfb\in\bN^k$
such that $\cO_{P_kV}(m)\otimes\cO_{P_kV}(-\bfb\cdot D^*)$
is an ample line bundle over~$P_kV$. Then $(X,V)$ is hyperbolic and
there exists $\varepsilon>0$ such that every closed curve $C\subset X$
satisfies
$$
\max\big(-\chi(\ol C)-\sum_{t\in\ol C}(m_{k-1}(t)-1),
\dev^\infty_\omega(C/V)\big)\ge\varepsilon\,\deg_\omega(C).
$$
\endclaim

Proposition 9.4 is likely to be true also if we assume more generally
that $(X,V)$ has non degenerate total $k$-jet curvature but, in this
case, some technical difficulties appear in the construction of the
required singular hermitian metric $h_k$ on $\cO_{P_kT_X}(1)$ (see the
proof below).

\proof. The hyperbolicity of $(X,V)$ follows from 8.7~i) and Theorem~8.8.
Now, the identity map defines a natural monomorphism $(X,V)\to(X,T_X)$ of
directed manifolds and therefore induces an embedding $P_kV\hookrightarrow
P_kT_X$ for each~$k$. With respect to this embedding, we have
$$
\eqalign{
&\cO_{P_kT_X}(1)_{\restriction P_kV}=\cO_{P_kV}(1),\cr
&\cO_{P_kT_X}(m)\otimes\cO_{P_kT_X}(-\bfb\cdot D^*)_{\restriction P_kV}=
\cO_{P_kV}(m)\otimes\cO_{P_kV}(-\bfb\cdot D^*)\cr}
$$
By our assumptions, $\cO_{P_kT_X}(m)\otimes\cO_{P_kT_X}(-\bfb\cdot D^*)$
is ample over $P_kV$ and over the fibers of the projection $P_kT_X\to X$.
Hence, we can find a smooth hermitian metric $h_{k,m,\bfb}$ on 
$\cO_{P_kT_X}(m)\otimes\cO_{P_kT_X}(-\bfb\cdot D^*)$ such that the
curvature form is positive definite on a neighborhood $U$
of $P_kV$ and satisfies
$$
\Theta\big(\cO_{P_kT_X}(m)\otimes\cO_{P_kT_X}(-\bfb\cdot D^*)\big)\ge
-C\pi_{k,0}^*\omega
$$
for some K\"ahler metric $\omega$ over~$X$.  This metric
$h_{k,m,\bfb}$ gives rise to a hermitian metric $h_k$ on
$\cO_{P_kT_X}(1)$ with singularity set $\Sigma_{h_k}\subset P^\sing_kT_X$
and similar curvature properties, that is
$$
\cases{
\Theta_{h_k}(\cO_{P_kT_X}(1))\ge-C\pi_{k,0}^*\omega&on $P_kT_X$,\cr
\Theta_{h_k}(\cO_{P_kT_X}(1))\ge\delta\omega_k
\ge\delta'\pi_{k,0}^*\omega\phantom{\Big(}\hfill
&on $U\supset P_kV$,\cr}\leqno(9.5)
$$
where $\omega_k$ is a hermitian metric on $P_kT_X$ and $\delta,\,\delta'>0$.
Now, assume that the conclusion of Prop.~9.4 is wrong. Then there
would exist a sequence of curves $(C_\ell)$ and a sequence of positive
numbers $\varepsilon_\ell$ converging to~$0$, such that
$$
\cO_{P_kT_X}(1)\cdot C_{\ell,[k]}\le\varepsilon_\ell\deg_\omega(C_\ell),\qquad
\dev^\infty_\omega(C_\ell/V)\big)\le\varepsilon_\ell\deg_\omega(C_\ell)
$$
where $C_{\ell,[k]}$ is the lifting of $C_\ell$ to $P_kT_X$ [indeed, we have
$\cO_{P_kT_X}(1)\cdot C_{\ell,[k]}=-\chi(\ol C_\ell)-\sum(m_{k-1}(t)-1)$].
Let $\nu_\ell:\ol C_\ell\to X$ be the normalization map. As 
$\dev^\infty_\omega(C_\ell/V)\big)=\sup \nu_\ell^*(\omega_{V^\perp})/
d\wt\sigma$ where $d\sigma$ is the Poincar\'e metric and $d\wt\sigma$ the 
associated normalized metric, the second condition means
$$
\sup\|\pr_{V^\perp}\nu_\ell'\|_{\sigma,\omega}^2
=\sup{\nu_\ell^*(\omega_{V^\perp})\over d\sigma}
\le{\varepsilon_\ell\deg_\omega(C_\ell)\over\int_{\ol C\ell}d\sigma}
=\varepsilon_\ell{\int_{\ol C\ell}\nu_\ell^*\omega\over
\int_{\ol C\ell}d\sigma}.
$$
In addition to this, we have
$$
{\int_{\ol C\ell}\nu_\ell^*\omega\over\int_{\ol C\ell}d\sigma}\le
R_\ell^2:=\sup\|\nu'_\ell\|_{\sigma,\omega}^2
$$
and $R=\sup R_\ell<+\infty$, otherwise the proof of Prop.~3.1 would produce
a non constant entire curve $g:\bC\to X$ tangent to $V$, contradicting the
hyperbolicity of $(X,V)$. An application of the Cauchy inequalities
to the components of $\pr_{V^\perp}$ on sufficiently small disks in the
universal covering of $\ol C_\ell$ and in suitable trivializations of $T_X/V$
shows that there is a constant $M_k\ge 0$ such that
$$
\sup_{1\le j\le k}\|\pr_{V^\perp}\nu_\ell^{(j)}\|_{\sigma,\omega}^2
\le M_k\sup\|\pr_{V^\perp}\nu_\ell'\|_{\sigma,\omega}^2
\le M_k\varepsilon_\ell{\int_{\ol C\ell}\nu_\ell^*\omega\over
\int_{\ol C\ell}d\sigma}.
$$
As $\int_{\ol C_\ell}\|\nu_\ell'\|_{\sigma,\omega}^{-2}\nu_\ell^*\omega
=\int_{\ol C_\ell}d\sigma$, we infer
$$
\int_{\ol C_\ell}{\sup_{1\le j\le k}\|\pr_{V^\perp}
\nu_\ell^{(j)}\|_{\sigma,\omega}^2\over\|\nu_\ell'\|_{\sigma,\omega}^2}
\nu_\ell^*\omega
\le M_k\varepsilon_\ell\int_{\ol C\ell}\nu_\ell^*\omega.
\leqno(9.6)
$$
Since $U$ is a neighborhood of $P_kV$, there exists a constant $\eta>0$ 
such that
$$
{\sup_{1\le j\le k}\|\pr_{V^\perp}\nu_\ell^{(j)}(t)\|_{\sigma,\omega}^2
\over\|\nu_\ell'(t)\|_{\sigma,\omega}^2}<\eta~~\Longrightarrow
\nu_{\ell,[k]}(t)\in U
$$
for any $t\in\ol C_\ell$. By the integral estimate (9.6), the set $S_\eta$ 
of ``bad points'' $t\in\ol C_\ell$ at which the left hand inequality does 
not hold has area${}<M_k\varepsilon_\ell\deg_\omega(C_\ell)/\eta$ with 
respect to $\nu_\ell^*\omega$. By (9.5), we then get
$$
\eqalign{
\cO_{P_kT_X}(1)\cdot C_{\ell,[k]}&=\int_{\ol C_\ell\ssm S_\eta}
\nu_{\ell,[k]}^*\Theta_{\cO_{P_kT_X}(1)}
+\int_{S_\eta}\nu_{\ell,[k]}^*\Theta_{\cO_{P_kT_X}(1)}\cr
&\ge\delta'\int_{\ol C_\ell\ssm S_\eta}\nu_\ell^*\omega-
C\int_{S_\eta}\nu_\ell^*\omega\cr
&=\big(\delta'(1-M_k\varepsilon_\ell/\eta)-CM_k\varepsilon_\ell/\eta\big)
\deg_\omega(C_\ell).\cr}
$$
This contradicts our initial hypothesis that 
$\cO_{P_kT_X}(1)\cdot C_{\ell,[k]}\le\varepsilon_\ell\deg_\omega(C_\ell)$
when $\varepsilon_\ell$ is small enough.\square

The above results lead in a natural way to the following questions,
dealing with the ``directed manifold case'' of Kleiman's criterion
(Kleiman's criterion states that a line bundle $L$ on $X$ is
ample if and only if there exists $\varepsilon>0$ such that $L\cdot
C\ge\varepsilon\deg_\omega C$ for every curve $C\subset X$).

\claim 9.7.~Questions|Let $(X,V)$ be a compact directed manifold and let
$L$ be a line bundle over~$X$. Fix $p\in[2,+\infty]$.
\smallskip
\item{\rm i)} Assume that
$$
\max\big(L\cdot C,\dev^p_\omega(C/V)\big)\ge\varepsilon\deg_\omega(C)
$$
for every algebraic curve $C\subset X$ $($and some
$\varepsilon>0)$. Does $L$ admit a smooth hermitian metric $h$ with
$(\Theta_{L,h})_{\restr V}$ positive definite$\,$?
\smallskip
\item{\rm ii)} Assume more generally that there is an analytic subset
$Y\supsetneq X$ such that {\rm i)} holds for all curves $C\not\subset Y$.
Does $L$ admit a singular hermitian metric $h$ with
$(\Theta_{L,h})_{\restr V}$ positive definite, and with degeneration
set $\Sigma_h\subset Y\,$?
\smallskip
\item{\rm iii)} 
Assume that there exists $\varepsilon>0$ such that
every closed curve $C\subset X$ satisfies
$$
\max\big(-\chi(\ol C)-\sum_{t\in\ol C}(m_{k-1}(t)-1),
\dev^p_\omega(C/V)\big)\ge\varepsilon\,\deg_\omega(C).
$$
Does it follow that $(X,V)$ admits non degenerate negative $k$-jet
$($total$)$ curvature$\,$?
\vskip0pt
\endclaim

The answer to 9.7~i) is positive if $V$ is the vertical tangent
sheaf of a smooth map $X\to S$, and in that case one can even restrict
oneself to curves that are tangent to~$V$ (i.e.\ vertical curves): this
is just the relative version of Kleiman's criterion. However, in
general, it is not sufficient to deal only with curves tangent to~$V$
(if $X$ is an abelian variety and $V$ is a constant line subbundle of $T_X$
with non closed leaves, the condition required for algebraic curves~$C$
is void, hence $L$ can be taken negative on $X\,$; then, of course,
the curvature cannot be made positive along~$V$.) 

\section{\S10. Proof of the 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
will be obtained as a simple consequence of the Ahlfors-Schwarz lemma 
(more specifically Theorem~8.8), combined with a jet bundle argument.
Our argument works in fact without any algebraicity assumption on the 
complex tori under consideration (only the case of abelian or semi-abelian 
varieties seems to have been treated earlier).

\claim 10.1.~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
$\ol{f(\bC)}{}^\Zar=a+Z'$.

\proof\ {\rm(based on the ideas of [GrGr80])}. Let $f:\bC\to Z$ be an
entire curve and let $X$ be the Zariski closure of~$f(\bC)$. We denote
by $Z_k=P_k(T_Z)$ the $k$-jet bundle of~$Z$ and by $X_k$ the closure
of $X^\reg_k= P_k(T_{X^\reg})$ in~$Z_k$. As $T_Z$ is trivial, we have
$Z_k=Z\times \bR_{n,k}$ where $\bR_{n,k}$ is the rational variety
introduced in~\S$\,$6. By Proposition~6.16~iii), there is a weight
$\bfa\in\bN^k$ such that $\cO_{Z_k}(\bfa)$ is relatively very ample.
This means that there is a very ample line bundle
$\cO_{\bR_{n,k}}(\bfa)$ over $\bR_{n,k}$ such that
$\cO_{Z_k}(\bfa)=\pr_2^*\cO_{\bR_{n,k}}(\bfa)$. Consider the map
$\Phi_k:X_k\to\bR_{n,k}$ which is the restriction to $X_k$ of the
second projection $Z_k\to\bR_{n,k}$. By fonctoriality, we have
$\cO_{X_k}(\bfa)=\Phi_k^*\cO_{\bR_{n,k}}(\bfa)$.

Define $B_k\subset X_k$ to be the set of points $x\in X_k$ such
that the fiber of $\Phi_k$ through~$x$ is positive dimensional.
Assume that $B_k\ne X_k$. By Proposition~8.2~ii), $\cO_{X_k}(\bfa)$ carries
a hermitian metric with degeneration set $B_k$ and with strictly positive 
definite curvature on~$X_k$ (if necessary, blow-up $X_k$ along the 
singularities and push the metric forward). Theorem 8.8 shows that
$f_{[k]}(\bC)\subset B_k$, and this is of course also true if $B_k=X_k$.
The inclusion $f_{[k]}(\bC)\subset B_k$ means that through every point
$f_{[k]}(t_0)$ there is a germ of positive dimensional variety in the fiber
$\Phi_k^{-1}(\Phi_k(f_{[k]}(t_0)))$, say a germ of curve $t'\mapsto u(t')=
(z(t'),j_k)\in X_k\subset Z\times\bR_{n,k}$ with $u(0)=f_{[k]}(t_0)=(z_0,j_k)$
and $z_0=f(t_0)$. Then $(z(t'),j_k)$ is the image of $f_{[k]}(t_0)$ by the 
$k$-th lifting of the translation $\tau_s:z\mapsto z+s$ defined by 
$s=z(t')-z_0$. Now, we have $f(\bC)\not\subset X^\sing$ since $X$ is the
Zariski closure of~$f(\bC)$, and we may therefore choose $t_0$ so that
$f(t_0)\in X^\reg$ and $f(t_0)$ is a regular point. Let us define
$$
A_k(f)=\big\{s\in Z\,:\;f_{[k]}(t_0)\in P_k(X)\cap P_k(\tau_{-s}(X))\big\}.
$$
Clearly $A_k(f)$ is an analytic subset of $Z$ containing the curve 
$t'\mapsto s(t')=z(t')-z_0$ through~$0$. Since
$$
A_1(f)\supset A_2(f)\supset\cdots\supset A_k(f)\supset\cdots,
$$
the Noetherian property shows that the sequence stabilizes at some~$A_k(f)$.
Therefore, there is a curve $D(0,r)\to Z$, $t'\mapsto s(t')$ such that 
the infinite jet~$j_\infty$ defined by $f$ at $t_0$ is $s(t')$-translation
invariant for all $t'$. By uniqueness of analytic continuation, we conclude
that $s(t')+f(t)\in X$ for all $t\in\bC$ and $t'\in D(0,r)$. As $X$ is the
Zariski closure of $f(\bC)$, we must have $s(t')+X\subset X$
for all $t'\in D(0,r)\,$; also, $X$ is irreducible, thus we have in fact
$s(t')+X=X$. Define
$$
W=\big\{s\in Z\,;~s+X=X\big\}.
$$
Then $W$ is a closed positive dimensional subgroup of~$Z$. Let $p:Z\to Z/W$
be the quotient map. As $Z/W$ is a complex torus with $\dim Z/W<\dim Z$, we 
conclude by induction on dimension that the curve $\wh f=p\circ f:\bC\to Z/W$
has its Zariski closure $\wh X:=\ol{\wh f(\bC)}{}^\Zar=p(X)$ equal to a
translate $\wh s+\wh T$ of some subtorus $\wh T\subset Z/W$. Since $X$
is \hbox{$W$-invariant}, we get $X=s+p^{-1}(\wh T)$, where 
$p^{-1}(\wh T)$ is a closed subgroup of~$Z$. This implies that $X$ is a
translate of a subtorus, as expected.\square

We now state two simple corollaries, and then the ``Bloch theorem'' itself
(see also [Och77], [Nog77,~81,~84], [Kaw80] for other approaches in the
algebraic case).

\claim 10.2.~Corollary|Let $X$ be a complex analytic subvariety in a
complex torus~$Z$. Then $X$ is hyperbolic if and only if $X$ does not
contain any translate of a subtorus.
\endclaim

\claim 10.4.~Corollary|Let $X$ be a complex analytic subvariety of a
complex torus~$Z$. Assume that $X$ is not a translate of a subtorus.
Then every entire curve drawn in $X$ is analytically degenerate.
\endclaim

\claim 10.4.~Bloch theorem|Let $X$ be a compact complex K\"ahler variety
such that the irregularity $q=h^0(X,\Omega^1_X)$ is larger than the
dimension $n=\dim X$. Then every entire curve drawn in $X$ is analytically
degenerate.
\endclaim

Here $X$ may be singular and $\Omega^1_X$ can be defined in any reasonable
way (direct image of the $\smash{\Omega^1_{\wh X}}$ of a desingularization 
$\wh X$ or direct image of $\Omega^1_U$ where $U$ is the set of regular
points in the normalization of~$X$).

\proof. By blowing-up, we may assume that $X$ is smooth. Then the Albanese
map $\alpha:X\to\Alb(X)$ sends $X$ onto a proper subvariety $Y\subset\Alb(X)$
(as $\dim Y\le\dim X<\dim\Alb(X)$), and $Y$ is not a translate of a subtorus
by the universal property of the Albanese map. Hence, for every entire curve
$f:\bC\to X$ we infer that $\alpha\circ f:\bC\to Y$ is analytically
degenerate; it follows that $f$ itself is analytically degenerate.\square


\section{\S11. Logarithmic jet bundles and a conjecture of Lang}

We discuss here an important question raised by 
S.~Lang, namely {\em whether the complement of an ample divisor in
an Abelian variety is Kobayashi hyperbolic\/}? This statement has been
first settled in the affirmative by Siu and Yeung [SiYe96b], using an
extension of some of the methods used to prove the Bloch theorem. We
will adopt here a slightly different approach of G.~Dethloff and
S.~Lu [DLu01], who followed a suggestion made during our Santa Cruz
lectures in July 1995. Namely, there should exist a theory of {\em
logarithmic jet bundles} extending Semple's construction, which
would allow to study the hyperbolicity properties of open varieties of
the form $X\ssm D$ ($D$ being a divisor in a projective variety~$X$).
We give here a short account of Dethloff and Lu's technique,
referring to [DLu01] for details, and to [SiYe96b], [Nog98] for 
alternative approaches.

Let $(X,V)$ be a compact directed manifold and $D$ a reduced divisor
in~$X$. Recall that the sheaf $\Omega^1_X\langle D\rangle$ of 
{\em holomorphic $1$-forms with logarithmic poles along $D$} is defined
to be the coherent sheaf generated by $\Omega^1_X$ and $ds_j/s_j$,
where~$s_j=0$ are local equations for the irreducible components of~$D$.
It is locally free as soon as $D$ is a normal crossing divisor (we may
always suppose that this is the case after blowing up $X$ along
smooth centers contained in~$D$). Similarly, one introduces the sheaf 
$\cO(V^*\langle D\rangle)$ to be the sheaf of holomorphic
$1$-forms along~$V$ with logarithmic poles along $D$ (this is just the
quotient of $\Omega^1_X\langle D\rangle$ by the conormal sheaf
$V^o\subset V^*$ of~$V$). It is locally free as soon as $D$ has normal
crossings and its components $D_{(j)}$ are everywhere tranversal to $V$ 
(by this we mean that $T_{D_{(j)}}+V=T_X$ along $D_{(j)}$). Under this 
assumption, we consider the dual (locally free) sheaves
$$
\cO(T_X\langle D\rangle):=(\Omega^1_X\langle D\rangle)^*,\qquad
\cO(V\langle D\rangle):=(V^*\langle D\rangle)^*.
\leqno(11.1)
$$
One easily checks that $\cO(T_X\langle D\rangle)$ (resp.\ 
$\cO(V\langle D\rangle)$) is the sheaf of germs of vector fields in
$\cO(T_X)$ (resp.\ $\cO(V)$) which are tangent to each component
of~$D$. Now, one defines a sequence
$$
(X_k,D_k,V_k)
\leqno(11.2)
$$
of logarithmic $k$-jet bundles exactly in the same way as we proceeded in
section \S$\,$5 and \S$\,$6: if $X_0=X$, $D_0=D$ and $V_0=V\langle D\rangle$,
one sets inductively $X_k=P(V_{k-1})$, $D_k=(\pi_{k,0})^{-1}(D)$, and
$V_k$ is the set of tangent vectors in $T_{X_k}\langle D_k\rangle$ which 
project into the line defined by the tautological line bundle 
$\cO_{X_k}(-1)\subset\pi_k^* V_{k-1}$. In this case, the direct 
image formula given in Theorem~6.8 reads
$$
(\pi_{k,0})_*\cO_{X_k}(m)=\cO(E_{k,m}V^*\langle D\rangle),
\leqno(11.3)
$$
where $\cO(E_{k,m}V^*\langle D\rangle)$ is the sheaf generated
by all polynomial differential operators in the derivatives of order
$1,2\ld k$ of the components $f_1\ld f_n$, together with the extra 
function $\log s_j(f)$ along the $j$-th component of $D$.

Just as before, a {\em logarithmic $k$-jet metric} is just a singular
hermitian metric on $\cO_{X_k}(-1)$. Dethloff and Lu [DLu01] state 
the following results 11.4--11.9, which extend our results of sections 
\S$\,$8 and \S$\,$10 (most of these results can already be derived
from [SiYe96b] as well).

\claim 11.4.~Theorem|Let $(X,D,V)$ be as above. Let $\Sigma_{k,m}$
be the union of the base locus of $\cO_{X_k}(m)$ and of the positive
dimensional fibers of the canonical map defined by the corresponding
linear system. Then
\smallskip
\item{\rm i)} If $\Sigma_{k,m}\ne X_k$, there exists a logarithmic $k$-jet
metric $h_k$ with strictly negative jet curvature and $\Sigma_{h_k}=
\Sigma_{k,m}$.
\smallskip
\item{\rm ii)} For every entire map $f:\bC\to X\ssm D$ tangent to~$V$, 
one has $f_{[k]}(\bC)\subset\Sigma_{k,m}$.
\smallskip
\item{\rm iii)} For every holomorphic map $f:\Delta^*\to X\ssm D$
tangent to $V$ $($where $\Delta^*$ is the punctured disk$)$,
one has: either $f$ extends to a holomorphic map $\ol f:\Delta\to X$ 
or $f_{[k]}(\Delta^*)\subset\Sigma_{k,m}$.
\vskip0pt
\endclaim

Consider now a semi-abelian variety $Z$ (that is, a commutative algebraic
group $\bC^n/\Gamma$), and let $D\subset Z$ be a reduced algebraic divisor.

\claim 11.5.~Theorem|Let $(Z,D)$ be as above.
\smallskip
\item{\rm i)} For every entire curve $f:\bC\to Z$, the Zariski closure
$\ol{f(\bC)}{}^\Zar$ is a translate of an algebraic subgroup of~$Z$.
\smallskip
\item{\rm ii)} For every entire curve $f:\bC\to Z\ssm D$, we have
$\ol{f(\bC)}{}^\Zar\cap D=\emptyset$.
\vskip0pt
\endclaim

\claim 11.6.~Corollary|If $D$ has non empty intersection with any
translate of an algebraic subgroup of~$Z$ of positive dimension,
then $Z\ssm D$ is Brody hyperbolic. This is true
e.g.\ if $Z$ is abelian and $D$ is ample.
\endclaim

\claim 11.7.~Remark|{\rm Theorem 11.5 and its corollary have been
obtained independently by Noguchi [Nog98], and also by Siu-Yeung [SiYe96b]
in the case of abelian varieties. Both of their proofs use value
distribution theory, whilst the present approach uses only negative
 curvature arguments. It is likely that Theorem~11.5 can be extended to
arbitrary commutative (non necessarily algebraic) Lie groups~$\bC^n/\Gamma$.}
\endclaim

\claim 11.8.~Theorem|The following properties hold true.
\smallskip
\item{\rm A)} Let $f:\Delta^*\to Z$ be a holomorphic map. Then either
it extends to a holomorphic map $\ol f:\Delta\to\ol Z$ or there exists a
maximal algebraic subgroup $Z'$ of $Z$ of positive dimension such that
$\ol{f(\Delta^*)}{}^\Zar$ is foliated by translates of~$Z'$.
\smallskip
\item{\rm B)} Let $f:\Delta^*\to Z\ssm D$ be a holomorphic map. Then
one of the following holds:
\smallskip
\itemitem{\rm i)} $f$ extends to a holomorphic map $\ol f:\Delta\to Z$.
\smallskip
\itemitem{\rm ii)} $\ol{f(\Delta^*)}{}^\Zar\cap D=\emptyset$.
\smallskip
\itemitem{\rm iii)} There exists an algebraic subgroup $Z''$ of $Z'$ of
positive dimension such that $\ol{f(\Delta^*)}{}^\Zar\cap D$ is
foliated by translates of~$Z''$.
\smallskip
\item{\rm C)} Assume here that $Z$ is an abelian variety and let
$f:\Delta^*\to Z\ssm D$ be a holomorphic map. Then one of the
following holds:
\smallskip
\itemitem{\rm i)} $f$ extends to a holomorphic map $\ol f:\Delta\to Z$.
\smallskip
\itemitem{\rm ii)} There exists an algebraic subgroup $Z''$ of $Z'$ of
positive dimension such that $D$ is foliated by translates of~$Z''$.
\vskip0pt
\endclaim

Part A) of Theorem 11.8 is due to Noguchi [Nog98] (again with a proof
based on Nevanlinna theory).

\claim 11.9.~Corollary|If $Z$ is abelian and $D$ is ample, then every
holomorphic map $f:\Delta^*\to Z\ssm D$ extends to a holomorphic map
$\ol f:\Delta\to Z$.
\endclaim

\section{\S12. Projective meromorphic connections 
and Wronskians}

We describe here an important method introduced by Siu [Siu87] and
later developped by Nadel [Nad89], which is powerful enough to provide
explicit examples of algebraic hyperbolic surfaces. It yields likewise
interesting results about the algebraic degeneration of entire curves
in higher dimensions.  The main idea is to use meromorphic connections
with low pole orders, and the associated Wronskian operators. In this
way, Nadel produced examples of hyperbolic surfaces in $\bP^3$ for any
degree of the form $p=6k+3\ge 21$. We present here a variation of
Nadel's method, based on the more general concept of {\em partial
  projective connection}, which allows us to extend his result to all
degrees $p\ge 11$. This approach is inspired from the PhD work of
J.~El Goul [EG96], and is in some sense a formalization of his
strategy.

Let $X$ be a complex $n$-dimensional manifold. A {\em meromorphic
connection} $\nabla$ on $T_X$ is a $\bC$-linear sheaf morphism
$$
\cM(U,T_X)\lra\cM(U,\Omega^1_X\otimes T_X)
$$
(where $\cM(U,\bu)$ stands for meromorphic sections over $U$), 
satisfying the Leibnitz rule
$$
\nabla(fs)=df\otimes s+ f\nabla s
$$
whenever $f\in\cM(U)$ (resp.\ $s\in\cM(U,T_X)$) is a meromorphic function
(resp.\ section of $T_X$). Let $(z_1,\ldots,z_n)$ be holomorphic local
coordinates on an open set~$U\subset X$. The {\em Christoffel symbols} of
$\nabla$ with respect to these coordinates are the coefficients
$\Gamma^\lambda_{j\mu}$ such that
$$
\Gamma^\lambda_\mu=\sum_{1\le j\le n}\Gamma^\lambda_{j\mu}dz_j=
\hbox{$\lambda$-th component of
$\displaystyle\nabla\Big({\partial\over\partial z_\mu}\Big).$}
$$
The associated {\em connection form} on $U$ is the tensor 
$$
\Gamma=\sum_{1\le j,\lambda,\mu\le n}\Gamma^\lambda_{j\mu}\,dz_j\otimes
dz_\mu \otimes{\partial\over\partial z_\lambda}\in\cM(U,T^*_X\otimes 
T^*_X\otimes T_X).
$$
Then, for all local sections $v=\sum_{1\le\lambda\le n}v_\lambda{\partial
\over\partial z_\lambda}$, $w=\sum_{1\le\lambda\le n}w_\lambda{\partial
\over\partial z_\lambda}$ of $\cM(U,T_X)$, we get
$$
\eqalign{
\nabla v&=\sum_{1\le\lambda\le n}\Big(dv_\lambda+\sum_{1\le\mu\le n}
\Gamma^\lambda_\mu v_\mu\Big){\partial\over\partial z_\lambda}
=dv+\Gamma\cdot v,\cr
\nabla_w v&=\sum_{1\le j,\lambda\le n}
\Big(w_j{\partial v_\lambda\over\partial z_j}+\sum_{1\le\mu\le n}
\Gamma^\lambda_{j\mu}w_jv_\mu\Big){\partial\over\partial z_\lambda}
=d_wv+\Gamma\cdot (w,v).\cr}
$$
The connection $\nabla$ is said to be {\em symmetric} if it satisfies
$\nabla_vw-\nabla_wv=[v,w]$, or equivalently, if the Christoffel symbols
$\Gamma_{j\mu}^\lambda=\Gamma_{\mu j}^\lambda$ are symmetric in $j,\,\mu$.

We now turn ourselves to the important concept of {\em Wronskian
operator}.  Let $B$ be the divisor of poles of $\nabla$, that is,
the divisor of the least common multiple of all denominators occuring
in the meromorphic functions~$\Gamma^\lambda_{j\mu}$. If $\beta\in
H^0(X,\cO(B))$ is the canonical section of divisor $B$, then the
operator $\beta\nabla$ has holomorphic coefficients. Given a
holomorphic curve $f:D(0,r)\to X$ whose image does not lie in the
support $|B|$ of~$B$, one can define inductively a sequence of
covariant derivatives
$$
f',\quad f''_\nabla=\nabla_{f'}(f'),~\ldots~,~f^{(k+1)}_\nabla:=
\nabla_{f'}(f^{(k)}_\nabla).
$$
These derivatives are given in local coordinates by the explicit
inductive formula
$$
f^{(k+1)}_\nabla(t)_\lambda={d\over dt}\big(f^{(k)}_\nabla(t)_\lambda\big)
+\sum_{1\le\mu\le n}(\Gamma^\lambda_{j\mu}\circ f)\,f'_j
\,f^{(k)}_\nabla(t)_\mu.
\leqno(12.1)
$$
Therefore, if $\Im f\not\subset|B|$, one can define the {\em Wronskian}
of $f$ relative to $\nabla$ as
$$
W_\nabla(f)=f'\wedge f''_\nabla\wedge\cdots\wedge f^{(n)}_\nabla.
\leqno(12.2)
$$
Clearly, $W_\nabla(f)$ is a meromorphic section of $f^*(\Lambda^nT_X)$.
By induction $\beta(f)^{k-1}f^{(k)}_\nabla$ is holomorphic for all
$k\ge 1$. We infer that $\beta(f)^{n(n-1)/2}W_\nabla(f)$ is holomorphic
and can be seen as a holomorphic section of the line
bundle $f^*(\Lambda^nT_X\otimes\cO_X({1\over 2} n(n-1)B)$.
From (12.1) and (12.2) we see that $P=\beta^{n(n-1)/2}W_\nabla$ is
a global holomorphic polynomial operator $f\mapsto P(f',f'',\ldots,f^{(n)})$
of order $n$ and total degree $n(n+1)/2$, with values in
$\Lambda^nT_X\otimes\cO_X({1\over 2}n(n-1)B)$. Moreover, if we take a
biholomorphic reparametrization $\varphi$, we get inductively
$$
(f\circ\varphi)^{(k)}_\nabla=(\varphi')^kf^{(k)}_\nabla\circ\varphi+
\hbox{$\,$linear combination of $f^{(j)}_\nabla\circ\varphi$,~ $j<k$}.
$$
Therefore
$$
W_\nabla(f\circ\varphi)=(\varphi')^{n(n+1)}W_\nabla(f)
$$
and $\beta^{n(n-1)/2}W_\nabla$ can be viewed as a section
$$
\beta^{n(n-1)/2}W_\nabla\in
H^0(X,E_{n,n(n+1)/2}T^*_X\otimes L^{-1}),
\leqno(12.3)
$$
where $L$ is the line bundle
$$
L=K_X\otimes\cO_X\Big(-{1\over 2}n(n-1)B\Big).
$$
From this, we get the following theorem, which is essentially due to
[Siu87] (with a more involved proof based on suitable generalizations of
Nevanlinna's second main theorem).

\claim 12.4.~Theorem {\rm(Y.T.~Siu)}|Let $X$ be a compact complex
manifold equipped with a meromorphic connection $\nabla$ of pole
divisor~$B$.  If $K_X\otimes\cO_X(-{1\over 2}n(n-1)B)$ is ample, then
for every non constant entire curve $f:\bC\to X$, one has either
$f(\bC)\subset|B|$ or $W_\nabla(f)\equiv 0$.
\endclaim

\proof. By Corollary~8.9 applied with $P=\beta^{n(n-1)/2}W_\nabla$, we
conclude that
$$\beta^{n(n-1)/2}(f)W_\nabla(f)\equiv 0,$$
whence the result.\square

\claim 12.5.~Basic observation|{\rm It is not necessary to know all
Christoffel coefficients of the meromorphic connection $\nabla$ in
order to be able to compute its Wronskian $W_\nabla$. In fact, assume
that $\wt\nabla$ is another connection such that there are meromorphic
$1$-forms $\alpha$, $\beta$ with
$$
\eqalign{
\wt\nabla&=\nabla+\alpha\otimes\Id_{T_X}+(\beta\otimes\Id_{T_X})_{\tau_{12}},
\qquad\hbox{i.e.,}\cr
\wt\nabla_wv&=\nabla_w v+\alpha(w)v+\beta(v)w,\cr}
$$ 
where $\tau_{12}$ means transposition of first and second arguments in 
the tensors of $T^*_X\otimes T^*_X\otimes T_X$. Then
$W_\nabla=W_{\wt\nabla}$. Indeed, the defining formula
$f^{(k+1)}_{\wt\nabla}=\swt\nabla|_{f'}(f^{(k)}_{\wt\nabla})$ implies that
$f^{(k+1)}_{\wt\nabla}=\nabla_{f'}(f^{(k)}_{\wt\nabla})+
\alpha(f')f^{(k)}_{\wt\nabla}+\beta(f^{(k)}_{\wt\nabla})f'$, and
an easy induction then shows that the $\swt\nabla|$ derivatives can be
expressed as linear combinations with meromorphic coefficients
$$
f^{(k)}_{\wt\nabla}(t)=f^{(k)}_\nabla(t)+\sum_{1\le j<k}\gamma_j(t)\,
f^{(j)}_\nabla(t).
\eqno\square
$$}
\endclaim

The essential consequence of Remark~12.5 is that we need only have a
``partial projective connection'' $\nabla$ on $X$, in the following
sense.

\claim 12.6.~Definition|A $($meromorphic$)$ partial projective 
connection $\nabla$ on $X$ is a section of the quotient sheaf of 
meromorphic connections modulo addition of meromorphic tensors in 
$(\Omega^1_X\otimes\Id_{T_X})\oplus(\Omega^1_X\otimes\Id_{T_X})_{\tau_{12}}$.
In other words, it can be defined as a collection of 
meromorphic connections $\nabla_j$ relative to an
open covering $(U_j)$ of~$X$, satisfying the compatibility conditions
$$
\nabla_k-\nabla_j=\alpha_{jk}\otimes\Id_{T_X}+(\beta_{jk}\otimes
\Id_{T_X})_{\tau_{12}}
$$
for suitable meromorphic $1$-forms $\alpha_{jk}$, $\beta_{jk}$ on
$U_j\cap U_k$.
\endclaim

If we have similar more restrictive compatibility relations with 
$\beta_{jk}=0$, the connection form $\Gamma$ is just
defined modulo $\Omega^1_X\otimes\Id_{T_X}$ and can thus be seen as a
$1$-form with values in the Lie algebra $\gpgl(n,\bC)=\gsl(n,\bC)$
rather than in $\ggl(n,\bC)$. Such objects are sometimes referred to 
as ``projective connections'', although this terminology has been
also employed in a completely different meaning. In any event, Proposition
12.4 extends (with a completely identical proof) to the more general
case where $\nabla$ is just a partial projective connection.
Accordingly, the pole divisor $B$ can be taken to be the pole divisor
of the trace free part
$$
\Gamma^0=\Gamma~~\mod~~(\Omega^1_X\otimes\Id_{T_X})\oplus(\Omega^1_X\otimes
\Id_{T_X})_{\tau_{12}}.
$$
Such partial projective connections occur in a natural way when one
considers quotient varieties under the action of a Lie group. Indeed,
let $W$ be a complex manifold in which a connected complex Lie group
$G$ acts freely and properly (on the left, say), and let $X=W/G$ be
the quotient complex manifold.  We denote by $\pi:W\to X$ the
projection. Given a connection $\swt\nabla|$ on $W$ and a local
section $\sigma:U\to W$ of $\pi$, one gets an induced connection on
$T_{X|U}$ by putting
$$
\nabla=\pi_*\circ(\sigma^*\wt\nabla),
\leqno(12.7)
$$
where $\sigma^*\wt\nabla$ is the induced connection on $\sigma^* 
T_W$ and $\pi_*:T_W\to \pi^* T_X$ is the projection. Of course, 
the connection $\nabla$ may depend on the choice of $\sigma$, but we 
nevertheless have the following simple criterion ensuring that it yields
an intrinsic partial projective connection.

\claim 12.8.~Lemma|Let $\swt\nabla|=d+\swt\Gamma|$ be a meromorphic
connection on $W$. Assume that $\swt\nabla|$ satisfies the following
conditions$\,:$
\smallskip
\item{\rm i)} $\swt\nabla|$ is $G$-invariant$\,;$
\smallskip
\item{\rm ii)} there are meromorphic $1$-forms $\alpha,\,\beta\in
\cM(W,T_{W/X})$ along the relative tangent bundle of~$X\to W$,
such that for all $G$-invariant holomorphic vector fields $v$, $\tau$ on~$W$ 
$($possibly only defined locally over~$X)$ such that $\tau$ is tangent to
the $G$-orbits, the vector fields
$$
\swt\nabla|_\tau v-\alpha(\tau)v,\qquad \swt\nabla|_v\tau-\beta(\tau)v
$$
are again tangent to the $G$-orbits $(\alpha$ and $\beta$ are thus 
necessarily $G$-invariant, and $\alpha=\beta$ if $\swt\nabla|$ is 
symmetric$)$.
\vskip0pt\noindent
Then Formula $(12.7)$ yields a partial projective connection $\nabla$
which is globally defined on~$X$ and independent of the choice of the
local sections~$\sigma$.
\endclaim

\proof. Since the expected conclusions are local with respect to~$X$, it is 
enough to treat the case when $W=X\times G$ and $G$ acts on the left
on the second factor. Then $W/G\simeq X$ and $\pi:W\to X$ is the first 
projection. If $d_G$ is the canonical left-invariant connection on $G$, 
we can write $\swt\nabla|$ as
$$
\swt\nabla|=d_X+d_G+\swt\Gamma|,\qquad \swt\Gamma|=\swt\Gamma|(x,g),\quad
x\in X,~g\in G,
$$
where $d_X$ is some connection on~$X$, e.g.\ the ``coordinate derivative''
taken with respect to given local coordinates
$(z_1,\ldots,z_n)$ on~$X$. Then $\swt\nabla|$ is left invariant on 
$W=X\times G$ if and only if $\swt\Gamma|(x,g)=\Gamma(x)$ is independent of
$g\in G$ (this is meaningful since the tangent bundle to~$G$ is trivial), 
and condition~ii) means that
$$
\Gamma(x)\cdot(\tau,v)-\alpha(\tau)v\quad\hbox{and}\quad
\Gamma(x)\cdot(v,\tau)-\beta(\tau)v
$$
are tangent to the $G$-orbits. A local section $\sigma:U\to W$ of $\pi$ can
be written $\sigma(x)=(x,h(x))$ for some holomorphic function $h:U\to G$. 
Formula (12.7) says more explicitly that
$$
\nabla_wv=\pi_*\big((\sigma^*\wt\nabla)_wv\big)=
\pi_*\big(d_{\sigma_* w}\sigma_* v+(\wt\Gamma\circ\sigma)
\cdot(\sigma_* w,\sigma_* v)\big).
$$
Let $v=\sum v_j(z)\partial/\partial z_j$, $w=\sum w_j(z)\partial/\partial 
z_j$ be local vector fields on~$U\subset X$. Since $\sigma_* v=v+dh(v)$,
we get
$$
\eqalign{(\sigma^*\wt\nabla)_wv
&=d_{w+dh(w)}(v+dh(v))+\wt\Gamma(x,h(x))\cdot\big(w+dh(w),v+dh(v)\big)\cr
&=d_wv+d^2h(w,v)+\Gamma(x)\cdot(w+dh(w),v+dh(v)).\cr}
$$
As $v$, $w$, $dh(v)$, $dh(w)$ depend only on~$X$, they can be seen as
$G$-invariant vector fields over~$W$, and $dh(v)$, $dh(w)$ are tangent to
the $G$-orbits. Hence 
$$
\Gamma(x)\cdot(dh(w),v)-\alpha(dh(w))v,\quad
\Gamma(x)\cdot(w,dh(v))-\beta(dh(v))w,\quad
\Gamma(x)\cdot(dh(w),dh(v))
$$
are tangent to the $G$-orbits, i.e., in the kernel of $\pi_*$.
We thus obtain
$$
\nabla_wv=\pi_*\big((\sigma^*\wt\nabla)_wv\big)=
d_wv+\Gamma(x)\cdot(w,v)+\alpha(dh(w))v+\beta(dh(v))w.
$$
From this it follows by definition that the local connections
$\nabla_{\restriction U_j}$ defined by various sections $\sigma_j:U_j\to W$
can be glued together to define a global partial projective connection
$\nabla$ on~$X$.\square

\claim 12.9.~Remark|{\rm Lemma~12.8 is also valid when $\swt\nabla|$ is a
partial projective connection. Hypothesis 12.8~ii) must then hold with 
local meromorphic $1$-forms $\alpha_j,\,\beta_j\in\cM(\swt U|_j,T_{W/X})$
relatively to some open covering $\swt U|_j$ of~$W$.\square}
\endclaim

\noindent
In the special case $\bP^n=(\bC^{n+1}\ssm\{0\})/\bC^*$, we get

\claim 12.10.~Corollary|Let $\swt\nabla|=d+\swt\Gamma|$ be a meromorphic
connection on $\bC^{n+1}$. Let $\varepsilon=\sum z_j\partial/\partial z_j$
be the Euler vector field on $\bC^{n+1}$ and $\pi:\bC^{n+1}\ssm\{0\}\to
\bP^n$ be the canonical projection. Then $\swt\nabla|$ induces a meromorphic
partial projective connection on $\bP^n$ provided that
\smallskip
\item{\rm i)} the Christoffel symbols $\Gamma_{j\mu}^\lambda$ are homogeneous
rational functions of degree $-1$ $($homothety invariance of the connection
$\swt\nabla|)\,;$
\smallskip
\item{\rm ii)} there are meromorphic functions $\alpha$, $\beta$ and 
meromorphic $1$-forms $\gamma$, $\eta$ such that
$$
\wt\Gamma\cdot(\varepsilon,v)=\alpha v+\gamma(v)\varepsilon,\qquad
\wt\Gamma\cdot(w,\varepsilon)=\beta w+\eta(w)\varepsilon
$$ 
for all vector fields $v,\,w$.\square
\vskip0pt
\endclaim

Now, our goal is to study certain hypersurfaces $Y$ of sufficiently high 
degree in $\bP^n$. Assume for the moment that $Y$ is an hypersurface in
some $n$-dimensional manifold $X$, and that $Y$ is defined locally
by a holomorphic equation $s=0$. We say that $Y$ is {\em totally
geodesic} with respect to a meromorphic connection $\nabla$ on~$X$ if
$Y$ is not contained in the pole divisor $|B|$ of $\nabla$, and for all
pairs $(v,w)$ of (local) vector fields tangent to $Y$ the covariant
derivative $\nabla_wv$ is again tangent to~$Y$. (Notice that this concept
also makes sense when $\nabla$ is a partial projective connection.) 
If~$Y$ is totally geodesic, the ambient connection $\nabla$ on $T_X$
induces by restriction a connection $\nabla_{\restriction Y}$ on~$T_Y$.

We now want to derive explicitly a condition for the hypersurface
\hbox{$Y=\{s=0\}$} to be totally geodesic in $(X,\nabla)$.
A vector field $v$ is tangent to $Y$ if and only if $ds\cdot v=0$
along $s=0$. By taking the differential of this identity along another
vector field $w$ tangent to $Y$, we find
$$
d^2s\cdot(w,v)+ds\cdot(d_wv)=0
\leqno(12.11)
$$
along $s=0$ (this is meaningful only with respect to some local
coordinates). On the other hand, the condition that $\nabla_wv=
d_wv+\Gamma\cdot(w,v)$ is tangent to $Y$ is 
$$
ds\cdot\nabla_wv=ds\cdot(d_wv)+ds\circ\Gamma\cdot(w,v)=0.
$$
By subtracting the above from (12.11), we get the following equivalent
condition: $(d^2s-ds\circ\Gamma)\cdot(w,v)=0$ for all vector fields $v,w$
in the kernel of $ds$ along $s=0$. Therefore we obtain the

\claim 12.12.~Characterization of totally geodesic hypersurfaces|The
hypersurface $Y=\{s=0\}$ is totally geodesic with respect to $\nabla$
if and only if there are holomorphic $1$-forms $a=\sum a_jdz_j$,
$b=\sum b_jdz_j$ and a $2$-form $c=\sum c_{j\mu}dz_j\otimes dz_\mu$
such that
$$
\nabla^*(ds)=d^2s-ds\circ\Gamma=a\otimes ds+ds\otimes b+s\,c
$$
in a neighborhood of every point of~$Y$ $($here $\nabla^*$ is the
induced connection on $T^*_Y)$.
\endclaim

\noindent
From this, we derive the following useful lemma.

\claim 12.14.~Lemma|Let $Y\subset X$ be an analytic hypersurface which
is totally geodesic with respect to a meromorphic connection~$\nabla$,
and let $n=\dim X=\dim Y+1$.  Let $f:D(0,R)\to X$ be a holomorphic curve
such that $W_\nabla(f)\equiv 0$. Assume that there is a point
$t_0\in D(0,R)$ such that
\smallskip
\item{\rm i)} $f(t_0)$ is not contained in the poles of $\nabla\,;$
\smallskip
\item{\rm ii)} the system of vectors $(f'(t),f''_\nabla(t),\ldots,
\smash{f^{(n-1)}_\nabla(t)})$ achieves its generic rank $($i.e.\ its maximal
rank$)$ at~$t=t_0\,;$
\smallskip
\item{\rm iii)}
$f(t_0)\in Y$ and $f'(t_0),~f''_\nabla(t_0),\ldots,f^{(n-1)}_\nabla(t_0)
\in T_{Y,f(t_0)}.$
\smallskip\noindent
Then $f(D(0,R))\subset Y$.
\endclaim

\proof. Since $W_\nabla(f)\equiv 0$, the vector fields $f'$, 
$f''_\nabla,\ldots,f^{(n)}_\nabla$ are linearly dependent and satisfy 
a non trivial relation
$$
u_1(t)f'(t)+u_2(t)f''_\nabla(t)+\cdots+u_n(t)f^{(n)}_\nabla(t)=0
$$
with suitable meromorphic coefficients $u_j(t)$ on~$D(0,R)$. If
$u_n$ happens to be${}\equiv 0$, we take $\nabla$-derivatives
in the above relation so as to reach another relation with
$u_n\not\equiv 0$. Hence we can always write
$$
f^{(n)}_\nabla=v_1f'+v_2f''_\nabla+\cdots+v_{n-1}f^{(n-1)}_\nabla
$$
for some meromorphic functions $v_1,\ldots,v_{n-1}$. We can even prescribe
the $v_j$ to be $0$ eXcept for indices $j=j_k\in\{1,\ldots,n-1\}$ such that
$\smash{(f^{(j_k)}_\nabla(t))}$ is a minimal set of generators at $t=t_0$.
Then the coefficients $v_j$ are uniquely defined and are holomorphic
near~$t_0$. By taking further derivatives, we conclude
that $f^{(k)}_\nabla(t_0)\in T_{X,f(\smash{t_0})}$ for all~$k$. We now
use the assumption that $X$ is totally geodesic to prove the
following claim: if $s=0$ is a local equation of $Y$, the
$k$-th derivative ${d^k\over dt^k}(s\circ f(t))$ can be expressed
as a holomorphic linear combination
$$
{d^k\over dt^k}\big(s\circ f(t)\big)=\gamma_{0k}(t)\,s\circ f(t)+
\sum_{1\le j\le k}\gamma_{jk}(t)\,ds_{f(t)}\cdot f^{(j)}_\nabla(t)
$$
on a neighborhood of $t_0$. This will imply ${d^k\over dt^k}(s\circ f)(t_0)
=0$ for all $k\ge 0$, hence $s\circ f\equiv 0$. Now, the above claim
is clearly true for $k=0,1$. By taking the derivative and arguing
inductively, we need only show that
$$
{d\over dt}\big(ds_{f(t)}\cdot f^{(j)}_\nabla(t)\big)
$$
is again a linear combination of the same type. However, Leibnitz's rule
for covariant differentiations together with 12.12 yield
$$
\eqalign{
{d\over dt}\big(ds_{f(t)}\cdot f^{(j)}_\nabla(t)\big)
&=ds_{f(t)}\cdot\Big({\nabla\over dt}f^{(j)}_\nabla(t)\Big)+
\nabla^*(ds)_{f(t)}\cdot\big(f'(t),f^{(j)}_\nabla(t)\big)\cr
&=ds\cdot f^{(j+1)}_\nabla(t)+(a\cdot f'(t))\big(ds\cdot
f^{(j)}_\nabla(t)\big)\cr
&\qquad{}+(ds\cdot f'(t))\big(b\cdot
f^{(j)}_\nabla(t)\big)+(s\circ f(t))\big(c\cdot(f'(t),f^{(j)}_\nabla(t))
\big),\cr}
$$
as desired.\square

If $Y=\{s=0\}\subset X$ is given and a connection $\nabla$ on $X$ is to
be found so that $Y$ is totally geodesic, condition 12.12
amounts to solving a highly underdetermined linear system of equations
$$
{\partial^2 s\over\partial z_j\partial z_\mu}-\sum_{1\le\lambda\le n}
\Gamma^\lambda_{j\mu}{\partial s\over\partial z_\lambda}
=a_j{\partial s\over\partial z_\mu}+b_\mu{\partial s\over\partial z_j}+
s\,c_{j\mu},\qquad 1\le j,\mu\le n,
$$
in terms of the unknowns $\Gamma^\lambda_{j\mu}$, $a_j$, $b_\mu$ and
$c_{j\mu}$. Nadel's idea is to take advantage of this indeterminacy to
achieve that all members in a large linear system $(Y_\alpha)$ of
hypersurfaces are totally geodesic with respect to $\nabla$. The following
definition is convenient.

\claim 12.14.~Definition|For any $(n+2)$-tuple of integers $(p,k_0,k_1\ldots,
k_n)$ with $0<k_j<p/2$, let $\cS_{p;\,k_0,\ldots,k_n}$ be the space 
of homogeneous polynomials
$s\in\bC[z_0,z_1,\ldots, z_n]$ of degree $p$ such that every monomial
of $s$ is a product of a power $z_j^{p-k_j}$ of one of the variables with
a lower degree monomial of degree~$k_j$. Any polynomial 
$s\in\cS_{p;\,k_0,\ldots,k_n}$ admits a unique decomposition
$$
s=s_0+s_1+\cdots+s_n,\qquad s_j\in\cS_{p;\,k_0,\ldots,k_n}
$$
where $s_j$ is divisible by $z_j^{p-k_j}$.
\endclaim

Given a homogeneous polynomial $s=s_0+s_1+\cdots+s_n\in
\cS_{p;\,k_0,\ldots,k_n}$, we consider the linear system
$$
Y_\alpha=\big\{\alpha_0s_0+\alpha_1s_1+\cdots+\alpha_ns_n=0\big\},
\quad \alpha=(\alpha_0,\ldots,\alpha_n)\in\bC^n.
\leqno(12.15)
$$
Our goal is to study smooth varieties $Z$ which arise as complete
intersections $Z=Y_{\alpha^1}\cap\cdots\cap Y_{\alpha^q}$ of members
in the linear system (the $\alpha^j$ being linearly independent
elements in~$\bC^{n+1}$).  For this, we want to construct a (partial
projective) meromorphic connection $\nabla$ on $\bP^n$ such that all
$Y_\alpha$ are totally geodesic. Corollary 12.10 shows that it is
enough to construct a meromorphic connection
$\swt\nabla|=d+\swt\Gamma|$ on $\bC^{n+1}$ satisfying 12.10~i) and ii),
such that the conic affine varieties $\swt Y|_\alpha\subset\bC^{n+1}$
lying over the $Y_\alpha$ are totally geodesic with respect to
$\swt\nabla|$.  Now, Characterization 12.12 yields a sufficient
condition in terms of the linear system of equations
$$
\sum_{0\le\lambda\le n}
\swt\Gamma|^\lambda_{j\mu}{\partial s_\kappa\over\partial z_\lambda}=
{\partial^2 s_\kappa\over\partial z_j\partial z_\mu},\qquad
0\le j,\kappa,\mu\le n.
\leqno(12.16)
$$
(We just fix the choice of $a_j$, $b_\mu$ and $c_{j\mu}$ to be $0$).
This linear system can be considered as a collection of decoupled
linear systems in the unknowns $(\swt\Gamma|^\lambda_{j\mu})_\lambda$, when
$j$ and $\mu$ are fixed. Each of these has format $(n+1)\times(n+1)$
and can be solved by Cramer's rule if the principal determinant
$$
\delta:=\det\Big({\partial s_\kappa\over\partial z_\lambda}\Big)_{
0\le\kappa,\lambda\le n}\not\equiv 0
\leqno(12.17)
$$
is not identically zero. We always assume in the sequel that this non
degene\-racy assumption is satisfied. As $\partial s_\kappa/\partial
z_\lambda$ is homogeneous of degree $p-1$ and $\partial^2 s_\kappa/
\partial z_j\partial z_\mu$ is homogeneous of degree $p-2$, 
the solutions $\swt\Gamma|^\lambda_{j\mu}(z)$ are homogeneous
rational functions of degree~$-1$ (condition 12.10~i)). Moreover, 
$\swt\nabla|$ is symmetric, for $\partial^2s/\partial z_j\partial z_\mu$ 
is symmetric in $j,\mu$. Finally, if we multiply (12.16) by $z_j$ and 
take the sum, Euler's identity yields
$$
\sum_{0\le j,\lambda\le n}
z_j\swt\Gamma|^\lambda_{j\mu}{\partial s_\kappa\over\partial z_\lambda}=
\sum_{0\le j\le n}z_j{\partial^2 s_\kappa\over\partial z_j\partial z_\mu}=
(p-1){\partial s_\kappa\over\partial z_\mu},\qquad 0\le\kappa,\mu\le n.
$$
The non degeneracy assumption implies 
$(\sum_jz_j\swt\Gamma|^\lambda_{j\mu})_{\lambda\mu}=(p-1)\Id$, hence
$$
\swt\Gamma|(\varepsilon,v)=\swt\Gamma|(v,\varepsilon)=(p-1)v
$$
and condition 12.10~ii) is satisfied. From this we infer

\claim 12.18.~Proposition|Let $s=s_0+\cdots+s_n\in\cS_{p;\,k_0,\ldots, 
k_n}$ be satisfying the non degeneracy condition $\delta:=\det(\partial 
s_\kappa/\partial z_\lambda)_{0\le\kappa,\lambda\le n} \not\equiv 0$. Then
the solution $\swt\Gamma|$ of the linear system $(12.16)$ provides
a partial projective meromorphic connection on $\bP^n$ such that
all hypersurfaces
$$
Y_\alpha=\{\alpha_0s_0+\cdots+\alpha_ns_n=0\}
$$
are totally geodesic. Moreover, the divisor of poles $B$ of $\nabla$ has
degree at most equal to~$n+1+\sum k_j$.
\endclaim

\proof. Only the final degree estimate on poles has to be checked. By 
Cramer's rule, the solutions are expressed in terms of ratios
$$
\swt\Gamma|^\lambda_{j\mu}={\delta^\lambda_{j\mu}\over\delta},
$$
where $\delta^\lambda_{j\mu}$ is the determinant obtained by replacing
the column of $\det(\partial s_\kappa/\partial z_\lambda)_{0\le\kappa,\lambda\le n}$of index $\lambda$ by the column
$(\partial^2 s_\kappa/\partial z_j\partial z_\mu)_{0\le\kappa\le n}$.
Now, $\partial s_\kappa/\partial z_\lambda$ is a homogeneous polynomial
of degree $p-1$ which is divisible by $z_\kappa^{p-k_\kappa-1}$, hence 
$\delta$ is a homogeneous polynomial of degree $(n+2)(p-1)$ which is divisible
by $\prod z_j^{p-k_j-1}$. Similarly,
$\partial^2 s_\kappa/\partial z_j\partial z_\mu$ has degree $p-2$ and
is divisible by $z_\kappa^{p-k_\kappa-2}$. This implies that
$\delta^\lambda_{j\mu}$ is divisible by $\prod z_j^{p-k_j-2}$.
After removing this common factor in the numerator and denominator,
we are left with a denominator of degree
$$
\sum_{0\le j\le n}\big((p-1)-(p-k_j-2)\big)=\sum(k_j+1)=n+1+\sum k_j,
$$
as stated.\square

An application of Theorem 12.4 then yields the following theorem on certain
complete intersections in projective spaces.

\claim 12.19.~Theorem|Let $s\in\cS_{p;\,k_0,\ldots,k_{n+q}}\subset
\bC[z_0,z_1,\ldots, z_{n+q}]$ be a homogeneous polynomial
satisfying the non degeneracy assumption $\det(\partial s_\kappa/
\partial z_\lambda)\not\equiv 0$ in $\bC^{n+q+1}$. Let 
$$
Y_\alpha=\big\{\alpha_0s_0+\alpha_1 s_1+\cdots+\alpha_{n+q}s_{n+q}=0\big\}
\subset \bP^{n+q}
$$ 
be the corresponding linear system, and let
$$
Z=Y_{\alpha^1}\cap\cdots\cap Y_{\alpha^q}\subset \bP^{n+q}
$$ 
be a smooth $n$-dimensional complete intersection, for some linearly
independent elements $\alpha^j\in\bC^{n+q+1}$ such that
$ds_{\alpha^1}\wedge\cdots\wedge ds_{\alpha^q}$ does not vanish
along~$Z$. Assume that $Z$ is not contained in the set of poles $|B|$
of the meromorphic connection $\nabla$ defined by $(12.16)$, nor in any 
of the coordinate hyperplanes $z_j=0$, and that
$$
pq>n+q+1+{1\over 2}n(n-1)\Big(n+q+1+\sum k_j\Big).
$$
Then every non\-cons\-tant entire curve $f:\bC\to Z$ is algebraically 
degenerate and satisfies either
\smallskip
\item{\rm i)} $f(\bC)\subset Z\cap|B|$~~  or
\smallskip
\item{\rm ii)} $f(\bC)\subset Z\cap Y_\alpha$ for some member $Y_\alpha$
which does not contain~$Z$.
\endclaim

\proof. By Proposition 12.18, the pole divisor of $\nabla$ has degree at
most equal to $n+q+1+\sum k_j$, hence, if we let $B=\cO(n+q+1+\sum k_j)$, 
we can find a section $\beta\in H^0(\bP^{n+q},B)$ such that the operator
$f\mapsto\beta^{n(n+1)/2}(f)\,W_{Z,\nabla}(f)$ is holomorphic. Moreover, 
as $Z$ is smooth, the adjunction formula yields
$$
K_Z=\big(K_{\bP^{n+q}}\otimes\cO(pq)\big)_{\restriction Z}=\cO_Z(pq-n-q-1).
$$
By (12.3), the differential operator $\beta^{n(n-1)/2}(f)\,W_{Z,\nabla}(f)$ 
defines a section in\break
\hbox{$H^0(Z,E_{n,n(n+1)/2}T^*_Z\otimes L^{-1})$} with
$$
\eqalign{L
&=K_Z\otimes\cO_Z\Big(-{1\over 2}n(n-1)B\Big)\cr
&=\cO_Z\Big(
pq-n-q-1-{1\over 2}n(n-1)\Big(n+q+1+\sum k_j\Big)\Big).\cr}
$$
Hence, if $f(\bC)\not\subset|B|$, we know by Theorem 12.4 that 
$W_{Z,\nabla}(f)\equiv 0$. Fix a point $t_0\in\bC$ such that 
$f(t_0)\notin|B|$ and $(f'(t_0),f''_\nabla(t_0),\ldots,
f^{(n)}_\nabla(t_0))$ is of maximal rank~$r<n$. There must exist an 
hypersurface $Y_\alpha\not\supset Z$ such that
$$
f(t_0)\in Y_\alpha,\quad
f'(t_0),~f''_\nabla(t_0),\ldots,~f^{(n)}_\nabla(t_0)\in T_{Y_\alpha,f(t_0)}.
$$
In fact, these conditions amount to solve a linear system of equations
$$
\sum_{0\le j\le n+q}\alpha_js_j(f(t_0))=0,\qquad \sum_{0\le j\le n+q}
\alpha_jds_j(f^{(j)}_\nabla(t_0))=0
$$
in the unknowns $(\alpha_0,\alpha_1,\ldots,\alpha_{n+q})=\alpha$, which 
has rank${}\le r+1\le n$. Hence the solutions form
a vector space $\Sol$ of dimension at least~$q+1$, and we can find a
solution $\alpha$ which is linearly independent from 
$\alpha^1,\ldots,\alpha^q$. We complete $(\alpha,\alpha^1,\ldots,\alpha^q)$
into a basis of $\bC^{n+q+1}$ and use the fact that the determinant
$\delta=\det(\partial s_\kappa/\partial s_\lambda)$ does not vanish 
identically on $Z$, since 
$$
Z\cap\{\delta=0\}\subset Z\cap\big(|B|\cup\{\prod z_j=0\}\big)
\subsetneq Z.
$$
From this we see that $\sum\alpha_jds_j$ does not vanish identically on~$Z$,
in particular $Z\not\subset Y_\alpha$. By taking a generic element 
$\alpha\in\Sol$, we get a smooth $n$-dimensional hypersurface
$Z_\alpha=Y_\alpha\cap Y_{\alpha^2}\cap\cdots\cap Y_{\alpha^q}$ in
$W=Y_{\alpha^2}\cap\cdots\cap Y_{\alpha^q}$. Lemma~12.13 applied to the
pair $(Z_\alpha,W)$ shows that $f(\bC)\subset Z_\alpha$, hence
$f(\bC)\subset Z\cap Z_\alpha=Z\cap Y_\alpha$, as desired.\square

If we want to decide whether $Z$ is hyperbolic, we are thus reduced
to decide whether the hypersurfaces $Z\cap|B|$ and $Z\cap Y_\alpha$ are 
hyperbolic. This may be a very hard problem, especially if $Z\cap|B|$
and $Z\cap Y_\alpha$ are singular. (In the case of a smooth intersection
$Z\cap Y_\alpha$, we can of course apply the theorem again to
$Z'=Z\cap Y_\alpha$ and try to argue by induction). However, when $Z$ is
a surface, $Z\cap|B|$ and $Z\cap Y_\alpha$ are curves and the problem can
in principle be solved directly through explicit genus calculations.

\claim 12.20.~Examples|{\rm\smallskip
\noindent{\rm i)} Consider the Fermat hypersurface of degree $p$
$$
Z=\big\{z_0^p+z_1^p+\cdots+z_{n+1}^p=0\big\}
$$
in $\bP^{n+1}$, which is defined by an element in $\cS_{p;\,0,\ldots,0}$.
A simple calculation shows that $\delta=p^{n+2}\prod z_j^{p-1}\not\equiv 0$
and that the Christoffel symbols are given by $\swt\Gamma|_{jj}^j=(p-1)/z_j$
(with all other coefficients being equal to~$0$).
Theorem 12.19 shows that all nonconstant entire curves $f:\bC\to Y$ are
algebraically degenerate when 
$$
p>n+2+{1\over2}n(n-1)(n+2).
$$ 
In fact the term ${1\over 2}n(n-1)(n+2)$ coming from the pole order estimate
of the Wronskian is by far too pessimistic. A more precise calculation shows 
in that case that $(z_0\cdots z_{n+1})^{n-1}$ can be taken as a denominator
for the Wronskian. Hence the algebraic degeneracy occurs for
$p>n+2+(n+2)(n-1)$, i.e., $p\ge(n+1)^2$. However, the Fermat hypersurfaces
are not hyperbolic. For instance, when $n=2$, they contain rational lines
$z_1=\omega z_0$, $z_3=\omega'z_2$ associated with any pair 
$(\omega,\omega')$ of $p$-th roots of~$-1$.
\medskip
\noindent{\rm ii)} Following J.~El Goul ([EG96,~97]), let us consider 
surfaces $Z\subset\bP^3$ of the form
$$
Z=\big\{z_0^p+z_1^p+z_2^p+z_3^{p-2}
(\varepsilon_0z_0^2+\varepsilon_1z_1^2+\varepsilon_2z_2^2+z_3^2)=0\big\},
$$
defined by the element in $\cS_{p;\,0,0,0,2}$ such that $s_3=z_3^{p-2}
(\varepsilon_0z_0^2+\varepsilon_1z_1^2+\varepsilon_2z_2^2+z_3^2)$ and
$s_j=z_j^p$ for $0\le j\le 2$. One can check that $Z$ is smooth provided
that
$$
\sum_{j\in J}\varepsilon_j^{\textstyle{p\over p-2}}\ne 
{2\over p-2}\Big(-{p\over 2}\Big)^{\textstyle{p\over p-2}},\qquad
\forall J\subset\{0,1,2\},
\leqno(12.21)
$$
for any choice of complex roots of order~$p-2$.  The connection
$\swt\nabla|=d+\swt\Gamma|$ is computed by solving linear systems with
principal determinant $\delta=\det(\partial s_\kappa/\partial z_\lambda)$
equal to
$$
\eqalign{
&\left\vert
\matrix{
pz_0^{p-1}     &      0     &     0       &     0     \cr
   0        & pz_1^{p-1}    &     0       &     0     \cr
   0        &     0     &    pz_2^{p-1}   &     0     \cr
2\varepsilon_0z_0z_3^{p-2} & 
2\varepsilon_1z_1z_3^{p-2} & 
2\varepsilon_2z_2z_3^{p-2} & 
(p-2)z_3^{p-3}\big(\varepsilon_0z_0^2+\varepsilon_1z_1^2+\varepsilon_2z_2^2+
{p\over p-2}z_3^2\big)\cr}
\right\vert\cr
&\quad{}=p^3(p-2)z_0^{p-1}z_1^{p-1}z_2^{p-1}z_3^{p-3}
\Big(\varepsilon_0z_0^2+\varepsilon_1z_1^2+\varepsilon_2z_2^2+
{p\over p-2}z_3^2\Big)\not\equiv 0.\cr}
$$
The numerator of $\swt\Gamma|^\lambda_{j\mu}$ is obtained by replacing
the column of index $\lambda$ of $\delta$ by\break
$(\partial^2s_\kappa/\partial z_j\partial z_\mu)_{0\le\kappa\le 3}$,
and $z_0^{p-2}z_1^{p-2}z_2^{p-2}z_3^{p-4}$ cancels in all terms. Hence
the pole order of $\swt\nabla|$ and of $W_{\swt\nabla|}$ is~$6$ (as given
by Proposition~12.18), with
$$
z_0z_1z_2z_3\Big(\varepsilon_0z_0^2+\varepsilon_1z_1^2+\varepsilon_2z_2^2+
{p\over p-2}z_3^2\Big)
$$
as the denominator, and its zero divisor as the divisor~$B$. The condition
on $p$ we get is $p>n+2+6=10$. An explicit calculation shows that all 
curves $Z\cap|B|$ and $Z\cap Y_\alpha$ have geometric genus${}\ge 2$ under
the additional hypothesis
$$
\left\{\eqalign{
&\hbox{none of the pairs $(\varepsilon_i,\varepsilon_j)$ is equal to 
$(0,0)$,}\cr
&\hbox{$\varepsilon_i/\varepsilon_j\ne-\theta^2$ whenever $\theta$ is a
root of $\theta^p=-1$.}\cr}\right.
\leqno(12.22)
$$
[(12.22) excludes the existence of lines in the intersections 
$Z\cap Y_\alpha$.]}
\endclaim

\claim 12.24.~Corollary|Under conditions $(12.21)$ and $(12.22)$, the
algebraic surface
$$
Z=\big\{z_0^p+z_1^p+z_2^p+z_3^{p-2}
(\varepsilon_0z_0^2+\varepsilon_1z_1^2+\varepsilon_2z_2^2+z_3^2)=0\big\}
\subset\bP^3
$$
is smooth and hyperbolic for all $p\ge 11$.\square
\endclaim

Another question which has raised considerable interest is to decide
when the complement $\bP^2\ssm C$ of a plane curve $C$ is hyperbolic.
If $C=\{\sigma=0\}$ is defined by a polynomial $\sigma(z_0,z_1,z_2)$
of degree~$p$, we can consider the surface $X$ in $\bP^3$ defined
by $z_3^p=\sigma(z_0,z_1,z_2)$. The projection 
$$
\rho:X\to\bP^2,\qquad (z_0,z_1,z_2,z_3)\mapsto(z_0,z_1,z_2)
$$
is a finite $p:1$ morphism, ramified along~$C$. It follows that
$\bP^2\ssm C$ is hyperbolic if and only if its unramified covering
$X\ssm\rho^{-1}(C)$ is hyperbolic; hence a sufficient condition is that
$X$ itself is hyperbolic. If we take $\varepsilon_2=0$ in Cor.~12.23
and exchange the roles of $z_2,\,z_3$, we get the following

\claim 12.24.~Corollary|Consider the plane curve
$$
C=\big\{z_0^p+z_1^p+z_2^{p-2}
(\varepsilon_0z_0^2+\varepsilon_1z_1^2+z_2^2)=0\big\}
\subset\bP^2,\qquad\varepsilon_0,\,\varepsilon_1\in\bC^*.
$$
Assume that neither of the numbers $\varepsilon_0,\,\varepsilon_1,\,
\varepsilon_0+\varepsilon_1$ is equal to
${2\over p-2}\big(-{p\over 2}\big)^{\textstyle{p\over p-2}}$
and that $\varepsilon_1/\varepsilon_0\ne-\theta^2$ whenever
$\theta^p=-1$. Then $\bP^2\ssm C$ is hyperbolic.\square
\endclaim

\section{\S13. Decomposition of jets in irreducible representations}

Let us first briefly recall the definition of the Schur fonctors
$\Gamma^\bullet$ (they are frequently denoted $S_\bullet$ in the
literature, but we want to avoid any confusion with ordinary symmetric
powers). Let $V$ be a complex vector space of dimension~$r$. To the
set of nonincreasing $r$-tuples $(a_1,a_2\ld a_r)\in\bZ^r$, $a_1\ge
a_2\ge\cdots\ge a_r$, one associates in a fonctorial way a collection
of vector spaces $\Gamma^{(a_1,a_2\ld a_r)}V$ which provide the list of
all irreducible representations of the linear group $\GL(V)$, up to
isomorphism (here, $(a_1\ld a_r)$ is the highest weight of the action of
a maximal torus $(\bC^*)^r\subset\GL(V))$. The Schur fonctors
can be defined in an elementary way as follows. Let
$\bU_r=\Big\{\pmatrix{1&0\cr *&1\cr}\Big\}$ be the group of lower
triangular unipotent $r\times r$ matrices. If all $a_j$ are nonnegative,
one defines
$$
\Gamma^{(a_1,a_2\ld a_r)}V\subset S^{a_1}V\otimes\cdots\otimes S^{a_r}V
$$
to be the set of polynomials $P(\xi_1\ld\xi_r )$ on $(V^*)^r$ which
are homogeneous of degree $a_j$ with respect to $\xi_j$ and which are
invariant under the left action of $\bU_r$ on $(V^*)^r=\Hom(V,\bC^r)$,
namely such that
$$
P(\xi_1\ld\xi_{j-1},\xi_j+\xi_k,\xi_{j+1}\ld\xi_r)=P(\xi_1\ld\xi_r)\qquad
\forall k<j.
$$
We agree that $\Gamma^{(a_1,a_2\ld a_r)}V=0$ unless $(a_1,a_2\ld a_r)$ is
nonincreasing. As a special case, we recover symmetric and exterior powers
$$
\leqalignno{S^kV&=\Gamma^{(k,0\ld 0)}V,&\cr
        \Lambda^kV&=\Gamma^{(1\ld 1,0\ld 0)}V,
\qquad\hbox{(with $k$ indices $1$)}&(13.1)\cr
        \det V&=\Gamma^{(1\ld 1)}V.\cr}
$$
The Schur fonctors satisfy the well-known formula
$$
\Gamma^{(a_1+\ell\ld a_r+\ell)}V=\Gamma^{(a_1\ld a_r)}V\otimes(\det V)^\ell.
\leqno(13.2)
$$
This formula can of course be used to define $\Gamma^{(a_1\ld a_r)}V$ if
any of the $a_j$'s happens to be negative.

Now, by what we saw in section \S$\,$7, the group $\bG_k'$ of
germs of reparametrizations $\varphi(t)=t+b_2t^2+\cdots+b_kt^k+O(t^{k+1})$
tangent to identity acts on $k$-tuples $(f',f''\ld f^{(k)})$ of derivatives
of $f$ at $0$ by the formulas
$$
(f\circ\varphi)'=f',\quad
(f\circ\varphi)''=f''+2b_2f',\quad
(f\circ\varphi)'''=f'''+3b_2f''+3b_3f',\,\ldots\,.
$$
This is clearly a unipotent action, induced by the action of $\bU_k$
through an embedding
$$
\bG'_k\lhra\bU_k,\qquad\varphi\longmapsto
\pmatrix{1&0&0&\cdots&0&0\cr
         2b_2&1&0&\cdots&0&0\cr
         3b_3&3b_2&1&\cdots&0&0\cr
         \vdots&&&&\vdots&\vdots\cr
         &&\cdots&&1&0\cr
         kb_k&&\cdots&&kb_2&1\cr}.
$$
By formula (6.5), we find that the graded bundle of $E_{k,m}V^*$ is
$$
\Gr^\bullet E_{k,m}V^*
=\left(\bigoplus_{\ell\in\bN^k,\,\ell_1+2\ell_2+\cdots+k\ell_k=m}
S^{\ell_1}V^*\otimes S^{\ell_2}V^*\otimes
\cdots\otimes S^{\ell_k}V^*\right)^{\bG'_k}.
$$
Since the action of $\bG'_k$ does not preserve each individual component
in the summation, the computation of the invariants is quite difficult in
general. We will see however that everything is easy if $k\le 2$. In fact,
if $k=1$, then
$$
E_{1,m}V^*=E^\GG_{1,m}V^*=S^mV^*.\leqno(13.3)
$$
If $k=2$, the effect of a parameter change
$(f',f'')\mapsto(f',f''+\lambda f')$ on a weighted homogeneous
polynomial $Q(f',f'')=\sum_{|\alpha_1|+2|\alpha_2|=m}
a_{\alpha_1\alpha_2}(f')^{\alpha_1}(f'')^{\alpha_2}$ is to replace each
monomial $(f')^{\alpha_1}(f'')^{\alpha_2}$ by a sum
$$
\sum_{\beta}C_\beta\lambda^{|\beta|}(f')^{\alpha_1+\beta}
(f'')^{\alpha_2-\beta}.
$$
It follows that terms $(f')^{\alpha_1}(f'')^{\alpha_2}$ corresponding to 
different values of the pair\break \hbox{$(|\alpha_1|,|\alpha_2|)=:(\ell_1,\ell_2)$}
cannot produce monomials with the same multidegree and the same exponent 
$|\beta|$ of $\lambda$. Hence the various components $S^{\ell_1}V^*
\otimes S^{\ell_2}V^*$ do not mix up and we get
$$
\Gr^\bullet E_{2,m}V^*=\bigoplus_{\ell_1+2\ell_2=m}
\big(S^{\ell_1}V^*\otimes S^{\ell_2}V^*\big)^{\bG'_k}
=\bigoplus_{\ell_1+2\ell_2=m}\Gamma^{(\ell_1,\ell_2,0\ld0)}V^*.
\leqno(13.4)
$$
In the special case when $r=\rank V=2$, (13.1) and (13.2) yield
$\Gamma^{(\ell_1,\ell_2)}V^*=S^{\ell_1-\ell_2}V^*
\otimes(\det V^*)^{\ell_2}$. Hence we get the simpler formula
$$
\Gr^\bullet E_{2,m}V^*=\bigoplus_{0\le j\le m/3}
S^{m-3j}V^*\otimes(\det V^*)^j
\qquad\quad(k=r=2).
\leqno(13.5)
$$
Similar calculations can be done for low values of $k$ and $m$, but it is
a major unsolved problem to compute the decomposition formula of
$\Gr^\bullet E_{k,m}V^*$ for arbitrary $k$ and~$m$.

\claim 13.6.~Special case|{\rm Assume that $X$ is a surface and consider
the absolute case $V=T_X$. We find
$$
\Gr^\bullet E_{2,m}T^*_X=\bigoplus_{0\le j\le m/3}
S^{m-3j}T^*_X\otimes K_X^j,
$$
where $E_{1,m}T^*_X=S^mT^*_X$ is a subbundle of $E_{2,m}
T^*_X$. We thus get an exact sequence
$$
0\to S^mT^*_X\to E_{2,m}T^*_X\to Q_m\to 0,
$$
and $Q_m$ admits a filtration with
$$
\Gr^\bu Q_m=\bigoplus_{1\le j\le m/3}S^{m-3j}T^*_X\otimes K_X^j.
$$
The simplest case is $m=3$, which yields the interesting exact sequence
$$
0\to S^3T^*_X\to E_{2,3}T^*_X\to K_X\to 0.
$$}
\endclaim

\claim 13.7.~Complement|{\rm Assume that $X$ is a surface of degree $d$
in $\bP^3$. Then $K_X=\cO_X(d-4)$. As $T_X^*$ is a quotient bundle 
of $T^*_{\bP^3|X}$ and as $T^*_{\bP^n|X}\otimes\cO(2)$ is 
generated by sections, we conclude that $S^{m-3j}T^*_X\otimes K_X^j$ 
is (very) ample whenever $j(d-4)>2(m-3j)$. This condition is most restrictive
when $j=1$. In particular, $Q_m$ is ample for $d>2m-2$, and we see that
there is at most a ``very small part'' of $E_{2,m}T^*_X$, namely
$S^mT^*_X$, which need not be ample when the degree $d$ is large. 
By contrast, the Green-Griffiths graded bundle 
$$
G^\bu E^\GG_{2,m}T^*_X=\sum_{\ell_1+2\ell_2=m}S^{\ell_1}T^*_X\otimes
S^{\ell_2}T^*_X
$$
does not such exhibit such strong positivity properties. This is one of
the reasons for which the invariant jet bundles
$E_{k,m}T^*_X$ frequently provide more accurate estimates in the study 
of hyperbolicity questions.}
\endclaim

\section{\S14. Riemann-Roch calculations and study of the base locus}

In view of the Green-Griffiths conjecture 4.7 concerning algebraic 
degeneration of entire curves, the main point is to compute the base loci
$$
B_k=\bigcap_{m>0}
\Bs\big(H^0(X_k,\cO_{X_k}(m)\otimes\pi_{k,0}^*\cO(-A))\big)\subset X_k
\leqno(14.1)
$$
where $X_k=P_kT_X$ and $A$ is an ample divisor over~$X$. By
corollary 8.9, every nonconstant entire curve $f:\bC\to X$ must satisfy
$f_{(k]}(\bC)\subset B_k$. If the set $Y=\bigcap_{k>0}\pi_{k,0}(B_k)$ is 
distinct from $X$, then $f(\bC)\subset Y\subsetneq X$ and every entire
curve is thus algebraically degenerate. We will call $Y$ the {\em 
Green-Griffiths locus of~$X$}, although Green and Griffiths did use
ordinary jet bundles in place of the Semple jet bundles.
Unfortunately, it turns out that $Y$ is extremely hard to compute,
especially in the case when $X$ is an hypersurface or complete
intersection in projective space. (However, an important breakthrough has 
been achieved in [SiYe96a] for the case of complements of curves in $\bP^2\,$;
noticeably, the authors obtain an explicit construction of global jet 
differentials of order $1$ and $2$, which allows them to show that the
base locus is small enough.) Here, we will derive a few sufficient 
conditions for the existence of sections, mostly based on Riemann-Roch
computations and a use of \hbox{(semi-)}stability inequalities. From now on,
we restrict ourselves to the case when $X$ is an algebraic surface of
general type.

The easiest case is the case of order $1$ jets 
$E_{1,m}T^*_X=S^mT^*_X$, namely symmetric differentials. 
The Riemann-Roch formula then gives
$$
\chi(X,S^mT^*_X\otimes \cO(-A))={m^3\over 6}(c_1^2-c_2)+O(m^2),
\leqno(14.2)
$$
where $c_1$ and $c_2$ are the Chern classes of~$X$. This can be seen
e.g.\ by computing $h^3$ for the hyperplane bundle first Chern class 
$h=c_1(\cO_{PT_X}(1))^3$ and using the identity $h^2+c_1h+c_2=0$.
By the Bogomolov vanishing theorem 17.1 of the Appendix, we get
$h^2(X,S^mT^*_X\otimes \cO(-A))=0$ for $m$ large, thus
$$
h^0(X,S^mT^*_X\otimes \cO(-A))\ge{m^3\over 6}(c_1^2-c_2)-O(m^2).
\leqno(14.3)
$$
As a consequence, if $c_1^2>c_2$, there are non trivial symmetric
differentials $\sigma$ with values in $\cO(-A)$, and every entire
curve must satisfy the corresponding order~$1$ differential equation
$\sigma(f')=0$. This is especially interesting in connection with the 
following result of Jouanolou [Jou78].

\claim 14.4. Theorem {\rm (Jouanolou)}|Let $Z$ be a compact complex 
manifold such that the Hodge spectral sequence degenerates in $E_2$,
and let $\cL\subset\Omega^1_Z$ be a rank $1$ coherent subsheaf such that
$\Omega^1_Z/\cL$ has no torsion. Let $\cV\subset\cO(T_X)$ be the dual
distribution of hyperplanes in~$T_Z$. Then either $\cV$ is the 
relative tangent sheaf of a meromorphic fibration from $Z$ to a curve, 
or there are only finitely many compact hypersurfaces tangent to~$\cV$. 
\endclaim

\noindent
(Jouanolou [Jou78] even obtains a precise upper bound for the number of
hypersurfaces which may occur in terms of $h^0(X,\Omega^2_X\otimes\cL^{-1})$
and of the Picard number of~$X$). As a consequence, one recovers the 
following result due to Bogomolov [Bog77].

\claim 14.5. Theorem {\rm (Bogomolov)}|On a surface $X$ of general type 
such that $c_1^2>c_2$, there are only finitely many rational or
elliptic curves.
\endclaim

\proof. By the results of \S$\,$7, these curves must be integral curves
of some multivalued distribution of lines in $X$, associated with
the zero divisor $Z\subset P(T_X)$ of any nonzero section in 
$$
H^0(P(T_X),\cO_{P(T_X)}(m)\otimes\pi_{1,0}^*\cO(-A)).
$$ 
At a generic point of $Z$ over a point $x\in X$, this distribution 
defines a unique line in~$T_{X,x}$, and we thus get a rank $1$ subsheaf
of $T_{\tilde Z}$ (or $\Omega^1_{\tilde Z}$) on any 
desingularization $\wt Z$ of~$Z$. By Jouanolou's result applied
to~$\smash{\wt Z}$, either these integral curves form a family or 
there are only a finite number of them. If they form a family, not all of them
can be rational or elliptic, otherwise $X$ would be a ruled or elliptic
surface; hence the general fiber has genus at least $2$.
In both cases, there are only finitely many rational or elliptic 
curves.\square

The above result of Bogomolov does not give information on transcendental
curves, essentially because very little is known on transcendental leaves
of a randomly chosen meromorphic foliation (e.g., one does not know how
to decide whether there are only finitely many integral curves of parabolic 
type). As observed by Lu and Yau [LuYa90], one can say more if the 
{\em topological index} $c_1^2-2c_2$ is positive, using the following
result of Schneider-Tancredi [ScTa85] (the special case when $E=T_X^*$
is due to Miyaoka [Miy82]).

\claim 14.6. Theorem {\rm(Schneider-Tancredi)}|Let $E$ be rank $2$ vector 
bundle over a projective algebraic surface~$X$. Assume that $\det E$
is nef and big $($i.e.\ $c_1(E)$ is numerically nonnegative and
$c_1(E)^2>0\,)$, that $E$ is $(\det E)$-semistable and that
$c_1(E)^2-2\,c_2(E)>0$. Then $E$ is almost ample in the sense that
$S^mE$ generates all $1$-jets of sections outside a finite union of curves
in~$X$, when $m$ is large enough.
\endclaim

\proof\ ({\em sketch}\/). Let $P=P(E^*)$ be the hyperplane bundle of $E$
and $H=\cO_P(1)$. Then $P$ is a ruled $3$-fold and the hypotheses imply
$c_1(H)^3=c_1(E)^2-c_2(E)>0$.
Hence by Riemann-Roch and Serre duality, either $h^0(X,S^mE)$ or 
$h^0(X,S^mE^*)$ grow fast. The latter case is impossible by the
assumption on semistability and the assumption $\det E$ nef. Therefore
$H$ is big. Fix an ample divisor $A$ on~$P$. We have to show that the 
base locus of $mH-A$ in $P$ projects to a curve in $X$ when $m$ is large. 
Otherwise, let $D$ be an irreducible component of a divisor in the linear 
system $|mH-A|$. In the Picard group $\Pic(P)=\Pic(X)\oplus\bZ[H]$ we then 
have $D=kH-\pi^* F$ for some integer $k>0$ and some
divisor $F$ on~$X$. Observing that the multiplication by the canonical 
section of $H^0(P,\cO(D))$ yields an injection of sheaves
$$
\cO(F)\hookrightarrow \pi_*\cO(kH)=\cO(S^kE),
$$
we find by semistability
$$
c_1(F)\cdot c_1(E)\le {1\over k+1}c_1(S^kE)\cdot c_1(E)={k\over 2}c_1(E)^2.
$$
From this, we infer
$$
\eqalign{
H^2\cdot D=H^2\cdot(kH-\pi^* F)
&=k(c_1(E)^2-c_2(E))-c_1(E)\cdot c_1(F)\cr
&\ge {k\over 2}\big(c_1(E)^2-2\,c_2(E)\big)>0,\cr}
$$
therefore $(mH-A)^2\cdot D>0$ for $m$ large. By Riemann-Roch, either
$$
h^0(D,p(mH-A)_{|D})\quad\hbox{or}\quad h^2(D,p(mH-A)_{|D})
$$ 
grows fast as $p$ goes to infinity. By stability again, the latter
case cannot occur, as we see by looking at the exact sequence
$$
0\to \cO(-D)\otimes \cO(p(mH-A)) \to \cO_P \otimes \cO(p(mH-A)) \to 
\cO_D \otimes \cO(p(mH-A)) \to 0,
$$
and descending everything at the $h^2$ and $h^3$
level down to $X$ by the Leray spectral sequence. Hence $H_{|D}$ is big, with
most of its sections extending to $P$. As $D\in|mH-A|$, we see that $H$
is strictly positive outside possibly some curve contained in $D$
and the claim follows.\square

\claim 14.7. Theorem {\rm([LuYa90])}|Let $X$ be a smooth algebraic surface
of general type such that $c_1^2-2c_2>0$. Then there are only finitely 
many rational or elliptic curves in $X$, and every non constant 
entire curve $f:\bC\to X$ maps to one of these.
\endclaim

\proof. One may assume that $X$ is minimal, i.e.\ that $K_X$ is nef (and big).
By the work of Bogomolov [Bog79], $T_X^*$ is semi-stable. The result
of Schneider and Tancredi now implies that $T_X^*$ is almost ample.
Theorem 8.8 concludes the proof.\square
\medskip

We now turn ourselves to the case of jet differentials of degree~$2$.
A simple Riemann-Roch computation based on Formula 13.6 shows that
$$
\chi(X,E_{2,m}T^*_X)=\chi(X,\Gr^\bu E_{2,m}T^*_X)
={m^4\over 648}(13\,c_1^2-9\,c_2)+O(m^3)
\leqno(14.8)
$$
where $c_1,\,c_2$ are the Chern classes of $X$ (only the terms of
bidegree $(2,2)$ in $\Ch(\Gr^\bu E_{2,m}T^*_X)$ play a role). 
This formula should be put in perspective with the one obtained
by Green and Griffiths [GrGr80] for the jet bundles $E_{k,m}^\GG T^*_X$.
In the case of surfaces, they obtain
$$
\chi(X,E_{k,m}^\GG T^*_X)={m^{2k+1}\over (k!)^2(2k+1)!}(\alpha_k\,
c_1^2-\beta_k\,c_2)+O(m^{2k}),
$$
where $\alpha_k\sim {1\over 2}(\log k)^2$ and $\beta_k=O(\log k)$ (especially
$\lim \beta_k/\alpha_k=0$). In the special case $n=k=2$, their formula
yields
$$
\chi(X,E_{2,m}^\GG T^*_X)={m^5\over 384}(7\,c_1^2-5\,c_2)+O(m^4).
$$
This is weaker than formula (14.8) in the sense that the ratio $5/7$ is
larger than $9/13$. In general, we expect analogous estimates of the form
$$
\chi(X,E_{k,m}T^*_X)\sim m^{k+2}(\gamma_k\,c_1^2-\delta_k\,c_2)+O(m^{k+1})
$$
with $\lim\delta_k/\gamma_k=0$ (and even similar higher dimensional
estimates with a leading term of the form $c_{n,k}m^{(n-1)k+n}(-c_1)^n$
when $m\gg k\gg 1$). Unfortunately, our lack of knowledge of the combinatorics
of the Schur representations involved makes the computation hard to achieve
(this will be overcome in \S15 by a direct use of holomorphic Morse inequalities).

In the special case when $X$ is a surface of degree $d$ in $\bP^3$, we have
$c_1=(4-d)h$ and $c_2=(d^2-4d+6)h^2$ where $h=c_1(\cO(1)_{|X})$,
$h^2=d$, thus
$$
\chi(X,E_{2,m}T^*_X)={m^4\over 648}\,d(4\,d^2-68\,d+154)+O(m^3).
$$
This estimate is especially useful in combination with vanishing theorems for 
holomorphic tensor fields (see Theorem 17.1 in the Appendix).

\claim 14.9.~Corollary|If $X$ is an algebraic surface of general type and
$A$ an ample line bundle over~$X$, then
$$
h^0(X,E_{2,m}T^*_X\otimes\cO(-A))\ge
{m^4\over 648}(13\,c_1^2-9\,c_2)-O(m^3).
$$
In particular, every smooth surface $X\subset\bP^3$ of degree $d\ge 15$
admits non trivial sections of $E_{2,m}T^*_X\otimes\cO(-A)$ for $m$
large, and every entire curve $f:\bC\to X$ must satisfy the corresponding
algebraic differential equations.
\endclaim

\proof. First note that the leading term in the Riemann-Roch estimate does
not depend on taking a tensor product by a line bundle $\cO(-A)$.
The claim will follow from the computation of the Euler characteristic
made in (14.8) if we check that $h^2(X,E_{2,m}T^*_X\otimes\cO(-A))=0$
for $m$ large. However
$$
H^2(X,E_{2,m}T^*_X\otimes\cO(-A))=
H^0(X,K_X\otimes (E_{2,m}T^*_X)^*\otimes\cO(A))
$$
by Serre duality. Since $K_X\otimes (E_{2,m}T^*_X)^*\otimes\cO(A)$
admits a filtration with graded pieces
$$
S^{m-3j}T_X\otimes K_X^{\otimes 1-j}\otimes\cO(A),
$$
we easily deduce the vanishing of global sections from Bogomolov's result 
17.1, using the fact that $K_X^{\otimes\nu}\otimes\cO(-A)$ is big for
$\nu\ge\nu_0$ large enough.\square

\claim Other approach using weighted line bundles $\cO_{X_k}(\bfa)$| 
{\rm We show here how a use of the weighted line bundles
$\cO_{X_k}(\bfa)$ may yield further information on the base locus.
Consider a directed manifold $(X,V)$ with $\dim X=n$ and $\rank V=r$.
We set $u_k=c_1(\cO_{X_k}(1))$ and let
$$
c_\bu^{[k]}=1+c_1^{[k]}+\cdots+c_r^{[k]}:=c_\bu(V_k)
$$
be the total chern class of $V_k$. Then the cohomology ring of 
$X_k=P(V_{k-1})$ is defined in terms of generators and relations as the
polynomial algebra $H^\bu(X)[u_1\ld u_k]$ with relations
$$
u_j^r+c_1^{[j-1]}u_j^{r-1}+\cdots+c_{r-1}^{[j-1]}u_j+c_r^{[j-1]}=0,\qquad
1\le j\le k
\leqno(14.10)
$$
(we omit all pull-backs $\pi_j^*$ for simplicity of notation). Moreover,
the exact sequences (5.4) and $(5.4')$ yield the inductive formula
$$
\eqalign{
&c_\bu^{[k]}=c_\bu(\cO_{X_k}(-1))\,c_\bu(T_{X_k/X_{k-1}})
 =(1-u_k)\,c_\bu(T_{X_k/X_{k-1}}),\cr
&c_\bu(T_{X_k/X_{k-1}})=c_\bu(\pi_k^* V_{k-1}\otimes\cO_{X_k}(1))
 =\sum_{0\le j\le r}c_j^{[k-1]}(1+u_k)^{r-j},\cr}
$$
in other words
$$
c_\bu^{[k]}=(1-u_k)\sum_{0\le j\le r}c_j^{[k-1]}(1+u_k)^{r-j}.
\leqno(14.11)
$$
In particular, if $r=\rank V=2$, we find
$$
\leqalignno{
&u_k^2+c_1^{[k-1]}u_1+c_2^{[k-1]}=0,&(14.12)\cr
&c_1^{[k]}=c_1^{[k-1]}+u_k,\qquad c_2^{[k]}=c_2^{[k-1]}-u_k^2,&\cr}
$$
hence
$$
c_1^{[k]}=c_1^{[0]}+u_1+\cdots+u_k,\qquad
c_2^{[k]}=c_2^{[0]}-u_1^2-\cdots-u_k^2.
\leqno(14.13)
$$
From now on, we concentrate again on the surface case. The $2$-jet bundle 
$$
X_2\to X_1\to X
$$ 
is a $2$-step tower of $\bP^1$-bundles over $X$ and therefore has
dimension~$4$. The exact sequence (5.4) shows that $V_1$ has splitting
type $V_{1\restriction F_1}=\cO(2)\oplus\cO(-1)$ along the fibers $F_1$ 
of $X_1\to X$, since $T_{X_1/X\restriction F_1}=\cO(2)$. Hence the fibers
$F_2$ of $X_2\to X$ are Hirzebruch surfaces $P(\cO(2)\oplus\cO(-1))\simeq
P(\cO\oplus\cO(-3))$ and
$$
\cO_{X_2}(1)_{\restriction F_2}=\cO_{P(\cO(2)\oplus\cO(-1))}(1).
$$
The weighted line bundle $\cO_{X_2}(2,1)$ is relatively nef over~$X$,
as follows from our general result (6.16~ii) or from the equality
$$
\cO_{X_2}(2,1)_{\restriction F_2}
=\cO_{P(\cO(2)\oplus\cO(-1))}(1)\otimes\pi^*\cO_{\bP^1}(2)
=\cO_{P(\cO\oplus\cO(-3))}(1).
$$
Its multiples have zero higher order direct images
$R^q(\pi_{2,0})_*\cO_{X_2}(2m,m)$, $q\ge 1$, and lower order direct 
images
$$
(\pi_{2,0})_*\cO_{X_2}(2m,m)=(\pi_{2,0})_*\cO_{X_2}(3m)=E_{2,3m}
T_X^*
$$
[either apply (6.16~i) or observe that 
$$
\eqalign{
&(\pi_{1,0})_*\big(\cO_{P(\cO\oplus\cO(-3))}(m)\big)=
S^m\big(\cO\oplus\cO(3)\big),\cr
&(\pi_{1,0})_*\big(\cO_{P(\cO(2)\oplus\cO(-1))}(3m)\big)=
S^{3m}\big(\cO(-2)\oplus\cO(1)\big)\cr}
$$
have the same sections over $\bP^1$]. By the Leray spectral sequence, we 
conclude that
$$
h^q(X_2,\cO_{X_2}(2m,m))=h^q(X,E_{2,3m}T_X^*),\qquad 0\le q\le 2,
$$
in particular the Euler characteristics are equal and grow as
${1\over 8}m^4(13\,c_1^2-9c_2)$ when $m\to+\infty$. This can also be 
checked directly by computing ${1\over 4!}(2u_1+u_2)^4$. In fact,
(14.12) and (14.13) easily provide
$$
u_1^4=0,\quad u_1^3u_2=c_1^2-c_2,\quad u_1^2u_2^2=c_2,\quad
u_1u_2^3=c_1^2-3c_2,\quad u_1^4=5c_2-c_1^2.
$$
The main difficulty when trying to check the
hyperbolicity of $X$ is to show that the base locus of
$\cO_{X_2}(2,1)$ is small enough. Proving that the base locus is one
dimensional would imply that $X$ only admits a finite number of rational and 
elliptic curves, and that every entire curve $f:\bC\to X$ maps into one of
these. A possibility for this would be to check that $(2u_1+u_2)^3\cdot Y>0$
for every $3$-fold $Y\subset X_2$ and $(2u_1+u_2)^2\cdot S>0$ for every
surface $S\subset X_2$. Unfortunately, such estimates are rather hard
to check, since we would need to evaluate the numerical cones of effective 
codimension $1$ and codimension $2$ cycles in the $4$-fold~$X_2$. The
codimension $1$ case, however, can be treated by using semi-stability 
arguments (although possibly the conditions obtained in this way are 
far from being optimal). The following computation is due to 
J.~El Goul [EG97].}
\endclaim

\claim 14.14. Proposition {\rm ([EG97])}|Let $X$ be a minimal algebraic 
surface of general type. If $c_1^2-{9\over 7}c_2>0$, then $\cO_{X_2}(2,1)$ 
is almost ample on $X_2$ with a base locus of dimension $2$ at most.
\endclaim

\proof\ {\em (sketch)}. We proceed as in the proof of the result by
Miyaoka and Schneider-Tancredi. Let $Y$ be a $3$-dimensional irreducible
component of the base locus, if any. In $\Pic(X_2)=\Pic(X)\oplus\bZ u_1\oplus
\bZ u_2$, we then find an equality
$Y=a_1u_1+a_2u_2-\pi^* F$ for some integers
$a_1,\,a_2\in\bZ$ and some divisor $F$ on $X$. As $Y$ is effective, we must
have $a_1\ge 0$, $a_2\ge 0$. Moreover, $\cO(F)$ can be viewed as a subsheaf
of $\pi_*(\cO_{X_2}(a_1,a_2))\subset E_{2,m}T^*_X$ where $m=a_1+a_2$.
Thus there is a non trivial morphism $\cO(F)\hookrightarrow S^{m-3j}T_X^*
\otimes K_X^j$ for some $j$, and the semistability inequality yields
$$
F\cdot K_X \le{m-j\over 2}K_X^2\le {m\over 2}c_1^2.
$$
A short computation now yields
$$
(2u_1+u_2)^2\cdot Y = (a_1+a_2)(13\,c_1^2-9\,c_2) - 12\,c_1\cdot F\ge
m(7\,c_1^2-9\,c_2).\eqno\square
$$

\claim Logarithmic case|{\rm 
Similar computations can be made in this situation. In fact, if
$X$ is a surface and $D$ is a smooth effective divisor in~$X$, the bundle
$E_{2,m}T_X^*\langle D\rangle$ of logarithmic jet differentials 
of order $2$ and degree $m$ admits a filtration with
$$
\Gr^\bullet E_{2,m}T^*_X\langle D\rangle=\bigoplus_{0\le j\le m/3}
S^{m-3j}(T^*_X\langle D\rangle)\otimes \det(T_X^*\langle D\rangle)^j.
\leqno(14.15)
$$
We thus get 
$$
\leqalignno{
h^0(X,E_{2,m}T^*_X\langle D\rangle)
&\ge\chi(X,E_{2,m}T^*_X\langle D\rangle)\cr
&\ge{m^4\over 648}\big(13\,c_1^2(T_X\langle D\rangle)
-9\,c_2(T_X\langle D\rangle)\big)-O(m^3).&(14.16)\cr}
$$
The exact sequence $0\to T_X\langle D\rangle\to T_X\to(i_D)_* N_{X/D}
\to 0$
yields 
$$
c_\bu(T_X\langle D\rangle)=c_\bu(T_X)c_\bu((i_D)_* N_{X/D})^{-1}
=(1+c_1+c_2)(1+\delta)^{-1},
$$ 
where $\delta=c_1(\cO_X(D))$ and $c_\bu((i_D)_* N_{X/D})=
c_\bu(\cO_X(D))=1+\delta$, thus
$$
c_1(T_X\langle D\rangle)=c_1-\delta,\qquad
c_2(T_X\langle D\rangle)=c_2-c_1\cdot\delta+\delta^2.
$$
Moreover, the expected vanishing theorem for $h^2(X,E_{2,m}T^*_X
\langle D\rangle)$ still holds since $T_X\langle D\rangle$ is a
subbundle of~$T_X$. In particular, if $X=\bP^2$ and $D$ is a smooth
curve of degree~$d$, we find $c_1(T_X\langle D\rangle)=3-d$,
$c_2(T_X\langle D\rangle)=3-3d+d^2$ and
$$
h^0(X,E_{2,m}T^*_{\bP^2}\langle D\rangle)\ge
{m^4\over 648}(4\,d^2-51\,d+90)-O(m^3).
$$
From this, one infers that every entire curve $f:\bC\to\bP^2\ssm D$
must satisfy a non trivial algebraic differential equation of order~$2$
if~$d\ge 11$.}
\endclaim

\section{\S15. 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].

\subsection \S15.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 15.1. 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 (15.1) that $(X,V)$ is hyperbolic if and only if
for every irreducible variety $Y\subset X$, the linear subspace
$$
V_{\wt Y}=\overline{\smash{T_{\wt Y\ssm E}}
\cap\mu_*^{-1}V}\subset T_{\wt Y}\leqno(15.2)
$$
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$\,$14, 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 15.3. 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~15.3 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 15.10 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.
$$
h^q(\wt X,\wt L^{\otimes m})-h^q(X,L^{\otimes m})=O(m^{n-1}).\leqno(15.4)
$$
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 15.5. 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
$$\Theta_{\cO_{P(V)}(1),h^*}\ge -C\omega\leqno(15.6)$$
for some smooth positive $(1,1)$-form on $P(V)$ and some constant $C>0\;$;
conversely, if (15.6) holds, then $\log h$ is quasi-psh.

\claim 15.7. 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~$15.5$. 
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, 
$$
K_{V,h}^m=\cK_V^m\otimes_{\cO_X}\ol\cJ_{h,m_0}^{m/m_0}\leqno(15.8)
$$
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 15.1 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 
$$
\big|P(z\,;\,\xi_1,\ldots,\xi_k)\big|\le 
C\Big(\sum\Vert \xi_j\Vert_h^{1/j}\Big)^m\leqno(15.9)
$$
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 (15.3) 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 15.10. 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 15.10~(SM) when $q=1$ yields a very useful
criterion for the existence of sections of large multiples of $L$.
\vskip2mm

\vbox{\claim 15.11. 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 \S15.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 $\bC^n$; also, for every holomorphic or meromorphic
section $\sigma$ of a Hermitian line bundle $(L,h)$  the Lelong-Poincar\'e
can be formulated
$$dd^c\log|\sigma|^2_h=[Z_\sigma]-\Theta_{L,h},$$ 
where $\Theta_{L,h}={\ii\over 2\pi}D_{L,h}^2$ is the $(1,1)$-curvature 
form of $L$ and $Z_\sigma$ 
the zero divisor of~$\sigma$.  The closed $(1,1)$-form $\Theta_{L,h}$ is
a representative of the first Chern class $c_1(L)$. 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:
$$
P(a_1,\ldots,a_k)=\bC^k\ssm\{0\}/\bC^*,\qquad
\lambda\cdot z =(\lambda^{a_1}z_1,\ldots,\lambda^{a_k}z_k).
\leqno(15.12)
$$
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
$$
\pi_a^*\omega_{a,p}=dd^c\varphi_{a,p},\qquad
\varphi_{a,p}(z)={1\over p}\log\sum_{1\le s\le k}|z_s|^{2p/a_s},
\leqno(15.13)
$$
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 $a_s$. It is at least $C^2$ if $p$ is real and $p\ge\max(a_s)$,
which will be more than sufficient for our purposes (but everything
would still work for any $p>0$). The resulting metric is in any case 
smooth and positive definite outside of the coordinate hyperplanes
$z_s=0$, and these hyperplanes will not matter here since they are of 
capacity zero with respect to all currents $(dd^c\varphi_{a,p})^\ell$.
In order to evaluate the volume $\int_{P(a_1,\ldots,a_k)}\omega_{a,p}^{k-1}$,
one can observe that
$$
\leqalignno{
\int_{P(a_1,\ldots,a_k)}\omega_{a,p}^{k-1}
&=\int_{z\in\bC^k,\,\varphi_{a,p}(z)=0}\pi_a^*\omega_{a,p}^{k-1}\wedge
d^c\varphi_{a,p}\cr
&=
\int_{z\in\bC^k,\,\varphi_{a,p}(z)=0}(dd^c\varphi_{a,p})^{k-1}\wedge
d^c\varphi_{a,p}\cr
&={1\over p^k}\int_{z\in\bC^k,\,\varphi_{a,p}(z)<0}(dd^ce^{p\varphi_{a,p}})^k.
&(15.14)\cr}
$$
The first equality comes from the fact that $\{\varphi_{a,p}(z)=0\}$ is a
circle bundle over\break $P(a_1,\ldots,a_k)$, together with the identities
$\varphi_{a,p}(\lambda\cdot z)=\varphi_{a,p}(z)+\log|\lambda|^2$
and\break \hbox{$\int_{|\lambda|=1}d^c\log|\lambda|^2=1$}. The third equality 
can be seen by Stokes formula applied to the $(2k-1)$-form
$$
(dd^ce^{p\varphi_{a,p}})^{k-1}\wedge
d^ce^{p\varphi_{a,p}}=
e^{p\varphi_{a,p}}(dd^c\varphi_{a,p})^{k-1}\wedge d^c\varphi_{a,p}
$$
on the pseudoconvex open set $\{z\in\bC^k\,;\;\varphi_{a,p}(z)<0\}$. 
Now, we find
$$
\leqalignno{\kern20pt
(dd^ce^{p\varphi_{a,p}})^k=\Big(dd^c\sum_{1\le s\le k}|z_s|^{2p/a_s}\Big)^k
&=\prod_{1\le s\le k}
\Big({p\over a_s}|z_s|^{{p\over a_s}-1}\Big)(dd^c|z|^2)^k,&(15.15)\cr
\int_{z\in\bC^k,\,\varphi_{a,p}(z)<0}
(dd^ce^{p\varphi_{a,p}})^k&=\prod_{1\le s\le k}{p\over a_s}=
{p^k\over a_1\ldots a_k}.&(15.16)\cr}
$$
In fact, (15.15) and (15.16) are clear when $p=a_1=\ldots=a_k=1$ (this is
just the standard calculation of the volume of the unit ball in $\bC^k$);
the general case follows by substituting formally $z_s\mapsto 
\smash{z_s^{p/a_s}}$, and using rotational invariance together with the
observation that the arguments of the complex
numbers $\smash{z_s^{p/a_s}}$ now run in the interval $[0,2\pi p/a_s[$ 
instead of $[0,2\pi[$ (say). As a consequence of
(15.14) and (15.16), we 
obtain the well known value
$$
\int_{P(a_1,\ldots,a_k)}\omega_{a,p}^{k-1}={1\over a_1\ldots a_k},\leqno(15.17)
$$
for the volume. 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 $p$).
When $p$ tends to $+\infty$, we have $\varphi_{a,p}(z)\mapsto
\varphi_{a,\infty}(z)=\log\max_{1\le s\le k}|z_s|^{2/a_s}$ and the volume form 
$\omega_{a,p}^{k-1}$ converges to a rotationally invariant measure 
supported by the image of the polycircle $\prod\{|z_s|=1\}$ in
$P(a_1,\ldots,a_k)$. This is so because not all $|z_s|^{2/a_s}$ are equal
outside of the image of the polycircle, thus $\varphi_{a,\infty}(z)$
locally depends only on~$k-1$ complex variables, and so 
$\omega_{a,\infty}^{k-1}=0$ there by log homogeneity.

Our later calculations will require a slightly more general setting.
Instead of looking at $\bC^k$, we consider the weighted $\bC^*$ action
defined by
$$
\bC^{|r|}=\bC^{r_1}\times\ldots\times\bC^{r_k},\qquad
\lambda\cdot z =(\lambda^{a_1}z_1,\ldots,\lambda^{a_k}z_k).
\leqno(15.18)
$$
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 
$$
\leqalignno{
&P(a_1^{[r_1]},\ldots,a_k^{[r_k]})=P(a_1,\ldots,a_1,\ldots,a_k,\ldots,a_k),\cr
&\pi_{a,r}:\bC^{r_1}\times\ldots\times\bC^{r_k}
\ssm\{0\}\longrightarrow P(a_1^{[r_1]},\ldots,a_k^{[r_k]})&(15.19)\cr}
$$
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
$$
\pi_{a,r}^*\omega_{a,r,p}=dd^c\varphi_{a,r,p},\qquad
\varphi_{a,r,p}(z)={1\over p}\log\sum_{1\le s\le k}|z_s|^{2p/a_s}
\leqno(15.20)
$$
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
$$
\int_{P(a_1^{[r_1]},\ldots,a_k^{[r_k]})}
\omega_{a,r,p}^{|r|-1}={1\over a_1^{r_1}\ldots a_k^{r_k}}.
\leqno(15.21)
$$
Since $(dd^c\log|z_s|^2)^{r_s}=0$ on $\bC^{r_s}\ssm\{0\}$ by homogeneity,
we conclude as before that the weak limit
$\lim_{p\to+\infty}\omega_{a,r,p}^{|r|-1}=\omega_{a,r,\infty}^{|r|-1}$ associated
with
$$
\varphi_{a,r,\infty}(z)=\log\max_{1\le s\le k}|z_s|^{2/a_s}\leqno(15.22)
$$
is a measure supported by the image of the product of unit spheres
$\prod S^{2r_s-1}$ in\break $\smash{P(a_1^{[r_1]},\ldots,a_k^{[r_k]})}$, which is
invariant under the action of 
\hbox{$U(r_1)\times\ldots\times U(r_k)$} on\break
\hbox{$\bC^{r_1}\times\ldots\times\bC^{r_k}$},
and thus coincides with the Hermitian area measure up to a constant 
determined by condition (15.21). In fact, outside of the product of
spheres, $\varphi_{a,r,\infty}$ locally depends only on at most
$k-1$ factors and thus, for dimension reasons, the top power 
$(dd^c\varphi_{a,r,\infty})^{|r|-1}$ must be zero there. In the next section,
the following change of variable formula will be needed. For simplicity
of exposition we restrict ourselves to continuous functions, but 
a standard density argument would easily extend the formula to all
functions that are Lebesgue integrable with respect to the volume 
form~$\smash{\omega_{a,r,p}^{|r|-1}}$.

\claim 15.24. 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

\proof. The area formula of the disc $\smash{\int_{|\lambda|<1}}
dd^c|\lambda|^2=1$ and a consideration of the unit disc bundle over
$\smash{P(a_1^{[r_1]},\ldots,a_k^{[r_k]})}$ imply that
$$
I_p:=\int_{P(a_1^{[r_1]},\ldots,a_k^{[r_k]})}\kern-2pt
f(z)\,\omega_{a,r,p}^{|r|-1}=\int_{z\in\bC^{|r|},\varphi_{a,r,p}(z)<0}
\kern-2pt f(z)\,
(dd^c\varphi_{a,r,p})^{|r|-1}\wedge dd^ce^{\varphi_{a,r,p}}.\kern-2pt
$$
Now, a straightforward calculation on $\bC^{|r|}$ gives
$$
\eqalign{
(dd^ce^{p\varphi_{a,r,p}})^{|r|}
&=\Big(dd^c\sum_{1\le s\le k}|z_s|^{2p/a_s}\Big)^{|r|}\cr
&=\prod_{1\le s\le k}\Big({p\over a_s}\Big)^{r_s+1}|z_s|^{2r_s(p/a_s-1)}
(dd^c|z|^2)^{|r|}.\cr}
$$
On the other hand, we have $(dd^c|z|^2)^{|r|}={|r|!\over r_1!\ldots r_k!}
\prod_{1\le s\le k}(dd^c|z_s|^2)^{r_s}$ and
$$
\leqalignno{
(dd^ce^{p\varphi_{a,r,p}})^{|r|}
&=\big(p\,e^{p\varphi_{a,r,p}}(dd^c\varphi_{a,r,p}+
p\,d\varphi_{a,r,p}\wedge d^c\varphi_{a,r,p})\big)^{|r|}\cr
&=|r|p^{|r|+1}e^{|r|p\varphi_{a,r,p}}(dd^c\varphi_{a,r,p})^{|r|-1}\wedge
d\varphi_{a,r,p}\wedge d^c\varphi_{a,r,p}\cr
&=|r|p^{|r|+1}e^{(|r|p-1)\varphi_{a,r,p}}(dd^c\varphi_{a,r,p})^{|r|-1}\wedge
dd^ce^{\varphi_{a,r,p}},\cr}
$$
thanks to the homogeneity relation $(dd^c\varphi_{a,r,p})^{|r|}=0$. Putting 
everything together, we find
$$
I_p=\int_{z\in\bC^{|r|},\,\varphi_{a,r,p}(z)<0}~~
{(|r|-1)!\,p^{k-1}f(z)\over (\sum_s|z_s|^{2p/a_s})^{|r|-1/p}}
\,\prod_s{(dd^c|z_s|^2)^{r_s}\over r_s!\,a_s^{r_s+1}|z_s|^{2r_s(1-p/a_s)}}.
$$
A standard calculation in polar coordinates with $z_s=\rho_su_s$,
$u_s\in S^{2r_s-1}$, yields
$$
{(dd^c|z_s|^2)^{r_s}\over |z_s|^{2r_s}}=2r_s{d\rho_s\over \rho_s}\,d\mu_s(u_s)
$$
where $\mu_s$ is the $U(r_s)$-invariant probability measure on 
$S^{2r_s-1}$. Therefore
$$
\eqalign{I_p
&=\int_{\varphi_{a,r,p}(z)<0}
{(|r|-1)!\,p^{k-1}f(\rho_1u_1,\ldots,\rho_ku_k)\over
(\sum_{1\le s\le k}\rho_s^{2p/a_s})^{|r|-1/p}}
\prod_s{2\rho_s^{2pr_s/a_s}{d\rho_s\over \rho_s}d\mu_s(u_s)
\over (r_s-1)!\,a_s^{r_s+1}}\cr
&=\int_{u_s\in S^{2r_s-1},\,\sum t_s<1}\kern-3pt
{(|r|-1)!\,p^{-1}f(t_1^{a_1/2p}u_1,\ldots,t_k^{a_k/2p}u_k)\over
(\sum_{1\le s\le k}t_s)^{|r|-1/p}}
\prod_s{t_s^{r_s-1}dt_s\,d\mu_s(u_s)
\over (r_s-1)!\,a_s^{r_s}}\cr}
$$
by putting $t_s=|z_s|^{2p/a_s}=\rho_s^{2p/a_s}$, i.e.\ 
$\rho_s=t_s^{a_s/2p}$, $t_s\in{}]0,1]$. We use still another change of variable 
$t_s=tx_s$ with $t=\sum_{1\le s\le k}t_s$ and $x_s\in{}]0,1]$,
$\sum_{1\le s\le k}x_s=1$. Then
$$dt_1\wedge\ldots\wedge dt_k=t^{k-1}\,dx\,dt\qquad
\hbox{where $dx=dx_1\wedge\ldots\wedge dx_{k-1}$}.
$$
The $\bC^*$ invariance of $f$ shows that
$$
\eqalign{
I_p&=\int_{u_s\in S^{2r_s-1}\atop{\scriptstyle \Sigma} x_s=1,\, t\in{}]0,1]}
(|r|-1)!f(x_1^{a_s/2p}u_1,\ldots,x_k^{a_k/2p}u_k)
\prod_{1\le s\le k}{x_s^{r_s-1}d\mu_s(u_s)\over (r_s-1)!\,a_s^{r_s}}
{dx\,dt\over p\,t^{1-1/p}}\cr
&=\int_{u_s\in S^{2r_s-1}\atop{\scriptstyle\Sigma}x_s=1\kern18pt}
(|r|-1)!f(x_1^{a_s/2p}u_1,\ldots,x_k^{a_k/2p}u_k)
\prod_{1\le s\le k}{x_s^{r_s-1}d\mu_s(u_s)\over (r_s-1)!\,a_s^{r_s}}\,dx.\cr}
$$
This is equivalent to the formula given in Proposition 15.24. We have
$x_s^{2a_s/p}\to 1$ as $p\to+\infty$, and by Lebesgue's bounded convergence 
theorem and Fubini's formula, we get
$$
\lim_{p\to+\infty}I_p=
{(|r|-1)!\over\prod_s a_s^{r_s}}
\int_{(x,u)\in\Delta_{k-1}\times\prod S^{2r_s-1}}
f(u)\prod_{1\le s\le k}{x_s^{r_s-1}\over(r_s-1)!}\,dx\,d\mu(u).
$$
It can be checked by elementary integrations by parts and induction on 
$k,\,r_1,\ldots,r_k$ that
$$
\int_{x\in\Delta_{k-1}}
\prod_{1\le s\le k}x_s^{r_s-1}dx_1\ldots dx_{k-1}
={1\over (|r|-1)!}\prod_{1\le s\le k}(r_s-1)!~.
\leqno(15.24)
$$
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 $\Delta_{k-1}$ 
and that
$$
\lim_{p\to+\infty}I_p=
{1\over\prod_s a_s^{r_s}}\int_{u\in\prod S^{2r_s-1}}f(u)\,d\mu(u).
$$
Even without an explicit check, Formula (15.24) also follows
from the fact that we must have equality for $f(z)\equiv 1$ in the
latter equality, if we take into account the volume\break
formula~(15.21).\square

\subsection \S15.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 
$$
X^\GG_k:=(J^kV\ssm\{0\})/\bC^*,\qquad \pi_k:\smash{X^\GG_k}\to X
\leqno(15.25)
$$
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
$$
\langle e_\alpha(z),e_\beta(z)\rangle =\delta_{\alpha\beta}+
\sum_{1\le i,j\le n,\,1\le\alpha,\beta\le r}c_{ij\alpha\beta}z_i\overline z_j+
O(|z|^3)\leqno(15.26)
$$
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
$$
\Theta_{V,h}(x_0)=-{\ii\over 2\pi}
\sum_{i,j,\alpha,\beta}
c_{ij\alpha\beta}\,dz_i\wedge d\overline z_j\otimes e_\alpha^*\otimes e_\beta.
\leqno(15.27)
$$
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 $f\in J^k_xV$
$$
\Psi_{h,p,\varepsilon}(f):=\Big(\sum_{\alpha\in I}
\theta_\alpha(x)\sum_{1\le s\le k}\varepsilon_s^{2p}\Vert\nabla^s_\alpha f(0)
\Vert_{h(x)}^{2p/s}\Big)^{1/p}
\leqno(15.28)
$$
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 
$$
\Psi_{h,p,\varepsilon}(\lambda\cdot f)=\Psi_{h,p,\varepsilon}(f)\,|\lambda|^2
\leqno(15.29)
$$
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
$$
\pi_k^*\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}=dd^c\log\Psi_{h,p,\varepsilon}
\leqno(15.30)
$$
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 15.31. 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 (15.26) 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 15.31, 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 (15.26) 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 (15.30), the curvature form of $L_k=\cO_{X_k^\GG}(1)$ 
is given at the central point $x_0$ by the following formula.

\claim 15.32. 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
$$
\gamma_k(z,\xi):={\ii\over 2\pi}
\sum_{1\le s\le k}{1\over s}{|\xi_s|^{2p/s}\over \sum_t|\xi_t|^{2p/t}}
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}(z)
{\xi_{s\alpha}\overline\xi_{s\beta}\over|\xi_s|^2}\,dz_i\wedge d\overline z_j
\leqno(15.33)
$$
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 15.24 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
$$
g_k(z,x,u)={\ii\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j
\leqno(15.34)
$$
and $\bOne_{g_k,q}(z,x,u)$ is the characteristic function of its $q$-index 
set. Here 
$$
d\nu_{k,r}(x)=(kr-1)!\,{(x_1\ldots x_k)^{r-1}\over (r-1)!^k}\,dx
\leqno(15.35)
$$
is a probability measure on $\Delta_{k-1}$, and we can rewrite
$$
\leqalignno{
&\int_{X^\GG_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}
={(n+kr-1)!\over n!(k!)^r(kr-1)!}~~\times\cr
&\qquad\int_{z\in X}
\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
\bOne_{g_k,q}(z,x,u)g_k(z,x,u)^n\,d\nu_{k,r}(x)\,d\mu(u).&(15.36)\cr}
$$
Now, formula (15.34) 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.
$$
{\bf E}(g_k(z,\bu,\bu))={1\over kr}
\Big(1+{1\over 2}+\ldots+{1\over k}\Big)\Theta_{\det(V^*),\det h^*},
\leqno(15.37)
$$
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 (15.36), 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, we will prove the following result.

\claim 15.38. 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
$q>n$. We start with three elementary lemmas.

\claim 15.39. Lemma| The integral
$$I_{k,r,n}=\int_{\Delta_{k-1}}
\bigg(\sum_{1\le s\le k}{x_s\over s}\bigg)^nd\nu_{k,r}(x)$$
is given by the expansion
$$
I_{k,r,n}=\sum_{1\le s_1,s_2,\ldots,s_n\le k}{1\over s_1s_2\ldots s_n}\;
{(kr-1)!\over  (r-1)!^k}\;{\prod_{1\le i\le k} (r-1+\beta_i)!\over(kr+n-1)!}.
\leqno{\rm(a)}
$$
where $\beta_i=\beta_i(s)=\card\{j\,;\;s_j=i\}$, $\sum\beta_i=n$, 
$1\le i\le k$. The quotient
$$
I_{k,r,n}\bigg/{r^n\over kr(kr+1)\ldots (kr+n-1)}
\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^n
$$
is bounded below by $1$ and bounded above by
$$
1+{1\over 3}\sum_{m=2}^n{2^mn!\over (n-m)!}
\bigg(1+{1\over 2}+\ldots+{1\over k}\bigg)^{-m}=1+O((\log k)^{-2})
\leqno{\rm(b)}
$$
As a consequence
$$
\leqalignno{
I_{k,r,n}&={1\over k^n}\Big(\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^n+
O((\log k)^{n-2})\Big)&{\rm(c)}\cr
&={(\log k+\gamma)^n+O((\log k)^{n-2})\over k^n}
\cr}
$$
where $\gamma$ is the Euler-Mascheroni constant.
\endclaim

\proof. Let us expand the $n$-th power 
$\big(\sum_{1\le s\le k}{x_s\over s}\big)^n$. This gives
$$
I_{k,r,n}=\sum_{1\le s_1,s_2,\ldots,s_n\le k}{1\over s_1s_2\ldots s_n}
\int_{\Delta_{k-1}}
x_1^{\beta_1}\ldots x_k^{\beta_k}\,d\nu_{k,r}(x)$$
and by definition of the measure $\nu_{k,r}$ we have
$$\int_{\Delta_{k-1}}
x_1^{\beta_1}\ldots x_k^{\beta_k}\,d\nu_{k,r}(x)=
{(kr-1)!\over (r-1)!^k}\int_{\Delta_{k-1}}
x_1^{r+\beta_1-1}\ldots x_k^{r+\beta_k-1}\,dx_1\ldots dx_k.
$$
By Formula (15.24), we find
$$\eqalign{
\int_{\Delta_{k-1}}
x_1^{\beta_1}\ldots x_k^{\beta_k}\,d\nu_{k,r}(x)&=
{(kr-1)!\over (r-1)!^k}\;{\prod_{1\le i\le k}(r+\beta_i-1)!\over (kr+n-1)!}\cr
&={r^n\prod_{i,\,\beta_i\ge 1}(1+{1\over r})(1+{2\over r})\ldots
(1+{\beta_i-1\over r})\over kr(kr+1)\ldots(kr+n-1)},\cr}
$$
and (15.39$\,$a) follows from the first equality. The final product is 
minimal \hbox{when $r=1$}, thus
$$
\leqalignno{
&{r^n\over kr(kr+1)\ldots(kr+n-1)}\le
\int_{\Delta_{k-1}}
x_1^{\beta_1}\ldots x_k^{\beta_k}\,d\nu_{k,r}(x)\cr
&\kern7cm\le{r^n\prod_{1\le i\le k}\beta_i!\over kr(kr+1)\ldots(kr+n-1)}.
&(15.40)\cr}
$$
Also, the integral is maximal when all $\beta_i$ vanish except one, in 
which case one gets
$$\int_{\Delta_{k-1}}
x_j^n\,d\nu_{k,r}(x)=
{r(r+1)\ldots(r+n-1)\over kr(kr+1)\ldots (kr+n-1)}.
\leqno(15.41)
$$
By (15.40), we find the lower and upper bounds
$$
\leqalignno{
I_{k,r,n}&\ge {r^n\over kr(kr+1)\ldots (kr+n-1)}
\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^n,&(15.42)\cr
I_{k,r,n}&\le {r^n\over kr(kr+1)\ldots (kr+n-1)}
\sum_{1\le s_1,\ldots,s_n\le k}{\beta_1!\ldots \beta_k!\over s_1\ldots s_n}.
&(15.43)\cr}
$$
In order to make the upper bound more explicit, we reorganize the
$n$-tuple $(s_1,\ldots,s_n)$ into those indices $t_1<\ldots<t_\ell$
which appear a certain number of times $\alpha_i=\beta_{t_i}\ge 2$, and 
those, say
$t_{\ell+1}<\ldots<t_{\ell+m}$, which appear only once. We have of
course $\sum\beta_i=n-m$, and each choice of the $t_i$'s corresponds
to $n!/\alpha_1!\ldots\alpha_\ell!$ possibilities for the $n$-tuple
$(s_1,\ldots,s_n)$. Therefore we get
$$
\sum_{1\le s_1,\ldots,s_n\le k}{\beta_1!\ldots \beta_k!\over s_1\ldots s_n}\le
n!\sum_{m=0}^n~~\sum_{\ell,\,\Sigma\alpha_i=n-m}~~\sum_{(t_i)}
{1\over t_1^{\alpha_1}\ldots t_\ell^{\alpha_\ell}}{1\over t_{\ell+1}
\ldots t_{\ell+m}}.
$$
A trivial comparison series vs.\ integral yields
$$
\sum_{s<t<+\infty}{1\over t^\alpha}\le {1\over \alpha-1}{1\over s^{\alpha-1}}
$$
and in this way, using successive integrations in $t_\ell$, $t_{\ell-1}$,
$\ldots$~, we get inductively
$$
\sum_{1\le t_1<\ldots<t_\ell<+\infty}{1\over t_1^{\alpha_1}\ldots t_\ell^{\alpha_\ell}}
\le{1\over \prod_{1\le i\le\ell}(\alpha_{\ell-i+1}+\ldots+\alpha_\ell-i)}
\le {1\over \ell!}\,,
$$
since $\alpha_i\ge 2$ implies $\alpha_{\ell-i+1}+\ldots+\alpha_\ell-i\ge i$. On 
the other hand
$$
\sum_{1\le t_{\ell+1}<\ldots<t_{\ell+m}\le k}{1\over t_{\ell+1}\ldots t_{\ell+m}}
\le{1\over m!}\kern-2pt\sum_{1\le s_1,\ldots,s_m\le k}\kern-2pt
{1\over s_1\ldots s_m}={1\over m!}\bigg(1+{1\over 2}+\ldots+{1\over k}\bigg)^m.
$$
Since partitions $\alpha_1+\ldots+\alpha_\ell=n-m$ satisfying the additional
restriction $\alpha_i\ge 2$ correspond to $\alpha_i'=\alpha_i-2$ satisfying 
$\sum \alpha'_i=n-m-2\ell$, their number is equal to
$$
{n-m-2\ell+\ell-1\choose \ell-1}={n-m-\ell-1\choose \ell-1}\le
2^{n-m-\ell-1}
$$
and we infer from this
$$
\sum_{1\le s_1,\ldots,s_n\le k}{\beta_1!\ldots \beta_k!\over s_1\ldots s_n}\le
\sum_{\ell\ge 1\atop 2\ell+m\le n}{2^{n-m-\ell-1}n!\over \ell!\,m!}
\bigg(1+{1\over 2}+\ldots+{1\over k}\bigg)^m
+\bigg(1+{1\over 2}+\ldots+{1\over k}\bigg)^n
$$
where the last term corresponds to the special case $\ell=0$, $m=n$. Therefore
$$
\eqalign{
\sum_{1\le s_i\le k}{\beta_1!\ldots \beta_k!\over s_1\ldots s_n}
&\le{e^{1/2}-1\over 2}\sum_{m=0}^{n-2}{2^{n-m}n!\over m!}
\bigg(1{+}{1\over 2}{+}\ldots{+}{1\over k}\bigg)^m\!+
\bigg(1{+}{1\over 2}{+}\ldots{+}{1\over k}\bigg)^n\cr
&\le {1\over 3}\sum_{m=2}^n{2^mn!\over (n-m)!}
\bigg(1{+}{1\over 2}{+}\ldots{+}{1\over k}\bigg)^{n-m}\!+
\bigg(1{+}{1\over 2}{+}\ldots{+}{1\over k}\bigg)^n.\cr}
$$
This estimate combined with (15.42,~15.43) implies the upper bound
(15.39~b) (the lower bound $1$ being now obvious). The asymptotic estimate
(15.39~c) follows immediately.\square

\claim 15.44. Lemma| If $A$ is a Hermitian $n\times n$ matrix, set
$\bOne_{A,q}$ to be equal to $1$ if $A$ has signature $(n-q,q)$ and
$0$ otherwise. Then for all $n\times n$ Hermitian matrices $A$,~$B$
we have the estimate
$$
\big|\bOne_{A,q}\det A-\bOne_{B,q}\det B\big|\le \Vert A-B\Vert~
\sum_{0\le i\le n-1}\Vert A\Vert^i\Vert B\Vert^{n-1-i},
$$
where $\Vert A\Vert$, $\Vert B\Vert$ are the Hermitian operator norms 
of the matrices.
\endclaim

\proof. We first check that the estimate holds for $|\det A-\det B|$.
Let $\lambda_1\le\ldots\le \lambda_n$ be the eigenvalues of $A$
and $\lambda'_1\le\ldots\le \lambda'_n$ be the eigenvalues of $B$.
We have $|\lambda_i|\le \Vert A\Vert$, $|\lambda'_i|\le \Vert B\Vert$
and the minimax principle implies that 
$|\lambda_i-\lambda'_i|\le\Vert A-B\Vert$.
We then get the desired estimate by writing
$$
\det A-\det B=\lambda_1\ldots\lambda_n-\lambda'_1\ldots\lambda'_n=
\sum_{1 \le i\le n}\lambda_1\ldots\lambda_{i-1}(\lambda_i-\lambda_i')
\lambda'_{i+1}\ldots\lambda'_n.
$$
This already implies (15.44) if $A$ or $B$ is degenerate. If $A$ and $B$ are 
non degenerate we only have to prove the result
when one of them (say $A$) has signature $(n-q,q)$ and the other
one (say $B$) has a different signature. If we put $M(t)=(1-t)A+tB$, the 
already established estimate for the determinant yields 
$$\Big|{d\over dt}\det M(t)\Big|\le n\Vert A-B\Vert\;\Vert M(t)\Vert
\le n\Vert A-B\Vert\big((1-t)\Vert A\Vert+t\Vert B\Vert\big)^{n-1}.
$$
However, since the signature of $M(t)$ is not the same for $t=0$ and $t=1$, 
there must exist $t_0\in{}]0,1[$ such that $(1-t_0)A+t_0B$ is degenerate.
Our claim follows by integrating the differential estimate on
the smallest such interval $[0,t_0]$, after observing that $M(0)=A$, 
$\det M(t_0)=0$, and that the integral of the right hand side on $[0,1]$ 
is the announced bound.\square

\claim 15.45. Lemma| Let $Q_A$ be the Hermitian quadratic form associated
with the Hermitian operator $A$ on $\bC^n$. If $\mu$ is the rotation 
invariant probability measure on the unit sphere $S^{2n-1}$ of $\bC^n$ 
and $\lambda_i$ are the eigenvalues of $A$, we have
$$
\int_{|\zeta|=1}|Q_A(\zeta)|^2d\mu(\zeta)={1\over n(n+1)}\Big(
\sum \lambda_i^2+\Big(\sum\lambda_i\Big)^2\Big).
$$
The norm $\Vert A\Vert=\max|\lambda_i|$ satisfies the estimate
$$
{1\over n^2}\Vert A\Vert^2 
\le\int_{|\zeta|=1}|Q_A(\zeta)|^2d\mu(\zeta)\le\Vert A\Vert^2.
$$
\endclaim

\proof. The first identity is an easy calculation, and the inequalities 
follow by computing the eigenvalues of the quadratic form 
$\sum \lambda_i^2+\big(\sum\lambda_i\big)^2-c\lambda_{i_0}^2$, $c>0$.
The lower bound is attained e.g.\
for $Q_A(\zeta)=|\zeta_1|^2-{1\over n}(|\zeta_2|^2+\ldots+|\zeta_n|^2)$
when we take $i_0=1$ and $c=1+{1\over n}$.\square

\noindent
{\em Proof of the Probabilistic estimate {\rm 15.38}}. Take a vector 
$\zeta\in T_{X,z}$,~ $\zeta=\sum\zeta_i{\partial\over\partial z_i}$,~
with $\Vert\zeta\Vert_\omega=1$, and introduce the trace free sesquilinear
quadratic form
$$Q_{z,\zeta}(u)=\sum_{i,j,\alpha,\beta}
\wt c_{ij\alpha\beta}(z)\,\zeta_i\overline\zeta_j\,u_\alpha
\overline u_\beta,\qquad
\wt c_{ij\alpha\beta} = c_{ij\alpha\beta} -
{1\over r}\eta_{ij}\delta_{\alpha\beta},\qquad u\in\bC^r
$$
where $\eta_{ij}=\sum_{1\le\alpha\le r}c_{ij\alpha\alpha}$. We consider the corresponding 
trace free curvature tensor
$$
\wt\Theta_V={\ii\over 2\pi}\sum_{i,j,\alpha,\beta}
\wt c_{ij\alpha\beta}\,dz_i\wedge d\overline z_j\otimes e_\alpha^*\otimes e_\beta.
\leqno(15.46)
$$
As a general matter of notation, we adopt here the convention that the cano\-nical 
correspondence between Hermitian forms and $(1,1)$-forms is 
normalized as $\sum a_{ij}dz_i\otimes d\overline z_j
\leftrightarrow {\ii\over 2\pi}\sum a_{ij}dz_i\wedge d\overline z_j$, and we take
the liberty of using the same symbols for both types of objects; we do so especially 
for $g_k(z,x,u)$ and $\eta(z)={\ii\over 2\pi}\sum \eta_{ij}(z)dz_i\wedge d\overline z_j=
\Tr \Theta_V(z)$. First observe that for all $k$-tuples of unit vectors 
$u=(u_1,\ldots,u_k)\in (S^{2r-1})^k$, $u_s=(u_{s\alpha})_{1\le\alpha\le r}$, we have
$$
\int_{(S^{2r-1})^k}\bigg|\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}\wt c_{ij\alpha\beta}(z)\,\zeta_i\overline\zeta_j
u_{s\alpha}\overline u_{s\beta}\bigg|^2d\mu(u)
=\sum_{1\le s\le k}{x_s^2\over s^2}{\bf V}(Q_{z,\zeta})
$$
where ${\bf V}(Q_{z,\zeta})$ is the variance of $Q_{z,\zeta}$
on $S^{2r-1}$. This is so because we have a sum over $s$ of
independent random variables on $(S^{2r-1})^k$, all of which have zero 
mean value (Lemma 15.45 shows that the variance ${\bf V}(Q)$
of a  trace free Hermitian quadratic form 
$Q(u)=\sum_{1\le\alpha\le r}\lambda_\alpha|u_\alpha|^2$ on the unit 
sphere $S^{2r-1}$ is equal to $\smash{{1\over r(r+1)}\sum\lambda_\alpha^2}\,$,
but we only give the formula to fix the ideas). Formula (15.41) yields
$$
\int_{\Delta_{k-1}}x_s^2d\nu_{k,r}(x)={r+1\over k(kr+1)}.
$$
Therefore, according to notation (15.34), we obtain 
the partial variance formula
$$
\eqalign{
\int_{\Delta_{k-1}\times (S^{2r-1})^k}\big|g_k(z,x,u)(\zeta)
&-\overline g_k(z,x)(\zeta)|^2d\nu_{k,r}(x)d\mu(u)\cr
&={(r+1)\over k(kr+1)}\bigg(\sum_{1\le s\le k}{1\over s^2}\bigg)
\sigma_h(\wt\Theta_V(\zeta,\zeta))^2\cr}
$$
in which
$$
\eqalign{
\overline g_k(z,x)(\zeta)&=\sum_{1\le s\le k}{1\over s}x_s{1\over r}
\sum_{ij\alpha}c_{ij\alpha\alpha}\zeta_i\overline\zeta_j=
\bigg(\sum_{1\le s\le k}{1\over s}x_s\bigg){1\over r}\eta(z)(\zeta),\cr
\sigma_h(\wt\Theta_V(\zeta,\zeta))^2&=
{\bf V}\big(u\mapsto\langle\wt\Theta_V(\zeta,\zeta)u,u\rangle_h\big)
=\int_{u\in S^{2r-1}}
\big|\langle\wt\Theta_V(\zeta,\zeta)u,u\rangle_h\big|^2d\mu(u).
\cr}
$$
By integrating over $\zeta\in S^{2n-1}\subset \bC^n$ and applying the 
left hand inequality in Lemma 15.45 we infer
$$
\leqalignno{
\int_{\Delta_{k-1}\times (S^{2r-1})^k}\big\Vert g_k(z,x,u)-
 \overline g_k(z,x)\Vert_\omega^2&d\nu_{k,r}(x)d\mu(u)\cr
&\le {n^2(r+1)\over k(kr+1)}\bigg(\sum_{1\le s\le k}{1\over s^2}\bigg)
\sigma_{\omega,h}(\wt\Theta_V)^2
&(15.47)\cr}
$$
where $\sigma_{\omega,h}(\wt\Theta_V)$ is the standard deviation of
$\wt\Theta_V$ on $S^{2n-1}\times S^{2r-1}$~:
$$
\sigma_{\omega,h}(\wt\Theta_V)^2=\int_{|\zeta|_\omega=1,\,
|u|_h=1}\big|\langle\wt\Theta_V(\zeta,\zeta)u,u\rangle_h\big|^2d\mu(\zeta)\,
d\mu(u).
$$
On the other hand, brutal estimates give the Hermitian operator norm
estimates
$$
\leqalignno{
\Vert\overline g_k(z,x)\Vert_\omega&\le 
\bigg(\sum_{1\le s\le k}{1\over s}x_s\bigg)
{1\over r}\Vert\eta(z)\Vert_\omega,&(15.48)\cr
\Vert g_k(z,x,u)\Vert_\omega&\le \bigg(\sum_{1\le s\le k}{1\over s}x_s
\bigg)\Vert\Theta_V\Vert_{\omega,h}&(15.49)\cr}
$$
where
$$
\Vert\Theta_V\Vert_{\omega,h}=\sup_{|\zeta|_\omega=1,\,|u|_h=1}
\big|\langle\Theta_V(\zeta,\zeta)u,u\rangle_h\big|.
$$
We use these estimates to evaluate the $q$-index integrals. The integral 
associated with $\overline g_k(z,x)$ is much easier to 
deal with than $g_k(z,x,u)$ since the characteristic function of
the $q$-index set depends only on $z$. By Lemma 15.44 we find
$$
\eqalign{
&\big|\bOne_{g_k,q}(z,x,u)\det g_k(z,x,u)-
\bOne_{\eta,q}(z)\det\overline g_k(z,x)\big|\cr
&\qquad{}\le
\big\Vert g_k(z,x,u)-\overline g_k(z,x)\big\Vert_\omega\sum_{0\le i\le n-1}
\Vert g_k(z,x,u)\Vert_\omega^i\Vert\overline g_k(z,x)\Vert_\omega^{n-1-i}.\cr}
$$
The Cauchy-Schwarz inequality combined with (15.47 -- 15.49) implies
$$
\eqalign{
&\int_{\Delta_{k-1}\times (S^{2r-1})^k}
\big|\bOne_{g_k,q}(z,x,u)\det g_k(z,x,u)-
\bOne_{\eta,q}(z)\det\overline g_k(z,x)\big|\,d\nu_{k,r}(x)d\mu(u)\cr
&\quad{}\le\bigg(\int_{\Delta_{k-1}\times (S^{2r-1})^k}\big\Vert g_k(z,x,u)-
\overline g_k(z,x)\big\Vert_\omega^2d\nu_{k,r}(x)d\mu(u)\bigg)^{1/2}\times\cr
&\kern22pt\bigg(\int_{\Delta_{k-1}\times (S^{2r-1})^k}
\kern-2pt\bigg(\sum_{0\le i\le n-1}\kern-4pt
\Vert g_k(z,x,u)\Vert_\omega^i\Vert\overline g_k(z,x)
\Vert_\omega^{n-1-i}\bigg)^2d\nu_{k,r}(x)d\mu(u)\kern-2pt\bigg)^{1/2}\cr
&\quad{}\le {n(1+1/r)^{1/2}\over (k(k+1/r))^{1/2}}
\bigg(\sum_{1\le s\le k}{1\over s^2}\bigg)^{1/2}\sigma_{\omega,h}(\wt\Theta_V)
\sum_{1\le i\le n-1}\!\Vert\Theta_V\Vert_{\omega,h}^i
\Big({1\over r}\Vert\eta(z)\Vert_\omega\Big)^{n-1-i}\cr
&\kern45pt{}\times\bigg(\int_{\Delta_{k-1}}
\bigg(\sum_{1\le s\le k}{x_s\over s}\bigg)^{2n-2}d\nu_{k,r}(x)\bigg)^{1/2}
{}={}~~O\Big({(\log k)^{n-1}\over k^n}\Big)\cr}
$$
by Lemma 15.39 with $n$ replaced by $2n-2$. This is the essential error 
estimate. As one can see, the growth of the error 
mainly depends on the final integral factor, since the initial
multiplicative factor is
uniformly bounded over $X$. In order to get the principal term, we compute
$$
\eqalign{
\int_{\Delta_{k-1}}\det \overline g_k(z,x)\,d\nu_{k,r}(x)
&={1\over r^n}\det\eta(z)\int_{\Delta_{k-1}}
\bigg(\sum_{1\le s\le k}{x_s\over s}\bigg)^nd\nu_{k,r}(x)\cr
&\sim{(\log k)^n\over r^nk^n}\det\eta(z).\cr}
$$
From there we conclude that
$$
\eqalign{
\int_{z\in X}\int_{(x,u)\in\Delta_{k-1}\times (S^{2r-1})^k}
\bOne_{g_k,q}(z,x,u) &g_k(z,x,u)^n\,d\nu_{k,r}(x)d\mu(u)\cr
&={(\log k)^n\over r^nk^n}
\int_X\bOne_{\eta,q}\eta^n+O\Big({(\log k)^{n-1}\over k^n}\Big)\cr}
$$
The probabilistic estimate 15.38 follows by (15.36).\square

\claim 15.50. Remark| {\rm If we take care of the precise bounds obtained
above, the proof gives in fact the explicit estimate
$$
\int_{X^\GG_k(L_k,q)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}=
{(n+kr-1)!\;I_{k,r,n}\over n!(k!)^r(kr-1)!}\bigg(
\int_X\bOne_{\eta,q}\eta^n+\varepsilon_{k,r,n}J\bigg)
$$
where
$$
J=n\,(1+1/r)^{1/2}\bigg(\sum_{s=1}^k{1\over s^2}\bigg)^{1/2}
\int_X\sigma_{\omega,h}(\wt\Theta_V)\sum_{i=1}^{n-1}
r^{i+1}\Vert\Theta_V\Vert_{\omega,h}^i
\Vert\eta(z)\Vert_\omega^{n-1-i}\omega^n
$$
and
$$
\eqalign{
|\varepsilon_{k,r,n}|&\le
{\displaystyle\bigg(\int_{\Delta_{k-1}}
\bigg(\sum_{s=1}^k{x_s\over s}\bigg)^{2n-2}d\nu_{k,r}(x)\bigg)^{1/2}\over
\displaystyle (k(k+1/r))^{1/2}\int_{\Delta_{k-1}}\phantom{\Bigg|}
\bigg(\sum_{s=1}^k{x_s\over s}\bigg)^nd\nu_{k,r}(x)}\cr
&\le
{\Big(1+{1\over 3}\sum_{m=2}^{2n-2}{2^m(2n-2)!\over(2n-2-m)!}
\big(1+{1\over 2}+\ldots+{1\over k}\big)^{-m}\Big)^{1/2}
\over 1+{1\over 2}+\ldots+{1\over k}}\sim{1\over \log k}\cr}
$$
by the lower and upper bounds of $I_{k,r,n}$, $I_{k,r,2n-2}$ obtained 
in Lemma 15.39. As\break $(2n-2)!/(2n-2-m)!\le (2n-2)^m$, one easily shows that
$$
|\varepsilon_{k,r,n}|\le {(31/15)^{1/2}\over \log k}\qquad\hbox{for $k\ge e^{5n-5}$}.\leqno(15.51)
$$
Also, we see that the error terms vanish if $\smash{\wt\Theta_V}$ is 
identically zero, but this is of course a rather unexpected 
circumstance. In general, since the form $\wt\Theta_V$ is trace free, 
Lemma~15.45 applied to the quadratic form
$u\mapsto\langle\wt\Theta_V(\zeta,\zeta)u,u\rangle$ on $\bC^r$ implies 
$$\sigma_{\omega,h}(\wt\Theta_V)\le (r+1)^{-1/2}\Vert\wt\Theta_V\Vert_{\omega,h}.$$
This yields the simpler bound
$$\kern15pt
J\le n\,r^{1/2}\bigg(\sum_{s=1}^k{1\over s^2}\bigg)^{1/2}
\int_X\Vert\wt\Theta_V\Vert_{\omega,h}\sum_{i=1}^{n-1}
r^i\Vert\Theta_V\Vert_{\omega,h}^i
\Vert\eta(z)\Vert_\omega^{n-1-i}\omega^n.
\leqno\hbox to 0.1pt{\rlap{\rlap{(15.52)}\kern\hsize\llap{$\square$}}}
$$}
\endclaim

It will be useful to extend the above estimates to the case of sections of
$$
L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big({1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)
\leqno(15.53)
$$
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
(15.36--15.38), the renormalized curvature $\eta_k(z,x,u)$ of $L_k$ takes 
the form
$$
\eta_k(z,x,u)={1\over {1\over kr}(1+{1\over 2}+\ldots+{1\over k})}g_k(z,x,u)+
\Theta_{F,h_F}(z),
\leqno(15.54)
$$
and by the same calculations its  expected value is
$$
\eta(z):={\bf E}(\eta_k(z,\bu,\bu))=\Theta_{\det V^*,\det h^*}(z)+
\Theta_{F,h_F}(z).
\leqno(15.55)
$$
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 15.38 is therefore still true in exactly the same form, provided
we use (15.53 -- 15.55) 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 15.56. 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 (15.56$\,$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~15.7).
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 15.7, 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~15.3.

\claim 15.57. 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 (15.57) follows from the fact that $\delta$ can be taken arbitrary 
small.\square

\claim 15.58. 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 
$$K_X=\cO_X(d_1+\ldots+d_s-n-s-1).$$
Assume that $X$ is of general type, i.e.\ $\sum d_j>n+s+1$.
Let us equip $V=T_X$ with the restriction of the Fubini-Study metric 
$h=\Theta_{\cO(1)}\,$;
a better choice might be the K\"ahler-Einstein metric but we want to 
keep the calculations as elementary as possible. The standard formula for
the curvature tensor of a submanifold gives
$$
\Theta_{T_X,h}=(\Theta_{T_{\bP^{n+s}},h})_{|X}+\beta^*\wedge\beta
$$
where
$\smash{\beta\in 
C^\infty\big(\Lambda^{1,0}T_X^*\otimes\Hom(T_X,\bigoplus\cO(d_j))\big)}$
is the second fundamental form. In other words, by the well known
formula for the curvature of projective space, we have
$$
\langle\Theta_{T_X,h}(\zeta,\zeta)u,u\rangle=|\zeta|^2|u|^2
+|\langle\zeta,u\rangle|^2-|\beta(\zeta)\cdot u|^2.
$$
The curvature $\rho$ of $(K_X,\det h^*)$ (i.e.\ the opposite of the 
Ricci form $\Tr\Theta_{T_X,h}$) is given by
$$
\rho=-\Tr\Theta_{T_X,h}=\Tr(\beta\wedge\beta^*)-(n+1)h\ge -(n+1)h.
\leqno(15.59)
$$
We take here $F=\cO_X(-a)$, $a\in\bQ_+$, and we want to determine 
conditions for the existence of sections
$$
H^0\bigg(X,E^\GG_{k,m}T_X^*\otimes\cO
\Big(-a{m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)\Big)\bigg),
\qquad m\gg 1.\leqno(15.60)
$$
We have to choose $K_X\otimes\cO_X(-a)$ ample, i.e.\
$\sum d_j>n+s+a+1$, and then (by an appropriate choice of the metric
of $F=\cO_X(-a)$), the form $\eta=\Theta_{K_X\otimes\cO_X(-a)}$ can be taken to
be any positive form cohomologous to $(\sum d_j-(n+s+a+1))h$.
We use remark 15.50 and estimate the error terms by considering
the K\"ahler metric
$$
\omega=\rho+(n+s+2)h\equiv \Big(\sum d_j+1\Big)h.
$$
Inequality (15.59) shows that $\omega\ge 2h$ and also that
$\omega\ge\Tr(\beta\wedge\beta^*)$. From this, one easily concludes 
that $\Vert\eta\Vert_\omega\le 1$ by an appropriate choice of $\eta$,
as well as $\Vert\Theta_{T_X,h}\Vert_{\omega,h}\le 1$ 
and $\Vert\wt\Theta_{T_X,h}\Vert_{\omega,h}\le 2$. By (15.52), we obtain
for $n\ge 2$
$$
J\le n^{3/2}{\pi\over \sqrt{6}}\times 2\,{n^n-1\over n-1}\int_X\omega^n
<{4\pi\over\sqrt{6}}\,n^{n+1/2}\int_X\omega^n
$$
where $\int_X\omega^n=\big(\sum d_j+1\big)^n\deg(X)$. On the other hand,
the leading term $\int_X\eta^n$ equals 
$\big(\sum d_j-n-s-a-1\big)^n\deg(X)$ with
$\deg(X)=d_1\ldots d_s$. By the bound (15.51) on the error term 
$\varepsilon_{k,r,n}$, we find that the leading coefficient of the
growth of our spaces of sections is strictly
controlled below by a multiple of
$$
\Big(\sum d_j-n-s-a-1\Big)^n-4\pi\Big({31\over 90}\Big)^{1/2}\,
{n^{n+1/2}\over \log k}\Big(\sum d_j+1\Big)^n
$$
if $k\ge e^{5n-5}$. A sufficient condition for the existence of
sections in (15.60) is thus
$$
k\ge\exp\Big(7.38\,n^{n+1/2}\Big({\sum d_j+1\over\sum d_j-n-s-a-1}\Big)^n\Big).
\leqno(15.61)
$$
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 \S15.D. Non probabilistic estimate of the Morse integrals|

We assume here that the curvature tensor $(c_{ij\alpha\beta})$ satisfies a lower
bound
$$
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}\xi_i\ol\xi_ju_\alpha\ol u_\beta\ge -\sum\gamma_{ij}\xi_i
\ol\xi_j\;|u|^2,
\qquad\forall\xi\in T_X,~u\in V\leqno(15.62)
$$
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
$$
\Theta_{V^*,h^*}+\gamma\otimes\Id_{V^*}\ge 0\leqno(15.62')
$$
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 (15.32), (15.53), (15.54) become instead
$$
\leqalignno{
&g_k^\gamma(z,x,u)={\ii\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}^\gamma(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j\ge 0,
&(15.63)\cr
&L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big(-{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)A\Big),
&(15.64)\cr
&\Theta_{L_k}=\eta_k(z,x,u)={1\over {1\over kr}(1+{1\over 2}+\ldots+{1\over k})}
g_k^\gamma(z,x,u)-(\Theta_{A,h_A}(z)+r\gamma(z)).&(15.65)\cr}
$$
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 15.66.~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 15.66 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 (15.36) 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 15.67.~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 15.11, 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~15.24 we have
$$
c_{n,r,k}={n!\,(kr-1)!\over (n+kr-1)!}\sum_{1\le s_1<\ldots<s_n\le k}{1\over s_1\ldots s_n}\ge
{(kr-1)!\over (n+kr-1)!},\leqno(15.68)
$$
(with equality for $k=n$), and Lemma 15.39~(b) provides the upper bound 
$$
{c'_{n,k,r}\over c_{n,k,r}}\le {(kr+n-1)r^{n-2}\over k/n}
\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^n\bigg[
1+{1\over 3}\sum_{m=2}^{n-1}{2^m(n-1)!\over(n-1-m)!}\Big(1+{1\over 2}+\ldots+{1\over k}
\Big)^{-m}\bigg].
$$
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
$$
{c'_{n,k,r}\over c_{n,k,r}}\le {n^2+n-1\over 3}n^{n-2}\exp\Big({2(n-1)\over H_n}+n\log H_n\Big)
\le {1\over 3} \big(n\log(n\log 24n)\big)^n.
\leqno(15.69)
$$
We will also need the particular values 
$$
\leqalignno{
&c_{2,2,2}={1\over 20},\kern4.8pt\qquad c'_{2,2,2}={9\over 16},\kern9.8pt\qquad
{c'_{2,2,2}\over c_{2,2,2}}={45\over 4},&(15.70_2)\cr
&c_{3,3,3}={1\over 990},\qquad c'_{3,3,3}={451\over 4860},\qquad
{c'_{3,3,3}\over c_{3,3,3}}={4961\over 54},&(15.70_3)\cr}
$$
which can be obtained by direct calculations.

\subsection \S15.E. On the base locus of sections of $k$-jet bundles|

The final step required for a complete solution of the Green-Griffiths
conjecture would be to calculate the base locus $B_k\subset X_k^{\GG}$
of the space of sections
$$
H^0(X_k^\GG,\cO_{X_k^\GG}(m)\otimes\pi_k^*\cO(-m\delta_kA)),\qquad
A~\hbox{ample on $X$,}~~\delta_k\le c{\log k\over k},~~c\ll 1,
$$
and to show that $Y_k=\pi_k(B_k)$ is a proper algebraic 
subvariety of~$X$ for $k$ large, under the assumption that $K_V$ is big. 
This does not look completely hopeless, 
since the statistics of curvature in the Morse inequalities do involve
currents for which the sets of poles depend only on the bigness of
$K_V$ and therefore project onto a proper subvariety $S$ of $X$ (see
the last step of the proof in section~$2$). It is not unreasonable to
think that a further analysis of the asymptotic behavior of sections,
e.g.\ through estimates of the Bergman kernel, might lead to such
results.

Even if the required property of the base locus cannot be obtained
directly, it would be enough, for a suitable irreducible analytic set
$Z\subset X_k^\GG$ contained in the base locus at some stage,
to construct non zero sections in
$$H^0(Z,\cO_{X_k^\GG}(m)_{|Z}\otimes\pi_k^*\cO(-m\delta_kA)_{|Z})$$
whenever $\pi_k(Z)=X$, and then to proceed inductively to cut-down
the base locus until one reaches some $Z'\subset Z$ with 
$\pi_k(Z')\subsetneq X$. Hence we have to estimate the cohomology groups
$H^0$ and $H^q$ not just on $X_k^\GG$, but also on all irreducible
subvarieties $Z\subset X_k^\GG$ such that $\pi_k(Z)=X$. We are not able
to do this in such a generality, but our method does provide
interesting results in this direction.

\claim (15.71) Theorem|Let $(X,V)$ be a compact directed $n$-dimensional
manifold, let \hbox{$r=\rank V$} and $F$ be a holomorphic line bundle on $X$. 
Fix an irreducible analytic set
$\smash{Z_{k_0}\subset X_{k_0}^\GG}$ or equivalently some
$\bC^*$-invariant set $\smash{Z'_{k_0}\subset J^{k_0}V}$, and assume that
$\smash{\pi_{k_0}(Z_{k_0})}=X$. For $k\gg k_0$,
denote by $Z_k\subset X_k^\GG$ the irreducible set corresponding to the
inverse image of $\smash{Z'_{k_0}}$ by the canonical morphism
\hbox{$J^kV\to J^{k_0}V$}. Let $h$ be an admissible metric on $V$, $h_F$ a
metric with analytic singularities on $F$ and
$$
\eqalign{
L_k&=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big({1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big),\cr
\eta&=\Theta_{K_V,\det h^*}+\Theta_{F,h_F},\qquad S=\Sing(\eta).\cr}
$$
Then for $m\gg k\gg k_0$ and $p_k=\dim Z_k=\dim Z_{k_0}+(k-k_0)r$ we have
$$\eqalign{
h^0(Z_k,&\cO(L_k^{\otimes m})_{|Z_k})\cr
&\ge {m^{p_k}\over p_k!}
{(\log k)^n\over n!}\deg_{X_k^\GG/X}(Z_k)\bigg(
\int_{X(\eta,\le 1)\ssm S}\eta^n-O((\log k)^{-1})\bigg)-o(m^{p_k})\cr}
$$
where $\deg_{X_k^\GG/X}(Z_k)=
\deg_{X_{k_0}^\GG/X}(Z_{k_0})\big({k_0!\over k!}\big)^r$
is the relative degree of $Z_k$ over $X$ with respect to the normalized
weighted ``K\"ahler metric'' $\omega_{a,r,p}$ introduced in~$(15.20)$.
\endclaim

We would also get similar upper and lower Morse bounds for the higher 
cohomology groups, provided that the sheaves $\smash{\cO_{X_k^\GG}(m)}$ are
twisted by the appropriate multiplier ideal sheaves $\cJ_{k,m}$ already 
described. The main trouble to proceed further in the analysis of the base
locus is that we have to take $k\gg k_0$ and that the $O(...)$ and $o(...)$
bounds depend on $Z_{k_0}$. Hence the newer sections can only be
constructed for higher and higher orders $k$, without any indication that
we can actually terminate the process somewhere, except possibly by
some extremely delicate uniform estimates which seem at present beyond 
reach.\medskip

\proof. The technique is a minor variation of what has been done in
paragraph~15.C, hence we will only indicate the basic idea. Essentially
the $k$-jet of $f$ is no longer completely random, its projection onto
the first $k_0$ components $(\nabla^jf(0))_{1\le j\le k_0}$ is assigned 
to belong to some given analytic set $\smash{Z'_{k_0}\subset J^{k_0}}V$.
This means that in the curvature formula (15.34)
$$
g_k(z,x,u)={i\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j
$$
only the sum $\sum_{k_0<s\le k}$ is perfectly random. The partial sum
$\sum_{1\le s\le k_0}$ remains bounded, while the harmonic
series diverges as $\log k$. This implies that the ``non randomness'' of the
initial terms perturbs the estimates merely by bounded quantities, and
in the end, the expected value is still similar to (15.37), i.e.
$$
{\bf E}(g_k(z,\bu,\bu))={1\over kr}
\Big(1+{1\over 2}+\ldots+{1\over k}+O(1)\Big)\big(\Theta_{K_V,\det h^*}+
\Theta_{F,h_F}\big).
$$
Once we are there, the calculation of standard deviation and the other
estimates are just routine, and Theorem~15.70 follows again from Proposition
15.38 when we integrate the Morse integrals over~$Z_k$ instead of the
whole $k$-jet space $X_k^\GG$.\square

Another possibility to analyze the base locus is to study the 
restriction maps
$$
\rho_{k,m}(x):H^0(X,E^\GG_{k,m}V^*\otimes\cO(-m\delta_kA))\to 
\big(E^\GG_{k,m}V^*\otimes\cO(-m\delta_kA)\big)_x\leqno(15.72)
$$
at generic points $x\in X$. If $\rho_{k,m}(x)$ can be shown to be surjective
at a generic point, then a fortiori the projection $Y_k=\pi_k(B_k)$ of the 
base locus does not contain $x$ and so $Y_k$ is a proper algebraic subvariety 
of~$X$. Now, proving the surjectivity of $\rho_{k,m}(x)$ could be done by
proving the vanishing of the $H^1$ group of our sheaf twisted by
the maximal ideal $\gm_{X,x}$. We cannot exactly reach such a precise
vanishing result, but Morse inequalities can be used to show that the 
$H^1$ groups do not grow too fast.

In fact assume that $A$ is an ample $\bQ$-divisor on $X$ which is chosen 
so small that $K_V\otimes\cO(-A)$ is still big. By our estimates, we 
can then take $\delta_k={1\over kr}(1+{1\over 2}+\ldots+{1\over k})$.
Pick a very ample divisor $G$ on $X$ and $n$ pencils
of sections $\sigma_{j,t}\in H^0(X,\cO(G))$, $1\le j\le n$, $t\in\bP^1_\bC$,
such that the divisors $\sigma_{j,t_j}(z)=0$ intersect transversally
at isolated points for generic choices of the parameters $t_j\in\bP^1_\bC$.
We select an admissible metric $h$ on $V$ which provides a strictly
positive curvature current on $K_V\otimes\cO(-A)$ and multiply it
by the additional weight factor $(e^\varphi)^{1/rm\delta_k}$ where 
$$\varphi(z)=\log\sum_{1\le j\le n}\prod_{t\in T_j}|\sigma_{j,t}(z)|^{2n}_{h_G}$$
and $T_j\subset \bP^1_\bC$ are generic finite subsets of given 
cardinality~$N$. The multiplier ideal sheaf of
$\varphi$ is precisely equal to the ideal $\cI_E$ of germs of
functions vanishing on a certain $0$-dimensional set 
$E=\{x_1,\ldots,x_s\}\subset X$ of cardinality $s=N^nG^n$. 
Also the resulting curvature form
$$
\eta=\Theta_{K_V,\det h^*}-\Theta_{A,h_A}+{1\over m\delta_k}dd^c\varphi
\ge \Theta_{K_V,\det h^*}-\Theta_{A,h_A}-{N\over m\delta_k}\Theta_{G,h_G}
$$
can be made to be strictly positive as a current provided that
$N\sim cm\delta_k$ with~$c\ll 1$. Then the corresponding multiplier
ideal sheaf of the induced hermitian metric on
$$
\cO_{X_k^\GG}(m)\otimes \pi_k^*\cO(-m\delta_k A)
$$
is the original multiplier sheaf $\cJ_{k,m}$ twisted 
by $\pi_k^*\cI_E$ above $x_j$, provided that the $x_j$ lie outside 
of $\Sing(V)$ and outside of the projection of the support $V(\cJ_{k,m})$. 
Consider the exact sequence
$$
\eqalign{
0&\lra \cO_{X_k^\GG}(m)\otimes \pi_k^*\cO(-m\delta_k A)\otimes\cJ_{k,m}
\otimes\pi_k^*\cI_E\cr
&\lra
\cO_{X_k^\GG}(m)\otimes \pi_k^*\cO(-m\delta_k A)\otimes\cJ_{k,m}\cr
&\lra \cO_{X_k^\GG}(m)\otimes \pi_k^*\cO(-m\delta_k A)\otimes\cJ_{k,m}
\otimes\pi_k^*(\cO_X/\cI_E) \lra 0.\cr}
$$
Its cohomology exact sequence yields an ``almost surjective arrow''
$$
H^0\big(\cO_{X_k^\GG}(m)\otimes
\pi_k^*\cO(-m\delta_k A)\otimes\cJ_{k,m}\big)
\lra\bigoplus_{1\le j\le s}\big(E^\GG_{k,m}V^*\otimes\cO(-m\delta_kA)\big)_{x_j},
$$
namely the image contains the kernel of the map
$$
\bigoplus_{1\le j\le s}\big(E^\GG_{k,m}V^*\otimes\cO(-m\delta_kA)\big)_{x_j}
\lra
H^1\big(\cO_{X_k^\GG}(m)\otimes
\pi_k^*\cO(-m\delta_k A)\otimes\cJ_{k,m}\otimes
\pi_k^*\cI_E\big).
$$
Now, we have a Morse upper bound
$$
h^1\big(\cO_{X_k^\GG}(m)\otimes
\pi_k^*\cO(-m\delta_k A)\otimes\cJ_{k,m}\otimes
\pi_k^*\cI_E\big)\le
{m^{n+kr-1}\over (n+kr-1)!}{(\log k)^n\over n!\,(k!)^r}O\big((\log k)^{-1}\big)
$$
since the $1$-index integral $\int_{X(\eta,1)}h^n$ is identically zero. At the
same time we have $s=N^nG^n\sim c'm^n(\log k)^n/k^n$, and 
it follows that 
$$\dim\bigoplus_{1\le j\le s}\big(E^\GG_{k,m}V^*\otimes
\cO(-m\delta_kA)\big)_{x_j}
\sim s{m^{kr-1}\over (kr-1)!(k!)^r}\sim
{c'm^{n+kr-1}\over (kr-1)!(k!)^r}{(\log k)^n\over k^n}.
$$
By selecting a suitable point $x_j$ and by using a trivial lower 
semi-continuity argument, we get the desired almost surjectivity.

\claim (15.73) Corollary|If $A$ is an ample $\bQ$-divisor on $X$ such that
$K_V\otimes\cO(-A)$ is big and $\delta_k=
{1\over kr}(1+{1\over 2}+\ldots+{1\over k})$, $r=\rank V$,
the restriction map
$$
\rho_{k,m}(x):H^0(X,E^\GG_{k,m}V^*\otimes\cO(-m\delta_kA))\to 
\big(E^\GG_{k,m}V^*\otimes\cO(-m\delta_kA)\big)_x
$$
has an image of dimension larger than $(1-O((\log k)^{-1}))\dim E^\GG_{k,m}V^*$
at a generic point $x\in X$ for $m\gg k\gg 1$.
\endclaim

Such a result puts an upper bound on the vanishing order that a generic
section may have on $X_k^\GG$ above a generic point of~$X$. Our hope is 
that one can then completely ``eliminate'' the base locus by taking 
vertical derivatives along the fibers of $J^kV\to X\,$; those derivations will
necessarily have some poles $\cO(pA)$ which we hope to get cancelled by 
the negative powers $\cO(-m\delta_kA)$. This strategy 
first devised by [Siu02, Siu04] has indeed been successful in some cases 
for the study of generic algebraic degeneracy (e.g.\ for hypersurfaces
of very large degree in $\bP^{n+1}_\bC$). This would work rather easily 
if the rough error term $O((\log k)^{-1})$ could be replaced e.g.\ by
$O(m^{-\varepsilon_k})$ in Corollary (15.73), but this is maybe too much 
to ask~for.

We finally discuss yet another approach. For this we have to introduce
invariant jet differentials along the lines of [Dem95]. In fact, to any 
directed manifold $(X,V)$ one can associate its tower of Semple 
$k$-jet spaces, which is a sequence of
directed pairs $(X_k,V_k)$ starting with $(X_0,V_0)=(X,V)$, together
with morhisms $\wt\pi_k:(X_k,V_k)\to (X_{k-1},V_{k-1})$. These spaces
are constructed inductively by putting $X_k=P(V_{k-1})$ and
$V_k=(\wt\pi_k)_*^{-1}(\cO_{X_k}(-1))$ where
$$
\cO_{X_k}(-1)\subset (\wt\pi_k)^*V_{k-1}\subset (\wt\pi_k)^*T_{X_{k-1}}
$$
is the tautological subbundle (cf.~[Dem95]). In the case where
$V$ is not a subbundle, we can first construct the
absolute tower $(\ol X_k,\ol V_k)$ by starting from $\ol V_0=T_X$, 
and then take $X_k$ to be the closure in $\ol X_k$ of the $k$-step $X'_k$ 
of the relative tower $(X'_k,V'_k)$ constructed 
over the dense Zariski open set $X'=X\ssm \Sing(V)$.
In this way, the tower $(X_k,V_k)$ is 
at least birationally well defined -- in such a birational context we 
can even assume that $X_k$ is
smooth after performing a suitable modification at each stage. Even if 
we start with $V=T_X$ (or an integrable subbundle $V\subset T_X$), the
$k$-jet lifting $V_k$ will not be integrable in general, the only
exception being when $\rank V_k=\rank V=1$. Now, if
$$
\pi_{k,0}=\wt\pi_k\circ\ldots\circ\wt\pi_1:X_k\to X_0=X,
$$
it is shown in [Dem95] that  the direct image sheaf
$$
\pi_{k,0}\cO_{X_k}(m):=E_{k,m}V^*\subset E_{k,m}^\GG V^*
$$
consists of algebraic differential operators $P(f^{(j)}_{j\le k})$ which 
satisfy the invariance property
$$
P((f\circ\varphi)^{(j)}_{j\le k})=
(\varphi')^mP(f^{(j)}_{j\le k})\circ \varphi
$$
when $\varphi\in \bG_k$ is in the group of $k$-jets of biholomorphisms
$\varphi:(\bC,0)\to(\bC,0)$. Since we already assume $\bC^*$ invariance,
it is enough to require invariance by the nilpotent subgroup
$\bG'_k\subset\bG_k$ of $k$-jets tangent to identity. The group
$\bG'_k$ is a semi-direct product of additive groups $(\bC,+)$ consisting
of biholomorphisms $\tau_{j,a}:t\mapsto t+at^j+O(t^{j+1})$, $2\le j\le k$,
$a\in\bC$. In this tower, the biholomorphisms $\tau_{k,a}$ actually
generate a normal subgroup of $\bG'_k$, and we have
$\bG'_k/\{\tau_{k,a}\}\simeq\bG'_{k-1}$. Now, assume that we have found
a section
$$
P\in H^0(X,E^\GG_{k,m}V^*\otimes\cO(-m\delta_kA))
$$
for some ample $\bQ$-divisor $A$ on $X$. Then we have an expansion
$$
P_a(f^{(j)}_{j\le k}):=P((f\circ\tau_{k,a})^{(j)}_{j\le k})=
\sum_{0\le s\le m/k}a^sP_s(f^{(j)}_{j\le k})
$$
and the highest  non zero term $P_s$ is $\{\tau_{k,a}\}$-invariant of 
weighted degree $m-(k-1)s\,$; this comes from the fact that the homothety 
$h_\lambda(t)=\lambda t$ satisfies 
$$\tau_{k,a}\circ h_\lambda=h_\lambda\circ\tau_{k,a\lambda^{k-1}}.$$
Then it makes sense to look at the action of 
$\{\tau_{k-1,a}\}$ on $P_s$,
and proceeding inductively we reach a non zero $\bG'_k$-invariant
(and thus $\bG_k$-invariant) polynomial
$$
Q\in H^0(X,E_{k,m'}V^*\otimes\cO(-m\delta_kA))
$$
of degree $m'\le m$ (and possibly of order $k'\le k$ but we can still
consider it to be of order $k$). By raising $Q$ to some power $p$ and using
the $\bQ$-ampleness of $A$, we obtain a genuine integral section
$$
Q^p\sigma_A^{p(m-m')\delta_k}\in H^0(X,E_{k,pm'}V^*\otimes\cO(-pm'\delta_kA)).
$$

\claim (15.74) Corollary|Let $(X,V)$ be a projective directed manifold such that
$K_V$ is big, and $A$ an ample $\bQ$-divisor on $X$ such that 
$K_V\otimes\cO(-A)$ is still big. Then, if we put $\delta_k={1\over kr}
(1+{1\over 2}+\ldots+{1\over k})$, $r=\rank V$, the space of global 
invariant jet differentials
$$
H^0(X,E_{k,m}V^*\otimes\cO(-m\delta_kA))
$$
has $($many$)$ non zero sections for $m\gg k\gg 1$.
\endclaim

If we have a directed projective variety $(X,V)$ with $K_V$ big, we conclude
that there exists $k\ge 1$ and a proper analytic set $Z\subset X_k$ such that
all entire curves have the image of their $k$-jet $f_{[k]}(\bC)$ contained
in $Z$. Let $Z'$ be an irreducible component of $Z$ such that 
$\pi_{k,0}(Z')=X$ (if $\pi_{k,0}(Z')\subsetneq X$ there is nothing more
to do). Consider the linear subspace
$V'=\overline{T_{Z'\ssm Z''}\cap V_k}$ where $Z''\subset Z'$ is chosen 
such that $Z'\ssm Z''$ is non singular and the intersection 
$T_{Z'\ssm Z''}\cap V_k$ is a subbundle of $T_{Z'\ssm Z''}$. 
If $f_{[k]}(\bC)$ is not
contained identically in $Z''$, then the curve $g=f_{[k]}$ is tangent
to $(Z',V')$. On the other hand, if $f_{[k]}(\bC)\subset Z''$ we can replace 
$Z'$ by $Z''$ and argue inductively on $\dim Z'$. What we have gained 
here is that we have
replaced the initial directed space $(X,V)$ with another one $(Z',V')$
such that $\rank V'<\rank V$, and we can try to argue by induction on
$r=\rank V$.

Observe that the generalized Green-Griffiths conjecture
is indeed trivial for $r=1$ (assuming $K_V=\cO(V^*)$ big)$\,$: in fact we get 
in this case a non zero section $P\in H^0(X,V^{*\otimes k}
\otimes\cO(-A))$ for some $k\gg 1$ and so $P(f)\cdot (f')^k$ must vanish 
for every entire curve $f:(\bC,T_\bC)\to (X,V)$. Therefore $f(\bC)\subset
Y:=\{P(z)=0\}\subsetneq X$. The main difficulty in this inductive
approach is that when we start with $(X,V)$ with $K_V$ big, it seems to 
be very hard to say anything about $K_{V'}$ on $(Z',V')$. Especially,
the singularities of $Z'$ and $V'$ do not seem to be under control. The only
hope would be to have enough control on the sections cutting out $Z'$, and this
requires anyway to understand much more precisely the behavior and 
vanishing order of generic sections
$P\in H^0(X,E_{k,m}V^*\otimes\cO(-m\delta_kA))$. One could try in this context
to take $A$ to approach the positive part in the Zariski decomposition
of~$K_V$, in such a way that the sections $P$ do not have much space to
move around statistically.

\section{\S16. Hyperbolicity properties of  hypersurfaces of high degree}

\subsection \S16.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 $J_k\cV$. Since the general calculations are extremely involved, we deal here with the
special case $n=2$, $k=2$, but the general case is similar by [Mer09].

We fix 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 $J_2\cV$. To begin with, observe that an affine change of coordinates $z\mapsto 1/z$ induces on jet variables the following transformation rules
$$
z'\mapsto -{z'\over z^2}\quad\hbox{and}\quad z''\mapsto {2(z')^2-zz''\over z^3}.
$$
Now, consider a general vector field in the vector space $\bC^3_{z_j}\times\bC^{N_d}_{a_\alpha}\times\bC^3_{z_j'}\times\bC^3_{z_j''}$; it is of the form
$$
\theta=\sum_{|\alpha|\le d,\alpha_1<d}v_\alpha\,{\partial\over\partial a_{\alpha}}
+\sum_{1\le j\le 3}v_j\,{\partial\over\partial z_j}+\sum_{1\le j\le 3}\xi_j^{(1)}\,{\partial\over\partial z_j'}
+\sum_{1\le j\le 3}\xi_j^{(2)}\,{\partial\over\partial z_j''}.
$$
Thus, the conditions to be satisfied by the coefficients of $\theta$ in order to belong to $J_2\cV_0$ are
$$
\eqalign{
&\sum_{|\alpha|\le d, \alpha_1<d}v_\alpha\,z^{\alpha}+\sum_{1\le j\le 3}~\sum_{|\alpha|\le d}a_\alpha\,{\partial z^\alpha\over\partial z_j}\,v_j=0,\cr
&\quad{}\sum_{1\le j\le 3}~\sum_{|\alpha|\le d,  \alpha_1<d}v_\alpha\,{\partial z^\alpha\over\partial z_j}\,z_j'
 \cr 
&\qquad{}+ \sum_{1\le j,k\le 3}~\sum_{|\alpha|\le d}a_\alpha\,{\partial^2 z^\alpha\over\partial z_k\partial z_j}\,v_kz_j'
+\sum_{1\le j\le 3}~\sum_{|\alpha|\le d}a_\alpha\,{\partial z^\alpha\over\partial z_j}\,\xi_j^{(1)}=0,\cr
&\sum_{|\alpha|\le d,  \alpha_1<d}\left(\sum_{1\le j\le 3}~{\partial z^\alpha\over\partial z_j}\,z_j''+
\sum_{j,k=1}^3{\partial^2 z^\alpha\over\partial z_k\partial z_j}\,z_j'z_k'\right)v_\alpha \cr
&\qquad{}+\sum_{1\le j\le 3}~\sum_{|\alpha|\le d}a_\alpha\left(\sum_{k=1}^3{\partial^2 z^\alpha\over\partial z_k\partial z_j}\,z_k''+\sum_{i,k=1}^3{\partial^3 z^\alpha\over\partial z_i\partial z_k\partial z_j}z_k'z_i'\right)v_j \cr
&\qquad{}+\sum_{|\alpha|\le d}\sum_{j,k=1}^3 a_\alpha\,{\partial^2 z^\alpha\over\partial z_k\partial z_j}\bigl(\xi_j^{(1)}z_k'+\xi_k^{(1)}z_j'\bigr)+\sum_{1\le j\le 3} a_\alpha\,{\partial z^\alpha\over\partial z_j}\,\xi_j^{(2)}=0.\cr}
$$
{\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
$$
\eqalign{
\theta_{\alpha}^{111}&={\partial\over\partial a_{\alpha}}-z_1\,{\partial\over\partial a_{\alpha-\delta_1}}-z_2\,{\partial\over\partial a_{\alpha-\delta_2}}-z_3\,{\partial\over\partial a_{\alpha-\delta_3}} \cr
&\qquad+z_1z_2\,{\partial\over\partial a_{\alpha-\delta_1-\delta_2}}+z_1z_3\,{\partial\over\partial a_{\alpha-\delta_1-\delta_3}}+z_2z_3\,{\partial\over\partial a_{\alpha-\delta_2-\delta_3}} \cr
&\qquad\quad -z_1z_2z_3{\partial\over\partial a_{\alpha-\delta_1-\delta_2-\delta_3}}.\cr}
$$
The pole order of these vector fields is equal to $3$, as a change of variables easily shows. Moreover, they are all tangent to $J_2\cV_0$ and invariant under the action of $\bG_2$ (because they do not contain any jet variable, on which the group acts). Of course, there are similarly defined vector fields constructed by permuting the $z$-variables, and changing the multiindex $\alpha$ as indicated by permutations: it is straightforward to see that all these vector fields together span a codimension $7$ vector space in $\ker (T_{J_2\cV}\to T_{J_2T_{\bP^3}})$. The vector fields which generate the remaining seven directions will be constructed at the end of this section.

\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
$$
{\partial\over\partial z_j}-\sum_{|\alpha+\delta_j|\le d}(\alpha_j+1)a_{\alpha+\delta_j}\,{\partial\over\partial a_{\alpha}}.
$$
It is immediate to check that these vector fields, once applied to the first defining equation of $J_2\cV_0$, make it identically vanish. Since the other equations of $J_2\cV_0$ are obtained by taking the derivative of thhe first just with respect to the $z_j$ and $z_j'$ variables, they make identically vanish the other two defining equations, too. Therefore they are tangent to $J_2\cV_0$. Their pole order is one in the $a_{\alpha}$'s variables and they are $\bG_2$-invariant since they do not contain jet variables.

\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. The additional condition on the Wronskian $z'\wedge z''\ne 0$ implies that the family $(\theta_B)$ spans all the $\partial/\partial z^{(k)}_j$-directions on $\cW_0$, as it is straightforward to see. We claim that one can choose the coefficients $p_{\alpha}(z,a,b)$ to be polynomials of degree at most $2$ in $z$ and at most one in $a$ in such a way that $\theta_B$ is tangent to $J_2\cV_0$. To see the invariance with respect to $\bG_2$, observe that the action is the following: if $\varphi: (\bC,0)\to(\bC,0)$ is a $2$-jet of biholomorphism of the origin then the action is
$$
\varphi\cdot(z,a,z',z'')\mapsto (z,a,\varphi'\cdot z',(\varphi')^2\cdot z''+\varphi''\cdot z')
$$
and the corresponding induced action on vector fields is
$$
{\partial\over\partial z}\mapsto{\partial\over\partial z},\quad
{\partial\over\partial a}\mapsto{\partial\over\partial a},\quad
{\partial\over\partial z'}\mapsto\varphi'{\partial\over\partial z'}+\varphi''{\partial\over\partial z''},\quad
{\partial\over\partial z''}\mapsto(\varphi')^2{\partial\over\partial z''}.
$$
For $\theta_B$, only the second addendum needs to be verified to be invariant: it is of the form
$$
z'\,{\partial\over\partial z'}+z''\,{\partial\over\partial z''}.
$$
On the one hand, letting $\varphi$ act on coordinates, one has
$$
z'\,{\partial\over\partial z'}+z''\,{\partial\over\partial z''}\mapsto 
\varphi'\cdot z'\,{\partial\over\partial z'}+\bigl((\varphi')^2\cdot z''+\varphi''\cdot z'\bigr)\,{\partial\over\partial z''};
$$
on the other hand, letting $\varphi$ act on vector fields by its differential, one has
$$
z'\,{\partial\over\partial z'}+z''\,{\partial\over\partial z''}\mapsto z'\,\left(\varphi'{\partial\over\partial z'}+\varphi''{\partial\over\partial z''}\right)+z''\,\left((\varphi')^2{\partial\over\partial z''}\right),
$$
and the invariance follows. Finally, as announced, we have to span the remaining directions in the vector space $\ker (T_{J_2\cV}\to T_{J_2T_{\bP^3}})$. So, consider a vector field with the following shape:
$$
\sum_{|\alpha|\le 2}v_\alpha\,{\partial\over\partial a_{\alpha}}.
$$
To be tangent to $J_2\cV_0$, its coefficients have to satisfy
$$
\sum_{|\alpha|\le 2}v_{\alpha}\,z^\alpha=0,
$$
$$
\sum_{|\alpha|\le 2}\sum_{1\le j\le 3}v_\alpha\,{\partial z^\alpha\over\partial z_j}\,z_j'=0
$$
and
$$
\sum_{\alpha\le 2}\left(\sum_{1\le j\le 3}{\partial z^\alpha\over\partial z_j}\,z_j''+\sum_{j,k=1}^3{\partial^2 z^\alpha\over\partial z_j\partial z_k}\,z_j'z_k'\right)v_\alpha.
$$
We place ourself outside $\cW_0$ and we suppose for simplicity that $z_1'z_2''-z_2'z_1''\ne 0$, the other cases being analogous. Then, we can solve this system with $v_{000}$, $v_{100}$ and $v_{010}$ as unknowns:
$$
\cases{
v_{000}+z_1\,v_{100}+z_2\,v_{010}\,=\cdots \cr
\quad\qquad z'_1\,v_{100}+z'_2\,v_{010}\,=\cdots \cr
\quad\qquad z''_1\,v_{100}+z''_2\,v_{010}=\cdots \cr}
$$
By the Cramer rule, we see that each of these quantities are linear combinations of the $v_\alpha$'s, where $|\alpha|\le 2$, $\alpha\ne (000), (100), (010)$, with coefficients rational functions in $z,z',z''$. The denominator of each such coefficient is just the Wronskian $z_1'z_2''-z_2'z_1''$ and the numerator is a polynomial whose monomials have either degree at most $2$ in $z$ and at most $1$ in $z'$ and $z''$, or degree $1$ in $z$ and three in $z'$; thus, the pole order here is at most $7$. Next, the system itself is $\bG_2$-invariant: letting $\varphi\in\bG_2$ act on it, we find
$$
\sum_{|\alpha|\le 2}v_{\alpha}\,z^\alpha=0,
$$
$$
\varphi'\sum_{|\alpha|\le 2}\sum_{1\le j\le 3}v_\alpha\,{\partial z^\alpha\over\partial z_j}\,z_j'=0
$$
and
$$
(\varphi')^2\sum_{\alpha\le 2}\left(\sum_{1\le j\le 3}{\partial z^\alpha\over\partial z_j}\,z_j''+\sum_{j,k=1}^3{\partial^2 z^\alpha\over\partial z_j\partial z_k}\,z_j'z_k'\right)v_\alpha
+\varphi''\underbrace{\sum_{\alpha\le 2}\sum_{1\le j\le 3}v_\alpha\,{\partial z^\alpha\over\partial z_j}\,z_j'}_{=0}=0.
$$
Therefore its solutions are invariant, too. Summing up, we have proved the following

\claim 16.1.~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 16.2.~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 16.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
(16.3) {\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
(16.4) {\em Assume also, according to Theorems $16.1$ and $16.2$, 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 16.5.~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 $(16.3)$ and $(16.4)$ 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 16.6.~Additional statement|In corollary $16.5$, under the same hypotheses
$(16.3)$ and $(16.4)$, 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

Now, suppose that we replace (16.4) by the weaker form:

\noindent $(16.4')$ {\em there are global $\bG_k$-invariant vector fields $\theta_i$ of
$T_{J_k\cV}\otimes p^*_kB$ which generate the fibers on the complement
of the Wronskian locus
$$
\cW=\big\{j_k(u)\,;\; u'\wedge u''\wedge\ldots\wedge
u^{(k)}(0)=0\big\},\qquad k\le n.
$$}%
Here the wedge product is the wedge product in $T_{\bP^{n+1}}$ calculated in any standard
affine open chart of $\bP^{n+1}$ (the condition is clearly invariant when we replace 
$u$ par $\lambda u$, so it does not depend on the affine chart chosen). The advantage 
of $(16.4)'$ over (16.4) is that we can possibly take $B$ to be smaller. Then, 
in the above arguments, we also have to consider the case where 
$j_k(f)\subset\cW$. This is dealt with by

\claim 16.7. Lemma|Let $f:\bC\to\bC^{N}$ be a holomorphic map $(N=n{+}1)$. 
If $f'\wedge f''\wedge\cdots\wedge f^{(k)}\equiv 0$, then $f(\bC)$ lies inside an affine 
linear subspace of codimension $N-k+1$.
\endclaim

\proof.
Without loss of generality, we can suppose $k>1$, $f'\wedge f''\wedge\cdots\wedge f^{(k-1)}\not\equiv 0$, $f'(0)\ne 0$ and $(f'\wedge f''\wedge\cdots\wedge f^{(k-1)})(0)\ne 0$. Then there exists an open neighborhood $\Omega\subset\bC$ of $0$ such that for each $t\in\Omega$ we have a linear combination
$$
f^{(k)}(t)=\sum_{j=1}^{k-1}\lambda_{j}(t)\,f^{(j)}(t)
$$
and the $\lambda_j$'s depend holomorphically on $t$. By taking
derivatives, one sees inductively that, in $\Omega$, every
$f^{(\ell)}$, $\ell\ge k$, is a linear combination of the $f^{(j)}$'s,
$1\le j\le k-1$. Thus, all the derivatives in $0$ of $f$ lie in the
linear space generated by $f'(0),\dots,f^{(k-1)}(0)$. The conclusion
follows by expanding $f$ as a power series at $0$.\square\medskip

Lemma 16.7 shows that the image of the entire curve $f$ lies in a subvariety $L_f$ of $X$ of codimension${}\ge 2$ (the intersection of $X$ with an arbitrary linear subspace of $\bP^{n+1}$ of codimension${}\ge2$ is also of codimension${}\ge 2$ provided that $X$ is generic). This is of course weaker than our earlier corollary 16.5 where $Y_s$ did not depend on $f$. However, if $n=3$ and $d\ge 7$, we know by Ein and Voisin that the very generic member $X_s$ does not contain rational or elliptic curves, so $f$ has to be constant anyway. This is also true when $n=2$, since $Y_s$ and $L_f$ must be $0$-dimensional in that case.

\claim 16.8. 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 $(16.3)$ and $(16.4')$ are met with $A-mB$ ample. Then the very generic hypersurface 
$X_s\subset\bP^{n+1}$ of degree $d$ is hyperbolic.
\endclaim

\subsection 16.C. Proof of the hyperbolicity of generic surfaces of high degree in $\bP^3$|

In this paragraph we treat the case of generic (hyper)surfaces in $\bP^3$. For this,
we need a slight extension of the Riemann-Roch calculation performed in \S$\,$14 (see also 
[Pau08]), combined with Bogomolov's vanishing theorem 17.1. Such results were
first obtained independently in [McQu99] and [DeEG00], using the results of [McQu98].

\claim 16.9.~Lemma|Let $X$ be a projective surface of general type. Then
$$
h^0(X,E_{2,m}T^*_X\otimes K_X^{-\delta m})\ge {m^4\over648}
\big((54\delta^2-48\delta+13)\,c_1(X)^2-9\,c_2(X)\big)+O(m^3),
$$
provided that $0\le\delta<1/3$.
\endclaim

\claim 16.10.~Theorem {\rm ([McQu99], [DeEG00], [Pau08])}|
Let $X\subset\bP^3$ be a very generic smooth surface of degree $d\ge 18$. Then 
$X$ is Kobayashi hyperbolic.
\endclaim

\proof. We will in fact only give the proof for the weaker bound $d\ge 90$. We refer to 
[Pau08] for the better bound $d\ge 18$; the argument then requires the full force
of McQuillan's deep results on parabolic leaves of algebraic foliations [McQu98].

Let us fix once again the notations. We consider $X\subset\bP^3$ a very generic smooth surface of degree $d$. Its canonical bundle is then expressed in term of the hyperplane bundle as $K_X=\cO_X(d-4)$; thus, $K_X^{\delta m}$ is the (ample) $\bQ$-line bundle $\cO_X(\delta m(d-4))$. The Chern classes of $X$ are given by
$$
c_1(X)=(4-d)\,h,\quad c_2(X)=(d^2-4d+6)\,h^2,
$$
so that the condition $(54\delta^2-48\delta+13)\,c_1(X)^2-9\,c_2(X)>0$ required by Lemma 16.9
becomes
$$
(54\delta^2 - 48\delta + 4)\,d^3 +(-432\delta^2 + 384\delta - 68)\,d^2  + (864\delta^2 - 768\delta + 154)\, d>0.\leqno(16.11)
$$
(In particular, if $0\le\delta<1/3$ and $54\delta^2 - 48\delta + 4>0$, Lemma~16.9 implies
the existence of a non zero global section of $E_{2,m}T^*_X\otimes K_X^{-\delta m}$ for
$m\gg d\gg 1$, but we have to take into account a constraint between $d$ and $\delta$ here). Consider the universal hypersurface $\cX\subset \bP^3\times\bP^{N_d}$ of degree $d$ in $\bP^3$ and the holomorphic subbundle $\cV\subset T_{\cX}$ given by the differential of the kernel of the second projection. By the results of \S$\,$16.A, we know that
$$
T_{J_2\cV}\otimes\cO_{\bP^3}(7)\otimes\cO_{\bP^{N_d}}(1)
$$
is globally generated by its global holomorphic sections over $J_2\cV\ssm\Sigma$ and moreover the generating sections can be chosen to be invariant by the action of $\bG_2$ on $J_2\cV$. In the general strategy discussed in \S$\,$16.B, we can take $A=K_X^{\delta m}=\cO_X(m\delta(d-4))$ and
$B=\cO_{\bP^3}(7)_{|X}$. The condition we need is
$$
A-mB=\cO_X(m\delta(d-4)-7m)\quad\hbox{ample,~~ i.e.\ $\delta(d-4)>7$}.
$$
If we plainly substitute $\delta=7/(d-4)$ in (16.11), we obtain the condition
$$
4\,d^3 - 404\,d^2 + 4144\,d>0,
$$
and (16.11) will then still be true for $\delta=7/(d-4)+\varepsilon$, $\varepsilon$ small.
As the largest root is a little smaller than $90$, we conclude that a very generic 
smooth projective surface of $\bP^3$ of degree $d\ge 90$ is hyperbolic; in that case the
proof does not rely on McQuillan's diophantine approximation technique, and one can check
that it is enough to take $X$ generic rather than very generic (thanks to the fact that
one needs Noether-Lefschetz only for curves of uniformly bounded degree.)\square

\subsection \S16.D. 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$\,$16.B,
combined with the earlier techniques of [Dem95]. Using our improved
bounds from~\S$\,$15.D, we obtain here a better estimate 
(actually of exponential order one $O(\exp(n^{1+\varepsilon}))$ rather 
than order $5$).

\claim 16.12.~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~15.67 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 16.2. 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 $n\ge
4$. Our estimates (15.69) and (15.70) give the expected bound~$d_n$.\square
\medskip

Thanks to 16.6, one also obtains the generic hyperbolicity of $2$ and $3$-dimensional 
hypersurfaces of large degree.

\claim 16.13.~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~$3$. One may also
wonder whether it is possible to use jets of order $k<n$ in the proof
of 16.12 and 16.13. Diverio [Div08] observed that the answer is negative.

\claim 16.14.~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

\proof. The bundle $E_{k,m}^\GG T^*_X$ admits a filtration whose
associated graded bundle is given by
$$
\Gr^\bullet E_{k,m}^\GG T^*_X
=\bigoplus_{l_1+2l_2+\dots+kl_k=m}S^{l_1}T^*_X\otimes
S^{l_2}T^*_X\otimes\dots\otimes S^{l_k}T^*_X.
$$
Now, the terms in the right hand side can be split into irreducible 
Gl$(T^*_X)$-representations of the type 
$\Gamma^{(\lambda_1,\dots,\lambda_n)}T^*_X$ with $\lambda_i=0$ for $i>k\,$:
this actually follows from the classical Pieri formula for Schur
fonctors (cf.\ [FH91]). However, by a result of Br\"uckmann-Rackwitz 
[BR90], we have $H^0(X,\Gamma^{(\lambda_1,\dots,\lambda_n)}T^*_X)=0$ as soon as
$t_1+\ldots+t_s<n$, where $t_i=\#\{\lambda_j\,;\;\lambda_j\ge i\}\le k$.
In particular, we always have $H^0(X,\Gamma^{(\lambda_1,\dots,\lambda_n)}T^*_X)=0$
for those representations when $sk<n$, and by an immediate filtration
argument we conclude that $H^0(X,E_{k,m}^\GG T^*_X)=0$ for $sk<n$.\square

\section{\S17. Appendix: a vanishing theorem for holomorphic
tensor fields}

In this appendix, we prove a basic vanishing theorem for holomorphic tensor 
fields on minimal varieties of general type. It has been observed since a 
long time that the existence of holomorphic tensor fields on a compact
K\"ahler manifold is governed in a rather precise way by the sign of the 
Ricci curvature (in case the Ricci curvature does admit some definite 
sign, semipositive or seminegative). See for instance the papers
[Li67,~71] by Lichnerowicz for the case of sections of $\Lambda^kT_X$ 
or $\Lambda^kT^*_X$, and S.~Kobayashi's articles [Kob80,~81] for 
the more general case of tensors in~$\Gamma^aT_X$. However, 
we want here to consider the situation of varieties of general type
(i.e.\ with $K_X$ big), and it is unknown whether $K_X$ should be 
semipositive even if $K_X$ is assumed to be big and nef. On the other
hand, it is a consequence of Bogomolov's work [Bog79] (dealing with 
the so-called ``Bogomolov stability'' concept), that such vanishing theorems
hold when $T_X$ is semistable; this is the case for instance if $X$ is 
a minimal surface of general type. Tsuji [Tsu88] has proved more generally 
that the tangent bundle $T_X$ is semistable for any minimal nonsingular 
projective variety of general type (here, $X$ ``minimal'' means that
$K_X$ is nef) . Thus, the following theorem 17.1 below can be obtained
as a combination of the above mentioned results of Bogomolov and
Tsuji. For the convenience of the reader, we give instead a direct
proof based on a use of approximate K\"ahler-Einstein metrics in
combination with the Bochner formula. Our hope is that similar a priori
estimates could produce as well vanishing theorems for higher degree
cohomology groups~$H^q$.

\claim 17.1.~Theorem|Let $X$ be a projective algebraic manifold, $n=\dim X$,
and let $L$ be a holomorphic line bundle over~$X$. Assume that $X$ is of
general type and minimal $($i.e.\ $K_X$ is big and nef$)$, and let 
$a=(a_1\ld a_n)\in\bZ^n$, $a_1\ge\cdots\ge a_n$, be a weight. If either
$L$ is pseudoeffective and $|a|=\sum a_j>0$, or $L$ is big and $|a|\ge 0$, 
then
$$
H^0(X,\Gamma^aT_X\otimes L^*)=0.
$$
\endclaim

Recall that a line bundle $L$ is said to be pseudoeffective if
$c_1(L)$ belongs to the closure of the cone of effective divisors, or
equivalently, if $L$ carries a singular hermitian metric $h$ with
curvature current $\Theta_h(L)\ge 0$. Also notice that the result is
invariant by modifications, hence it extends to the case when $X$ is
of general type and possesses a smooth minimal model $\swt X|\,$; this
is always the case when $X$ is a surface. On the other hand, it is
likely that the result holds for all varieties $X$ of general type, 
in view of Mori's minimal model conjecture (however, the differential
geometric proof given below might be difficult to extend to the case
when the minimal model is singular).

\proof\ {\em of Theorem~17.1}. We will use the following notation: if all
$a_j$ are nonnegative integers, $\Gamma^aT_X$ is viewed as a subbundle of
$(T_X)^{\otimes p}$ with $p=|a|$. In particular, given coordinates
$(z_1\ld z_n)$ on $X$, any tensor of $\Gamma^aT_X$ can be expressed as
a linear combination of the elements
$$
(\partial/\partial z)^I:={\partial\over\partial z_{i_1}}\otimes\cdots\otimes
{\partial\over\partial z_{i_p}},\qquad I=(i_1\ld i_p),\quad 1\le i_k\le n,
$$
which form a basis of $(T_X)^{\otimes p}$. If some $a_j$ is negative, we 
use instead the identity
$$
\Gamma^{(a_1\ld a_r)}T_X=\Gamma^{(a_1+\ell\ld a_r+\ell)}T_X\otimes
(\det T_X)^{-\ell}
$$
with $\ell=\max(-a_j)$, and consider the basis elements 
$(\partial/\partial z)^J\otimes(dz_1\wedge\cdots\wedge dz_n)^\ell$
with $|J|=p+n\ell$. Same notation with the elements of $\Gamma^aT^*_X$
in terms of the basis $(dz)^I=dz_{i_1}\otimes\cdots\otimes dz_{i_p}$, resp.\
$(dz)^J\otimes(dz_1\wedge\cdots\wedge dz_n)^{-\ell}$.

\claim 17.2.~Lemma|Let $L$ be a holomorphic line bundle over $X$
equipped with a smooth hermitian metric $h$, and let $\omega$ be a
K\"ahler metric over~$X$. We denote by $\#$ the conjugate linear
$C^\infty$-isomorphism $T_X\to T^*_X$, $v\mapsto i\ol v\ort\omega$, 
defined by contracting $(0,1)$-vectors with the K\"ahler
metric~$\omega$.  Denote also by $\#:\Gamma^aT_X\otimes
L\to\Gamma^aT^*_X\otimes L^*$ the induced $C^\infty$
isomorphism on the Schur tensor powers of $T_X$ and $T^*_X$, combined
with the conjugate linear $($metric$)$ isomorphism $L\to L^*$. 
Then for an arbitrary smooth section $v$ of $\Gamma^aT_X\otimes L$ we have
$$
\int_X \Vert\dbar(\#\,v)\Vert^2dV_\omega
=\int_X\Vert\dbar v\Vert^2dV_\omega+\int_X\langle\cR_a(v),v\rangle
+\gamma|v|^2\,dV_\omega
$$
where $dV_\omega$ is the K\"ahler element of volume, $\gamma$ the 
trace $({}={}$sum of eigenvalues$)$ of $\Theta_h(L)$ with respect to
$\omega$, and $\cR_a$ is the hermitian operator
$$
\eqalign{
&v=\sum_{|I|=p}v_I(\partial/\partial z)^I\otimes s\longmapsto
\cR_a(v)=\sum_{|I|=p}\Big(\sum_{1\le k\le p}\rho_{i_k}\Big)
v_I(\partial/\partial z)^I\otimes s,\cr
&\hbox{$($resp.\ $\cR_av=\cR_{a+\ell(1\ld 1)}v-\ell(\sum_j\rho_j)v$ with 
$\ell=\max(-a_j)$, if $a\notin\bN^n)$}\cr}
$$
associated with the Ricci curvature form: $\rho_k$ denotes the eigenvalues
of $\Ricci(\omega)$ and $(\partial/\partial z_k)$, $s$ are supposed to be
orthonormal frames of $(T_X,\omega)$ and $(L,h)$.
\endclaim

\proof. We first make a pointwise calculation of
$\dbar^*\dbar v$ and $\dbar^*\dbar(\#\,v)$ in a normal coordinate
system for the K\"ahler metric~$\omega$ and in a normalized holomorphic 
frame $(s)$ for $(L,h)$. In suitable such coordinates we can write
$$
\eqalign{
\omega&=i\sum_{1\le m\le n}dz_m\wedge d\ol z_m-i\sum_{1\le j,k,\ell,m\le n}
c_{jk\ell m}z_j\ol z_k dz_\ell\wedge d\ol z_m+O(|z|^3),\cr
|s|^2&=1-\sum_{1\le j\le n}\gamma_{jk}z_j\ol z_k+O(|z|^3)\cr}
$$
where $(c_{jk\ell m})$ is the curvature tensor of $T_X$ with respect
to~$\omega$, and the $\gamma_{jk}$'s are the coefficients of $\Theta_h(L)$. 
The K\"ahler property shows that we have the symmetry
relations $c_{jk\ell m}=c_{\ell kjm}=c_{jm\ell k}$, and the Ricci
tensor $R=\sum R_{\ell m}dz_\ell\wedge d\ol z_m$ is obtained as the
trace: $R_{\ell m}=\sum_j c_{jj\ell m}$. Since~$\omega$
is tangent of order $2$ to a flat metric at the center $x_0$ of the chart,
we easily see that the first order operator $\dbar^*$ has the same
formal expression at $x_0$ as in the case of the flat metric on $\bC^n\,$: 
if~$w$ if a smooth $(0,q)$-form with values in a holomorphic vector bundle
$E$ trivialized locally by a holomorphic frame $(e_\lambda)$ such that
$(e_\lambda(x_0))$ is orthonormal and $De_\lambda(x_0)=0$, we have at
$x_0$ the formula
$$
w=\sum_{|J|=q,\,1\le\lambda\le r}w_{J,\lambda}\,d\ol z_J\otimes e_\lambda,
\qquad
\dbar^* w=-\sum_{|J|=q,\,\lambda,\,k}{\partial w_{J,\lambda}\over
\partial z_k}\,\Big({\partial\over\partial\ol z_k}\ort d\ol z_J\Big)
\otimes e_\lambda.
$$
We apply this to smooth sections $v=\sum v_I\,(\partial/\partial z)^I\otimes 
s$ of $\Gamma^aT_X\otimes L$ and $w=\sum w_I\,(dz)^I\otimes s^*$ of
$\Gamma^aT^*_X\otimes L^*$ where $s^*$ denotes the holomorphic 
section of $L^*$ such that $s^*(s)=1$. We get
$$
\dbar^*\dbar v=-\sum_{I,k}{\partial^2v_I\over\partial z_k\partial\ol z_k}
\,(\partial/\partial z)^I\otimes s,\qquad
\dbar^*\dbar w=-\sum_{I,k}{\partial^2w_I\over\partial z_k\partial\ol z_k}
\,(dz)^I\otimes s^*
$$
at~$x_0$. Now, we find
$$
\eqalign{
&\#\,{\partial\over\partial z_m}=i{\partial\over\partial\ol z_m}\ort\omega
 =dz_m-\sum_{j,k,\ell}c_{jk\ell m}z_j\ol z_kdz_\ell+O(|z|^3),\cr
&\#\,s
 =\Big(1-\sum_{1\le j,k\le n}\gamma_{jk}z_j\ol z_k+O(|z|^3)\Big)s^*\cr
&\#\,v
 =\sum_I\ol v_I(dz)^I\otimes s^*-\sum_{I,j,k,\ell,m}\ol v_Ic_{jk\ell m}
 z_j\ol z_k\Big(dz_\ell\otimes{\partial\over\partial z_m}\Big)\ort 
 (dz)^I\otimes s^*,\cr
&\phantom{\#\,v=\sum_I\ol v_I(dz)^I\otimes s^*}
 -\sum_{I,j,k}\ol v_I\,\gamma_{jk}z_j\ol z_k(dz)^I\otimes s^*
 +O(|z|^3)\cr}
$$
where (by definition)
$$
\Big(dz_\ell\otimes{\partial\over\partial z_m}\Big)\ort (dz)^I:=
\sum_{1\le k\le p,\,i_k=m}dz_{i_1}\otimes\cdots\otimes dz_{i_{k-1}}\otimes
dz_\ell\otimes dz_{i_{k+1}}\otimes\cdots\otimes dz_{i_p}.
$$
Computing $\dbar^*\dbar(\#\,v)$ at $x_0$ we obtain
$$
\eqalign{\dbar^*\dbar(\#\,v)
&=-\sum_{I,k}{\partial^2\ol v_I\over\partial z_k\partial\ol z_k}(dz)^I
  \otimes s^*+\sum_{I,k,\ell,m}\ol v_Ic_{kk\ell m}\Big(dz_\ell\otimes
  {\partial\over\partial z_m}\Big)\ort (dz)^I\otimes s^*\cr
&\phantom{{}=-\sum_{I,k}{\partial^2\ol v_I\over\partial z_k\partial\ol z_k}
  (dz)^I\otimes s^*}+\sum_j\gamma_{jj}\sum_I\ol v_I
  (dz)^I\otimes s^*\cr
&=\#\,(\dbar^*\dbar v)+\sum_{I,\ell,m}\ol v_IR_{\ell m}
  \Big(dz_\ell\otimes{\partial\over\partial z_m}\Big)\ort 
  (dz)^I\otimes s^*+\gamma(\#\,v)\cr
&=\#\,(\dbar^*\dbar v)+\#\,\cR_a(v)+\gamma(\#\,v)\cr}
$$
where $\gamma=\sum_j\gamma_{jj}$. Lemma 17.2 then follows from this 
identity by writing
$$
\int_X\Vert\dbar(\#\,v)\Vert^2dV_\omega=
\int_X\langle\dbar^*\dbar(\#\,v),\#\,v\rangle\,dV_\omega=
\int_X\langle\dbar^*\dbar v+\cR_a(v)+\gamma v,v\rangle\,dV_\omega.
\eqno\square
$$

\proof\ {\em of Theorem~17.1 (end)}. Our goal is to apply the Bochner formula 
of Lemma~17.2 to show that every section $v$ of $H^0(X,\Gamma^aT_X\otimes
L^*)$ must vanish. We first make a reduction to the case when $L$ is
ample. In fact, by raising $v$ to some tensor power, we get a section
$v^m\in H^0(X,\Gamma^{ma}T_X\otimes L^{\otimes -m})$. If $L$ is big,
some power $L^{\otimes m}$ can be written as $\cO(A+D)$ where $A$ is an
ample divisor and $D$ an effective divisor. It is then enough to prove
the vanishing of $H^0(\Gamma^{ma}T_X\otimes\cO(-A))$. If $L$ is just 
pseudoeffective, then $|a|>0$ by hypothesis and we write
$$
\Gamma^{ma}T_X\otimes L^{\otimes -m}=\Gamma^{ma-(1\ld 1)}T_X\otimes 
(L^{\otimes m}\otimes K_X)^{-1}
$$
where $|ma-(1\ld 1)|=m|a|-n>0$ for $m>n$ and $L^{\otimes m}\otimes K_X$ is 
big. We can now proceed as before to reduce the situation to the case
of an ample~$L$.

If, in addition to this, $K_X$ is also ample, the statement is a 
straightforward consequence of the Aubin-Calabi-Yau theorem ([Aub77], 
[Yau77]). In fact, we can choose $\omega$ to be K\"ahler-Einstein, i.e., 
$\Ricci(\omega)=-\omega$.  Then, for any holomorphic section 
$v\in H^0(X,\Gamma^aT_X\otimes L^*$, Lemma~17.2 yields
$$
\eqalign{
0&=\int_X\Vert\dbar v\Vert^2dV_\omega=\int_X \Vert\dbar(\#\,v)\Vert^2dV_\omega
\int_X-\langle\cR_a(v),v\rangle+\gamma|v|^2\,dV_\omega\cr
&\ge\int_X(|a|+\gamma)|v|^2\,dV_\omega\cr}
$$
($\gamma$ becomes $-\gamma$ since we changed $L$ into $L^*$, and
all Ricci eigenvalues are equal to $-1$ in that case). As $|a|\ge 0$
and $\gamma>0$ by the ampleness of~$L$, we get the desired conclusion.

If $K_X$ is only big and nef, we take $\omega$ to be a K\"ahler form in
the positive class $c_1(K_X)+\varepsilon c_1(L)=-c_1(X)+\varepsilon c_1(L)$, 
such that
$$
\Ricci(\omega)=-\omega+\varepsilon\theta
$$
where $\theta=\Theta_h(L)>0$ (the existence of such $\omega$ is is a 
well-known consequence of the theory of Monge-Amp\`ere equations). Then
the Ricci curvature eigenvalues satisfy $\rho_j=-1+\varepsilon\gamma_j\le
-1+\varepsilon\gamma$ and we get
$$
\langle-\cR_a(v),v\rangle+\gamma|v|^2\ge(|a|+\gamma-N\varepsilon\gamma)|v|^2
$$
where $N$ is an integer depending only on the weight $a=(a_1\ld a_n)$~; for
instance, $N=|a|$ works if all $a_j$ are nonnegative, otherwise we can take
\hbox{$N=|a|+n\max({-}a_j)$}.\square


\section{References}
\bigskip

{\eightpoint\parskip=3pt plus 1pt minus 1pt

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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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