% Algebraic criteria for Kobayashi hyperbolic projective varieties
% and jet differentials
%
% Lecture Notes of a 5 hour series of courses given at Santa Cruz (CA, USA)
% at the AMS Summer Research Institute in Algebraic Geometry (July 1995)

% Jean-Pierre Demailly 
%
% Universit\'e de Grenoble I, Institut Fourier
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\topmatter
\title 
Algebraic Criteria for Kobayashi Hyperbolic Projective Varieties and
Jet Differentials
\endtitle
\rightheadtext{Hyperbolic projective varieties and jet differentials}
\author
Jean-Pierre Demailly
\endauthor
\affil
Universit\'e de Grenoble I, Institut Fourier
\endaffil
\address
UMR 5582 du C.N.R.S., BP 74, 38402 Saint-Martin d'H\`eres, France
\endaddress
\email 
demailly\@fourier.ujf-grenoble.fr
\endemail
\subjclass
32H20, 32L10, 53C55, 14J40
\endsubjclass
\keywords
Kobayashi hyperbolic variety, directed manifold,
genus of curves, jet bundle, jet differential, jet metric, Chern connection
and curvature, negativity of jet curvature, variety of general type
\endkeywords
\abstract
These notes are an expanded version of five lectures delivered at the 
Santa Cruz AMS Summer School on Algebraic Geometry. The main goal
of the notes is to study complex varieties (mostly compact or projective
algebraic ones), through a few geometric questions related to hyperbolicity in
the sense of Kobayashi. A convenient framework for this 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~$T_X$. 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~$V$ (Brody's criterion). We describe
a construction of projectivized $k$-jet bundles $P_kV$, which generalizes a
construction made by Semple in 1954 and allows to analyze hyperbolicity in
terms of negativity properties of the curvature. More precisely, $\pi_k:P_kV
\to X$ 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 
$(\pi_k)_\star\cO_{P_kV}(m)$ 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, we establish a basic Ahlfors-Schwarz lemma in the situation
when $\cO_{P_kV}(-1)$ has a (possibly singular) metric of negative curvature,
and we infer that every nonconstant entire curve $f:\bC\to V$ tangent to 
$V$ must be contained in the base locus of the  metric. This basic result is 
then used to obtain a proof of the Bloch theorem, according to which the
Zariski closure of an entire curve in a complex torus is a translate of a 
subtorus. Our hope, supported by explicit Riemann-Roch calculations
and other geometric considerations, is that the Semple bundle
construction should be an efficient tool to calculate the base locus.
Necessary or sufficient algebraic criteria for hyperbolicity are then
obtained in terms of inequalities relating genera of algebraic curves
drawn on the variety, and singularities of such curves. We finally
describe some techniques introduced by Siu in value distribution
theory, based on a use of meromorphic connections.  These techniques
have been developped later by Nadel to produce elegant examples
of hyperbolic surfaces of low degree in projective $3$-space; thanks 
to a suitable concept of ``partial projective projection'' and the 
associated Wronskian operators, substantial improvements on Nadel's 
degree estimate will be achieved here.
\endabstract
\toc
\eightpoint
\head 0. Introduction\sectref{0}\endhead
\head 1. Hyperbolicity Concepts and Directed Manifolds\sectref{1}\endhead
\head 2. Hyperbolicity and Bounds for the Genus of Curves\sectref{2}\endhead
\head 3. The Ahlfors-Schwarz Lemma for Metrics of Negative 
Curvature\sectref{3}\endhead
\head 4. Projectivization of a Directed Manifold\sectref{4}\endhead
\head 5. Jets of Curves and Semple Jet Bundles\sectref{5}\endhead
\head 6. Jet Differentials\sectref{6}\endhead
\head 7. $k$-Jet Metrics with Negative Curvature\sectref{7}\endhead
\head 8. Algebraic Criterion for the Negativity of Jet 
Curvature\sectref{8}\endhead
\head 9. Proof of the Bloch Theorem\sectref{9}\endhead
\head 10\rlap{.} Logarithmic Jet Bundles and a Conjecture of 
Lang\sectref{10}\endhead
\head 11\rlap{.} Projective Meromorphic Connections and 
Wronskians\sectref{11}\endhead
\head 12\rlap{.} Decomposition of Jets in Irreducible 
Representations\sectref{12}\endhead
\head 13\rlap{.} Riemann-Roch Calculations and Study of the Base 
Locus\sectref{13}\endhead
\head 14\rlap{.} Appendix: a Vanishing Theorem for Holomorphic Tensor
Fields\sectref{14}\endhead
\head \phantom{0} References\sectref{15}\endhead
\endtoc
\endtopmatter

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% Main text

\head 0. Introduction
\writetoaux\endhead

In these notes, we investigate some geometric questions related to the
concept of hyperbolic variety in the sense of Kobayashi [Kob70]. 
Hyperbolic algebraic varieties have attracted considerable attention, in
part because of their conjectured diophantine properties. For instance,
[Lang86] has conjectured (among other things) that any hyperbolic
complex projective variety over a number field $K$ can contain only
finitely many rational points over~$K$; this conjecture, which seems at
present far beyond reach, may be regarded as a higher dimensional
analogue of the Mordell conjecture. The reader can consult P.~Vojta
[Voj87] for aspects connected to diophantine problems.

We will be concerned here only with the geometric aspects of the theory
which, although a priori more tractable than the diophantine aspects,
are still conjectural for a major part; in fact very few satisfactory
general purpose theorems are available. We hope that some of the ideas
presented here will prove useful to achieve substantial progress. The
reader is referred to S.~Lang's survey [Lang86] and book [Lang87] for
an overview of the theory until the mid 80's, and to
J.~Noguchi-T.~Ochiai [NoOc90], P.M.~Wong [Wong93] and
M.~Zaidenberg [Zai93] for a good exposition of more
recent problems. Our goal here is not to provide an exhaustive
compilation of known results, but rather to emphasize two or three
important ideas around the concepts of jet bundles and jet metrics. 
Similar ideas have been applied successfully in a somewhat special 
situation in the recent work [SiYe96a] by
Siu and Yeung, where the authors prove the hyperbolicity of the complement
of an irreducible generic curve of high degree $d\ge 10^{13}$ 
in~$\bP^2$. Let us fix here our terminology: the word ``generic'' will 
refer to a property which holds true in the complement of a global 
algebraic or analytic subset in the parameter space, and the expression 
``very generic'' will be used when the exceptional set 
of parameters is a countable union of algebraic or analytic subsets. As we 
will see in several instances, the geometry of jets conveys many natural
interesting problems concerning the relationship between hyperbolicity 
and jet curvature negativity.

We now give a short outline of the contents. Recall that a complex variety
is {\em hyperbolic in the sense of Kobayashi} if the family of holomorphic
maps $f:\Delta\to X$ from the unit disk into $X$ is a normal family. If
$X$ is compact (e.g.\ projective algebraic), it is well known that $X$ is
Kobayashi hyperbolic if and only if it is {\em Brody hyperbolic}, that is, 
if there are no nonconstant entire holomorphic curve $f:\bC\to X$. In
particular $X$ has no rational or elliptic curves, and more generally
every holomorphic map $f:Z\to X$ from an abelian variety (or complex
torus) to $X$ must be constant. Conversely, it has been suggested by
Kobayashi [Kob70] and [Lang86] that these algebraic properties could be
equivalent to hyperbolicity. To prove this, one would have to construct
a torus $Z$ and a nontrivial holomorphic map $f:Z\to X$ whenever $X$
is non hyperbolic. A hint that this should be true is given by the
following observation: if $X$ is hyperbolic, there is an absolute
constant $\varepsilon>0$ such that the genus of any compact curve of $X$
is bounded below by $\varepsilon$ times the degree; conversely, this
property fails to be true in many examples of nonhyperbolic projective
varieties. Our belief, supported by some heuristic arguments,
is that any sequence of compact curves $(C_\ell)$ with
genus$(C_\ell)/{}$degree$(C_\ell)\to 0$ should have a cluster set swept
out by the image of a map $f:Z\to X$ from a complex torus $Z$, such that
the limit of some subsequence of the sequence of universal covering maps
$\Delta\to C_\ell\to X$ (suitably reparametrized) coincides
with the image of a (non necessarily compact) straight line of $Z$
into~$X$. A related conjecture of [Lang86]
states that a projective variety is hyperbolic if and only if all its
irreducible algebraic varieties are of general type. The most elementary
step would be to exclude the case of manifolds with $c_1\equiv 0$
(say, Calabi-Yau manifolds and symplectic manifolds), by showing for
instance that they do admit a sequence of compact curves $(C_\ell)$ with
genus$(C_\ell)/{}$degree$(C_\ell)\to 0$.

We next introduce jet bundles and jet differentials, extending some
ideas of Green and Griffiths [GrGr80] (actually, the idea of using jet
differentials can be traced back to the work of A.~Bloch [Blo26, $26'$],
H.~Cartan [Car28], L.~Ahlfors [Ahl41] and T.~Ochiai [Och77]). The basic
idea is to introduce a bundle $E_{k,m}$ of algebraic differential
operators $Q(f',f''\ld f^{(k)})$ of weighted degree $m$, acting on germs
of holomorphic curves (``{\em jet differentials of order $k$ and degree
$m$}''). When no other restrictions are made on $Q$, one obtains a ``huge''
bundle which will be denoted $E^\GG_{k,m}$ in reference to Green-Griffiths'
work. In our case, the main goal is to determine the conformal type of 
entire curves drawn on the variety. Therefore, the way curves are 
parametrized is irrelevant. For this reason, one is led to introduce
a subbundle $E_{k,m}\subset E^\GG_{k,m}$ of ``special jet differential 
operators'' $Q(f',f''\ld f^{(k)})$, namely operators which have the 
property of being invariant by reparametrization, i.e.\
$$
Q((f\circ\varphi)',(f\circ\varphi)''\ld (f\circ\varphi)^{(k)})=
(\varphi')^mQ(f',f''\ld f^{(k)})
$$
for any local reparametrization $\varphi:(\bC,0)\to(\bC,0)$ of the curve.
For instance, all Wronskian determinants appearing in the wedge products
$f'$, $f'\wedge f''$, $f'\wedge f''\wedge f'''$, $\ldots$~, 
$f'\wedge f''\wedge\cdots\wedge f^{(\ell)}$
(computed in any coordinate system), and all polynomial combinations
of these give rise to local sections of $E_{k,m}$. The subbundle $E_{k,m}$
turns out to have better positivity properties than $E^\GG_{k,m}$
(see section \S$\,$12). The 
bundles $E^\GG_{k,m}$ and $E_{k,m}$ have natural filtrations
for which the graded pieces split in irreducible tensor product
representations of $T^\star_X$ (Schur fonctor representations).
However, in the case of $E_{k,m}$, it seems to be a highly non
trivial question to decide which representations are actually involved,
except for the simpler cases of jet differentials of order $k\le 2$.

A basic observation is that any entire curve $f:\bC\to X$ must automatically
satisfy all algebraic differential equations $Q(f',f''\ld f^{(k)})=0$ 
arising from global jet differential operators $Q\in H^0(X,E_{k,m}\otimes 
\cO(-A))$ which vanish on some ample divisor~$A$. Our proof is based
on a strong pointwise version of the Ahlfors-Schwarz lemma (Lemma 3.2 and 
Theorem 7.8), and actually completes the scheme of proof
suggested in [GrGr80] in the case of invariant jet differentials;
the general pointwise case of the Ahlfors-Schwarz lemma for non necessarily
invariant jet differentials, however, seems to be still unsettled. Let us 
mention, although we will not need it here,
that the above vanishing theorem is still true with sections of 
$E^\GG_{k,m}\otimes \cO(-A)$ in place of $E_{k,m}\otimes \cO(-A)$
(see Remark~7.11 and [SiYe96c]). The vanishing theorem provides a way of 
investigating the hyperbolicity of $X$ by trying to compute the base 
locus of sections in $H^0(X,E_{k,m}\otimes\cO(-A))$ for large $k$ and $m$
(we will call this set the ``{\em Green-Griffiths locus}'' of $X$, although
[GrGr80] deals rather with sections of $E^\GG_{k,m}\otimes\cO(-A)$).
We believe that the use of $E_{k,m}$ in place of $E^\GG_{k,m}$ should
make easier to understand the structure of the base locus (especially
in the case of higher values of $k$ and $n=\dim X$), since the
dimension of the projectivized jet bundles under consideration is smaller
and $E_{k,m}$ is ``more positive'' than $E^\GG_{k,m}$.
At least in the case of surfaces of general type, the existence of sections
in $E_{k,m}$ and $E^\GG_{k,m}$ can be obtained by Riemann-Roch computations
and suitable vanishing theorems proved by Bogomolov [Bog79] (here again, the
conditions involved for the existence of sections in $E_{k,m}$ are better
than those for $E^\GG_{k,m}$). It is reasonable to hope
that suitable refinements of these ideas could lead in the future to a
complete proof of the conjecture that every surface of general type only
has finitely many rational and elliptic curves. Such a result is indeed
obtained in [Bog77] for the class of surfaces $X$ satisfying
$c_1(X)^2>c_2(X)$, thanks to a finiteness theorem for integral curves 
of foliations on surfaces (cf.\ Jouanolou [Jou78]). When 
$c_1(X)^2>2\,c_2(X)$, Lu-Yau [LuYa90] proved the 
additional result that the transcendental curves are algebraically 
degenerate; the argument is based on a result of Miyaoka
asserting that the base locus of the first order jet differentials
in $H^0(X,S^kT^\star_X\otimes\cO(-A))$ is at most $1$-dimensional
for $c_1(X)^2>2\,c_2(X)$ and~$k\gg 0$.
Recently, S.~Lu [Lu96] also obtained a proof for the limit
case $c_1(X)^2=2\,c_2(X)$. By a different method, Lu-Miyaoka [LuMi95]
investigate the case of arbitrary surfaces of general type, and get
the desired finiteness results under suitable restrictions on the curve
singularities.

In a similar manner, for varieties of general type of arbitrary
dimension, it is conjectured that all entire curves $f:\bC\to X$
are contained in a proper algebraic subset. One of the most celebrated
result in this direction is the proof of the ``Bloch theorem'', i.e.\ the
special case of the above conjecture when $X$ has irregularity
$q=h^0(X,\Omega^1_X)>\dim X\,$; the subject was revived by Ochiai
[0ch77], who considerably clarified the ideas introduced in [Blo26], 
and formulated a technical result that would yield what he termed
the ``Bloch conjecture''. The Bloch theorem was finally settled in the
affirmative, by means of various techniques, in fundamental papers by
Noguchi [Nog77,~81,~84], Kawamata [Kaw80] and Green-Griffiths [GrGr80]
(to avoid the slight technical difficulty with the proof of the 
Ahlfors-Schwarz lemma in [GrGr80], we will prove here Bloch's theorem
by replacing the Green-Griffiths jet differentials $E^\GG_{k,m}$ 
with the invariant jet differentials $E_{k,m}$). A very interesting
related result is the proof by Siu-Yeung [SiYe96b] of the hyperbolicity of
complements of ample divisors in abelian varieties (see also
Noguchi [Nog96a] for an extension to the semi-abelian case). In \S$\,$10,
we give a brief overview of these results, following an alternative
approach suggested by Dethloff and Lu [DLu96], which makes use of
a ``logarithmic version'' of our jet bundles. An earlier interesting 
result is the construction by Nadel [Nad89] of explicit hyperbolic 
algebraic surfaces in~$\bP^3$. Nadel's method is based
on a use of meromorphic connections with low pole order, according to
ideas introduced by Siu [Siu87]. In our setting, Nadel's technique is
just the very special case when the jet differential operators under
consideration are Wronskian operators associated with a
meromorphic connection. Unfortunately, the method seems to work only 
for a restricted class of hypersurfaces or complete intersections
defined by polynomials with few monomials. Thanks to a new flexible
concept of {\it partial projective connection}, we have been able to
improve Nadel's bounds and show the existence of hyperbolic algebraic
surfaces in $\bP^3$ of any degree${}\ge 11$ (the conjectured optimal
bound should be~$5$); see \S11 and [EG96,~97], [DeEG97]%
\bottomnote{*}{{After these notes were completed, similar results have
been announced independently in [SiYe96c].}}.

Contrary to most prior methods, including J.~Noguchi's ``jet
projection method'' (see [NoOc90], [Nog96]), our method does not use
any Nevanlinna theory at all. Our approach is to construct suitable
jet metrics of negative curvature, following original ideas of
Grauert-Reckziegel [GRec65], [Gra89], Kobayashi [Kob75],
Cowen-Griffiths [CoGr76] and Green-Griffiths [GrGr80]. In fact, the
sheaves $\cO(E_{k,m})$ of jet differentials defined above are the
direct image sheaves of some canonical invertibles sheaves
$\cO_{X_k}(m)$ defined over suitable ``projectivized $k$-jet bundles''
$X_k\to X$. The $k$-jet bundle $X_k$ is a tower of projective bundles,
and can be obtained by iterating a natural fonctorial construction
$(X,V)\rsa (X_1,V_1)$ in the category of ``{\em directed manifolds}''.
By definition, objects of this category are pairs $(X,V)$, where $X$ is
a complex manifold and $V$ a holomorphic subbundle of $T_X$, and the 
arrows are holomorphic morphisms preserving the $V$ subbundles. We show 
in \S$\,$6 that the bundle $X_k$
is a canonical smooth compactification of the bundle of ``geometric''
$k$-jets of regular curves (by ``geometric jets'', we mean that one
does not pay attention to the way curves are parametrized). Such
bundles $X_k$ appear to be a natural generalization of a construction
introduced by Semple [Sem54] in 1954, which has been used recently as
a tool for establishing enumerative formulas dealing with the order of
contact of plane curves (see [Coll88], [ASS92], [CoKe94]).

In fact, almost all concepts pertaining to hyperbolicity can be
extended in the general framework of directed manifolds $(X,V)$, which
we may think of as a ``relative'' situation (in fact, it is not
necessary to assume that $V$ is an integrable subbundle of $T_X$, but
the case when $V=T_{X/S}$ is the relative tangent bundle of a smooth
map $X\to S$ is of special interest). For instance, $(X,V)$ is said to be
Brody hyperbolic if there are no global holomorphic curve $f:\bC\to X$
tangent to $V$. In the inductive definition of the $k$-jet spaces
$(X_k,V_k)$, $X_k$ is simply the projectivization $P(V_{k-1})$ of
$V_{k-1}\to X_{k-1}$, and $\cO_{X_{k-1}}(-1)$ is the tautological line
subbundle of the inverse image of $V_{k-1}$ over $X_k$. Now, we say
that $X$ has $k$-jet negative curvature if $\cO_{X_k}(-1)$ can be
equipped with a hermitian metric (for which some type of singularities
have to be allowed), such that the $(1,1)$ curvature form of the
metric is negative along~$V_k$. The negativity property of the $k$-jet
curvature is closely related to the existence of sections of large
degree in $H^0(X,E_{k,m})$, exactly in the same way positivity and
ampleness are related. A variant of the Ahlfors-Schwarz lemma shows
that the negativity of $k$-jet curvature implies hyperbolicity.
Conversely, Kobayashi [Kob70] and [Lang86] raised the question whether
hyperbolicity is equivalent to $1$-jet negativity (in our
terminology). We show that this optimistic picture is unfortunately
wrong. In fact the $k$-jet negativity property yields the following
necessary algebraic condition: there exists a constant $\varepsilon>0$
such that every algebraic curve $C\subset X$ satisfies
$$
2g(\ol C)-2\ge\varepsilon\deg(C)+\sum_{x\in C_{k-1}}(m_{C_{k-1}}(x)-1),
$$
where $\ol C$ is the normalization of $C$ and $m_{C_{k-1}}(x)$ are the
multiplicities of the singular points in the $(k-1)$-st jet lifting
of $C$. Using this criterion, we construct for every integer $k_0$
an hyperbolic algebraic surface which cannot have any $k$-jet metric
of negative curvature when $k\le k_0$. It is nevertheless reasonable
to expect that hyperbolicity is equivalent to the existence of a
sufficiently large integer $k_1$ such that $X$ has $k$-jet negative
curvature for $k\ge k_1$.

We want to stress that many important questions have been left out in
these notes, especially Nevanlinna theory and its applications to
hyperbolicity theory [CaGr72], [Nog83], [Siu87], [Wong89], [RuSt91],
[Nog91]. Especially noticeable in this respect is the work of
Dethloff-Wong-Schumacher [DSW92, 94] on the hyperbolicity of
complements of 3 or more generic curves in the projective plane, 
and the construction by Masuda-Noguchi [MaNo93] of hyperbolic hypersurfaces
of large degree in $\bP^n$. Also, in a more algebraic setting, there is
an extensive literature dealing with the question of computing genus of
curves in algebraic surfaces, bearing an intimate connection with
hyperbolicity ([Bog77], [Cle86], [CKM88], [LuYa90], [Lu91], [LuMi95], [Lu96]
[Xu94]). Last but not least, there are several important questions
of Number Theory which either depend on Nevanlinna theory or suggest new
tools for the study of differential geometric problems. The reader may
profitably consult McQuillan's paper [McQu96], in which the method of
Vojta-Faltings is adapted to give a completely new proof of the Bloch
theorem.

I wish to express my gratitude to the organizers of the AMS Summer
Institute held at Santa Cruz in July 1995 for giving me the opportunity
of making a series of lectures on hyperbolicity theory. I warmly
thank Gerd Dethloff, Siegmund Kosarew, Steven Lu, Bernard Shiffman,
Yum-Tong Siu and Mikhail Zaidenberg for formal or informal discussions
which got me started in the subject and helped me to improve these notes.

\head 1. Hyperbolicity Concepts and Directed Manifolds
\writetoaux\endhead

We first recall a few basic facts concerning the concept of
hyperbolicity, according to S.~Kobayashi [Kob70, Kob76]. Let $X$ be
a complex $n$-dimensional manifold. We denote by $f:\Delta\to X$ an
arbitrary holomorphic map from the unit disk $\Delta\subset\bC$ to~$X$.
The {\em Kobayashi-Royden infinitesimal pseudometric} on $X$ is the
Finsler pseudometric on the tangent bundle $T_X$ defined by
$$
\bfk_X(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\Delta\to X,\,f(0)=x,\,
\lambda f'(0)=\xi\big\},\qquad x\in X,~\xi\in T_{X,x}
$$
(see H.~Royden [Roy71], [Roy74]). 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,
$$
N(\lambda\xi)=|\lambda|\,N(\xi)\qquad
\hbox{for all $\lambda\in\bC$ and $\xi\in E$.}
$$
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 pseudodistance}
$d_K(x,y)$ is the geodesic pseudodistance obtained by integrating the
Kobayashi-Royden infinitesimal metric. The manifold $X$ is said to be
{\em hyperbolic} (in the sense of Kobayashi) if $d_K$ is actually a
distance, namely if $d_K(x,y)>0$ for all pairs of distinct points
$(x,y)$ in~$X$. In this context, we have the following well-known
results of Brody [Bro78].

\proclaim{1.1.~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
$$
\|(f\circ\psi)'(0)\|_\omega=1,\qquad \|(f\circ\psi)'(t)\|_\omega
\le{1\over 1-|t|^2/R^2}\quad\hbox{for every $t\in D(0,R)$.}
$$
In particular, if $X$ is compact, given any sequence of holomorphic
mappings\break $f_\nu:\Delta\to X$ 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 $\sup_{t\in\bC}\|g'(t)\|_\omega\le 1$.
\endproclaim

\demo{Proof}
The first assertion of Brody's lemma is obtained by
selecting $t_0\in\Delta$ 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
\enddemo

\proclaim{1.2.~Corollary {\rm(Brody's theorem)}}A compact complex manifold
$X$ is hyperbolic if and only if there are no non constant entire holomorphic
maps $g:\bC\to X$. 
\endproclaim

\demo{Proof} The arguments are rather standard and will be developped in more 
detail in the proof of Prop.~1.5 below.\square
\enddemo

Now, more generally, let $(X,V)$ be a complex manifold equipped with a
{\em holomorphic subbundle $V\subset T_X$}. We will refer to such a pair
as being a {\em complex directed manifold}. A morphism
$\Phi:(X,V)\to (Y,W)$ in the category of complex directed manifolds is
a holomorphic map such that $\Phi_\star(V)\subset W$.
Our philosophy is that directed manifolds are also useful to study the
``absolute case'', i.e.\ the case $V=T_X$, because there are fonctorial
constructions which work better in the category of directed manifolds
(see e.g.\ \S$\,$4,~5,~6). We think of directed manifolds as a kind
of ``relative situation'', covering e.g.\ the case when $V$ is the
relative tangent sheaf to a smooth map $X\to S$. We want
to stress here that no assumption need be made on the Lie bracket
tensor $[~,~]:V\times V\to T_X/V$, and the rank $r=\rank V$ may be an
arbitrary integer in the range $1\le r\le n:=\dim_\bC X$. For the sake
of generality, one might also wish to allow singularities in the
subbundle~$V$: for this, one can take $V$ to be given by
an arbitrary coherent subsheaf $\cV\subset\cO(T_X)$ such that
$\cO(T_X)/\cV$ has no torsion; then $V$ is a subbundle
outside an analytic subset of codimension at least~$2$ (it is
however somewhat safer to view $\cV^\star$ as given by a quotient sheaf
morphism $\Omega^1_X\to\cV^\star$ and let $V^\star$ be the associated
linear space, see Remark 3.10 below). For the sake of simplicity, we
will assume most of the time that $V$ is actually a subbundle of~$T_X$.
In this situation, we generalize the notion of hyperbolicity as follows.

\proclaim{1.3.~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
$$
\bfe^p_{(X,V)}(\xi)=\inf\big\{\lambda>0\,;\,\exists f:\bB_p\to X,\,f(0)=x,\,
\lambda f_\star(\tau_0)=\xi,\,f_\star(T_{\bB_p})\subset V\big\}
$$
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
\endproclaim

If $\Phi:(X,V)\to (Y,W)$ is a morphism of directed manifolds, it is
immediate to check that we have the monotonicity property
$$
\leqalignno{
&\bfk_{(Y,W)}(\Phi_\star\xi)\le \bfk_{(X,V)}(\xi),
\qquad\forall\xi\in V,&(1.4)\cr
&\bfe^p_{(Y,W)}(\Phi_\star\xi)\le \bfe^p_{(X,V)}(\xi),
\qquad\forall\xi=\xi_1\wedge\cdots\wedge\xi_p\in\Lambda^pV.&(1.4^p)\cr}
$$
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}.

\proclaim{1.5.~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~$V$.
\endproclaim

\demo{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 $(X',V')=
(X\times\Delta,\pr_1^\star V\oplus\pr_2^\star T_\Delta)$, $f$ 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~$\Omega$. 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 $V$, Brody's
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
\enddemo

We conclude this section by showing that hyperbolicity is an open
property.

\proclaim{1.6. 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 a holomorphic subbundle $\cV\subset T_{\cX/S}$
of the relative tangent bundle, defining a deformation $(X_t,V_t)_{t\in S}$
of the fibers$)$. Then the set of $t\in S$ such that the fiber $(X_t,V_t)$
is hyperbolic is open in $S$ with respect to the euclidean topology.
\endproclaim

\demo{Proof} Take a sequence of non hyperbolic fibers $(X_{t_\nu},V_{t_\nu})$
with $t_\nu\to t$ and fix a hermitian metric $\omega$ on $\cX$. By
Brody's lemma, there is a sequence of entire holomorphic maps
$g_\nu:\bC\to X_{t_\nu}$ tangent to $V_{t_\nu}$, 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 $g:\bC\to X_t$, tangent to $V_t$, with $\|g'(0)\|_\omega=1$.
Hence $(X_t,V_t)$ is not hyperbolic, and the collection of non
hyperbolic fibers is closed in~$S$.\square
\enddemo

\head 2. Hyperbolicity and Bounds for the Genus of Curves
\writetoaux\endhead

In the case of projective algebraic varieties, hyperbolicity is
expected to be related to other properties of a more algebraic
nature. Theorem~2.1 below is a first step in this direction.

\proclaim{2.1.~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
$\Phi_\star(T_Z)\subset V$.
\smallskip\noindent
Then {\rm i)$\,\Rightarrow\,$ii)$\,\Rightarrow\,$iii)}.
\endproclaim

\demo{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
\hbox{$\rho^\star \bfk_{\ol C}=\bfk_\Delta$}. 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
$$
\int_{\ol C}d\sigma_{\ol C}=-{1\over 4}\int_{\ol C}\text{\rm curv}
(\bfk_{\ol C})=-{\pi\over 2}\chi(\ol C)
$$
On the other hand, if $j:C\to X$ is the inclusion, the monotonicity
property (1.4) applied to the holomorphic map $j\circ\nu:\ol C\to X$
shows that
$$
\bfk_{\ol C}(t)\ge \bfk_{(X,V)}\big((j\circ\nu)_\star t\big)
\ge \varepsilon_0\big\|(j\circ\nu)_\star t\big\|_\omega,
\qquad\forall t\in T_{\ol C}.
$$
From this, we infer $d\sigma_{\ol C}\ge\varepsilon_0^2(j\circ\nu)^\star
\omega$, thus
$$
-{\pi\over 2}\chi(\ol C)=\int_{\ol C}d\sigma_{\ol C}\ge
\varepsilon_0^2\int_{\ol C}(j\circ\nu)^\star\omega=
\varepsilon_0^2\int_C\omega.
$$
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 $\Phi_\star(T_Z)\subset V$. We must have
$\dim\Phi(Z)\ge 2$, otherwise $\Phi(Z)$ would be a
curve covered by images of holomorphic maps \hbox{$\bC\to\Phi(Z)$},
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
$$
\int_{C_m}\omega=\int_\Gamma (\Phi\circ u_m)^\star\omega=
\int_Z[\Gamma]\wedge u_m^\star(\Phi^\star\omega),
$$
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
$$
\int_{s\in Z}\left(\int_{C_{m,s}}\omega\right)d\mu(s)=\int_Z\left(\int_{s
\in Z}[\Gamma+s]\,d\mu(s)\right)\wedge u_m^\star(\Phi^\star\omega).
$$
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
$(u_m)_\star\gamma=m^2\gamma$. Therefore we get
$$
\int_{s\in Z}d\mu(s)\int_{C_{m,s}}\omega=m^2\int_Z \gamma\wedge
\Phi^\star\omega.
$$
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
$$
\deg_\omega(C_{m,s_m})=\int_{C_{m,s_m}}\omega\ge m^2
\int_Z \gamma\wedge\Phi^\star\omega.
$$
As $\int_Z \gamma\wedge\Phi^\star\omega>0$, the curves $C_{m,s_m}$ have
bounded genus and their degree is growing quadratically with $m$,
contradiction to property ii).\square
\enddemo

\proclaim{2.2.~Definition}We say that a projective directed manifold
$(X,V)$ is ``algebraically hyperbolic'' if it satisfies property
$2.1$~{\rm ii)}, namely, if there exists $\varepsilon>0$ such that
every algebraic curve $C\subset X$ tangent to $V$ satisfies
$$
2g(\ol C)-2\ge\varepsilon\,\deg_\omega(C).
$$
\endproclaim

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

\proclaim{2.3.~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$)$.
\endproclaim

\demo{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
\enddemo

\proclaim{2.4.~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~2.2.}
\endproclaim

\proclaim{2.5.~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 Zariski 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] recently 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. M.~Green [Green77] proved the hyperbolicity
of the complement of $(2n+1)$ generic hyperplanes in $\bP^n$. Zaidenberg
[Zai89] showed the existence of curves with hyperbolic complement for
every degree $d\ge 5$. In response to a conjecture of [Zai89], [DSW92,
94] showed that the complement of the union of at least 3 generic
curves is hyperbolic, when the sum of degrees is at least 5. More
recently, Siu and Yeung [SiYe96a] proved the harder fact that the
complement of a generic irreducible curve of high degree in $\bP^2$ 
is hyperbolic. Their approach uses jet bundle techniques, and it is one of
our goals to explain some of the underlying geometric ideas.\square}
\endproclaim 

In the ``absolute case''~$V=T_X$, it seems reasonable to expect that all
three properties 2.1~i), ii), iii) are equivalent, in particular that
Kobayashi and algebraic hyperbolicity coincide. In fact, S.~Lang
made the following conjecture.

\proclaim{2.6.~Conjecture {\rm([Lang 86, 87])}}A compact complex manifold
$X$ is hyperbolic if and only if there are no nontrivial holomorphic
maps $Z\to X$ where $Z=\bC^p/\Lambda$ is a compact complex torus.
\endproclaim

The ``only if'' part of the conjecture is of course clearly true. On
the other hand, if $X$ is projective algebraic, every holomorphic map
$Z\to X$ of a compact complex torus $Z$ to $X$ admits a factorization
$Z\to Z'\to X$ where $Z'$ is an abelian variety (see e.g.\ A.~Weil
[We57]). Thus, for $X$ projective algebraic and $V=T_X$, a positive
solution to Conjecture 2.5 would imply the equivalence of properties
i), ii), iii) in Theorem~2.1.

\proclaim{2.7.~Hint of heuristic proof of Lang's conjecture}{\rm
Although Lang's conjecture seems at present far beyond reach, we
would like to present here a heuristic argument showing how things could
possibly work. These ideas arose through discussions with S.~Kosarew.
Let $\cB_X$ be the space of all Brody curves, that
is, the set of all entire holomorphic curves $f:\bC\to X$ with
$\|f'\|_\omega\le 1$ for some given hermitian metric $\omega$ on $X$.
For each $\varepsilon\ge0$, we define a distance $\delta_\varepsilon$ on
$\cB_X$ by putting
$$
\delta_\varepsilon(f,g)=\sup_{t\in\bC}d_\omega(f(t),g(t))
e^{-\varepsilon|t|}
$$
where $d_\omega$ is the geodesic distance on $X$. For all $\varepsilon>0$,
$d_\varepsilon$ defines the topology of uniform convergence of compact sets,
whilst $d_0$ is the topology of uniform convergence up to infinity.
Hence $(\cB_X,d_\varepsilon)$ is a compact metric space for $\varepsilon>0$.
Assume that this is still true for $\varepsilon=0$ and assume moreover
that $X$ is not hyperbolic, i.e.\ $\cB_X\ne\emptyset$. We then consider
the compact topological group $G_\cB$ of isometries of $(\cB_X,d_0)$ and
look at the group homomorphism
$$
\Psi:(\bC,+)\to(G_\cB,\circ),\qquad a\mapsto f_a\quad
\hbox{where $f_a(t)=f(t-a)$}.
$$
We claim that $G_\cB$ should be a finite dimensional Lie group
and $Z=\ol{\Psi(\bC)}\subset G_\cB$ a compact commutative complex subgroup,
thus a complex torus. In fact, a compact Banach Lie group is finite
dimensional, and the ``Lie algebra'' of $G_\cB$ seems to be
interpretable as a closed subspace of the Banach
space of bounded holomorphic sections in $H^0(\cB_X\times\bC,\text{\rm 
ev}^\star T_X)$ where ev${}:\cB_X\times\bC\to X$ is the evaluation map
$(f,t)\mapsto f(t)$. The complex structure on $Z$ should arise from the
complex structure on that Banach space. Now, we obtain a non trivial
holomorphic map $\Phi:Z\to X$ by selecting an $f\in\cB_X$ which is not
a fixed point of $Z$ and putting $\Phi(\gamma)=\gamma(f)(0)$,
$\gamma\in Z$.\square}
\endproclaim

In the general context of directed manifolds, algebraic hyperbolicity
can be strictly weaker than Kobayashi hyperbolicity. The simplest example
is provided by a $2$-dimensional abelian variety $X=\bC^2/\Lambda$
equipped with a constant subbundle $V\subset T_X$ given by a complex
line $V_0\subset\bC^2$ such that $V_0\cap\Lambda=\{0\}$. Then all
leaves of $V$ are isomorphic to complex lines, in particular there are
no compact curves tangent to~$V$, and thus 2.1~ii) and 2.1~iii) are
satisfied (as void conditions). This pathology can somehow be corrected
by observing that the hyperbolicity of $(X,V)$ implies a statement
analogue to 2.1~ii) but substantially stronger, namely a lower
bound of $-\chi(\ol C)$ for curves $C$ which are almost tangent to~$V$,
in the sense that their ``deviation with respect to $V$'' is small.

\proclaim{2.8.~Definition}Let $X$ be equipped with a hermitian $(1,1)$-form
$\omega$, and let $C\subset X$ be a compact curve in~$X$.
We define the $L^2$-deviation of $C$ with respect to $V$ by
$$
\dev^2_\omega(C/V)=\int_C\omega_{V^\perp}
$$
where $\omega=\omega_V\oplus\omega_{V^\perp}$ is the orthogonal
decomposition of $\omega$ on $V\oplus V^\perp$. Similarly, if
$\nu:\ol C\to X$ is the normalization map and $\ol C\not\simeq\bP^1$,
we define the $L^\infty$-deviation $($resp.\ the $L^p$-deviation$)$ to be
$$
\eqalign{
\dev^\infty_\omega(C/V)&=\sup_{t\in\ol C}{\nu^\star\omega_{V^\perp}(t)
\over d\wt\sigma(t)}=\sup_{t\in\ol C}\Vert \nu'(t)^\perp
\Vert_{\wt\sigma,\omega_{V^\perp}}^2,\cr
\dev^p_\omega(C/V)&=\Big[\int_{t\in\ol C}\Big({\nu^\star\omega_{V^\perp}(t)
\over d\wt\sigma(t)}\Big)^{p/2}d\wt\sigma(t)\Big]^{2/p}
=\Big[\int_{t\in\ol C}\Vert \nu'(t)^\perp\Vert^p
_{\wt\sigma,\omega_{V^\perp}}d\wt\sigma(t)\Big]^{2/p},\cr}
$$
where $d\wt\sigma$ is the normalized Poincar\'e metric on $\ol C$
$($hermitian metric of constant curvature with $\int_{\ol C}d\wt\sigma=1)$,
and $\nu'(t)^\perp$ is the projection of the tangent vector $\nu'(t)$
on~$V^\perp$. If $\ol C\simeq\bP^1$, we set instead
$$
\eqalign{
\dev^\infty_\omega(C/V)&=\inf_{\gamma\in\PGL_2(\bC)}~\,\sup_{t\in\ol C}~\,
{\nu^\star\omega_{V^\perp}(t)\over\gamma^\star d\wt\sigma(t)}\cr
\dev^p_\omega(C/V)&=\inf_{\gamma\in\PGL_2(\bC)}~\,
\Big[\int_{t\in\ol C}\Vert \nu'(t)^\perp\Vert^p_{\gamma^\star
\wt\sigma,\omega_{V^\perp}}d\gamma^\star\wt\sigma(t)\Big]^{2/p}.\cr}
$$
\endproclaim

\proclaim{2.9.~Proposition}Let $(X,V)$ be a compact directed manifold equipped
with a hermitian metric~$\omega$. If $(X,V)$ is hyperbolic, there exists
a constant $\varepsilon>0$ such that
$$
\max\big(-\chi(\ol C),\dev^\infty_\omega(C/V)\big)\ge
\varepsilon\deg_\omega(C),
$$
for every compact curve $C\subset X$.
\endproclaim

\demo{Proof} Otherwise, there would exist a sequence of curves $(C_\ell)$ and
a sequence of positive numbers $\varepsilon_\ell$ converging to~$0$,
such that
$$
-\chi(\ol C_\ell)\le\varepsilon_\ell\deg_\omega(C_\ell),\qquad
\dev^\infty_\omega(C_\ell/V)\big)\le\varepsilon_\ell\deg_\omega(C_\ell).
$$
First assume that all curves $C_\ell$ have geometric genus
$g(\ol C_\ell)\ge 2$. Let $\nu_\ell:\ol C_\ell\to X$ be the
normalization map of $C_\ell$, and let $d\sigma_\ell$ be the
area measure associated with the Poincar\'e metric on $\ol C_\ell$ and
$d\swt\sigma|_\ell=\lambda_\ell^{-1}d\sigma_\ell$ the
normalized Poincar\'e metric with $\lambda_\ell=
\int_{\ol C_\ell}d\sigma_\ell={\pi\over 2}(-\chi(C_\ell))$.
Select a point $x_\ell\in\ol C_\ell$ where the ratio
$\nu_\ell^\star\omega/d\sigma_\ell$ is maximum. Since
$\int_{\ol C_\ell}\nu_\ell^\star\omega=\deg_\omega(C_\ell)$, we have
$$
R_\ell^2:={\nu_\ell^\star\omega(x_\ell)\over d\sigma_\ell(x_\ell)}\ge
{\int_{\ol C_\ell}\nu_\ell^\star\omega\over
\int_{\ol C_\ell}d\sigma_\ell}={\deg_\omega(C_\ell)\over\lambda_\ell}
\ge{2\over\pi\varepsilon_\ell}\to+\infty.
$$
Let $\rho_\ell:\Delta\to\ol C_\ell$ be the universal covering map
of $\ol C_\ell$, chosen such that $\rho_\ell(0)=x_\ell$. 
We get a holomorphic map $f_\ell=\nu_\ell\circ\rho_\ell:\Delta\to X$
such that $\|f_\ell'(0)\|_\omega=R_\ell\to+\infty$.
By Brody's reparametrization lemma, we can reparametrize $f_\ell$ as
$g_\ell(t)=f_\ell(t/R_\ell)$ so that some
subsequence of $(g_\ell)$ converges uniformly on every compact set to a
limit $g:\bC\to X$ with $\|g'(0)\|_\omega=1$. We claim that $g$ must be
tangent to~$V$. In fact, by definition of the $L^\infty$ deviation, we get
$\nu_\ell^\star\omega_{V^\perp}\le\varepsilon_\ell\deg_\omega(C_\ell)
d\swt\sigma|_\ell$, thus $f_\ell^\star\omega_{V^\perp}=
\rho_\ell^\star(\nu_\ell^\star\omega_{V^\perp})$ and
$g_\ell^\star\omega_{V^\perp}$ satisfy
$$
\eqalign{
f_\ell^\star\omega_{V^\perp}&\le{1\over\lambda_\ell}
\varepsilon_\ell\deg_\omega(C_\ell){|dt|^2\over(1-|t|^2)^2},\cr
g_\ell^\star\omega_{V^\perp}&\le{\deg_\omega(C_\ell)\over\lambda_\ell}
\varepsilon_\ell{R_\ell^{-2}|dt|^2\over(1-|t|^2/R_\ell^2)^2}
\le\varepsilon_\ell{|dt|^2\over(1-|t|^2/R_\ell^2)^2}.\cr}
$$
From this we conclude that $g^\star\omega_{V^\perp}=0$, hence $g$ is
tangent to~$V$, contradiction. If the curves $C_\ell$ are of genus
$0$ or $1$, the arguments are similar and will be left to the reader.\square
\enddemo

\head 3. The Ahlfors-Schwarz Lemma for Metrics of Negative Curvature
\writetoaux\endhead

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 $T^\star_X$) 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. 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 $\Theta_h(V)={i\over 2\pi}
\nabla_h^2$ its Chern curvature tensor.

\proclaim{3.1.~Proposition}Let $(X,V)$ be a compact directed
manifold. Assume that $V^\star$ is ample. Then $(X,V)$ is hyperbolic.
\endproclaim

\demo{Proof {\rm(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 $E^\star$ by putting
$$
N(\xi)=\Big(\sum_{1\le j\le N}|\sigma_j(x)\cdot\xi^m|^2\Big)^{1/2m},\qquad
\xi\in E^\star_x,
$$
and $N$ is then a strictly plurisubharmonic function on the total space
of $E^\star$ minus the zero section (in other words, the line
bundle $\cO_{P(E^\star)}(1)$ has a metric of positive curvature). By
the ampleness assumption on $V^\star$, we thus have a Finsler
metric $N$ on $V$ which is strictly plurisubharmonic outside the zero
section. By Brody's 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
\enddemo

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

\proclaim{3.2.~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
$$
\det(\gamma)\le\Big({p+1\over AR^2}\Big)^p{1\over(1-|t|^2/R^2)^{p+1}}.
$$
\endproclaim

\demo{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
$$
u(t)=(1-|t|^2/R^2)^{p+1}\exp\big(\rho_\varepsilon\star\log\det\gamma(t)
\big)
$$
where $(\rho_\varepsilon)$ is a family of regularizing kernels.
The argument goes through because
$$
i\,\ddbar(\rho_\varepsilon\star\log\det\gamma)\ge A\,\rho_\varepsilon
\star\gamma
$$
and $\log\det(\rho_\varepsilon\star\gamma)\ge\rho_\varepsilon\star
\log\det\gamma$ by concavity of the $\log\det$ function.\square
\enddemo

\proclaim{3.3.~Proposition}Let $(X,V)$ be a compact directed
manifold. Assume that $V^\star$ 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 $H^0(X,S^mV^\star\otimes L^{-1})$
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''$]$.
\endproclaim

\demo{Proof} Let $\sigma_1\ld\sigma_N\in H^0(X,S^mV^\star\otimes L^{-1})$ be
a basis of sections generating $S^mV^\star\otimes L^{-1}$ 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 
$$
i\,\ddbar\log\gamma_0\ge {2\pi\over m}f^\star\Theta(L)
$$
where $\Theta(L)$ is the curvature form. The positivity assumption combined
with an obvious homogeneity argument yield
$$
{2\pi\over m}f^\star\Theta(L)\ge\varepsilon\|f'(t)\|_\omega^2\,|dt|^2\ge
\varepsilon'\,\gamma(t)
$$
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
\enddemo

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

\proclaim{3.4.~Proposition}Let $(X,V)$ be a compact directed manifold.
Assume that $\Lambda^pV^\star$ is ample. Then $(X,V)$ is infinitesimally
$p$-measure hyperbolic. More generally, assume that $\Lambda^pV^\star$
is very big with base locus contained in $Y\subsetneq X$ $($see $3.3)$.
Then $\bfe^p$ is non degenerate over $X\ssm Y$.
\endproclaim

\demo{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 $\swt\gamma|=f^\star\omega$. 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
$$
i\,\ddbar\log\det\gamma=i\,\ddbar\log\delta\ge (f,\Lambda^pf')^\star
(i\,\ddbar\log N)\ge\varepsilon f^\star\omega\ge\varepsilon'\,\gamma.
$$
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 $\Lambda^pV^\star$ 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
\enddemo

\proclaim{3.5.~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
$Y\subsetneq X$.
\endproclaim

The converse of Corollary 3.5 is 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]):

\proclaim{3.6.~Conjecture {\rm(Green-Griffiths [GrGr80])}}A projective
algebraic variety $X$ is almost measure hyperbolic $($i.e.\ $\bfe^n$
degenerates only along a proper algebraic subvariety$)$ if and only if
$X$ is of general type.
\endproclaim

In the same vein, Green-Griffiths and Lang proposed the following
conjectures.

\proclaim{3.7.~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~$Y$.
\endproclaim

The most outstanding result in the direction of Conjecture 3.7 is the
proof of the Bloch theorem, as proposed by Bloch [Blo26] and Ochiai
[Och77]. The Bloch theorem is the special case of 3.7 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$\,$9 for a proof based
on jet bundle techniques.

\proclaim{3.8.~Conjecture {\rm([Lang86, 87])}}A projective algebraic
variety $X$ is hyperbolic if and only if all its algebraic subvarieties
$($including $X$ itself$)$ are of general type.
\endproclaim

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 symplectic
manifolds, all of which have $c_1(X)=0$) are not hyperbolic, e.g.\ by
exhibiting a sequence of curves $C_\ell$ such that
$(2g(\ol C_\ell)-2)/\deg(C_\ell)\to 0$. In fact, it is even expected that
there are covering families of such curves, whereby proving that
such manifolds are not measure hyperbolic. An analogous conjecture in
the relative situation might stand as follows.

\proclaim{3.9.~Conjecture}Let $(X,V)$ be a projective directed manifold.
Assume that $\det V^\star$ is big and that $V$ is semistable in some
sense $($e.g.\ with respect to $\det V^\star$, if $\det V^\star$ is
ample$)$. Then there is a proper algebraic subset $Y\subsetneq X$ such
that every entire curve $f:\bC\to X$ tangent to $V$ satisfies
$f(\bC)\subset Y$.
\endproclaim

Recall that a line bundle $L$ is said to be {\em big} if it has maximal
Kodaira dimension, in other words, if $H^0(X,L^{\otimes m})\ge c\,
m^{\dim X}$ for some constant $c>0$, when $m\ge m_0$ is sufficiently large.
Some sort of semistability condition is clearly required, otherwise one
might take $X$ to be an abelian variety of dimension${}\ge 3$ and
$V=L+L'$ where $L\subset T_X$ is a constant line subbundle with dense
trajectories, and $L'\subset T_X$ a ``generic'' sufficiently negative
line bundle (also take a blow-up $\swt X|\to X$ to resolve the
singularities of $V$, so as to obtain a subbundle $\swt V|\subset
T_{\swt X|}$). In the absolute case $V=T_X$, the semistability condition
is probably not needed since $T_X$ tends to be always semistable in some
sense (if $K_X$ is ample, there is always a K\"ahler-Einstein metric,
hence $T_X$ is $K_X$-semistable).

\proclaim{3.10.~Remark}{\rm One should take care of the fact that
Propositions 3.1, 3.3 and 3.4 cannot be extended without modifications
to the case when $V$ admits singularities. For instance, take $X=\bP^n$
and let $\ell=\bP^1\subset X$ be a line. Take a section of
$T_\ell\otimes\cO_\ell(d)\simeq\cO_\ell(d+2)$ admitting only one zero
$z_0$ of multiplicity $d+2$, and extend it as a section $\sigma$ of
$T_{\bP^n}\otimes\cO(d)$ admitting only isolated zeroes (this is always
possible for arbitrary $d>0$). Then $\sigma$ defines a sheaf injection
$\sigma:\cO(-d)\lhra\cO(T_{\bP^n})$ with
$\cV=\sigma(\cO(-d))\simeq\cO(-d)$ negative, nevertheless $\cV$ admits
a complex line $\ell\ssm\{z_0\}\simeq\bC$ as one of its integral
curves. The correct assumption guaranteeing the hyperbolicity of
$(X,V)$ is that $\cW=\Im(\Omega_X^1\to\cV^\star)$ should be ample.}
\endproclaim

\head 4. Projectivization of a Directed Manifold
\writetoaux\endhead

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 $v\in V_x\ssm\{0\}$,
$$
\swt V|_{(x,[v])}=\big\{\xi\in T_{\swt X|,\,(x,[v])}\,;\,\pi_\star\xi\in
\bC v\big\},\qquad\bC v\subset V_x\subset T_{X,x},\leqno(4.1)
$$
where $\pi:\swt X|=P(V)\to X$ is the natural projection and $\pi_\star:
T_{\swt X|}\to\pi^\star T_X$ is its differential. On $\swt X|=P(V)$
we have a tautological line bundle $\cO_{\swt X|}(-1)\subset\pi^\star V$
such that $\cO_{\swt X|}(-1)_{(x,[v])}=\bC v$. The bundle $\swt V|$ is
characterized by the two exact sequences
$$
\leqalignno{
&0\lra T_{\swt X|/X}\lra\swt V|\build\lra|{\pi_\star}||\cO_{\swt X|}(-1)
\lra 0,&(4.2)\cr
&0\lra\cO_{\swt X|}\lra \pi^\star V\otimes\cO_{\swt X|}(1)
\lra T_{\swt X|/X}\lra 0,&(4.2')\cr}
$$
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
$$
\dim\swt X|=n+r-1,\qquad \rank\swt V|=\rank V=r,\leqno(4.3)
$$
and by taking determinants we find $\det(T_{\swt X|/X})=
\pi^\star\det V\otimes\cO_{\swt X|}(r)$, thus
$$
\det\swt V|=\pi^\star\det V\otimes\cO_{\swt X|}(r-1).\leqno(4.4)
$$
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
$$
\eqalign{(\wt X,\wt V)&\build\lra|{\textstyle\pi}||(X,V)\cr
&\kern-27pt\wt\Phi\vdasharrow\kern45pt\Big\downarrow\Phi\cr
(\wt Y,\wt W)&\build\lra|{\textstyle\pi}||(Y,W)\cr}\leqno(4.5)
$$
where the left vertical arrow is the meromorphic map $P(V)\dasharrow P(W)$
induced by the differential $\Phi_\star:V\to\Phi^\star W$ ($\swt\Phi|$ is
actually holomorphic if $\Phi_\star:V\to\Phi^\star W$ is injective).

Now, 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 {\em tangent trajectory} 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
$$
\swt f|:\Delta_R\to\swt X|,\qquad
t\mapsto\swt f|(t):=(f(t),[f'(t)])\leqno(4.6)
$$
is holomorphic (at a stationary point $t_0$, we just write
$f'(t)=(t-t_0)^su(t)$ with $s\in\bN^\star$ 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
$$
f':T_{\Delta_R}\to\swt f|^\star\cO_{\swt X|}(-1).\leqno(4.7)
$$
Moreover $\pi\circ\swt f|=f$, therefore
$$
\pi_\star\swt f|'(t)=f'(t)\in\bC u(t)\Longrightarrow
\swt f|'(t)\in\swt V|_{(f(t),u(t))}=\swt V|_{\swt f|(t)}
$$
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
$$
V_z=\Big\{\xi=\sum_{1\le j\le n}\xi_j{\partial\over\partial z_j}\,;\,
\xi_j= \sum_{1\le k\le r}a_{jk}(z)\xi_k~\hbox{\rm for $j=r+1\ld n$}\Big\},
\leqno(4.8)
$$
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
$$
\Big(z_1\ld z_n;{\xi_1\over\xi_j}\ld{\xi_{j-1}\over\xi_j},
{\xi_{j+1}\over\xi_j}\ld{\xi_r\over\xi_j}\Big).\leqno(4.9)
$$
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
$$
f_j'(t)=\sum_{1\le k\le r}a_{jk}(f(t))f_k'(t),\qquad j>r,\leqno(4.10)
$$
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 $m\in\bN^\star$
such that $f_j^{(m)}(t_0)\ne 0$ for some~$j$. By (4.10), 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
$$
\swt f|\simeq\Big(f_1\ld f_n;{f_1'\over f_r'}\ld{f_{r-1}'\over f_r'}\Big).
\leqno(4.11)
$$
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 $\Theta_h(V)={i\over 2\pi}\nabla_h^2$ 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
$$
\langle e_\lambda,e_\mu\rangle_h=\delta_{\lambda\mu}-
\sum_{1\le j,k\le n}c_{jk\lambda\mu}z_j\ol z_k+O(|z|^3),\leqno(4.12)
$$
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
$$
\leqalignno{\nabla'_h e_\lambda
&=-\sum_{j,k,\mu}c_{jk\lambda\mu}\ol z_k\,dz_j\otimes e_\mu+O(|z|^2),\cr
\Theta_h(V)_{x_0}
&={i\over 2\pi}\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}dz_j\wedge
d\ol z_k\otimes e_\lambda^\star\otimes e_\mu.&(4.13)\cr}
$$
The above curvature tensor can also be viewed as a hermitian form
on $T_X\otimes V$. In fact, one associates with $\Theta_h(V)$ the hermitian
form $\langle\Theta_h(V)\rangle$ on $T_X\otimes V$ defined for all
$(\zeta,v)\in T_X\times_X V$ by
$$
\langle\Theta_h(V)\rangle(\zeta\otimes v)=
\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\zeta_j\ol\zeta_kv_\lambda\ol v_\mu.\leqno(4.14)
$$
Let $h_1$ be the hermitian metric on the tautological line bundle
$\cO_{P(V)}(-1)\subset\pi^\star V$ 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 $\Theta_h(V)$. For
simplicity, we suppose that the frame $(e_\lambda)_{1\le\lambda\le r}$ has
been chosen in such a way that $[e_r(x_0)]=[v_0]$ 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 (4.12) for $h$, we find
$$
\leqalignno{
|\eta|_{h_1}^2=|\eta|_h^2=1+|\xi|^2&{}-\sum_{1\le j,k\le n}c_{jkrr}
z_j\ol z_k+O((|z|+|\xi|)^3),\cr
\Theta_{h_1}(\cO_{P(V)}(-1))_{(x_0,[v_0])}
&=-{i\over 2\pi}d'd''\log|\eta|_{h_1}^2\cr
&={i\over 2\pi}\Big(\sum_{1\le j,k\le n}c_{jkrr}dz_j\wedge d\ol z_k
-\sum_{1\le\lambda\le r-1}d\xi_\lambda\wedge d\ol\xi_\lambda\Big).
&(4.15)}
$$
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, (4.15) can be rewritten as
$$
\langle\Theta_{h_1}(\cO_{P(V)}(-1))\rangle_{(x_0,[v_0])}(\tau)
=\langle\Theta_h(V)\rangle_{x_0}({}^H\!\tau\otimes v_0)-|{}^V\!\tau|_\FS^2
\leqno(4.16)
$$
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
$$
|w|^2_{\widetilde h_\rho}=|{}^H\!w|^2_h+\rho^2|{}^V\!w|^2_h\qquad
\hbox{for $w\in\swt V|_{(x_0,[v_0])}$},
$$
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
$$
\leqalignno{
\langle\Theta_{\swt h|_\rho}(\swt V|)\rangle_{(x_0,[v_0])}(\tau\otimes w)
&=\langle\Theta_h(V)\rangle_{x_0}({}^H\!\tau\otimes v_0)~
(|{}^H\!w|_h^2-\rho^2|{}^V\!w|_h^2)\cr
&\phantom{{}={}}{+}\;
\rho^2\langle\Theta_h(V)\rangle_{x_0}({}^H\!\tau\otimes{}^V\!w)\cr
&\phantom{{}={}}{+}\;
\rho^2\big(|\langle{}^V\!\tau,{}^V\!w\rangle_h|^2+|{}^V\!\tau|_h^2
|{}^V\!w|_h^2\big)-|{}^V\!\tau|_h^2|{}^H\!w|_h^2&(4.17)\cr
&\phantom{{}={}}{+}\;
O(\rho)|\tau|_\omega^2|w|^2_{\widetilde h_\rho},\qquad
\tau\in T_{\swt X|},~w\in\swt V|,\cr}
$$
where $|\tau|_\omega^2$ is computed from a fixed hermitian metric $\omega$
on~$T_X$.

\head 5. Jets of Curves and Semple Jet Bundles
\writetoaux\endhead

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 $(f'(0)\ld f^{(k)}(0))\in(\bC^n)^k$.
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 (6.2)
in~\S$\,$6).

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.

\proclaim{5.1.~Definition}Let $(X,V)$ be a complex directed manifold.
We define the bundle $J_kV\to X$ to be the set of all $k$-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~$X$.
\endproclaim

It is easy to check that $J_kV$ is actually a subbundle of~$J_k$. In
fact, by using (4.8) and (4.10), 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~$J_k$.\square

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{*}{{Very recently, a preprint [LaTh96] by Laksov and Thorup
has also appeared, dealing in depth with 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
$$
(X_0,V_0)=(X,V),\qquad (X_k,V_k)=(\swt X|_{k-1},\swt V|_{k-1}).
\leqno(5.2)
$$
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$\,$4. By (4.2--4.7), we find
$$
\dim P_kV=n+k(r-1),\qquad\rank V_k=r,\leqno(5.3)
$$
together with exact sequences
$$
\leqalignno{
&0\lra T_{P_kV/P_{k-1}V}\lra V_k\build{\vlra11|}|{(\pi_k)_\star}||
\cO_{P_kV}(-1)\lra 0,&(5.4)\cr
&0\lra\cO_{P_kV}\lra\pi_k^\star V_{k-1}\otimes\cO_{P_kV}(1)
\lra T_{P_kV/P_{k-1}V}\lra 0.
&(5.4')\cr}
$$
where $\pi_k$ is the natural projection $\pi_k:P_kV\to P_{k-1}V$ and
$(\pi_k)_\star$ its differential. Formula (4.4) yields
$$
\det V_k=\pi_k^\star\det V_{k-1}\otimes\cO_{P_kV}(r-1).\leqno(5.5)
$$
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
$$
f_{[k-1]}':T_{\Delta_R}\to f_{[k]}^\star\cO_{P_kV}(-1).\leqno(5.6)
$$
In coordinates, one can compute $f_{[k]}$ in terms of its components in
the various affine charts (4.9) occurring at each step: we get inductively
$$
f_{[k]}=(F_1\ld F_N),\qquad f_{[k+1]}=\Big(F_1\ld F_N,
{F_{s_1}'\over F_{s_r}'}\ld{F_{s_{r-1}}'\over F_{s_r}'}\Big)\leqno(5.7)
$$
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 $\cO_{P_kV}(-1)
\hookrightarrow\pi_k^\star V_{k-1}$, and a composition with the
projection $(\pi_{k-1})_\star$ (analogue for order $k-1$ of the
arrow~$(\pi_k)_\star$ in sequence (5.4)) yields for all $k\ge 2$ a
canonical line bundle morphism
$$
\cO_{P_kV}(-1)\lhra\pi_k^\star V_{k-1}\build{\vlra16|}|
{(\pi_k)^\star(\pi_{k-1})_\star}||\pi_k^\star\cO_{P_{k-1}V}(-1),\leqno(5.8)
$$
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
$$
\cO_{P_kV}(1)=\pi_k^\star\cO_{P_{k-1}V}(1)\otimes\cO(D_k).\leqno(5.9)
$$
Now, we consider the composition of projections
$$
\pi_{j,k}=\pi_{j+1}\circ\cdots\circ\pi_{k-1}\circ\pi_k:P_kV\lra P_jV.
\leqno(5.10)
$$
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~6.8 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
$s\in\bN^\star$ 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~$k$.

\proclaim{5.11.~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~$j\in\{0\ld k-1\}$.
\endproclaim

\demo{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
$f_{[j-1]}'(t)=t^{m_{j-1}-1}u_{j-1}(t)\in V_{j-1}$, \hbox{$u_{j-1}(0)\ne 0$}.
By taking the composition with the differential of the projection
\hbox{$\pi_{j-1}:P_{j-1}V\to P_{j-2}V$}, we find $f_{[j-2]}'(t)=t^{m_{j-1}-1}
(\pi_{j-1})_\star u_{j-1}(t)$. Therefore
$$
m_{j-2}=m_{j-1}+\ord_{t=0}(\pi_{j-1})_\star u_{j-1}(t),
$$
and so $m_{j-2}>m_{j-1}$ if and only if $(\pi_{j-1})_\star u_{j-1}(0)=0$,
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 (4.9)
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
(5.7) 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 (4.10).
We then get the desired Taylor expansion of order $d_k$ for~$f$.\square
\enddemo

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

\proclaim{5.12.~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~$w$.
\endproclaim

\demo{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
\enddemo

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~5.11 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
$$
\eqalign{
P_kV^\reg&=\bigcap_{2\le j\le k}\pi_{j,k}^{-1}(P_jV\ssm D_j),\cr
P_kV^\sing&=\bigcup_{2\le j\le k}\pi_{j,k}^{-1}(D_j)=P_kV\ssm P_kV^\reg,
\cr}
\leqno(5.13)
$$
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.

\proclaim{5.14.~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 $\smash{f_{[k-1]}'(0)}\ne 0$.
\endproclaim

\head 6. Jet Differentials
\writetoaux\endhead

Following Green-Griffiths [GrGr80], we now introduce the concept of jet
differential. This concept gives an intrinsic way of describing
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 $f$ 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
$$
t\mapsto\varphi(t)=a_1t+a_2t^2+\cdots+a_kt^k,\qquad
a_1\in\bC^\star,~a_j\in\bC,~j\ge 2,
$$
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
$$
1\to \bG'_k\to \bG_k\to\bC^\star\to 1
$$
where $\bG_k\to\bC^\star$ 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 $\bH\simeq\bC^\star$ 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 $E^\GG_{k,m}V^\star\to X$
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
$\bC^\star$ action defined by $H$, that is, such that
$$
Q(\lambda f',\lambda^2 f'',\ldots,\lambda^k f^{(k)})=\lambda^m
Q(f',f'',\ldots,f^{(k)})\leqno(6.1)
$$
for all $\lambda\in\bC^\star$ 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
$$
(\Psi\circ f)^{(j)}=\Psi'(f)\cdot f^{(j)}+\sum_{s=2}^{s=j}{~}
\sum_{j_1+j_2+\cdots+j_s=j}c_{j_1\ldots j_s}\Psi^{(s)}(f)\cdot
(f^{(j_1)}\ld f^{(j_s)})\leqno(6.2)
$$
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 $E^\GG_{k,m}T^\star_X=E^\GG_{k,m}$. 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
$$
E^\GG_{k,m}\to E^\GG_{k,m}W^\star\to E^\GG_{k,m}V^\star,
$$
in particular $E^\GG_{k,m}V^\star$ can be seen as a quotient of $E^\GG_{k,m}$.
(The notation $V^\star$ is used here to make the contravariance property
implicit from the notation).

If $Q\in E^\GG_{k,m}V^\star$ 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 $E^\GG_{k,m}V^\star$ will be called the {\em bundle of jet
differentials of order $k$ and weighted degree~$m$}. It is clear from (6.2)
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
$E^\GG_{k,m}V^\star$ as follows:
$$
F^p_s(E^\GG_{k,m}V^\star)=\left\{
{\displaystyle
\hbox{\rm $Q(f',f''\ld f^{(k)})\in E^\GG_{k,m}V^\star$ involving}
\atop\hbox{\rm only monomials $(f^{(\bullet)})^\ell$ with $|\ell|_s\ge p$}
\hfill}\right\},\qquad
\forall p\in\bN.\leqno(6.3)
$$
The graded terms $\Gr^p_{k-1}(E^\GG_{k,m}V^\star)$ associated with the
filtration $F^p_{k-1}(E^\GG_{k,m}V^\star)$ 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 $\Gr^p_{k-1}(E^\GG_{k,m}V^\star)=0$ 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 (6.2), 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
$$
G_{k-1}^{m-k\ell_k}(E^\GG_{k,m}V^\star)=E^\GG_{k-1,m-k\ell_k}V^\star\otimes
S^{\ell_k}V^\star.
\leqno(6.4)
$$
Combining all filtrations $F_s^\bullet$ together, we find inductively a
filtration $F^\bullet$ on $E^\GG_{k,m}V^\star$ such that the graded terms are
$$
\Gr^\ell(E^\GG_{k,m}V^\star)=S^{\ell_1}V^\star\otimes S^{\ell_2}V^\star\otimes
\cdots\otimes S^{\ell_k}V^\star,\qquad\ell\in\bN^k,\quad
|\ell|_k=m.\leqno(6.5)
$$

The bundles $E^\GG_{k,m}V^\star$ have other interesting properties. In fact,
$$
E^\GG_{k,\bu}V^\star:=\bigoplus_{m\ge 0}E^\GG_{k,m}V^\star
$$
is in a natural way a bundle of graded algebras (the product is
obtained simply by taking products of polynomials). There are
natural inclusions \hbox{$E^\GG_{k,\bu}V^\star\subset E^\GG_{k+1,\bu}
V^\star$} of algebras, hence $E^\GG_{\infty,\bu}V^\star=\bigcup_{k\ge 0}
E^\GG_{k,\bu}V^\star$ is also an algebra. Moreover, the sheaf of
holomorphic sections $\cO(E^\GG_{\infty,\bu} V^\star)$ admits a
canonical derivation $\nabla$ given by a collection of $\bC$-linear maps
$$
\nabla:\cO(E^\GG_{k,m}V^\star)\to\cO(E^\GG_{k+1,m+1}V^\star),\leqno(6.6)
$$
constructed in the following way. A holomorphic section of
$E^\GG_{k,m}V^\star$ 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
$$
Q(f)=\sum_{|\alpha_1|+2|\alpha_2|+\cdots+k|\alpha_k|=m}
a_{\alpha_1\ldots\alpha_k}(f)\,(f')^{\alpha_1}(f'')^{\alpha_2}\cdots
(f^{(k)})^{\alpha_k}
$$
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 $E^\GG_{k,\bu}V^\star$, we 
have an analytic fiber bundle
$\Proj(E^\GG_{k,\bu}V^\star)=J_kV^\nc/\bC^\star$ over $X$, which has
weighted projective spaces $\bP(r\ld r\,;\,1,2\ld k)$ as fibers (these
weighted projective spaces are singular for $k>1$, but they only have
quotient singularities, see [Dol81]$\,$; here $J_kV^\nc$ denotes 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). However, we are not
really interested in the bundles $J_kV^\nc/\bC^\star$ themselves, but
rather on their quotients $J_kV^\nc/\bG_k$ (would such nice complex
space quotients exist!). We will see that the Semple bundle $P_kV$ 
constructed in \S$\,$5 plays the role of such a quotient. First we 
introduce a canonical bundle subalgebra of~$E^\GG_{k,\bu}V^\star$.

\proclaim{6.7.~Definition}We introduce a subbundle
$E_{k,m}V^\star\subset E^\GG_{k,m}V^\star$, called the bundle of
invariant jet differentials of order~$k$ and degree $m$, defined as
follows: $E_{k,m}V^\star$ 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 $\varphi\in \bG_k$
$$
Q\big((f\circ\varphi)',(f\circ\varphi)'',\ldots,(f\circ\varphi)^{(k)})=
\varphi'(0)^m Q(f',f'',\ldots,f^{(k)}).
$$
\endproclaim

Alternatively, $E_{k,m}V^\star=(E^\GG_{k,m}V^\star)^{\bG'_k}$ is the set of
invariants of $E^\GG_{k,m}V^\star$ under the action of~$\bG'_k$. Clearly,
$E_{\infty,\bu}V^\star=\bigcup_{k\ge 0}\bigoplus_{m\ge0}E_{k,m}V^\star$
is a subalgebra of $E^\GG_{k,m}V^\star$ (observe however that this algebra is
not invariant under the derivation~$\nabla$, since e.g.\ $f_j''=\nabla f_j$
is not an invariant polynomial). In addition to this, there are natural
induced filtrations
$F^p_s(E_{k,m}V^\star)=E_{k,m}V^\star\cap F^p_s(E^\GG_{k,m}V^\star)$
(all locally trivial over~$X$). These induced filtrations will play an
important role in Section~12.

\proclaim{6.8.~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~$5$,
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
$$
(\pi_{0,k})_\star\cO_{P_kV}(m)\simeq\cO(E_{k,m}V^\star)
$$
can be identified with the sheaf of holomorphic sections of
$E_{k,m}V^\star$.
\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
\endproclaim

\demo{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 (4.10)].
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
$f_i^{(j)}(t)=d^jf_i/dt^j$
$$
\leqalignno{
(g_1'\ld g_{r-1}')&=\Big({f'_1\over f'_r}\ld{f'_{r-1}\over f'_r}\Big);\cr
(g_1''\ld g_{r-1}'')&=\Big({f''_1f'_r-f''_rf'_1\over f^{\prime 3}_r}\ld
{f''_{r-1}f'_r-f''_rf'_{r-1}\over f^{\prime 3}_r}\Big);~\ldots~;&(6.9)\cr
\qquad(g_1^{(k)}\ld g_{r-1}^{(k)})&=\Big({f^{(k)}_1f'_r-f^{(k)}_rf'_1\over
f^{\prime k+1}_r}\ld{f^{(k)}_{r-1}f'_r-f^{(k)}_rf'_{r-1}\over
f^{\prime k+1}_r}\Big)+(\hbox{order}<k).\cr}
$$
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 $E_{k,m}V^\star$ 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
$E_{k,m}V^\star_x$, at any point  $x\in X$. Let $f\in J_kV_x^\reg$
be a regular $k$-jet at~$x$. By (5.6), 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
$$
Q(f',f''\ld f^{(k)})=\sigma(f_{[k]}(0))\cdot(f_{[k-1]}'(0))^m.
\leqno(6.10)
$$
Clearly, $Q$ is holomorphic on $J_kV_x^\reg$ (by the holomorphicity
of~$\sigma$), and the $\bG_k$-invariance condition of Def.~6.7 is satisfied
since $f_{[k]}(0)$ does not depend on repara\-me\-trization 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 (6.7) implies in particular that $Q$ must be
multihomogeneous in the sense of (6.1), and thus $Q$ must be a
polynomial. We conclude that $Q\in E_{k,m}V^\star_x$, as desired.

Conversely, Corollary~5.12 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 $Q\in E_{k,m}V^\star_x$ yields a
holomorphic section $\sigma$ of $\cO_{P_kV}(m)$ over the fiber $P_kV_x$
by putting
$$
\sigma(w)=Q(f_w',f_w''\ld f_w^{(k)})(0)\,\big((f_w)_{[k-1]}'(0)\big)^{-m}.
\leqno(6.11)
$$

\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
$Q\in E_{k,m}V^\star_x$, and this polynomial can be expressed
on the Zariski open chart $f'_r\ne 0$ of $P_kV^\reg_x$ as
$$
Q(f',f''\ld f^{(k)})=f_r^{\prime m}\swh Q|(g',g''\ld g^{(k)}),
\leqno(6.12)
$$
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 (6.9).

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 $Q(f'\ld f^{(k)})=f_r^{\prime 2k-1}g_i^{(j)}$, $1\le i\le r-1$,
\hbox{$1\le j\le k$}; 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~5.14). 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 (6.11).\square
\enddemo

Theorem (6.8~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
$$
\cO_{P_kV}(\bfa)=\pi_{1,k}^\star\cO_{P_1V}(a_1)\otimes
\pi_{2,k}^\star\cO_{P_2V}(a_2)\otimes\cdots\otimes\cO_{P_kV}(a_k).
\leqno(6.13)
$$
By (5.9), we have $\pi_{j,k}^\star\cO_{P_jV}(1)=
\cO_{P_kV}(1)\otimes\cO_{P_kV}
(-\pi_{j+1,k}^\star D_{j+1}-\cdots-D_k)$, thus by putting $D^\star_j=
\pi_{j+1,k}^\star D_{j+1}$ for $1\le j\le k-1$ and $D^\star_k=0$, we find
an identity
$$
\leqalignno{
&\cO_{P_kV}(\bfa)=\cO_{P_kV}(b_k)\otimes\cO_{P_kV}(-\bfb\cdot D^\star),
\qquad\hbox{\rm where}&(6.14)\cr
&\bfb=(b_1\ld b_k)\in\bZ^k,\quad b_j=a_1+\cdots+a_j,&\cr
&\bfb\cdot D^\star=\sum_{1\le j\le k-1}b_j\,\pi_{j+1,k}^\star D_{j+1}.&\cr}
$$
In particular, if $\bfb\in\bN^k$, i.e., $a_1+\cdots+a_j\ge 0$, we get a
morphism
$$
\cO_{P_kV}(\bfa)=\cO_{P_kV}(b_k)\otimes\cO_{P_kV}(-\bfb\cdot D^\star)
\to\cO_{P_kV}(b_k).\leqno(6.15)
$$

\proclaim{6.16.~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
$$
(\pi_{0,k})_\star\cO_{P_kV}(\bfa)\simeq\cO(\ol F^\bfa E_{k,m}V^\star)
\subset\cO(E_{k,m}V^\star)
$$
where $\ol F^\bfa E_{k,m}V^\star$ is the subbundle of polynomials
\hbox{$Q(f',f''\ld f^{(k)})\in E_{k,m}V^\star$\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
\endproclaim

\demo{Proof} i) By (6.15), we find a sheaf injection
$$
(\pi_{0,k})_\star\cO_{P_kV}(\bfa)\hookrightarrow
(\pi_{0,k})_\star\cO_{P_kV}(m)=\cO(E_{k,m}V^\star).
$$
Given a section $\sigma$ of $\cO_{P_kV}(\bfa)$ over a fiber $P_kV_x$,
the associated polynomial
$$
Q(f',f''\ld f^{(k)})\in E_{k,m}V^\star_x
$$
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 (6.10) and from the fact that $f_{[k-1]}'(0)$
is mapped to $f_{[j-1]}'(0)$ by the projection morphism
$$
(\pi_{j-1,k-1})_\star:\cO_{P_kV}(-1)\to\pi_{j,k}^\star\cO_{P_jV}(-1)
$$
(cf.\ (5.8)), which is dual to the corresponding morphism (6.15). Now,
we prove the inclusion $(\pi_{0,k})_\star\cO_{P_kV}(\bfa)
\subset\cO(\ol F^\bfa E_{k,m}V^\star)$ 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
$\pi_{2,k}^\star\cO_{P_2V}(a_2)\otimes\cdots \otimes\cO_{P_kV}(a_k)$
on the fibers of the projection $P_kV=P_{k-1}V_1\to X_1=P_1V$.
Then we get a polynomial $Q_1\in E_{k-1,m-a_1}V_1^\star$ 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
(6.16$\,$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 (6.9), we get $Q(f',f''\ld f^{(k)})=(f_r')^m
Q(g',g''\ld g^{(k)})$, and the desired inequality (6.16$\,$i)
for the monomials $(f^{(\bu)})^\ell$ follows easily. In the opposite
direction, if we are given a section $Q(f',f''\ld f^{(k)})\in\cO(\ol F^\bfa
E_{k,m}V^\star)$, we see by induction on $k$ that $Q$ defines
a section of
$$
\cO_{P_1V}(a_1)\otimes(\pi_{1,k})_\star
\big(\pi_{2,k}^\star\cO_{P_2V}(a_2)\otimes\cdots\otimes\cO_{P_kV}(a_k)\big)
$$
on $P_1V$, and we conclude that we get a section of
$(\pi_{0,k})_\star\cO_{P_kV}(\bfa)$ by taking the direct image by
$(\pi_1)_\star$.
\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
$\cO_{P_kV}(1)\otimes\pi_k^\star L_{k-1}$ is relatively nef; by
definition, this is equivalent to saying that the vector bundle
$V_{k-1}^\star\otimes L_{k-1}$ 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 (5.4) yields an exact sequence
$$
0\lra\cO_{P_kV}(1)\lra V_k^\star\lra T_{P_kV/P_{k-1}V}^\star\lra 0.
$$
As an extension of nef vector bundles is nef, it is enough to select $L_k$
in such a way that
$$
\cO_{P_kV}(1)\otimes L_k\quad\hbox{and}\quad
T^\star_{P_kV/P_{k-1}V}\otimes L_k\quad\hbox{are relatively nef.}\leqno(6.17)
$$
By taking the second wedge power of the central term in $(5.4')$, we get an
injection
$$
0\lra T_{P_kV/P_{k-1}V}\lra \Lambda^2\big(\pi_k^\star V_{k-1}\otimes
\cO_{P_kV}(1)\big).
$$
By dualizing and twisting with $\cO_{P_{k-1}V}(2)\otimes\pi_k^\star
L_{k-1}^{\otimes 2}$, we find a surjection
$$
\pi_k^\star\Lambda^2(V_{k-1}^\star\otimes L_{k-1})\lra T_{P_kV/P_{k-1}V
}^\star\otimes\cO_{P_kV}(2)\otimes\pi_k^\star L_{k-1}^{\otimes 2}\lra 0.
$$
As $V_{k-1}^\star\otimes L_{k-1}$ is relatively nef by the induction
hypothesis, we obtain that its quotient $T_{P_kV/P_{k-1}V}^\star\otimes
\cO_{P_kV}(2)\otimes\pi_k^\star L_{k-1}^{\otimes 2}$
is also relatively nef. Hence Condition (6.17) is achieved if we take
$L_k\ge\pi_k^\star L_{k-1}$ and
$L_k\ge\cO_{P_kV}(2)\otimes\pi_k^\star L_{k-1}^{\otimes 2}$ (the ordering
relation $\ge$ is the one given by the cone of relatively nef line bundles).
We need only define $L_k$ inductively by
$$
L_k=\cO_{P_kV}(2)\otimes\pi_k^\star L_{k-1}^{\otimes 3}.
$$
The relative ampleness of $L_k$ is then clear by induction, since
$\cO_{P_kV}(1)\otimes\pi_k^\star L_{k-1}$ 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
$$
\cO_{P_kV}(1)\otimes\pi_k^\star L_{k-1}=
\cO_{P_kV}\big((2\cdot 3^{k-2},2\cdot 3^{k-3}\ld 6,2,1)\big)
\qquad\hbox{relatively nef}.
$$
These properties imply ii) and iii) by taking suitable convex
combinations.\square
\enddemo 

\proclaim{6.18.~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
$$
\Phi^\star:H^0(Y,E^\GG_{k,m})\to H^0(X,E^\GG_{k,m}),\qquad
\Phi^\star:H^0(Y,E_{k,m})\to H^0(X,E_{k,m}).
$$
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 $F_s^\bullet$.}
\endproclaim

\head 7. $k$-Jet Metrics with Negative Curvature
\writetoaux\endhead

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. However, 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 7.14 below). In the present situation, it is
important to allow also hermitian metrics possessing some singularities
(``singular hermitian metrics'' in the sense of [Dem90]).

\proclaim{7.1.~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
$\Theta_h(L)={i\over 2\pi}\ddbar\varphi$, 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~$\Sigma$.
\endproclaim

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 $\Theta_h(L)\ge\varepsilon
\omega$ in the sense of currents, and we write in this case
$\Theta_h(L)\gg0$. 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 the case 
of general complex tori in \S$\,$9).

\proclaim{7.2.~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 $\Theta_h(L)\gg0$ 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 $\Theta_h(L)$
  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~$\Theta_h(L)$, 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.
\smallskip
\item{\rm iv)} Assume that $L$ admits a singular hermitian metric $h$ 
with positive definite curvature current, such that the degeneration set 
$\Sigma$ is an analytic subset of $X$. Assume moreover that for each 
irreducible component $\Sigma_j$ of $\Sigma$, $L_{|\Sigma_j}$ admits a 
singular hermitian metric $h_j$ 
with positive definite curvature current on $\Sigma_j$ and degeneration
set $\Sigma_{j,k}\subset \Sigma_j$. Then $L$ admits a singular
hermitian metric $\wt h$ of positive curvature current on $X$, with
degeneration set $\wt\Sigma=\bigcup_{j,k}\Sigma_{j,k}$.
\vskip0pt 
\endproclaim

\demo{Proof} i) is proved e.g.\ in [Dem90,~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 $\Theta_h(L)$ 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 $\wt L=\mu^\star(L^{\otimes m})
\otimes \cO_{\wt X}(-E)$ $($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
\hbox{$D\cap\Phi_m^{-1}(\Phi_m(x))=\emptyset$}. 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 $\Phi_m^\star\sigma$ can be
viewed as a section of $\Phi_m^\star\cO_{\bP^N}(k)\otimes\cO_X(-D)=
L^{\otimes km}\otimes G^{-1}$, and $\Phi_m^\star\sigma$ 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.

\noindent
iv) is obtained by extending the metric $h_j$ to a metric
$\wt h_j$ on a neighborhood of $\Sigma_j$ (it is maybe necessary
to modify $\wt h_j$ slightly by adding some ``transversally 
convex terms'' in the weight, so as to obtain positive curvature in 
all directions of $T_X$, on a suitable neighborhood of $\Sigma_j$),
and then taking $\wt h=\min(h,\varepsilon\wt h_j)$ with $\varepsilon>0$
small enough.\square
\enddemo

We now come to the main definitions. By (5.6), 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 $\cO_{P_kV}(-1)$. 

\proclaim{7.3.~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.
\endproclaim

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.

\proclaim{7.4.~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.$)$
\endproclaim

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.~7.2~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.~7.4.

\proclaim{7.5.~Special case of $1$-jet metrics}{\rm 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, $V^\star$~ample.
In the absolute case $V=T_X$, there are only few examples of varieties
$X$ such that $T^\star_X$ is ample, mainly quotients of the ball
$\bB_n\subset\bC^n$ by a discrete cocompact group of automorphisms.
The $1$-jet negativity condition considered in Definition~7.4 is much
weaker. For example, if the hermitian metric $h_1$ comes from a 
(smooth) hermitian metric $h$ on $V$, then formula (4.16) 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 $\langle\Theta_h(V)\rangle
(\zeta\otimes v)<0$ 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 $\Theta_h(V)$ satisfies
the much weaker condition that the {\em holomorphic sectional curvature}
$\langle\Theta_h(V)\rangle(v\otimes v)$ is negative on every complex
line.\square}
\endproclaim

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.

\proclaim{7.6.~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$)$.
\endproclaim

\demo{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
$$
\langle\Theta_{\wt h}(\cO_{P_kV}(D_k))\rangle(\xi)\ge
\delta|\xi|_{\omega_k}^2-C|(\pi_k)_\star\xi|_{\omega_{k-1}}^2,\qquad
\forall\xi\in T_{P_kV}
$$
for some constants $\delta,C>0$. Combining both inequalities (the second one
being applied to $\xi\in V_k$ and the first one to $(\pi_k)_\star\xi\in
V_{k-1}$), we get
$$
\eqalign{
\langle\Theta_{(\pi_k^\star h_{k-1})^{-p}\wt h\,}(\pi_k^\star\cO_{P_{k-1}V}(p)
\otimes{}&\cO_{P_kV}(D_k))\rangle(\xi)\ge\cr
&\ge\delta|\xi|_{\omega_k}^2+(p\varepsilon-C)
|(\pi_k)_\star\xi|_{\omega_{k-1}}^2,\qquad\forall\xi\in V_k.\cr}
$$
Hence, for $p$ large enough, $(\pi_k^\star h_{k-1})^{-p}\wt h$ has positive
definite curvature along~$V_k$. Now, by (5.9), there is a sheaf injection
$$
\cO_{P_kV}(-p)=\pi_k^\star\cO_{P_{k-1}V}(-p)\otimes\cO_{P_kV}(-pD_k)
\hookrightarrow\big(\pi_k^\star\cO_{P_{k-1}V}(p)\otimes\cO_{P_kV}(D_k)
\big)^{-1}
$$
obtained by twisting with $\cO_{P_kV}((p-1)D_k)$.  Therefore
$h_k:=((\pi_k^\star h_{k-1})^{-p}\wt h)^{-1/p}=(\pi_k^\star
h_{k-1})\wt h^{-1/p}$ 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
$$
\Theta_{h_k^{-1}}(\cO_{P_kV}(1))={1\over p}\Theta_{(\pi_k^\star 
h_{k-1})^{-p}\wt h}+{p-1\over p}[D_k]
$$
is positive definite along~$V_k$. The same proof works in the case of
negative total jet curvature.\square
\enddemo

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

\proclaim{7.7.~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 $\cO_{P_kV}(m)\otimes\cO_{P_kV}(-\bfb\cdot D^\star)$
is ample over~$P_kV$. Set $A=\cO_{P_kV}(m)\otimes\cO_{P_kV}(-\bfb\cdot
D^\star)$. 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 $h_k=(\mu^\star
h_A^{-1})^{1/m}$ on $\cO_{P_kV}(-1)$ which is degenerate on the support of
the zero divisor $\div(\mu)=\bfb\cdot D^\star$. Hence
$\Sigma_{h_k}=\Supp(\bfb\cdot D^\star)\subset P_kV^\sing$ and
$$
\Theta_{h_k^{-1}}(\cO_{P_kV}(1))={1\over m}\Theta_{h_A}(A)+{1\over m}
[\bfb\cdot D^\star]\ge{1\over m}\Theta_{h_A}(A)>0.
$$
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
$H^0(P_kV,\cO_{P_kV}(m)\otimes\pi_{0,k}^\star L^{-1})$ 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 6.8~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
$$
\Theta_{h_k^{\prime\,-1}}(\cO_{P_kV}(1))\ge{1\over m}\pi_{0,k}^\star\Theta(L).
$$
By (6.15) and 6.16~iii), there exists $\bfb\in\bQ^k_+$ such
that $\cO_{P_kV}(1)\otimes\cO_{P_kV}(-\bfb\cdot D^\star)$ is relatively ample
over~$X$. Hence $A=\cO_{P_kV}(1)\otimes\cO_{P_kV}(-\bfb\cdot D^\star)\otimes
\pi_{0,k}^\star L^{\otimes p}$ 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 $\Sigma_{h_k''}=\Supp(\bfb\cdot D^\star)=P_kV^\sing$ and
$$
\Theta_{h_k^{\prime\prime\,-1}}(\cO_{P_kV}(1))=\Theta(A)+[\bfb\cdot D^\star]-
p\,\pi_{0,k}^\star\Theta(L),
$$
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 $E_{k,m}V^\star$ is ample, there is an ample
line bundle $L$ and a sufficiently high symmetric power such that
$S^p(E_{k,m}V^\star)\otimes L^{-1}$ is generated by sections. These
sections can be viewed as sections of $\cO_{P_kV}(mp)\otimes\pi_{0,k}^\star
L^{-1}$ over $P_kV$, and their base locus is exactly $Z=P_kV^\sing$ by
6.8~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}
\endproclaim

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.

\proclaim{7.8.~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.
\endproclaim

\demo{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~7.8. 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 (5.4), $(\pi_k)_\star$ 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
$$
|(\pi_k)_\star\xi|_{h_k}^2\le C|\xi|_{\omega_k}^2,\qquad\forall\xi\in V_k.
$$
Combining these inequalities, we find
$$
\langle\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\rangle(\xi)\ge
{\varepsilon\over C}|(\pi_k)_\star\xi|_{h_k}^2,\qquad\forall\xi\in V_k.
$$
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 (5.6)
$$
F=f_{[k-1]}':T_{\Delta_R}\to f_{[k]}^\star\cO_{P_kV}(-1)
$$
to obtain a pullback metric
$$
\gamma=\gamma_0(t)\,dt\otimes d\ol t=F^\star h_k\qquad
\hbox{on $T_{\Delta_R}$}.
$$
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
$$
{i\,\ddbar\log\gamma_0(t)\over\gamma(t)}
={-2\pi\,(f_{[k]})^\star\Theta_{h_k}(\cO_{P_kV}(-1))\over F^\star h_k}
={\langle\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\rangle(f_{[k]}'(t))\over
|f_{[k-1]}'(t)|_{h_k}^2}\ge {\varepsilon\over C},
$$
since $f_{[k-1]}'(t)=(\pi_k)_\star f_{[k]}'(t)$. The Ahlfors-Schwarz lemma
3.2 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
\enddemo

Combining Theorem~7.8 with 7.7~ii) and iii), we get the following
consequences.

\proclaim{7.9.~Corollary}Assume that there exist integers $k,m>0$ and an
ample line bundle $L$ on $X$ such that $H^0(P_kV,\cO_{P_kV}(m)\otimes
\pi_{0,k}^\star L^{-1})\simeq H^0(X,E_{k,m}(V^\star)\otimes L^{-1})$
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
\endproclaim

\proclaim{7.10.~Corollary}Let $(X,V)$ be a compact complex directed
manifold. If $E_{k,m}V^\star$ is ample for some positive integers
$k,m$, then $(X,V)$ is hyperbolic.\square
\endproclaim

\proclaim{7.11.~Remark}{\rm Green and Griffiths [GrGr80] stated that Corollary
7.9 is even true with sections $\sigma_j\in H^0(X,E^\GG_{k,m}(V^\star)
\otimes L^{-1})$, in the special case $V=T_X$ they consider. We refer
to the recent preprint [SiYe96c] 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 Brody's
theorem, this is enough if one is merely interested in proving
hyperbolicity, thus Corollary 7.10 will be valid with $E^\GG_{k,m}V^\star$
in place of $E_{k,m}V^\star$). 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 $f^\star
L^{-1}$. 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}
\endproclaim

The above results justify the following definition and problems.

\proclaim{7.12.~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
$\Sigma_{h_k}\subset P_kV^\sing$.
\endproclaim

\proclaim{7.13.~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.
\endproclaim

This is probably a hard problem. In fact, we will see in the next
section that the smallest admissible integer $k$ must depend on the
geometry of $X$ and need not be uniformly bounded as soon as $\dim X\ge
2$ (even in the absolute case~$V=T_X$). On the other hand, if $(X,V)$ is
hyperbolic, we get for each integer $k\ge 1$ a generalized
Kobayashi-Royden metric $\bfk_{(P_{k-1}V,V_{k-1})}$ on $V_{k-1}$ (see
Definition~1.3), 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 $\pi_k^\star h_k\le h_{k+1}$ 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.

\proclaim{7.14.~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 $+\infty$.
\endproclaim

\head 8. Algebraic Criterion for the Negativity of Jet Curvature
\writetoaux\endhead

Our goal is to show that the negativity of $k$-jet curvature
implies strong restrictions of an algebraic nature, similar to
property 2.1~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.

\proclaim{8.1.~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 $\nu$.
\endproclaim

\demo{Proof} By (5.6), we get a lifting $\nu_{[k]}:\ol C\to P_kV$ of the
normalization map~$\nu$, and there is a canonical map
$$
\nu_{[k-1]}':T_{\ol C}\to\nu_{[k]}^\star\cO_{P_kV}(-1).
$$
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
$$
T_{\ol C}\simeq\nu_{[k]}^\star\cO_{P_kV}(-1)\otimes\cO_{\ol C}
\Big(-\sum(m_j-1)p_j\Big).
$$
Taking any $k$-jet metric $h_k$ with negative jet curvature on
$\cO_{P_kV}(-1)$, the Gauss-Bonnet formula yields
$$
2g(\ol C)-2=\int_{\ol C}\Theta(T_{\ol C}^\star)=
\sum(m_j-1)+\int_{\ol C}\nu_{[k]}^\star\Theta_{h_k^{-1}}(\cO_{P_kV}(1)).
$$
Now, the curvature hypothesis implies
$$
\langle\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\rangle(\xi)\ge\varepsilon'
|\xi|_{\omega_k}^2\ge\varepsilon''|(\pi_{0,k})_\star\xi|_\omega^2
\qquad\forall\xi\in V_k,
$$
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 $\nu_{[k]}^\star\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\ge\nu^\star\omega$,
hence
$$
\int_{\ol C}\nu_{[k]}^\star\Theta_{h_k^{-1}}(\cO_{P_kV}(1))\ge
{\varepsilon''\over 2\pi}\int_{\ol C}\nu^\star\omega=
\varepsilon\,\deg_\omega(C)
$$
with $\varepsilon=\varepsilon''/2\pi$. Theorem~8.1 follows.\square
\enddemo

\proclaim{8.2.~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.
\endproclaim

\demo{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~8.1. 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
\enddemo

If we make an assumption on the total jet curvature (as is the case
with the algebraic sufficient conditions~7.7), Theorem~8.1 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.

\proclaim{8.3.~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
$$
\Theta_h(L)\ge\alpha,\qquad \alpha_{\restriction V} \ge 
\delta\omega_{\restriction V}
$$
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
$$
\max\big(L\cdot C,\dev^2_\omega(C/V)\big)\ge \varepsilon\,\deg_\omega(C).
$$
\endproclaim

\demo{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 $V^\star\otimes(V^\perp)^\star$ in a hermitian form on~$V$).
In particular, we find
$$
\Theta_h(L)+M\,\omega^{V^\perp} \ge {\delta\over 2}\omega
$$
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
\enddemo

\proclaim{8.4.~Proposition}Let $(X,V)$ be a compact complex directed 
manifold. Assume that there are integers $k,m>0$, $\bfb\in\bN^k$
such that \hbox{$\cO_{P_kV}(m)\otimes\cO_{P_kV}(-\bfb\cdot D^\star)$}
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
$$
\max\big(-\chi(\ol C)-\sum_{t\in\ol C}(m_{k-1}(t)-1),
\dev^\infty_\omega(C/V)\big)\ge\varepsilon\,\deg_\omega(C).
$$
\endproclaim

Proposition 8.4 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).

\demo{Proof} The hyperbolicity of $(X,V)$ follows from 7.7~i) and Theorem~7.8.
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
$$
\eqalign{
&\cO_{P_kT_X}(1)_{\restriction P_kV}=\cO_{P_kV}(1),\cr
&\cO_{P_kT_X}(m)\otimes\cO_{P_kT_X}(-\bfb\cdot D^\star)_{\restriction P_kV}=
\cO_{P_kV}(m)\otimes\cO_{P_kV}(-\bfb\cdot D^\star)\cr}
$$
By our assumptions, $\cO_{P_kT_X}(m)\otimes\cO_{P_kT_X}(-\bfb\cdot D^\star)$
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 
$\cO_{P_kT_X}(m)\otimes\cO_{P_kT_X}(-\bfb\cdot D^\star)$ such that the
curvature form is positive definite on a neighborhood $U$
of $P_kV$ and satisfies
$$
\Theta\big(\cO_{P_kT_X}(m)\otimes\cO_{P_kT_X}(-\bfb\cdot D^\star)\big)\ge
-C\pi_{k,0}^\star\omega
$$
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
$$
\cases
\Theta_{h_k}(\cO_{P_kT_X}(1))\ge-C\pi_{k,0}^\star\omega&
\text{on $P_kT_X$,}\cr
\Theta_{h_k}(\cO_{P_kT_X}(1))\ge\delta\omega_k
\ge\delta'\pi_{k,0}^\star\omega\phantom{\Big)}&
\text{on $U\supset P_kV$,}\cr
\endcases
\leqno(8.5)
$$
where $\omega_k$ is a hermitian metric on $P_kT_X$ and $\delta,\,\delta'>0$.
Now, assume that the conclusion of Prop.~8.4 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 
$\dev^\infty_\omega(C_\ell/V)\big)=\sup \nu_\ell^\star(\omega_{V^\perp})/
d\wt\sigma$ where $d\sigma$ is the Poincar\'e metric and $d\wt\sigma$ the 
associated normalized metric, the second condition means
$$
\sup\|\pr_{V^\perp}\nu_\ell'\|_{\sigma,\omega}^2
=\sup{\nu_\ell^\star(\omega_{V^\perp})\over d\sigma}
\le{\varepsilon_\ell\deg_\omega(C_\ell)\over\int_{\ol C\ell}d\sigma}
=\varepsilon_\ell{\int_{\ol C\ell}\nu_\ell^\star\omega\over
\int_{\ol C\ell}d\sigma}.
$$
In addition to this, we have
$$
{\int_{\ol C\ell}\nu_\ell^\star\omega\over\int_{\ol C\ell}d\sigma}\le
R_\ell^2:=\sup\|\nu'_\ell\|_{\sigma,\omega}^2
$$
and $R=\sup R_\ell<+\infty$, otherwise the proof of Prop.~2.9 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
$$
\sup_{1\le j\le k}\|\pr_{V^\perp}\nu_\ell^{(j)}\|_{\sigma,\omega}^2
\le M_k\sup\|\pr_{V^\perp}\nu_\ell'\|_{\sigma,\omega}^2
\le M_k\varepsilon_\ell{\int_{\ol C\ell}\nu_\ell^\star\omega\over
\int_{\ol C\ell}d\sigma}.
$$
As $\int_{\ol C_\ell}\|\nu_\ell'\|_{\sigma,\omega}^{-2}\nu_\ell^\star\omega
=\int_{\ol C_\ell}d\sigma$, we infer
$$
\int_{\ol C_\ell}{\sup_{1\le j\le k}\|\pr_{V^\perp}
\nu_\ell^{(j)}\|_{\sigma,\omega}^2\over\|\nu_\ell'\|_{\sigma,\omega}^2}
\nu_\ell^\star\omega
\le M_k\varepsilon_\ell\int_{\ol C\ell}\nu_\ell^\star\omega.
\leqno(8.6)
$$
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 (8.6), 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 $\nu_\ell^\star\omega$. By (8.5), we then get
$$
\eqalign{
\cO_{P_kT_X}(1)\cdot C_{\ell,[k]}&=\int_{\ol C_\ell\ssm S_\eta}
\nu_{\ell,[k]}^\star\Theta(\cO_{P_kT_X}(1))
+\int_{S_\eta}\nu_{\ell,[k]}^\star\Theta(\cO_{P_kT_X}(1))\cr
&\ge\delta'\int_{\ol C_\ell\ssm S_\eta}\nu_\ell^\star\omega-
C\int_{S_\eta}\nu_\ell^\star\omega\cr
&=\big(\delta'(1-M_k\varepsilon_\ell/\eta)-CM_k\varepsilon_\ell/\eta\big)
\deg_\omega(C_\ell).\cr}
$$
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
\enddemo

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 $C\subset X$).

\proclaim{8.7.~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
$\Theta_h(L)_{\restr V}$ 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
$\Theta_h(L)_{\restr V}$ 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$)$ curva\-ture$\,$?
\vskip0pt
\endproclaim

The answer to 8.7~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~$V$.) 

\head 9. Proof of the Bloch Theorem
\writetoaux\endhead

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~7.8), 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).

\proclaim{9.1.~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.
\endproclaim

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

\demo{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$\,$5. 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
$\cO_{Z_k}(\bfa)=\pr_2^\star\cO_{\bR_{n,k}}(\bfa)$. 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
$\cO_{X_k}(\bfa)=\Phi_k^\star\cO_{\bR_{n,k}}(\bfa)$.

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~7.2~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 7.8 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
\enddemo

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).

\proclaim{9.2.~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.
\endproclaim

\proclaim{9.3.~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.
\endproclaim

\proclaim{9.4.~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.
\endproclaim

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~$X$).

\demo{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
\enddemo

\head 10. Logarithmic Jet Bundles and a Conjecture of Lang
\writetoaux\endhead

We want to report here briefly about an important question raised by 
S.~Lang, namely {\it whether the complement of an ample divisor in
an Abelian variety is Koba\-yashi 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 Bloch's theorem. We
will adopt here a slightly different approach of G.~Dethloff and
S.~Lu [DLu96], 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 [DLu96] for details, and to [SiYe96b], [Nog96a] 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 
$\cO(V^\star\langle D\rangle)$ 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
$V^o\subset V^\star$ 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
$$
\cO(T_X\langle D\rangle):=(\Omega^1_X\langle D\rangle)^\star,\qquad
\cO(V\langle D\rangle):=(V^\star\langle D\rangle)^\star.
\leqno(10.1)
$$
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
$$
(X_k,D_k,V_k)
\leqno(10.2)
$$
of logarithmic $k$-jet bundles exactly in the same way as we proceeded in
section \S$\,$4 and \S$\,$5: 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 
$\cO_{X_k}(-1)\subset\pi_k^\star V_{k-1}$. In this case, the direct 
image formula given in Theorem~6.8 reads
$$
(\pi_{k,0})_\star\cO_{X_k}(m)=\cO(E_{k,m}V^\star\langle D\rangle),
\leqno(10.3)
$$
where $\cO(E_{k,m}V^\star\langle D\rangle)$ 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 [DLu96] state 
the following results 10.4--10.9, which extend our results of sections 
\S$\,$7 and \S$\,$9 (most of these results can already be derived
from [SiYe96b] as well).

\proclaim{10.4.~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 $f:\Delta^\star\to X\ssm D$
tangent to $V$ $($where $\Delta^\star$ is the punctured disk$)$,
one has: either $f$ extends to a holomorphic map $\ol f:\Delta\to X$ 
or $f_{[k]}(\Delta^\star)\subset\Sigma_{k,m}$.
\vskip0pt
\endproclaim

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.

\proclaim{10.5.~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
\endproclaim

\proclaim{10.6.~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.
\endproclaim

\proclaim{10.7.~Remark}{\rm Theorem 10.5 and its corollary have been
obtained independently by Noguchi [Nog96a], 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~10.5 can be extended to
arbitrary commutative (non necessarily algebraic) Lie groups~$\bC^n/\Gamma$.}
\endproclaim

\proclaim{10.8.~Theorem}The following properties hold true.
\smallskip
\item{\rm A)} Let $f:\Delta^\star\to Z$ 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
$\ol{f(\Delta^\star)}{}^\Zar$ is foliated by translates of~$Z'$.
\smallskip
\item{\rm B)} Let $f:\Delta^\star\to Z\ssm D$ 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)} $\ol{f(\Delta^\star)}{}^\Zar\cap D=\emptyset$.
\smallskip
\itemitem{\rm iii)} There exists an algebraic subgroup $Z''$ of $Z'$ of
positive dimension such that $\ol{f(\Delta^\star)}{}^\Zar\cap D$ is
foliated by translates of~$Z''$.
\smallskip
\item{\rm C)} Assume here that $Z$ is an abelian variety and let
$f:\Delta^\star\to Z\ssm D$ 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
\endproclaim

Part A) of Theorem 10.8 is due to Noguchi [Nog96a] (again with a proof
based on Nevanlinna theory).

\proclaim{10.9.~Corollary}If $Z$ is abelian and $D$ is ample, then every
holomorphic map $f:\Delta^\star\to Z\ssm D$ extends to a holomorphic map
$\ol f:\Delta\to Z$.
\endproclaim

\head 11. Projective Meromorphic Connections and Wronskians
\writetoaux\endhead

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 a recent 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 
$$
\Gamma=\sum_{1\le j,\lambda,\mu\le n}\Gamma^\lambda_{j\mu}\,dz_j\otimes
dz_\mu \otimes{\partial\over\partial z_\lambda}\in\cM(U,T^\star_X\otimes 
T^\star_X\otimes T_X).
$$
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
$$
f^{(k+1)}_\nabla(t)_\lambda={d\over dt}\big(f^{(k)}_\nabla(t)_\lambda\big)
+\sum_{1\le\mu\le n}(\Gamma^\lambda_{j\mu}\circ f)\,f'_j
\,f^{(k)}_\nabla(t)_\mu.
\leqno(11.1)
$$
Therefore, if $\Im f\not\subset|B|$, one can define the {\em Wronskian}
of $f$ relative to $\nabla$ as
$$
W_\nabla(f)=f'\wedge f''_\nabla\wedge\cdots\wedge f^{(n)}_\nabla.
\leqno(11.2)
$$
Clearly, $W_\nabla(f)$ is a meromorphic section of $f^\star(\Lambda^nT_X)$.
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 $f^\star(\Lambda^nT_X\otimes\cO_X({1\over 2} n(n-1)B)$.
From (11.1) and (11.2) 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
$$
\beta^{n(n-1)/2}W_\nabla\in
H^0(X,E_{n,n(n+1)/2}T^\star_X\otimes L^{-1}),
\leqno(11.3)
$$
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).

\proclaim{11.4.~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 $W_\nabla(f)\equiv 0$.
\endproclaim

\demo{Proof} By Corollary~7.9 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
\enddemo

\proclaim{11.5.~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 \hbox{$T^\star_X\otimes T^\star_X\otimes T_X$}. 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
$$
f^{(k)}_{\wt\nabla}(t)=f^{(k)}_\nabla(t)+\sum_{1\le j<k}\gamma_j(t)\,
f^{(j)}_\nabla(t).
\eqno\square
$$}
\endproclaim

The essential consequence of Remark~11.5 is that we need only have a
``partial projective connection'' $\nabla$ on $X$, in the following
sense.

\proclaim{11.6.~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
$U_j\cap U_k$.
\endproclaim

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
11.4 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
$$
\nabla=\pi_\star\circ(\sigma^\star\wt\nabla),
\leqno(11.7)
$$
where $\sigma^\star\wt\nabla$ is the induced connection on $\sigma^\star 
T_W$ and $\pi_\star:T_W\to \pi^\star T_X$ 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.

\proclaim{11.8.~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 $(11.7)$ yields a partial projective connection $\nabla$
which is globally defined on~$X$ and independent of the choice of the
local sections~$\sigma$.
\endproclaim

\demo{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 (11.7) says more explicitly that
$$
\nabla_wv=\pi_\star\big((\sigma^\star\wt\nabla)_wv\big)=
\pi_\star\big(d_{\sigma_\star w}\sigma_\star v+(\wt\Gamma\circ\sigma)
\cdot(\sigma_\star w,\sigma_\star v)\big).
$$
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 $\sigma_\star v=v+dh(v)$,
we get
$$
\eqalign{(\sigma^\star\wt\nabla)_wv
&=d_{w+dh(w)}(v+dh(v))+\wt\Gamma(x,h(x))\cdot\big(w+dh(w),v+dh(v)\big)\cr
&=d_wv+d^2h(w,v)+\Gamma(x)\cdot(w+dh(w),v+dh(v)).\cr}
$$
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 $\pi_\star$.
We thus obtain
$$
\nabla_wv=\pi_\star\big((\sigma^\star\wt\nabla)_wv\big)=
d_wv+\Gamma(x)\cdot(w,v)+\alpha(dh(w))v+\beta(dh(v))w.
$$
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
\enddemo

\proclaim{11.9.~Remark}{\rm Lemma~11.8 is also valid when $\swt\nabla|$ is a
partial projective connection. Hypothesis 11.8~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}
\endproclaim

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

\proclaim{11.10.~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
\endproclaim

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
$$
d^2s\cdot(w,v)+ds\cdot(d_wv)=0
\leqno(11.11)
$$
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 (11.11), 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

\proclaim{11.12.~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
$$
\nabla^\star(ds)=d^2s-ds\circ\Gamma=a\otimes ds+ds\otimes b+s\,c
$$
in a neighborhood of every point of~$Y$ $($here $\nabla^\star$ is the
induced connection on $T^\star_Y)$.
\endproclaim

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

\proclaim{11.13.~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 $f(D(0,R))\subset Y$.
\endproclaim

\demo{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 11.12 yield
$$
\eqalign{
{d\over dt}\big(ds_{f(t)}\cdot f^{(j)}_\nabla(t)\big)
&=ds_{f(t)}\cdot\Big({\nabla\over dt}f^{(j)}_\nabla(t)\Big)+
\nabla^\star(ds)_{f(t)}\cdot\big(f'(t),f^{(j)}_\nabla(t)\big)\cr
&=ds\cdot f^{(j+1)}_\nabla(t)+(a\cdot f'(t))\big(ds\cdot
f^{(j)}_\nabla(t)\big)\cr
&\qquad{}+(ds\cdot f'(t))\big(b\cdot
f^{(j)}_\nabla(t)\big)+(s\circ f(t))\big(c\cdot(f'(t),f^{(j)}_\nabla(t))
\big),\cr}
$$
as desired.\square
\enddemo

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 11.12
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.

\proclaim{11.14.~Definition}For any $(n+2)$-tuple of integers 
$(p,k_0,k_1\ld k_n)$ with \hbox{$0<k_j<p/2$}, 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 $p$ on $\bC^{n+1}$ 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 $z_j^{p-k_j}$.
\endproclaim

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
$$
Y_\alpha=\big\{\alpha_0s_0+\alpha_1s_1+\cdots+\alpha_ns_n=0\big\},
\quad \alpha=(\alpha_0,\ldots,\alpha_n)\in\bC^n.
\leqno(11.15)
$$
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 11.10 shows that it is
enough to construct a meromorphic connection
$\swt\nabla|=d+\swt\Gamma|$ on $\bC^{n+1}$ satisfying 11.10~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 11.12 yields a sufficient
condition in terms of the linear system of equations
$$
\sum_{0\le\lambda\le n}
\swt\Gamma|^\lambda_{j\mu}{\partial s_\kappa\over\partial z_\lambda}=
{\partial^2 s_\kappa\over\partial z_j\partial z_\mu},\qquad
0\le j,\kappa,\mu\le n.
\leqno(11.16)
$$
(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
$$
\delta:=\det\Big({\partial s_\kappa\over\partial z_\lambda}\Big)_{
0\le\kappa,\lambda\le n}\not\equiv 0
\leqno(11.17)
$$
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 11.10~i)). Moreover, 
$\swt\nabla|$ is symmetric, for $\partial^2s/\partial z_j\partial z_\mu$ 
is symmetric in $j,\mu$. Finally, if we multiply (11.16) 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 11.10~ii) is satisfied. From this we infer

\proclaim{11.18.~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 $(11.16)$ 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~$n+1+\sum k_j$.
\endproclaim

\demo{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 index $\lambda$ in $\det(\partial s_\kappa/\partial
z_\lambda)_{0\le\kappa,\lambda\le n}$ 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
\enddemo

An application of Theorem 11.4 then yields the following theorem on certain
complete intersections in projective spaces.

\proclaim{11.19.~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 $(11.16)$, 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~$Z$.
\endproclaim

\demo{Proof} By Proposition 11.18, 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 (11.3), the differential operator $\beta^{n(n-1)/2}(f)\,W_{Z,\nabla}(f)$ 
defines a section in\break 
$H^0(Z,E_{n,n(n+1)/2}T^\star_Z\otimes L^{-1})$ 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 11.4 that 
$W_{Z,\nabla}(f)\equiv 0$. Fix a point \hbox{$t_0\in\bC$} 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
\hbox{$\delta=\det(\partial s_\kappa/\partial s_\lambda)$} 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~11.13 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
\enddemo

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.

\proclaim{11.20.~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 11.19 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
$$
\sum_{j\in J}\varepsilon_j^{\textstyle{p\over p-2}}\ne 
{2\over p-2}\Big(-{p\over 2}\Big)^{\textstyle{p\over p-2}},\qquad
\forall J\subset\{0,1,2\},
\leqno(11.21)
$$
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
$$
\eqalign{
&\left\vert
{\matrix
pz_0^{p-1}     &      0     &     0       &     0     \cr
   0        & pz_1^{p-1}    &     0       &     0     \cr
   0        &     0     &    pz_2^{p-1}   &     0     \cr
2\varepsilon_0z_0z_3^{p-2} & 
2\varepsilon_1z_1z_3^{p-2} & 
2\varepsilon_2z_2z_3^{p-2} & 
(p-2)z_3^{p-3}\big(\varepsilon_0z_0^2+\varepsilon_1z_1^2+\varepsilon_2z_2^2+
{p\over p-2}z_3^2\big)\cr
\endmatrix}
\right\vert\cr
&\quad{}=p^3(p-2)z_0^{p-1}z_1^{p-1}z_2^{p-1}z_3^{p-3}
\Big(\varepsilon_0z_0^2+\varepsilon_1z_1^2+\varepsilon_2z_2^2+
{p\over p-2}z_3^2\Big)\not\equiv 0.\cr}
$$
The numerator of $\swt\Gamma|^\lambda_{j\mu}$ is obtained by replacing
the column of index $\lambda$ of $\delta$ by
$(\partial^2s_\kappa/\partial z_j\partial z_\mu)_{0\le\kappa\le 3}$,
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~11.18), 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
$$
\left\{\eqalign{
&\hbox{none of the pairs $(\varepsilon_i,\varepsilon_j)$ is equal to 
$(0,0)$,}\cr
&\hbox{$\varepsilon_i/\varepsilon_j\ne-\theta^2$ whenever $\theta$ is a
root of $\theta^p=-1$.}\cr}\right.
\leqno(11.22)
$$
[(11.22) excludes the existence of lines in the intersections 
$Z\cap Y_\alpha$.]}
\endproclaim

\proclaim{11.23.~Corollary}Under conditions $(11.21)$ and $(11.22)$, 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
\endproclaim

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.~11.23
and exchange the roles of $z_2,\,z_3$, we get the following

\proclaim{11.24.~Corollary}Consider the plane curve
$$
C=\big\{z_0^p+z_1^p+z_2^{p-2}
(\varepsilon_0z_0^2+\varepsilon_1z_1^2+z_2^2)=0\big\}
\subset\bP^2,\qquad\varepsilon_0,\,\varepsilon_1\in\bC^\star.
$$
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
\endproclaim

\head 12. Decomposition of Jets in Irreducible Representations
\writetoaux\endhead

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 $(\bC^\star)^r\subset\GL(V))$. The Schur fonctors
can be defined in an elementary way as follows. Let
$\bU_r=\Big\{\pmatrix 1&0\cr \star&1\cr\endpmatrix\Big\}$ 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 $(V^\star)^r$ which
are homogeneous of degree $a_j$ with respect to $\xi_j$ and which are
invariant under the left action of $\bU_r$ on $(V^\star)^r=\Hom(V,\bC^r)$,
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
$$
\leqalignno{S^kV&=\Gamma^{(k,0\ld 0)}V,&\cr
        \Lambda^kV&=\Gamma^{(1\ld 1,0\ld 0)}V,
\qquad\hbox{(with $k$ indices $1$)}&(12.1)\cr
        \det V&=\Gamma^{(1\ld 1)}V.\cr}
$$
The Schur fonctors satisfy the well-known formula
$$
\Gamma^{(a_1+\ell\ld a_r+\ell)}V=\Gamma^{(a_1\ld a_r)}V\otimes(\det V)^\ell.
\leqno(12.2)
$$
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 \S6, 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
$$
\bG'_k\lhra\bU_k,\qquad\varphi\longmapsto
\pmatrix 1&0&0&\cdots&0&0\cr
         2b_2&1&0&\cdots&0&0\cr
         3b_3&3b_2&1&\cdots&0&0\cr
         \vdots&&&&\vdots&\vdots\cr
         &&\cdots&&1&0\cr
         kb_k&&\cdots&&kb_2&1\cr
\endpmatrix.
$$
By formula (6.5), we find that the graded bundle of $E_{k,m}V^\star$ is
$$
\Gr^\bullet E_{k,m}V^\star
=\left(\bigoplus_{\ell\in\bN^k,\,\ell_1+2\ell_2+\cdots+k\ell_k=m}
S^{\ell_1}V^\star\otimes S^{\ell_2}V^\star\otimes
\cdots\otimes S^{\ell_k}V^\star\right)^{\bG'_k}.
$$
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
$$
E_{1,m}V^\star=E^\GG_{1,m}V^\star=S^mV^\star.\leqno(12.3)
$$
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 \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 $S^{\ell_1}V^\star
\otimes S^{\ell_2}V^\star$ do not mix up and we get
$$
\Gr^\bullet E_{2,m}V^\star=\bigoplus_{\ell_1+2\ell_2=m}
\big(S^{\ell_1}V^\star\otimes S^{\ell_2}V^\star\big)^{\bG'_k}
=\bigoplus_{\ell_1+2\ell_2=m}\Gamma^{(\ell_1,\ell_2,0\ld0)}V^\star.
\leqno(12.4)
$$
In the special case when $r=\rank V=2$, (12.1) and (12.2) yield
$\Gamma^{(\ell_1,\ell_2)}V^\star=S^{\ell_1-\ell_2}V^\star
\otimes(\det V^\star)^{\ell_2}$. Hence we get the simpler formula
$$
\Gr^\bullet E_{2,m}V^\star=\bigoplus_{0\le j\le m/3}
S^{m-3j}V^\star\otimes(\det V^\star)^j
\qquad\quad(k=r=2).
\leqno(12.5)
$$
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
$\Gr^\bullet E_{k,m}V^\star$ for arbitrary $k$ and~$m$.

\proclaim{12.6.~Special case}{\rm Assume that $X$ is a surface and consider
the absolute case $V=T_X$. We find
$$
\Gr^\bullet E_{2,m}T^\star_X=\bigoplus_{0\le j\le m/3}
S^{m-3j}T^\star_X\otimes K_X^j,
$$
where $E_{1,m}T^\star_X=S^mT^\star_X$ is a subbundle of $E_{2,m}
T^\star_X$. We thus get an exact sequence
$$
0\to S^mT^\star_X\to E_{2,m}T^\star_X\to Q_m\to 0,
$$
and $Q_m$ admits a filtration with
$$
\Gr^\bu Q_m=\bigoplus_{1\le j\le m/3}S^{m-3j}T^\star_X\otimes K_X^j.
$$
The simplest case is $m=3$, which yields the interesting exact sequence
$$
0\to S^3T^\star_X\to E_{2,3}T^\star_X\to K_X\to 0.
$$}
\endproclaim

\proclaim{12.7.~Complement}{\rm Assume that $X$ is a surface of degree $d$
in $\bP^3$. Then $K_X=\cO_X(d-4)$. As $T_X^\star$ is a quotient bundle 
of $T^\star_{\bP^3|X}$ and as $T^\star_{\bP^n|X}\otimes\cO(2)$ is 
generated by sections, we conclude that $S^{m-3j}T^\star_X\otimes K_X^j$ 
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 $E_{2,m}T^\star_X$, namely
$S^mT^\star_X$, which need not be ample when the degree $d$ is large. 
By contrast, the Green-Griffiths graded bundle 
$$
G^\bu E^\GG_{2,m}T^\star_X=\sum_{\ell_1+2\ell_2=m}S^{\ell_1}T^\star_X\otimes
S^{\ell_2}T^\star_X
$$
does not such exhibit such strong positivity properties. This is one of
the main reasons for which we believe that the invariant jet bundles
$E_{k,m}T^\star_X$ are more appropriate to the study of hyperbolicity
questions.}
\endproclaim

\head 13. Riemann-Roch Calculations and Study of the Base Locus
\writetoaux\endhead

In view of the Green-Griffiths conjecture 3.7 concerning algebraic 
degeneration of entire curves, the main point is to compute the base loci
$$
B_k=\bigcap_{m>0}
\Bs\big(H^0(X_k,\cO_{X_k}(m)\otimes\pi_{k,0}^\star\cO(-A))\big)\subset X_k
\leqno(13.1)
$$
where $X_k=P_kT_X$ and $A$ is an ample divisor over~$X$. By
corollary 7.9, 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 (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 
$E_{1,m}T^\star_X=S^mT^\star_X$, namely symmetric differentials. 
The Riemann-Roch formula then gives
$$
\chi(X,S^mT^\star_X\otimes \cO(-A))={m^3\over 6}(c_1^2-c_2)+O(m^2),
\leqno(13.2)
$$
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 14.1 of the Appendix, we get
$h^2(X,S^mT^\star_X\otimes \cO(-A))=0$ for $m$ large, thus
$$
h^0(X,S^mT^\star_X\otimes \cO(-A))\ge{m^3\over 6}(c_1^2-c_2)-O(m^2).
\leqno(13.3)
$$
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].

\proclaim{13.4. 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~$\cV$. 
\endproclaim

\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].

\proclaim{13.5. Theorem {\rm (Bogomolov)}}On a surface $X$ of general type 
such that \hbox{$c_1^2>c_2$}, there are only finitely many rational or
elliptic curves.
\endproclaim

\demo{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 
$$
H^0(P(T_X),\cO_{P(T_X)}(m)\otimes\pi_{1,0}^\star\cO(-A)).
$$ 
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
\enddemo

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 $E=T_X^\star$
is due to Miyaoka [Miy82]).

\proclaim{13.6. 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.
\endproclaim

\demo{Proof (sketch)} Let $P=P(E^\star)$ be the hyperplane bundle
and \hbox{$H=\cO_P(1)$}. 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 
$h^0(X,S^mE^\star)$ 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 $D=kH-\pi^\star F$ 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
$$
\cO(F)\hookrightarrow \pi_\star\cO(kH)=\cO(S^kE),
$$
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
$$
\eqalign{
H^2\cdot D=H^2\cdot(kH-\pi^\star F)
&=k(c_1(E)^2-c_2(E))-c_1(E)\cdot c_1(F)\cr
&\ge {k\over 2}\big(c_1(E)^2-2\,c_2(E)\big)>0,\cr}
$$
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
and the claim follows by 7.2~iv).\square
\enddemo

\proclaim{13.7. 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.
\endproclaim

\demo{Proof} One may assume that $X$ is minimal, i.e.\ that $K_X$ is nef (and big).
By the work of Bogomolov [Bog79], $T_X^\star$ is semi-stable. The result
of Schneider and Tancredi now implies that $T_X^\star$ is almost ample.
Theorem 7.8 concludes the proof.\square
\enddemo

We now turn ourselves to the case of jet differentials of degree~$2$.
A simple Riemann-Roch computation based on Formula 12.6 shows that
$$
\chi(X,E_{2,m}T^\star_X)=\chi(X,\Gr^\bu E_{2,m}T^\star_X)
={m^4\over 648}(13\,c_1^2-9\,c_2)+O(m^3)
\leqno(13.8)
$$
where $c_1,\,c_2$ are the Chern classes of $X$ (only the terms of
bidegree $(2,2)$ in $\Ch(\Gr^\bu E_{2,m}T^\star_X)$ play a role). 
This formula should be put in perspective with the one obtained
by Green and Griffiths [GrGr80] for the jet bundles $E_{k,m}^\GG T^\star_X$.
In the case of surfaces, they obtain
$$
\chi(X,E_{k,m}^\GG T^\star_X)={m^{2k+1}\over (k!)^2(2k+1)!}(\alpha_k\,
c_1^2-\beta_k\,c_2)+O(m^{2k}),
$$
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
$$
\chi(X,E_{2,m}^\GG T^\star_X)={m^5\over 384}(7\,c_1^2-5\,c_2)+O(m^4).
$$
This is weaker than formula (13.8) in the sense that the ratio $5/7$ is
larger than $9/13$. In general, we expect analogous estimates of the form
$$
\chi(X,E_{k,m}T^\star_X)\sim m^{k+2}(\gamma_k\,c_1^2-\delta_k\,c_2)+O(m^{k+1})
$$
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.

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
$$
\chi(X,E_{2,m}T^\star_X)={m^4\over 648}\,d(4\,d^2-68\,d+154)+O(m^3).
$$
This estimate is especially useful in combination with vanishing theorems for 
holomorphic tensor fields (see Theorem 14.1 in the Appendix).

\proclaim{13.9.~Corollary}If $X$ is an algebraic surface of general type and
$A$ an ample line bundle over~$X$, then
$$
h^0(X,E_{2,m}T^\star_X\otimes\cO(-A))\ge
{m^4\over 648}(13\,c_1^2-9\,c_2)-O(m^3).
$$
In particular, every smooth surface $X\subset\bP^3$ of degree $d\ge 15$
admits non trivial sections of $E_{2,m}T^\star_X\otimes\cO(-A)$ for $m$
large, and every entire curve $f:\bC\to X$ must satisfy the corresponding
algebraic differential equations.
\endproclaim

\demo{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 (13.8) if we check that $h^2(X,E_{2,m}T^\star_X\otimes\cO(-A))=0$
for $m$ large. However
$$
H^2(X,E_{2,m}T^\star_X\otimes\cO(-A))=
H^0(X,K_X\otimes (E_{2,m}T^\star_X)^\star\otimes\cO(A))
$$
by Serre duality. Since $K_X\otimes (E_{2,m}T^\star_X)^\star\otimes\cO(A)$
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 
14.1, using the fact that $K_X^{\otimes\nu}\otimes\cO(-A)$ is big for
$\nu\ge\nu_0$ large enough.\square
\enddemo

\proclaim{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
$$
u_j^r+c_1^{[j-1]}u_j^{r-1}+\cdots+c_{r-1}^{[j-1]}u_j+c_r^{[j-1]}=0,\qquad
1\le j\le k
\leqno(13.10)
$$
(we omit all pull-backs $\pi_j^\star$ for simplicity of notation). Moreover,
the exact sequences (5.4) and $(5.4')$ yield the inductive formula
$$
\eqalign{
&c_\bu^{[k]}=c_\bu(\cO_{X_k}(-1))\,c_\bu(T_{X_k/X_{k-1}})
 =(1-u_k)\,c_\bu(T_{X_k/X_{k-1}}),\cr
&c_\bu(T_{X_k/X_{k-1}})=c_\bu(\pi_k^\star V_{k-1}\otimes\cO_{X_k}(1))
 =\sum_{0\le j\le r}c_j^{[k-1]}(1+u_k)^{r-j},\cr}
$$
in other words
$$
c_\bu^{[k]}=(1-u_k)\sum_{0\le j\le r}c_j^{[k-1]}(1+u_k)^{r-j}.
\leqno(13.11)
$$
In particular, if $r=\rank V=2$, we find
$$
\leqalignno{
&u_k^2+c_1^{[k-1]}u_1+c_2^{[k-1]}=0,&(13.12)\cr
&c_1^{[k]}=c_1^{[k-1]}+u_k,\qquad c_2^{[k]}=c_2^{[k-1]}-u_k^2,&\cr}
$$
hence
$$
c_1^{[k]}=c_1^{[0]}+u_1+\cdots+u_k,\qquad
c_2^{[k]}=c_2^{[0]}-u_1^2-\cdots-u_k^2.
\leqno(13.13)
$$
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
$$
\cO_{X_2}(2,1)_{\restriction F_2}
=\cO_{P(\cO(2)\oplus\cO(-1))}(1)\otimes\pi^\star\cO_{\bP^1}(2)
=\cO_{P(\cO\oplus\cO(-3))}(1).
$$
Its multiples have zero higher order direct images
$R^q(\pi_{2,0})_\star\cO_{X_2}(2m,m)$, $q\ge 1$, and lower order direct 
images
$$
(\pi_{2,0})_\star\cO_{X_2}(2m,m)=(\pi_{2,0})_\star\cO_{X_2}(3m)=E_{2,3m}
T_X^\star
$$
[either apply (6.16~i) or observe that 
$$
\eqalign{
&(\pi_{1,0})_\star\big(\cO_{P(\cO\oplus\cO(-3))}(m)\big)=
S^m\big(\cO\oplus\cO(3)\big),\cr
&(\pi_{1,0})_\star\big(\cO_{P(\cO(2)\oplus\cO(-1))}(3m)\big)=
S^{3m}\big(\cO(-2)\oplus\cO(1)\big)\cr}
$$
have the same sections over $\bP^1$]. By the Leray spectral sequence, we 
conclude that
$$
h^q(X_2,\cO_{X_2}(2m,m))=h^q(X,E_{2,3m}T_X^\star),\qquad 0\le q\le 2,
$$
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,
(13.12) and (13.13) 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].}
\endproclaim

\proclaim{13.14. 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.
\endproclaim

\demo{Proof (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
$Y=a_1u_1+a_2u_2-\pi^\star F$ 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 $\pi_\star(\cO_{X_2}(a_1,a_2))\subset E_{2,m}T^\star_X$ where $m=a_1+a_2$.
Thus there is a non trivial morphism $\cO(F)\hookrightarrow S^{m-3j}T_X^\star
\otimes K_X^j$ 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
$$
(2u_1+u_2)^2\cdot Y = (a_1+a_2)(13\,c_1^2-9\,c_2) - 12\,c_1\cdot F\ge
m(7\,c_1^2-9\,c_2).\eqno\square
$$
\enddemo

\proclaim{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
$E_{2,m}T_X^\star\langle D\rangle$ of logarithmic jet differentials 
of order $2$ and degree $m$ admits a filtration with
$$
\Gr^\bullet E_{2,m}T^\star_X\langle D\rangle=\bigoplus_{0\le j\le m/3}
S^{m-3j}(T^\star_X\langle D\rangle)\otimes \det(T_X^\star\langle D\rangle)^j.
\leqno(13.15)
$$
We thus get 
$$
\leqalignno{
h^0(X,E_{2,m}T^\star_X\langle D\rangle)
&\ge\chi(X,E_{2,m}T^\star_X\langle D\rangle)\cr
&\ge{m^4\over 648}\big(13\,c_1^2(T_X\langle D\rangle)
-9\,c_2(T_X\langle D\rangle)\big)-O(m^3).&(13.16)\cr}
$$
The exact sequence $0\to T_X\langle D\rangle\to T_X\to(i_D)_\star N_{X/D}
\to 0$
yields 
$$
c_\bu(T_X\langle D\rangle)=c_\bu(T_X)c_\bu((i_D)_\star N_{X/D})^{-1}
=(1+c_1+c_2)(1+\delta)^{-1},
$$ 
where $\delta=c_1(\cO_X(D))$ and $c_\bu((i_D)_\star N_{X/D})=
c_\bu(\cO_X(D))=1+\delta$, 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 $h^2(X,E_{2,m}T^\star_X
\langle D\rangle)$ 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
$$
h^0(X,E_{2,m}T^\star_{\bP^2}\langle D\rangle)\ge
{m^4\over 648}(4\,d^2-51\,d+90)-O(m^3).
$$
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~$d\ge 11$.}
\endproclaim

\head 14. Appendix: a Vanishing Theorem for Holomorphic Tensor Fields
\writetoaux\endhead

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 $\Lambda^kT^\star_X$, 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 14.1 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~$H^q$.

\proclaim{14.1.~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
$$
H^0(X,\Gamma^aT_X\otimes L^\star)=0.
$$
\endproclaim

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).

\demo{Proof of Theorem~14.1} 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 $\Gamma^aT^\star_X$
in terms of the basis $(dz)^I=dz_{i_1}\otimes\cdots\otimes dz_{i_p}$, resp.\
$(dz)^J\otimes(dz_1\wedge\cdots\wedge dz_n)^{-\ell}$.
\enddemo

\proclaim{14.2.~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 $T_X\to T^\star_X$, $v\mapsto i\ol v\ort\omega$, 
defined by contracting $(0,1)$-vectors with the K\"ahler
metric~$\omega$.  Denote also by $\#:\Gamma^aT_X\otimes
L\to\Gamma^aT^\star_X\otimes L^\star$ the induced $C^\infty$
isomorphism on the Schur tensor powers of $T_X$ and $T^\star_X$, combined
with the conjugate linear $($metric$)$ isomorphism $L\to L^\star$. 
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 $(L,h)$.
\endproclaim

\demo{Proof} We first make a pointwise calculation of
$\dbar^\star\dbar v$ and $\dbar^\star\dbar(\#\,v)$ 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 $\dbar^\star$ 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
$$
w=\sum_{|J|=q,\,1\le\lambda\le r}w_{J,\lambda}\,d\ol z_J\otimes e_\lambda,
\qquad
\dbar^\star w=-\sum_{|J|=q,\,\lambda,\,k}{\partial w_{J,\lambda}\over
\partial z_k}\,\Big({\partial\over\partial\ol z_k}\ort d\ol z_J\Big)
\otimes e_\lambda.
$$
We apply this to smooth sections $v=\sum v_I\,(\partial/\partial z)^I\otimes 
s$ of $\Gamma^aT_X\otimes L$ and\break $w=\sum w_I\,(dz)^I\otimes s^\star$
of $\Gamma^aT^\star_X\otimes L^\star$ where $s^\star$ denotes the holomorphic 
section of $L^\star$ such that $s^\star(s)=1$. We get
$$
\dbar^\star\dbar v=-\sum_{I,k}{\partial^2v_I\over\partial z_k\partial\ol z_k}
\,(\partial/\partial z)^I\otimes s,\qquad
\dbar^\star\dbar w=-\sum_{I,k}{\partial^2w_I\over\partial z_k\partial\ol z_k}
\,(dz)^I\otimes s^\star
$$
at~$x_0$. Now, we find
$$
\eqalign{
&\#\,{\partial\over\partial z_m}=i{\partial\over\partial\ol z_m}\ort\omega
 =dz_m-\sum_{j,k,\ell}c_{jk\ell m}z_j\ol z_kdz_\ell+O(|z|^3),\cr
&\#\,s
 =\Big(1-\sum_{1\le j,k\le n}\gamma_{jk}z_j\ol z_k+O(|z|^3)\Big)s^\star\cr
&\#\,v
 =\sum_I\ol v_I(dz)^I\otimes s^\star-\sum_{I,j,k,\ell,m}\ol v_Ic_{jk\ell m}
 z_j\ol z_k\Big(dz_\ell\otimes{\partial\over\partial z_m}\Big)\ort 
 (dz)^I\otimes s^\star,\cr
&\phantom{\#\,v=\sum_I\ol v_I(dz)^I\otimes s^\star}
 -\sum_{I,j,k}\ol v_I\,\gamma_{jk}z_j\ol z_k(dz)^I\otimes s^\star
 +O(|z|^3)\cr}
$$
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 $\dbar^\star\dbar(\#\,v)$ at $x_0$ we obtain
$$
\eqalign{\dbar^\star\dbar(\#\,v)
&=-\sum_{I,k}{\partial^2\ol v_I\over\partial z_k\partial\ol z_k}(dz)^I
  \otimes s^\star+\sum_{I,k,\ell,m}\ol v_Ic_{kk\ell m}\Big(dz_\ell\otimes
  {\partial\over\partial z_m}\Big)\ort (dz)^I\otimes s^\star\cr
&\phantom{{}=-\sum_{I,k}{\partial^2\ol v_I\over\partial z_k\partial\ol z_k}
  (dz)^I\otimes s^\star}+\sum_j\gamma_{jj}\sum_I\ol v_I
  (dz)^I\otimes s^\star\cr
&=\#\,(\dbar^\star\dbar v)+\sum_{I,\ell,m}\ol v_IR_{\ell m}
  \Big(dz_\ell\otimes{\partial\over\partial z_m}\Big)\ort 
  (dz)^I\otimes s^\star+\gamma(\#\,v)\cr
&=\#\,(\dbar^\star\dbar v)+\#\,\cR_a(v)+\gamma(\#\,v)\cr}
$$
where $\gamma=\sum_j\gamma_{jj}$. Lemma 14.2 then follows from this 
identity by writing
$$
\int_X\Vert\dbar(\#\,v)\Vert^2dV_\omega=
\int_X\langle\dbar^\star\dbar(\#\,v),\#\,v\rangle\,dV_\omega=
\int_X\langle\dbar^\star\dbar v+\cR_a(v)+\gamma v,v\rangle\,dV_\omega.
\eqno\square
$$
\enddemo

\demo{Proof of Theorem~14.1 (end)} Our goal is to apply the Bochner formula 
of Lemma~14.2 to show that every section $v$ of $H^0(X,\Gamma^aT_X\otimes
L^\star)$ 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 
$v\in H^0(X,\Gamma^aT_X\otimes L^\star$, Lemma~14.2 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 $L^\star$, 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, the choice $N=|a|$ works if all $a_j$ are nonnegative, otherwise 
we can take \hbox{$N=|a|+n\max({-}a_j)$}.\square
\enddemo

\Refs
\writetoaux
\widestnumber\key{MoMu82}

\ref\key Ahl41
\by L.\ Ahlfors
\paper The theory of meromorphic curves
\jour Acta Soc.\ Sci.\ Finn.\ N.S., A
\vol 3
\yr 1941 
\pages 1--31
\endref

\ref\key ASS92
\by E.\ Arrondo, I.\ Sols, R.\ Speiser
\paper Global moduli for contacts
\jour Preprint 1992, to appear
\endref

\ref\key Blo26
\by A.\ Bloch
\paper Sur les syst\`emes de fonctions uniformes
satisfaisant \`a l'\'equation d'une vari\'et\'e alg\'ebrique dont
l'irr\'egularit\'e d\'epasse la dimension
\jour Jour.\ de Math.
\vol 5
\yr 1926
\pages 19--66
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\endRefs
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