% Morse cohomology estimates for Semple jet bundles 
%
%
% Jean-Pierre Demailly 
% Universit\'e Grenoble Alpes, Institut Fourier
%
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% Main text

\title[Morse cohomology estimates for Semple jet bundles]{Morse
cohomology estimates for Semple jet bundles}


\author{Jean-Pierre Demailly}
\date{}

\begin{document}

\begin{abstract}
In 2016, the conjecture has been settled in a different
way by Damian Brotbek, making a more direct use of Wronskian differential
operators and associated multiplier ideals; shortly afterwards, Ya Deng 
showed how the proof could be modified to yield an explicit value of $d_n$.
We give here a short proof based on a drastic simplification of their
ideas,  along with a further improvement of Deng's bound, namely
$d_n=\lfloor\frac{1}{5}(en)^{2n+2}\rfloor$.
\vskip10pt
\noindent
{\bf Key words:} Kobayashi hyperbolic variety, directed manifold,
genus of a curve, jet bundle, jet differential, jet metric, Chern connection
and curvature, negativity of jet curvature, variety of general type, 
Kobayashi conjecture, Green-Griffiths conjecture, Lang conjecture.
\vskip10pt
\noindent
{\bf MSC~Classification (2010):} 32H20, 32L10, 53C55, 14J40
\end{abstract}

\maketitle

\tableofcontents

\setcounter{section}{-1}
\section{Introduction}

The goal of these lectures is to study the conjecture of Kobayashi
[Kob70, Kob78] on the hyperbolicity of generic hypersurfaces of high
degree in projective space, and the related conjecture by
Green-Griffiths [GG79] and Lang [Lan86] on the structure of
entire curve loci. 
\medskip

\section{Semple tower associated to a directed manifold}

\plainsubsection 1.A. Category of directed manifolds|

We start by recalling the main definitions concerning the category of
directed varieties. For the sake of simplicity, we first assume that the
objects under consideration are nonsingular.

\claim 1.1. Definition|A $($complex$)$ {\rm directed manifold} is a pair
$(X,V)$ consisting of a $n$-dimensional complex manifold $X$ equipped with a
A morphism $\Phi:(X,V)\to (Y,W)$ in the category of directed manifolds is
a holomorphic map such that $\Phi_*(V)\subset W$.
\endclaim

It is eventually interesting to allow singularities for $V$. We then assume
that there exists a dense Zariski open set $X'=X\ssm Y\subset X$ such that
$V_{|X'}$ is a subbundle of $(T_X)){|X'}$ and the closure
$\overline{V_{|X'}}$ in the total space of $T_X$ is an anaytic subset.
The rank $r\in\{0,1,\ldots,n\}$ of $V$ is by definition the dimension
of $V_x$ at points $x\in X'\,$; the dimension may be larger at points
$x\in Y$. This happens e.g.\ on $X=\bC^n$ for the rank~$1$
linear space $V$ generated by the Euler vector field:
$V_z=\bC\sum_{1\le j\le n} z_j{\partial\over\partial z_j}$ for
$z\ne 0$, and $V_0=\bC^n$.  The absolute situation is the case $V=T_X$
and the relative situation is the case when $V=T_{X/S}$ is the
relative tangent space to a smooth holomorphic map $X\to S$. In
general, we can associate to $V$ a sheaf $\cV=\cO(V)\subset\cO(T_X)$
of holomorphic sections. No assumption need be made on the Lie bracket tensor
$[\bu,\bu]:\cV\times \cV\to \cO(T_X)/\cV$, i.e.\ we do not assume any
kind of integrability for~$\cV$. One of the most central conjectures
in the theory is the

\claim 1.2. Generalized Green-Griffiths-Lang conjecture|
Let $(X,V)$ be a projective directed manifold where $V\subset T_X$ is
nonsingular $($i.e.\ a subbundle of $T_X)$. Assume that $(X,V)$ is of
``general type'' in the sense that $K_V:=\det V^*$ is a big line bundle.
Then there should exist a proper algebraic subvariety $Y\subsetneq X$
containing the images $f(\bC)$ of all entire curves $f:\bC\to X$
tangent to~$V$.
\endclaim

A similar statement can be made when $V$ is singular, but then $K_V$ has
to be replaced by a certain (nonnecessarily invertible) rank 1 sheaf of
``locally bounded'' forms of $\cO(\det V^*)$, with respect to a smooth
hermitian form $\omega$ on $T_X$. The reader will find a more precise
definition in [Dem18].
    

\plainsubsection 1.B. The $1$-jet fonctor|
The basic idea is to introduce a fonctorial process which produces a new
complex directed manifold $(\swt{X},\swt{V})$ from a given one~$(X,V)$.
The new structure $(\swt{X},\swt{V})$ plays the role of a space of $1$-jets
over~$X$.  We let
$$
\swt{X}=P(V),\qquad \swt{V}\subset T_{\swt{X}}
\leqno(1.3)
$$
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 $v\in V_x\ssm\{0\}$,
$$
\swt{V}_{(x,[v])}=\big\{\eta\in T_{\swt{X},\,(x,[v])}\,;\,d\pi_x(\eta)\in
\bC v\big\},\qquad\bC v\subset V_x\subset T_{X,x},\leqno(1.3')
$$
where $\pi:\swt{X}=P(V)\to X$ is the natural projection and $\pi_*:
T_{\swt{X}}\to\pi^* T_X$ is its differential. On $\swt{X}=P(V)$
we have a tautological line bundle $\cO_{\swt{X}}(-1)\subset\pi^* V
\subset\pi^* T_X$ such that $\cO_{\swt{X}}(-1)_{(x,[v])}=\bC v$.
The bundle $\swt{V}$ is characterized by the exact sequences
$$
\leqalignno{
&0\lra T_{\swt{X}/X}\lra T_{\swt{X}}\build\lra^{d\pi}_{}\pi^*T_X\lra 0,\cr
&\kern 40pt ||\kern32pt\cup\kern40pt\cup\cr
&0\lra T_{\swt{X}/X}\lra\swt{V}\build\lra^{d\pi}_{}\cO_{\swt{X}}(-1)
\lra 0,&(1.4)\cr
\noalign{\vskip6pt}
&0\lra\cO_{\swt{X}}\lra \pi^* V\otimes\cO_{\swt{X}}(1)
\lra T_{\swt{X}/X}\lra 0,&(1.4')\cr}
$$
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
$$
\dim\swt{X}=n+r-1,\qquad \rank\swt{V}=\rank V=r,\leqno(1.5)
$$
and by taking determinants we find $\det(T_{\swt{X}/X})=
\pi^*\det V\otimes\cO_{\swt{X}}(r)$, hence
$$
\det\swt{V}=\pi^*\det V\otimes\cO_{\swt{X}}(r-1).
\leqno(1.6)
$$
Clearly $\pi:(\swt{X},\swt{V})\to(X,V)$ is a morphism of
complex directed manifolds and this construction is fonctorial with
respect to morphisms $\Phi:(X,V)\to(Y,W)$ for which $\Phi_*$ is injective.

\plainsubsection 1.C. Lifting of curves to the $1$-jet bundle|

Suppose that we are given a holomorphic curve $f:D(0,R)\to X$
parametrized by the disk $D(0,R)$ of centre $0$ and radius $R$
in the complex plane, and that $f$ is a tangent curve of the
directed manifold, i.e., $f'(t)\in V_{f(t)}$ for every $t\in D(0,R)$.
If $f$ is nonconstant, there is a well defined and unique tangent line
$[f'(t)]$ for every~$t$, even at stationary points, and the map
$$
\swt{f}:D(0,R)\to\swt{X},\qquad
t\mapsto\swt{f}(t):=(f(t),[f'(t)])\leqno(1.7)
$$
is holomorphic (at a stationary point $t_0$, we just write
$f'(t)=(t-t_0)^su(t)$ with $s\in\bN^*$ and $u(t_0)\ne 0$,
and we define the tangent line at $t_0$ to be $[u(t_0)]$, hence
$\swt{f}(t)=(f(t),[u(t)])$ near $t_0\,$; even for $t=t_0$, we still denote
$[f'(t_0)]=[u(t_0)]$ for simplicity of notation). By definition
$f'(t)\in\cO_{\swt{X}}(-1)_{\swt{f}(t)}=\bC\,u(t)$, hence the derivative
$f'$ defines a section
$$
f':T_{D(0,R)}\to\swt{f}^*\cO_{\swt{X}}(-1).\leqno(1.8)
$$
Moreover $\pi\circ\swt{f}=f$, therefore
$$
\pi_*\swt{f}'(t)=f'(t)\in\bC u(t)\Longrightarrow
\swt{f}'(t)\in\swt{V}_{(f(t),u(t))}=\swt{V}_{\swt{f}(t)}
$$
and we see that $\swt{f}$ is a tangent trajectory of $(\swt{X},\swt{V})$.
We say that $\swt{f}$ is the {\em canonical lifting} of $f$ to~$\swt{X}$.
Conversely, if $g:D(0,R)\to\swt{X}$ 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
$$
V_z=\Big\{v=\sum_{1\le j\le n}v_j{\partial\over\partial z_j}\,;\,
v_j= \sum_{1\le k\le r}a_{jk}(z)v_k~\hbox{\rm for $j=r+1\ld n$}\Big\},
\leqno(1.9)
$$
where $(a_{jk})$ is a holomorphic $(n-r)\times r$ matrix. It follows that
a vector $v\in V_z$ is completely determined by its first $r$ components
$(v_1\ld v_r)$, and the affine chart $v_j\ne 0$ of $P(V)_{\restriction
\Omega}$ can be described by the coordinate system
$$
\Big(z_1\ld z_n;{v_1\over v_j}\ld{v_{j-1}\over v_j},
{v_{j+1}\over v_j}\ld{v_r\over v_j}\Big).\leqno(1.10)
$$
Let $f\simeq(f_1\ld f_n)$ be the components of $f$ in the coordinates
$(z_1\ld z_n)$ (we suppose here $R$ so small that $f(D(0,R))\subset\Omega$).
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
$$
f_j'(t)=\sum_{1\le k\le r}a_{jk}(f(t))f_k'(t),\qquad j>r,\leqno(1.11)
$$
on a neighborhood of~$0$, with initial data $f(0)=x$.
We denote by $m=m(f,t_0)$ the {\em multiplicity} of $f$ at any point
$t_0\in D(0,R)$, that is, $m(f,t_0)$ is the smallest integer $m\in\bN^*$
such that $f_j^{(m)}(t_0)\ne 0$ for some~$j$. By (1.11), we can always
suppose $j\in\{1\ld r\}$, for example $f_r^{(m)}(t_0)\ne 0$. Then
$f'(t)=(t-t_0)^{m-1}u(t)$ with $u_r(t_0)\ne 0$, and the lifting $\swt{f}$
is described in the coordinates of the affine chart $v_r\ne 0$ of
$P(V)_{\restriction\Omega}$ by
$$
\swt{f}\simeq\Big(f_1\ld f_n;{f_1'\over f_r'}\ld{f_{r-1}'\over f_r'}\Big).
\leqno(1.12)
$$

\plainsubsection 1.D. The Semple tower|

Following [Dem95], we define inductively the
{\em projectivized $k$-jet bundle $X_k$}
(or {\em Semple $k$-jet bundle}) and the associated subbundle 
$V_k\subset T_{X_k}$ by
$$
(X_0,V_0)=(X,V),\qquad (X_k,V_k)=(\swt{X}_{k-1},\swt{V}_{k-1}).
\leqno(1.13)
$$
In other words, $(X_k,V_k)$ is obtained from $(X,V)$ by iterating
$k$-times the lifting construction $(X,V)\mapsto(\swt{X},\swt{V})$ described
in~\S$\,$1.B. By (1.3--1.5), we find
$$
\dim X_k=n+k(r-1),\qquad\rank V_k=r,\leqno(1.14)
$$
together with exact sequences
$$
\leqalignno{
&0\lra T_{X_k/X_{k-1}}\lra V_k\build{\vlra{11}}^{(\pi_k)_*}_{}
\cO_{X_k}(-1)\lra 0,&(1.15)\cr
&0\lra\cO_{X_k}\lra\pi_k^* V_{k-1}\otimes\cO_{X_k}(1)
\lra T_{X_k/X_{k-1}}\lra 0.
&(1.15')\cr}
$$
where $\pi_k$ is the natural projection $\pi_k:X_k\to X_{k-1}$ and
$(\pi_k)_*$ its differential. Formula (1.6) yields
$$
\det V_k=\pi_k^*\det V_{k-1}\otimes\cO_{X_k}(r-1).\leqno(1.16)
$$
Every nonconstant tangent trajectory
\hbox{$f:D(0,R)\to X$} of $(X,V)$ lifts to a well defined and
unique tangent trajectory $f_{[k]}:D(0,R)\to X_k$ of $(X_k,V_k)$.
Moreover, the derivative $f_{[k-1]}'$ gives rise to a section
$$
f_{[k-1]}':T_{D(0,R)}\to f_{[k]}^*\cO_{X_k}(-1).\leqno(1.17)
$$
In coordinates, one can compute $f_{[k]}$ in terms of its components in
the various affine charts (1.10) occurring at each step: we get inductively
$$
f_{[k]}=(F_1\ld F_N),\qquad f_{[k+1]}=\Big(F_1\ld F_N,
{F_{s_1}'\over F_{s_r}'}\ld{F_{s_{r-1}}'\over F_{s_r}'}\Big)\leqno(1.18)
$$
where $N=n+k(r-1)$ and $\{s_1\ld s_r\}\subset\{1\ld N\}$. If $k\ge 1$,
$\{s_1\ld s_r\}$ contains the last $r-1$ indices of $\{1\ld N\}$
corresponding to the ``vertical'' components of the projection
$X_k\to X_{k-1}$, and in general, $s_r$ is an index such that
$m(F_{s_r},0)=m(f_{[k]},0)$, that is, $F_{s_r}$ has the smallest
vanishing order among all components $F_s$ ($s_r$ may be vertical
or not, and the choice of $\{s_1\ld s_r\}$ need not be unique).

By definition, there is a canonical injection $\cO_{X_k}(-1)
\hookrightarrow\pi_k^* V_{k-1}$, and a composition with the
projection $(\pi_{k-1})_*$ (analogue for order $k-1$ of the
arrow~$(\pi_k)_*$ in the sequence (1.15)) yields for all $k\ge 2$ a
canonical line bundle morphism
$$
\cO_{X_k}(-1)\lhra\pi_k^* V_{k-1}\build{\vlra{16}}^
{(\pi_k)^*(d\pi_{k-1})}_{}\pi_k^*\cO_{X_{k-1}}(-1),\leqno(1.19)
$$
which admits precisely \hbox{$D_k=P(T_{X_{k-1}/X_{k-2}})
\subset P(V_{k-1})=X_k$} as its zero divisor (clearly, $D_k$ is a
hyperplane subbundle of~$X_k$). Hence we find
$$
\cO_{X_k}(1)=\pi_k^*\cO_{X_{k-1}}(1)\otimes\cO(D_k).\leqno(1.20)
$$
Now, we consider the composition of projections
$$
\pi_{k,j}=\pi_{j+1}\circ\cdots\circ\pi_{k-1}\circ\pi_k:X_k\lra X_j.
\leqno(1.21)
$$
Then $\pi_{k,0}:X_k\to X_0=X$ is a locally trivial holomorphic fiber bundle
over~$X$, and the fibers $X_{k,x}=\pi_{k,0}^{-1}(x)$ are $k$-stage
towers of $\bP^{r-1}$-bundles. Since we have (in both directions) morphisms 
$(\bC^r,T_{\bC^r})\leftrightarrow(X,V)$ of directed manifolds which
are bijective on the level of bundle morphisms, the fibers are
all isomorphic to a ``universal'' non singular projective algebraic 
variety of dimension $k(r-1)$ which we will denote by~$\bR_{r,k}\,$; it is
not hard to see that $\bR_{r,k}$ is rational, since (1.18) provides
affine charts of $\bR_{r,k}$ that are isomorphic to $\bC^{k(r-1)}$.

\claim 1.22. Remark|{\rm When $(X,V)$ is singular, one can easily
extend the construction of the Semple tower by fonctoriality.
In fact, assume that $X$ is a closed analytic subset of some open
set $Z\subset\bC^N$, and that $X'\subset X$ is a Zariski open subset on
which $V_{\restriction X'}$ is a subbundle of $T_{X'}$. Then we consider the
injection of the nonsingular directed manifold $(X',V')$ into
the absolute structure $(Z,W)$, $W=T_Z$. This yields an injection
$(X'_k,V'_k)\hookrightarrow (Z_k,W_k)$, and we simply define
$(X_k,V_k)$ to be the closure of $(X'_k,V'_k)$ into $(Z_k,W_k)$. It is
not hard to see that this is indeed a closed analytic subset of
the same dimension $n+k(r-1)$, where $r=\rank V'$.}
\endclaim

\plainsubsection 1.E. Jet bundles and jet differentials|

Following Green-Griffiths [GrGr79], we consider the bundle $J_kX\to X$
of $k$-jets of germs of para\-metrized curves in~$X$, i.e., the set
of equivalence
classes of holomorphic maps $f:(\bC,0)\to(X,x)$, with the equivalence
relation $f\sim g$ if and only if all derivatives $f^{(j)}(0)=g^{(j)}(0)$
coincide for $0\le j\le k$, when computed in some local coordinate system
of $X$ near~$x$. The projection map $J_kX\to X$ is simply $f\mapsto f(0)$.
If $(z_1\ld z_n)$ are local holomorphic coordinates on an open set
$\Omega\subset X$, the elements $f$ of any fiber $J_kX_x$,
$x\in\Omega$, can be seen as $\bC^n$-valued maps
$$
f=(f_1\ld f_n):(\bC,0)\to\Omega\subset\bC^n,
$$
and they are 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 $J_kX_x$ can thus be identified with the
set of $k$-tuples of vectors 
$(\xi_1,\ldots,\xi_k)=(f'(0)\ld f^{(k)}(0))\in(\bC^n)^k$.
It follows that $J_kX$ is a holomorphic fiber bundle with typical fiber
$(\bC^n)^k$ over $X$. However, $J_kX$ is not a vector bundle for $k\ge 2$,
because of the nonlinearity of coordinate changes: a coordinate change
$z\mapsto w=\Psi(z)$ on $X$ induces a polynomial
transition automorphism on the fibers of $J_kX$, given by a formula
$$
(\Psi\circ f)^{(j)}=\Psi'(f)\cdot f^{(j)}+\sum_{s=2}^{s=j}{~}
\sum_{j_1+j_2+\cdots+j_s=j}c_{j_1\ldots j_s}\Psi^{(s)}(f)\cdot
(f^{(j_1)}\ld f^{(j_s)})\leqno(1.23)
$$
with suitable integer constants $c_{j_1\ldots j_s}$ (this is easily
checked by induction on~$s$). According to the above philosophy,
we introduce the concept of jet bundle in the
general situation of complex directed manifolds.

\claim 1.24.~Definition|Let $(X,V)$ be a complex directed manifold.
We define $J_kV\to X$ to be the bundle of $k$-jets of curves
\hbox{$f:(\bC,0)\to X$} which are tangent to $V$, i.e., such that
$f'(t)\in V_{f(t)}$ for all $t$ in a neighborhood of~$0$, together with
the projection map $f\mapsto f(0)$ onto~$X$.
\endclaim

It is easy to check that $J_kV$ is actually a subbundle of~$J_kX$. In
fact, by using (1.11), we see that the fibers $J_kV_x$ are
parametrized by 
$$
\big((f_1'(0)\ld f_r'(0));(f_1''(0)\ld f_r''(0));\ldots;
(f_1^{(k)}(0)\ld f_r^{(k)}(0))\big)\in(\bC^r)^k
$$
for all $x\in\Omega$, hence $J_kV$ is a locally trivial
$(\bC^r)^k$-subbundle of~$J_kX$. Alternatively, we can pick a
local holomorphic connection $\nabla$ on $V$ such that for
any germs $w=\sum_{1\leq j\leq n}w_j{\partial\over \partial z_j}
\in\cO(T_{X,x})$ and $v=\sum_{1\leq\lambda\leq r}v_\lambda e_\lambda\in\cO(V)_x$
in a local trivializing frame $(e_1,\ldots,e_r)$ of~$V_{\restriction \Omega}$
we have
$$
\nabla_wv(x)=\sum_{1\leq j\leq n,\,1\leq\lambda\leq r}
w_j{\partial v_\lambda\over\partial z_j}e_\lambda(x)+
\sum_{1\leq j\leq n,\,1\leq\lambda,\mu\leq r}
\Gamma_{j\lambda}^\mu(x)w_jv_\lambda\,e_\mu(x).
\leqno(1.25)
$$
We can of course take the frame obtained from (1.9) by lifting
the vector fields $\partial/\partial z_1,\ldots,\partial/\partial z_r$,
and the ``trivial connection'' given by the zero Christoffel symbolds
$\Gamma=0$. One then obtains a
trivialization $J^kV_{\restriction \Omega}\simeq
V^{\oplus k}_{\restriction \Omega}$ by considering
$$
J_kV_x\ni f\mapsto(\xi_1,\xi_2,\ldots,\xi_k)=(\nabla f(0),\nabla^2f(0),\ldots,\nabla^k f(0))\in V_x^{\oplus k}
\leqno(1.26)
$$
and computing inductively the successive derivatives $\nabla f(t)=f'(t)$ and
$\nabla^sf(t)$ via
$$
\nabla^sf=(f^*\nabla)_{d/dt}(\nabla^{s-1}f)
=\sum_{1\leq\lambda\leq r}
{d\over dt}\Big(\nabla^{s-1}f\Big)_\lambda e_\lambda(f)+
\sum_{1\leq j\leq n,\,1\leq\lambda,\mu\leq r}
\Gamma_{j\lambda}^\mu(f)f'_j\Big(\nabla^{s-1}f\Big)_\lambda e_\mu(f).
$$
This identification depends of course on the choice of $\nabla$ and cannot 
be defined globally in general (unless we are in the rare situation
where $V$ has a global holomorphic connection).\qed
\medskip

Let $\bG_k$ be the group of germs of $k$-jets of biholomorphisms of $(\bC,0)$,
that is, the group of germs of biholomorphic maps
$$
t\mapsto\varphi(t)=a_1t+a_2t^2+\cdots+a_kt^k,\qquad
a_1\in\bC^*,~a_j\in\bC,~j\ge 2,
$$
in which the composition law is taken modulo terms $t^j$ of degree $j>k$.
Then $\bG_k$ is a $k$-dimensional nilpotent complex Lie group,
which admits a natural fiberwise right action on $J_kV$
$$
J_kV\times\bG_k\to J_kV,\quad (f,\varphi)\mapsto f\circ\varphi.
\leqno(1.27)
$$
There is a semidirect decomposition $\bG_k=\bG'_k\ltimes\bC^*$ given by
a split exact sequence
$$
1\to \bG'_k\to \bG_k\to\bC^*\to 1
$$
where $\bG_k\to\bC^*$ is the obvious morphism $\varphi\mapsto\varphi'(0)$,
the commutator group $\bG'_k=[\bG_k,\bG_k]$ is the group of $k$-jets
of biholomorphisms tangent to the identity, and $\bC^*\subset\bG_k$ is
the (nonnormal) subgroup of homotheties $\varphi(t)=\lambda t$.
The corresponding action of $\bC^*$ on $k$-jets is described
in coordinates by
$$
\lambda\cdot(\xi_1,\xi_2,\ldots,\xi_k)=
(\lambda \xi_1,\lambda^2\xi_2,\ldots,\lambda^k\xi_k),\quad \xi_s=\nabla^sf(0).
$$
Following [GrGr79], we introduce the bundle $E^\GG_{k,m}V^*\to X$ 
of polynomials $P(x;\xi_1,\ldots,\xi_k)$ that are homogeneous
on the fibers of $J_kV$ of weighted degree $m$ with respect to the
$\bC^*$ action, i.e.
$$
P(x;\lambda\xi_1,\ldots,\lambda^k\xi_k)=\lambda^m P(x;\xi_1,\ldots\xi_k),
\leqno(1.28)
$$
in other words they are polynomials of the form
$$
P(x;\xi_1,\ldots\xi_k)=
\sum_{|\alpha_1|+2|\alpha_2|+\cdots+k|\alpha_k|=m}
a_{\alpha_1\ldots\alpha_k}(x)\,\xi_1^{\alpha_1}\xi_2^{\alpha_2}\cdots
\xi_k^{\alpha_k}\leqno(1.29)
$$
where $\xi_s=(\xi_{s,1},\ldots,\xi_{s,r})\in\bC^r\simeq V_x$ and
$\xi_s^{\alpha_s}=\xi_{s,1}^{\alpha_{s,1}}\ldots\xi_{s,r}^{\alpha_{s,r}}$,
$|\alpha_s|=\sum_{1\leq j\leq r}\alpha_{s,j}$. Sections of the
sheaf $\cO(E^\GG_{k,m}V^*)$ can also be viewed as algebraic differential
operators acting on germs of curves $f:(\bC,0)\to X$ tangent to~$V$, by putting
$$
P(f)(t)=\sum_{|\alpha_1|+2|\alpha_2|+\cdots+k|\alpha_k|=m}
a_{\alpha_1\ldots\alpha_k}(f(t))\,(\nabla f(t))^{\alpha_1}(\nabla^2 f(t))^{\alpha_2}\cdots
(\nabla^kf(t))^{\alpha_k}\leqno(1.29')
$$
where the $a_{\alpha_1\ldots\alpha_k}(x)$ are holomorphic in~$x$.
With the graded algebra bundle $E^\GG_{k,\bu}V^*=\bigoplus_mE^\GG_{k,m}V^*$
we associate an analytic fiber bundle
$$X_k^\GG:=\Proj(E^\GG_{k,\bu}V^*)=(J_kV\ssm\{0\})/\bC^*\leqno(1.30)$$
over $X$, which has weighted projective spaces
$\bP(1^{[r]},2^{[r]}\ld k^{[r]})$ as fibers; here $J_kV\ssm\{0\}$ is the set
of nonconstant jets of order~$k$.
As such, it possesses a tautological sheaf $\cO_{X^\GG_k}(1)$ [the reader
should observe however that $\cO_{X^\GG_k}(m)$ is invertible only when
$m$ is a multiple of $\lcm(1,2,\ldots,k)$].

\claim 1.31. Proposition|By construction, if $\pi_k:X_k^\GG\to X$ is the
natural projection, we have the direct image formula
$$
(\pi_k)_*\cO_{X^\GG_k}(m)=\cO(E_{k,m}^\GG V^*)
$$
for all $k$ and $m$.
\endclaim

In the geometric context, we are not really interested in the 
bundles $(J_kV\ssm\{0\})/\bC^*$ themselves, but
rather on their quotients $(J_kV\ssm\{0\})\mathrel{/\!/}\bG_k$
(would such nice complex space quotients exist!). In fact the following
fundamental result from [Dem95] shows that the Semple bundle $X_k$ 
constructed above plays the role of such a quotient.

\claim 1.32. Theorem and Definition|Let $E_{k,m}V^*\subset E^\GG_{k,m}V^*$
be the set of polynomial differential operators $f\mapsto P(f)$
that are invariant under arbitrary changes of parametrization, i.e.,
such that for every $\varphi\in \bG_k$
$$
P(f\circ\varphi)=(\varphi')^mP(f)\circ\varphi\leqno(*)
$$
$[$the weighted degree condition $(1.28)$ being the special case when
$\varphi(t)=\lambda t$, $\lambda\in\bC^*]$.\vskip5pt\noindent
Let $\pi_{k,0}:X_k\to X$ be the Semple jet bundles defined above
and let $J_kV^\reg$ be the bundle of regular $k$-jets of maps
$f:(\bC,0)\to (X,V)$, that is, jets $f$ such that $f'(0)\ne 0$. Then
\plainitem{\rm(i)} The quotient $J_kV^\reg/\bG_k$ has the structure of a
locally trivial bundle over~$X$, and there is a holomorphic embedding
$J_kV^\reg/\bG_k\hookrightarrow X_k$ over $X$, which identifies
$J_kV^\reg/\bG_k$ with $X_k^\reg$ $($thus $X_k$ is a relative
compactification of $J_kV^\reg/\bG_k$ over~$X)$.
\plainitem{\rm(ii)} The direct image sheaf
$$
(\pi_{k,0})_*\cO_{X_k}(m)\simeq\cO(E_{k,m}V^*)
$$
can be identified with the sheaf of holomorphic sections of
$E_{k,m}V^*$.
\plainitem{\rm(iii)} For every $m\geq 1$, the relative base locus of the linear
system $|\cO_{X_k}(m)|$ is equal to the set $X_k^\sing$
of singular $k$-jets $[$one has $X_k^\sing=\emptyset$ for $k=1]$. Moreover,
$\cO_{X_k}(1)$ is relatively big over~$X$.
\vskip0pt
\endclaim

\noindent
{\em Sketch of proof.} We refer to [Dem95] for details. In order to prove (i)
and (ii), the main point is
that the lifts $f_{[k]}:D(0,R)\to (X_k,V_k)$ of a curve $f:D(0,R)\to(X,V)$
are defined inductively by $f_{[k]}=(f_{[k-1]},[f'_{[k-1]}])$, hence
for any change of variable $\varphi:D(0,R')\to D(0,R)$, they satisfy the
relations
$$
(f\circ\varphi)_{[k]}=f_{[k]}\circ\varphi,\quad
(f\circ\varphi)'_{[k-1]}=\varphi'f'_{[k-1]}\circ\varphi\in
\cO_{X_k}(-1)\subset \pi_{k,k-1}^*V_{k-1}.
$$
We conclude that there is a well defined set-theoretic map
$$
J_kV^\reg/\bG_k\to X_k^\reg,\qquad f~\mod~\bG_k\mapsto f_{[k]}(0).
\leqno(1.33)
$$
Given a holomorphic section $\sigma\in H^0(\pi_{k,0}^{-1}(U),\cO_{X_k}(m))$,
we can then associate a differential operator
$$
P(f)=\sigma(f_{[k]})\cdot(f_{[k-1]}')^m.
\leqno(1.34)
$$
Clearly, condition $(*)$ is satisfied and in particular $P$ is homogeneous
of degree $m$ on $J_kV^\reg$; such a holomorphic function must be a homogeneous
polynomial on the fibers.\qed


\section{Algebraic properties of the algebra of
 differential operators}

\plainsubsection 2.A. Green-Griffiths and Semple algebras|

By construction, the {\em Green-Griffiths graded algebra}
$$
\cA^\GG_kV^\star=\bigoplus_{m\in\bZ}E^\GG_{k,m}V^\star
\leqno(2.1)
$$
of differential operators is fiberwise isomorphic to the polynomial ring
$$
\bC[f'_1,\ldots,f'_r,f''_1,\ldots,f''_r,\ldots,f^{(k)}_1,\ldots,f^{(k)}_r]
$$
and in particular it is finitely generated. More geometrically, we get
a holomorphic filtration of $E^\GG_{k,m}V^\star$ by considering the partial
degree of $P(f)$ in terms of the last derivative $f^{(k)}$ and putting
$$
F^a(E^\GG_{k,m}V^\star)=\big\{
P\in E^\GG_{k,m}V^\star\,;\;\deg_{f^{(k)}}P(f)\leq a\big\}.
$$
Then the graded pieces are polynomials of the
form $Q(f',\ldots,f^{(k-1)})(f^{(k)})^{\alpha_k}$, $ |\alpha_k|=a$, i.e.
$$
G^a(E^\GG_{k,m}V^\star)\simeq E^\GG_{k-1,m-ka}V^\star\otimes S^aV^*.
$$
We can then inductively combine the successive filtrations obtained
via the partial degrees in $f^{(k)}$, $f^{(k-1)},\ldots,f^{(1)}=f'$ to get a
full decomposition
$$
G^\bu(E^\GG_{k,m}V^\star)\simeq\bigoplus_{a=(a_1,\ldots,a_k)\in \bN^k\atop
  a_1+2a_2+\cdots+ka_k=m}S^{a_1}V^*\otimes \cdots\otimes S^{a_k}V^*.
\leqno(2.2)
$$
Hence $\cA^\GG_kV^\star$ is just locally isomorphic to a $k$-fold
tensor product of symmetric algebras $S^\bu V^*$. We define
the {\em Semple algebra} to be the graded subalgebra of $\cA^\GG_k$
such that
$$
\cA_kV^\star=(\cA^\GG_k)^{\bG'_k}=\bigoplus_{m\in\bZ}
E_{k,m}V^\star,
\leqno(2.3)
$$
in particular $\cA_1V^\star=\cA^\GG_1V^\star=S^\bu V^*$. As $\bG'_k$ is a
non reductive group, it is a priori unclear whether
$\cA_kV^\star$ is finitely generated for $k\ge 2$.

the subalgebra of $\bG'_k$-invariant differential operators
is finitely generated. This can be checked by hand
([Dem07a], [Dem07b]) for $n=2$ and $k\le 4$. Rousseau [Rou06] also checked
the case $n=3$, $k=3$, and then Merker [Mer08, Mer10] proved the finiteness
for $n=2,3,4$, $k\leq 4$ and $n=2$, $k=5$. Recently, B\'erczi and 
Kirwan [BeKi12] made an attempt to prove the finiteness in full generality,
but it appears that the general case is still unsettled.

Fix coordinates $(z_1\ld z_n)$ near a point $x_0\in X$, such that
$V_{x_0}=\Vect(\partial/\partial z_1\ld\partial/\partial z_r)$. Let
$f=(f_1\ld f_n)$ be a regular $k$-jet tangent to~$V$. Then there exists
$i\in\{1,2\ld r\}$ such that $f_i'(0)\ne 0$, and there is a unique
reparametrization $t=\varphi(\tau)$ such that $f\circ\varphi=g
=(g_1,g_2\ld g_n)$ with $g_i(\tau)=\tau$ (we just express the curve as
a graph over the $z_i$-axis, by means of a change of parameter $\tau=f_i(t)$,
i.e.\ $t=\varphi(\tau)=f_i^{-1}(\tau)$). Suppose $i=r$ for the simplicity of
notation. The space $X_k$ is a $k$-stage tower of $\bP^{r-1}$-bundles.
In~the corresponding inhomogeneous coordinates on these $\bP^{r-1}$'s,
the point $f_{[k]}(0)$ is given by the collection of derivatives
$$
\big((g_1'(0)\ld g_{r-1}'(0));(g_1''(0)\ld g_{r-1}''(0));\ldots;
(g_1^{(k)}(0)\ld g_{r-1}^{(k)}(0))\big).
$$
[Recall that the other components $(g_{r+1}\ld g_n)$ can be recovered from
$(g_1\ld g_r)$ by integrating the differential system (5.10)].
Thus the map $J_kV^\reg/\bG_k\to X_k$ is a bijection onto $X_k^\reg$, and the
fibers of these isomorphic bundles can be seen as unions of $r$ affine
charts ${}\simeq(\bC^{r-1})^k$, associated with each choice of the axis
$z_i$ used to describe the curve as a graph. The change of parameter formula
${d\over d\tau}={1\over f_r'(t)}{d\over dt}$ expresses all derivatives
$g_i^{(j)}(\tau)=d^jg_i/d\tau^j$ in terms of the derivatives
$f_i^{(j)}(t)=d^jf_i/dt^j$
$$
\leqalignno{
(g_1'\ld g_{r-1}')&=\Big({f'_1\over f'_r}\ld{f'_{r-1}\over f'_r}\Big);\cr
(g_1''\ld g_{r-1}'')&=\Big({f''_1f'_r-f''_rf'_1\over f^{\prime 3}_r}\ld
{f''_{r-1}f'_r-f''_rf'_{r-1}\over f^{\prime 3}_r}\Big);~\ldots~;&(3.12)\cr
\qquad(g_1^{(k)}\ld g_{r-1}^{(k)})&=\Big({f^{(k)}_1f'_r-f^{(k)}_rf'_1\over
f^{\prime k+1}_r}\ld{f^{(k)}_{r-1}f'_r-f^{(k)}_rf'_{r-1}\over
f^{\prime k+1}_r}\Big)+(\hbox{order}<k).\cr}
$$

\plainsubsection 2.B. Wronskians|

Let $U$ be an open set of $X$, $\dim X=n$, and
$s_0,\ldots,s_k\in\cO_X(U)$ be holomorphic functions. To these
functions, we can associate a Wronskian operator of order $k$ defined
by
$$
W_k(s_0,\ldots,s_k)(f)=\left|
\plainmatrix{
  s_0(f) & s_1(f) & \ldots &s_k(f)\cr
  D(s_0(f)) & D(s_1(f)) & \ldots &D(s_k(f))\cr
  \vdots &  &  &\vdots\cr
  D^k(s_0(f)) & D^k(s_1(f)) & \ldots &D^k(s_k(f))\cr}\right|\leqno(4.1)
$$
where $f:t\mapsto f(t)\in U\subset X$ is a germ of holomorphic curve
(or a $k$-jet of curve), and $D=\frac{d}{dt}$. For a biholomorphic change of
variable $\varphi$ of $(\bC,0)$, we find by induction on $\ell$ a
polynomial differential operator $Q_{\ell,s}$ of order${}\leq\ell$ acting on
$\varphi$ satisfying
$$
D^\ell(s_j(f\circ \varphi))=\varphi^{\prime\ell}D^\ell(s_j(f))\circ\varphi
+\sum_{s<\ell}p_{\ell,s}(\varphi)D^s(s_j(f))\circ\varphi.
$$
It follows easily from there that
$$
W_k(s_0,\ldots,s_k)(f\circ\varphi)=(\varphi')^{1+2+\cdots+k}
W_k(s_0,\ldots,s_k)(f)\circ\varphi,
$$ 
hence $W_k(s_0,\ldots,s_k)(f)$ is an invariant
differential operator of degree $k'=\frac{1}{2}k(k+1)$. Especially, we get
in this way a section that we denote
$$
W_k(s_0,\ldots,s_k)=
\left|
\plainmatrix{
  s_0 & s_1 & \ldots &s_k\cr
  D(s_0) & D(s_1) & \ldots &D(s_k)\cr
  \vdots &  &  &\vdots\cr
  D^k(s_0) & D^k(s_1) & \ldots &D^k(s_k)\cr}\right|\in
H^0(U,E_{k,k'}T^*_X).\leqno(4.2)
$$

\claim 2.2. Proposition|These Wronskian operators satisfy the following
properties.
\vskip2pt
\plainitem{\rm(a)} $W_k(s_0,\ldots,s_k)$ is $\bC$-multilinear and
alternate in $(s_0,\ldots,s_k)$.
\vskip2pt\plainitem{\rm(b)} For any $g\in\cO_X(U)$, we have
$$W_k(gs_0,\ldots,gs_k)=g^{k+1}W_k(s_0,\ldots,s_k).$$
\endclaim

\noindent
Property 2.2~(b) is an easy consequence of the Leibniz formula
$$
D^\ell(g(f)s_j(f))=\sum_{k=0}^{\ell}{\ell\choose k}D^k(g(f))
D^{\ell-k}(s_j(f)),
$$
by performing linear combinations of rows in the determinants. This property
implies in its turn that one can define more generally an operator
$$
W_k(s_0,\ldots,s_k)\in H^0(U,E_{k,k'}T^*_X\otimes L^{k+1})
\leqno(2.3)
$$
for any $(k+1)$-tuple of sections $s_0,\ldots,s_k\in H^0(U,L)$ of
a holomorphic line bundle $L\to X$. In fact, when we compute the Wronskian
in a local trivialization of $L_{\restriction U}$, Property 4.3~(b)
shows that the determinant is independent of the trivialization.
Moreover, if $g\in H^0(U,G)$ for some line bundle $G\to X$, we have
$$
W_k(gs_0,\ldots,gs_k)=g^{k+1}W_k(s_0,\ldots,s_k)\in H^0(U,E_{k,k'}T^*_X
\otimes L^{k+1}\otimes G^{k+1}).\leqno(2.4)
$$

\plainsubsection 2.C. Brackets|

If $P$ is a differential operator given by a section of
$\cO(E^\GG_{k,m}V^\star)$, we define $DP$ to be its ``obvious'' derivative
$$
DP(f)=P(f)',\quad\hbox{i.e.}\quad DP(f)(t)={d\over dt}P(f)(t).
\leqno(2.5)
$$
The operator $DP$ is then a section of $\cO(E^\GG_{k,m}V^\star)$.
If $P$ is a $\bG'_k$-invariant operator in $\cO(E_{k,m}V^\star)$, the
relation $P(f\circ\varphi)=\varphi^{\prime\,m}P(f)\circ \varphi$ implies
$$
DP(f\circ\varphi)=(\varphi^{\prime\,m}P(f)\circ \varphi)'=
\varphi^{\prime\,m+1}DP(f)\circ \varphi+m
\varphi^{\prime\,m-1}\varphi''P(f)\circ\varphi,
$$
therefore $DP$ is no longer an invariant operator (unless $m=0$, in  which
case $DP$ is of degree $1$). However, if $P,Q$ are $\bG'_k$-invariant
operators of respective degrees $\delta_P,\delta_Q$; it is easy to check
that their {\em bracket} defined as
$$
[P,Q]=\delta_PP(DQ)-\delta_QQ(DP)\leqno(2.6)
$$
is again $\bG'_k$-invariant, of degree $\delta_P+\delta_Q+1$. This can be seen
by observing that the terms $\varphi''(\ldots)$ coming from
$DP$ and $DQ$ cancel, or by noticing that
$$
[P,Q]=P^{\delta_Q+1}Q^{-\delta_P+1}D\bigg({Q^{\delta_P}\over P^{\delta_Q}}\bigg)
\leqno(2.6')
$$
where $Q^{\delta_P}/P^{\delta_Q}$ is homogeneous of degree $0$ (i.e. $\bG_k$-invariant).
A~straightforward albeit tedious calculation shows that this bracket
satisfies the usual Jacobi identity
$$
[P,[Q,R]]+[Q,[R,P]]+[R,[P,Q]]=0.\leqno(2.7)
$$
Formula $(2.6')$ has the advantage that for any line bundle $L$
(or even any $\bQ$-line bundle $L$) and
$$
P\in H^0(X,E_{k,\delta_P}V^*\otimes L^{\delta_P}),~
Q\in H^0(X,E_{k,\delta_Q}V^*\otimes L^{\delta_Q})
$$
we also get globally defined brackets
$$[P,Q]\in H^0(X,E_{k+1,\delta_P+\delta_Q+1}V^*\otimes L^{\delta_P+\delta_Q}).
$$
Another special case is the ``degree $0$'' bracket associated with
two sections $\sigma_1,\sigma_2\in H^0(X,L)$
$$
\tau=[\sigma_1,\sigma_2]=\sigma_1D\sigma_2-\sigma_2 D\sigma_1=
\sigma_1^2D\bigg({\sigma_2\over \sigma_1}\bigg)
\in H^0(X,E_{1,1}T^*_X\otimes L^2)=H^0(X,T^*_X\otimes L^2).
\leqno(2.8)
$$
In a similar way, given integers $a,b\geq 1$ and families of sections
$\sigma_j\in H^0(X,L^a)$, $1\leq j\leq k-2$,
$\tau_j\in H^0(X,V^*\otimes L^b)=
H^0(X,E_{1,1}V^*\otimes L^b)$, $1\leq j\leq k$,
we define inductively iterated brackets
$$
B_k(\sigma_1,\ldots,\sigma_{k-2}\,;\tau_1,\ldots,\tau_k)\in
H^0(X,E_{k,2k-1}V^*\otimes L^{(k-2)a+kb}),\quad k\geq 2,\leqno(2.9)
$$
by putting
$$
B_2(\tau_1,\tau_2)=[\tau_1,\tau_2]=
\tau_1^2D\bigg({\tau_2\over \tau_1}\bigg),
\leqno(2.9')
$$
and, for $k\geq 3$ and $c_k=(k-3)-(k-2)b/a\in\bQ$,
$$
B_k(\sigma_1,\ldots,\sigma_{k-2}\,;
\tau_1,\ldots,\tau_k)=\sigma_1^{c_k+1}\tau_1^{2k-2}\,
D\bigg({B_{k-1}(\sigma_2,\ldots,\sigma_{k-2}\,;
\tau_2,\ldots,\tau_k)\over\sigma_1^{c_k}\tau_1^{2k-3}}\bigg).
\leqno(2.9'')
$$
If $L$ is very ample, by (2.8), there are many such sections when we
take $a=1$, $b=2$, $c_k=1-k$, and we then get sections
$B_k(\sigma_1,\ldots,\sigma_{k-2}\,;\tau_1,\ldots,\tau_k)\in 
H^0(X,E_{k,2k-1}V^*\otimes L^{3k-2})$ whose degrees $m=2k-1$ grow linearly
with~$k$, as well as the correcting twist $L^{3k-2}$.

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

\plainsubsection 3.A. Statement of Morse inequalities|

One of the main purpose of holomorphic Morse inequalities is to provide
estimates of cohomology groups with values in high tensor powers
of a given line bundle $L$, once a smooth hermitian metric $h$ on $L$ is given.
We denote by $\Theta_ {L,h}=-{i\over 2\pi}\ddbar\log h$ the $(1,1)$-curvature
form of $h$.

\claim 3.1. Holomorphic Morse inequalities {\rm ([Dem85])}|Let $X$
be a compact complex  manifolds, $E\to X$ a holomorphic vector bundle of
rank $r$, and $(L,h)$ a hermitian line bundle. The dimensions
$h^q(X,E\otimes L^m)$ of cohomology groups of the tensor powers 
$E\otimes L^m$ satisfy the following asymptotic estimates as 
$m\to +\infty\,:$\vskip4pt
\noindent {\rm(3.1~WM)} Weak Morse inequalities$\,:$
$$h^q(X,E\otimes L^m)\le r {m^n\over n!}\int_{X(L,h,q)} (-1)^q \Theta_{L,h}^n + o(m^n)~.$$
\noindent {\rm(3.1~SM)} Strong Morse inequalities$\,:$
$$\sum_{0\le j\le q} (-1)^{q-j}h^j(X,E\otimes L^m) \le r {m^n\over n!}
\int_{X(L,h,\le q)}(-1)^q\Theta_{L,h}^n+o(m^n)~.$$
\noindent {\rm(3.1~RR)} Asymptotic Riemann-Roch formula$\,:$
$$\chi(X,E\otimes L^m) := \sum_{0\le j\le n} (-1)^jh^j(X,E\otimes L^m)
= r{m^n\over n!}\int_X \Theta_{L,h}^n + o(m^n)~.$$
\endclaim

\noindent Moreover $($cf.\ Bonavero's PhD thesis {\rm [Bon93]}$)$, 
if $h=e^{-\varphi}$ is a singular hermitian metric with analytic 
singularities of pole set $\Sigma=\varphi^{-1}(-\infty)$, the estimates 
still hold provided all cohomology groups
are replaced by cohomology groups 
$H^q(X,E\otimes L^m\otimes\cI(h^m))$ twisted with the 
corresponding $L^2$ multiplier ideal sheaves 
$$
\cI(h^m)=\cI(k\varphi)=\big\{f\in\cO_{X,x},\;\;\exists V\ni x,~
\int_V|f(z)|^2e^{-m\varphi(z)}d\lambda(z)<+\infty\big\},
$$
and provided the Morse integrals are computed on the regular locus of $h$, 
namely restricted to $X(L,h,q)\ssm\Sigma\,$:
$$
\int_{X(L,h,q)\ssm\Sigma}(-1)^q\Theta_{L,h}^n.
$$
The special case of 3.1~(SM) when $q=1$ yields a very useful
criterion for the existence of sections of large multiples of $L$.

\claim 3.2. Corollary|Let $L\to X$ be a holomorphic line bundle 
equipped with a singular hermitian metric $h=e^{-\varphi}$ 
with analytic singularities of pole set $\Sigma=\varphi^{-1}(-\infty)$.
Then we have the following lower bounds
\vskip3pt
\plainitem{\rm(a)} at the $h^0$ level~$:$
$$
\eqalign{
h^0(X,E\otimes L^m)&\geq h^0(X,E\otimes L^m\otimes\cI(h^m))\cr
&\geq h^0(X,E\otimes L^m\otimes\cI(h^m))-h^1(X,E\otimes L^m\otimes\cI(h^m))\cr
&\geq r{k^n\over n!}\int_{X(L,h,\le 1)\ssm\Sigma}\Theta_{L,h}^n -o(k^n)~.\cr}
$$
Especially $L$ is big as soon as $\int_{X(L,h,\le 1)\ssm\Sigma}\Theta_{L,h}^n>0$
for some singular hermitian metric $h$ on~$L$.
\vskip3pt
\plainitem{\rm(b)} at the $h^q$ level~$:$
$$
h^q(X,E\otimes L^m\otimes\cI(h^m))\geq r{k^n\over n!}
\sum_{j=q-1,q,q+1}(-1)^q\int_{X(L,h,j)\ssm\Sigma}\Theta_{L,h}^n -o(k^n)~.
$$
\endclaim


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

\plainsubsection 3.B. Positively curved hermitian metric on the
Semple tautological bundle|

To start with, we consider the directed variety $(\bC^r,T_{\bC^r})$ when
$\bC^r$ is equipped with its standard hermitian metric. Pick
a random $k$-jet
$$
f(t)=x+t\xi_1+t^2\xi_2+\cdots+t^k\xi_k+O(t^{k+1}),\quad
\xi_s\in\bC^r,~~1\leq s\leq k.\leqno(3.3)
$$
Given $\varepsilon_1,\ldots,\varepsilon_k>0$, 
we get a natural Finsler metric on the tautological Green-Griffiths bundle
$\cO_{\GG,(\bC^r)_k}(-1)$ of $(\bC^r,T_{\bC^r})$ by putting
$$
\Vert f\Vert_{\varepsilon,p}^\GG=\bigg(\sum_{s=1}^k
(\varepsilon_s|\xi_s|)^{2p/s}\bigg)^{1/2p}.\leqno(3.4)
$$
In fact if $\lambda\cdot f$ denotes the $t\mapsto f(\lambda t)$, we do have
the required homogeneity property under the $\bC^*$ action, namely
$\Vert\lambda\cdot f\Vert^\GG_{\varepsilon,p}=|\lambda|\,
\Vert f\Vert_{\varepsilon,p}^\GG$. The choice of a suitable  integer $p$
(e.g.\ $p=\lcm(1,2,\ldots,k)$ or a multiple) yields a smooth metric
on $\cO_{\GG,(\bC^r)_k}(-1)$. However, formula (3.4) is not $\bG'_k$ invariant
and therefore cannot be used to construct a metric on the Semple
tautological bundle $\cO_{(\bC^r)_k}(-1)$ of $(\bC^r,T_{\bC^r})$.
Let us assume that we have a regular $k$-jet, i.e. that $\xi_1\neq 0$.
By composing $f$ with a suitable
element $\varphi(t)=t+a_2t^2+\ldots+a_kt^k+O(t^{k+1})$ and applying
a Gram-Schmidt orthogonalization argument, we can always obtain
$\xi_s\in(\xi_1)^\perp$ for $s\geq 2$ (proceeding inductively
with changes of variables $t\mapsto t+a_st^s$). In fact, there is a unique
$\varphi\in\bG'_k$ achieving this condition. In view of (??), it is
natural to define a Finsler metric on $f$ (taken $\mod\bG'_k$) by
$$
\Vert f\Vert_{\varepsilon,\beta,p}=\Bigg(\sum_{s=1}^k
\big(\varepsilon_s|\xi_1|^{1-s\beta_s}|\xi_s|^{\beta_s}\big)^{2p}\Bigg)^{1/2p},
\leqno(3.5)
$$
under the assumption that $\xi_s\in(\xi_1)^\perp$ for $s\geq 2\,$;
here $\beta=(\beta_1,\ldots,\beta_s)$ is a positive weight with
$\beta_1=1$ and $0<\beta_s\leq 1/s$ for $s\geq 2$
(the first term in the sum is equal to $(\varepsilon_1|\xi_1|)^{2p}$
and we will in fact take $\varepsilon_1=1$). The curvature of 
$\cO_{(\bC^r)_k}(1)$ fiberwise (i.e.\ on the rational variety~$\bR_{r,k}$)
is given by ${i\over 2\pi}\ddbar\log\Vert f\Vert_{\varepsilon,p}^2$, provided
holomorphic coordinates are used. We compute the curvature
form at a point \hbox{$\xi^0=(\xi_s^0)_{1\leq s\leq k}$} such that
$\xi^0_1\neq 0$. By applying a dilation $t\mapsto f(\lambda t)$,
$\lambda\in\bC^*$, we can assume that $|\xi^0_1|=1$. A nearby point
$\xi=(\xi_s)_{1\leq s\leq k}$ can be written as
$\xi_1=\xi_1^0+\zeta$ and $\xi_s$, $s\geq 2$, with
$\zeta,\xi_s\in (\xi^0_1)^\perp$. Notice that $((\xi^0_1)^\perp)^k$ is
a $k(r-1)$-dimensional complex subspace that defines an affine chart
of $\bR_{r,k}$ containing $\xi^0$. The difficulty
is that we do not necessarily have $\langle\xi_s,\xi_1\rangle=0$ any more,
however $\langle\xi_s,\xi_1\rangle=\langle\xi_s,\zeta\rangle=O(|\zeta|)$
for $s\geq 2$, and we have to correct this by applying an element
$\varphi\in\bG'_k$ close to identity. When computing $i\ddbar(...)$
at $\zeta=0$, all terms $O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3)$
can be neglected, and the calculations performed below will be made modulo
such terms. In particular, higher powers of
$\langle\xi_s,\zeta\rangle$ can be be neglected as they are of the form
$O(\smash{\ol\zeta}^{\kern 1pt 2})$. A suitable choice is
$$
\varphi(t)=t-\sum_{s=2}^k\langle\xi_s,\zeta\rangle t^s.
$$
Then
$$
f\circ\varphi(t)=x+\sum_{s=1}^kt^s\tilde\xi_s+O(t^{k+1})
$$
where $\tilde\xi_1=\xi_1$ and
$$
\tilde\xi_s=\xi_s- \langle\xi_s,\zeta\rangle\xi_1-\sum_{j=2}^{s-1}j
\langle\xi_{s-j+1},\zeta\rangle\xi_j~~\mod
O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3),\quad s\geq 2
$$
(the final summation is obtained by expanding the
terms $(t-\sum_{\ell\geq 2}\langle\xi_\ell,\zeta\rangle t^\ell)^j\xi_j$
for $2\leq j\leq s-1$ and $\ell=s-j+1$). Observe that
$|\xi_1|^2=1+ |\zeta|^2$ and
$\langle\xi_j,\xi_1\rangle=\langle\xi_j,\xi^0_1\rangle+
\langle\xi_j,\zeta\rangle=\langle\xi_j,\zeta\rangle$ for $j\geq 2$, hence
$$
\langle\tilde\xi_s,\xi_1\rangle=\langle \xi_s,\zeta\rangle
-\langle \xi_s,\zeta\rangle(1+|\zeta|^2)
-\sum_{j=2}^{s-1}j\langle\xi_{s-j+1},\zeta\rangle\langle \xi_j,\zeta\rangle
=O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3),
$$
so we do not need any more accurate correction. Moreover, by a straightforward
calculation
$$
|\tilde\xi_s|^2=
\bigg|\xi_s-\sum_{j=2}^{s-1}j\langle\xi_{s-j+1},\zeta\rangle\xi_j\bigg|^2
-|\langle\xi_s,\zeta\rangle|^2
~~\mod O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3).
$$
Therefore, for $\varepsilon_1=1$, we find
$$
\leqalignno{
\Vert f\Vert_{\varepsilon,\beta,p}
&=\Bigg(|\xi_1|^{2p}+\sum_{s=2}^k\big(\varepsilon_s|\xi_1|^{1-s\beta_s}
|\tilde\xi_s|^{\beta_s}\big)^{2p}\Bigg)^{1/2p}\cr
&=\Bigg((1+|\zeta|^2)^p+\sum_{s=2}^k\big(\varepsilon_s^2(1+|\zeta|^2)^{1-s\beta_s}
|\tilde\xi_s|^{2\beta_s}\big)^p\Bigg)^{1/2p}&(3.6)\cr}
$$
We take profit of the constants $\varepsilon_s$ to perform a rescaling
and set $u_s=\varepsilon_s\xi_s$. Then our formula becomes
$$
\Vert f\Vert_{\varepsilon,\beta,p}^2=
\bigg((1+|\zeta|^2)^p+\sum_{s=2}^k\big((1+|\zeta|^2)^{1-s\beta_s}
|\tilde u_s|^{2\beta_s}\big)^p\bigg)^{1/2p}\leqno(3.7)
$$
where (modulo negligible terms) the $u_s$ and $\tilde u_s$ are defined by
$u_1=u^0_1+\zeta$, $|u^0_1|^2=1$,
$$
\eqalign{
\tilde u_s&=u_s- \langle u_s,\zeta\rangle u_1-\sum_{j=2}^{s-1}
{\varepsilon_s\over \varepsilon_j\varepsilon_{s-j+1}}
j\langle u_{s-j+1},\zeta\rangle u_j\cr
|\tilde u_s|^2&=
\bigg|u_s-\sum_{j=2}^{s-1}{\varepsilon_s\over \varepsilon_j\varepsilon_{s-j+1}}
j\langle u_{s-j+1},\zeta\rangle u_j\bigg|^2-|\langle u_s,\zeta\rangle|^2.\cr}
$$
By taking $\varepsilon_k\ll \varepsilon_{k-1}\ll\cdots\ll\varepsilon_2\ll 1$,
all terms $\varepsilon_s/(\varepsilon_j\varepsilon_{s-j+1})$ will become
negligible (e.g.\ equal to $\varepsilon$ if we put
$\varepsilon_s=\varepsilon^{2s-3}$, $s\geq 2$). However, we have to estimate the
errors more precisely and for this, we observe that for all complex numbers
$x,y$ and all $\gamma>0$, we have
$$
|xy|\leq |x|^{1+\gamma}+|y|^{1+1/\gamma}
$$
(consider the disjoint cases $|y|\leq |x|^\gamma$ and $|x|<|y|^{1/\gamma}$).
In particular
$$
(|u_{s_j+1}||u_j|)^{\beta_s}\leq |u_{s-j+1}|^{\beta_s(1+\gamma)}+
|u_j|^{\beta_s(1+1/\gamma)}.
$$
In this circumstance, we want that $\beta_s(1+\gamma)=\beta_{s-j+1}$
and $\beta_s(1+1/\gamma)=\beta_j$, whence
$\gamma=\beta_{s-j+1}/\beta_j$, and so we get the condition
$$
\beta_s(1+\beta_{s-j+1}/\beta_j)=\beta_{s-j+1},\qquad
\hbox{i.e}\quad {1\over \beta_j}+{1\over \beta_{s-j+1}}={1\over \beta_s}
$$
for all $j,s\geq 2$. This leads us to take $1/\beta_s=\hbox{Const}\,(s-1)$,
i.e.\ $\beta_s={\alpha/(s-1)}$ for $s\geq 2$, and we want to take the
largest possible value of $\alpha>0$. In any case, with the above
choise of the $\beta_s$ and with $|\zeta|\leq 1$, we find 
$$
\Vert f\Vert_{\varepsilon,\beta,p}^2=
\bigg((1+|\zeta|^2)^p+\sum_{s=2}^k(1+|\zeta|^2)^{p(1-s\beta_s)}
\big(|u_s|^2-|\langle u_s,\zeta\rangle|^2\big)^{p\beta_s}\bigg)^{1/2p}
(1\pm O(\varepsilon))
\leqno(3.8)
$$
by expanding the $\varepsilon$ terms in the squares~$|\tilde u_s|^2$.
Modulo negligible terms $O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3)$,
we have
$$
|u_s|^2-|\langle u_s,\zeta\rangle|^2=
|u_s|^2+|u_s\wedge\zeta|^2-|u_s|^2|\zeta|^2
={|u_s|^2+|u_s\wedge\zeta|^2\over 1+|\zeta|^2}.
$$
hence
$$
\Vert f\Vert_{\varepsilon,\beta,p}^2=
\bigg((1+|\zeta|^2)^p+\sum_{s=2}^k(1+|\zeta|^2)^{p(1-(s+1)\beta_s)}
\big(|u_s|^2+|u_s\wedge\zeta|^2\big)^{p\beta_s}\bigg)^{1/2p}
(1\pm O(\varepsilon)).
\leqno(3.8')
$$
In order to ensure the plurisubharmonicity  of the summation between
the big parentheses (so as to get a singular hermitian metric of
$\cO_{X_k}(1)$ of nonnegative relative curvature form), we want
$\beta_s=\alpha/(s-1)\leq 1/(s+1)$.
For $s\geq 2$, this leads us to take $\alpha=1/3$, hence $\beta_s=1/3(s-1)$.

A suitable choice is
$$
\varphi(t)=t-|\xi^0_1|^{-2}\sum_{s=2}^k\langle\xi_s,\zeta\rangle t^s.
$$
Then
$$
f\circ\varphi(t)=x+\sum_{s=1}^kt^s\tilde\xi_s+O(t^{k+1})
$$
where $\tilde\xi_1=\xi_1$ and
$$
\tilde\xi_s=\xi_s- |\xi^0_1|^{-2}\Bigg(
\langle\xi_s,\zeta\rangle\xi_1+\sum_{j=2}^{s-1}j
\langle\xi_{s-j+1},\zeta\rangle\xi_j\Bigg)~~\mod
O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3),\quad s\geq 2
$$
(the final summation is obtained by expanding the
terms $(t-\langle\xi_\ell,\zeta\rangle t^\ell)^j\xi_j$
for $\ell=s-j+1$, $j\leq s-1$. As
$\langle\tilde\xi_s,\xi_1\rangle=|\xi^0_1|^2+|\zeta|^2$, we have
$$
\langle\tilde\xi_s,\xi_1\rangle=\langle \xi_s,\zeta\rangle
-\langle \xi_s,\zeta\rangle{|\xi^0_1|^2+|\zeta|^2\over |\xi^0_1|^2}
-|\xi^0_1|^{-2}\sum_{j=2}^{s-1}j
\langle\xi_{s-j+1},\zeta\rangle\langle \xi_j,\zeta\rangle
=O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3),
$$
so we do not need any more accurate correction. Moreover, by a straightforward
calculation
$$
|\tilde\xi_s|^2=
\bigg|\xi_s-\sum_{j=2}^{s-1}{j\langle\xi_{s-j+1},\zeta\rangle\over
  |\xi^0_1|^2}\xi_j\bigg|^2-{|\langle\xi_s,\zeta\rangle|^2\over |\xi^0_1|^2}
~~\mod O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3).
$$
Therefore, we find
$$
\leqalignno{
\Vert f\Vert_{\varepsilon,p}
&=\bigg(\sum_{s=1}^k\varepsilon_s\big(|\xi_1|^{s-1}
|\tilde\xi_s|\big)^{2p/(2s-1)}\bigg)^{1/2p}\cr
&=\bigg(\sum_{s=1}^k\varepsilon_s\big((1+|\zeta|^2)^{s-1}
|\tilde\xi_s|\big)^{p/(2s-1)}\bigg)^{1/2p}&(3.6)\cr}
$$









By rotating the coordinates and using
a dilation, we can always reduce ourselves to computing the curvature
at a point \hbox{$\xi^0=(\xi_s^0)_{1\leq s\leq k}$} such that
$\xi^0_1=(1,0,\ldots,0)$ and
$\langle\xi^0_s,\xi^0_1\rangle=0$, i.e.\ $\xi^0_{s,1}=0$ for $s\geq 2$.
A nearby point $(\xi_s)_{1\leq s\leq k}$ is given by
$\xi_1=\xi_1^0+\zeta=(1,\zeta_2,\ldots,\zeta_r)$ and 
$\xi_s=(0,\xi_{s,2},\ldots,\xi_{s,r})$, and the $k(r-1)$ coordinates
$\zeta_j$ and $\xi_{s,j}$,
$2\leq s,j\leq r$, precisely define the holomorphic structure. The difficulty
is that we do not necessarily have $\langle\xi_s,\xi_1\rangle=0$ any more,
however $\langle\xi_s,\xi_1\rangle=\langle\xi_s,\zeta\rangle=O(|\zeta|)$
for $s\geq 2$, and we have to correct this by applying an element
$\varphi\in\bG'_k$ close to identity. When computing $i\ddbar(...)$
at $\zeta=0$, all terms $O(\zeta^2,\smash{\ol\zeta}^{\kern 1pt 2},|\zeta|^3)$
can be neglected, and the calculations performed below will be made modulo
such terms. In particular, higher powers of
$\langle\xi_2,\zeta\rangle$ can be be neglected (they are of the form
$O(\smash{\ol\zeta}^{\kern 1pt 2})$).


\plainsubsection 9.A. Introduction|

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

\claim 9.1. Theorem| Let $(X,V)$ be a directed projective variety
such that $K_V$ is big and let $A$ be an ample divisor. Then 
for $k\gg 1$ and $\delta\in\bQ_+$ small enough, $\delta\le c(\log k)/k$,
the number of sections
$h^0(X,E^\GG_{k,m}V^*\otimes\cO(-m\delta A))$ has maximal growth, 
i.e.\ is larger that $c_km^{n+kr-1}$ for some~$m\ge m_k$, 
where $c,\,c_k>0$, $n=\dim X$ and $r=\rank V$. In particular, entire curves
$f:(\bC,T_\bC)\to(X,V)$ satisfy $($many$)$ algebraic differential 
equations.
\endclaim

The statement is very elementary to check when $r=\rank V=1$, and 
therefore when $n=\dim X=1$. In higher dimensions  $n\ge 2$, only 
very partial results were known before Theorem 9.1 was obtained in [Dem11],
[and they dealt merely with the absolute case $V=T_X$]. In dimension~$2$,
Theorem~9.1 is a consequence of the Riemann-Roch calculation of 
Green-Griffiths [GrGr79], 
combined with a vanishing theorem due to Bogomolov [Bog79] -- the latter 
actually only applies to the top cohomology group $H^n$, and things 
become much more delicate when extimates of intermediate cohomology
groups are needed. In higher dimensions, Diverio [Div08, Div09] proved the 
existence of sections of $H^0(X,E^\GG_{k,m}V^*\otimes\cO(-1))$
whenever $X$ is a hypersurface of $\bP^{n+1}_\bC$ of high degree $d\ge
d_n$, assuming $k\ge n$ and $m\ge m_n$. More recently, Merker [Mer15]
was able to treat the case of arbitrary hypersurfaces of general type,
i.e.\ $d\ge n+3$, assuming this time $k$ to be very large.  The latter
result is obtained through explicit algebraic calculations of the
spaces of sections, and the proof is computationally very
intensive. B\'erczi [Ber15, Ber18] also obtained related results with a
different approach based on residue formulas, assuming e.g.\ $d\ge n^{9n}$.

All these approaches are algebraic in nature. Here,
however, our techniques are based on more elaborate curvature estimates in 
the spirit of Cowen-Griffiths [CoGr76]. They require holomorphic Morse 
inequalities (see 9.10 below) -- and we
do not know how to translate our method in an algebraic setting.
Notice that holomorphic Morse inequalities are essentially
insensitive to singularities, as we can pass to non singular models
and blow-up $X$ as much as we want: if $\mu:\wt X\to X$ is a
modification then $\mu_*\cO_{\wt X}=\cO_X$ and $R^q\mu_*\cO_{\wt X}$
is supported on a codimension $1$ analytic subset (even codimension $2$ if
$X$ is smooth). It~follows from the
Leray spectral sequence that the cohomology estimates for $L$ on $X$ or for
$\wt L=\mu^*L$ on $\wt X$ differ by negligible terms, i.e.
$$
h^q(\wt X,\wt L^{\otimes m})-h^q(X,L^{\otimes m})=O(m^{n-1}).\leqno(9.2)
$$
Finally, singular holomorphic Morse inequalities (in the form obtained 
by L.~Bonavero 
[Bon93]) allow us to work with singular Hermitian metrics $h$; this is the
reason why we will only require to have big line bundles rather than
ample line bundles. In the case of linear subspaces $V\subset T_X$, we
introduce singular Hermitian metrics as follows.

\claim 9.3. Definition| A singular hermitian metric on a linear
subspace $V\subset T_X$ is a metric $h$ on the fibers of $V$ such that
the function $\log h:\xi\mapsto\log|\xi|_h^2$ is locally integrable 
on the total space of $V$.
\endclaim

Such a metric can also be viewed as a singular Hermitian metric on
the tauto\-logical line bundle $\cO_{P(V)}(-1)$ on the projectivized bundle
$P(V)=V\ssm\{0\}/\bC^*$, and therefore its dual metric $h^*$ defines a 
curvature current $\Theta_{\cO_{P(V)}(1),h^*}$ of type $(1,1)$ on
$P(V)\subset P(T_X)$, such that
$$
p^*\Theta_{\cO_{P(V)}(1),h^*} ={\ii\over 2\pi}\ddbar\log h,\qquad
\hbox{where $p:V\ssm\{0\}\to P(V)$}.\leqno(9.4)
$$
If $\log h$ is quasi-plurisubharmonic $($or quasi-psh, which means psh 
modulo addition of a smooth function$)$ on $V$, then $\log h$ is indeed
locally integrable, and we have moreover
$$\Theta_{\cO_{P(V)}(1),h^*}\ge -C\omega\leqno(9.5)$$
for some smooth positive $(1,1)$-form on $P(V)$ and some constant $C>0\;$;
conversely, if (9.5) holds, then $\log h$ is quasi-psh.

\claim 9.6. Definition| We will say that a singular Hermitian metric $h$ on
$V$ is {\rm admissible} if $h$ can be written as $h=e^\varphi h_{0|V}$ where
$h_0$ is a smooth positive definite Hermitian on $T_X$ and $\varphi$
is a quasi-psh weight with analytic singularities on $X$, as in 
Definition~$9.3$. 
Then $h$ can be seen as a singular hermitian metric on $\cO_{P(V)}(1)$,
with the property that it induces a smooth positive definite metric
on a Zariski open set $X'\subset X\ssm \Sing(V)\,;$ we will denote by
$\Sing(h)\supset\Sing(V)$ the complement of the largest such Zariski
open set $X'$.
\endclaim

If $h$ is an admissible metric, we define 
$\cO_h(V^*)$ to be the sheaf of germs of holomorphic sections 
sections of $V^*_{|X\ssm\Sing(h)}$ which are $h^*$-bounded 
near $\Sing(h)$; 
by the assumption on the analytic singularities, 
this is a coherent sheaf (as the direct image of some coherent sheaf on
$P(V)$), and actually, since $h^*=e^{-\varphi}h_0^*$, it is 
a subsheaf of the
sheaf $\cO(V^*):=\cO_{h_0}(V^*)$ associated with a smooth positive
definite metric $h_0$  on $T_X$. If $r$ is the generic rank of $V$ and
$m$ a positive integer, we define similarly 
$$
\leqalignno{
\bddK_{V,h}^{[m]}={}&\hbox{sheaf of germs of holomorphic sections of 
$(\det V^*_{|X'})^{\otimes m}=(\Lambda^rV^*_{|X'})^{\otimes m}$}&(9.7)\cr
\noalign{\vskip-2pt}
&\hbox{which are $\det h^*$-bounded},\cr}
$$
so that $\bddK_V^{[m]}:=\bddK_{V,h_0}^{[m]}$ according to Def.~2.7.
For a given admissible Hermitian structure $(V,h)$, we define similarly
the sheaf $E^\GG_{k, m}V^*_h$ to be the sheaf of 
polynomials defined over $X\ssm\Sing(h)$ which are \hbox{``$h$-bounded''}.
This means that when they are viewed as polynomials $P(z\,;\,\xi_1,\ldots,
\xi_k)$ in terms of $\xi_j=(\nabla_{h_0}^{1,0})^jf(0)$ where
$\nabla_{h_0}^{1,0}$ is the $(1,0)$-component of the induced Chern 
connection on $(V,h_0)$, there is a uniform bound 
$$
\big|P(z\,;\,\xi_1,\ldots,\xi_k)\big|\le 
C\Big(\sum\Vert \xi_j\Vert_h^{1/j}\Big)^m\leqno(9.8)
$$
near points of $X\ssm X'$ (see section~2 for more details on
this). Again, by a direct image argument, one sees that
$E^\GG_{k, m}V^*_h$ is always a coherent sheaf. The sheaf
$E^\GG_{k, m}V^*$ is defined to be $E^\GG_{k, m}V^*_h$ 
when $h=h_0$ (it is actually independent of the choice of $h_0$, as
follows from arguments similar to those given in section~2). Notice
that this is exactly what is needed to extend the proof of the vanishing
theorem~8.15 to the case of a singular linear space $V\,$; the value 
distribution theory argument can only work when the functions 
$P(f\,;\;f',\ldots,f^{(k)})(t)$ do not exhibit poles, and this is guaranteed
here by the boundedness assumption.

Our strategy can be described as follows. We consider the
Green-Griffiths bundle of $k$-jets $X_k^\GG=J^kV\ssm\{0\}/\bC^*$,
which by (9.3) consists of a fibration in {\em weighted projective
spaces}, and its associated tautological sheaf 
$$L=\cO_{X_k^\GG}(1),$$
viewed rather as a virtual $\bQ$-line bundle $\cO_{X_k^\GG}(m_0)^{1/m_0}$
with $m_0=\lcm(1,2,\,...\,,k)$.
Then, if $\pi_k:X_k^\GG\to X$ is the natural projection, we have
$$
E^\GG_{k,m}=(\pi_k)_*\cO_{X_k^\GG}(m)\quad\hbox{and}\quad
R^q(\pi_k)_*\cO_{X_k^\GG}(m)=0~\hbox{for $q\ge 1$}.
$$
Hence, by the Leray spectral sequence we get for every invertible sheaf
$F$ on $X$ the isomorphism
$$
H^q(X,E^\GG_{k,m}V^*\otimes F)\simeq 
H^q(X_k^\GG,\cO_{X_k^\GG}(m)\otimes\pi_k^*F).\leqno(9.9)
$$
The latter group can be evaluated thanks to holomorphic Morse inequalities.
Let us recall the main statement.

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

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

\noindent Moreover $($cf.\ Bonavero's PhD thesis {\rm [Bon93]}$)$, 
if $h=e^{-\varphi}$ is a singular hermitian metric with analytic 
singularities of pole set $P=\varphi^{-1}(-\infty)$, the estimates 
still hold provided all cohomology groups
are replaced by cohomology groups 
$H^q(X,E\otimes L^m\otimes\cI(h^m))$ twisted with the 
corresponding $L^2$ multiplier ideal sheaves 
$$
\cI(h^m)=\cI(k\varphi)=\big\{f\in\cO_{X,x},\;\;\exists V\ni x,~
\int_V|f(z)|^2e^{-m\varphi(z)}d\lambda(z)<+\infty\big\},
$$
and provided the Morse integrals are computed on the regular locus of $h$, 
namely restricted to $X(L,h,q)\ssm\Sigma\,$:
$$
\int_{X(L,h,q)\ssm\Sigma}(-1)^q\Theta_{L,h}^n.
$$
The special case of 9.10~(SM) when $q=1$ yields a very useful
criterion for the existence of sections of large multiples of $L$.

\claim 9.11. Corollary|Let $L\to X$ be a holomorphic line bundle 
equipped with a singular hermitian metric $h=e^{-\varphi}$ 
with analytic singularities of pole set $\Sigma=\varphi^{-1}(-\infty)$.
Then we have the following lower bounds
\vskip3pt
\plainitem{\rm(a)} at the $h^0$ level~$:$
$$
\eqalign{
h^0(X,E\otimes L^m)&\geq h^0(X,E\otimes L^m\otimes\cI(h^m))\cr
&\geq h^0(X,E\otimes L^m\otimes\cI(h^m))-h^1(X,E\otimes L^m\otimes\cI(h^m))\cr
&\geq r{k^n\over n!}\int_{X(L,h,\le 1)\ssm\Sigma}\Theta_{L,h}^n -o(k^n)~.\cr}
$$
Especially $L$ is big as soon as $\int_{X(L,h,\le 1)\ssm\Sigma}\Theta_{L,h}^n>0$
for some singular hermitian metric $h$ on~$L$.
\vskip3pt
\plainitem{\rm(b)} at the $h^q$ level~$:$
$$
h^q(X,E\otimes L^m\otimes\cI(h^m))\geq r{k^n\over n!}
\sum_{j=q-1,q,q+1}(-1)^q\int_{X(L,h,j)\ssm\Sigma}\Theta_{L,h}^n -o(k^n)~.
$$
\endclaim

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

\plainsubsection 9.B. Hermitian geometry of weighted projective spaces|

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

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

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

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

\plainsubsection 9.C. Probabilistic estimate of the curvature of $k$-jet bundles|

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

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

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

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

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

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

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

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

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

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

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

In fact, thanks to Bonavero's singular holomorphic Morse inequalities 
[Bon93], everything works almost unchanged 
in the case where $V\subset T_X$
has singularities and $h$ is an admissible metric on $V$ (see Definition~9.6).
We only have to find a blow-up $\mu:\wt X_k\to X_k$ so that
the resulting pull-backs $\mu^*L_k$ and $\mu^*V$ are locally free,
and $\mu^*\det h^*$, $\mu^*\Psi_{h,p,\varepsilon}$ only have divisorial
singularities. Then $\eta$ is a $(1,1)$-current with logarithmic poles,
and we have to deal with smooth metrics on $
\mu^*L_k^{\otimes m}\otimes\cO(-mE_k)$ where $E_k$ is a certain effective 
divisor on $X_k$ (which, by our assumption in 9.6, does not project onto
$X$). The cohomology groups involved are then the twisted
cohomology groups
$$
H^q(X_k^\GG,\cO(L_k^{\otimes m})\otimes\cJ_{k,m})
$$
where $\cJ_{k,m}=\mu_*(\cO(-mE_k))$ is the corresponding multiplier ideal sheaf,
and the Morse integrals need only be evaluated in the complement of the 
poles, that is on $X(\eta,q)\ssm S$ where $S=\Sing(V)\cup\Sing(h)$. Since
$$
(\pi_k)_*\big(\cO(L_k^{\otimes m})\otimes\cJ_{k,m}\big)\subset
E_{k,m}^\GG V^*\otimes
\cO\Big({m\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)F\Big)\Big)
$$
we still get a lower bound for the $H^0$ of the latter sheaf (or for the $H^0$
of the un-twisted line bundle $\cO(L_k^{\otimes m})$ on $X_k^\GG$).
If we assume that $K_V\otimes F$ is big, these considerations
also allow us to obtain a strong estimate in terms of the volume, by
using an approximate Zariski decomposition on a suitable blow-up of~$(X,V)$.
The following corollary implies in particular Theorem~9.1.

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

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

The following corollary implies Theorem~0.12.

\claim 9.40. Corollary|Let $(X,V)$ be a projective directed manifold such that
$K_V^\bullet$ is big, and $A$ an ample $\bQ$-divisor on $X$ such that 
$K_V^\bullet\otimes\cO(-A)$ is still big. Then, if we put 
$\delta_k={1\over kr}
(1+{1\over 2}+\ldots+{1\over k})$, $r=\rank V$, the space of global 
invariant jet differentials
$$
H^0(X,E_{k,m}V^*\otimes\cO(-m\delta_kA))
$$
has $($many$)$ non zero sections for $m\gg k\gg 1$ and $m$ sufficiently 
divisible.
\endclaim

\plainproof. Corollary 9.39 produces a non zero section 
$P\in H^0(E_{k,m}^\GG V^*\otimes\cO_X(-m\delta_kA))$ for $m\gg k\gg 1$, and
the arguments given in subsection 7.D (cf.\ (7.30)) yield a non zero section
$$Q\in H^0(E_{k,m'}V^*\otimes\cO_X(-m\delta_kA)),\quad m'\le m.$$
By raising $Q$ to some power $p$ and using
a section $\sigma\in H^0(X,\cO_X(dA))$, we obtain a section
$$
Q^p\sigma^{mq}\in H^0(X,E_{k,pm'}V^*\otimes\cO(-m(p\delta_k-qd)A)).
$$
One can adjust $p$ and $q$ so that $m(p\delta_k-qd)=pm'\delta_k$ and
$pm'\delta_kA$ is an integral divisor.\qed

\claim 9.41. Example| {\rm In some simple cases, the above estimates can 
lead to very explicit results. Take for instance $X$ to be a smooth
complete intersection of multidegree $(d_1,d_2,\ldots,d_s)$ in $\bP^{n+s}_\bC$
and consider the absolute case $V=T_X$. Then 
$K_X=\cO_X(d_1+\ldots+d_s-n-s-1)$ and one can check via
explicit bounds of the error terms (cf.\ [Dem11], [Dem12])
that a sufficient condition for the existence of
sections is
$$
k\ge\exp\Big(7.38\,n^{n+1/2}\Big({\sum d_j+1\over\sum d_j-n-s-a-1}\Big)^n\Big).
$$
This is good in view of the fact that we can cover arbitrary smooth 
complete intersections of general type. On the other hand, even when the
degrees $d_j$ tend to $+\infty$, we still get a large lower bound
$k\sim \exp(7.38\,n^{n+1/2})$ on the order of jets, and this is far 
from being optimal$\,$: Diverio [Div08, Div09] has shown e.g.\ that one can take
$k=n$ for smooth hypersurfaces of high degree, using the 
algebraic Morse inequalities of Trapani [Tra95].
The next paragraph uses essentially the same idea, in our more analytic
setting.\qed}
\endclaim

\plainsubsection 9.D. Non probabilistic estimate of the Morse integrals|

We assume here that the curvature tensor $(c_{ij\alpha\beta})$ satisfies a lower
bound
$$
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}\xi_i\ol\xi_ju_\alpha\ol u_\beta\ge -\sum\gamma_{ij}\xi_i
\ol\xi_j\;|u|^2,
\qquad\forall\xi\in T_X,~u\in V\leqno(9.42)
$$
for some semipositive $(1,1)$-form $\gamma={i\over 2\pi}\sum\gamma_{ij}(z)\,
dz_i\wedge d\ol z_j$ on~$X$. This is the
same as assuming that the curvature tensor of $(V^*,h^*)$ satisfies
the semipositivity condition
$$
\Theta_{V^*,h^*}+\gamma\otimes\Id_{V^*}\ge 0\leqno(9.42')
$$
in the sense of Griffiths, or equivalently $\Theta_{V,h}-\gamma\otimes\Id_V\le 0$.
Thanks to the compactness of~$X$, such a form $\gamma$ always
exists if $h$ is an admissible metric on~$V$. Now, instead of replacing $\Theta_V$
with its trace free part $\wt\Theta_V$ and exploiting a Monte Carlo convergence
process, we replace $\Theta_V$ with $\Theta_V^\gamma=\Theta_V-\gamma\otimes\Id_V\le 0$, 
i.e.\ $c_{ij\alpha\beta}$ by 
$c_{ij\alpha\beta}^\gamma=c_{ij\alpha\beta}+\gamma_{ij}\delta_{\alpha\beta}$. Also, we take
a line bundle $F=A^{-1}$ with $\Theta_{A,h_A}\ge 0$, i.e.\ $F$ seminegative.
Then our earlier formulas (9.28), (9.35), (9.36) become instead
$$
\leqalignno{
&g_k^\gamma(z,x,u)={\ii\over 2\pi}\sum_{1\le s\le k}{1\over s}x_s
\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}^\gamma(z)\,
u_{s\alpha}\overline u_{s\beta}\,dz_i\wedge d\overline z_j\ge 0,
&(9.43)\cr
&L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big(-{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)A\Big),
&(9.44)\cr
&\Theta_{L_k}=\eta_k(z,x,u)={1\over {1\over kr}(1+{1\over 2}+\ldots+{1\over k})}
g_k^\gamma(z,x,u)-(\Theta_{A,h_A}(z)+r\gamma(z)).&(9.45)\cr}
$$
In fact, replacing $\Theta_V$ by $\Theta_V-\gamma\otimes\Id_V$ has the effect of
replacing $\Theta_{\det V^*}=\Tr\Theta_{V^*}$ by $\Theta_{\det V^*}+r\gamma$. The major
gain that we have is that $\eta_k=\Theta_{L_k}$ is now expressed as a difference
of semipositive $(1,1)$-forms, and we can exploit the following simple lemma, which is
the key to derive algebraic Morse inequalities from their analytic form
(cf.\ [Dem94], Theorem~12.3).

\claim 9.46.~Lemma|Let $\eta=\alpha-\beta$ be a difference of semipositive 
$(1,1)$-forms on an $n$-dimensional complex manifold~$X$, 
and let $\bOne_{\eta,\le q}$ be the characteristic function of the
open set where $\eta$ is non degenerate with a number of negative eigenvalues 
at most equal to~$q$.
Then
$$
(-1)^q\bOne_{\eta,\le q}~\eta^n\le \sum_{0\le j\le q}(-1)^{q-j}\alpha^{n-j}\beta^j,
$$
in particular
$$
\bOne_{\eta,\le 1}~\eta^n\ge \alpha^n-n\alpha^{n-1}\wedge \beta\qquad\hbox{for $q=1$.}
$$
\endclaim

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

We apply here Lemma 9.46 with
$$
\alpha=g_k^\gamma(z,x,u),\qquad\beta=\beta_k=
{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)(\Theta_{A,h_A}+r\gamma),
$$
which are both semipositive by our assumption. The analogue of (9.32) leads to
$$
\eqalign{
&\int_{X^\GG_k(L_k,\le 1)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}\cr
&\quad{}={(n+kr-1)!\over n!(k!)^r(kr-1)!}\int_{z\in X}
\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
\bOne_{g_k^\gamma-\beta_k,\le 1}\;(g_k^\gamma-\beta_k)^n\,d\nu_{k,r}(x)\,d\mu(u)\cr
&\quad{}\ge{(n+kr-1)!\over n!(k!)^r(kr-1)!}\int_{z\in X}
\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
((g_k^\gamma)^n-n(g_k^\gamma)^{n-1}\wedge\beta_k)\,d\nu_{k,r}(x)\,d\mu(u).\cr}
$$
The resulting integral now produces a ``closed formula'' which can be
expressed solely in terms of Chern classes (at least if we assume that
$\gamma$ is the Chern form of some semipositive line bundle). It is
just a matter of routine to find a sufficient condition for the
positivity of the integral. One can first observe that $g_k^\gamma$ is
bounded from above by taking the trace of $(c_{ij\alpha\beta})$, in
this way we get
$$
0\le g_k^\gamma\le\bigg(\sum_{1\le s\le k}{x_s\over s}\bigg)\big(\Theta_{\det V^*}+r\gamma\big)
$$
where the right hand side no longer depends on $u\in (S^{2r-1})^k$. 
Also, $g_k^\gamma$ can be written as a sum of semipositive $(1,1)$-forms
$$
g_k^\gamma =\sum_{1\le s\le k}{x_s\over s}\theta^\gamma(u_s),\qquad
\theta^\gamma(u)=\sum_{i,j,\alpha,\beta}c_{ij\alpha\beta}^\gamma
u_\alpha\ol u_\beta\,dz_i\wedge d\ol z_j,
$$
hence for $k\ge n$ we have
$$
(g_k^\gamma)^n\ge n!\sum_{1\le s_1<\ldots<s_n\le k}
{x_{s_1}\ldots x_{s_n}\over s_1\ldots s_n}\,
\theta^\gamma(u_{s_1})\wedge \theta^\gamma(u_{s_2})\wedge\ldots\wedge \theta^\gamma(u_{s_n}).
$$
Since $\int_{S^{2r-1}}\theta^\gamma(u)\,d\mu(u)={1\over r}\Tr(\Theta_{V^*}+\gamma)=
{1\over r}\Theta_{\det V^*}+\gamma$, we infer from this
$$
\eqalign{
&\int_{(x,u)\in\Delta_{k-1}\times(S^{2r-1})^k}
(g_k^\gamma)^n\,d\nu_{k,r}(x)\,d\mu(u)\cr
&\qquad\ge 
n!\sum_{1\le s_1<\ldots<s_n\le k}
{1\over s_1\ldots s_n}\Big(\int_{\Delta_{k-1}}x_1\ldots x_n\,d\nu_{k,r}(x)\Big)
\Big({1\over r}\Theta_{\det V^*}+\gamma\Big)^n.\cr}
$$
By putting everything together, we conclude:

\claim 9.47.~Theorem|Assume that $\Theta_{V^*}\ge-\gamma\otimes\Id_{V^*}$ with a
semipositive $(1,1)$-form $\gamma$ on $X$. Then the Morse integral of the line bundle
$$
L_k=\cO_{X_k^\GG}(1)\otimes
\pi_k^*\cO\Big(-{1\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)A\Big),\qquad
A\ge 0
$$
satisfies for $k\ge n$ the inequality
$$
\leqalignno{
&{1\over (n+kr-1)!}\int_{X^\GG_k(L_k,\le 1)}\Theta_{L_k,\Psi^*_{h,p,\varepsilon}}^{n+kr-1}\cr
&\qquad{}\ge{1\over n!(k!)^r(kr-1)!}
\int_Xc_{n,r,k}\big(\Theta_{\det V^*}+r\gamma\big)^n-c'_{n,r,k}
\big(\Theta_{\det V^*}+r\gamma\big)^{n-1}\wedge
\big(\Theta_{A,h_A}+r\gamma\big)&(*)\cr}
$$
where
$$
\eqalign{
c_{n,r,k}&={n!\over r^n}\Big(\sum_{1\le s_1<\ldots<s_n\le k}
{1\over s_1\ldots s_n}\Big)\int_{\Delta_{k-1}}x_1\ldots x_n\,d\nu_{k,r}(x),\cr
c'_{n,r,k}&={n\over kr}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)
\int_{\Delta_{k-1}}\Big(\sum_{1\le s\le k}{x_s\over s}\Big)^{n-1}\,d\nu_{k,r}(x).
\cr}
$$
Especially we have a lot of sections in $H^0(X_k^\GG,mL_k)$, $m\gg 1$, as soon
as the difference occurring in $(*)$ is positive.
\endclaim

The statement is also true for $k<n$, but then $c_{n,r,k}=0$ and the lower bound $(*)$ cannot
be positive. By Corollary 9.11, it still provides a non trivial lower bound for
$h^0(X_k^\GG,mL_k)-h^1(X_k^\GG,mL_k)$, though. For $k\ge n$ we have $c_{n,r,k}>0$ and
$(*)$ will be positive if $\Theta_{\det V^*}$ is large enough. By Formula~9.20 we have
$$
c_{n,r,k}={n!\,(kr-1)!\over (n+kr-1)!}
\smash{\sum_{1\le s_1<\ldots<s_n\le k}{1\over s_1\ldots s_n}\ge
{(kr-1)!\over (n+kr-1)!}},\leqno(9.48)
$$
(with equality for $k=n$), and by ([Dem11], Lemma 2.20~(b)) we get
the upper bound 
$$
{c'_{n,k,r}\over c_{n,k,r}}\le {(kr+n-1)r^{n-2}\over k/n}
\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^n\smash{\bigg[
1+{1\over 3}\sum_{m=2}^{n-1}{2^m(n-1)!\over(n-1-m)!}\Big(1+{1\over 2}+\ldots+{1\over k}\Big)^{-m}\bigg]}.
$$
The case $k=n$ is especially interesting. For $k=n\ge 2$ one can show
(with $r\le n$ and $H_n$ denoting the harmonic sequence) that
$$
{c'_{n,k,r}\over c_{n,k,r}}\le {n^2+n-1\over 3}n^{n-2}\exp\Big({2(n-1)\over H_n}+n\log H_n\Big)
\le {1\over 3} \big(n\log(n\log 24n)\big)^n.
\leqno(9.49)
$$
We will later need the particular values that can be obtained by
direct calculations (cf.\ Formula (9.20 and [Dem11, Lemma 2.20]).
$$
\leqalignno{
\noalign{\vskip-3pt}
&c_{2,2,2}={1\over 20},\kern4.8pt\qquad c'_{2,2,2}={9\over 16},\kern9.8pt\qquad
{c'_{2,2,2}\over c_{2,2,2}}={45\over 4},&(9.50_2)\cr
&c_{3,3,3}={1\over 990},\qquad c'_{3,3,3}={451\over 4860},\qquad
{c'_{3,3,3}\over c_{3,3,3}}={4961\over 54}.&(9.50_3)\cr
\noalign{\vskip-5pt}}
$$


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\endgroup

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(version of February 24, 2018, printed on \today, \timeofday)
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Jean-Pierre Demailly\hfil\break
Universit\'e de Grenoble-Alpes, Institut Fourier (Math\'ematiques)\hfil\break
UMR 5582 du C.N.R.S., 100 rue des Maths, 38610 Gi\`eres, France\hfil\break
{\em e-mail:}\/ jean-pierre.demailly@univ-grenoble-alpes.fr

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