\input broccod
\input ams
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\font\pench=cmssi10

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\def\glr{{\rm Gl}_r}
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\def\Isom{{\rm Isom}}
\def\ad{{\rm ad}}
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            \kern-5.1pt\raise0.5pt\hbox{$\scriptscriptstyle\cup\kern0.6pt$}}
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             \raise4pt\hbox{$\scriptscriptstyle\check :$}}
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\centerline{\bf VANISHING THEOREMS FOR TENSOR POWERS}
\centerline{\bf OF AN AMPLE VECTOR BUNDLE}
\vskip 0.5cm
{\noindent\hangindent4cm\hangafter-1
{\sl Jean-Pierre DEMAILLY

}}

{\noindent\hangindent4cm\hangafter-4{\sl
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
Laboratoire associ\'e au C.N.R.S.
n\kern-2pt{\rm\char'27}188,\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\vskip 30pt
{\bf Abstract}\pointir Let $E$ be a holomorphic vector bundle of rank $r$ on
a compact complex manifold $X$ of dimension $n$~. It is shown that the
cohomology groups $H^{p,q}(X,E^{\otimes k}\otimes(\det E)^l)$ vanish if $E$ is
ample and $p+q\ge n+1$~, $l\ge n-p+r-1$~. The proof rests on the well-known 
fact that every tensor power $E^{\otimes k}$ splits into irreducible 
representations of $\gl(E)$~. By Bott's theory, each component is canonically 
isomorphic to the direct image on $X$ of a homogeneous line bundle over a flag
manifold of $E$~. The proof is then reduced to the Kodaira-Akizuki-Nakano
vanishing theorem for line bundles by means of the Leray spectral sequence, 
using backward induction on $p$~. We also obtain a generalization of Le 
Potier's isomorphism theorem and a counterexample to a vanishing conjecture of
Sommese.

\vskip 30pt
\titre 0. Statement of results.|
 
Many problems and results of contemporary algebraic geometry involve vanishing
theorems for holomorphic vector bundles. Furthermore, tensor powers of
such bundles are often introduced by natural geometric constructions. The 
aim of this work is to prove a rather general vanishing theorem for cohomology 
groups of tensor powers of a holomorphic vector bundle.

 Let $X$ be a complex compact $n$--dimensional manifold and $E$ a holomorphic
vector bundle of rank $r$ on $X$~. If $E$ is ample and $r>1$~, only
very few general and optimal vanishing results are available for the Dolbeault
cohomology groups $H^{p,q}$ of tensor powers of $E$~. For example,
the famous Le Potier vanishing theorem~[13]~:
  $$E~~{\rm ample}~~\Longrightarrow~~H^{p,q}(X,E)=0~~{\rm for}~~p+q\ge n+r$$
does not extend to symmetric powers $S^kE$~, even when $p=n$ and $q=n-2$ 
(\cf [11])~. Nevertheless, we will see that the vanishing property is true for 
tensor powers involving a sufficiently large power of $\det E$~. In all the
sequel, we let $L$ be a holomorphic line bundle on $X$ and we assume~:

\tha Hypothesis 0.1|$E$ is ample and $L$ semi-ample, or $E$ is semi-ample and
$L$ ample.\fintha

The precise definitions concerning ampleness are given in \S 1~. Under this
hypothesis, we prove the following two results.

\tha Theorem 0.2|Let us denote by $\Gamma^aE$ the irreducible tensor power 
representation of $\gl(E)$ of highest weight $a\in\bbz^r$~. If 
$h\in\{1\ld r-1\}$ and $a_1\ge a_2\ge\ldots\ge a_h>a_{h+1}=\ldots=a_r=0$~, then
$$H^{n,q}(X,\Gamma^aE\otimes(\det E)^h\otimes L)=0~~{\sl for}~~q\ge 1~~.$$
\fintha

Since $S^kE=\Gamma^{(k,0,\ldots,0)}E$~, the special case $h=1$ is Griffiths'
vanishing theorem~[8]~:
$$H^{n,q}(X,S^kE\otimes\det E\otimes L)=0~~~{\rm for}~~q\ge 1~~.$$

\tha Theorem 0.3|For all integers $p+q\ge n+1$~, $k\ge 1$~, $l\ge 
n-p+r-1$\break then~ $H^{p,q}(X,E^{\otimes k}\otimes(\det E)^l\otimes L)=0$~.
\fintha

For $p=n$ and arbitrary $r\,,\,k_0\ge 2$~, Peternell-Le Potier and Schneider 
[11] have constructed an example of an ample vector bundle $E$ of rank $r$ on a
manifold $X$ of dimension $n=2r$ such that
$$H^{n,n-2}(X,S^kE)\ne 0~~,~~2\le k\le k_0~~.\leqno(0.4)$$
This result shows that $\det E$ cannot be omitted in theorem 0.2 when $h=1$~.
More generally, the following example shows that the exponent $h$ is optimal.

\tha Example|Let $X=G_r(V)$ be the Grassmannian of subspaces of codimension
$r$ of a vector space $V$ of dimension $d$~, and $E$ the tautological quotient
vector bundle of rank $r$ on $X$ (then $E$ is spanned and $L=\det E$ very ample)~. Let $h\in\{1\ld r-1\}$ and $a\in\bbz^r$~, $\beta\in\bbz^d$ be such that
$$\eqalign{&a_1\ge\ldots\ge a_h\ge d-r~~,~~a_{h+1}=\ldots=a_r=0~~,\cr
           &\beta=(a_1-d+r\ld a_h-d+r,0\ld 0)~~.\cr}$$
Set $n=\dim X=r(d-r)$~, $q=(r-h)(d-r)$~. Then
$$H^{n,q}\big(X,\Gamma^aE\otimes(\det E)^h\big)=\Gamma^\beta V\otimes(\det V)^h
\ne 0~~.\leqno(0.5)$$
\fintha


Our approach is based on three well-known facts. First, every tensor power
$E^{\otimes k}$ splits into irreducible representations $\Gamma^aE$ of the 
linear group $\gl(E)$ (\cf (2.16)). Secondly, every irreducible tensor bundle
$\Gamma^aE$ appears in a natural way as the direct image on $X$ of an 
{\sl ample line bundle} on a suitable flag manifold of $E$~. This follows 
from Bott's theory of homogeneous vector bundles [3]. The third fact is the 
isomorphism theorem of Le Potier [13], which relates the cohomology groups of 
$E$ on $X$ to those of the line bundle ${\rm O}_E(1)$ on $P(E^\star)$~. 
In \S 3, we generalize this isomorphism to the case of arbitrary flag bundles 
associated to $E$~; theorem 0.2 is an immediate consequence.

The proof of theorem 0.3 rests on a generalization of the Borel-Le Potier 
spectral sequence, but we have avoided to make it explicit at this point in 
order to simplify the exposition. The main argument is a backward induction on 
$p$, based on the usual Leray spectral sequence and on the 
Kodaira-Akizuki-Nakano vanishing theorem for line bundles. A by-product of 
these methods is the following isomorphism result, already contained in the 
standard Borel-Le Potier spectral sequence, but which seems to have been 
overlooked.\eject

\tha Theorem 0.6|For every vector bundle $E$ and every line bundle $L$ one can
define a canonical morphism 
$$H^{p,q}(X,\Lambda^2E\otimes L)\lw H^{p+1,q+1}(X,S^2E\otimes L)~~.$$
Under hypothesis 0.1, this morphism is one-to-one for $p+q\ge n+r-1$ and 
surjective for $p+q=n+r-2$~.\fintha
 
Combining theorem 0.6 and example (0.4), we get $H^{n-1,n-3}(X,\Lambda^2
E)\ne 0$~. This shows that Sommese's conjecture ([15], conjecture (4.2)) 
$$E~~{\rm ample}~~\Longrightarrow~~H^{p,q}(X,\Lambda^kE)=0~~~{\rm for}~~
p+q\ge n+r-k+1$$
is false for $n=2r\ge 6$~.

The following related problem is interesting, but its complete solution 
certainly requires a better understanding of the Borel-Le Potier spectral 
sequence for flag bundles.

\tha Problem|Given any dominant weight $a\in\bbz^r$ with $a_r=0$ and $p,q$
such that $p+q\ge n+1$~, determine the smallest exponent $l_0=l_0(n,p,q,r,a)$ 
such that $H^{p,q}(X,\Gamma^aE\otimes(\det E)^l\otimes L)=0$ for $l\ge l_0$~.
\fintha

   We show in \S 5 that if the Borel-Le Potier spectral sequence degenerates at
the $E_2$ level, it is always sufficient to take $l\ge r-1+\min\{n-p,n-q\}$~. 
In that case, theorem 0.6 appears also as a special case of a fairly general 
exact sequence. The $E_2$--degeneracy of the Borel-Le Potier spectral sequence
is thus an important feature which would be interesting to investigate.

  Some of the above results have been annouced in the note [4] and 
corresponding detailed proofs are given in [5]~. However, the method of [5] is
based on differential geometry  and leads to 
results which overlap the present ones only in part. The author wishes to thank
warmly Prof. Michel Brion, Friedrich Knopp, Thomas Peternell and Michael 
Schneider for valuable remarks which led to substantial improvements of this 
work.\vfill\eject

\titre 1. Basic definitions and tools.|

   We recall here a few basic facts and definitions which will be used 
repeatedly in the sequel.\vskip2pt

\noindent(1.1) {\pench Ample vector bundles} (compare with Hartshorne [9]).

A vector bundle $E$ on $X$ is said to be {\sl spanned} if the canonical map 
$H^0(X,E)\lw E_x$ is onto for every $x\in X$~, and {\sl semi-ample} if the
symmetric powers $S^kE$ are spanned for $k\ge k_0$ large enough.

$E$ is said to be {\sl very ample} if the canonical maps
$$\left\{\hbox{$\eqalign{\displaystyle
&H^0(X,E)\lw E_x\oplus E_y~~~,~~~\forall x\ne y\in X~~,\cr
&H^0(X,E)\lw {\cal O}(E)\otimes {\cal O}_x/{\cal M}_x^2~~,~~\forall x\in X~~,
\cr}$}\right.$$
are onto, ${\cal M}_x$ denoting the maximal ideal at $x\in X$~. If $E$ is very
ample, then the holomorphic map from $X$ to the Grassmannian of 
$r$--codimensional subspaces of $V=H^0(X,E)$ given by
$$\Phi~:~X\lw G_r(V)~~,~~x\longmapsto W_x=\{\sigma\in V;\sigma(x)=0\}$$
is an embedding, and $E$ is the pull-back by $\Phi$ of the canonical quotient
vector bundle of rank $r$ on $G_r(V)$~. The embedding condition is in fact
equivalent to $E$ being very ample if $r=1$~, but weaker if $r\ge 2$~.

At last, $E$ is said to be {\sl ample} if the symmetric powers $S^kE$ are very 
ample for $k\ge k_0$ large enough. Denoting $\rmo_E(1)$ the canonical line 
bundle on $Y=P(E^\star)$ associated to $E$ and $\pi~:~Y\lw X$ the projection, 
it is well-known that $\pi_\star\rmo_E(k)=S^kE$~. One gets then easily
$$\eqalign{
&E~~{\rm spanned~on~}X~~~\Longleftrightarrow
~~~\rmo_E(1)~~{\rm spanned~on~}Y~~,\cr
&E~~{\rm (semi\hbox{-})ample~on~}X~~~\Longleftrightarrow
~~~\rmo_E(1)~~{\rm (semi\hbox{-})ample~on~}Y~~.\cr}$$
Moreover, for any line bundle $L$ on $X$~, ampleness is equivalent to the 
existence of a smooth hermitian metric on the fibers of $L$ with positive
curvature form $ic(L)$~. Since $S^k(E\otimes L)=S^kE\otimes L^k$~, it is clear 
that
$$\eqalign{
&E,L~~{\rm semi\hbox{-}ample}~~~\Longrightarrow
~~~E\otimes L~~{\rm semi\hbox{-}ample}~~,\cr
&E,L~~{\rm spanned,~one~of~them~very~ample}~~~\Longrightarrow
~~~E\otimes L~~{\rm very~ample}~~,\cr
&E,L~~{\rm semi\hbox{-}ample,~one~of~them~ample}~~~\Longrightarrow
~~~E\otimes L~~{\rm ample}~~.\cr}$$


\noindent(1.2) {\pench Kodaira-Akizuki-Nakano vanishing theorem} [1].

If $L$ is an ample (or positive) line bundle on $X$~, then
$$H^{p,q}(X,L)=H^q(X,\Omega^p_X\otimes L)=0~~~{\rm for}~~p+q\ge n+1~~.$$

\noindent(1.3) {\pench Leray spectral sequence} (\cf Godement [7]).

Let $\pi~:~Y\lw X$ be a continuous map between topological spaces, $\calS$ a 
sheaf of abelian groups on $Y$~, and $R^q\pi_\star\calS$ the direct image 
sheaves of $\calS$ on $X$~. Then there exists a spectral sequence such that 
$$E_2^{p,q}=H^p(X,R^q\pi_\star\calS)$$
and such that the limit term $E_\infty^{p,q-p}$ is the $p$--graded module 
associated to a decreasing filtration of $H^q(Y,\calS)$~.

\noindent(1.4) {\pench Cohomology of a filtered sheaf}.

We will need also the following elementary result. Let $\cal F$ be a sheaf of 
abelian groups on $X$ and
$${\cal F}={\cal F}^0\supset{\cal F}^1\supset\ldots\supset{\cal F}^p\supset
\ldots\supset{\cal F}^N=0$$
be a filtration of $\cal F$ such that the graded sheaf $\bigoplus{\cal G}^p$~,
${\cal G}^p={\cal F}^p/{\cal F}^{p+1}$~, satisfies $H^q(X,{\cal G}^p)=0$ for
$q\ge q_0$~. Then $H^q(X,{\cal F})=0$ for $q\ge q_0$~.

Indeed, it is immediately verified by induction on $p\ge 1$ that $H^q(X,
{\cal F}/{\cal F}^p)$ vanishes for $q\ge q_0$~, using the cohomology long exact
sequence associated to
$$0\lw{\cal G}^p\lw{\cal F}/{\cal F}^{p+1}\lw{\cal F}/{\cal F}^p\lw 0~~.$$

\vskip 20pt
\centerline{\bf 2. Homogeneous line bundles on flag manifolds and}
\centerline{\bf irreducible representations of the linear group.}
\vskip5pt
   The aim of this section is to settle notations and to recall a few basic 
results on homogeneous line bundles on flag manifolds (\cf Borel-Weil [2] 
and R. Bott [3]).

   Let $B_r$ be the Borel subgroup of $\glr=\gl(\bbc^r)$ of lower triangular 
matrices, $U_r\subset B_r$ the subgroup of unipotent matrices, and $T^r$ the 
complex torus $(\bbc^\star)^r$ of diagonal matrices. Let $V$ be a complex vector space of dimension $r$~. We denote by $M(V)$ the manifold of complete flags
$$V=V_0\supset V_1\supset\ldots\supset V_r=\{0\}~~,~~{\rm codim}_\bbc\,
V_\lambda=\lambda~~.$$
To every linear isomorphism $\zeta\in\Isom(\bbc^r,V)~:~(u_1\ld u_r)\longmapsto
\sum_{1\le\lambda\le r}u_\lambda\zeta_\lambda$~,\break one can associate the 
flag $[\zeta]\in M(V)$ defined by $V_\lambda={\rm Vect}(\zeta_{\lambda+1}\ld
\zeta_r)$~, $1\le\lambda\le r$~. This leads to the identification
$$M(V)=\Isom(\bbc^r,V)/B_r$$
where $B_r$ acts on the right side. We denote simply by $V_\lambda$ the 
tautological vector bundle of rank $r-\lambda$ on $M(V)$~, and we consider the 
canonical quotient line bundles
$$\left\{\eqalign{\displaystyle
          Q_\lambda=V_{\lambda-1}/V_\lambda~~,\qquad\qquad 
                &1\le \lambda\le r~~,\cr
          Q^a=Q_1^{a_1}\otimes\ldots\otimes Q_r^{a_r}~~,~~
                &a=(a_1\ld a_r)\in{\Bbb Z}^r~~.\cr}\right.\leqno(2.1)$$
The linear group $\gl(V)$ acts on $M(V)$ on the left, and there exist natural
equivariant left actions of $\gl(V)$ on all bundles $V_\lambda$~, $Q_\lambda$~,
$Q^a$~. If $\rho~:~B_r\to\gl(E)$ is a representation of $B_r$~, we may 
associate to $\rho$ the manifold
    $$E_{V,\rho}=\Isom(\bbc^r,V)\times_{B_r}E=\Isom(\bbc^r,V)\times E/\sim$$
where $\sim$ denotes the equivalence relation $(\zeta g,u)\sim(\zeta,
\rho(g)u)$~, $g\in B_r$~. Then $E_{V,\rho}$ is a $\gl(V)$--equivariant bundle 
over $M(V)$~, and all such bundles are obtained in this way. It is clear that 
$Q^a$ is isomorphic to the line bundle $\bbc_{V,\underline a}$ arising from the
1--dimensional representation of weight $\underline a$~:
  $$\underline a~~:~~T^r=B_r/U_r\lw\bbc^\star~~~,~~~(t_1\ldots t_r)
       \longmapsto t_1^{a_1}\ldots~ t_r^{a_r}~~;$$
the isomorphism $\displaystyle\bbc_{V,\underline a}~\tilde{\lw}~
Q^a$ is induced by the map 
$$\Isom(\bbc^r,V)\times\bbc\ni(\zeta,u)\longmapsto u\tilde\zeta_1^{a_1}\otimes
\ldots\otimes\tilde\zeta_r^{a_r}\in Q^a~~,~~\tilde\zeta_\lambda=\zeta_\lambda~
{\rm mod}~V_\lambda~~.$$
We compute now the tangent and cotangent vector bundles of $M(V)$~. The action
of $\gl(V)$ on $M(V)$ yields
$$TM(V)=\hom(V,V)/W\leqno(2.2)$$
where $W$ is the subbundle of endomorphisms $g\in\hom(V,V)$ such that
$g(V_\lambda)\subset V_\lambda$~, $1\le\lambda\le r$~. Using the self-duality 
of $\hom(V,V)$ given by the Killing form\break $(g_1,g_2)\mapsto
{\rm tr}(g_1g_2)$~, we find
$$\left\{\hbox{$\eqalign{&T^\star M(V)=\big(\hom(V,V)/W\big)^\star=W^\perp\cr
&W^\perp=\{g\in\hom(V,V)\ ;\ g(V_{\lambda-1})\subset V_\lambda~,
~1\le\lambda\le r\}~~.\cr}$}\right.\leqno(2.3)$$
If ad denotes the adjoint representation of $B_r$ on the Lie algebras
$\gothgl_r$~, $\gothb_r$~, $\gothu_r$~, then
$$TM(V)=(\gothgl_r/\gothb_r)_{V,\ad}~~,~~T^\star M(V)=(\gothu_r)_{V,\ad}
\leqno(2.4)$$
because $W_{[\zeta]}=\zeta\,\gothb_r\zeta^{-1}~,~W^\perp_{[\zeta]}=
\zeta\,\gothu_r\zeta^{-1}$ at every point $[\zeta]\in M(V)$~. There exists a 
filtration of $T^\star M(V)$ by subbundles of the type 
   $$\{g\in\hom(V,V)\ ;\ g(V_\lambda)\subset V_{\mu(\lambda)}~,
          ~1\le\lambda\le r\}$$
in such a way that the corresponding graded bundle is the direct sum of the
line bundles $\hom(Q_\lambda,Q_\mu)=Q_\lambda^{-1}\otimes Q_\mu$~, 
$\lambda<\mu$~; their tensor product is thus isomorphic to the canonical line 
bundle $K_{M(V)}=\det(T^\star M(V))$~:
$$K_{M(V)}=Q_1^{1-r}\otimes\ldots\otimes Q_\lambda^{2\lambda-r-1}\otimes\ldots
          \otimes Q_r^{r-1}=Q^c\leqno(2.5)$$
where $c=(1-r\ld r-1)$~; $\underline c$ will be called the canonical weight of
$M(V)$~.

\vskip3pt
\noindent$\bullet$ {\pench Case of incomplete flag manifolds.}

   More generally, given any sequence of integers $s=(s_0\ld s_m)$ such that
$0=s_0<s_1<\ldots<s_m=r$~, we may consider the manifold $M_s(V)$ of 
incomplete flags
$$V=V_{s_0}\supset V_{s_1}\supset\ldots\supset V_{s_m}=\{0\}~~,~~
  {\rm codim}_\bbc\, V_{s_j}=s_j~~.$$
On $M_s(V)$ we still have tautological vector bundles $V_{s,j}$ of rank
$r-s_j$ and line bundles 
$$Q_{s,j}=\det(V_{s,j-1}/V_{s,j})~~,~~
1\le j\le m~~.\leqno(2.6)$$
For any $r$--tuple $a\in\bbz^r$ such that $a_{s_{j-1}+1}=\ldots=a_{s_j}$~,
$1\le j\le m$~, we set
   $$Q_s^a=Q_{s,1}^{a_{s_1}}\otimes\ldots\otimes Q_{s,m}^{a_{s_m}}~~.$$
If $\eta:M(V)\to M_s(V)$ is the natural projection, then
$$\eta^\star\, V_{s,j}=V_{s_j}~~,~~\eta^\star Q_{s,j}=
Q_{s_{j-1}+1}\otimes\ldots\otimes Q_{s_j}~~,~~\eta^\star Q_s^a=Q^a~~.
\leqno(2.7)$$
On the other hand, one has the identification
   $$M_s(V)=\Isom(\bbc^r,V)/B_s$$
where $B_s$ is the parabolic subgroup of matrices $(z_{\lambda\mu})$ with
$z_{\lambda\mu}=0$ for all $\lambda,\mu$ such that there exists an integer
$j=1\ld m-1$ with $\lambda\le s_j$ and $\mu>s_j$~. We define $U_s$
as the unipotent subgroup of lower triangular matrices 
$(z_{\lambda\mu})$ with $z_{\lambda\mu}=0$ for all $\lambda,\mu$ such that 
$s_{j-1}<\lambda\ne\mu\le s_j$ for some $j$ (hence $\gothu_s=
\gothb_s^\perp$~). In the same way as above, we get
$$\leqalignno{TM_s(V)&=\hom(V,V)/W_s~~,~~W_s=\{g~;~g(V_{s_{j-1}})
           \subset V_{s_j}\}~~,&(2.8)\cr
T^\star M_s(V)&=W_s^\perp=(\gothu_s)_{V,\ad}~~,&(2.9)\cr
K_{M_s(V)}&=Q_{s,1}^{s_1-r}\otimes\ldots\otimes Q_{s,j}
  ^{s_{j-1}+s_j-r}\otimes\ldots\otimes Q_{s,m}^{s_{m-1}}=Q_s^{c(s)}
&(2.10)\cr}$$
where $c(s)=(s_1-r\ld s_1-r\ld s_{j-1}+s_j-r\ld s_{j-1}+s_j-r,\,\ldots)$ is 
the canonical weight of $M_s(V)$~. 

\tha Lemma 2.11|$Q_s^a$ enjoys the following properties~:
\vskip0.5pt\noindent
{\rm (a)} If $a_{s_j}<a_{s_{j+1}}$ for some $j=1\ld m-1$~, then 
$H^0(M_s(V),Q^a_s)=0$~.
\vskip0.5pt\noindent
{\rm (b)} $Q^a_s$ is spanned if and only if $a_{s_1}\ge a_{s_2}\ge\ldots\ge 
a_{s_m}$~.
\vskip0.5pt\noindent
{\rm (c)} $Q^a_s$ is (very) ample if and only if $a_{s_1}>a_{s_2}>\ldots>
a_{s_m}$~.\fintha

{\sl Proof}\pointir

\noindent
(a) Let $(V^0_{s_0}\supset V^0_{s_1}\supset\ldots\supset V^0_{s_m})\in M_s(V)$
be an arbitrary flag and $F,F'$ subspaces of $V$ such that 
$$V^0_{s_{j-1}}\supset F\supset V^0_{s_j}\supset F'\supset V^0_{s_{j+1}}~~
,~~\dim F=s_j+1~~,~~\dim F'=s_j-1~~.$$ 
Let us consider the projective line $P(F/F')\subset M_s(V)$ of flags
$$V^0_{s_0}\supset\ldots\supset V^0_{s_{j-1}}\supset V_{s_j}\supset 
V^0_{s_{j+1}}\supset V^0_{s_m}$$
such that $F\supset V_{s_j}\supset F'$~. Then
$$\eqalign{\displaystyle
&Q_{s,j\upharpoonright P(F/F')}=\det(V^0_{s_{j-1}}/V_{s_j})\simeq F/V_{s_j}
 \simeq\rmo(1)~~,\cr
&Q_{s,j+1\upharpoonright P(F/F')}=\det(V_{s_j}/V^0_{s_{j+1}})\simeq V_{s_j}/F'
 \simeq\rmo(-1)\cr}$$
thus $Q^a_{s\upharpoonright P(F/F')}\simeq\rmo(a_{s_j}-a_{s_{j+1}})$~, which
implies (a) and the ``only if" part of assertions (b), (c).

\noindent(b) Since $Q_{s,1}\otimes\ldots\otimes Q_{s,j}=\det(V/V_{s_j})$~, we
see that this line bundle is a quotient of the trivial bundle 
$\Lambda^{s_j}V$~. Hence
$$Q^a_s=\bigotimes_{1\le j\le m-1}\det(V/V_{s_j})^{a_{s_j}-a_{s_{j+1}}}\otimes
(\det V)^{a_r}\leqno(2.12)$$
is spanned by sections arising from elements of $\bigotimes
S^{a_{s_j}-a_{s_{j+1}}}(\Lambda^{s_j}V)\otimes(\det V)^{a_r}$ as soon as
$a_{s_1}\ge\ldots\ge a_{s_m}$~.

\noindent(c) If $a_{s_1}>\ldots>a_{s_m}$~, it is elementary to verify that the
sections of $Q^a_s$ arising from (2.12) suffice to make $Q^a_s$ very ample
(this is in fact a generalization of Pl\"ucker's imbedding of Grassmannians).
\finpr

\vskip5pt
\noindent$\bullet$ {\pench Cohomology groups of $Q^a$~.}

It remains now to compute~ $H^0(M_s(V),Q_s^a)\simeq H^0(M(V),Q^a)$ when\break
$a_1\ge\ldots\ge a_r$~. Without loss of generality we may assume that 
$a_r\ge 0$~, because $Q_1\otimes\ldots\otimes Q_r=\det V$ is a trivial bundle.
\eject

\tha Proposition 2.13|For all integers $a_1\ge a_2\ge\ldots\ge a_r\ge 0$~, 
there is a canonical isomorphism
$$H^0(M(V),Q^a)=\Gamma^aV$$
where $\Gamma^aV\subset S^{a_1}V\otimes\ldots\otimes S^{a_r}V$ is the set of
polynomials $f(\zeta_1^\star\ld \zeta_r^\star)$ on $(V^\star)^r$ which are 
homogeneous of degree $a_\lambda$ with respect to $\zeta_\lambda^\star$ and
invariant under the left action of $U_r$ on $(V^\star)^r=\hom(V,\bbc^r)$~:
$$f(\zeta_1^\star\ld\zeta_{\lambda-1}^\star,\zeta_\lambda^\star+\zeta_\nu^\star
\ld\zeta_r^\star)=f(\zeta_1^\star\ld\zeta_r^\star)~~,~~\forall \nu<\lambda~~.$$
\fintha

{\sl Proof}\pointir To any section $\sigma\in H^0(M(V),Q^a)$ we associate 
the holomorphic function $f$ on $\Isom(V,\bbc^r)\subset(V^\star)^r$ defined by
$$f(\zeta_1^\star\ld\zeta_r^\star)=(\zeta_1^\star)^{a_1}\otimes\ldots\otimes
(\zeta_r^\star)^{a_r}.\,\sigma([\zeta_1\ld\zeta_r])$$
where $(\zeta_1\ld\zeta_r)$ is the dual basis of $(\zeta_1^\star\ld
\zeta_r^\star)$~, and where the linear form induced by $\zeta_\lambda^\star$ on
$Q_\lambda=V_{\lambda-1}/V_\lambda\simeq\bbc\zeta_\lambda$ is still denoted
$\zeta_\lambda^\star$~. Let us observe that $f$
is homogeneous of degree $a_\lambda$ in $\zeta_\lambda^\star$ and locally
bounded in a neighborhood of every $r$--tuple of $(V^\star)^r\setminus\Isom
(V,\bbc^r)$ (because $M(V)$ is compact and $a_\lambda\ge 0$)~. Therefore $f$ 
can be extended to a polynomial on all $(V^\star)^r$~. The invariance of $f$ 
under $U_r$ is clear. Conversely, such a polynomial $f$ obviously defines a 
unique section $\sigma$ on $M(V)$~. \finpr

From the definition of $\Gamma^aV$~, we see that
$$\leqalignno{S^kV&=\Gamma^{(k,0\ld 0)}V~~,&(2.14)\cr
        \Lambda^kV&=\Gamma^{(1\ld 1,0\ld 0)}V~~.&(2.15)\cr}$$
For arbitrary $a\in\bbz^r$~, proposition 2.13 remains true if we set
$$\eqalign{\Gamma^aV&=\Gamma^{(a_1-a_r\ld a_{r-1}-a_r,0)}V\otimes(\det V)^{a_r}
                           ~~{\rm when}~~a~~\hbox{\rm is non-increasing}~~,\cr
           \Gamma^aV&=0~~~~~~{\rm otherwise}~~.\cr}$$
The weights will be ordered according to their usual partial ordering~:
  $$\underline a\succcurlyeq\underline b~~~{\rm iff}~~~
    \sum_{1\le\lambda\le\mu}a_\lambda\ge\sum_{1\le\lambda\le\mu}b_\lambda~~,~~
    1\le\mu\le r~~.$$
Bott's theorem [3] shows that $\Gamma^aV$ is an irreducible representation of 
$\gl(V)$ of highest weight $\underline a$~; all irreducible representations
of $\gl(V)$ are in fact of this type (\cf Kraft [10]). In particular, since
the weights of the action of a maximal torus $T^r\subset\gl(V)$ on 
$V^{\otimes k}$ verify $a_1+\ldots+a_r=k$ and $a_\lambda\ge 0$~, we have a
canonical $\gl(V)$--isomorphism
$$V^{\otimes k}=\bigoplus_{\scriptstyle a_1+\ldots+a_r=k\atop \scriptstyle
             a_1\ge\ldots\ge a_r\ge 0}\mu(a,k)\,\Gamma^aV\leqno(2.16)$$
where $\mu(a,k)>0$ is the multiplicity of the isotypical factor $\Gamma^aV$ in
$V^{\otimes k}$~.

   Bott's formula (\cf also Demazure [6] for a very simple proof) gives in fact
the expression of all cohomology groups $H^q(M(V),Q^a)$~. A weight 
$\underline a$ is called {\sl regular} if all the elements of $a$ are distinct,
and {\sl singular} otherwise. We will denote by $a^{\scriptscriptstyle\ge}$ 
the reordered non-increasing sequence associated to $a$~, by $l(a)$ the number 
of strict inversions of the order, and we will set
$$\widehat a=\Big(a-\csur\Big)^{\ge}+\csur\in\bbz^r~~;$$
the sequence $\widehat a$ is non-increasing if and only if $a-\csur$ is 
regular.

\tha Proposition 2.17 {\rm (Bott [3])}|One has
  $$H^q(M(V),Q^a)=\cases{0&if~~$q\ne l(a-\csur)$~~,\cr
                         \Gamma^{\hat a}V&if~~$q=l(a-\csur)$~~.\cr}$$
In particular all $H^q$ vanish if and only if $a-\csur$ is singular.\fintha

  We will be particularly interested by those line bundles $Q^a$ such that the
cohomology groups $H^{p,q}$ vanish for all $q\ge 1$~, $p$ being given. The 
following proposition is one of the main steps in the proof of our results.

\tha Proposition 2.18|Set $N=\dim M(V)$~, $N(s)=\dim M_s(V)$~. Then
\vskip2pt\noindent
{\rm (a)}\quad $H^{p,q}(M_s(V),Q_s^a)=0$~~~for all~~$p+q\ge N(s)+1$
\hfill\break 
\phantom{{\rm (a)}}\quad as soon as~~$a_{s_j}-a_{s_{j+1}}\ge 1~~,~~
                         1\le j\le m-1~~.$
\vskip2pt\noindent
{\rm (b)}\quad $H^q(M_s(V),Q_s^a)=0$~~~for all~~$q\ge 1$
\hfill\break
\phantom{{\rm (b)}}\quad as soon as~~$a_{s_j}-a_{s_{j+1}}\ge1-(s_{j+1}-s_{j-1})
                    ~~,~~1\le j\le m-1~~.$ 
\vskip2pt\noindent
{\rm (c)}\quad In general, $H^{p,q}(M_s(V),Q_s^{a-c(s)})$ is isomorphic to a
direct sum of irredu-\break 
\phantom{{\rm (c)}}\quad cible $\gl(V)$--modules $\Gamma^bV$ with
$$b_1\ge\ldots\ge b_r\ge\min\{a_\lambda\}-(N(s)-p)~~.$$ 
{\rm (d)}\quad $H^{p,q}(M_s(V),Q_s^a)=0$~~~for all~~$q\ge 1$~~as soon as
\hfill\break
\phantom{{\rm (d)}}\quad $a_{s_j}-a_{s_{j+1}}\ge \min\{p\,,\,
N(s)-p+(s_{j+1}-s_{j-1})-1\,,\,r+1-(s_{j+1}-s_{j-1})\}~~.$
\vskip2pt\noindent
{\rm (e)}\quad Under the assumption of {\rm (d)}, there is a 
$\gl(V)$--isomorphism\hfill\break
\phantom{{\rm (e)}}\quad $\displaystyle H^{p,0}(M_s(V),Q_s^a)=
\bigoplus_{u\in\bbz^r}\nu_s(\underline u,p)\,\Gamma^{a+u}V~~,$
\vskip2pt\noindent
where $\nu_s(\underline u,p)$ is the multiplicity of the weight 
$\underline u$ in $\Lambda^p\gothu_s$~~.
\fintha

{\sl Proof}\pointir Under the assumption of (a), $Q_s^a$ is ample by 
lemma~2.11~(c)~. The result follows therefore from the Kodaira-Nakano-Akizuki 
theorem. Now (b) is a consequence of (a) since $c(s)_{s_j}-c(s)_{s_{j+1}}
=-(s_{j+1}-s_{j-1})$ and
$$H^q(M_s(V),Q_s^a)=H^{N(s),q}(M_s(V),Q_s^{a-c(s)})~~.$$
Let us now observe that for every vector bundle $E$ on $M_s(V)$ there are 
isomorphisms
$$\leqalignno{\displaystyle
&H^q(M_s(V),E)\simeq H^q(M(V),\eta^\star E)~~,&(2.19)\cr
&H^q(M_s(V),E)\simeq H^{q+N-N(s)}(M(V),Q^{c-c(s)}\otimes\eta^\star E)~~,&(2.20)
}$$
where $Q^{c-c(s)}$ is the relative canonical bundle along the fibers of the 
projection $\eta~:~M(V)\lw M_s(V)$~. In fact, the fibers
of $\eta$ are products of flag manifolds. For such a fiber $F$~, we have
$H^q(F,K_F)=\big(H^{\dim F-q}(F,{\cal O})\big)^\star=\bbc$ when\break
$q=\dim F=N-N(s)$ and $H^q(F,K_F)=0$ otherwise (apply (b) with $a=0$ and 
K\"unneth's formula). We get thus direct image sheaves
$$\eqalign{\displaystyle
R^q\eta_\star(\eta^\star E)\big)&=\hbox{$
          \cases{0&if~~$q\ne 0$\cr
          {\cal O}(E)&if~~$q=0$~~.\cr}$}\cr
R^q\eta_\star\big(Q^{c-c(s)}\otimes\eta^\star E\big)&=\hbox{$
          \cases{0&if~~$q\ne N-N(s)$\cr
          {\cal O}(E)&if~~$q=N-N(s)$~~.\cr}$}\cr}$$
Formulas (2.19) and (2.20) are thus immediate consequences of the Leray 
spectral sequence. When applied to $E=\Lambda^pT^\star M_s(V)\otimes 
Q_s^{a-c(s)}$~, formula (2.19) yields
$$\eqalign{H^{p,q}(M_s(V),Q^{a-c(s)}_s)&=H^q\big(M(V),Q^{a-c(s)}
\otimes\eta^\star\Lambda^pT^\star M_s(V)\big)\cr
                                        &=H^q\big(M(V),Q^a
\otimes\eta^\star\Lambda^{N(s)-p}TM_s(V)\big)~~,\cr}$$
using the isomorphism $\Lambda^pT^\star M_s(V)\simeq K_{M_s(V)}\otimes
\Lambda^{N(s)-p}TM_s(V)$~. In the same way, (2.20) implies
$$\eqalign{H^{p,q}(M_s(V),Q^a_s)&=H^{q+N-N(s)}\big(M(V),Q^{a+c-c(s)}
\otimes\eta^\star\Lambda^pT^\star M_s(V)\big)\cr
                                        &=H^{q+N-N(s)}\big(M(V),Q^{a+c}
\otimes\eta^\star\Lambda^{N(s)-p}TM_s(V)\big)~~.\cr}$$
The bundle $\eta^\star T^\star M_s(V)=(\gothu_s)_{V,\ad}$
(resp. $\eta^\star TM_s(V)=(\gothgl_r/\gothb_s)_{V,\ad}$~) has a
filtration with associated graded bundle
$$\bigoplus Q_\lambda^{-1}\otimes Q_\mu~~~\big({\rm resp.}~
  \bigoplus Q_\lambda\otimes Q_\mu^{-1}~~\big)$$
where the indices $\lambda,\mu$ are such that $\lambda\le s_j<\mu$ for some 
$j$~. It follows that $\eta^\star\Lambda^pT^\star M_s(V)$ $\big($ 
resp. $\eta^\star\Lambda^{N(s)-p}TM_s(V)~\big)$ has a filtration with
associated graded bundle
$$\bigoplus_{u\in\bbz^r}\nu_s(\underline u,p)\,Q^u~~~\big({\rm resp.}~
  \bigoplus_{u'\in\bbz^r}\nu'_s(\underline u',N(s)-p)\,Q^{u'}~~\big)$$
where the weights $\underline u$ (resp. $\underline u'$) and multiplicities $
\nu_s(\underline u,p)$ (resp. $\nu'_s(\underline u',N(s)-p))$ are those
of $\Lambda^p\gothu_s$ (resp. $\Lambda^{N(s)-p}\gothgl_r/\gothb_s~)$~.
We need a lemma.

\tha Lemma 2.21|The weights $u$ of $\Lambda^p\gothu_s$ verify
$$u_\mu-u_\lambda\le\min\{p+1\,,\,r+1-(\mu-\lambda)\,,\,
N(s)-p+(s_{j+1}-s_{j-1})\}$$
for $s_{j-1}<\lambda\le s_j<\mu\le s_{j+1}$~, $1\le j\le m-1$~.\fintha

{\sl Proof}\pointir
Let us denote by $(\varepsilon_\lambda)_{1\le\lambda\le r}$ the canonical basis
of $\bbz^r$~. The weights $\underline u$ (resp. $\underline u'$) are the sums 
of $p$ (resp. $N(s)-p$) distinct elements of the set of weights 
$-\varepsilon_\lambda+\varepsilon_\mu$ (resp.
$\varepsilon_\lambda-\varepsilon_\mu$~) where $\lambda\le s_j<\mu$ for some 
$j$~. It follows that $u_\mu-u_\lambda\le r+1-(\mu-\lambda)$~, with equality
for the weight
$$u=(-\varepsilon_1+\varepsilon_\mu)+\ldots+(-\varepsilon_\lambda+
  \varepsilon_\mu)+(-\varepsilon_\lambda+\varepsilon_{\mu+1})+\ldots+
  (-\varepsilon_\lambda+\varepsilon_r)~~.$$
It is clear also that $u_\mu-u_\lambda\le p+1$ and $u'_\mu-u'_\lambda\le 
N(s)-p$~. Since the weights $\underline u$~, $\underline u'$ are related by
$u=u'+c(s)$~, lemma 2.21 follows. \finpr

{\sl Proof of} (c)\pointir By the filtration property (1.4) and the above 
formulas, we see that $H^{p,q}(M_s(V),Q_s^{a-c(s)})$ is a direct sum of
certain irreducible $\gl(V)$--modules
$$H^q\big(M(V),Q^{a+u'}\big)=\Gamma^bV~~,~~b=(a+u')^\wedge=\Big(a-\csur+u'
\Big)^\ge+\csur~~.$$
Clearly $u'_\lambda\ge-(N(s)-p)$~, thus
$$b_r\ge\min\{a_\lambda\}-\max\Big\{{c_\lambda\over2}\Big\}-(N(s)-p)+
   {c_r\over2}=\min\{a_\lambda\}-(N(s)-p)~~.$$

{\sl Proof of} (d)\pointir Similarly, (d) is reduced by (1.4) to proving
        $$H^{q+N-N(s)}(M(V),Q^{a+u+c-c(s)})=0$$
for all the above weights $\underline u$ and all $q\ge 1$~. By proposition 2.17
it is sufficient to get
       $$l\big(a+u+\csur-c(s)\big)\le N-N(s)~~.$$
Let us observe that a weight $\underline h$ will be such that $l(h)\le N-N(s)$
as soon as $h_\lambda-h_\mu\ge 0$ whenever $s_{j-1}<\lambda\le s_j<\mu\le 
s_{j+1}$~, $1\le j\le m-1$~. For $h=a+u+\csur-c(s)$ this condition yields,
thanks to lemma~2.21~:
$$a_\lambda-a_\mu\ge\min\{p+1\,,\,r+1-(\mu-\lambda)\,,\,
N(s)-p+(s_{j+1}-s_{j-1})\}+(\mu-\lambda)-(s_{j+1}-s_j)~~.$$
Taking $\mu-\lambda$ as large as possible, \ie $\mu-\lambda
=s_{j+1}-s_{j-1}-1$~, we get the asserted condition.

{\sl Proof of} (e)\pointir Because of the vanishing of $H^1$ of the graded
quotients, we obtain
$$H^{p,0}(M_s(V),Q^a_s)=\bigoplus_u \nu_s(\underline u,p)
\,H^{N-N(s)}\big(M(V),Q^{a+u+c-c(s)}\big)~~.$$
Applying proposition 2.17, we get $(a+u+c-c(s))^\wedge=a+\widetilde u$~, where
$\widetilde u$ is the partial reordering of $u$ such that only the coefficients
in each interval $[s_{j-1}+1,s_j]$ have been reordered (in non-increasing 
order). The number of inversions is always $\le N-N(s)$~, with equality if and 
only if $u$ is non-decreasing in each interval. In that case
$$H^{N-N(s)}\big(M(V),Q^{a+u+c-c(s)}\big)=\Gamma^{a+\tilde u}V~~,$$
and $=0$ otherwise. Since $\Lambda^p\gothu_s$ is a $B_s$--module we 
have $\nu_s(\underline u,p)=\nu_s(\underline{\widetilde u},p)$~. 
Formula (e) is proved, and we see that non-zero terms correspond to 
weights $\underline u$ which are non-increasing in each interval 
$[s_{j-1}+1,s_j]$~. \finpr

{\sl Remark} 2.22\pointir If the manifold $M_s(V)$ is a Grassmannian, then
$m=2$~, $s=(0,s_1,r)$~. The condition required in proposition 2.21 is 
therefore always satisfied when $a_r-a_{s_1}\ge 1$~, \ie when $Q^a_s$ is
ample.\vskip20pt

\titre 3. An isomorphism theorem|

  Our aim here is to generalize Griffiths and Le Potier's isomorphism theorems
([8], [13]) in the case of arbitrary flag bundles, following the simple method 
of Schneider [14]~.

Let $X$ be a $n$--dimensional compact complex manifold and $E\lw X$
a holo\-mor\-phic vector bundle of rank $r$~. For every sequence
$0=s_0<s_1<\ldots<s_m=r$~,\break we associate to $E$ its flag bundle
$Y=M_s(E)\lw X$~. If $a\in\bbz^r$ is such that $a_{s_{j-1}+1}=
\ldots=a_{s_j}~$~, $1\le j\le m$~, we may define a line bundle 
$Q_s^a\lw Y$ just as we did in \S 2~. Let us set
$$\Omega^p_X=\Lambda^pT^\star X\quad,\quad\Omega^p_Y=\Lambda^pT^\star Y~~.$$
One has an exact sequence
$$0\lw \pi^\star\Omega^1_X\lw\Omega^1_Y\lw
\Omega^1_{Y/X}\lw 0\leqno(3.1)$$
where $\Omega^1_{Y/X}$ is by definition the bundle of relative differential
1--forms along the fibers of the projection $\pi:Y=M_s(E)\lw X$~. 
One may then define a decreasing filtration of $\Omega_Y^t$ as follows~:
$$F^{p,t}=F^p(\Omega_Y^t)=\pi^\star(\Omega^p_X)\wedge\Omega^{t-p}_Y~~.
        \leqno(3.2)$$
The corresponding graded bundle is given by
$$G^{p,t}=F^{p,t}/F^{p+1,t}=\pi^\star(\Omega^p_X)\otimes\Omega^{t-p}_{Y/X}~~.
\leqno(3.3)$$
Over any open subset of $X$ where $E$ is a trivial bundle $X\times V$ with
$\dim{}_{\Bbb C}\,V=r$~, the exact sequence (3.1) splits as well as the
filtration (3.2). Using proposition 2.18 (a),~(d), we obtain the following
lemma.

\tha Lemma 3.4|For every weight $\underline a$ such that
$$\left\{\hbox{$\eqalign{&a_{s_j}-a_{s_{j+1}}\ge 1~~~{\sl if}~~t=N(s)~~~
               {\sl and~otherwise}\cr
               &a_{s_j}-a_{s_{j+1}}\ge\min\{t,N(s)-t+(s_{j+1}-s_j)-1,
                r+1-(s_{j+1}-s_{j-1})\}\cr}$}\right.\leqno(3.5)$$
the sheaf of sections of $\Omega^t_{Y/X}\otimes Q^a_s$ has direct images
$$\left\{\hbox{$\eqalign{
R^q\pi_\star\big(\Omega^t_{Y/X}\otimes Q^a_s\big)&=0~~~{\sl for}~~q\ge 1\cr
\pi_\star\big(\Omega^t_{Y/X}\otimes Q^a_s\big)&=
\bigoplus_u \nu_s(\underline u,t)~\Gamma^{a+u}E~~.\cr}$}
\right.\leqno(3.6)$$\fintha

We have in particular
$$\pi_\star(Q^a_s)=\Gamma^aE~~,~~\pi_\star\big(\Omega^{N(s)}_{Y/X}\otimes
  Q^a_s\big)=\Gamma^{a+c(s)}E~~.\leqno(3.7)$$
Let $L$ be an arbitrary line bundle on $X$~. Under assumption (3.5), 
formulas (3.3) and (3.6) yield
$$\eqalign{
R^q&\pi_\star(G^{p,p+t}\otimes Q^a_s\otimes\pi^\star L)=0
                 ~~~{\rm for}~~q\ge 1~~,\cr
&\pi_\star(G^{p,p+t}\otimes Q^a_s\otimes\pi^\star L)=
                 \bigoplus_u\nu_s(\underline u,t)~
\Omega^p_X\otimes\Gamma^{a+u}E\otimes L~~.\cr}$$
The Leray spectral sequence implies therefore~:

\tha Theorem 3.8|Under assumption (3.5), one has for all  $q\ge 0$~:
$$H^q(Y,G^{p,p+t}\otimes Q^a_s\otimes\pi^\star L)\simeq
 \bigoplus_u\nu_s(\underline u,t)~H^{p,q}(X,\Gamma^{a+u}E\otimes L)~~.$$
\fintha
\vskip-15pt\noindent The special case $t=N(s)$ gives :

\tha Corollary 3.9|If~~$a_{s_j}-a_{s_{j+1}}\ge 1$~, then for all $q\ge 0$
$$H^q(Y,G^{p,p+N(s)}\otimes Q^a_s\otimes\pi^\star L)\simeq
H^{p,q}(X,\Gamma^{a+c(s)}E\otimes L)~~.$$
\fintha

When $p=n$~, $G^{n,n+N(s)}$ is the only non-vanishing quotient in the 
filtration of the canonical line bundle $\Omega_Y^{n+N(s)}$~. We thus obtain 
the following generalization of Griffiths' isomorphism theorem [8]~:
$$H^{n+N(s),q}(M_s(E),Q^a_s\otimes\pi^\star L)\simeq
H^{n,q}(X,\Gamma^{a+c(s)}E\otimes L)~~.\leqno(3.10)$$
\vskip15pt

\titre 4. Vanishing theorems.|

  In order to carry over results for line bundles to vector bundles, one needs
the following simple lemma. 

\tha Lemma 4.1|Assume that $a_{s_1}>a_{s_2}>\ldots>a_{s_m}\ge 0$~. Then
\vskip0.1pt\noindent
{\rm (a)}\qquad $E$ semi-ample~~~$\Longrightarrow~~~Q^a_s$ semi-ample~;
\vskip0.1pt\noindent
{\rm (b)}\qquad $E$ ample~~~$\Longrightarrow~~~Q^a_s$ ample~;
\vskip0.1pt\noindent
{\rm (c)}\qquad $\,E$ semi-ample and $L$ ample~~~$\Longrightarrow~~~Q^a_s
\otimes\pi^\star L$~~ample.
\fintha

{\sl Proof}\pointir\vskip0.1pt\noindent
(a) If $E$ is semi-ample, then by definition (1.1) $S^kE$ is 
spanned for $k\ge k_0$~ large enough. Hence $\Gamma^{ka}E$, which is a direct
summand in \hbox{$S^{ka_1}E\otimes\ldots\otimes S^{ka_r}E$~,} is also spanned
for $k\ge k_0$~. Since the fibers of $\pi_\star(Q^{ka}_s)=\Gamma^{ka}E$ 
generate $Q^{ka}_s$~, we conclude that $Q^{ka}_s$ is spanned for $k\ge k_0$~.
\vskip3pt\noindent
(b) Similar proof, replacing ``semi-ample" by ``ample" and ``spanned" by
``very ample". One needs moreover the fact that $Q_s^{ka}$ is very ample
along the fibers of $\pi$ (lemma 2.11).\vskip3pt\noindent
(c) If $L^k$ is very ample for $k\ge k_1$~, then $Q_s^{ka}\otimes\pi^\star L^k$
is very ample for $k\ge\max(k_0,k_1)$~, because $Q_s^{ka}$ is spanned on $Y$ 
and very ample along the fibers, whereas $H^0(Y,\pi^\star L^k)=H^0(X,L^k)$ 
separates points of $Y$ which lie in distinct fibers. \finpr

We are now ready to attack the proof of the main theorems. 

{\sl Proof of theorem} 0.2\pointir Let $a\in\bbz^r$ be such that
$$a_1\ge a_2\ge\ldots\ge a_h>a_{h+1}=\ldots=a_r=0~~.$$
Define $s_1<s_2<\ldots<s_{m-1}$ as the sequence of indices $\lambda=1\ld r-1$
such that $a_\lambda>a_{\lambda+1}$ and set $a'=a+(h\ld h)-c(s)$~. The
canonical weight $c(s)$ is non-decreasing and $c(s)_r=s_{m-1}=h$~, $s_m=r$~,
hence
$$a'_{s_1}>a'_{s_2}>\ldots>a'_{s_m}=h-c(s)_r=0~~,$$
so $Q_s^{a'}\otimes\pi^\star L$ is ample by lemma 4.1 and $\Gamma^{a'+c(s)}E=
\Gamma^aE\otimes(\det E)^h$~. Formula (3.10) yields
$$H^{n,q}(X,\Gamma^aE\otimes(\det E)^h\otimes L)\simeq
  H^{n+N(s),q}(M_s(E),Q^{a'}_s\otimes\pi^\star L)~~.$$
Since $\dim M_s(E)=n+N(s)$~, the group in the right hand side is zero for
$q\ge 1$ by the Kodaira-Akizuki-Nakano vanishing theorem (1.2) .\finpr

{\sl Proof of theorem} 0.3\pointir The proof proceeds by backward induction
on $p$~. The case $p=n$ is already settled by theorem 0.2. The decomposition
formula (2.16) shows that the result is equivalent to
$$H^{p,q}(X,\Gamma^aE\otimes(\det E)^l\otimes L)=0~~~{\rm for}~~p+q\ge n+1~,~
  l\ge n-p+r-1$$
and $a_1\ge\ldots\ge a_r=0$ not all zero. Define $s$ as above and
$a'=a+(l\ld l)-c(s)$~. Then $Q_s^{a'}$ is ample since $a_r=l-h\ge 0$~, and
corollary 3.9 implies
$$H^{p,q}(X,\Gamma^aE\otimes(\det E)^l\otimes L)\simeq
  H^q(Y,G^{p,p+N(s)}\otimes Q^{a'}_s\otimes\pi^\star L)~~.\leqno(4.2)$$
Now, it is clear that $F^{p,p+N(s)}=\Omega_Y^{p+N(s)}$~. We get thus
an exact sequence
  $$0\lw F^{p+1,p+N(s)}\lw \Omega_Y^{p+N(s)}
     \lw G^{p,p+N(s)}\lw 0~~.$$
The Kodaira-Akizuki-Nakano vanishing theorem (1.2) applied to 
$Q^{a'}_s\otimes\pi^\star L$ with  $\dim Y=n+N(s)$ yields
 $$H^q(Y,\Omega_Y^{p+N(s)}\otimes Q^{a'}_s\otimes\pi^\star L)=0~~~{\rm for}~~
     p+q\ge n+1~~.$$
The cohomology groups in (4.2) will therefore vanish if and only if
   $$H^{q+1}(Y,F^{p+1,p+N(s)}\otimes Q^{a'}_s\otimes\pi^\star L)=0~~.
     \leqno(4.3)$$
Let us observe that $F^{p+1,p+N(s)}$ has a filtration with associated graded  
bundle $\bigoplus_{t\ge 1}G^{p+t,p+N(s)}$~. In order to verify (4.3), it is 
thus enough to get
$$H^{q+1}(Y,G^{p+t,p+N(s)}\otimes Q^{a'}_s\otimes\pi^\star L)=0~~,~~t\ge1~~.
  \leqno(4.4)$$
At this point, the Leray spectral sequence will be used in an essential way.
Since $G^{p+t,p+N(s)}=\pi^\star(\Omega_X^{p+t})\otimes\Omega_{Y/X}^{N(s)-t}$~,
we get
$$R^k\pi_\star(G^{p+t,p+N(s)}\otimes Q_s^{a'}\otimes\pi^\star L)=
  \Omega_X^{p+t}\otimes L\otimes R^k\pi_\star\big(\Omega_{Y/X}^{N(s)-t}\otimes
  Q^{a+(l\ld l)-c(s)}\big)~~.$$
Proposition 2.18 (a) and (c) yields
$$\eqalign{\displaystyle
R^k\pi_\star\big(\Omega_{Y/X}^{N(s)-t}\otimes Q^{a+(l\ld l)-c(s)}\big)&=
\cases{0~~~{\rm for}~~k\ge t+1~,~{\rm otherwise}&\cr
       {\scriptstyle\bigoplus}_b~\Gamma^bE~~,~~b_r\ge l-t~~,&\cr}\cr
R^k\pi_\star(G^{p+t,p+N(s)}\otimes Q_s^{a'}\otimes\pi^\star L)&=
\cases{0~~~{\rm for}~~k\ge t+1~,~{\rm otherwise}&\cr
       {\scriptstyle\bigoplus}_{b,m}~\Omega_X^{p+t}\otimes\Gamma^bE\otimes
       (\det E)^m\otimes L~~,&\cr}\cr}$$
where the last sum runs over weights $\underline b$ such that $b_r=0$ and
integers $m$ such that $m\ge l-t\ge n-(p+t)+r-1$~.
Therefore $H^{q+1}(Y,G^{p+t,p+N(s)}\otimes Q^{a'}_s\otimes\pi^\star L)$ has a
filtration whose graded module is the limit term $\bigoplus_j 
E_\infty^{j,q+1-j}$ of a spectral sequence such that
$$E_2^{j,k}=\cases{0~~~{\rm for}~~k\ge t+1~,~{\rm otherwise}&\cr
       {\scriptstyle\bigoplus}_{b,m}~H^{p+t,j}(X,\Gamma^bE\otimes
       (\det E)^m\otimes L)~~,~~m\ge n-(p+t)+r-1~~.&\cr}$$
We have thus $E_2^{j,q+1-j}=0$ for $j\le q-t$ by the first case, and also 
for $j\ge q-t$\break by the second case and the induction hypothesis. Hence
$E_\infty^{j,q+1-j}=0$ for all $j$~, and (4.4) is proved.\finpr

{\sl Proof of theorem 0.6}\pointir\kern-2pt
Let us take here $a=(2,0\ld 0)$ and 
$s=(0,1,r)$~, so that $Y=M_s(E)=P(E^\star)$~. Since $\Gamma^aE=S^2E$ and
$\Gamma^{(1,1,0\ld 0)}E=\Lambda^2E$~, theorem 3.8 yields
$$\left\{\hbox{$\eqalign{\displaystyle
H^q(Y,G^{p+1,p+1}\otimes Q^2_{s,1}\otimes\pi^\star L)&\simeq
H^{p+1,q}(X,S^2E)~~,\cr
H^q(Y,G^{p,p+1}\otimes Q^2_{s,1}\otimes\pi^\star L)&\simeq
H^{p,q}(X,\Lambda^2E)~~,\cr
H^q(Y,G^{k,p+1}\otimes Q^2_{s,1}\otimes\pi^\star L)&=0~~~
{\rm for}~~k<p~~,\cr}$}\right.\leqno(4.5)$$
because $u=-\varepsilon_1+\varepsilon_2$ is the only weight of $\Lambda^1
\gothu_s$ such that $a+u$ is non-increasing, and because no such weights
exist for higher exterior powers $\Lambda^j\gothu_s$~. Now,
$\Omega^{p+1}_Y/F^{p,p+1}$ has a filtration with graded quotients $G^{k,p+1}$,
$k<p$~, therefore
$$H^q(Y,\Omega^{p+1}_Y/F^{p,p+1}\otimes Q^2_{s,1}\otimes\pi^\star L)=0~~~
{\rm for~all}~q~~.$$
Considering the exact sequences
$$\eqalign{&0\lw G^{p,p+1}\lw\Omega^{p+1}_Y/F^{p+1,p+1}\lw
\Omega^{p+1}_Y/F^{p,p+1}\lw 0~~,\cr
&0\lw G^{p+1,p+1}\lw\Omega^{p+1}_Y\lw\Omega^{p+1}_Y/F^{p+1,p+1}\lw 0~~,\cr}$$
we get an isomorphism
$$H^q(Y,G^{p,p+1}\otimes Q^2_{s,1}\otimes\pi^\star L)\simeq
H^q(Y,\Omega^{p+1}_Y/F^{p+1,p+1}\otimes Q^2_{s,1}\otimes\pi^\star L)$$
and a canonical coboundary morphism
$$H^q(Y,\Omega^{p+1}_Y/F^{p+1,p+1}\otimes Q^2_{s,1}\otimes\pi^\star L)\lw
H^{q+1}(Y,G^{p+1,p+1}\otimes Q^2_{s,1}\otimes\pi^\star L)$$
which by Kodaira-Akizuki-Nakano is onto for $p+1+q+1\ge n+N(s)+1$~, \ie
$p+q\ge n+r-2$~, and bijective for $p+q\ge n+r-1$~. Combining this with
the first two isomorphisms (4.5) achieves the proof of theorem 0.6. \finpr

  As promised in the introduction, we show now that the condition $l\ge h$
in theorem 0.2 is best possible.

{\sl Example} 4.6\pointir Let $X=G_r(V)$ be the Grassmannian of subspaces of 
codimension $r$ of a vector space $V$~, $\dim_\bbc V=d$~, and $E$ the 
tautological quotient vector bundle of rank $r$ over $X$~. Then $E$ is spanned
(hence semi-ample) and \hbox{$L=\det E$} is very ample. According
to the notations of \S 2~, we have
$$\eqalign{X&=M_s(V)~~,~~s=(s_0,s_1,s_2)=(0,r,d)~~,\cr
           E&=V/V_r~~,~~\det E=Q_{s,1}~~,~~\det V_r=Q_{s,2}~~,\cr
           K_X&=Q^{r-d}_{s,1}\otimes Q^r_{s,2}~~.\cr}$$
Furthermore, for any sequence $a_1\ge\ldots\ge a_r\ge 0$~, we have
$$\Gamma^aE=\Gamma^a(V/V_r)=\eta_\star Q^{(a,0)}$$
where $\eta~:~M(V)\lw M_s(V)$ is the projection. If
$n=\dim X=r(d-r)$~, this implies
$$\eqalign{H^{n,q}\big(X,\Gamma^aE\otimes(\det E)^h\big)&=H^q\big(X,
Q_{s,1}^{h+r-d}\otimes Q^r_{s,2}\otimes\eta_\star Q^{(a,0)}\big)\cr
&=H^q(X,\eta_\star Q^\alpha)\cr}$$
where $\alpha=(a_1+h+r-d\ld a_r+h+r-d,r\ld r)\in\bbz^d$~. The fiber of 
$\eta$ is $M(V/V_r)\times M(V_r)$~, hence $R^q\eta_\star Q^\alpha=0$
for $q\ge 1$ by K\"unneth's formula and proposition 2.18~(b)~. We obtain
therefore
$$H^{n,q}\big(X,\Gamma^aE\otimes(\det E)^h\big)=H^q(M(V),Q^\alpha)~~.$$
Assume now that $a_1\ge\ldots\ge a_h\ge d-r$ and $a_h=\ldots=a_r=0$~. Define
${\bf 1}=(1\ld 1)\in\bbz^d$~. Then
$$\alpha-\csur={1-d\over 2}\cdot{\bf 1}+(a_1+r+h-1\ld a_h+r\,;\,r-1\ld h\,;\,
d-1\ld r)$$
where dots indicate a decreasing sequence of consecutive integers. There are
exactly $(r-h)(d-r)$ inversions of the ordering, corresponding to the inversion
of the last two blocks between semicolons. Then one finds easily
$$\Big(\alpha-\csur\Big)^\ge+\csur=(a_1+h+r-d\ld a_h+h+r-d\,;\,h\ld h)=
\beta+h{\bf 1}$$
where $\beta=(a_1+r-d\ld a_r+r-d\,;\,0\ld 0)$~. Proposition 2.17 yields 
therefore
$$H^{n,q}\big(X,\Gamma^aE\otimes(\det E)^h\big)=\cases{
0&if~~$q\ne(r-h)(d-r)$\cr
\Gamma^\beta V\otimes (\det V)^h&if~~$q=(r-h)(d-r)$~~.\cr}$$

\vskip15pt
\titre 5. On the Borel-Le Potier spectral sequence.|

Denote as before $\pi~:~Y=M_s(E)\lw X$ the projection. To every integer $t$
and every coherent analytic sheaf $\calS$ on $Y$~, one may associate the 
filtration of $\Omega_Y^t\otimes\calS$ by its subsheaves
$$F^{p,t}\otimes\calS\subset\Omega_Y^t\otimes\calS~~,$$
the corresponding graded sheaf being of course $\bigoplus_p G^{p,t}\otimes
\calS$~. This gives rise to a spectral sequence which we shall name after Borel
and Le Potier, whose $E_1$ term is given by
$$E_1^{p,q-p}=H^q(Y,G^{p,t}\otimes\calS)~~.\leqno(5.1)$$
The limit term $E_\infty^{p,q-p}$ is the $p$--graded module corresponding to
the filtration of\break $H^q({\cal K}^\blt)=H^q(Y,\Omega^t_Y\otimes\calS)$ by 
the canonical images of the groups $H^q({\cal K}^\blt_p)$~. Assume\break 
that the spectral sequence degenerates in $E_2$~, \ie \hbox{$d_r\,:\,
E_r^{p,q-p}\to E_r^{p+r,q+1-(p+r)}$} is zero for all $r\ge 2$ (by Peternell, 
Le Potier and Schneider [12], the spectral sequence does not degenerate in 
general in $E_1$). Then $E_2^{p,q-p}=E_\infty^{p,q-p}$~. This equality means 
that the $q$--th cohomology group of the $E_1$--complex
$$d_1~:~H^q(Y,G^{p,t}\otimes\calS)\lw H^{q+1}(Y,G^{p+1,t}\otimes\calS)$$
is the $p$-graded module corresponding to a filtration of $H^q(Y,\Omega^t_Y
\otimes\calS)$~. By Kodaira-Akizuki-Nakano, we get therefore~:

\tha Proposition 5.1|Assume that $E$ is ample and $L\ge 0$~, or $E\ge 0$ and
$L$ ample, and that the $E_2$--degeneracy occurs for the ample invertible sheaf
$\calS=Q^a_s\otimes\pi^\star L$ on $Y$~. Then the complex
$$d_1~:~H^q(Y,G^{p,t}\otimes Q^a_s\otimes\pi^\star L)\lw H^{q+1}(Y,
G^{p+1,t}\otimes Q^a_s\otimes\pi^\star L)$$
is exact in degree $q\ge n+N(s)+1-t$~.
\fintha

 Our hope is that the $E_2$--degeneracy can be proved in all cases by means of
harmonic forms and Hodge theory, but we have been unable to do so. 
Since $G^{p,t}=0$ for $t>p+N(s)$~, proposition 5.1 yields for all $p+q\ge n+1$
an exact sequence
$$0\lw H^q(Y,G^{p,p+N(s)}\otimes Q^a_s\otimes\pi^\star L)\lw
H^{q+1}(Y,G^{p+1,p+N(s)}\otimes Q^a_s\otimes\pi^\star L)~~.$$
Replacing $a$ by $a'=a+(l\ld l)-c(s)$ as in the proof of theorem 0.3, and using
formula (3.10) and theorem 3.8, one gets an exact sequence
$$\eqalign{0\lw H^{p,q}&(X,\Gamma^aE\otimes(\det E)^l\otimes L)\cr
&\lw\bigoplus_u\nu_s
(\underline u,N(s)-1)\,H^{p+1,q+1}(X,\Gamma^{a-c(s)+u}E\otimes(\det E)^l\otimes
L)\cr}$$
because assumption (3.5) is satisfied for the weight $\underline b$ and for 
$t=N(s)-1$~. But the above direct sum can be rewritten
$$\bigoplus_{u'}\nu'_s(\underline u',1)\,H^{p+1,q+1}(X,\Gamma^{a+u'}E
\otimes(\det E)^l\otimes L)$$
in terms of the weights $u'=\varepsilon_\lambda-\varepsilon_\mu$~, $\lambda\le
s_j<\mu$~, of $\Lambda^1\gothgl_r/\gothb_s$~. A backward induction on 
$\max\{p,q\}$ yields then immediately~:

\tha Corollary 5.2|If the $E_2$--degeneracy occurs for all ample line bundles
$Q^a_s\otimes\pi^\star L$~, $a_1\ge\ldots\ge a_r=0$~, on all flag
manifolds of $E$~, then
$$H^{p,q}(X,\Gamma^aE\otimes(\det E)^l\otimes L)=0$$
for $p+q\ge n+1$ and $l\ge r-1+\min\{n-p,n-q\}$~.
\fintha

Another interesting consequence of proposition 5.1 in the case
$Y=P(E^\star)$ would be the following generalization of theorem 0.6.

\tha Corollary 5.3|Set $Z^{t,k}E=\Gamma^{(k-t,1,\ldots,1,0,\ldots,0)}E
\subset S^{k-t}E\otimes\Lambda^tE$\break if $0\le t\le k-1$ and $Z^{t,k}E=0$
otherwise. Then there is a canonical complex
$$\ldots\lw H^{p,q}(X,Z^{t,k}E\otimes L)\lw H^{p+1,q+1}(X,Z^{t-1,k}E\otimes L)
\lw\ldots$$
Under the hypotheses of proposition 5.1 for $\calS=Q^{(k,0\ld 0)}_s
\otimes\pi^\star L$~, this complex is exact in each degree $q$ such that 
$p+q+t\ge n+r$~.
\fintha 

{\sl Proof}\pointir 
The only possible weight $\underline u$ of $\Lambda^t\gothu_s$ such that 
$(k,0\ld 0)+u$ be non-increasing is $u=(-t,1\ld 1,0\ld 0)$~. Theorem 3.8 
yields therefore
$$H^q(Y,G^{p,p+t}\otimes Q^{(k,0\ld 0)}_s\otimes\pi^\star L)=
  H^{p,q}(X,Z^{t,k}E\otimes L)~~.~\finpr$$

Note that the special case $k=1$~, $t=0$ is Le Potier's theorem, and that the
special case $k=2$~, $t=1$ is theorem 0.6. These two cases do not depend on 
any degeneracy assumption.
\vfill\eject

\titre References|
\vskip15pt
\eightpoint
\article 1|y. Akizuki {\rm and} s. Nakano|Note on Kodaira-Spencer's proof of
Lefschetz theorems|Proc. Jap. Acad.|30|1954|266--272|
\divers 2|a. Borel {\rm and} a. Weil|{\sl Repr\'esentations lin\'eaires et
espaces homog\`enes k\"ahl\'eriens des groupes de Lie compacts}|{\rm 
S\'eminaire Bourbaki (expos\'e \no 100 par J.-P. Serre),\break (mai 
{\oldstyle 1954}), 8 pages}|
\article 3|r. Bott|Homogeneous vector bundles|Ann. of Math.|66|1957|203--248|
\article 4|j.-p. Demailly|Th\'eor\`emes d'annulation pour la cohomologie des
puissances tensorielles d'un fibr\'e positif|C. R. Acad. Sci. Paris S\'er. I
Math.|305|1987|\`a para\^\i tre|
\divers 5|j.-p. Demailly|Vanishing theorems for tensor powers of a positive
vector bundle|{\rm Proceedings of the Conference Geometry and Analysis on 
Manifolds held in Katata, Japan (august {\oldstyle 1987}), edited by T. Sunada,
Lecture Notes in Math. n\omini~1339, Springer}|
\article 6|b. Demazure|A very simple proof of Bott's theorem|Invent. 
Math.|33|1976|271--272|
\livre 7|r. Godement|Th\'eorie des faisceaux|Hermann, Paris|1958|
\article 8|p.a. Griffiths|Hermitian differential geometry, Chern classes
and positive vector bundles|Global Analysis, Papers in honor of K. Kodaira,
Princeton Univ. Press|{\rm Princeton}|1969|185--251|
\article 9|r. Hartshorne|Ample vector bundles|Publ. Math. 
I.H.E.S.|29|1966|63--94|
\livre 10|h. Kraft|Geometrische Methoden in der Invariantentheorie|Aspekte
der Mathematik, Band D1, Braunschweig, Vieweg \& Sohn|1985|
\article 11|t{\rm h}. Peternell, j. le Potier {\rm and} m. Schneider|Vanishing 
theorems, linear and quadratic normality|Invent. Math.|87|1987|573--586|
\article 12|t. Peternell, j. le Potier {\rm and} m. Schneider|{\sl Direct 
images of sheaves of differentials and the Atiyah class}|Math.
Z.|196|1987|75--85|
\article 13|j. Le Potier|Annulation de la cohomologie \`a valeurs dans un
fibr\'e vectoriel holomorphe de rang quelconque|Math. Ann.|218|1975|35--53|
\article 14|m. Schneider|Ein einfacher Beweis des Verschwindungssatzes f\"ur
positive holomorphe Vektorraumb\"undel|Manuscripta Math.|11|1974|95--101|
\article 15|a.j. Sommese|Submanifolds of abelian varieties|Math. 
Ann.|233|1978|229--256|

\end
