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\centerline{\bf VANISHING THEOREMS FOR TENSOR POWERS}
\centerline{\bf OF A POSITIVE 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 $r$ over
a compact complex manifold $X$ of dimension $n$~. It is shown that the 
Dolbeault cohomology groups $H^{p,q}(X,E^{\otimes k}\otimes(\det E)^l)$ 
vanish if $E$ is positive in the sense of Griffiths and $p+q\ge n+1$~,
$l\ge r+C(n,p,q)$~. The proof rests on the well-known fact that every tensor 
power $E^{\otimes k}$ splits into irreducible representations of
$\gl(E)$~, each component being canonically isomorphic to the direct image on 
$X$ of a positive homogeneous line bundle over a flag manifold of $E$~. The 
vanishing property is then obtained by a suitable generalization of Le Potier's
isomorphism theorem, combined with a new curvature estimate for the bundle of 
$X$--relative differential forms on the flag manifold of $E$~.

\vskip 30pt
\titre 0. Statement of results.|
 
The aim of this work is to prove a rather general vanishing theorem for 
cohomology groups of tensor powers of a positive vector bundle.

 Let $X$ be a complex compact $n$--dimensional manifold and $E$ a hermitian
vector bundle of rank $r$ over $X$~. We denote by $C^\infty_{p,q}(X,E)$ the
space of smooth $E$--valued differential forms of type $(p,q)$ on $X$ and by
$$D_E=D'_E+D_E''~:~C^\infty_{p,q}(X,E)\lw C^\infty_{p+1,q}(X,E)
\oplus C^\infty_{p,q+1}(X,E)$$
(or simply $D=D'+D''$ ) the Chern connection of $E$~. Let 
$(x_1 \ld x_n)$
be holo\-morphic coordinates on $X$ and $(e_1 \ld e_n)$ a local $C^\infty$ 
orthonormal frame of $E$~. The Chern curvature tensor $c(E)$ is defined by 
$D^2=c(E)\wedge\blt$ and may be written
  $$c(E)=\sum_{i,j,\lambda ,\mu }c_{ij\lambda \mu }\,dx_i\wedge
    d{\overline x}_j\otimes e^\star_\lambda \otimes e_\mu~~,~~
    1\le i,j\le n~,~1\le\lambda,\mu \le r~.$$
The curvature tensor $ic(E)$ is in fact a (1,1)--form with values in the bundle
\ ${\rm Herm}(E,E)$ \ of hermitian endomorphisms of $E$~, \ie 
$~c_{ij\lambda\mu}={\overline c}_{ji\mu \lambda}$~; thus $ic(E)$ can be 
identified with a hermitian form on $TX\otimes E$~.

   Let us recall that the bundle $E$ is said to be {\sl positive}, resp. {\sl
semi-positive}\break (in the sense of Griffiths [8]) if $E$ can be endowed with
a hermitian metric such that at every point $x\in X$ one has
$$ic(E)_x(\zeta\otimes v,\zeta\otimes v)=\sum_{i,j,\lambda,\mu }
c_{ij\lambda\mu}(x)\,\zeta_i\overline\zeta_jv_\lambda {\overline v}_\mu >0~~,
~~{\rm resp.}~~\ge 0 $$
for all non zero vectors $\zeta=\sum \zeta_i\,\partial/\partial z_i \in T_xX~,$
$v=\sum v_\lambda e_\lambda\in E_x$~. Every vector bundle $E$ generated by 
sections is $\ge 0$~.

   Another important concept is that of ampleness, for which we refer to
Hartshorne [9]~; a vector bundle $E$ is ample if and only if the line bundle 
$O_E(1)$ over $P(E^\star)$ is ample (or equivalently $>0$)~. It is also
well-known that $E>0$ implies\break $E$ ample, but the converse is unknown.

  In the case of a positive or ample vector bundle $E$ of rank $r>1$~, 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\break 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, the following result shows that the 
vanishing property is true for tensor powers involving a sufficiently large
power of $\det E$~.

\tha Theorem|Let $L$ be a holomorphic line bundle over $X$~. Assume that $E>0$
and $L\ge 0$~, or $E\ge 0$ and $L>0$~. For all integers $p,q$ such that
$p+q\ge n$~, set
  $$A(n,p,q)={n(n+1)(p+1)(q+1)\over 4(p+q-n+1)}~~~~{\sl if}~p<n$$
and~ $A(n,p,q)=0$~ if $p=n$~. Let $h\in\{1\ld r-1\}$ and let $\Gamma^aE$ be 
the\break irreducible tensor power representation of $\gl(E)$ of highest weight
$a\in\bbz^r$~, with
  $$a_1\ge a_2\ge\ldots\ge a_h>a_{h+1}=\ldots=a_r=0~~.$$
If $p+q\ge n+1$ then
$$H^{p,q}(X,\Gamma^aE\otimes(\det E)^l\otimes L)=0~~~
            {\sl for}~~l\ge h+A(n,p,q)~~.$$
\fintha

The proof of this theorem is based on analysis and differential geometry, but
an analogous result can be obtained in a purely algebraic way (\cf [5]); in
that case the positivity hypothesis can be replaced by ampleness, the
semi-positivity hypothesis by the fact that the bundle is generated by its 
global sections; then, the condition required on $l$ is $l\ge n-p+r-1$~.
Both results overlap in most cases, but the above analytic result can be 
better if $r-h$ is very large. 

Observing that $S^kE$ is the irreducible representation of 
highest weight $(k,0\,\ld 0)$ and that $E^{\otimes k}$ splits into irreducible 
representations of the type $\Gamma^aE\otimes(\det E)^l$ with $h\le\min\{k,r-1
\}$ (\cf formula (2.17)), we obtain~:

\tha Corollary|Under the positivity hypotheses of the theorem, then for all 
$p,q$ such that $p+q\ge n+1$ one has
\vskip 2pt\noindent
{\rm (0.1)}\quad $H^{p,q}(X,S^kE\otimes(\det E)^l\otimes L)=0$~~~if~~
$l\ge 1+A(n,p,q)$~~;
\vskip 2pt\noindent
{\rm (0.2)}\quad $H^{p,q}(X,E^{\otimes k}\otimes(\det E)^l\otimes L)=0$~~~if~~
$l\ge\min\{k,r-1\}+A(n,p,q)$~~.
\fintha 

   The special case $p=n$ of (0.1) is due to P. Griffiths [8]~. For $p=n$ and 
arbitrary $r\,,\,k_0\ge 2$~, Peternell-Le Potier and Schneider [11] have 
constructed an example of a vector bundle $E>0$ of rank $r$ over a manifold
$X$ of dimension $n=2r$ such that
$$H^{n,n-2}(X,S^kE)\ne 0~~,~~2\le k\le k_0~~.\leqno(0.3)$$
This result shows that the lower bound $l\ge 1$ in (0.1) cannot be improved. 
More generally, the following example (for which we refer to [5]) 
shows that our condition $l\ge h$ in the theorem is optimal. This example gives
a negative answer to a question of Sommese [15].

\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 $r$ over $X$ (then $E\ge 0$ and $L=\det E$ is 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
$$H^{n,q}\big(X,\Gamma^aE\otimes(\det E)^h\big)=\Gamma^\beta V\otimes(\det V)^h
\ne 0~~.\leqno(0.4)$$
\fintha

Our approach is based on three well-known facts. First, every tensor power
of $E$ splits into irreducible representations of the linear group $\gl(E)$~.
It is thus sufficient to consider ``irreducible'' tensor powers of $E$~. 
Secondly, every irreducible tensor power of $E$ appears in a natural way as the
direct image on $X$ of a {\sl positive line bundle} over 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 $E$ over $X$ to those of the line bundle
${\rm O}_E(1)$ over $P(E^\star)$~. We generalize here this isomorphism
to the case of arbitrary flag bundles associated to $E$~.

  When $p=n$ the above-mentioned algebraic facts suffice to prove the theorem.
However, when $p<n$~, the generalized Borel-Le Potier spectral sequence
does not degenerate at the $E_1$ level (\cf [12]). A possible way in order to 
overcome this difficulty is to establish a curvature estimate for the bundle 
of $X$--relative differential forms on the flag manifold of $E$~, using
the standard Kodaira-Akizuki-Nakano inequality [1]~. Our estimate (\cf 
\S 4) measures in some sense how far is the spectral sequence from being
$E_1$--degenerate. 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 a dominant weight $a\in\bbz^r$ with $a_r=0$~, determine the
smallest admissible constant $A(n,p,q)$ in the theorem.
\fintha

It is shown in [5] that if the Borel-Le Potier spectral sequence degenerates
in $E_2$~, then it is always sufficient to take $l\ge r-1+\min
\{n-p,n-q\}$~. In view of the above main theorem, one may suspect that the 
correct answer could be $l\ge h+\min\{n-p,n-q\}$~.

  The above results have been annouced in the note [4].
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. Kodaira-Akizuki-Nakano vanishing theorem.|

   We recall here the basic Akizuki-Nakano inequality [1] which will be used 
several times in the sequel. Assume that $X$ carries a K\"ahler metric 
$\omega$~, and let $L$ be a hermitian line bundle over $X$~. At each point
$x\in X$~, one can write
  $$ic(L)=i\sum_{1\le j\le n}\gamma_j\, dz_j\wedge d\overline z_j$$
where $(dz_1 \ld dz_n)$ is an $\omega$--orthonormal basis of $T^\star X$ and
where $\gamma_1\le\,\ldots\,\le\gamma_n$ are the curvature eigenvalues of $L$~.
For every $v\in C^\infty_{p,q}(X,L)$ we have
$$||D_L''v||^2+||D_L''{}^\star v||^2\ge\langle\Theta_Lv\,,\,v\rangle~~,
\leqno(1.1)$$
where $\Theta_L$ is the hermitian endomorphism defined by
$$\langle\Theta_Lv\,,\,v\rangle=\sum_{|I|=p,|J|=q}\big(\gamma_I+\gamma_J
-\sum_{1\le j\le n}\gamma_j\big)|v_{I,J}|^2~~,\leqno(1.2)$$
with $\gamma_I=\sum_{m\in I}\gamma_m$~. When $L$ is $>0$~, one can choose
$\omega=ic(L)$ as the K\"ahler metric on $X$~; in that case $\gamma_1=\ldots=
\gamma_n=1$ and therefore
$$||D_L''v||^2+||D_L''{}^\star v||^2\ge(p+q-n)||v||^2~~.\leqno(1.3)$$

\tha Corollary {\rm (Kodaira-Akizuki-Nakano)}|One has\bid\hfill\break
{\rm(1.4)}\qquad $L>0~~\Longrightarrow~~H^{p,q}(X,L)=0$~~~for~~ $p+q\ge n+1$~~,
\hfill\break
{\rm(1.5)}\qquad $L<0~~\Longrightarrow~~H^{p,q}(X,L)=0$~~~for~~
$p+q\le n-1\bid$~~.\fintha

\vskip 20pt
\centerline{\bf 2. Homogeneous line bundles over 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 over flag manifolds. The classical
foundation works on this subject are Borel-Weil [2] and R. Bott [3]~, which
contain all the required material (\cf also Demazure [6] for a very
simple proof of Bott's formula). We will give however an independent 
self-contained exposition in order to prepare the tools needed in the 
differential geometric approach of \S 4~.

   Let $B_r$ (resp. $B^r$ ) be the Borel subgroup of $\glr=\gl(\bbc^r)$ of
lower (resp. upper) triangular matrices, $U_r\subset B_r$~, $U^r\subset B^r$
the subgroups of unipotent matrices, and $T^r=B_r\cap B^r$ 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 $V_\lambda$~, $Q_\lambda$~,
$Q^a$~. 

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
$$\left\{\hbox{$\eqalign{&T^\star M(V)=\big(\hom(V,V)/W\big)^\star=W^\perp\cr
&W^\perp=\{g\in\hom(V,V)\ ;\ g(V_{\lambda-1})\subset V_\lambda~,
~1\le\lambda\le r\}~~.\cr}$}\right.\leqno(2.3)$$
There exists a filtration of $T^\star M(V)$ by subbundles of the type 
   $$\{g\in\hom(V,V)\ ;\ g(V_\lambda)\subset V_{\mu(\lambda)}~,
           ~\lambda<\mu(\lambda)~,~1\le\lambda\le r\}$$
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 $K_{M(V)}=\det(T^\star M(V))$~:
$$K_{M(V)}=Q_1^{1-r}\otimes\ldots\otimes Q_\lambda^{2\lambda-r-1}\otimes\ldots
          \otimes Q_r^{r-1}=Q^c\leqno(2.4)$$
where $c=(1-r\ld r-1)$~; $c$ will be called the canonical weight of
$M(V)$~.

\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 
$$Q_{s,j}=\det(V_{s,j-1}/V_{s,j})~~,~~
1\le j\le m~~.\leqno(2.5)$$
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
$$\eta^\star\, V_{s,j}=V_{s_j}~~,~~\eta^\star Q_{s,j}=
Q_{s_{j-1}+1}\otimes\ldots\otimes Q_{s_j}~~,~~\eta^\star Q_s^a=Q^a~~.
\leqno(2.6)$$
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 $j$~, and we set $B^s={}^tB_s$~, 
$U^s={}^tU_s$~. In the same way as above, we get
$$\leqalignno{TM_s(V)&=\hom(V,V)/W_s~~,~~W_s=\{g~;~g(V_{s_{j-1}})
           \subset V_{s_j}\}~~,&(2.7)\cr
T^\star M_s(V)&=W_s^\perp~~,&(2.8)\cr
K_{M_s(V)}&=Q_{s,1}^{s_1-r}\otimes\ldots\otimes Q_{s,j}
  ^{s_{j-1}+s_j-r}\otimes\ldots\otimes Q_{s,m}^{s_{m-1}}=Q_s^{c(s)}
&(2.9)\cr}$$
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 $M_s(V)$~. 

\noindent$\bullet$ {\pench Curvature form of the line bundle $Q^a$~.}

Assume now that $V$ is a {\sl hermitian} vector space. Then all our bundles 
carry a natural hermitian metric. We are going to compute the curvature of 
$Q^a$ at any point $[e]\in M(V)$~. Choose an orthonormal basis $(e_1 \ld e_r)$
of $V$ which corresponds to the given point $[e]$~. It is clear that
$eB_re^{-1}\subset \gl(V)$ is the isotropy subgroup of $[e]$~, whereas
$eU^re^{-1}.[e]=[eU^r]$ is an affine open subset of $M(V)$~, corresponding
to bases $(\zeta_1\ld \zeta_r)$ of $V$ such that
$$\zeta_\mu=e_\mu+\sum_{\lambda<\mu} z_{\lambda\mu}e_\lambda~~,~~
    1\le\mu\le r~~,~~z_{\lambda\mu}\in\bbc~~.$$
Then $(z_{\lambda\mu})_{1\le\lambda<\mu\le r}$ is a coordinate system on
$[eU^r]$ and the map
$$M(V)\ni[\zeta]\longmapsto\tilde\zeta_\mu:=\zeta_\mu~~{\rm mod}~V_\mu$$
is a local section of $Q_\mu=V_{\mu-1}/V_\mu$~. Hence
$$c(Q_\mu)=-d'd''\log|\tilde\zeta_\mu|^2~~.$$
Let us identify $\tilde\zeta_\mu$ with the orthogonal projection of $\zeta_\mu$
on $V_{\mu-1}\cap V_\mu^\perp$~. Then Gram-Schmidt's orthogonalization process
yields
$$\tilde\zeta_\mu=\zeta_\mu-\sum_{\nu>\mu}{\langle\zeta_\mu,\tilde\zeta_\nu
\rangle\over|\tilde\zeta_\nu|^2}\,\tilde\zeta_\nu~~,~~|\tilde\zeta_\mu|^2=
|\zeta_\mu|^2-\sum_{\nu>\mu}{|\langle\zeta_\mu,\tilde\zeta_\nu\rangle|^2\over
|\tilde\zeta_\nu|^2}~~.$$
Since $\langle\zeta_\mu,\zeta_\nu\rangle=\overline z_{\mu\nu}+\sum_{\lambda
<\mu}z_{\lambda\mu}\overline z_{\lambda\nu}$~, it follows by backward induction
on $\nu$ that $\langle\zeta_\mu,\tilde\zeta_\nu\rangle=\overline z_{\mu\nu}+
\rmo(|z|^2)$ for $\nu>\mu$~, hence
$$|\tilde\zeta_\mu|^2=1+\sum_{\lambda<\mu}|z_{\lambda\mu}|^2-\sum_{\nu>\mu}
|z_{\mu\nu}|^2+\rmo(|z|^3)~~.$$
We obtain therefore
$$\leqalignno{\displaystyle
c(Q_\mu)_{[e]}&=-\sum_{\lambda<\mu}dz_{\lambda\mu}\wedge d\overline z_{\lambda
\mu}+\sum_{\nu>\mu}dz_{\mu\nu}\wedge d\overline z_{\mu\nu}~~,&(2.10)\cr
c(Q^a)_{[e]}&=\sum_\mu a_\mu c(Q_\mu)=\sum_{\lambda<\mu}(a_\lambda-a_\mu)
dz_{\lambda\mu}\wedge d\overline z_{\lambda\mu}~~.&(2.11)\cr}$$

\tha Corollary 2.12|$Q^a$ is $\ge 0$ if and only if
$a_1\ge a_2\ge\ldots\ge a_r$~.\break If there exists an index $j$ such that 
$a_j<a_{j+1}$~, then $H^0(M(V),Q^a)=0$~.
\fintha

{\sl Proof}\pointir Only the second statement and the ``only if'' part of the
first remain to be proved. Let us observe that the projection 
$\eta~:~M(V)\lw M_s(V)$~, $s=(0\ld j-1,j+1\ld r)$~, is a bundle with
fibers $P(V_{j-1}/V_{j+1})\simeq {\Bbb P}^1$~.\break The restriction of 
$Q_\lambda$ to each fiber is trivial if $\lambda \ne j,j+1$ whereas 
$Q_{j\upharpoonright{\Bbb P}^1}\simeq\rmo(1)$ and $Q_{j+1\upharpoonright
{\Bbb P}^1}\simeq\rmo(-1)$~. 
Therefore $Q^a_{\upharpoonright{\Bbb P}^1}\simeq\rmo(a_j-a_{j+1})$
cannot have any non-zero section or any semi-positive metric if
$a_j<a_{j+1}$~. \finpr

When $a_1\ge\ldots\ge a_r$~, the bundle $Q^a$ is not necessarily $>0$ on
$M(V)$~; in fact one can write $Q^a$ as the induced bundle $\eta^\star
Q_s^a$ where $s_1<\ldots<s_{m-1}$ is the sequence of integers $\lambda=
1\ld r-1$ such that $a_{\lambda+1}>a_\lambda$~. The affine open subset 
$[eU^s]\subset M_s(V)$ is a neighborhood of $[e]$~, and $M_s(V)$ 
has local coordinates $(z_{\lambda\mu})$ where $\lambda,\mu$ are such that 
$\lambda\le s_{j-1}<s_j\le\mu$ for some $j$~, \ie $a_\lambda>a_\mu$~. The 
curvature of $Q_s^a$ is given formally by the same expression as (2.11)~:
$$c(Q_s^a)_{[e]}=\sum_{a_\lambda>a_\mu}(a_\lambda-a_\mu)
dz_{\lambda\mu}\wedge d\overline z_{\lambda\mu}~~.\leqno(2.13)$$
We see therefore that $Q_s^a>0$ on $M_s(V)$~.
\vskip8pt

\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.

\tha Proposition 2.14|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
$$\leqalignno{S^kV&=\Gamma^{(k,0\ld 0)}V~~,&(2.15)\cr
        \Lambda^kV&=\Gamma^{(1\ld 1,0\ld 0)}V~~.&(2.16)\cr}$$
For arbitrary $a\in\bbz^r$~, proposition 2.14 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 elements $a\in\bbz^r$ will be ordered according to the partial ordering~:
  $$a\succcurlyeq b~~~{\rm iff}~~~
    \sum_{1\le\lambda\le\mu}a_\lambda\ge\sum_{1\le\lambda\le\mu}b_\lambda~~,~~
    1\le\mu\le r~~.$$
Bott's theorem [3] shows that $\Gamma^aV$ is an irreducible representation of 
$\gl(V)$ of highest weight $a$~; 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
$$V^{\otimes k}=\bigoplus_{\scriptstyle a_1+\ldots+a_r=k\atop \scriptstyle
             a_1\ge\ldots\ge a_r\ge 0}\mu(a,k)\,\Gamma^aV\leqno(2.17)$$
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 $H^q(M(V),Q^a)$~, but we will need
them here only in the case of {\sl dominant} weights $a_1\ge\ldots\ge a_r$~.

\tha Proposition 2.18|Set $N=\dim M(V)$~, $N(s)=\dim M_s(V)$~.\break If
$a_{s_j}-a_{s_{j+1}}\ge 1$~, then
\vskip2pt\noindent
{\rm (a)}\quad $H^{N(s),q}(M_s(V),Q_s^a)=0$~~~for all~~$q\ge 1$
\vskip2pt\noindent
{\rm (b)}\quad $H^{N(s),0}(M_s(V),Q_s^a)=\Gamma^{a+c(s)}V$~.
\fintha

{\sl Proof}\pointir Under the assumption of (a), $Q_s^a$ is $>0$ by 
(2.13). The result follows therefore from the Kodaira-Akizuki-Nakano theorem.
Now (b) is a consequence of proposition 2.14 since
$$H^{N(s),q}(M_s(V),Q_s^a)=H^q(M_s(V),K_{M_s(V)}\otimes Q_s^a)=
                           H^q(M_s(V),Q_s^{a+c(s)})~~.\finpr$$

\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, we obtain the following
lemma.

\tha Lemma|For every weight $a$ such that $a_{s_j}-a_{s_{j+1}}\ge 1$~, $1\le j
\le m-1$~, the sheaf of sections of $\Omega^{N(s)}_{Y/X}\otimes Q^a_s$ has 
direct images
$$\left\{\hbox{$\eqalign{
R^q\pi_\star\big(\Omega^{N(s)}_{Y/X}\otimes Q^a_s\big)&=0~~~{\sl for}~~q\ge1\cr
\pi_\star\big(\Omega^{N(s)}_{Y/X}\otimes Q^a_s\big)&=\Gamma^{a+c(s)}E~~.\cr}$}
\right.\leqno(3.4)$$\fintha

Let $L$ be an arbitrary line bundle on $X$. Under the hypothesis
$a_{s_j}-a_{s_{j+1}}\ge 1$,\break formulas (3.3) and (3.4) yield
$$\eqalign{
R^q&\pi_\star(G^{p,p+N(s)}\otimes Q^a_s\otimes\pi^\star L)=0
                 ~~~{\rm for}~~q\ge 1~~,\cr
&\pi_\star(G^{p,p+N(s)}\otimes Q^a_s\otimes\pi^\star L)=
\Omega^p_X\otimes\Gamma^{a+c(s)}E\otimes L~~.\cr}$$
The Leray spectral sequence implies therefore~:

\tha Theorem 3.5|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]~:
$$H^{n+N(s),q}(M_s(E),Q^a_s\otimes\pi^\star L)\simeq
H^{n,q}(X,\Gamma^{a+c(s)}E\otimes L)~~.\leqno(3.6)$$
  
  In order to carry results for line bundles over to vector bundles, one needs
the following lemma. 

\tha Lemma 3.7|Assume that $a_{s_1}>a_{s_2}>\ldots>a_{s_m}\ge 0$~. Then
\vskip0.1pt\noindent
{\rm (a)}\qquad $E\ge 0~~~($resp. $>0)~~~\Longrightarrow~~~Q^a_s\ge 0~~~
{\sl (resp.}~>0)~~;$
\vskip0.1pt\noindent
{\rm (b)}\qquad $E\ge 0$~~and~~$L>0~~~\Longrightarrow~~~Q^a_s
\otimes\pi^\star L>0~~;$
\vskip0.1pt\noindent
{\rm (c)}\qquad $\,E$ ample~~~$\Longrightarrow~~~Q^a_s$~~ample~.
\fintha

{\sl Proof}\pointir Part (a) will be proved in \S 4 (\cf formula (4.9)) and (b)
follows from the fact that $c(Q^a_s)>0$ along the fibers of $\pi$~.

\noindent
(c) By definition of an ample vector bundle (Hartshorne [9])~, $S^kE$ is very
ample 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 very 
ample for $k\ge k_0$~. Now, formula (2.14) shows that $Q^a_s>0$ along the 
fibers of $\pi$~, hence $Q^{ka}_s$ is very ample along each fiber for 
$k\ge k_1$~. Since $\pi_\star(Q^{ka}_s)=\Gamma^{ka}E$~, we conclude that 
$Q^{ka}_s$ is very ample for $k\ge\max(k_0,k_1)$~. \finpr

We are now ready to attack the proof of the main theorem. We study first the
special case $p=n$~.

\tha Theorem 3.8|Let $a\in\bbz^r$ be such that
$$a_1\ge a_2\ge\ldots\ge a_h>a_{h+1}=\ldots=a_r=0~~,~~1\le h\le r-1~~.$$
Assume that $E$ is ample and $L\ge 0$~, or $E\ge 0$ and $L$ ample. Then 
$$H^{n,q}(X,\Gamma^aE\otimes(\det E)^l\otimes L)=0~~~
        {\sl for}~~q\ge1~~,~~l\ge h~~.$$
\fintha

{\sl Proof of theorem 3.8 and of the main theorem}\pointir 

Let $s_1>\ldots>s_{m-1}$ be the
sequence of integers $\lambda=1\ld r-1$ such that $a_{\lambda+1}>a_\lambda$~.
Then theorem 3.5 implies
$$H^{p,q}(X,\Gamma^aE\otimes(\det E)^l\otimes L)\simeq
  H^q(Y,G^{p,p+N(s)}\otimes Q^b_s\otimes\pi^\star L)\leqno(3.9)$$
where $b=a-c(s)+(l\ld l)$~. The canonical weight $c(s)$ is non-decreasing and 
$c(s)_r=s_{m-1}=h$~, hence
 $$b_{s_1}>\ldots>b_{s_m}=l-h~~.$$
Lemma 3.7 shows that $Q^b_s\otimes\pi^\star L>0$ if 
$l\ge h$~. Now, it is clear that $F^{p,p+N(s)}=\Omega_Y^{p+N(s)}$~. One gets 
thus an exact sequence
  $$0\lw F^{p+1,p+N(s)}\lw \Omega_Y^{p+N(s)}
     \lw G^{p,p+N(s)}\lw 0~~.\leqno(3.10)$$
The Kodaira-Akizuki-Nakano vanishing theorem (1.4) applied to 
$Q^b_s\otimes\pi^\star L$ with  $\dim Y=n+N(s)$ yields
 $$H^q(Y,\Omega_Y^{p+N(s)}\otimes Q^b_s\otimes\pi^\star L)=0~~~{\rm for}~~
     p+q\ge n+1~~.$$
The cohomology groups in (3.9) will therefore vanish if and only if
   $$H^{q+1}(Y,F^{p+1,p+N(s)}\otimes Q^b_s\otimes\pi^\star L)=0~~.
     \leqno(3.11)$$
This is obvious if $p=n$~, for $F^{n+1,n+N(s)}=0$~. In the general case $p<n$~,
we will establish in \S 4 that (3.11) holds for $p+q\ge n$ and 
$b_r=l-h\ge A(n,p,q)$~. This will be done by means of a curvature
estimate for the bundle $F^{p+1,p+N(s)}$~. \finpr

{\sl Remark} 3.12\pointir If $p+q=n$~, we still obtain some result, namely 
that the canonical map
$$H^{p+N(s),q}(Y,Q^b_s\otimes\pi^\star L)\lw 
H^{p,q}(X,\Gamma^aE\otimes(\det E)^l\otimes L)$$
is onto when $l\ge h+A(n,p,q)$~.

{\sl Remark} 3.13\pointir If the exact sequence (3.10) splits, then (3.11) is
an immediate consequence of the Kodaira-Nakano-Akizuki theorem. However, 
Peternell-Le Potier and Schneider [11]~, [12] have shown that in general the 
filtration $F^\blt(\Omega^t_Y)$ does not split, and this is the reason why we
have to introduce additional consi\-derations in order to prove (3.11).

\vskip15pt
\titre 4. A curvature estimate for the subbundle $F^{p+1,p+N(s)}$~.|
 
   We assume here that $E,L\lw X$ are hermitian vector bundles of 
respective ranks $r,1$ and that $E\ge 0$ and $L>0$~, or $E>0$ and $L\ge 0$~.
Let $a\in\bbz^r$ be such that $a_1\ge\ldots\ge a_r\ge 0$ and let $s_1<\ldots<
s_{m-1}$ be the sequence of integers $\lambda=1\ld r-1$ such that 
$a_{\lambda+1}>a_\lambda$~. We set for simplicity 
$$Y=M_s(E)~~,~~\Omega=\Omega_Y^{p+N(s)}~~,~~F=F^{p+1,p+N(s)}~~,~~
G=G^{p,p+N(s)}=\Omega/F~~.$$ 
Our aim is to prove that the analogue of (3.11)~:
$$H^{q+1}(Y,F\otimes Q^a_s\otimes\pi^\star L)=0\leqno(4.1)$$
holds when $p+q\ge n$ and $a_r\ge A(n,p,q)$~. Let us consider the exact 
sequences of vector bundles over $Y$~:
$$\left\{\hbox{$\displaystyle
  \eqalign{0&\lw \ \ F\ \cr
           0&\lw  F(a)\cr}
  \eqalign{&\lw \ \ \Omega\ \cr
           &\lw \Omega(a) \cr} 
  \eqalign{&\lw \ \ G\ \cr
           &\lw G(a) \cr}
  \eqalign{&\lw 0~~, \cr
           &\lw 0~~, \cr}$}\right.\leqno(4.2)$$
where the second sequence arises from the first one after taking tensor 
products with the line bundle $Q^a_s\otimes \pi^\star L$~. Then $Y$  can be
equipped with the K\"ahler metric $\omega=ic(Q^a_s\otimes\pi^\star L)$~; 
the positivity of $\omega$ is a consequence of (4.9) below. To~every smooth 
form $v$ of type $(p+N(s),q+1)$ with values in $Q^a_s\otimes\pi^\star L$~,
let us apply the Akizuki-Nakano inequality (1.3)~, where $\dim Y=n+N(s)$~:
     $$||D_{\Omega(a)}''v||^2+||D^{\pr\pr\star}_{\Omega(a)}v||^2
                        \ge (p+q-n+1))\ ||v||^2~~.\leqno(4.3)$$
With respect to the orthogonal $C^\infty$--splitting $\Omega\simeq F\oplus G$~,
the Chern connections of $\Omega$~, $F$~, $G$ are related by the well-known
formula (\cf [8])~:
$$D_\Omega=\pmatrix{D_F & -\beta^\star\wedge\blt\cr
                    \beta\wedge\blt & D_G\cr}~~,~~
  \beta\in C^{\infty}(\Lambda^{1,0}T^\star Y\otimes\hom(F,G))~;$$
$\beta^\star$ is a $D''$--closed (0,1)--form with values in $\hom(G,F)$~, and
its cohomology class is the obstuction to the existence of a global splitting
of (4.2). We obtain therefore
$$D_{\Omega(a)}''=
   \pmatrix{D_{F(a)}'' & -\beta^\star\wedge\blt\cr 0 & D_{G(a)}''\cr}\quad,
                   \quad D^{\pr\pr\star}_{\Omega(a)}=
   \pmatrix{D^{\pr\pr\star}_{F(a)}& 0\cr
                  -\beta\ort\blt & D^{\pr\pr\star}_{G(a)}\cr}~~,$$
where $\ort$ denotes the interior product of differential forms combined with 
the evaluation map $\hom(F,G)\times F\to G$ (note that $\beta\ort\blt=
(\beta^\star\wedge\blt)^\star$).  
For every $(0,q+1)$--form $f$ with values in $F(a)$ we get
$$D_{F(a)}''f=D_{\Omega(a)}''f~~,~~||D^{\pr\pr\star}_{F(a)}f||^2=
||D^{\pr\pr\star}_{\Omega(a)}f||^2-||\beta \ort f||^2~~.\leqno(4.4)$$
From (4.3) and (4.4), we see that the vanishing property (4.1) will hold if
  $$|\beta\ort f|^2<(p+q-n+1)\,|f|^2\leqno(4.5)$$
at every point of $Y$~. We are going to compute $\beta$ explicitly in suitable 
coordinate systems on $Y$~. Let $y^0\in Y$ be an arbitrary point and 
$(x_1\ld x_n)$ local coordinates on $X$ centered at the point $x^0=\pi(y^0)$~.

\tha Lemma|There exists a local holomorphic frame $(e_1\ld e_r)$ of $E$
such that $y^0$ coincides with the flag $[e_1(x^0)\ld e_r(x^0)]$ and
$$\langle e_\lambda(x)\,,e_\mu(x)\rangle=\delta_{\lambda\mu}-\sum_{i,j}c_{ij
\lambda\mu}\,x_i\overline x_j+\rmo(|x|^3)~~,\leqno(4.6)$$
where $(c_{ij\lambda\mu})$ is the curvature tensor of $E$~.
\fintha 

{\sl Proof}\pointir Choose a holomorphic frame~ $(\varepsilon_1\ld\varepsilon
_r)$~ of $E$ such that\break $(\varepsilon_1(x^0)\ld\varepsilon_r(x^0))$
is orthonormal and $[\varepsilon_1(x^0)\ld\varepsilon_r(x^0)]=y^0$~. Then the
inner product $\langle\varepsilon_\lambda(x),\varepsilon_\mu(x)\rangle$ has
a Taylor expansion of the type
$$\eqalign{\displaystyle
\langle\varepsilon_\lambda(x),\varepsilon_\mu(x)\rangle=\delta_{\lambda\mu}
&+\sum_i(\gamma_{i\lambda\mu}x_i+\overline\gamma_{i\mu\lambda}\overline x_i)\cr
&+\sum_{i,j}(\gamma_{ij\lambda\mu}x_i\overline x_j+\gamma'_{ij\lambda\mu}x_ix_j
 +\overline\gamma'_{ij\mu\lambda}\overline x_i\overline x_j)+\rmo(|x|^3)~~.\cr}
$$
This expansion can be reduced to (4.6) (with suitable coefficients 
$c_{ij\lambda\mu}$) if one sets
$$e_\lambda(x)=\varepsilon_\lambda(x)-\sum_{i,\mu}\gamma_{i\lambda\mu}x_i\,
            e_\mu-\sum_{i,j,\mu}\gamma'_{ij\lambda\mu}x_ix_j\,e_\mu~~.$$
Now (4.6) implies
$$\eqalign{\displaystyle
De_\lambda&=-\sum_{i,j,\mu}c_{ij\lambda\mu}\overline x_jdx_i\otimes e_\mu
            +\rmo(|x|^2)~~,\cr
D^2e_\lambda&=\sum_{i,j,\mu}c_{ij\lambda\mu}dx_i\wedge d\overline x_j
            \otimes e_\mu+\rmo(|x|)~~,\cr}$$
showing that the $c_{ij\lambda\mu}$ 's are precisely the curvature 
coefficients at $x^0$~. \finpr

Let us denote by $z=(z_{\lambda\mu})$ the affine coordinates on 
the fiber $M_s(E_x)\subset Y$ associated to the basis $(e_1(x)\ld
e_r(x))$~. Then $(x_1\ld x_n,z_{\lambda\mu})$ define local coordinates on $Y$
in a neighborhood of $y^0$~. Assume first that $Y=M_s(E)=M(E)$ is the
manifold of complete flags of $E$~. Then we have tautological subbundles
$V_\lambda={\rm Vect}(e_{\lambda+1}\ld e_r)\subset\pi^\star E$ and the map
$$Y\ni(x,z)\longmapsto\zeta_\mu=e_\mu(x)+\sum_{\lambda<\mu}z_{\lambda\mu}
e_\lambda(x)\leqno(4.7)$$
is a local section of $V_{\mu-1}$~. Let us denote by $\widetilde\zeta_\mu$
the image of $\zeta_\mu$ in $Q_\mu=V_{\mu-1}/V_\mu$~, represented by the
orthogonal projection of $\zeta_\mu$ on $V_{\mu-1}\cap(V_\mu)^\perp$~. As
in \S 2, one finds
$$\left\{\hbox{$\eqalign{\displaystyle
&\widetilde\zeta_\lambda=\zeta_\lambda-\sum_{\mu>\lambda}{\langle\zeta_\lambda,
\widetilde\zeta_\mu\rangle\over|\widetilde\zeta_\mu|^2}\,\widetilde\zeta_\mu~~,
~~|\widetilde\zeta_\lambda|^2=|\zeta_\lambda|^2-\sum_{\mu>\lambda}
{|\langle\zeta_\lambda,\widetilde\zeta_\mu\rangle|^2\over|\widetilde
\zeta_\mu|^2}~~,\cr
&\langle\zeta_\lambda,\widetilde\zeta_\mu\rangle=\overline z_{\lambda\mu}-
\sum_{i,j}c_{ij\lambda\mu}x_i\overline x_j~~~{\rm mod}(z^2,x^3,x^2z)~~
{\rm for}~~\lambda<\mu~~.\cr}$}\right.\leqno(4.8)$$
We need a Taylor expansion of $c(Q_\mu)=-d'd''\log|\widetilde\zeta_\mu|^2$ 
up to order 1, hence of $|\widetilde\zeta_\mu|^2$ up to order 3. Moreover, pure
terms $x^3$~, $z^3$ will not play any role because the K\"ahler property of
$c(Q_\mu)$ enables one to get rid of the terms $\rmo(|x|dx\wedge d\overline 
x)$~, $\rmo(|z|dz\wedge d\overline z)$~. Therefore, we are interested only
in terms of degree $\le 2$ and in mixed terms $xz^2$~, $x^2z$~. Thanks to 
formulas (4.7) and (4.8), we get the following equalities modulo 
the ideal $(x^3,z^3,x^2z^2)$~:
$$\eqalign{\displaystyle
|\widetilde\zeta_\lambda|^2 &\sim |\zeta_\lambda|^2-\sum_{\mu>\lambda}
|\langle\zeta_\lambda,\widetilde\zeta_\mu\rangle|^2~~,\cr
|\zeta_\lambda|^2&\sim 1-\sum_{i,j}c_{ij\lambda\lambda}x_i\overline x_j+\sum_
{\mu<\lambda}|z_{\mu\lambda}|^2\cr
&\phantom{=1}\,-\sum_{i,j,\mu<\lambda}c_{ij\lambda\mu}x_i\overline x_j\overline
z_{\mu\lambda}-\sum_{i,j,\mu<\lambda}\overline c_{ij\lambda\mu}\overline x_i 
x_jz_{\mu\lambda}~~,\cr
|\langle\zeta_\lambda,\widetilde\zeta_\mu\rangle|^2&\sim|z_{\lambda\mu}|^2-
\sum_{i,j}c_{ij\lambda\mu}x_i\overline x_jz_{\lambda\mu}-\sum_{i,j}
\overline c_{ij\lambda\mu}\overline x_ix_j\overline z_{\lambda\mu}~~.\cr}$$
We have now
$$c(Q^a)=\sum_\lambda a_\lambda c(Q_\lambda)=d'd''\Big(-\sum_\lambda a_\lambda
\log|\widetilde\zeta_\lambda|^2\Big)~~,$$
\vskip-10pt\noindent
$$\eqalign{-\sum_\lambda a_\lambda\log|\widetilde\zeta_\lambda|^2&\sim
\sum_{i,j,\lambda}a_\lambda c_{ij\lambda\lambda}x_i\overline x_j+
\sum_{\lambda<\mu}(a_\lambda-a_\mu)|z_{\lambda\mu}|^2\cr
&-\sum_{i,j,\lambda<\mu}(a_\lambda-a_\mu)c_{ij\lambda\mu}x_i\overline x_j
z_{\lambda\mu}\cr
&-\sum_{i,j,\lambda<\mu}(a_\lambda-a_\mu)\overline c_{ij\lambda\mu}\overline 
x_ix_j\overline z_{\lambda\mu}~~.\cr}$$
We find therefore
$$\eqalign{c(Q^a)&=
\sum_{i,j}\Big(\sum_\lambda a_\lambda c_{ij\lambda\lambda}+\rmo(|x|)\Big)
dx_i\wedge d\overline x_j\cr
&+\sum_{\lambda<\mu}(a_\lambda-a_\mu)dz_{\lambda\mu}\wedge d\overline 
z_{\lambda\mu}+\rmo(|z|dz\wedge d\overline z)\cr
&-\sum_{i,j,\lambda<\mu}(a_\lambda-a_\mu)c_{ij\lambda\mu}
\big(z_{\lambda\mu}dx_i\wedge d\overline x_j+x_idz_{\lambda\mu}\wedge
d\overline x_j\big)\cr
&-\sum_{i,j,\lambda<\mu}(a_\lambda-a_\mu)\overline c_{ij\lambda\mu}\big(
\overline z_{\lambda\mu}dx_j\wedge d\overline x_i+\overline x_idx_j\wedge
d\overline z_{\lambda\mu}\big)+\rmo(|x|^2+|z|^2)~~.\cr}$$
Since $Q^a=\pi_s^\star Q_s^a$~, the same identity holds for
$Q_s^a$~. At the point $y^0$ we get
$$c(Q^a_s)_{y^0}=\sum_{i,j,\lambda}a_\lambda c_{ij\lambda\lambda}dx_i\wedge
d\overline x_j+\sum_{a_\lambda>a_\mu}(a_\lambda-a_\mu)dz_{\lambda\mu}\wedge
d\overline z_{\lambda\mu}~~.\leqno(4.9)$$
Now, $\omega=i\big(\pi^\star c(L)+c(Q^a_s)\big)$ is K\"ahler
on $Y=M_s(E)$~, thus in particular along the fiber $x=0$ and along the 
local section $z=0$~. It follows that one can find coordinate changes
$x\mapsto x'$~, $z'_{\lambda\mu}=\sqrt{a_\lambda-a_\mu}\,z_{\lambda\mu}~{\rm
mod}~z^2$ such that the terms $\rmo(|x|dx\wedge d\overline x)$ and
$\rmo(|z|dz\wedge d\overline z)$ disappear in the expansion of $\omega$~,
and such that
$$\omega_{ij}(y^0)=c(L)_{ij}(x^0)+\sum_\lambda a_\lambda c_{ij\lambda\lambda}
=\delta_{ij}~~.\leqno(4.10)$$
We obtain therefore
$$\eqalign{{1\over i}\omega&=
\sum_j dx'_j\wedge d\overline x'_j+\sum_{\lambda<\mu}dz'_{\lambda\mu}\wedge
d\overline z'_{\lambda\mu}\cr
&-\sum_{i,j,\lambda<\mu}\sqrt{a_\lambda-a_\mu}~c_{ij\lambda\mu}
\big(z'_{\lambda\mu}dx'_i\wedge d\overline x'_j+x'_idz'_{\lambda\mu}\wedge
d\overline x'_j\big)\cr
&-\sum_{i,j,\lambda<\mu}\sqrt{a_\lambda-a_\mu}~\overline c_{ij\lambda\mu}\big(
\overline z'_{\lambda\mu}dx'_j\wedge d\overline x'_i+\overline x'_idx'_j\wedge
d\overline z'_{\lambda\mu}\big)
+\rmo(|x'|^2+|z'|^2)~~.\cr}$$
Omitting the primes in the coordinates $x'$~, $z'$ for simplicity, we see that
the norms of the basis elements of $TY$ with respect to $\omega$ are given 
modulo $\rmo(|x|^2+|z|^2)$ by
$$\eqalign{
\langle{\partial\over\partial x_i},{\partial\over\partial x_j}\rangle &\sim
  \delta_{ij}-\sum_{\lambda<\mu}\sqrt{a_\lambda-a_\mu}\,\big(c_{ij\lambda\mu}
z_{\lambda\mu}+c_{ij\mu\lambda}\overline z_{\lambda\mu}\big)~~,\cr
\langle{\partial\over\partial z_{\lambda\mu}},{\partial\over\partial
z_{\lambda'\mu'}}\rangle&\sim\delta_{\lambda\lambda'}\delta_{\mu\mu'}~~,
\phantom{\sum_i}\cr
\langle{\partial\over\partial z_{\lambda\mu}},{\partial\over\partial x_j}
\rangle &\sim-\sum_i\sqrt{a_\lambda-a_\mu}c_{ij\lambda\mu}x_i~~.\cr}$$
By duality, we get
$$\eqalign{
\langle dx_i,dx_j\rangle &\sim
\delta_{ij}+\sum_{\lambda<\mu}\sqrt{a_\lambda-a_\mu}~\big(
\overline c_{ij\lambda\mu}\overline z_{\lambda\mu}+\overline c_{ij\mu\lambda}
z_{\lambda\mu}\big)~~,\cr
\langle dz_{\lambda\mu},dz_{\lambda'\mu'}\rangle&\sim
\delta_{\lambda\lambda'}\delta_{\mu\mu'}~~,\phantom{\sum_i}\cr
\langle dz_{\lambda\mu},dx_j\rangle &\sim\sum_i\sqrt{a_\lambda-a_\mu}~
\overline c_{ij\lambda\mu}\overline x_i~~.\cr}$$
Taking the exterior derivative in the above estimates, we find that the 
Chern connection $D$ on $\Omega^1_Y=T^\star Y$ is given in terms of the
basis vectors $dx_i$~, $dz_{\lambda\mu}$ by
$$\eqalign{
D(dx_j)&=\sum_{i,\lambda<\mu}\sqrt{a_\lambda-a_\mu}~c_{ij\lambda\mu}(
dz_{\lambda\mu}\otimes dx_i+dx_i\otimes dz_{\lambda\mu})+\rmo(|x|+|z|)~~,\cr
D(dz_{\lambda\mu})&=~~0~~+\rmo(|x|+|z|)~~.}$$
The subbundle $F=F^{p+1,N(s)}$ (resp. the quotient bundle $G=G^{p,p+N(s)}$) 
admits at $y^0$ the orthonormal basis
$$dx_I\wedge dz_J~~{\rm with}~~|I|+|J|=p+N(s)~,~|I|\ge p+1~~\big({\rm resp.}~
|I|\le p~,~|J|=N(s)\big)~~.$$
Let $v=\sum v_{I,J}\,dx_I\wedge dz_J$ be a $C^\infty$ section of $F$~. The
$(1,0)$--form $\beta\wedge v$ is nothing else than the projection of $Dv$ on
$G=\Omega/F$~. From this observation, one obtains the expression of $\beta$
at $y^0$~: 
 $$\beta\wedge v=\sum_{i,j,\lambda<\mu}\sqrt{a_\lambda-a_\mu}~
     c_{ij\lambda\mu}\,dx_i\otimes
     \big(dz_{\lambda\mu}\wedge\big({\partial\over\partial x_j}\ort v\big)\big)
     ~~{\rm mod}\,F~~,\leqno(4.11)$$
where $\xi\ort v$ means contraction of the differential form $v$ by the tangent
vector $\xi$~. In fact any differentiation of a factor $dx_j$ in a term
$D(v_{I,J}\,dx_I\wedge dz_J)$ decreases of one unity the partial degree $|I|$
when $dx_j$ is differentiated into $c_{ij\lambda\mu}\,dx_i\otimes 
dz_{\lambda\mu}$~. The corresponding part of the differential is thus in $G$ 
if $|I|=p+1$~. For every $(0,q+1)$--form~ $f=\sum f_{I,J,K,L}\,dx_I\wedge dz_J
\wedge d\overline x_K\wedge d\overline z_L$~ with values in $F(a)$~, 
$|I|+|J|=p+N(s)$~, $|I|\ge p+1$~, $|K|+|L|=q+1$~, we obtain consequently
   $$\beta\ort f=\sum_{i,j,\lambda<\mu}\sqrt{a_\lambda-a_\mu}~
             c_{ij\lambda\mu}\,{\partial\over\partial\overline x_i}
             \ort\big(dz_{\lambda\mu}\wedge\big(
             {\partial\over\partial x_j}\ort f\big)\big)~~
             {\rm mod}\,F(a)~~.\leqno(4.12)$$
The only terms of $f$ that contribute to the expression of $\beta\ort f$ 
are those for which $|I|=p+1$ and $|J|=N(s)-1$~. Let us write $g=\beta\ort f$ 
under the form
$$g=\sum g_{I',J',K',L'}\,dx_{I'}\wedge dz_{J'}\wedge d\overline x_{K'}\wedge
d\overline z_{L'}~~,$$
where $|I'|=p~,~|J'|=N(s)~,~|K'|+|L'|=q$~. Formula (4.12) implies
$$\eqalign{
g_{I',J',K',L'}&=\sum_{i,j,\lambda<\mu}\pm\sqrt{a_\lambda-a_\mu}~
c_{ij\lambda\mu}\,f_{jI',J'\setminus\{\lambda\mu\},iK',L'}~~,\cr
|g_{I',J',K',L'}|^2&\le\Big(\sum_{i,j,\lambda<\mu}(a_\lambda-a_\mu)
|c_{ij\lambda\mu}|^2\Big)\sum_{i,j,\lambda<\mu}|f_{jI',J'\setminus
\{\lambda\mu\},iK',L'}|^2~~,\cr}$$
and $\displaystyle\sum_{I',J',K',L'}~~\sum_{i,j,\lambda<\mu}|f_{jI',J'\setminus
\{\lambda\mu\},iK',L'}|^2\le(p+1)(q+1)\sum_{I,J,K,L}|f_{I,J,K,L}|^2$~.
We obtain therefore the inequality
$$|\beta\ort f|^2\le (p+1)(q+1)\Big(\sum_{i,j,\lambda<\mu}(a_\lambda-a_\mu)
|c_{ij\lambda\mu}|^2\Big)\,|f|^2~~.\leqno(4.13)$$
The main point now is to find an estimate of the sum $\sum_{i,j,\lambda<\mu}
(a_\lambda-a_\mu)|c_{ij\lambda\mu}|^2$ under condition (4.10).

\tha Lemma 4.14|Let $(h_{\lambda\mu})_{1\le\lambda,\mu\le r}$ be a
semi-positive hermitian matrix and let $\alpha_1\le\ldots\le\alpha_r$ be real 
numbers. Then
$$\sum_{\lambda<\mu}(\alpha_\mu-\alpha_\lambda)|h_{\lambda\mu}|^2\le
 {1\over 4}(\alpha_r-\alpha_1)\Big(\sum_\lambda h_{\lambda\lambda}\Big)^2~~.$$
\fintha

{\sl Proof}\pointir Use Cauchy-Schwarz inequality $|h_{\lambda\mu}|^2\le 
h_{\lambda\lambda}h_{\mu\mu}$ and take\break $t_\lambda=h_{\lambda\lambda}$ in
the identity
$$\eqalign{\displaystyle
{1\over 4}(\alpha_r&-\alpha_1)\Big(\sum_\lambda t_\lambda\Big)^2-
\sum_{\lambda<\mu}(\alpha_\mu-\alpha_\lambda)\,t_\lambda t_\mu\cr
&={1\over 4}\sum_{1\le\lambda<r}(\alpha_{\lambda+1}-\alpha_\lambda)
(t_1+\ldots+t_\lambda-t_{\lambda+1}-\ldots-t_r)^2\ge 0~~.~\finpr\cr}$$

\tha Lemma 4.15|Under condition (4.10) one has
$$\sum_{i,j,\lambda<\mu}(a_\lambda-a_\mu)|c_{ij\lambda\mu}|^2\le
{1\over 4}n(n+1)\Big({1\over a_r}-{1\over a_1}\Big)~~.$$
\fintha

{\sl Proof}\pointir Let us apply lemma 4.14 to 
$$h_{\lambda\mu}=\sqrt{a_\lambda a_\mu}\,\sum_{i,j}c_{ij\lambda\mu}
  t_i\overline t_j~~,~~\alpha_\lambda={1\over a_\lambda}~~,$$ 
where $t=(t_1\ld t_n)$ are arbitrary complex numbers. The Griffiths
semi-positivity assumption on $c(E)$ means that $(h_{\lambda\mu})$ is
semi-positive for all $t$~. We~get  
$$\sum_\lambda h_{\lambda\lambda}=\sum_{i,j,\lambda} a_\lambda 
c_{ij\lambda\lambda}t_i\overline t_j\le |t|^2$$
by condition (4.10), thus
$$\sum_{\lambda<\mu}(a_\lambda-a_\mu)\Big|\sum_{i,j}c_{ij\lambda\mu}t_i
\overline t_j\Big|^2\le {1\over4}\Big({1\over a_r}-{1\over a_1}\Big)|t|^4~~.
\leqno(4.16)$$
Apply now inequality (4.16) to $t=(e^{i\theta_1}\ld e^{i\theta_n})
\in{\Bbb T}^n$ and integrate the result over ${\Bbb T}^n$~. Parseval's
identity for Fourier series yields
$$\sum_{\lambda<\mu}(a_\lambda-a_\mu)\Big(\sum_{i\ne j}|c_{ij\lambda\mu}|^2+
  \big|\sum_i c_{ii\lambda\mu}\big|^2\Big)\le{1\over 4}\Big({1\over a_r}-
   {1\over a_1}\Big)n^2~~.$$ 
Inequality (4.16) applied to each vector of the standard basis of ${\Bbb C}^n$
yields in the same way
 $$\sum_{\lambda<\mu}(a_\lambda-a_\mu)|c_{ii\lambda\mu}|^2\le{1\over 4}
    \Big({1\over a_r}-{1\over a_1}\Big)$$
for all $i$~, and lemma 4.15 follows. \finpr

Combining inequality (4.13) with lemma 4.15 we get
$$|\beta\ort f|^2<{1\over 4a_r}\,n(n+1)(p+1)(q+1)|f|^2~~,$$
and using criterion (4.5) we see that $H^{q+1}(Y,F\otimes Q^a_s\otimes
\pi^\star L)=0$ for
$$a_r\ge {n(n+1)(p+1)(q+1)\over 4(p+q-n+1)}~~,~~p+q\ge n~~.$$
The proof of the main theorem is therefore achieved.

\vskip12pt
\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 
complex
$$D''~:~{\cal K}^q=\Gamma\big(Y,{\cal C}^\infty_Y(\Omega^t_Y\otimes\overline
\Omega^q_Y)\otimes_{{\cal O}_Y}\calS\big)\lw{\cal K}^{q+1}$$
of $C^\infty$--differential forms of type $(t,q)$ with values in $\calS$~.
This Dolbeault complex is filtered by the decreasing sequence of subcomplexes
$$D''~:~{\cal K}^q_p=\Gamma\big(Y,{\cal C}^\infty_Y(F^p(\Omega^t_Y)\otimes
\overline\Omega^q_Y)\otimes_{{\cal O}_Y}\calS\big)\lw{\cal K}^{q+1}_p~~.$$
This gives rise to a spectral sequence which we shall name after Borel and
Le Potier, whose $E_0$~, $E_1$ terms are
$$\left\{\eqalign{
E_0^{p,q-p}&=\Gamma\big(Y,{\cal C}^\infty_Y(G^{p,t}\otimes\overline\Omega^q_Y)
\otimes_{{\cal O}_Y}\calS\big)\cr
E_1^{p,q-p}&=H^q(Y,G^{p,t}\otimes\calS)~~.\cr}\right.\leqno(5.1)$$
The limit term $E_\infty^{p,q-p}$ is the $p$--graded module corresponding to
the filtration of $H^q({\cal K}^\blt)=H^q(Y,\Omega^t_Y\otimes\calS)$ by the
canonical images of the groups $H^q({\cal K}^\blt_p)$~. Assume 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

This result would be a considerable help for the proof of vanishing theorems.
For example, it is shown in [5] that the main vanishing theorem would be true 
with $l\ge r-1+\min\{n-p,n-q\}$~.

Since $d_1$ is the coboundary operator associated to the exact sequence\break
$0\lw G^{p+1,t}\lw F^{p,t}/F^{p+2,t}\lw G^{p,t}\lw 0$~, it is easy to see that
$d_1=-\beta^\star\wedge\blt$ where $\beta^\star$ is the $D''$--closed 
(0,1)--form of \S 4, reinterpreted as a (0,1)--section of $\hom(G^{p,t},
G^{p+1,t})$~. Our hope is that the $E_2$--degeneracy can be proved in all cases
by a suitable deepening of the analytic method of \S 4. 
\vfill\eject

\titre References|
\vskip15pt
\eightpoint
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\divers 2|a. Borel {\rm and} a. Weil|{\sl Repr\'esentations lin\'eaires et
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\divers 5|j.-p. Demailly|{\sl Vanishing theorems for tensor powers of an
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images of sheaves of differentials and the Atiyah class}|{\rm preprint
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positive holomorphe Vektorraumb\"undel|Manuscripta Math.|11|1974|95--101|
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\end
