VANISHING THEOREMS FOR TENSOR POWERS OF AN AMPLE

18 Pages·2007·0.21 MB·English

VANISHING THEOREMS FOR TENSOR POWERS
OF AN AMPLE VECTOR BUNDLE
Jean-Pierre DEMAILLY
Universit´e de Grenoble I,
Institut Fourier, BP 74,
Laboratoire associ´e au C.N.R.S. n˚188,
F-38402 Saint-Martin d’H`eres
Abstract. — LetE beaholomorphicvectorbundleofrankr onacompact
complex manifold X of dimension n . It is shown that the cohomology groups
Hp,q(X,E⊗k⊗(detE)l) vanish if E is ample and p+q ≥ n+1 , l ≥ n−p+r−1 .
The proof rests on the well-known fact that every tensor power E⊗k splits
into irreducible representations of Gl(E) . By Bott’s theory, each component is
canonicallyisomorphic to thedirect imageon X ofa homogeneous linebundle 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
isomorphismtheorem anda counterexample to avanishing conjecture of Sommese.
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 Hp,q
of tensor powers of E . For example, the famous Le Potier vanishing theorem [13] :
E ample =⇒ Hp,q(X,E) = 0 for p+q ≥ n+r
does not extend to symmetric powers SkE , 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 detE . In all the sequel, we let L be
a holomorphic line bundle on X and we assume :
Hypothesis 0.1. — E is ample and L semi-ample, or E is semi-ample
and L ample.
The precise definitions concerning ampleness are given in §1 . Under this
hypothesis, we prove the following two results.
1
Theorem 0.2. — Let us denote by ΓaE the irreducible tensor power
representation of Gl(E) of highest weight a ∈ Zr . If h ∈ {1, ... ,r − 1} and
a ≥ a ≥ ... ≥ a > a = ... = a = 0 , then
1 2 h h+1 r
Hn,q(X,ΓaE ⊗(detE)h ⊗L) = 0 for q ≥ 1 .
Since SkE = Γ(k,0,...,0)E , the special case h = 1 is Griffiths’ vanishing
theorem [8] :
Hn,q(X,SkE ⊗detE ⊗L) = 0 for q ≥ 1 .
Theorem 0.3. — For all integers p+q ≥ n+1 , k ≥ 1 , l ≥ n−p+r−1
then Hp,q(X,E⊗k ⊗(detE)l ⊗L) = 0 .
For p = n and arbitrary r, k ≥ 2 , Peternell-Le Potier and Schneider [11]
0
have constructed an example of an ample vector bundle E of rank r on a manifold
X of dimension n = 2r such that
(0.4) Hn,n−2(X,SkE) 6= 0 , 2 ≤ k ≤ k .
0
This result shows that detE cannot be omitted in theorem 0.2 when h = 1 . More
generally, the following example shows that the exponent h is optimal.
Example. — Let X = G (V) be the Grassmannian of subspaces of
r
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 = detE very ample) . Let
h ∈ {1, ... ,r−1} and a ∈ Zr , β ∈ Zd be such that
a ≥ ... ≥ a ≥ d−r , a = ... = a = 0 ,
1 h h+1 r
β = (a −d+r, ... ,a −d+r,0, ... ,0) .
1 h
Set n = dimX = r(d−r) , q = (r−h)(d−r) . Then
(0.5) Hn,q X,ΓaE ⊗(detE)h = ΓβV ⊗(detV)h 6= 0 .
(cid:0) (cid:1)
Our approach is based on three well-known facts. First, every tensor
power E⊗k splits into irreducible representations ΓaE of the linear group Gl(E)
(cf. (2.16)). Secondly, every irreducible tensor bundle ΓaE appears in a natural
way as the direct image on X of an 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 O (1) on P(E⋆) . In §3, we generalize
E
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.
2
Theorem 0.6. — For every vector bundle E and every line bundle L one
can define a canonical morphism
Hp,q(X,Λ2E ⊗L) −→ Hp+1,q+1(X,S2E ⊗L) .
Under hypothesis 0.1, this morphism is one-to-one for p + q ≥ n + r − 1 and
surjective for p+q = n+r −2 .
Combining theorem 0.6 and example (0.4), we get Hn−1,n−3(X,Λ2E) 6= 0 .
This shows that Sommese’s conjecture ([15], conjecture (4.2))
E ample =⇒ Hp,q(X,ΛkE) = 0 for p+q ≥ n+r −k +1
is false for n = 2r ≥ 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.
Problem. — Given any dominant weight a ∈ Zr with a = 0 and p,q
r
such that p+q ≥ n+1 , determine the smallest exponent l = l (n,p,q,r,a) such
0 0
that Hp,q(X,ΓaE ⊗(detE)l ⊗L) = 0 for l ≥ l .
0
We show in §5 that if the Borel-Le Potier spectral sequence degenerates at
the E level, it is always sufficient to take l ≥ r −1+min{n−p,n−q} . In that
2
case, theorem 0.6 appears also as a special case of a fairly general exact sequence.
The E –degeneracy of the Borel-Le Potier spectral sequence is thus an important
2
feature which would be interesting to investigate.
Some of the above results have been annouced in the note [4] and corre-
sponding 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 sub-
stantial improvements of this work.
3
1. Basic definitions and tools.
Werecallhere a few basic factsand definitionswhich willbeused repeatedly
in the sequel.
(1.1) Ample vector bundles (compare with Hartshorne [9]).
A vector bundle E on X is said to be spanned if the canonical map
H0(X,E) −→ E is onto for every x ∈ X , and semi-ample if the symmetric
x
powers SkE are spanned for k ≥ k large enough.
0
E is said to be very ample if the canonical maps
H0(X,E) −→ E ⊕E , ∀x 6= y ∈ X ,
x y
(H0(X,E) −→ O(E)⊗Ox/M2x , ∀x ∈ X ,
are onto, M denoting the maximal ideal at x ∈ X . If E is very ample, then the
x
holomorphic map from X to the Grassmannian of r–codimensional subspaces of
V = H0(X,E) given by
Φ : X −→ G (V) , x 7−→ W = {σ ∈ V;σ(x) = 0}
r x
is an embedding, and E is the pull-back by Φ of the canonical quotient vector
bundle of rank r on G (V) . The embedding condition is in fact equivalent to E
r
being very ample if r = 1 , but weaker if r ≥ 2 .
At last, E is said to be ample if the symmetric powers SkE are very
ample for k ≥ k large enough. Denoting O (1) the canonical line bundle on
0 E
Y = P(E⋆) associated to E and π : Y −→ X the projection, it is well-known
that π O (k) = SkE . One gets then easily
⋆ E
E spanned on X ⇐⇒ O (1) spanned on Y ,
E
E (semi-)ample on X ⇐⇒ O (1) (semi-)ample on Y .
E
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 Sk(E ⊗L) = SkE ⊗Lk , it is clear that
E,L semi-ample =⇒ E ⊗L semi-ample ,
E,L spanned, one of them very ample =⇒ E ⊗L very ample ,
E,L semi-ample, one of them ample =⇒ E ⊗L ample .
(1.2) Kodaira-Akizuki-Nakano vanishing theorem [1].
If L is an ample (or positive) line bundle on X , then
Hp,q(X,L) = Hq(X,Ωp ⊗L) = 0 for p+q ≥ n+1 .
X
(1.3) Leray spectral sequence (cf. Godement [7]).
Let π : Y −→ X be a continuous map between topological spaces, ⌣⌣S a
sheaf of abelian groups on Y , and Rqπ ⌣⌣S the direct image sheaves of⌣⌣S on X .
⋆
Then there exists a spectral sequence such that
Ep,q = Hp(X,Rqπ ⌣⌣S)
2 ⋆
and such that the limit term Ep,q−p is the p–graded module associated to a
∞
decreasing filtration of Hq(Y,⌣⌣S) .
4
(1.4) Cohomology of a filtered sheaf.
Wewillneedalsothefollowingelementaryresult.Let F beasheafofabelian
groups on X and
F = F0 ⊃ F1 ⊃ ... ⊃ Fp ⊃ ... ⊃ FN = 0
be a filtration of F such that the graded sheaf Gp , Gp = Fp/Fp+1 , satisfies
Hq(X,Gp) = 0 for q ≥ q . Then Hq(X,F) = 0 for q ≥ q .
0 0
L
Indeed, it is immediately verified by induction on p ≥ 1 that Hq(X,F/Fp)
vanishes for q ≥ q , using the cohomology long exact sequence associated to
0
0 −→ Gp −→ F/Fp+1 −→ F/Fp −→ 0 .
2. Homogeneous line bundles on flag manifolds and
irreducible representations of the linear group.
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 be the Borel subgroup of Gl = Gl(Cr) of lower triangular matrices,
r r
U ⊂ B the subgroup of unipotent matrices, and Tr the complex torus (C⋆)r of
r r
diagonal matrices. Let V be a complex vector space of dimension r . We denote
by M(V) the manifold of complete flags
V = V0 ⊃ V1 ⊃ ... ⊃ Vr = {0} , codimCVλ = λ .
To every linear isomorphism ζ ∈ Isom(Cr,V) : (u , ... ,u ) 7−→ u ζ ,
1 r 1≤λ≤r λ λ
one can associate the flag [ζ] ∈ M(V) defined by V = Vect(ζ , ... ,ζ ) ,
λ λ+1 r
P
1 ≤ λ ≤ r . This leads to the identification
M(V) = Isom(Cr,V)/B
r
where B acts on the right side. We denote simply by V the tautological vector
r λ
bundle of rank r−λ on M(V) , and we consider the canonical quotient line bundles
Q = V /V , 1 ≤ λ ≤ r ,
λ λ−1 λ
(2.1)
( Qa = Qa11 ⊗...⊗Qarr , a = (a1, ... ,ar) ∈ Zr .
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 , Q , Qa . If ρ : B → Gl(E)
λ λ r
is a representation of B , we may associate to ρ the manifold
r
E = Isom(Cr,V)× E = Isom(Cr,V)×E/ ∼
V,ρ Br
where ∼ denotes the equivalence relation (ζg,u) ∼ (ζ,ρ(g)u) , g ∈ B . Then E
r V,ρ
is a Gl(V)–equivariant bundle over M(V) , and all such bundles are obtained in
this way. It is clear that Qa is isomorphic to the line bundle C arising from the
V,a
1–dimensional representation of weight a :
a : Tr = B /U −→ C⋆ , (t ...t ) 7−→ ta1 ... tar ;
r r 1 r 1 r
the isomorphism C −˜→ Qa is induced by the map
V,a
Isom(Cr,V)×C ∋ (ζ,u) 7−→ uζ˜a1 ⊗...⊗ζ˜ar ∈ Qa , ζ˜ = ζ mod V .
1 r λ λ λ
5
We compute now the tangent and cotangent vector bundles of M(V) . The action
of Gl(V) on M(V) yields
(2.2) TM(V) = Hom(V,V)/W
whereW isthesubbundleofendomorphismsg ∈ Hom(V,V)suchthatg(V ) ⊂ V ,
λ λ
1 ≤ λ ≤ r . Using the self-duality of Hom(V,V) given by the Killing form
(g ,g ) 7→ tr(g g ) , we find
1 2 1 2
T⋆M(V) = Hom(V,V)/W ⋆ = W⊥
(2.3)
(W⊥ = {g ∈(cid:0)Hom(V,V) ; g((cid:1)Vλ−1) ⊂ Vλ , 1 ≤ λ ≤ r} .
If ad denotes the adjoint representation of B on the Lie algebras gˇl: , R , U ,
r r ∪∪r r
then
(2.4) TM(V) = (gˇl: /R ) , T⋆M(V) = (U )
r ∪∪r V,ad r V,ad
because W = ζR ζ−1 , W⊥ = ζU ζ−1 at every point [ζ] ∈ M(V) . There exists
[ζ] ∪∪r [ζ] r
a filtration of T⋆M(V) by subbundles of the type
{g ∈ Hom(V,V) ; g(V ) ⊂ V , 1 ≤ λ ≤ r}
λ µ(λ)
in such a way that the corresponding graded bundle is the direct sum of the line
bundles Hom(Q ,Q ) = Q−1⊗Q , λ < µ; theirtensorproduct isthus isomorphic
λ µ λ µ
to the canonical line bundle K = det(T⋆M(V)) :
M(V)
(2.5) K = Q1−r ⊗...⊗Q2λ−r−1 ⊗...⊗Qr−1 = Qc
M(V) 1 λ r
where c = (1−r, ... ,r−1); c will be called the canonical weight of M(V) .
• Case of incomplete flag manifolds.
More generally, given any sequence of integers s = (s , ... ,s ) such that
0 m
0 = s < s < ... < s = r , we may consider the manifold M (V) of incomplete
0 1 m s
flags
V = Vs0 ⊃ Vs1 ⊃ ... ⊃ Vsm = {0} , codimCVsj = sj .
On M (V) we still have tautological vector bundles V of rank r − s and line
s s,j j
bundles
(2.6) Q = det(V /V ) , 1 ≤ j ≤ m .
s,j s,j−1 s,j
For any r–tuple a ∈ Zr such that a = ... = a , 1 ≤ j ≤ m , we set
sj−1+1 sj
Qa = Qas1 ⊗...⊗Qasm .
s s,1 s,m
If η : M(V) → M (V) is the natural projection, then
s
(2.7) η⋆V = V , η⋆Q = Q ⊗...⊗Q , η⋆Qa = Qa .
s,j sj s,j sj−1+1 sj s
On the other hand, one has the identification
M (V) = Isom(Cr,V)/B
s s
where B is the parabolic subgroup of matrices (z ) with z = 0 for all λ,µ such
s λµ λµ
that there exists an integer j = 1, ... ,m−1 with λ ≤ s and µ > s . We define
j j
U as the unipotent subgroup of lower triangular matrices (z ) with z = 0 for
s λµ λµ
6
all λ,µ such that s < λ 6= µ ≤ s for some j (hence U = R⊥ ). In the same
j−1 j s ∪∪s
way as above, we get
(2.8) TM (V) = Hom(V,V)/W , W = {g ; g(V ) ⊂ V } ,
s s s sj−1 sj
(2.9) T⋆M (V) = W⊥ = (U ) ,
s s s V,ad
(2.10) K = Qs1−r ⊗...⊗Qsj−1+sj−r ⊗...⊗Qsm−1 = Qc(s)
Ms(V) s,1 s,j s,m s
where c(s) = (s −r, ... ,s −r, ... ,s +s −r, ... ,s +s −r, ...) is the
1 1 j−1 j j−1 j
canonical weight of M (V) .
s
Lemma 2.11. — Qa enjoys the following properties :
s
(a) If a < a for some j = 1, ... ,m−1 , then H0(M (V),Qa) = 0 .
sj sj+1 s s
(b) Qa is spanned if and only if a ≥ a ≥ ... ≥ a .
s s1 s2 sm
(c) Qa is (very) ample if and only if a > a > ... > a .
s s1 s2 sm
Proof. —
(a) Let (V0 ⊃ V0 ⊃ ... ⊃ V0 ) ∈ M (V) be an arbitrary flag and F,F′ subspaces
s0 s1 sm s
of V such that
V0 ⊃ F ⊃ V0 ⊃ F′ ⊃ V0 , dimF = s +1 , dimF′ = s −1 .
sj−1 sj sj+1 j j
Let us consider the projective line P(F/F′) ⊂ M (V) of flags
s
V0 ⊃ ... ⊃ V0 ⊃ V ⊃ V0 ⊃ V0
s0 sj−1 sj sj+1 sm
such that F ⊃ V ⊃ F′ . Then
sj
Q = det(V0 /V ) ≃ F/V ≃ O(1) ,
s,j↾P(F/F′) sj−1 sj sj
Q = det(V /V0 ) ≃ V /F′ ≃ O(−1)
s,j+1↾P(F/F′) sj sj+1 sj
thus Qa ≃ O(a − a ) , which implies (a) and the “only if” part of
s↾P(F/F′) sj sj+1
assertions (b), (c).
(b) Since Q ⊗...⊗Q = det(V/V ) , we see that this line bundle is a quotient
s,1 s,j sj
of the trivial bundle ΛsjV . Hence
(2.12) Qas = det(V/Vsj)asj−asj+1 ⊗(detV)ar
1≤j≤m−1
O
is spanned by sections arising from elements of Sasj−asj+1(ΛsjV) ⊗ (detV)ar
as soon as a ≥ ... ≥ a .
s1 sm N
(c)Ifa > ... > a ,itiselementarytoverifythatthesectionsofQa arisingfrom
s1 sm s
(2.12) suffice to make Qa very ample (this is in fact a generalization of Plu¨cker’s
s
imbedding of Grassmannians).
• Cohomology groups of Qa .
It remains now to compute H0(M (V),Qa) ≃ H0(M(V),Qa) when
s s
a ≥ ... ≥ a . Without loss of generality we may assume that a ≥ 0 , because
1 r r
Q ⊗...⊗Q = detV is a trivial bundle.
1 r
7
Proposition 2.13. — For all integers a ≥ a ≥ ... ≥ a ≥ 0 , there is a
1 2 r
canonical isomorphism
H0(M(V),Qa) = ΓaV
where ΓaV ⊂ Sa1V ⊗...⊗SarV is the set of polynomials f(ζ⋆, ... ,ζ⋆) on (V⋆)r
1 r
which are homogeneous of degree a with respect to ζ⋆ and invariant under the
λ λ
left action of U on (V⋆)r = Hom(V,Cr) :
r
f(ζ⋆, ... ,ζ⋆ ,ζ⋆ +ζ⋆, ... ,ζ⋆) = f(ζ⋆, ... ,ζ⋆) , ∀ν < λ .
1 λ−1 λ ν r 1 r
Proof. — To any section σ ∈ H0(M(V),Qa) we associate the holomorphic
function f on Isom(V,Cr) ⊂ (V⋆)r defined by
f(ζ⋆, ... ,ζ⋆) = (ζ⋆)a1 ⊗...⊗(ζ⋆)ar.σ([ζ , ... ,ζ ])
1 r 1 r 1 r
where (ζ , ... ,ζ ) is the dual basis of (ζ⋆, ... ,ζ⋆) , and where the linear form
1 r 1 r
induced by ζ⋆ on Q = V /V ≃ Cζ is still denoted ζ⋆ . Let us observe that f
λ λ λ−1 λ λ λ
is homogeneous of degree a in ζ⋆ and locally bounded in a neighborhood of every
λ λ
r–tuple of (V⋆)r\Isom(V,Cr) (because M(V) is compact and a ≥ 0) . Therefore
λ
f can be extended to a polynomial on all (V⋆)r . The invariance of f under U
r
is clear. Conversely, such a polynomial f obviously defines a unique section σ on
M(V) .
From the definition of ΓaV , we see that
(2.14) SkV = Γ(k,0,...,0)V ,
(2.15) ΛkV = Γ(1,...,1,0,...,0)V .
For arbitrary a ∈ Zr , proposition 2.13 remains true if we set
ΓaV = Γ(a1−ar,...,ar−1−ar,0)V ⊗(detV)ar when a is non-increasing ,
ΓaV = 0 otherwise .
The weights will be ordered according to their usual partial ordering :
a < b iff a ≥ b , 1 ≤ µ ≤ r .
λ λ
1≤λ≤µ 1≤λ≤µ
X X
Bott’s theorem [3] shows that ΓaV 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
Tr ⊂ Gl(V) on V⊗k verify a + ...+ a = k and a ≥ 0 , we have a canonical
1 r λ
Gl(V)–isomorphism
(2.16) V⊗k = µ(a,k)ΓaV
a1+.M..+ar=k
a1≥...≥ar≥0
where µ(a,k) > 0 is the multiplicity of the isotypical factor ΓaV in V⊗k .
Bott’s formula (cf. also Demazure [6] for a very simple proof) gives in fact
the expression of all cohomology groups Hq(M(V),Qa) . A weight a is called
regular if all the elements of a are distinct, and singular otherwise. We will denote
by a≥ the reordered non-increasing sequence associated to a , by l(a) the number
of strict inversions of the order, and we will set
c ≥ c
a = a− + ∈ Zr ;
2 2
(cid:16) (cid:17)
b 8
the sequence a is non-increasing if and only if a− c is regular.
2
Proposition 2.17 (Bott [3]). — One has
b
0 if q 6= l(a− c) ,
Hq(M(V),Qa) = 2
ΓaˆV if q = l(a− c) .
(cid:26) 2
In particular all Hq vanish if and only if a− c is singular.
2
We will be particularly interested by those line bundles Qa such that the
cohomology groups Hp,q vanish for all q ≥ 1 , p being given. The following
proposition is one of the main steps in the proof of our results.
Proposition 2.18. — Set N = dimM(V) , N(s) = dimM (V) . Then
s
(a) Hp,q(M (V),Qa) = 0 for all p+q ≥ N(s)+1
s s
as soon as a −a ≥ 1 , 1 ≤ j ≤ m−1 .
sj sj+1
(b) Hq(M (V),Qa) = 0 for all q ≥ 1
s s
as soon as a −a ≥ 1−(s −s ) , 1 ≤ j ≤ m−1 .
sj sj+1 j+1 j−1
(c) In general, Hp,q(M (V),Qa−c(s)) is isomorphic to a direct sum of irredu-
s s
cible Gl(V)–modules ΓbV with
b ≥ ... ≥ b ≥ min{a }−(N(s)−p) .
1 r λ
(d) Hp,q(M (V),Qa) = 0 for all q ≥ 1 as soon as
s s
a −a ≥ min{p, N(s)−p+(s −s )−1, r+1−(s −s )} .
sj sj+1 j+1 j−1 j+1 j−1
(e) Under the assumption of (d), there is a Gl(V)–isomorphism
Hp,0(M (V),Qa) = ν (u,p)Γa+uV ,
s s s
u∈Zr
M
where ν (u,p) is the multiplicity of the weight u in ΛpU .
s s
Proof. — Under the assumption of (a), Qa is ample by lemma 2.11 (c) .
s
The result follows therefore from the Kodaira-Nakano-Akizuki theorem. Now (b)
is a consequence of (a) since c(s) −c(s) = −(s −s ) and
sj sj+1 j+1 j−1
Hq(M (V),Qa) = HN(s),q(M (V),Qa−c(s)) .
s s s s
Let us now observe that for every vector bundle E on M (V) there are isomor-
s
phisms
(2.19) Hq(M (V),E) ≃ Hq(M(V),η⋆E) ,
s
(2.20) Hq(M (V),E) ≃ Hq+N−N(s)(M(V),Qc−c(s) ⊗η⋆E) ,
s
where Qc−c(s) is the relative canonical bundle along the fibers of the projection
η : M(V) −→ M (V) . In fact, the fibers of η are products of flag manifolds.
s
For such a fiber F , we have Hq(F,K ) = HdimF−q(F,O) ⋆ = C when
F
q = dimF = N − N(s) and Hq(F,K ) = 0 otherwise (apply (b) with a = 0
F
(cid:0) (cid:1)
and Ku¨nneth’s formula). We get thus direct image sheaves
0 if q 6= 0
Rqη (η⋆E) =
⋆ O(E) if q = 0 .
(cid:26)
(cid:1)
0 if q 6= N −N(s)
Rqη Qc−c(s) ⊗η⋆E =
⋆ O(E) if q = N −N(s) .
(cid:26)
(cid:0) (cid:1)
9
Formulas (2.19) and (2.20) are thus immediate consequences of the Leray spectral
sequence. When applied to E = ΛpT⋆M (V)⊗Qa−c(s) , formula (2.19) yields
s s
Hp,q(M (V),Qa−c(s)) = Hq M(V),Qa−c(s)⊗η⋆ΛpT⋆M (V)
s s s
= Hq(cid:0)M(V),Qa ⊗η⋆ΛN(s)−pTMs(V)(cid:1) ,
using the isomorphism ΛpT⋆M (V) (cid:0)≃ K ⊗ ΛN(s)−pTM (V) . I(cid:1)n the same
s Ms(V) s
way, (2.20) implies
Hp,q(M (V),Qa) = Hq+N−N(s) M(V),Qa+c−c(s)⊗η⋆ΛpT⋆M (V)
s s s
= Hq+N−N(s)(cid:0)M(V),Qa+c ⊗η⋆ΛN(s)−pTMs(V)(cid:1) .
The bundle η⋆T⋆Ms(V) = (Us)V,ad(cid:0)(resp. η⋆TMs(V) = (gˇl:r/R∪∪s)V,ad(cid:1)) has a
filtration with associated graded bundle
Q−1 ⊗Q resp. Q ⊗Q−1
λ µ λ µ
where the indices λ,Mµ are such that(cid:0) λ ≤ Ms < µ for som(cid:1)e j . It follows that
j
η⋆ΛpT⋆M (V) resp. η⋆ΛN(s)−pTM (V) has a filtration with associated graded
s s
bundle
(cid:0) ν (u,p)Qu resp. (cid:1) ν′(u′,N(s)−p)Qu′
s s
u∈Zr u′∈Zr
M (cid:0) M (cid:1)
where the weights u (resp. u′) and multiplicities ν (u,p) (resp. ν′(u′,N(s)− p))
s s
are those of ΛpU (resp. ΛN(s)−pgˇl: /R ) . We need a lemma.
s r ∪∪s
Lemma 2.21. — The weights u of ΛpU verify
s
u −u ≤ min{p+1, r +1−(µ−λ), N(s)−p+(s −s )}
µ λ j+1 j−1
for s < λ ≤ s < µ ≤ s , 1 ≤ j ≤ m−1 .
j−1 j j+1
Proof. — Let us denote by (ε ) the canonical basis of Zr . The
λ 1≤λ≤r
weights u (resp. u′) are the sums of p (resp. N(s)−p) distinct elements of the set
of weights −ε +ε (resp. ε −ε ) where λ ≤ s < µ for some j . It follows that
λ µ λ µ j
u −u ≤ r +1−(µ−λ) , with equality for the weight
µ λ
u = (−ε +ε )+...+(−ε +ε )+(−ε +ε )+...+(−ε +ε ) .
1 µ λ µ λ µ+1 λ r
It is clear also that u −u ≤ p+1 and u′ −u′ ≤ N(s)−p . Since the weights
µ λ µ λ
u , u′ are related by u = u′ +c(s) , lemma 2.21 follows.
Proof of (c). — By the filtration property (1.4) and the above formulas, we
seethatHp,q(M (V),Qa−c(s))isadirectsumofcertainirreducibleGl(V)–modules
s s
c ≥ c
Hq M(V),Qa+u′ = ΓbV , b = (a+u′)∧ = a− +u′ + .
2 2
(cid:16) (cid:17)
Clearly u′(cid:0)≥ −(N(s)−p(cid:1)) , thus
λ
c c
λ r
b ≥ min{a }−max −(N(s)−p)+ = min{a }−(N(s)−p) .
r λ λ
2 2
n o
Proof of (d). — Similarly, (d) is reduced by (1.4) to proving
Hq+N−N(s)(M(V),Qa+u+c−c(s)) = 0
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