NRS n 188 F38402 SaintMartin dHeres Abstract Let be a holomorphic vector bundle of rank on a compact complex manifold of dimension It is shown that the cohomology groups pq X E det vanish if is ample and 1 1 The proof rests on the wellknown fac ID: 35513 Download Pdf

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NRS n 188 F38402 SaintMartin dHeres Abstract Let be a holomorphic vector bundle of rank on a compact complex manifold of dimension It is shown that the cohomology groups pq X E det vanish if is ample and 1 1 The proof rests on the wellknown fac

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VANISHING THEOREMS FOR TENSOR POWERS OF AN AMPLE VECTOR BUNDLE Jean-Pierre DEMAILLY Universite de Grenoble I, Institut Fourier, BP 74, Laboratoire associe au C.N.R.S. n 188, F-38402 Saint-Martin d’H`eres Abstract . — Let be a holomorphic vector bundle of rank on a compact complex manifold of dimension . It is shown that the cohomology groups p,q X, E (det ) vanish if is ample and +1 , 1 . The proof rests on the well-known fact that every tensor powe splits into irreducible representations of Gl( ) . By Bott’s theory, each component is canonically isomorphic to

the direct image on of a homogeneous line bundle over a ﬂag manifold of . The proof is then reduced to the Kodaira-Akizuki-Nakano vanishing theorem for line bundles by means of the Leray spec tral sequence, using backward induction on . We also obtain a generalization of Le Potier’s isomorphism theorem and a counterexample to a vanishing con jecture of Sommese. 0. Statement of results. Many problems and results of contemporary algebraic geomet ry involve vanishing theorems for holomorphic vector bundles. Furthe rmore, tensor powers of such bundles are often introduced by natural

geometric co nstructions. The aim of this work is to prove a rather general vanishing theorem fo r cohomology groups of tensor powers of a holomorphic vector bundle. Let be a complex compact –dimensional manifold and a holomorphic vector bundle of rank on . If is ample and r > 1 , only very few general and optimal vanishing results are available for the Dolbeault c ohomology groups p,q of tensor powers of . For example, the famous Le Potier vanishing theorem [13] : ample = p,q X, E ) = 0 for does not extend to symmetric powers , even when and cf. [11]) . Nevertheless, we will see that the

vanishing propert y is true for tensor powers involving a suﬃciently large power of det . In all the sequel, we let be a holomorphic line bundle on and we assume : Hypothesis 0.1. is ample and semi-ample, or is semi-ample and ample. The precise deﬁnitions concerning ampleness are given in 1 . Under this hypothesis, we prove the following two results.

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Theorem 0.2. Let us denote by the irreducible tensor power representation of Gl( of highest weight . If ∈{ , . . . , r and . . . > a +1 . . . = 0 , then n,q X, (det ) = 0 for Since = k, ,..., 0) , the special

case = 1 is Griﬃths’ vanishing theorem [8] : n,q X, S det ) = 0 for Theorem 0.3. For all integers + 1 then p,q X, E (det ) = 0 For and arbitrary r , k 2 , Peternell-Le Potier and Schneider [11] have constructed an example of an ample vector bundle of rank on a manifold of dimension = 2 such that (0 4) n,n X, S = 0 This result shows that det cannot be omitted in theorem 0.2 when = 1 . More generally, the following example shows that the exponent is optimal. Example. Let be the Grassmannian of subspaces of codimension of a vector space of dimension , and the tautological quotient vector

bundle of rank on (then is spanned and = det very ample) . Let ∈{ , . . . , r and be such that . . . r , a +1 . . . = 0 = ( r, . . . , a r, , . . . , 0) Set = dim = ( )( . Then (0 5) n,q X, (det = (det = 0 Our approach is based on three well-known facts. First, ever y tensor power splits into irreducible representations of the linear group Gl( cf. (2.16)). Secondly, every irreducible tensor bundle appears in a natural way as the direct image on of an ample line bundle on a suitable ﬂag manifold of . This follows from Bott’s theory of homogeneous vector bund les [3]. The third fact

is the isomorphism theorem of Le Potier [13], which rela tes the cohomology groups of on to those of the line bundle O (1) on ) . In 3, we generalize this isomorphism to the case of arbitrary ﬂag bundles associ ated to ; theorem 0.2 is an immediate consequence. The proof of theorem 0.3 rests on a generalization of the Bore l-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 i nduction on based on the usual Leray spectral sequence and on the Kodaira -Akizuki-Nakano vanishing theorem for

line bundles. A by-product of these me thods is the following isomorphism result, already contained in the standard Bore l-Le Potier spectral sequence, but which seems to have been overlooked.

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Theorem 0.6. For every vector bundle and every line bundle one can deﬁne a canonical morphism p,q X, +1 ,q +1 X, S Under hypothesis 0.1, this morphism is one-to-one for and surjective for Combining theorem 0.6 and example (0.4), we get ,n X, = 0 . This shows that Sommese’s conjecture ([15], conjecture (4. 2)) ample = p,q X, ) = 0 for + 1 is false for = 2 6 . The following related

problem is interesting, but its compl ete solution certainly requires a better understanding of the Borel-Le P otier spectral sequence for ﬂag bundles. Problem. Given any dominant weight with = 0 and p, q such that + 1 , determine the smallest exponent n, p, q, r, a such that p,q X, (det ) = 0 for We show in 5 that if the Borel-Le Potier spectral sequence degenerates at the level, it is always suﬃcient to take 1 + min p, n . In that case, theorem 0.6 appears also as a special case of a fairly ge neral exact sequence. The –degeneracy of the Borel-Le Potier spectral sequence is thu

s an important feature which would be interesting to investigate. Some of the above results have been annouced in the note [4] an d corre- sponding detailed proofs are given in [5] . However, the meth od 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, F riedrich Knopp, Thomas Peternell and Michael Schneider for valuable remark s which led to sub- stantial improvements of this work.

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1. Basic deﬁnitions and tools. We recall here a few basic facts and

deﬁnitions which will be u sed repeatedly in the sequel. (1.1) Ample vector bundles (compare with Hartshorne [9]). A vector bundle on is said to be spanned if the canonical map X, E is onto for every , and semi-ample if the symmetric powers are spanned for large enough. is said to be very ample if the canonical maps X, E X , X, E ⊗→O ⊗O X , are onto, denoting the maximal ideal at . If is very ample, then the holomorphic map from to the Grassmannian of –codimensional subspaces of X, E ) given by Φ : , x 7 ) = 0 is an embedding, and is the pull-back by Φ of the

canonical quotient vector bundle of rank on ) . The embedding condition is in fact equivalent to being very ample if = 1 , but weaker if 2 . At last, is said to be ample if the symmetric powers are very ample for large enough. Denoting O (1) the canonical line bundle on ) associated to and the projection, it is well-known that ) = . One gets then easily spanned on (1) spanned on Y , (semi-)ample on (1) (semi-)ample on Y . Moreover, for any line bundle on , ampleness is equivalent to the existence of a smooth hermitian metric on the ﬁbers of with positive curvature form ic ) . Since ) =

, it is clear that E, L semi-ample = semi-ample E, L spanned one of them very ample = very ample E, L semi-ample one of them ample = ample (1.2) Kodaira-Akizuki-Nakano vanishing theorem [1]. If is an ample (or positive) line bundle on , then p,q X, L ) = X, ) = 0 for + 1 (1.3) Leray spectral sequence cf. Godement [7]). Let be a continuous map between topological spaces, sheaf of abelian groups on , and the direct image sheaves of on Then there exists a spectral sequence such that p,q X, R and such that the limit term p,q is the –graded module associated to a decreasing ﬁltration of Y, )

.

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(1.4) Cohomology of a ﬁltered sheaf We will need also the following elementary result. Let be a sheaf of abelian groups on and ⊃F . . . ⊃F . . . ⊃F = 0 be a ﬁltration of such that the graded sheaf +1 , satisﬁes X, ) = 0 for . Then X, ) = 0 for Indeed, it is immediately veriﬁed by induction on 1 that X, vanishes for , using the cohomology long exact sequence associated to ⊗→G ⊗→F +1 ⊗→F 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 f ew basic results on homogeneous line bundles on ﬂag manifolds ( cf. Borel-Weil [2] and R. Bott [3]). Let be the Borel subgroup of Gl = Gl( ) of lower triangular matrices, the subgroup of unipotent matrices, and the complex torus ( of diagonal matrices. Let be a complex vector space of dimension . We denote by ) the manifold of complete ﬂags . . . codim λ . To every linear isomorphism Isom( , V ) : ( , . . . , u 7 one can associate the ﬂag [ ) deﬁned by = Vect( +1 , . . . , ) , . This leads to the

identiﬁcation ) = Isom( , V /B where acts on the right side. We denote simply by the tautological vector bundle of rank on ) , and we consider the canonical quotient line bundles (2 1) /V r , . . . , a = ( , . . . , a The linear group Gl( ) acts on ) on the left, and there exist natural equivariant left actions of Gl( ) on all bundles . If Gl( is a representation of , we may associate to the manifold V, = Isom( , V = Isom( , V E/ where denotes the equivalence relation ( ζg, u ζ, ) , . Then V, is a Gl( )–equivariant bundle over ) , and all such bundles are obtained in this way.

It is clear that is isomorphic to the line bundle V,a arising from the 1–dimensional representation of weight /U . . .t 7 . . . t the isomorphism V,a is induced by the map Isom( , V ζ, u 7 . . . mod

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We compute now the tangent and cotangent vector bundles of ) . The action of Gl( ) on ) yields (2 2) TM ) = Hom( V, V /W where is the subbundle of endomorphisms Hom( V, V ) such that . Using the self-duality of Hom( V, V ) given by the Killing form , g 7 tr( ) , we ﬁnd (2 3) ) = Hom( V, V /W Hom( V, V ) ; If ad denotes the adjoint representation of on the Lie algebras

ˇ: then (2 4) TM ) = ( ˇ: V, ad , T ) = ( V, ad because , W at every point [ ) . There exists a ﬁltration of ) by subbundles of the type Hom( V, V ) ; in such a way that the corresponding graded bundle is the dire ct sum of the line bundles Hom( , Q ) = λ < ; their tensor product is thus isomorphic to the canonical line bundle = det( )) : (2 5) . . . . . . where = (1 r, . . . , r 1); will be called the canonical weight of ) . Case of incomplete ﬂag manifolds. More generally, given any sequence of integers = ( , . . . , s ) such that 0 = < s < . . . < s , we may

consider the manifold ) of incomplete ﬂags . . . codim On ) we still have tautological vector bundles s,j of rank and line bundles (2 6) s,j = det( s,j /V s,j m . For any –tuple such that +1 . . . , 1 , we set s, . . . s,m If ) is the natural projection, then (2 7) s,j , s,j +1 . . . , On the other hand, one has the identiﬁcation ) = Isom( , V /B where is the parabolic subgroup of matrices ( ) with = 0 for all λ, such that there exists an integer = 1 , . . . , m 1 with and > s . We deﬁne as the unipotent subgroup of lower triangular matrices ( ) with = 0 for

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all λ, such that < for some (hence ). In the same way as above, we get TM ) = Hom( V, V /W , W (2 8) ) = = ( V, ad (2 9) s, . . . s,j . . . s,m (2 10) where ) = ( r, . . . , s r, . . . , s r, . . . , s r, . . . ) is the canonical weight of ) . Lemma 2.11. enjoys the following properties : (a) If < a +1 for some = 1 , . . . , m , then , Q ) = 0 (b) is spanned if and only if . . . (c) is (very) ample if and only if > a > . . . > a Proof . (a) Let ( . . . ) be an arbitrary ﬂag and F, F subspaces of such that +1 dim + 1 dim Let us consider the projective line F/F ) of

ﬂags . . . +1 such that . Then s,j F/F = det( /V F/V O(1) s,j +1 F/F = det( /V +1 /F O( 1) thus F/F O( +1 ) , which implies (a) and the “only if” part of assertions (b), (c). (b) Since s, . . . s,j = det( V/V ) , we see that this line bundle is a quotient of the trivial bundle . Hence (2 12) det( V/V +1 (det is spanned by sections arising from elements of +1 ( (det as soon as . . . (c) If > . . . > a , it is elementary to verify that the sections of arising from (2.12) suﬃce to make very ample (this is in fact a generalization of Plucker’s imbedding of Grassmannians).

Cohomology groups of It remains now to compute , Q , Q ) when . . . . Without loss of generality we may assume that 0 , because . . . = det is a trivial bundle.

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Proposition 2.13. For all integers . . . , there is a canonical isomorphism , Q ) = where . . . is the set of polynomials , . . . , on which are homogeneous of degree with respect to and invariant under the left action of on = Hom( V, , . . . , , , . . . , ) = , . . . , ν < λ . Proof . — To any section , Q ) we associate the holomorphic function on Isom( V, deﬁned by , . . . , ) = ( . . . . ([ , . . .

, ]) where ( , . . . , ) is the dual basis of ( , . . . , ) , and where the linear form induced by on /V is still denoted . Let us observe that is homogeneous of degree in and locally bounded in a neighborhood of every –tuple of ( Isom( V, ) (because ) is compact and 0) . Therefore can be extended to a polynomial on all ( . The invariance of under is clear. Conversely, such a polynomial obviously deﬁnes a unique section on ) . From the deﬁnition of , we see that = k, , ... , 0) V , (2 14) = (1 , ... , , ... , 0) V . (2 15) For arbitrary , proposition 2.13 remains true if we set =

, ... ,a 0) (det when is non-increasing = 0 otherwise The weights will be ordered according to their usual partial ordering : i r . Bott’s theorem [3] shows that is an irreducible representation of Gl( ) of highest weight ; all irreducible representations of Gl( ) are in fact of this type cf. Kraft [10]). In particular, since the weights of the action o f a maximal torus Gl( ) on verify . . . and 0 , we have a canonical Gl( )–isomorphism (2 16) ... ... a, k ) where a, k 0 is the multiplicity of the isotypical factor in Bott’s formula ( cf. also Demazure [6] for a very simple proof) gives in

fact the expression of all cohomology groups , Q ) . A weight is called regular if all the elements of are distinct, and singular otherwise. We will denote by the reordered non-increasing sequence associated to , by ) the number of strict inversions of the order, and we will set

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the sequence is non-increasing if and only if is regular. Proposition 2.17 (Bott [3]) . One has , Q ) = if if In particular all vanish if and only if is singular. We will be particularly interested by those line bundles such that the cohomology groups p,q vanish for all 1 , being given. The following

proposition is one of the main steps in the proof of our result s. Proposition 2.18. Set = dim ) = dim . Then (a) p,q , Q ) = 0 for all ) + 1 as soon as +1 (b) , Q ) = 0 for all as soon as +1 +1 (c) In general, p,q , Q is isomorphic to a direct sum of irredu- cible Gl( –modules with . . . min } (d) p,q , Q ) = 0 for all as soon as +1 min p , N + ( +1 , r + 1 +1 (e) Under the assumption of (d) , there is a Gl( –isomorphism p, , Q ) = , p ) V , where , p is the multiplicity of the weight in Proof . — Under the assumption of (a), is ample by lemma 2.11 (c) . The result follows therefore from the

Kodaira-Nakano-Akiz uki theorem. Now (b) is a consequence of (a) since +1 +1 ) and , Q ) = ,q , Q Let us now observe that for every vector bundle on ) there are isomor- phisms , E , (2 19) , E , Q (2 20) where is the relative canonical bundle along the ﬁbers of the proje ction ) . In fact, the ﬁbers of are products of ﬂag manifolds. For such a ﬁber , we have F, K ) = dim F, when = dim ) and F, K ) = 0 otherwise (apply (b) with = 0 and Kunneth’s formula). We get thus direct image sheaves 0 if = 0 ) if = 0 . 0 if ) if ) .

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Formulas (2.19) and

(2.20) are thus immediate consequences of the Leray spectral sequence. When applied to = , formula (2.19) yields p,q , Q ) = , Q , Q TM using the isomorphism TM ) . In the same way, (2.20) implies p,q , Q ) = , Q , Q TM The bundle ) = ( V, ad (resp. TM ) = ( ˇ: V, ad ) has a ﬁltration with associated graded bundle resp where the indices λ, are such that < for some . It follows that resp. TM has a ﬁltration with associated graded bundle , p resp , N where the weights (resp. ) and multiplicities , p ) (resp. , N )) are those of (resp. ˇ: ) . We need a lemma. Lemma

2.21. The weights of verify min + 1 , r + 1 , N + ( +1 for < < +1 Proof . — Let us denote by ( the canonical basis of . The weights (resp. ) are the sums of (resp. ) distinct elements of the set of weights (resp. ) where < for some . It follows that + 1 ) , with equality for the weight = ( ) + . . . + ( ) + ( +1 ) + . . . + ( It is clear also that + 1 and . Since the weights are related by ) , lemma 2.21 follows. Proof of (c). — By the ﬁltration property (1.4) and the above formulas , we see that p,q , Q ) is a direct sum of certain irreducible Gl( )–modules , Q = V , b = ( Clearly ) ,

thus min } max ) + = min } Proof of (d). — Similarly, (d) is reduced by (1.4) to proving , Q ) = 0 10

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for all the above weights and all 1 . By proposition 2.17 it is suﬃcient to get Let us observe that a weight will be such that ) as soon as 0 whenever < < +1 , 1 1 . For ) this condition yields, thanks to lemma 2.21 : min +1 , r +1 , N +( +1 +( +1 Taking as large as possible, i.e. +1 1 , we get the asserted condition. Proof of (e). — Because of the vanishing of of the graded quotients, we obtain p, , Q ) = , p , Q Applying proposition 2.17, we get ( )) , where is the

partial reordering of such that only the coeﬃcients in each interval [ +1 , s have been reordered (in non-increasing order). The number o f inversions is always ) , with equality if and only if is non-decreasing in each interval. In that case , Q = + V , and = 0 otherwise. Since is a –module we have , p ) = , p ) . Formula (e) is proved, and we see that non-zero terms corresp ond to weights which are non-increasing in each interval [ + 1 , s ] . Remark 2.22. — If the manifold ) is a Grassmannian, then = 2 , = (0 , s , r ) . The condition required in proposition 2.21 is therefore a lways

satisﬁed when 1 , i.e. when is ample. 3. An isomorphism theorem Our aim here is to generalize Griﬃths and Le Potier’s isomorp hism theorems ([8], [13]) in the case of arbitrary ﬂag bundles, following t he simple method of Schneider [14] . Let be a –dimensional compact complex manifold and a holo- morphic vector bundle of rank . For every sequence 0 = < s < . . . < s we associate to its ﬂag bundle . If is such that +1 . . . , 1 , we may deﬁne a line bundle just as we did in 2 . Let us set = X , = Y . One has an exact sequence (3 1) 0 Y/X where Y/X is by

deﬁnition the bundle of relative diﬀerential 1–forms a long the ﬁbers of the projection . One may then deﬁne a decreasing ﬁltration of as follows : (3 2) p,t ( ) = ( 11

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The corresponding graded bundle is given by (3 3) p,t p,t /F +1 ,t ( Y/X Over any open subset of where is a trivial bundle with dim the exact sequence (3.1) splits as well as the ﬁltration (3.2 ). Using proposition 2.18 (a), (d), we obtain the following lemma. Lemma 3.4. For every weight such that (3 5) +1 if and otherwise +1 min t, N + ( +1 , r + 1 +1 the sheaf of

sections of Y/X has direct images (3 6) Y/X = 0 for Y/X , t ) E . We have in particular (3 7) ) = E , Y/X = E . Let be an arbitrary line bundle on . Under assumption (3.5), formulas (3.3) and (3.6) yield p,p ) = 0 for p,p ) = , t ) L . The Leray spectral sequence implies therefore : Theorem 3.8. Under assumption (3.5), one has for all Y, G p,p , t p,q X, The special case ) gives : Corollary 3.9. If +1 , then for all Y, G p,p p,q X, When n,n is the only non-vanishing quotient in the ﬁltration of the canonical line bundle . We thus obtain the following generalization of Griﬃths’

isomorphism theorem [8] : (3 10) ,q , Q n,q X, 12

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4. Vanishing theorems. In order to carry over results for line bundles to vector bund les, one needs the following simple lemma. Lemma 4.1. Assume that > a > . . . > a . Then (a) semi-ample semi-ample; (b) ample ample; (c) semi-ample and ample ample. Proof . (a) If is semi-ample, then by deﬁnition (1.1) is spanned for large enough. Hence ka , which is a direct summand in ka . . . ka , is also spanned for . Since the ﬁbers of ka ) = ka generate ka , we conclude that ka is spanned for (b) Similar proof, replacing

“semi-ample” by “ample” and “s panned” by “very ample”. One needs moreover the fact that ka is very ample along the ﬁbers of (lemma 2.11). (c) If is very ample for , then ka is very ample for max( , k ) , because ka is spanned on and very ample along the ﬁbers, whereas Y, ) = X, L ) separates points of which lie in distinct ﬁbers. We are now ready to attack the proof of the main theorems. Proof of theorem 0.2. — Let be such that . . . > a +1 . . . = 0 Deﬁne < s < . . . < s as the sequence of indices = 1 , . . . , r 1 such that > a +1 and set + ( h, . . . , h ) . The

canonical weight ) is non-decreasing and , hence > a > . . . > a = 0 so is ample by lemma 4.1 and = (det . Formula (3.10) yields n,q X, (det ,q , Q Since dim ) = ) , the group in the right hand side is zero for by the Kodaira-Akizuki-Nakano vanishing theorem (1.2) . Proof of theorem 0.3. — The proof proceeds by backward induction on The case is already settled by theorem 0.2. The decomposition formul (2.16) shows that the result is equivalent to p,q X, (det ) = 0 for + 1 , l and . . . = 0 not all zero. Deﬁne as above and +( l, . . . , l ) . Then is ample since 0 , and corollary 3.9

implies (4 2) p,q X, (det Y, G p,p 13

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Now, it is clear that p,p = . We get thus an exact sequence +1 ,p p,p The Kodaira-Akizuki-Nakano vanishing theorem (1.2) appli ed to with dim ) yields Y, ) = 0 for + 1 The cohomology groups in (4.2) will therefore vanish if and o nly if (4 3) +1 Y, F +1 ,p ) = 0 Let us observe that +1 ,p has a ﬁltration with associated graded bundle t,p . In order to verify (4.3), it is thus enough to get (4 4) +1 Y, G t,p ) = 0 , t At this point, the Leray spectral sequence will be used in an e ssential way. Since t,p ( Y/X , we get t,p ) = Y/X +(

l, ... ,l Proposition 2.18 (a) and (c) yields Y/X +( l, ... ,l 0 for + 1 otherwise E , b t , t,p ) = 0 for + 1 otherwise b,m (det L , where the last sum runs over weights such that = 0 and integers such that ) + 1 . Therefore +1 Y, G t,p has a ﬁltration whose graded module is the limit term j,q +1 of a spectral sequence such that j,k 0 for + 1 otherwise b,m t,j X, (det , m ) + We have thus j,q +1 = 0 for by the ﬁrst case, and also for by the second case and the induction hypothesis. Hence j,q +1 = 0 for all and (4.4) is proved. Proof of theorem 0.6 . — Let us take here = (2 , . .

. , 0) and = (0 , r ) , so that ) = ) . Since and (1 , ... , 0) = theorem 3.8 yields (4 5) Y, G +1 ,p +1 s, +1 ,q X, S Y, G p,p +1 s, p,q X, Y, G k,p +1 s, ) = 0 for k < p , because is the only weight of such that is non- increasing, and because no such weights exist for higher ext erior powers Now, +1 /F p,p +1 has a ﬁltration with graded quotients k,p +1 k < p , therefore Y, +1 /F p,p +1 s, ) = 0 for all q . 14

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Considering the exact sequences p,p +1 +1 /F +1 ,p +1 +1 /F p,p +1 +1 ,p +1 +1 +1 /F +1 ,p +1 we get an isomorphism Y, G p,p +1 s, Y, +1 /F +1 ,p +1 s, and a

canonical coboundary morphism Y, +1 /F +1 ,p +1 s, +1 Y, G +1 ,p +1 s, which by Kodaira-Akizuki-Nakano is onto for + 1 + + 1 ) + 1 , i.e. 2 , and bijective for 1 . Combining this with the ﬁrst two isomorphisms (4.5) achieves the proof of theorem 0. 6. As promised in the introduction, we show now that the conditi on in theorem 0.2 is best possible. Example 4.6. — Let ) be the Grassmannian of subspaces of codimension of a vector space , dim , and the tautological quotient vector bundle of rank over . Then is spanned (hence semi-ample) and = det is very ample. According to the notations of

2 , we have , s = ( , s , s ) = (0 , r, d V/V det s, det s, s, s, Furthermore, for any sequence . . . 0 , we have = V/V ) = a, 0) where ) is the projection. If = dim ) , this implies n,q X, (det X, Q s, s, a, 0) X, where = ( d, . . . , a d, r, . . . , r . The ﬁber of is V/V ) , hence = 0 for 1 by Kunneth’s formula and proposition 2.18 (b) . We obtain therefore n,q X, (det , Q Assume now that . . . and . . . = 0 . Deﬁne = (1 , . . . , 1) . Then + ( , . . . , a , . . . , h , . . . , r where dots indicate a decreasing sequence of consecutive in tegers. There are exactly )( )

inversions of the ordering, corresponding to the inversio n of the last two blocks between semicolons. Then one ﬁnds easily = ( d, . . . , a h, . . . , h ) = where = ( d, . . . , a ; 0 , . . . , 0) . Proposition 2.17 yields therefore n,q X, (det 0 if = ( )( (det if = ( )( ) . 15

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5. On the Borel-Le Potier spectral sequence. Denote as before the projection. To every integer and every coherent analytic sheaf on , one may associate the ﬁltration of ⊗S by its subsheaves p,t ⊗S ⊗S the corresponding graded sheaf being of course p,t ⊗S . This

gives rise to a spectral sequence which we shall name after Borel and Le Poti er, whose term is given by (5 1) p,q Y, G p,t ⊗S The limit term p,q is the –graded module corresponding to the ﬁltration of ) = Y, ⊗S ) by the canonical images of the groups ) . Assume that the spectral sequence degenerates in i.e. p,q r,q +1 is zero for all 2 (by Peternell, Le Potier and Schneider [12], the spectral s equence does not degenerate in general in ). Then p,q p,q . This equality means that the –th cohomology group of the –complex Y, G p,t ⊗S +1 Y, G +1 ,t ⊗S is the -graded

module corresponding to a ﬁltration of Y, ⊗S ) . By Kodaira- Akizuki-Nakano, we get therefore : Proposition 5.1. Assume that is ample and , or and ample, and that the –degeneracy occurs for the ample invertible sheaf on . Then the complex Y, G p,t +1 Y, G +1 ,t is exact in degree ) + 1 Our hope is that the –degeneracy can be proved in all cases by means of harmonic forms and Hodge theory, but we have been unable to do so. Since p,t = 0 for t > p ) , proposition 5.1 yields for all + 1 an exact sequence Y, G p,p +1 Y, G +1 ,p Replacing by + ( l, . . . , l ) as in the proof of

theorem 0.3, and using formula (3.10) and theorem 3.8, one gets an exact sequence p,q X, (det , N 1) +1 ,q +1 X, )+ (det because assumption (3.5) is satisﬁed for the weight and for 1 . But the above direct sum can be rewritten 1) +1 ,q +1 X, (det in terms of the weights < , of ˇ: . A backward induction on max p, q yields then immediately : 16

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Corollary 5.2. If the –degeneracy occurs for all ample line bundles . . . = 0 , on all ﬂag manifolds of , then p,q X, (det ) = 0 for + 1 and 1 + min p, n Another interesting consequence of proposition 5.1 in the c ase

would be the following generalization of theorem 0.6. Corollary 5.3. Set t,k = t, ,..., ,..., 0) if and t,k = 0 otherwise. Then there is a canonical complex . . . p,q X, Z t,k +1 ,q +1 X, Z ,k . . . Under the hypotheses of proposition 5.1 for k, , ... , 0) , this complex is exact in each degree such that Proof . — The only possible weight of such that ( k, , . . . , 0)+ be non-increasing is = ( t, , . . . , , . . . , 0) . Theorem 3.8 yields therefore Y, G p,p k, , ... , 0) ) = p,q X, Z t,k Note that the special case = 1 , = 0 is Le Potier’s theorem, and that the special case = 2 , = 1 is

theorem 0.6. These two cases do not depend on any degeneracy assumption. 17

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References [1] y. Akizuki and s. Nakano . Note on Kodaira-Spencer’s proof of Lefschetz theorems , Proc. Jap. Acad., 30 ), 266–272. [2] a. Borel and a. Weil . Representations lineaires et espaces homog`enes kahle riens des groupes de Lie compacts, Seminaire Bourbaki (expose n˚100 par J.-P. Serre), (mai ), 8 pages [3] r. Bott . Homogeneous vector bundles , Ann. of Math., 66 ), 203–248. [4] j.-p. Demailly . Theor`emes d’annulation pour la

cohomologie des puissanc es ten- sorielles d’un ﬁbre positif , C. R. Acad. Sci. Paris Ser. I Math., 305 ), `a paraˆıtre. [5] j.-p. Demailly . — Vanishing theorems for tensor powers of a positive vector bu ndle, Proceedings of the Conference Geometry and Analysis on Manifol ds held in Katata, Japan (august ), edited by T. Sunada, Lecture Notes in Math. n 1339, Springer. [6] b. Demazure . A very simple proof of Bott’s theorem , Invent. Math., 33 ), 271–272. [7] r. Godement . Theorie des faisceaux , Hermann, Paris, [8] p.a. Griffiths . Hermitian

diﬀerential geometry, Chern classes and positive vec tor bundles , Global Analysis, Papers in honor of K. Kodaira, Princeton Univ . Press, Princeton ), 185–251. [9] r. Hartshorne . Ample vector bundles , Publ. Math. I.H.E.S., 29 ), 63–94. [10] h. Kraft . Geometrische Methoden in der Invariantentheorie , Aspekte der Mathe- matik, Band D1, Braunschweig, Vieweg & Sohn, [11] . Peternell, j. le Potier and m. Schneider . Vanishing theorems, linear and quadratic normality , Invent. Math., 87 ), 573–586. [12] t. Peternell, j. le Potier and m. Schneider . Direct images of sheaves of

diﬀerentials and the Atiyah class , Math. Z., 196 ), 75–85. [13] j. Le Potier . Annulation de la cohomologie `a valeurs dans un ﬁbre vectorie holomorphe de rang quelconque , Math. Ann., 218 ), 35–53. [14] m. Schneider . Ein einfacher Beweis des Verschwindungssatzes fur positiv e holomor- phe Vektorraumbundel , Manuscripta Math., 11 ), 95–101. [15] a.j. Sommese . Submanifolds of abelian varieties , Math. Ann., 233 ), 229–256. 18

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