VANISHING THEOREMS FOR TENSOR POWERS OF AN AMPLE VECTOR BUNDLE

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 be a holomorphic vector bundle of rankron a compact complex manifold X of dimension n . It is shown that the cohomology groups H^p,q(X, E^⊗k⊗(detE)^l) vanish ifE 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 canonically isomorphic to the direct image onXof a homogeneous line bundle over a flag manifold of E . The proof is then reduced to the Kodaira-Akizuki-Nakano vanishing theorem for line bundles by means of the Leray spectral sequence, using backward induction on p . We also obtain a generalization of Le Potier’s isomorphism theorem and a counterexample to a vanishing conjecture of Sommese.

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.

LetX be a complex compact n–dimensional manifold and E a holomorphic vector bundle of rank r onX . IfE is ample andr >1 , only very few general and optimal vanishing results are available for the Dolbeault cohomology groups H^p,q of tensor powers ofE . For example, the famous Le Potier vanishing theorem [13] :

E ample =⇒ H^p,q(X, E) = 0 for p+q≥n+r

does not extend to symmetric powers S^kE , even when p = n and q = n−2 (cf. [11]) . Nevertheless, we will see that the vanishing property is true for tensor powers involving a sufficiently large power of 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.

Theorem 0.2. — Let us denote by Γ^aE the irreducible tensor power representation of Gl(E) of highest weight a ∈ Z^r . If h ∈ {1, . . . , r −1} and a1 ≥a2 ≥. . .≥ah > ah+1 =. . .=ar = 0 , then

H^n,q(X,Γ^aE⊗(detE)^h⊗L) = 0 for q ≥1 .

Since S^kE = Γ(k,0,...,0)E , the special case h = 1 is Griffiths’ vanishing theorem [8] :

H^n,q(X, S^kE ⊗detE⊗L) = 0 for q ≥1 .

Theorem 0.3. — For all integers p+q ≥n+ 1 ,k ≥1 ,l ≥n−p+r−1 then H^p,q(X, E^⊗k⊗(detE)^l⊗L) = 0 .

For p =n and arbitrary r , k₀ ≥2 , Peternell-Le Potier and Schneider [11]

have constructed an example of an ample vector bundle E of rank r on a manifold X of dimension n= 2r such that

(0.4) H^n,n−2(X, S^kE)6= 0 , 2≤k ≤k₀ .

This result shows that detE cannot be omitted in theorem 0.2 whenh= 1 . More generally, the following example shows that the exponent h is optimal.

Example. — Let X = Gr(V) be the Grassmannian of subspaces of codimension rof a vector spaceV of dimensiond, andE the tautological quotient vector bundle of rank ron X (then E is spanned and L= detE very ample) . Let h ∈ {1, . . . , r−1} and a∈Z^r , β ∈Z^d be such that

a1 ≥. . .≥ah ≥d−r , ah+1 =. . .=ar = 0 , β = (a1−d+r, . . . , ah−d+r,0, . . . ,0) . Set n= dimX =r(d−r) , q = (r−h)(d−r) . Then

(0.5) H^n,q X,Γ^aE ⊗(detE)^h

= Γ^βV ⊗(detV)^h 6= 0 .

Our approach is based on three well-known facts. First, every tensor power E^⊗k splits into irreducible representations Γ^aE of the linear group Gl(E) (cf. (2.16)). Secondly, every irreducible tensor bundle Γ^aE appears in a natural way as the direct image onX of anample line bundleon 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 ofE onX to those of the line bundle O_E(1) onP(E^⋆) . In§3, we generalize this isomorphism to the case of arbitrary flag bundles associated to E; theorem 0.2 is an immediate consequence.

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

Theorem 0.6. — For every vector bundle E and every line bundle Lone can define a canonical morphism

H^p,q(X,Λ²E⊗L) −→H^p+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 getH^n−1,n−3(X,Λ²E)6= 0 . This shows that Sommese’s conjecture ([15], conjecture (4.2))

E ample =⇒ H^p,q(X,Λ^kE) = 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 ∈ Z^r with a_r = 0 and p, q such that p+q ≥n+ 1, determine the smallest exponent l₀ =l₀(n, p, q, r, a)such that H^p,q(X,Γ^aE ⊗(detE)^l⊗L) = 0 for l≥l₀ .

We show in §5 that if the Borel-Le Potier spectral sequence degenerates at the E₂ level, it is always sufficient to take l ≥r−1 + min{n−p, n−q} . In that case, theorem 0.6 appears also as a special case of a fairly general exact sequence.

The E₂–degeneracy of the Borel-Le Potier spectral sequence is thus an important feature which would be interesting to investigate.

Some of the above results have been annouced in the note [4] and 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.

1. Basic definitions and tools.

We recall here a few basic facts and definitions which will be used 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_x is onto for every x ∈ X , and semi-ample if the symmetric powers S^kE are spanned for k ≥k₀ large enough.

E is said to be very ample if the canonical maps

(H⁰(X, E)−→E_x⊕E_y , ∀x6=y ∈X , H⁰(X, E)−→ O(E)⊗ O_x/M²_x , ∀x∈X ,

are onto, Mx denoting the maximal ideal at x∈ X . IfE is very ample, then the holomorphic map from X to the Grassmannian of r–codimensional subspaces of V =H⁰(X, E) given by

Φ : X −→G_r(V) , x7−→W_x ={σ ∈V;σ(x) = 0}

is an embedding, and E is the pull-back by Φ of the canonical quotient vector bundle of rank r on G_r(V) . The embedding condition is in fact equivalent to E being very ample if r = 1 , but weaker ifr ≥2 .

At last, E is said to be ample if the symmetric powers S^kE are very ample for k ≥ k0 large enough. Denoting OE(1) the canonical line bundle on Y = P(E^⋆) associated to E and π : Y −→ X the projection, it is well-known that π⋆OE(k) =S^kE . One gets then easily

E spanned on X ⇐⇒ O_E(1) spanned on Y ,

E (semi-)ample on X ⇐⇒ O_E(1) (semi-)ample onY .

Moreover, for any line bundle Lon X , ampleness is equivalent to the existence of a smooth hermitian metric on the fibers of L with positive curvature formic(L) . Since S^k(E⊗L) =S^kE⊗L^k , 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].

IfL is an ample (or positive) line bundle onX , then

H^p,q(X, L) =H^q(X,Ω^p_X ⊗L) = 0 for p+q ≥n+ 1 . (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 R^qπ_⋆^⌣⌣S the direct image sheaves of^⌣⌣S on X . Then there exists a spectral sequence such that

E₂^p,q =H^p(X, R^qπ_⋆^⌣⌣S)

and such that the limit term E_∞^p,q−p is the p–graded module associated to a decreasing filtration of H^q(Y,^⌣⌣S) .

(1.4) Cohomology of a filtered sheaf.

We will need also the following elementary result. LetF be a sheaf of abelian groups on X and

F =F⁰ ⊃ F¹ ⊃. . .⊃ F^p ⊃. . .⊃ F^N = 0 be a filtration of F such that the graded sheaf L

G^p , G^p = F^p/F^p+1 , satisfies H^q(X,G^p) = 0 for q ≥q₀ . Then H^q(X,F) = 0 for q ≥q₀ .

Indeed, it is immediately verified by induction onp≥1 thatH^q(X,F/F^p) vanishes for q ≥q0 , using the cohomology long exact sequence associated to

0−→ G^p −→ F/F^p+1 −→ F/F^p −→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]).

LetB_r be the Borel subgroup of Gl_r = Gl(C^r) of lower triangular matrices, U_r ⊂ B_r the subgroup of unipotent matrices, and T^r the complex torus (C^⋆)^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 =V₀ ⊃V₁ ⊃. . .⊃V_r ={0} , codim^CV_λ=λ . To every linear isomorphism ζ ∈ Isom(C^r, V) : (u₁, . . . , u_r) 7−→P

1≤λ≤ru_λζ_λ , one can associate the flag [ζ] ∈ M(V) defined by Vλ = Vect(ζλ+1, . . . , ζr) , 1≤λ ≤r . This leads to the identification

M(V) = Isom(C^r, V)/Br

where B_r acts on the right side. We denote simply by V_λ the tautological vector bundle of rankr−λonM(V) , and we consider the canonical quotient line bundles (2.1)

(Q_λ=V_λ−1/V_λ , 1≤λ≤r ,

Q^a =Q^a₁¹ ⊗. . .⊗Q^a_r^r , a= (a₁, . . . , a_r)∈Z^r .

The linear group Gl(V) acts on M(V) on the left, and there exist natural equivariant left actions of Gl(V) on all bundlesV_λ ,Q_λ ,Q^a . Ifρ : B_r →Gl(E) is a representation of Br , we may associate toρ the manifold

E_V,ρ = Isom(C^r, V)×_Br E = Isom(C^r, V)×E/∼

where ∼denotes the equivalence relation (ζg, u)∼(ζ, ρ(g)u) , g∈B_r . ThenE_V,ρ is a Gl(V)–equivariant bundle over M(V) , and all such bundles are obtained in this way. It is clear that Q^a is isomorphic to the line bundleC_V,a arising from the 1–dimensional representation of weight a :

a : T^r =B_r/U_r −→C^⋆ , (t₁. . . t_r)7−→t^a₁¹. . . t^a_r^r ; the isomorphism C_V,a −→˜ Q^a is induced by the map

Isom(C^r, V)×C∋(ζ, u)7−→uζ˜₁^a1 ⊗. . .⊗ζ˜_r^ar ∈Q^a , ζ˜_λ =ζ_λ mod V_λ .

We compute now the tangent and cotangent vector bundles of M(V) . The action of Gl(V) on M(V) yields

(2.2) T M(V) = Hom(V, V)/W

whereW is the subbundle of endomorphismsg∈Hom(V, V) such thatg(V_λ)⊂V_λ, 1 ≤ λ ≤ r . Using the self-duality of Hom(V, V) given by the Killing form (g₁, g₂)7→tr(g₁g₂) , we find

(2.3)

(T^⋆M(V) = Hom(V, V)/W⋆

=W^⊥

W^⊥ ={g∈Hom(V, V) ; g(V_λ−1)⊂V_λ , 1≤λ ≤r} .

If ad denotes the adjoint representation of B_r on the Lie algebras gl^ˇ:_r , R^∪∪_r , U_r , then

(2.4) T M(V) = (g^ˇl^:r/R∪∪r)V,ad , T^⋆M(V) = (U_r)V,ad

because W_[ζ] =ζR^∪∪_rζ⁻¹ , W_[ζ]^⊥ =ζU_rζ⁻¹at every point [ζ]∈M(V) . There exists a filtration of T^⋆M(V) by subbundles of the type

{g∈Hom(V, V) ; g(V_λ)⊂V_µ(λ) , 1≤λ≤r}

in such a way that the corresponding graded bundle is the direct sum of the line bundles Hom(Q_λ, Q_µ) =Q⁻_λ¹⊗Q_µ,λ < µ; their tensor product is thus isomorphic to the canonical line bundle K_M(V) = det(T^⋆M(V)) :

(2.5) K_M(V) =Q^1−r₁ ⊗. . .⊗Q^2λ−r−1_λ ⊗. . .⊗Q^r−1_r =Q^c

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 = (s0, . . . , sm) such that 0 = s₀ < s₁ < . . . < s_m =r , we may consider the manifold M_s(V) of incomplete flags

V =V_s0 ⊃V_s1 ⊃. . .⊃V_sm ={0} , codim^CV_sj =s_j .

On M_s(V) we still have tautological vector bundles V_s,j of rank r−s_j and line bundles

(2.6) Qs,j = det(Vs,j−1/Vs,j) , 1≤j ≤m .

For any r–tuple a∈Z^r such that a_sj−1+1 =. . .=a_sj , 1≤j ≤m , we set Q^a_s =Q^a_s,1^s1 ⊗. . .⊗Q^as,m^sm .

If η :M(V)→M_s(V) is the natural projection, then

(2.7) η^⋆V_s,j =V_sj , η^⋆Q_s,j =Q_sj−1+1⊗. . .⊗Q_sj , η^⋆Q^a_s =Q^a . On the other hand, one has the identification

M_s(V) = Isom(C^r, V)/B_s

whereB_s is the parabolic subgroup of matrices (z_λµ) withz_λµ = 0 for allλ, µ such that there exists an integer j = 1, . . . , m−1 with λ ≤sj and µ > sj . We define U_s as the unipotent subgroup of lower triangular matrices (z_λµ) with z_λµ = 0 for

all λ, µ such that sj−1 < λ 6= µ ≤ sj for some j (hence U_s = R^∪∪⊥_s ). In the same way as above, we get

T M_s(V) = Hom(V, V)/W_s , W_s ={g ; g(V_sj−1)⊂V_sj} , (2.8)

T^⋆M_s(V) =W_s^⊥ = (U_s)_V,ad , (2.9)

K_Ms(V)=Q^s_s,1^1−r⊗. . .⊗Q^s_s,j^j−1+sj−r⊗. . .⊗Q^s_s,m^m−1 = Q^c(s)_s (2.10)

where c(s) = (s₁−r, . . . , s₁−r, . . . , s_j−1 +s_j −r, . . . , s_j−1+s_j−r, . . .) is the canonical weight of Ms(V) .

Lemma 2.11. — Q^a_s enjoys the following properties :

(a) If a_sj < a_sj+1 for some j = 1, . . . , m−1 , then H⁰(M_s(V), Q^a_s) = 0 . (b) Q^a_s is spanned if and only if a_s1 ≥a_s2 ≥. . .≥a_sm .

(c) Q^a_s is (very) ample if and only if a_s1 > a_s2 > . . . > a_sm . Proof. —

(a) Let (V_s⁰₀ ⊃V_s⁰₁ ⊃. . .⊃V_s⁰_m)∈M_s(V) be an arbitrary flag andF, F^′ subspaces of V such that

V_s⁰_j−1 ⊃F ⊃V_s⁰_j ⊃F^′ ⊃V_s⁰_j+1 , dimF =s_j + 1 , dimF^′ =s_j−1 . Let us consider the projective line P(F/F^′)⊂Ms(V) of flags

V_s⁰₀ ⊃. . .⊃V_s⁰_j−1 ⊃V_sj ⊃V_s⁰_j+1 ⊃V_s⁰_m such that F ⊃Vsj ⊃F^′ . Then

Q_s,j↾P(F/F^′₎= det(V_s⁰_j−1/V_sj)≃F/V_sj ≃O(1) , Q_{s,j+1↾P(F/F}^′₎ = det(Vsj/V_s⁰_j+1)≃Vsj/F^′ ≃O(−1)

thus Q^a_s↾P(F/F′) ≃ O(a_sj −a_sj+1) , which implies (a) and the “only if” part of assertions (b), (c).

(b) Since Qs,1⊗. . .⊗Qs,j = det(V /Vsj) , we see that this line bundle is a quotient of the trivial bundle Λ^sjV . Hence

(2.12) Q^a_s = O

1≤j≤m−1

det(V /V_sj)^asj−asj+1 ⊗(detV)^ar is spanned by sections arising from elements of N

S^asj−asj+1(Λ^sjV)⊗(detV)^ar as soon as as1 ≥. . .≥asm .

(c) Ifa_s1 > . . . > a_sm , it is elementary to verify that the sections ofQ^a_s arising from (2.12) suffice to make Q^a_s very ample (this is in fact a generalization of Pl¨ucker’s imbedding of Grassmannians).

• Cohomology groups of Q^a .

It remains now to compute H⁰(M_s(V), Q^a_s) ≃ H⁰(M(V), Q^a) when a1 ≥ . . . ≥ ar . Without loss of generality we may assume that ar ≥ 0 , because Q₁⊗. . .⊗Q_r = detV is a trivial bundle.

Proposition 2.13. — For all integers a1 ≥a2 ≥. . .≥ar≥0 , there is a canonical isomorphism

H⁰(M(V), Q^a) = Γ^aV

where Γ^aV ⊂S^a1V ⊗. . .⊗S^arV is the set of polynomialsf(ζ₁^⋆, . . . , ζ_r^⋆) on (V^⋆)^r which are homogeneous of degree a_λ with respect to ζ_λ^⋆ and invariant under the left action of U_r on (V^⋆)^r = Hom(V,C^r) :

f(ζ₁^⋆, . . . , ζ_λ^⋆₋₁, ζ_λ^⋆+ζ_ν^⋆, . . . , ζ_r^⋆) =f(ζ₁^⋆, . . . , ζ_r^⋆) , ∀ν < λ .

Proof. — To any sectionσ ∈H⁰(M(V), Q^a) we associate the holomorphic function f on Isom(V,C^r)⊂(V^⋆)^r defined by

f(ζ₁^⋆, . . . , ζ_r^⋆) = (ζ₁^⋆)^a1 ⊗. . .⊗(ζ_r^⋆)^ar. σ([ζ1, . . . , ζr])

where (ζ₁, . . . , ζ_r) is the dual basis of (ζ₁^⋆, . . . , ζ_r^⋆) , and where the linear form induced by ζ_λ^⋆ on Qλ=Vλ−1/Vλ ≃Cζλ is still denotedζ_λ^⋆ . Let us observe that f is homogeneous of degreea_λin ζ_λ^⋆ and locally bounded in a neighborhood of every r–tuple of (V^⋆)^r\Isom(V,C^r) (becauseM(V) is compact andaλ≥0) . Therefore 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 Γ^aV , we see that S^kV = Γ(k,0, ... ,0)V , (2.14)

Λ^kV = Γ(1, ... ,1,0, ... ,0)V . (2.15)

For arbitrary a∈Z^r , proposition 2.13 remains true if we set

Γ^aV = Γ^{(a1−ar, ... ,ar−1−ar,0)}V ⊗(detV)^ar when a is non-increasing , Γ^aV = 0 otherwise .

The weights will be ordered according to their usual partial ordering : a<b iff X

1≤λ≤µ

a_λ≥ X

1≤λ≤µ

b_λ , 1≤µ≤r .

Bott’s theorem [3] shows that Γ^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 ⊂ Gl(V) on V^⊗k verify a1+. . .+ar = k and aλ ≥ 0 , we have a canonical Gl(V)–isomorphism

(2.16) V^⊗k= M

a1+...+ar=k a1≥...≥ar≥0

µ(a, k) Γ^aV

where µ(a, k)>0 is the multiplicity of the isotypical factor Γ^aV in V^⊗k .

Bott’s formula (cf. also Demazure [6] for a very simple proof) gives in fact the expression of all cohomology groups H^q(M(V), Q^a) . A weight a is called regular if all the elements ofa are distinct, andsingular 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

ba= a− c

2 ≥

+ c

2 ∈Z^r ;

the sequence ba is non-increasing if and only if a− ₂^c is regular.

Proposition 2.17 (Bott [3]). — One has H^q(M(V), Q^a) =

0 if q 6=l(a− ^c₂) , Γ^ˆaV if q =l(a− ^c₂) . In particular all H^q vanish if and only ifa− ^c₂ is singular.

We will be particularly interested by those line bundles Q^a such that the cohomology groups H^p,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. — SetN = dimM(V) , N(s) = dimM_s(V) . Then (a) H^p,q(M_s(V), Q^a_s) = 0 for all p+q≥N(s) + 1

as soon as a_sj −a_sj+1 ≥1 , 1≤j ≤m−1 . (b) H^q(M_s(V), Q^a_s) = 0 for all q ≥1

as soon as a_sj −a_sj+1 ≥1−(s_j+1−s_j−1) , 1≤j ≤m−1 .

(c) In general, H^p,q(M_s(V), Q^a−c(s)_s ) is isomorphic to a direct sum of irredu- cible Gl(V)–modules Γ^bV with

b1≥. . .≥br ≥min{aλ} −(N(s)−p) . (d) H^p,q(M_s(V), Q^a_s) = 0 for all q≥1 as soon as

asj −asj+1 ≥min{p , N(s)−p+ (sj+1−sj−1)−1, r+ 1−(sj+1−sj−1)} . (e) Under the assumption of (d), there is aGl(V)–isomorphism

H^p,0(M_s(V), Q^a_s) = M

u∈Zr

ν_s(u, p) Γ^a+uV ,

where ν_s(u, p) is the multiplicity of the weight u in Λ^pU_s .

Proof. — Under the assumption of (a), Q^a_s is ample by lemma 2.11 (c) . The result follows therefore from the Kodaira-Nakano-Akizuki theorem. Now (b) is a consequence of (a) since c(s)sj −c(s)sj+1 =−(sj+1−sj−1) and

H^q(M_s(V), Q^a_s) =H^N(s),q(M_s(V), Q^a−c(s)_s ) .

Let us now observe that for every vector bundle E on M_s(V) there are isomor- phisms

H^q(M_s(V), E)≃H^q(M(V), η^⋆E) , (2.19)

H^q(M_s(V), E)≃H^q+N−N(s)(M(V), Q^c−c(s)⊗η^⋆E) , (2.20)

where Q^c−c(s) is the relative canonical bundle along the fibers of the projection η : M(V) −→ M_s(V) . In fact, the fibers of η are products of flag manifolds.

For such a fiber F , we have H^q(F, KF) = H^dimF−q(F,O)⋆

= C when q = dimF = N −N(s) and H^q(F, K_F) = 0 otherwise (apply (b) with a = 0 and K¨unneth’s formula). We get thus direct image sheaves

R^qη_⋆(η^⋆E)

=

0 if q 6= 0 O(E) if q = 0 . R^qη_⋆ Q^c−c(s)⊗η^⋆E

=

0 if q 6=N −N(s) O(E) if q =N −N(s) .

Formulas (2.19) and (2.20) are thus immediate consequences of the Leray spectral sequence. When applied to E = Λ^pT^⋆Ms(V)⊗Q^a−c(s)s , formula (2.19) yields

H^p,q(M_s(V), Q^a−c(s)_s ) =H^q M(V), Q^a−c(s)⊗η^⋆Λ^pT^⋆M_s(V)

=H^q M(V), Q^a⊗η^⋆Λ^N(s)−pT M_s(V) ,

using the isomorphism Λ^pT^⋆M_s(V) ≃ K_Ms(V) ⊗Λ^N(s)−pT M_s(V) . In the same way, (2.20) implies

H^p,q(M_s(V), Q^a_s) =H^q+N−N(s) M(V), Q^a+c−c(s)⊗η^⋆Λ^pT^⋆M_s(V)

=H^q+N−N(s) M(V), Q^a+c⊗η^⋆Λ^N(s)−pT Ms(V) . The bundle η^⋆T^⋆Ms(V) = (Us)V,ad (resp. η^⋆T Ms(V) = (gl^ˇ:r/R^∪∪s)V,ad ) has a filtration with associated graded bundle

MQ⁻¹_λ ⊗Q_µ resp. M

Q_λ⊗Q⁻_µ¹

where the indices λ, µ are such that λ ≤ s_j < µ for some j . It follows that η^⋆Λ^pT^⋆M_s(V) resp.η^⋆Λ^N(s)−pT M_s(V)

has a filtration with associated graded

bundle M

u∈Zr

ν_s(u, p)Q^u resp. M

u^′∈Zr

ν_s^′(u^′, N(s)−p)Q^u′

where the weights u (resp. u^′) and multiplicities ν_s(u, p) (resp. ν_s^′(u^′, N(s)−p)) are those of Λ^pU_s (resp. Λ^N(s)−pgl^ˇ:_r/R^∪∪_s ) . We need a lemma.

Lemma 2.21. — The weights u of Λ^pU_s verify

u_µ−u_λ≤min{p+ 1, r+ 1−(µ−λ), N(s)−p+ (s_j+1−s_j−1)}

for s_j−1 < λ ≤s_j < µ≤s_j+1 , 1≤j ≤m−1 .

Proof. — Let us denote by (ε_λ)₁≤λ≤r the canonical basis of Z^r . The weights u (resp.u^′) are the sums ofp (resp. N(s)−p) distinct elements of the set of weights −ε_λ+ε_µ (resp.ε_λ−ε_µ ) whereλ≤s_j < µ for some j . It follows that u_µ−u_λ ≤r+ 1−(µ−λ) , with equality for the weight

u = (−ε1+εµ) +. . .+ (−ελ+εµ) + (−ελ+εµ+1) +. . .+ (−ελ+εr) . It is clear also that u_µ−u_λ ≤ p+ 1 and u^′_µ−u^′_λ ≤ N(s)−p . Since the weights u , u^′ are related byu =u^′+c(s) , lemma 2.21 follows.

Proof of(c). — By the filtration property (1.4) and the above formulas, we see thatH^p,q(M_s(V), Q^a−c(s)s ) is a direct sum of certain irreducible Gl(V)–modules

H^q M(V), Q^a+u′

= Γ^bV , b= (a+u^′)^∧= a− c

2 +u^′≥

+ c 2 . Clearly u^′_λ≥ −(N(s)−p) , thus

b_r ≥min{a_λ} −maxnc_λ 2

o−(N(s)−p) + c_r

2 = min{a_λ} −(N(s)−p) . Proof of (d). — Similarly, (d) is reduced by (1.4) to proving

H^q+N−N(s)(M(V), Q^a+u+c−c(s)) = 0

for all the above weights uand all q≥1 . By proposition 2.17 it is sufficient to get l a+u+ c

2 −c(s)

≤N −N(s) .

Let us observe that a weight h will be such that l(h) ≤ N −N(s) as soon as h_λ − h_µ ≥ 0 whenever s_j−1 < λ ≤ s_j < µ ≤ s_j+1 , 1 ≤ j ≤ m− 1 . For h =a+u+ ₂^c −c(s) this condition yields, thanks to lemma 2.21 :

a_λ−a_µ ≥min{p+1, r+1−(µ−λ), N(s)−p+(s_j+1−s_j−1)}+(µ−λ)−(s_j+1−s_j) . Taking µ−λ as large as possible,i.e.µ−λ=s_j+1−s_j−1−1 , we get the asserted condition.

Proof of(e). — Because of the vanishing of H¹ of the graded quotients, we obtain

H^p,0(Ms(V), Q^a_s) =M

u

νs(u, p)H^N−N(s) M(V), Q^a+u+c−c(s) .

Applying proposition 2.17, we get (a +u +c−c(s))^∧ = a+ue , where ue is the partial reordering of u such that only the coefficients in each interval [s_j−1+ 1, s_j] have been reordered (in non-increasing order). The number of inversions is always

≤ N −N(s) , with equality if and only if u is non-decreasing in each interval. In that case

H^N−N(s) M(V), Q^a+u+c−c(s)

= Γ^a+˜uV ,

and = 0 otherwise. Since Λ^pU_s is a Bs–module we have νs(u, p) = νs(u, p) .e Formula (e) is proved, and we see that non-zero terms correspond to weights u which are non-increasing in each interval [sj−1+ 1, sj] .

Remark 2.22. — If the manifold Ms(V) is a Grassmannian, then m = 2 , s = (0, s₁, r) . The condition required in proposition 2.21 is therefore always satisfied when ar−as1 ≥1 , i.e.when Q^a_s is ample.

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] .

LetX be an–dimensional compact complex manifold andE −→X a holo- morphic vector bundle of rankr . For every sequence 0 =s0 < s1 < . . . < sm =r, we associate to E its flag bundle Y = M_s(E) −→ X . If a ∈ Z^r is such that asj−1+1 =. . .= asj , 1 ≤j ≤ m , we may define a line bundle Q^a_s −→ Y just as we did in §2 . Let us set

Ω^p_X = Λ^pT^⋆X , Ω^p_Y = Λ^pT^⋆Y . One has an exact sequence

(3.1) 0−→π^⋆Ω¹_X −→Ω¹_Y −→Ω¹_{Y /X} −→0

where Ω¹_{Y /X} is by definition the bundle of relative differential 1–forms along the fibers of the projection π :Y =M_s(E)−→X . One may then define a decreasing filtration of Ω^t_Y as follows :

(3.2) F^p,t=F^p(Ω^t_Y) =π^⋆(Ω^p_X)∧Ω^t−p_Y .

The corresponding graded bundle is given by

(3.3) G^p,t =F^p,t/F^p+1,t =π^⋆(Ω^p_X)⊗Ω^t−p_{Y /X} .

Over any open subset of X where E is a trivial bundleX ×V with dimCV =r , the exact sequence (3.1) splits as well as the filtration (3.2). Using proposition 2.18 (a), (d), we obtain the following lemma.

Lemma 3.4. — For every weight a such that

(3.5)

(a_sj −a_sj+1 ≥1 if t =N(s) and otherwise

a_sj −a_sj+1 ≥min{t, N(s)−t+ (s_j+1−s_j)−1, r+ 1−(s_j+1−s_j−1)}

the sheaf of sections of Ω^t_{Y /X}⊗Q^a_s has direct images

(3.6)







R^qπ⋆ Ω^t_{Y /X} ⊗Q^a_s

= 0 for q≥1 π⋆ Ω^t_{Y /X} ⊗Q^a_s

=M

u

νs(u, t) Γ^a+uE .

We have in particular

(3.7) π_⋆(Q^a_s) = Γ^aE , π_⋆ Ω^N_{Y /X}^(s)⊗Q^a_s

= Γ^a+c(s)E .

Let L be an arbitrary line bundle on X . Under assumption (3.5), formulas (3.3) and (3.6) yield

R^qπ⋆(G^p,p+t⊗Q^a_s⊗π^⋆L) = 0 for q≥1 , π_⋆(G^p,p+t⊗Q^a_s⊗π^⋆L) =M

u

ν_s(u, t) Ω^p_X⊗Γ^a+uE ⊗L . The Leray spectral sequence implies therefore :

Theorem 3.8. — Under assumption (3.5), one has for all q ≥0 : H^q(Y, G^p,p+t⊗Q^a_s ⊗π^⋆L)≃M

u

νs(u, t) H^p,q(X,Γ^a+uE ⊗L) . The special case t=N(s) gives :

Corollary 3.9. — If asj −asj+1 ≥1 , then for all q ≥0 H^q(Y, G^p,p+N(s)⊗Q^a_s ⊗π^⋆L)≃H^p,q(X,Γ^a+c(s)E ⊗L) .

Whenp=n ,G^n,n+N(s) is the only non-vanishing quotient in the filtration of the canonical line bundle Ω^n+N(s)_Y . We thus obtain the following generalization of Griffiths’ isomorphism theorem [8] :

(3.10) H^n+N(s),q(M_s(E), Q^a_s⊗π^⋆L)≃H^n,q(X,Γ^a+c(s)E⊗L) .

4. Vanishing theorems.

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

Lemma 4.1. — Assume that a_s1 > a_s2 > . . . > a_sm ≥0 . Then (a) E semi-ample =⇒ Q^a_s semi-ample;

(b) E ample =⇒ Q^a_s ample;

(c) E semi-ample and L ample =⇒ Q^a_s ⊗π^⋆L ample.

Proof. —

(a) If E is semi-ample, then by definition (1.1) S^kE is spanned for k ≥ k₀ large enough. Hence Γ^kaE, which is a direct summand in S^ka1E⊗. . .⊗S^karE , is also spanned fork ≥k₀ . Since the fibers ofπ_⋆(Q^ka_s ) = Γ^kaE generateQ^ka_s , we conclude that Q^ka_s is spanned for k ≥k₀ .

(b) Similar proof, replacing “semi-ample” by “ample” and “spanned” by “very ample”. One needs moreover the fact that Q^ka_s is very ample along the fibers of π (lemma 2.11).

(c) If L^k is very ample for k ≥ k₁ , then Q^ka_s ⊗ π^⋆L^k is very ample for k ≥ max(k0, k1) , because Q^ka_s is spanned on Y and very ample along the fibers, whereas H⁰(Y, π^⋆L^k) = H⁰(X, L^k) separates points of Y which lie in distinct fibers.

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

Proof of theorem 0.2. — Let a∈Z^r be such that a1 ≥a2 ≥. . .≥ah > ah+1 =. . .=ar = 0 .

Define s₁ < s₂ < . . . < s_m−1 as the sequence of indices λ = 1, . . . , r−1 such that aλ > aλ+1 and set a^′ = a+ (h, . . . , h)−c(s) . The canonical weight c(s) is non-decreasing and c(s)_r =s_m−1 =h , s_m =r , hence

a^′_s1 > a^′_s2 > . . . > a^′_sm =h−c(s)_r = 0 ,

so Q^a_s^′ ⊗π^⋆L is ample by lemma 4.1 and Γ^a′+c(s)E = Γ^aE ⊗(detE)^h . Formula (3.10) yields

H^n,q(X,Γ^aE⊗(detE)^h⊗L)≃H^n+N(s),q(M_s(E), Q^a_s^′⊗π^⋆L) .

Since dimMs(E) =n+N(s) , the group in the right hand side is zero for q ≥1 by the Kodaira-Akizuki-Nakano vanishing theorem (1.2) .

Proof of theorem0.3. — The proof proceeds by backward induction onp. The case p = n is already settled by theorem 0.2. The decomposition formula (2.16) shows that the result is equivalent to

H^p,q(X,Γ^aE⊗(detE)^l⊗L) = 0 for p+q ≥n+ 1 , l ≥n−p+r−1 anda1 ≥. . .≥ar = 0 not all zero. Defines as above anda^′ =a+(l, . . . , l)−c(s) . Then Q^a_s^′ is ample since a_r =l−h≥0 , and corollary 3.9 implies

(4.2) H^p,q(X,Γ^aE⊗(detE)^l⊗L)≃H^q(Y, G^p,p+N(s)⊗Q^a_s^′ ⊗π^⋆L) .

Now, it is clear that F^p,p+N(s) = Ω^p+N(s)_Y . We get thus an exact sequence 0−→F^p+1,p+N(s) −→Ω^p+N(s)_Y −→G^p,p+N(s) −→0 .

The Kodaira-Akizuki-Nakano vanishing theorem (1.2) applied to Q^a_s^′ ⊗π^⋆L with dimY =n+N(s) yields

H^q(Y,Ω^p+N(s)_Y ⊗Q^a_s^′ ⊗π^⋆L) = 0 for p+q ≥n+ 1 . The cohomology groups in (4.2) will therefore vanish if and only if (4.3) H^q+1(Y, F^p+1,p+N(s)⊗Q^a_s^′⊗π^⋆L) = 0 .

Let us observe that F^p+1,p+N(s) has a filtration with associated graded bundle L

t≥1G^p+t,p+N(s) . In order to verify (4.3), it is thus enough to get (4.4) H^q+1(Y, G^p+t,p+N(s)⊗Q^a_s^′ ⊗π^⋆L) = 0 , t≥1 .

At this point, the Leray spectral sequence will be used in an essential way. Since G^p+t,p+N(s) =π^⋆(Ω^p+t_X )⊗Ω^N_{Y /X}^(s)−t , we get

R^kπ_⋆(G^p+t,p+N(s)⊗Q^a_s^′⊗π^⋆L) = Ω^p+t_X ⊗L⊗R^kπ_⋆ Ω^N(s)_{Y /X}^−t ⊗Qa+(l, ... ,l)−c(s) . Proposition 2.18 (a) and (c) yields

R^kπ_⋆ Ω^N(s)−t_{Y /X} ⊗Qa+(l, ... ,l)−c(s)

=

0 for k≥t+ 1 , otherwise L

b Γ^bE , br ≥l−t , R^kπ⋆(G^p+t,p+N(s)⊗Q^a_s^′ ⊗π^⋆L) =

0 for k≥t+ 1 , otherwise L

b,m Ω^p+t_X ⊗Γ^bE ⊗(detE)^m⊗L , where the last sum runs over weights bsuch that br = 0 and integersm such that m ≥ l−t ≥ n−(p+t) +r −1 . Therefore H^q+1(Y, G^p+t,p+N(s) ⊗Q^a_s^′ ⊗π^⋆L) has a filtration whose graded module is the limit term L

jE_∞^j,q+1−j of a spectral sequence such that

E₂^j,k =

0 for k≥t+ 1 , otherwise L

b,m H^p+t,j(X,Γ^bE⊗(detE)^m⊗L) , m≥n−(p+t) +r−1 . We have thus E₂^j,q+1−j = 0 for j ≤ q−t by the first case, and also for j ≥ q−t by the second case and the induction hypothesis. Hence E_∞^j,q+1−j = 0 for all j , and (4.4) is proved.

Proof of theorem 0.6. — Let us take herea = (2,0, . . . ,0) ands= (0,1, r) , so that Y = Ms(E) = P(E^⋆) . Since Γ^aE = S²E and Γ(1,1,0, ... ,0)E = Λ²E , theorem 3.8 yields

(4.5)







H^q(Y, G^p+1,p+1⊗Q²_s,1⊗π^⋆L)≃H^p+1,q(X, S²E) , H^q(Y, G^p,p+1⊗Q²_s,1⊗π^⋆L)≃H^p,q(X,Λ²E) , H^q(Y, G^k,p+1⊗Q²_s,1⊗π^⋆L) = 0 for k < p ,

because u = −ε₁ + ε₂ is the only weight of Λ¹U_s such that a + u is non- increasing, and because no such weights exist for higher exterior powers Λ^jU_s . Now, Ω^p+1_Y /F^p,p+1 has a filtration with graded quotientsG^k,p+1,k < p , therefore

H^q(Y,Ω^p+1_Y /F^p,p+1⊗Q²_s,1⊗π^⋆L) = 0 for all q .

Considering the exact sequences

0−→G^p,p+1 −→Ω^p+1_Y /F^p+1,p+1 −→Ω^p+1_Y /F^p,p+1 −→0 , 0−→G^p+1,p+1 −→Ω^p+1_Y −→Ω^p+1_Y /F^p+1,p+1 −→0 , we get an isomorphism

H^q(Y, G^p,p+1⊗Q²_s,1⊗π^⋆L)≃H^q(Y,Ω^p+1_Y /F^p+1,p+1⊗Q²_s,1⊗π^⋆L) and a canonical coboundary morphism

H^q(Y,Ω^p+1_Y /F^p+1,p+1⊗Q²_s,1⊗π^⋆L)−→H^q+1(Y, G^p+1,p+1⊗Q²_s,1⊗π^⋆L) which by Kodaira-Akizuki-Nakano is onto for p+ 1 +q+ 1 ≥ n+N(s) + 1 , i.e.

p+q ≥n+r−2 , and bijective for p+q ≥n+r−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 condition l ≥h in theorem 0.2 is best possible.

Example 4.6. — Let X = G_r(V) be the Grassmannian of subspaces of codimension r of a vector space V , dim^CV =d , andE the tautological quotient vector bundle of rank r over X . Then E is spanned (hence semi-ample) and L = detE is very ample. According to the notations of§2 , we have

X =M_s(V) , s= (s₀, s₁, s₂) = (0, r, d) , E =V /V_r , detE =Q_s,1 , detV_r =Q_s,2 , KX =Q^r−d_s,1 ⊗Q^r_s,2 .

Furthermore, for any sequence a₁ ≥. . .≥a_r ≥0 , we have Γ^aE = Γ^a(V /V_r) =η_⋆Q^(a,0)

where η : M(V) −→ Ms(V) is the projection. If n = dimX = r(d −r) , this implies

H^n,q X,Γ^aE⊗(detE)^h

=H^q X, Q^h+r−d_s,1 ⊗Q^r_s,2⊗η_⋆Q^(a,0)

=H^q(X, η⋆Q^α)

where α = (a1 +h +r−d, . . . , ar +h+r−d, r, . . . , r) ∈ Z^d . The fiber of η is M(V /V_r)×M(V_r) , hence R^qη_⋆Q^α = 0 for q ≥ 1 by K¨unneth’s formula and proposition 2.18 (b) . We obtain therefore

H^n,q X,Γ^aE⊗(detE)^h

=H^q(M(V), Q^α) .

Assume now that a₁ ≥ . . . ≥ a_h ≥ d −r and a_h = . . . = a_r = 0 . Define 1 = (1, . . . ,1)∈Z^d . Then

α− c

2 = 1−d

2 ·1+ (a₁+r+h−1, . . . , a_h+r; r−1, . . . , h; d−1, . . . , r) where dots indicate a decreasing sequence of consecutive integers. There are exactly (r−h)(d−r) inversions of the ordering, corresponding to the inversion of the last two blocks between semicolons. Then one finds easily

α− c 2

≥

+ c

2 = (a₁+h+r−d, . . . , a_h+h+r−d; h, . . . , h) =β+h1 where β = (a1+r−d, . . . , ar+r−d; 0, . . . ,0) . Proposition 2.17 yields therefore

H^n,q X,Γ^aE⊗(detE)^h

=

0 if q 6= (r−h)(d−r) Γ^βV ⊗(detV)^h if q = (r−h)(d−r) .