On branched coverings of some homogeneous spaces

Meeyoung Kim, Laurent Manivel

Dedicated to the memory of Prof. Michael Schneider

Abstract. We study nonsingular branched coverings of a homogeneous space X. There is a vector bundle associated with such a covering which was conjectured by O. Debarre to be ample when the Picard number of X is one. We prove this conjecture, which implies Barth-Lefschetz type theorems, for lagrangian grassmannians, and for quadrics up to dimension six. We propose a conjectural extension to homogeneous spaces of Picard number larger than one and prove a weaker version.

1 Introduction and main results

Let f : Y → ℙ^{n} be a branched covering of degree d of a complex projective space where Y is a nonsingular connected complex projective variety. A celebrated result of R. Lazarsfeld [11] states the following:

the induced morphism f_{∗} : H^{i}(ℙ^{n}, ℂ) → H^{i}(Y, ℂ) is an isomorphism for i ≤ n − d + 1.

There is a natural vector bundle E on ℙ^{n} of rank d − 1 associated with f, defined by the splitting

f_{∗}O_{Y} ≅ O_{ℙ^{n}} ⊕ E^{∗}

induced by the trace homomorphism. The map f factors through an embedding of Y in the total space |E| of E. Lazarsfeld proved that E is always ample.

This result implies in particular W. Barth’s theorem [2] for the cohomology of small codimensional subvarieties of projective spaces. Since Barth’s fundamental paper, much attention has been paid to small codimension subvarieties of more general complex projective homogeneous spaces [19, 5, 1]. This and the result of Lazarsfeld lead to the question of understanding low degree branched coverings of such spaces X = G/P, where G is a semisimple complex Lie group and P is a parabolic subgroup. Note that the codimension cod (X, |E|) = d−1. A recent paper of O. Debarre [4] gives strong evidence to the following conjecture:

Conjecture. Let f : Y → X = G/P be a branched covering where Y is a nonsingular connected complex projective variety, and P ic(X) = ℤ. Then the associated vector bundle E on X is ample.

The assumption P ic(X) = ℤ simply means that P is a maximal parabolic subgroup. For G = SL(n, ℂ), the homogeneous space G/P is then a grassmannian. The first half of the way to the conjecture in that case was made in [9], the second half in [14]. The purpose of this paper is to prove the conjecture for a few other cases. Our main result is the following:

Theorem A. Let 𝕃𝔾_{n} be the lagrangian grassmannian of maximal isotropic subspaces of a 2n-dimensional symplectic vector space. Then if f : Y → 𝕃𝔾_{n} is a branched covering where Y is a nonsingular connected complex projective variety, then the associated vector bundle E on 𝕃𝔾_{n} is ample.

Here, 𝕃𝔾_{n} is homogeneous with G = Sp(2n, ℂ), the symplectic group; with the exception of ℙ^{2n−1}, 𝕃𝔾_{n} is the only homogeneous space for the symplectic group which is a hermitian symmetric space. Using Lazarsfeld’s ideas, we immediately deduce from Theorem A the following Barth-Lefschetz type theorem:

Theorem B. Let f : Y → 𝕃𝔾_{n} be a branched covering of degree d where Y is a nonsingular connected complex projective variety. Then the induced morphism

f^{∗} : H^{i}(𝕃𝔾_{n}, ℂ) → H^{i}(Y, ℂ)

is an isomorphism for i ≤ dim 𝕃𝔾_{n} − d + 1.

The cohomological range of A. Sommese’s Barth-Lefschetz type theorem [18, 19] concerning a submanifold of a homogeneous space X = G/P depends on the k-ampleness (this is a suitable weakening of the notion of ampleness) of the normal bundle of the submanifold. To boot, by considering N. Goldstein’s computation [7] of the k-ampleness of the tangent bundle of X, we can see that the codimension above which one loses all cohomological information depends linearly on the rank of G, that is on the dimension of a maximal torus contained in G; this is reminiscent of G. Faltings’ connectedness result for homogeneous spaces (see [5, 7] for the details). Here, our condition involves the dimension of X = G/P, which is in general much larger, although in the case of ℙ^{n} this makes no difference.

After the one of projective spaces, the easiest case should be the case of quadrics. Surprisingly enough, it does not seem to be so, despite partial results of the first author [10]. Actually, we have been able to prove the conjecture only for quadrics of very small dimension:

Theorem C. Let ℚ_{n} be the n-dimensional nonsingular quadric with 3 ≤ n ≤ 6. If f : Y → ℚ_{n} is a branched covering where Y is a nonsingular connected complex projective variety, then the associated vector bundle E on ℚ_{n} is ample.

Finally, let us mention that the conjecture by Debarre should be extended to the cases of homogeneous spaces X = G/P with P not necessarily maximal. In this case, one cannot expect the vector bundle associated with a branched covering to be ample. Nevertheless, it will be k-ample for a suitable value k; this implies Barth-Lefschetz type theorems. In section 5.3 we propose a related conjecture and give some evidence to it by proving the following result:

Theorem D. Let f : Y → X = G/P be a branched covering where Y is a nonsingular connected complex projective variety, and G = SL(n, ℂ) or Sp(2n, ℂ). Then the associated vector bundle E on X is generated by its global sections.

Actually, as in [11] we prove a stronger positivity property for E on X = G/P with Picard number one: E⊗ O_{X}(−1) is generated by its global sections where O_{X}(1) is the (very) ample generator of P ic(X). To show this we set up an appropriate Castelnuovo-Mumford type criterion on X and use vanishing theorems.

In the case of lagrangian grassmannians, we have been obliged to get out of the algebraic category, and instead we use vanishing theorems “à la Nakano” for hermitian vector bundles. In section 2, we will recall a few elementary facts on the curvature of hermitian bundles, and state a vanishing theorem that will be suitable to our purposes. With the help of a formula due to D. Snow for the curvature of homogeneous bundles, we will show in section 3 how to apply this theorem to the proof of Theorem A. In section 4 we deal with low dimensional quadrics, where we make use of the so-called spinor bundles to obtain Theorem C. Section 5 is devoted to the proof of Theorem D, which is simple on ordinary flags, and much more involved in the symplectic case.

The first author thanks the Max-Planck-Institut für Mathematik in Bonn for its warm hospitality during the preparation of part of this paper.

2 On the curvature of hermitian vector bundles

2.1 Griffiths and Nakano positivity

Let E be a hermitian vector bundle on an n-dimensional nonsingular complex variety X. The curvature Θ_{E} of the associated Chern connection is a (1, 1)-form with values in the vector space of hermitian endomorphisms of E. For every point z ∈ X and every local coordinate system (z_{j})_{1≤j≤n} at z, the curvature can be written as

Θ_{E} = ∑_{jkλμ} c_{jkλμ}dz_{j} ∧ dz̄_{k}⊗ e_{λ}^{∗}⊗ e_{μ},

with c_{jkλμ} = c̄_{kjμλ}, where (e_{1}, …, e_{r}) is a local frame of E in a neighborhood of z. One may consider Θ_{E} as a hermitian form on T X⊗ E:

Θ_{E}(u) = ∑_{jkλμ} c_{jkλμ}(z)u_{jλ}ū_{kμ}

for u = ∑_{jλ} u_{jλ}(∂/∂z_{j})⊗ e_{λ} ∈ T_{z}X⊗ E_{z}.

Definition. The hermitian vector bundle E is Nakano (semi-)positive if Θ_{E} defines a (semi-)positive definite hermitian form on T X⊗ E. It is Griffiths (semi-)positive if Θ_{E} is (semi-)positive on non-zero decomposable tensors in T X⊗ E.

Nakano vanishing theorem states that the adjoint to a Nakano positive vector bundle on a compact Kähler manifold has no higher cohomology:

Theorem 2.1 (cf. [15]) Let E be a hermitian vector bundle on a compact Kähler manifold. Suppose that E is Nakano semi-positive, and that Θ_{E} is positive definite at least at one point. Then

H^{q}(X, K_{X}⊗ E) = 0 ∀q > 0.

2.2 A variant of Griffiths’ vanishing theorem

We will need in the sequel of this paper a consequence of the Nakano vanishing theorem, which can also be seen as a variant of Griffiths’ vanishing theorem [8]. We will state it in terms of Schur powers.

Let V be a finite dimensional complex vector space. To each partition λ, i.e. to every non-increasing sequence of non-negative integers λ_{1} ≥ ⋯ ≥ λ_{m} ≥ 0 (=:the parts of λ), one associates a Schur power S_{λ}V, which is a polynomial Gl(V)-module [13]. This module reduces to zero if and only if the number of nonzero parts of λ (=:the length l(λ)) is larger than the dimension of V. Otherwise, S_{λ}V is an irreducible Gl(V)-module.

The usual symmetric powers are a special case of Schur powers. They correspond to partitions with a single nonzero part, or length one partitions. Also the wedge powers correspond to partitions with all nonzero parts equal to one. The tensor product of any Schur power with a symmetric or a wedge power is described by the classical Pieri’s rules [13], which imply, in particular, that each Schur power S_{λ}V is an irreducible component of the tensor product of l(λ) symmetric powers (the exponents of which can be chosen as the parts of the partition λ).

Proposition 2.2 Let X be a compact Kähler manifold, and E, F be hermitian vector bundles on X. Suppose that E is Nakano semi-positive, and that Θ_{E} is positive definite at least at one point. Suppose moreover that F is Griffiths semi-positive. Then for any partition λ,

H^{q}(X, K_{X}⊗ S_{λ}F ⊗ (det F)^{l(λ)}⊗ E) = 0 ∀q > 0.

Proof. We first note that it is enough to treat the case of l(λ) = 1, that is, of symmetric powers. Indeed, if we can prove the vanishing for one symmetric power, then by replacing F by F^{⊕l} where l = l(λ), which is also Griffiths semi-positive, we deduce the vanishing for a tensor product of l symmetric powers. Note that any Schur power S_{λ}E is a direct summand of such a tensor product.

Now let π : Y = ℙ(F^{∗}) ⟶ X be the bundle of hyperplanes of F, and O_{F} (1) be the tautological quotient line bundle on Y. By the isomorphisms [8]

H^{q}(X, K_{X}⊗ S^{l}F ⊗ det F ⊗ E) ≅ H^{q}(Y, K_{Y} ⊗ O_{F} (l + r)⊗ π^{∗}E).

We want to show that O_{F} (m)⊗ π^{∗}E is Nakano semi-positive, and its curvature is positive definite at least at one point, for any m > 0, so that we can apply Nakano vanishing theorem. Of course we endow this vector bundle with two obvious metrics: on the one hand by the given metric on E , which we simply pull back to one on π^{∗}E, on the other hand by the pulled-back metric on π^{∗}F, which provides a metric on its quotient line bundle O_{F} (1). Note that, since Griffiths semi-positivity is preserved by pull-backs and quotients, this metric on O_{F} (1) has Griffiths semi-positive curvature. Moreover, the Fubini-Study metric being positive on the projective space, it has positive curvature on the fibers of π, that is, on “vertical” tangent vectors.

Now the induced metric on O_{F} (m)⊗ π^{∗}E has curvature

Θ_{O_{F} (m)⊗ π^{∗}E} = mΘ_{O_{F} (1)}⊗ Id_{π^{∗}E} + π^{∗}Θ_{E}.

Let us choose a local frame (e_{1}, ..., e_{r}) of E in a neighborhood of x = π(y) where y ∈ Y. Then if u = ∑_{i=1}^{r} u_{i}⊗ e_{i} ∈ T_{y}Y ⊗ E_{π(y)}, we have

Θ_{O_{F} (m)⊗ π^{∗}E}(u) = m ∑_{i=1}^{r} Θ_{O_{F} (1)}(u_{i}) + Θ_{E}(π_{∗}u).

This is a sum of non-negative terms. Moreover, suppose that Θ_{E} is positive definite at x = π(y). Then if Θ_{O_{F} (m)⊗ π^{∗}E}(u) is equal to zero, we must have π_{∗}u = 0. This means that u has only vertical components, and since Θ_{O_{F} (1)} is positive definite on the fibers of π, we must have u_{i} = 0 for each i, hence u = 0. Thus Θ_{O_{F} (m)⊗ π^{∗}E} is positive definite at y, which proves our claim. □

We will use the following vanishing theorem in the sequel, which is a slight generalization of Le Potier vanishing theorem [12]:

Proposition 2.3 Let E_{1}, …, E_{m} be globally generated vector bundles of ranks e_{1}, …, e_{m}, and L be an ample line bundle, on a nonsingular complex projective variety X. Then

H^{q}(X, K_{X}⊗ ∧^{e_{1}−k_{1}}E_{1}⊗ ⋯ ⊗ ∧^{e_{m}−k_{m}}E_{m}⊗ L) = 0

for q > k_{1} + ⋯ + k_{m}.

2.3 The curvature of homogeneous bundles

We now consider the very specific case of a homogeneous vector bundle E = G ×_{P} E_{0} of rank r on a generalized flag manifold X = G/P, where P is a parabolic subgroup of a semisimple complex Lie group G. Here E_{0} is a finite dimensional P -module.

Assume that E is generated by its global sections so that the evaluation P -module homomorphism

ϕ : V = H^{0}(X, E) → E_{0}

is surjective. For a given hermitian metric on V, we have an induced metric on the fiber E_{0}, and a left-invariant hermitian metric on E.

The curvature of this metric was computed by D. Snow [17]. We need some notation to state his formula. Firstly, let us fix a maximal torus and a Borel subgroup in G, from which we deduce a root system for its Lie algebra. Up to conjugacy, the parabolic subgroup P of G is then generated by this Borel subgroup, and a closed set of certain positive roots. The opposite of these roots form a set I of simple roots, and Φ_{X}, called the roots of X, is then defined as the set of positive roots which, when decomposed as a linear combination of simple roots, have positive coefficients on the roots in I.

Let v_{1}, …, v_{m} be a basis of root vectors for V and e_{1}, …, e_{r} be a basis for E_{0}, such that

ϕ(v_{k}) = \{ \begin{matrix}e_{k} & for 1 ≤ k ≤ r, \\ 0 & for r < k ≤ m. \end{matrix}

Proposition 2.4 The curvature of the hermitian metric induced on E by the evaluation morphism ϕ, is given at the identity coset by the following formula:

Θ_{E} = ∑_{α,β∈Φ_{X}} ∑_{k>r} dx_{α}∧dx̄_{β}⊗ ϕ(X_{α} · v_{k})⊗ \overline{ϕ(X_{β} · v_{k})^{t}},

where transposition is taken with respect to the given basis.

Remark. Being induced by a constant metric on a trivial vector bundle, this metric on E is Griffiths semi-positive. But in general, is neither Griffiths positive, nor Nakano semipositive and a fortiori not even Nakano positive. It would be useful to find conditions on E_{0} ensuring that the induced metric on E has some of these properties.

3 Symplectic grassmannians

In this section, we consider a symplectic grassmannian 𝕃𝔾_{n}, defined as the subvariety of the usual grassmannian 𝔾_{n,2n}, consisting of maximal isotropic spaces in ℂ^{2n} with respect to a fixed symplectic form. For example, 𝕃𝔾_{1} = ℙ^{1}, and 𝕃𝔾_{2} is a hyperplane section of 𝔾_{2,4} in its Plücker embedding, hence a three dimensional quadric. In general, 𝕃𝔾_{n} is a hermitian symmetric space of dimension n(n + 1)/2, homogeneous under the action of the symplectic group Sp(2n, ℂ). If we write 𝕃𝔾_{n} = Sp(2n, ℂ)/P, then the parabolic subgroup P is defined (up to conjugacy) by the (unique) long simple root.

3.1 A Castelnuovo-Mumford criterion

We will need a Castelnuovo-Mumford criterion on the symplectic grassmannian 𝕃𝔾_{n}, that is, a cohomological criterion for a coherent sheaf on 𝕃𝔾_{n} to be generated by its global sections. On the usual grassmannian, such a criterion is deduced in [9] and [14] from the simple fact that a general section of a some homogeneous bundle (precisely, the sum of a suitable number of copies of the tautological quotient vector bundle) vanishes precisely at one point. Unfortunately, we cannot give the same argument on symplectic grassmannians (see [6] for a discussion of closely related questions). Moreover, previously mentioned Castelnuovo-Mumford criterion on 𝔾_{n,2n}, which we may also use on the subvariety 𝕃𝔾_{n}, will not be good enough for our purposes.

Let Q_{n} be the tautological quotient bundle of rank n on 𝕃𝔾_{n}. We define

L_{i}^{n,j} = \{ \begin{matrix}∧^{i}Q_{n}^{∗} & if & i < j, \\ O_{𝕃𝔾_{n}}(−1) & if & i = j, \\ 0 & if & i > j. \end{matrix}

Our Castelnuovo-Mumford criterion on 𝕃𝔾_{n} will be the following:

Proposition 3.1 Let F be a coherent sheaf on 𝕃𝔾_{n}. Then F is generated by its global sections as soon as

H^{q}(𝕃𝔾_{n}, L_{i_{1}}^{n,1}⊗ ⋯ ⊗ L_{i_{n}}^{n,n}⊗ F) = 0

for q ≥ i_{1}+ ⋯ +i_{n} > 0 and q > i_{1}+ ⋯ +i_{n} = 0.

Remark. In the following proof and in the sequel, we will repeatedly use the following simple observation: for a coherent sheaf G on a variety X, let R_{•} → G → 0 be a finite resolution of G indexed by non-negative integers. Then the cohomology group H^{q}(X, G) vanishes as soon as

H^{q+m}(X, R_{m}) = 0

for all m ≥ 0. This can be easily seen by breaking the resolution into short exact sequences, and inspecting the corresponding long exact sequences of cohomology groups. Similarly, if we have a resolution 0 → G → S_{•}, then H^{q}(X, G) vanishes as soon as H^{q−m}(X, S_{m}) = 0 for all m ≥ 0.

Proof. We proceed by induction on n. The space of global sections of Q_{n} is ℂ^{2n}, and the zero-locus of the section of Q_{n} corresponding to some non-zero vector v is the space of maximal isotropic subspaces of ℂ^{2n} containing v. This locus can be identified with the lagrangian grassmannian 𝕃𝔾_{n−1} of maximal isotropic spaces for the induced symplectic form on v^{⊥}/v ≃ ℂ^{2n−2}. Moreover, these loci cover 𝕃𝔾_{n} v as varies. To prove that F is generated by its global sections on 𝕃𝔾_{n}, it is therefore enough to check that its restriction to any such 𝕃𝔾_{n−1} is generated by its global sections, and the surjectivity of the restriction map

H^{0}(𝕃𝔾_{n}, F) ⟶ H^{0}(𝕃𝔾_{n−1}, F|_{𝕃𝔾_{n−1}}).

By the induction hypothesis, F|_{𝕃𝔾_{n−1}} will be generated by its global sections as soon as

H^{q}(𝕃𝔾_{n−1}, L_{i_{1}}^{n−1,1}⊗ ⋯ ⊗ L_{i_{n−1}}^{n−1,n−1}⊗ F|_{𝕃𝔾_{n−1}}) = 0

for q ≥ i_{1}+ ⋯ +i_{n−1} > 0 and q > i_{1}+ ⋯ +i_{n−1} = 0. From the short exact sequence

0 → O_{𝕃𝔾_{n−1}} → Q_{n}^{∗}|_{𝕃𝔾_{n−1}} → Q_{n−1}^{∗} → 0,

we get long exact sequences

0 → O_{𝕃𝔾_{n−1}} → Q_{n}^{∗}|_{𝕃𝔾_{n−1}} → ∧^{2} Q_{n}^{∗}|_{𝕃𝔾_{n−1}} → ⋯ → ∧^{k} Q_{n}^{∗}|_{𝕃𝔾_{n−1}} → ∧^{k} Q_{n−1}^{∗} → 0,

from which we deduce that the proceeding vanishing will hold if

H^{q}(𝕃𝔾_{n−1}, (L_{j_{1}}^{n,1}⊗ ⋯ ⊗ L_{j_{n−1}}^{n,n−1}⊗ F)|_{𝕃𝔾_{n−1}}) = 0

for q ≥ j_{1}+ ⋯ +j_{n−1} > 0 and q > j_{1}+ ⋯ +j_{n−1} = 0. Now we compare cohomology groups of coherent sheaves on 𝕃𝔾_{n}, and of their restriction to 𝕃𝔾_{n−1}, thanks to the exact Koszul complex of the section of Q_{n} defining 𝕃𝔾_{n−1}:

0 → O_{𝕃𝔾_{n}}(−1) → ∧^{n−1}Q_{n}^{∗} → ⋯ → Q_{n}^{∗} → O_{𝕃𝔾_{n}} → O_{𝕃𝔾_{n−1}} → 0.

The above vanishing on 𝕃𝔾_{n−1} will thus be a consequence of the following ones on 𝕃𝔾_{n}:

H^{q}(𝕃𝔾_{n}, L_{j_{1}}^{n,1}⊗ ⋯ ⊗ L_{j_{n−1}}^{n,n−1}⊗ L_{j_{n}}^{n,n}⊗ F) = 0

for q ≥ j_{1}+ ⋯ +j_{n} > 0 and q > j_{1}+ ⋯ +j_{n} = 0. This is precisely our criterion. Finally, again, by the use of the above Koszul complex, the surjectivity of the restriction map

H^{0}(𝕃𝔾_{n}, F) ⟶ H^{0}(𝕃𝔾_{n−1}, F|_{𝕃𝔾_{n−1}})

is a consequence of the vanishing of H^{q}(𝕃𝔾_{n}, L_{q}^{n,n}⊗ F) for q > 0, which is also included in our criterion. □

3.2 The key observation

If we try to apply the previous Castelnuovo-Mumford criterion on 𝕃𝔾_{n} to E(−1) where E is the vector bundle associated to some finite covering, the classical vanishing theorem of Griffiths or Le Potier will not suffice. Actually, we will need our variant of Griffiths vanishing 2.2, which involves a Nakano positive vector bundle. Our key observation is the following:

Proposition 3.2 The vector bundle Q_{n}(1) on 𝕃𝔾_{n} is Nakano positive.

Remark. Note that the corresponding statement for the tautological quotient bundle Q on a usual grassmannian is false. However, since Q is a quotient bundle of a trivial bundle which is Griffiths semi-positive, it follows from general properties of hermitian bundles that Q(1) = Q⊗ det Q is Nakano semi-positive.

Proof. Let J = ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ) be the 2n × 2n matrix associated with a symplectic form

on ℂ^{2n}, where I_{n} is the identity matrix. We follow the conventions and notations of [3]. In particular, we denote by (e_{1}, …, e_{n}, e_{−n}, …, e_{−1}) the canonical basis of ℂ^{2n}. We have the roots of 𝕃𝔾_{n},

Φ_{𝕃𝔾_{n}} = {ε_{i} + ε_{j} : 1 ≤ i ≤ j ≤ n},

and the the following corresponding root vectors:

X_{ij} = X_{ε_{i}+ε_{j}} = E_{i,−j} + E_{j,−i} for 1 ≤ i < j ≤ n,
X_{ii} = X_{2ε_{i}} = E_{i,−i}  for 1 ≤ i ≤ n.

Let us apply Snow’s formula 2.4 to Q_{n}, at the point of 𝕃𝔾_{n} given by the maximal isotropic subspace of ℂ^{2n} generated by e_{1}, …, e_{n}. The vectors v_{m} = e_{−m}, for 1 ≤ m ≤ n, form a basis of the kernel of the evaluation map, and we have

∑_{1≤i≤j≤n} ϕ(X_{ij} · v_{m}) dx_{ij} = ∑_{1≤i<j≤n}(δ_{jm}e_{i} + δ_{im}e_{j}) dx_{ij} + e_{m} dx_{mm}

= ∑_{i=1}^{n} e_{i} dx_{im}

by adopting the notation dx_{ij} = dx_{ji}. We then obtain the curvature forms

Θ_{Q_{n}} = ∑_{1≤i,j,k≤n} dx_{ik} ∧ dx̄_{jk}⊗ e_{i}⊗ e_{j}^{∗},
Θ_{O_{𝕃𝔾_{n}}(1)} = Trace Θ_{Q_{n}} = ∑_{1≤i,k≤n} dx_{ik} ∧ dx̄_{ik}.

The second of these formulae is a consequence of the first since O_{𝕃𝔾_{n}}(1) = det Q_{n}. Then we have the hermitian form

Θ_{Q_{n}(1)}(u) = ∑_{1≤i,j,k≤n}( u_{jk}^{i} \overline{u}_{ik}^{j} + |u_{ik}^{j}|^{2} )

where u_{jk}^{i} = u_{kj}^{i}. We can rewrite this form as

Θ_{Q_{n}(1)}(u) = ∑_{i<j<k} Θ_{ijk}(u) + ∑_{i≠j} Θ_{ij}(u) + ∑_{i} Θ_{i}(u),

where the different terms of this sum are hermitian forms involving disjoint sets of components of u. Specifically,

Θ_{ijk}(u) = 2 (|u_{jk}^{i}|^{2} + |u_{ik}^{j}|^{2} + |u_{ij}^{k}|^{2})+
+u_{jk}^{i}ū_{ik}^{j} + ū_{jk}^{i}u_{ik}^{j} + u_{jk}^{i}ū_{ij}^{k} + ū_{jk}^{i}u_{ij}^{k} + u_{ik}^{j}ū_{ij}^{k} + ū_{ik}^{j}u_{ij}^{k}
= |u_{jk}^{i} + u_{ik}^{j} + u_{ij}^{k}|^{2} + |u_{jk}^{i}|^{2} + |u_{ik}^{j}|^{2} + |u_{ij}^{k}|^{2}.

It is a positive definite hermitian form in the three variables u_{ij}^{k}, u_{jk}^{i} and u_{ik}^{j}, while

Θ_{ij}(u) = |u_{jj}^{i} + u_{ij}^{j}|^{2} + 2 |u_{ij}^{j}|^{2}, Θ_{i}(u) = 2 |u_{ii}^{i}|^{2}

are positive definite hermitian forms in u_{ij}^{j}, u_{jj}^{i} and u_{ii}^{i}, respectively. This clearly implies that the hermitian form Θ_{Q_{n}(1)} is positive definite, and the proposition is proved. □

3.3 A lemma on tensor products

Before proving Theorem A, we need the following simple remark on Schur powers, which is an easy consequence of Pieri’s rules:

Lemma 3.3 Let S_{λ}V be an irreducible component of a tensor product of m symmetric powers S^{k_{1}}V ⊗ ⋯ ⊗ S^{k_{m}}V. Then the length of λ is at most m, and if it is equal to m, S_{λ}V is also an irreducible component of the tensor product ∧^{m}V ⊗ S^{k_{1}−1}V ⊗ ⋯ ⊗ S^{k_{m}−1}V.

Remark. More precisely, S_{λ}V has the same multiplicity in both tensor products, but we do not need this fact in this context.

Proof. Because of Pieri’s rules, S_{λ}V is a component of S^{k_{1}}V ⊗ ⋯ ⊗ S^{k_{m}}V if and only if there exists a semistandard tableau T of shape λ and weight κ = (k_{1}, ..., k_{m}): that is, a numbering of the diagram of the partition λ, weakly increasing on rows, strictly increasing on columns such that each integer i occurs k_{i} times; see [13].

The length of λ is the length of its first column. Since this column is numbered in a strictly increasing way by integers not exceeding m, its length is certainly bounded by m. If it is equal to m, this column must be numbered by 1, …, m. If we remove the first column, we get a new semistandard tableau S, of shape μ and weight (k_{1} − 1, …, k_{m} − 1). Since S_{λ}V is an irreducible component of ∧^{m}V ⊗ S_{μ}V, the lemma is proved. □

3.4 Proof of Theorem A

Let f : Y ⟶ 𝕃𝔾_{n} be a degree d covering, and E be the associated vector bundle on 𝕃𝔾_{n} of rank d−1. We want to show that E(−1) is generated by its global sections. Using Serre duality and the fact that K_{𝕃𝔾_{n}} = O_{𝕃𝔾_{n}}(−n − 1), we see that our Castelnuovo-Mumford criterion is equivalent to the vanishings of

H^{N−q}(𝕃𝔾_{n}, (L_{i_{1}}^{n,1}⊗ ⋯ ⊗ L_{i_{n}}^{n,n})^{∗}⊗ E^{∗}(−n))

for q ≥ i_{1}+ ⋯ +i_{n} > 0 or q > i_{1}+ ⋯ +i_{n} = 0, where N = dim 𝕃𝔾_{n}. Because of the definition of E, Serre duality again and Leray spectral sequence, these vanishings are equivalent to the isomorphisms

H^{q}(𝕃𝔾_{n}, K_{𝕃𝔾_{n}}⊗ L_{i_{1}}^{n,1}⊗ ⋯ ⊗ L_{i_{n}}^{n,n}(n)) ≅ H^{q}(Y, K_{Y} ⊗ f^{∗}(L_{i_{1}}^{n,1}⊗ ⋯ ⊗ L_{i_{n}}^{n,n}(n)))

for q ≥ i_{1}+ ⋯ +i_{n} > 0 or q > i_{1}+ ⋯ +i_{n} = 0.

If i_{k} = k for 1 ≤ k ≤ n, and q = i_{1}+ ⋯ +i_{n} = N, both cohomology groups are equal to ℂ. If i_{k} = 0 for 1 ≤ k ≤ n, then both groups vanish by Kodaira vanishing theorem. We shall show that all the other groups also vanish.

We first deal with cohomology groups on 𝕃𝔾_{n}. Note that they could be computed using representation theory and applying the Borel-Weil-Bott theorem, but we will give a quite different argument mainly using Griffiths’ vanishing theorem and its variant in section 2.2.

Let us consider a given n-tuple i_{1}, …, i_{n}, with 0 ≤ i_{k} ≤ k. Let a be the number of indices k such that i_{k} < k, which is supposed to be positive. Note that the sum of the other i_{k}’s is bigger than or equal to (n − a)(n − a + 1)/2. It will therefore be enough to prove that

H^{q}(𝕃𝔾_{n}, K_{𝕃𝔾_{n}}⊗ ∧^{l_{1}}Q_{n}^{∗}⊗ ⋯ ⊗ ∧^{l_{a}}Q_{n}^{∗}(a)) = 0

for q ≥ l_{1}+ ⋯ +l_{a} +(n−a)(n−a+1)/2, and for all sequences l_{1}, …, l_{a} with l_{k} < n−a+k. (This condition is equivalent to the existence of a strictly increasing sequence 1 ≤ m_{1} < ⋯ < m_{a} ≤ n such that l_{k} < m_{k} for all k.)

We then use the tautological exact sequence 0 → Q_{n}^{∗} → V → Q_{n} → 0 on 𝕃𝔾_{n} where V = ℂ^{2n} (identified with the corresponding trivial bundle on 𝕃𝔾_{n}; recall that if W is maximal isotropic subspace of V, then the quotient V/W is canonically identified with the dual W^{∗}), and its induced long exact sequences of wedge powers

0 → ∧^{l}Q_{n}^{∗} → ∧^{l}V → ⋯ → ∧^{l−m}V ⊗ S^{m}Q_{n} → ⋯ → S^{l}Q_{n} → 0.

First case: a < n. Using the above long exact sequences for each wedge power, we are reduced to prove that

H^{q−m_{1}−···−m_{a}}(𝕃𝔾_{n}, K_{𝕃𝔾_{n}}⊗ S^{m_{1}}Q_{n}⊗ ⋯ ⊗ S^{m_{a}}Q_{n}(a)) = 0

for q ≥ l_{1}+ ⋯ +l_{a} + (n − a)(n − a + 1)/2, and for all sequences m_{1}, …, m_{a} with m_{k} ≤ l_{k} for all k.

Let S_{λ}Q_{n} be an irreducible component of S^{m_{1}}Q_{n}⊗ ⋯ ⊗ S^{m_{a}}Q_{n}. If l(λ) < a, then the vanishing follows from Griffiths’ vanishing theorem, more precisely from its variant 2.2 by setting F = Q_{n}, which is Griffiths semi-positive as a quotient of a trivial bundle, and by setting E = O_{𝕃𝔾_{n}}(1), which is positive. If l(λ) = a, then lemma 3.3 implies that it is enough to prove that

H^{q−m_{1}−···−m_{a}}(𝕃𝔾_{n}, K_{𝕃𝔾_{n}}⊗ S^{m_{1}−1}Q_{n}⊗ ⋯ ⊗ S^{m_{a}−1}Q_{n}⊗ ∧^{a}Q_{n}(a)) = 0.

Because ∧^{a}Q_{n} = ∧^{n−a}Q_{n}^{∗}(1), we can use the above complexes again to reduce to the vanishings of

H^{q−m_{1}−···−m_{a}−m_{a+1}}(𝕃𝔾_{n}, K_{𝕃𝔾_{n}}⊗ S^{m_{1}−1}Q_{n}⊗ ⋯ ⊗ S^{m_{a}−1}Q_{n}⊗ S^{m_{a+1}}Q_{n}(a + 1)),

with m_{a+1} ≤ n − a. We then use the same argument, repeated n − a − 1 times, and we are finally reduced to prove the vanishings of

\begin{matrix}H^{q−m_{1}−···−m_{n−1}}(𝕃𝔾_{n}, K_{𝕃𝔾_{n}}⊗ S^{m_{1}−n+a+1}Q_{n}⊗ ⋯ ⊗ S^{m_{a}−n+a+1}Q_{n}⊗ \\ ⊗ S^{m_{a+1}−n+a+2}Q_{n}⊗ ⋯ ⊗ S^{m_{n−1}}Q_{n}(n − 1)), \end{matrix}

where m_{a+k} ≤ n − a − k + 1. Firstly note that q − m_{1} − ⋯ − m_{n−1} is positive since m_{1}+ ⋯ +m_{a} ≤ l_{1}+ ⋯ +l_{a} and m_{a+1}+ ⋯ +m_{n−1} < (n − a)(n − a + 1)/2. Recall moreover that l_{1} ≤ n − a, so that m_{1} − n + a + 1 ≤ 1. If it is equal to zero, we can apply Griffiths’ vanishing theorem as we did above. If it is equal to one, we can factor out a Q_{n}(1) and apply 2.2 by setting F = Q_{n} and E = Q_{n}(1).

Second case: a = n. We need to prove that

H^{q}(𝕃𝔾_{n}, K_{𝕃𝔾_{n}}⊗ ∧^{l_{1}}Q_{n}^{∗}⊗ ⋯ ⊗ ∧^{l_{n}}Q_{n}^{∗}(n)) = 0

for q ≥ l_{1}+ ⋯ +l_{n} > 0, and for all sequences l_{1}, …, l_{n} with l_{k} < k. Note that in particular l_{1} = 0. Let us choose p such that l_{p} > 0. We then use our long exact sequences for each wedge power other than ∧^{l_{p}}Q_{n}^{∗}. We are thus reduced to prove that

H^{q}(𝕃𝔾_{n}, K_{𝕃𝔾_{n}}⊗ S^{m_{2}}Q_{n}⊗ ⋯ ⊗ S^{m_{n}}Q_{n}⊗ ∧^{n−l_{p}}Q_{n}(n − 1)) = 0

for q ≥ l_{p}, and for all sequences m_{1}, …, m_{p−1}, m_{p+1}, …, m_{n} with m_{k} ≤ l_{k}. Note that we have only n−2 symmetric powers in this tensor product. Let S_{λ}Q_{n} be some component of it. If l(λ) < n − 2, or l(λ) = n, we can apply Griffiths’ vanishing theorem. If l(λ) = n − 1, than the last non-zero part of the partition λ must be equal to one. Let μ be the partition obtained by deletion of this last part: then l(μ) = n − 2, and S_{λ}Q_{n} is a component of S_{μ}Q_{n}⊗ Q_{n}. We can therefore apply 2.2 by setting F = Q_{n} and E = Q_{n}(1) to get the required vanishing.

There remains to prove the similar vanishings on the finite covering Y of 𝕃𝔾_{n}. But the very same argument “pulls back” to Y. Indeed, we used nothing more on 𝕃𝔾_{n} than the tautological exact sequence, the Griffiths semi-positivity of Q_{n} and the Nakano positivity of O_{𝕃𝔾_{n}}(1) and Q_{n}(1). But the tautological sequence is clearly preserved on Y, as well as the Griffiths and Nakano semi-positivity of pulled-back bundles. Strict positivity, of course, is not preserved by pull-back. But for a finite covering, it is preserved outside the ramification locus, since outside this locus we have a local isomorphism for the complex topology. Since only positivity at one point is required to apply the vanishing theorem 2.2, we conclude our theorem. □

Remark. By proceeding in the same way as in the proof of [9, Proposition 4.4], we obtain the following homotopy result: let Y and f be as in Theorem A; for any fixed point y ∈ Y, f_{∗} : π_{i}(Y, y) → π_{i}(𝕃𝔾_{n}, f(y)) is an isomorphism for i ≤ n − 2d + 2.

4 Quadrics

4.1 Spinor bundles

The n-dimensional nonsingular quadric ℚ_{n} is a homogeneous space SO(n+2, ℂ)/P, where P is the maximal parabolic subgroup associated, in the notation of [3], with the first simple root of so(n + 2, ℂ). It is a simple Lie algebra of type B if n is odd, and of type D if n is even. Note that the semisimple part of Lie(P) is isomorphic to so(n, ℂ). If n = 2m+1, this semisimple Lie algebra has a spinor representation which defines a homogeneous bundle S on ℚ_{n} of rank 2^{m}. If n = 2m, there are two non isomorphic half spinor representations, giving rise to two homogeneous vector bundles S′ and S′′ of rank 2^{m−1}.

These spinor bundles have been extensively studied by Ottaviani [16]. Among other things, he showed the following: On ℚ_{2m+1}, we have a tautological exact sequence

0 → S → V → S(1) → 0

and an isomorphism S^{∗} ≅ S(1) where V is trivial of rank 2^{m+1}. On ℚ_{2m}, we have two tautological exact sequences (dual to each other)

0 → S′ → U → S′′(1) → 0, 0 → S′′ → U^{∗} → S′(1) → 0,

where U is trivial of rank 2^{m}. If m ≡ 0 (mod 2) then S^{′∗} ≅ S′(1) and S^{′′∗} ≅ S′′(1), and if m ≡ 1 (mod 2) then S^{′∗} ≅ S′′(1) and S^{′′∗} ≅ S′(1). Furthermore, spinor bundles behave nicely under restriction to generic hyperplane sections: namely, the restriction of S to ℚ_{2m} is S′⊕ S′′, while the restriction of S′ or S′′ to ℚ_{2m−1} is the spinor bundle on that quadric.

4.2 Quadrics of dimension five

On ℚ_{5}, we have the following Castelnuovo-Mumford criterion:

Proposition 4.1 A coherent sheaf F on ℚ_{5} is generated by its global sections as soon as

H^{i+j}(ℚ_{5}, F(−l)) = 0  for (i, l) = (1, 1), (2, 2), (5, 3), j ≥ 0,
H^{i+j}(ℚ_{5}, F⊗ S(−l)) = 0 for (i, l) = (1, 0),  j ≥ 0, and
H^{i}(ℚ_{5}, F⊗ ∧^{2}S(−l)) = 0 for (i, l) = (3, 1), (4, 2).

Proof. Let x be a point in ℚ_{5}, and ℚ_{4} a four dimensional quadric in ℚ_{5} containing x. Since ℚ_{4} is a hyperplane section, we have an exact sequence

0 → O_{ℚ_{5}}(−1) → O_{ℚ_{5}} → O_{ℚ_{4}} → 0,

showing in particular that the restriction map

H^{0}(ℚ_{5}, F) ⟶ H^{0}(ℚ_{4}, F|_{ℚ_{4}})

is surjective, since H^{1}(ℚ_{5}, F(−1)) = 0 by hypothesis.

On ℚ_{4}, we have two half spinor bundles S′ and S′′ of rank 2, and x can be realized as the zero-locus of a section of (S′⊕ S′)^{∗}. Indeed, we can identify ℚ_{4} with the grassmannian 𝔾_{2,4} in such a way that S^{′∗} identifies with the tautological quotient bundle on that grassmannian. Using the Koszul complex associated with a section of (S′⊕ S′)^{∗}, we see that F|_{ℚ_{4}} is globally generated at x as soon as

H^{q}(ℚ_{4}, ∧^{q}(S′⊕ S′)⊗ F|_{ℚ_{4}}) = 0 ∀q > 0.

Now recall that S|_{ℚ_{4}} = S′⊕ S′′, so that for q = 1, this vanishing is a consequence of the following conditions on ℚ_{5}:

H^{1}(ℚ_{5}, F⊗ S) ≅ H^{2}(ℚ_{5}, F⊗ S(−1)) = 0,

where the isomorphism is obtained by the tautological exact sequence on ℚ_{5} and our hypothesis. For q = 2, we want to show that H^{2}(ℚ_{4}, S′⊗ S′⊗ F|_{ℚ_{4}}) = 0. Tensoring the second tautological sequence by S′(−1)⊗ F|_{ℚ_{4}} on ℚ_{4}, we reduce to the vanishings of H^{2}(ℚ_{4}, S′(−1)⊗ F|_{ℚ_{4}}) and H^{3}(ℚ_{4}, S′⊗ S′′(−1)⊗ F|_{ℚ_{4}}). The first of these groups is dealt with as above. For the second one, we note that S′⊗ S′′ is a summand of ∧^{2}S|_{ℚ_{4}}, so that our vanishing condition is a consequence of

H^{3}(ℚ_{5}, F⊗ ∧^{2}S(−1)) = H^{4}(ℚ_{5}, F⊗ ∧^{2}S(−2)) = 0.

Finally, the cases q = 3 and q = 4 are similar to that of q = 1. □

Proof of Theorem C for 3 ≤ n ≤ 5. Recall that ℚ_{3} ≃ 𝕃𝔾_{2} and ℚ_{4} ≃ 𝔾_{2,4}, so that Theorem C follows for n = 3 from Theorem A, and for n = 4 from [14]. Let now n = 5.

Let f : Y → ℚ_{5} be a branched covering and E be the associated vector bundle. We prove that E(−1) is generated by global sections, using our Castelnuovo-Mumford criterion on ℚ_{5}. Using Serre duality, the definition of E and the fact that K_{ℚ_{5}} = O_{ℚ_{5}}(−5), we see that what we need to show is that certain cohomology groups of type

H^{i}(ℚ_{5}, K_{ℚ_{5}}⊗ ∧^{j}S(4 − l))

are equal to the corresponding cohomology groups of the pulled-back bundles on Y. We show that all these groups vanish, with one exception.

This is clear for j = 0 since l is at most two, which allows to use Kodaira vanishing. For j = 1, we can use the tautological exact sequence 0 → S → V → S^{∗} = S(1) → 0, which easily implies that

H^{i}(ℚ_{5}, K_{ℚ_{5}}⊗ S(m)) ≅ H^{i−n}(ℚ_{5}, K_{ℚ_{5}}⊗ S(m + n)),

provided that m, m + n and i − n are positive. Since S = S^{∗}(−1), det S^{∗} = O_{ℚ_{5}}(2) and S^{∗} is globally generated, the Griffiths vanishing theorem then gives

H^{i}(ℚ_{5}, K_{ℚ_{5}}⊗ S(4 − l)) = H^{1}(ℚ_{5}, K_{ℚ_{5}}⊗ S^{∗}⊗ det S^{∗}(i − l)) = 0

for 0 ≤ l < i ≤ 4, which is the case in our criterion.

Finally for j = 2, we have two conditions. The first one concerns

H^{3}(ℚ_{5}, K_{ℚ_{5}}⊗ ∧^{2}S(3)) ≅ H^{3}(ℚ_{5}, K_{ℚ_{5}}⊗ ∧^{2}S^{∗}(1))

which is zero by the Le Potier vanishing theorem. For the second one, we shall prove that

H^{4}(ℚ_{5}, K_{ℚ_{5}}⊗ ∧^{2}S(2)) ≅ ℂ.

We first notice that S^{∗} is globally generated, but c_{4}(S^{∗}) = 0; see [16]. This implies that a generic section of S^{∗} does not vanishes anywhere. Choosing such a section, we get an exact sequence

0 → O_{ℚ_{5}} → S^{∗} → Q^{∗} → 0,

where Q^{∗} is a globally generated vector bundle and det Q^{∗} = O_{ℚ_{5}}(2). Moreover, there is an exact sequence 0 → ∧^{2}Q → ∧^{2}S → Q → 0. For i ≥ 3, we have H^{i}(K_{ℚ_{5}}⊗ Q(2)) ≅ H^{i}(K_{ℚ_{5}}⊗ S(2)), which we proved to be zero. Hence

H^{4}(ℚ_{5}, K_{ℚ_{5}}⊗ ∧^{2}S(2)) ≅ H^{4}(ℚ_{5}, K_{ℚ_{5}}⊗ ∧^{2}Q(2)) ≅ H^{4}(ℚ_{5}, K_{ℚ_{5}}⊗ Q^{∗}).

And since H^{i}(ℚ_{5}, K_{ℚ_{5}}⊗ S^{∗}) = H^{i}(ℚ_{5}, K_{ℚ_{5}}⊗ S(1)) = 0 for i ≥ 4, we have

H^{4}(ℚ_{5}, K_{ℚ_{5}}⊗ Q^{∗}) ≅ H^{5}(ℚ_{5}, K_{ℚ_{5}}) = ℂ,

as claimed. Since the argument “pulls back” to Y by f, we are done. □

4.3 Quadrics of dimension six

On ℚ_{6} the situation is very close to that of ℚ_{5}, except that we have now two half spinor bundles S′ and S′′, both of rank four, both having for restriction to ℚ_{5} the spinor bundle S on this subquadric. Since ℚ_{5} is a hyperplane section of ℚ_{6}, the exact sequence 0 → O_{ℚ_{6}}(−1) → O_{ℚ_{6}} → O_{ℚ_{5}} → 0 allows to deduce almost immediately from 4.1, a Castelnuovo-Mumford criterion on ℚ_{6}: a coherent sheaf F on ℚ_{6} will be generated by its global sections as soon as

H^{i+j}(ℚ_{6}, F(−l)) = 0  for (i, l) = (1, 1), (2, 2), (4, 3), (6, 4) j ≥ 0,
H^{i+j}(ℚ_{6}, F⊗ S′(−l)) = 0 for (i, l) = (1, 0), (2, 1),  j ≥ 0, and
H^{i}(ℚ_{6}, F⊗ ∧^{2}S′(−l)) = 0 for (i, l) = (3, 1), (4, 2), (5, 3).

Proof of Theorem C for n = 6. The argument is almost exactly the same as on ℚ_{5}. Indeed, we have c_{4}(S′) = c_{4}(S′′) = 0 [16]. Choosing a nowhere vanishing section of S^{′∗}, we get an exact sequence

0 → O_{ℚ_{6}} → S^{′∗} → Q^{′∗} → 0,

where Q^{′∗} is a globally generated vector bundle and det Q^{′∗} = O_{ℚ_{6}}(2). Using this sequence, we check that

H^{5}(ℚ_{6}, K_{ℚ_{6}}⊗ ∧^{2}S′(2)) = H^{5}(ℚ_{6}, K_{ℚ_{6}}⊗ Q^{′∗}) = H^{5}(ℚ_{6}, K_{ℚ_{6}}) = ℂ.

This is enough to get the result that if f : Y → ℚ_{6} is a branched covering with its associated vector bundle E, then E(−1) is generated by its global sections, since the other cohomology groups involved in our Castelnuovo-Mumford criterion on ℚ_{6} are easily checked to vanish.

□

5 Global generation

In this section we prove that for any branched covering of ordinary and symplectic flag manifolds, the associated vector bundle E is generated by its global sections. Of course, E cannot be ample in general, since flag manifolds that are not grassmannians fiber over smaller flag manifolds, from which one could pull-back any branched covering. Nevertheless, they should have intermediate positivity properties, presumably some k-ampleness in the sense of Sommese. This would again imply Barth-Lefschetz type theorems, but unfortunately we have been unable to prove such positivity properties.

5.1 Ordinary flag manifolds

Let 𝔽_{n} denote the variety of complete flags in V = ℂ^{n}, with universal flag

W_{•} : 0 = W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{n−1} ⊂ V ⊗ O_{𝔽_{n}}

of subbundles W_{i} of rank i of the trivial vector bundle V ⊗ O_{𝔽_{n}}. We denote by Q_{i} = V/W_{n−i} the quotient bundle of rank i, and

O_{𝔽_{n}}(a_{1}, …, a_{n−1}) = O_{𝔽_{n}}(∑_{i=1}^{n−1} a_{i}ε_{i}) = ⨂_{i=1}^{n−1}(det Q_{i})^{a_{i}}.

This line bundle is generated by its global sections (ample, respectively) if and only if a_{i} ≥ 0 (a_{i} > 0, respectively) for all i. The canonical line bundle of 𝔽_{n} is K_{𝔽_{n}} = O_{𝔽_{n}}(−2, …, −2).

Let x be a point of 𝔽_{n}, given by some complete flag 0 = U_{0} ⊂ U_{1} ⊂ ⋯ ⊂ U_{n−1} ⊂ V. Let us choose a compatible basis u_{1}, …, u_{n} of V, that is such that u_{1}, …, u_{i} is a basis of U_{i} for all i. Each vector u_{n−i} defines a global section of the quotient vector bundle Q_{i}. We therefore have a global section of Q_{1} ⊕ ⋯ ⊕ Q_{n−1}, vanishing precisely at x. The corresponding Koszul complex is

0 → ∧^{\frac{n(n−1)}{2}} (Q_{1}^{∗} ⊕ ⋯ ⊕ Q_{n−1}^{∗}) → ⋯ → Q_{1}^{∗} ⊕ ⋯ ⊕ Q_{n−1}^{∗} → O_{𝔽_{n}} → O_{x} → 0.

Hence we obtain a very simple Castelnuovo-Mumford criterion on 𝔽_{n}:

Proposition 5.1 A coherent sheaf F on 𝔽_{n} is generated by its global sections as soon as

H^{q}(𝔽_{n}, ∧^{i_{1}}Q_{1}^{∗}⊗ ⋯ ⊗ ∧^{i_{n−1}}Q_{n−1}^{∗}⊗ F) = 0

for q = i_{1} + ⋯ + i_{n−1} > 0.

We will apply this criterion to the vector bundle associated with a finite covering of 𝔽_{n}. With the generalized Le Potier vanishing theorem 2.3 in mind, it will be easy to prove the first half of Theorem D:

Proposition 5.2 Let f : Y → 𝔽_{n} be a finite surjective morphism where Y is a nonsingular connected complex projective variety. Then the associated vector bundle E is generated by its global sections.

Proof. We need to prove that

H^{q}(𝔽_{n}, K_{𝔽_{n}}⊗ ∧^{i_{1}}Q_{1}^{∗}⊗ ⋯ ⊗ ∧^{i_{n−1}}Q_{n−1}^{∗}(2, …, 2)⊗ E) = 0

for q = i_{1} + ⋯ + i_{n−1} > 0. Using Serre duality, the definition of E, and Leray spectral sequence, this vanishing is seen to be equivalent to the equality between the cohomology group

H^{q}(𝔽_{n}, K_{𝔽_{n}}⊗ ∧^{i_{1}}Q_{1}^{∗}⊗ ⋯ ⊗ ∧^{i_{n−1}}Q_{n−1}^{∗}(2, …, 2)),

and the corresponding cohomology group on Y,

H^{q}(Y, K_{Y} ⊗ f^{∗}(∧^{i_{1}}Q_{1}^{∗}⊗ ⋯ ⊗ ∧^{i_{n−1}}Q_{n−1}^{∗}(2, …, 2)))

for q = i_{1} + ⋯ + i_{n−1} > 0. We shall prove that both groups are equal to zero. We begin with the first one.

If i_{1} > 0, then since Q_{1} is of rank 1 we can suppose that i_{1} = 1, and we apply the generalized Le Potier vanishing theorem 2.3. If i_{1} = 0, let k be the smallest integer such that i_{k} > 0. We proceed by induction on k. We have a short exact sequence

0 → O_{𝔽_{n}}(ε_{k} − ε_{k−1}) → Q_{k} → Q_{k−1} → 0,

from which we get an exact sequence for wedge powers:

0 → ∧^{i_{k}}Q_{k−1}^{∗} → ∧^{i_{k}}Q_{k}^{∗} → ∧^{i_{k}−1}Q_{k−1}^{∗}(ε_{k−1} − ε_{k}) → 0.

Let us tensor this exact sequence by K_{𝔽_{n}}⊗ ∧^{i_{k+1}}Q_{1}^{∗}⊗ ⋯ ⊗ ∧^{i_{n−1}}Q_{n−1}^{∗}(2, …, 2). The q-th cohomology group of the first vector bundle of the exact sequence we get, then vanishes by induction hypothesis. Also, for the last vector bundle we obtain the cohomology group

H^{q}(𝔽_{n}, K_{𝔽_{n}}⊗ ∧^{i_{k−1}}Q_{k−1}^{∗}⊗ ∧^{i_{k+1}}Q_{k+1}^{∗} ⋯ ⊗ ∧^{i_{n−1}}Q_{n−1}^{∗}(2, …, 3, 1, 2, …, , 2)),

where 3 and 1 occupy the (k − 1)-th and k-th position, respectively. But this is zero by the generalized Le Potier vanishing theorem. The q-th cohomology group of the middle vector bundle therefore vanishes, which is what we wanted to prove.

Since the very same argument applies to the pulled-back vector bundles on Y, the proposition is proved. □

Corollary 5.3 Let f : Y → 𝔽 be a finite covering of any variety of incomplete flags 𝔽, where Y is a nonsingular connected complex projective variety. Then the associated vector bundle E is generated by its global sections.

Proof. Since a variety of complete flags fibers on any variety of incomplete flags of the same vector space, we have a smooth fibration π : 𝔽_{n} → 𝔽. Pulling back by f, we get a finite covering f′ : Y′ → 𝔽_{n} with its associated vector bundle E′ = π^{∗}E. Moreover, since π has connected fibers, π_{∗}O_{𝔽_{n}} = O_{𝔽}, hence

H^{0}(Y′, E′) = H^{0}(Y, π_{∗}E′) = H^{0}(Y, E).

Since, by the proposition above, E′ is generated by its global sections, all of which are pull-backs of sections of E, then E itself must be globally generated. □

5.2 Symplectic flag manifolds

We denote by 𝕃𝔽_{n} the variety of complete lagrangian flags in V = ℂ^{2n}, with universal flag of subbundles

W_{•} : 0 = W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{n} = W_{n}^{⊥} ⊂ W_{n−1}^{⊥} ⊂ ⋯ ⊂ W_{1}^{⊥} ⊂ O_{𝕃𝔽_{n}}⊗ V,

where dim W_{i} = i and dim W_{i}^{⊥} = 2n − i. This variety has dimension n^{2}, and is homogeneous under the natural action of the symplectic group Sp(2n, ℂ). We will consider it as a subvariety of 𝔽_{n,2n}, the variety of incomplete flags in ℂ^{2n} consisting of subspaces of dimensions 1 to n.

For 1 ≤ i ≤ n, we denote by Q_{i} = V/W_{i} the quotient bundle of rank 2n − i, and

O_{𝕃𝔽_{n}}(a_{1}, …, a_{n}) = O_{𝔽_{n}}(∑_{i} a_{i}ε_{i}) = ⨂_{i=1}^{n}(det Q_{i})^{a_{i}}.

In particular, the canonical line bundle of 𝕃𝔽_{n} is K_{𝕃𝔽_{n}} = O_{𝕃𝔽_{n}}(−2, …, −2).

On 𝔽_{n,2n}, a generic section of Q_{1} ⊕ ⋯ ⊕ Q_{n} vanishes at a simple point, and the associated Koszul complex provides us with a Castelnuovo-Mumford criterion, which can be restricted to 𝕃𝔽_{n}. Since we have Q_{i}^{∗} ≃ W_{n−i+1}^{⊥} on 𝕃𝔽_{n}, we get:

Proposition 5.4 A coherent sheaf F on 𝕃𝔽_{n} is generated by its global sections as soon as

H^{q}(𝕃𝔽_{n}, ∧^{i_{1}}W_{1}^{⊥}⊗ ⋯ ⊗ ∧^{i_{n}}W_{n}^{⊥}⊗ F) = 0

for q = i_{1} + ⋯ + i_{n} > 0.

In the sequel, we will need to understand the cohomology of certain homogeneous bundles, which will be tensor products of wedge powers of the tautological vector bundles. We call such a bundle a wedge bundle, and say that it is of type t = (t_{1}, …, t_{n}) if it has t_{i} factors that are wedge powers of W_{i} or W_{i}^{⊥}, and weight w = (w_{1}, …, w_{n}) if the sum of the exponents of these wedge powers of W_{i} and W_{i}^{⊥} is equal to w_{i}. The type can of course be decomposed into subtypes t′ and t′′, given by the number of factors of involving W_{i}, and W_{i}^{⊥}, respectively. We first need the following refined statement of the generalized Le Potier vanishing theorems for wedge bundles on 𝕃𝔽_{n}.

Proposition 5.5 Let W be a wedge bundle on 𝕃𝔽_{n}, of type t and weight w. Then

H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ W(l_{1}, …, l_{n})) = 0

for q > w_{1} + ⋯ + w_{n}, if either :

• l_{k} > t_{k} for 1 ≤ k ≤ n,

• l_{k} > t_{k} for k ≠ p, q, where p < q are such that t′′_{q} > 0, l_{q} ≥ t_{q} and l_{p} > t_{p} + 1, or

• l_{k} > t_{k} for k ≠ p, q, where p > q are such that t′_{q} > 0, l_{q} ≥ t_{q} and l_{p} > t_{p} + 1.

Proof. The first assertion is an immediate consequence of the generalized Le Potier vanishing theorem. Indeed, we can rewrite each wedge power ∧^{m}W_{i} as ∧^{n−i+1−m}W_{i}^{∗}(−1), and similarly each ∧^{m}W_{i}^{⊥} as ∧^{n−i+1−m}(W_{i}^{⊥})^{∗}(−1). If l_{k} > t_{k} for 1 ≤ k ≤ n, we get a tensor product of wedge powers of globally generated vector bundles, by the ample line bundle O_{𝕃𝔽_{n}}(l_{1} − t_{1}, …, l_{n} − t_{n}). Hence we can apply the generalized Le Potier vanishing theorem 2.3, and we obtain the desired statement.

For the second assertion, which is not a direct consequence of Le Potier’s theorem, we will proceed by induction on q. Since t′′_{q} is non zero, then we have at least a factor ∧^{m}W_{q}^{⊥} in W. Therefore we can use the exact sequence

0 → ∧^{m}W_{q}^{⊥} → ∧^{m}W_{q−1}^{⊥} → ∧^{m−1}W_{q}^{⊥}(ε_{q} − ε_{q−1}) → 0.

Let us tensor this exact sequence by the other factors of W. We get a short exact sequence involving wedge bundles,

0 → W → W′ → W′′(ε_{q} − ε_{q−1}) → 0.

Here W′ has weight (w_{1}, …, w_{q−1} + m, w_{q} − m, …, w_{n}) and type (t_{1}, …, t_{q−1} + 1, t_{q} − 1, …, w_{n}). Hence

H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ W′(l_{1}, …, l_{n})) = 0

for q > w_{1} + ⋯ + w_{n}. Indeed, this is the case by induction hypothesis if p < q − 1, while if p = q − 1 we are in the situation of the first part of the proposition. On the other hand, W′′ has weight (w_{1}, …, w_{q−1}, w_{q} − 1, …, w_{n}) and type (t_{1}, …, t_{q−1}, t_{q}, …, w_{n}) (except if m = 1, in which case t_{q} is replaced by t_{q} − 1, which is even better). This again allows us to use our induction hypothesis, which gives

H^{q−1}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ W′′(l_{1}, …, l_{q−1} − 1, l_{q} + 1, …, l_{n})) = 0

for q > w_{1} + ⋯ + w_{n}. And these vanishings for W′ and W′′ imply our claim for W.

The third assertion is proved in the very same way using the other series of exact sequences,

0 → ∧^{m}W_{q} → ∧^{m}W_{q+1} → ∧^{m−1}W_{q}(ε_{q} − ε_{q+1}) → 0,

and a similar, but now descending, induction on q. □

With the help of this vanishing theorem, we can now extend 5.3 to the case of symplectic flag manifolds and prove the second half of Theorem D. As for ordinary flags, it will suffice to consider complete symplectic flags, that is, coverings of 𝕃𝔽_{n}.

Proposition 5.6 Let f : Y → 𝕃𝔽_{n} be a finite covering, where Y is a nonsingular connected complex projective variety. Then the associated vector bundle E is generated by its global sections.

Proof. We apply our Castelnuovo-Mumford criterion on 𝕃𝔽_{n} to E. Because of Serre duality and the definition of E, we need to prove that the cohomology group

H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ ∧^{i_{1}}W_{1}^{⊥}⊗ ⋯ ⊗ ∧^{i_{n}}W_{n}^{⊥}(2, …, 2)),

for q = i_{1} + ⋯ + i_{n} > 0, is equal to the corresponding group obtained by pull-back to Y. We shall prove that both groups are equal to zero. Note that on 𝕃𝔽_{n}, this would follow from the first part of the proceeding proposition if we had W_{1} instead of W_{1}^{⊥}, at least for i_{1} > 0. Indeed, since W_{1} has rank one, this would force i_{1} = 1 and we could replace W_{1} by O_{𝕃𝔽_{n}}(−ε_{1}). Our strategy will therefore be to use the tautological exact sequences on 𝕃𝔽_{n}, first to replace W_{1}^{⊥} by W_{n}^{⊥} = W_{n}, then W_{n} by W_{1}.

Let k be the smallest integer such that i_{k} > 0. For notational simplicity, let us suppose that k = 1: the argument would be the same for k > 1. We have an exact sequence

0 → ∧^{i_{1}}W_{2}^{⊥} → ∧^{i_{1}}W_{1}^{⊥} → ∧^{i_{1}−1}W_{2}^{⊥}(ε_{2} − ε_{1}) → 0.

Let us tensor it by W = ∧^{i_{2}}W_{2}^{⊥}⊗ ⋯ ⊗ ∧^{i_{n}}W_{n}^{⊥}. The wedge bundle W⊗ ∧^{i_{1}−1}W_{2}^{⊥}(ε_{2} − ε_{1}) has type (0, 2, 1, ..., 1); so that by the second part of proposition 5.5,

H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ W⊗ ∧^{i_{1}−1}W_{2}^{⊥}(1, 3, 2, …, 2) = 0.

Our vanishing will therefore follow from that of H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ ∧^{i_{1}}W_{2}^{⊥}⊗ W(2, …, 2)). Using the same argument n − 1 times, we are then reduced to prove that

H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ ∧^{i_{1}}W_{n}^{⊥}⊗ W(2, …, 2)) = 0.

But since W_{n}^{⊥} = W_{n}, we can now use the exact sequence

0 → ∧^{i_{1}}W_{n−1} → ∧^{i_{1}}W_{n} → ∧^{i_{1}−1}W_{n−1}(ε_{n−1} − ε_{n}) → 0.

Using the third part of proposition 5.5, we see that

H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ ∧^{i_{1}}W_{n−1}⊗ W(2, …, 2, 3, 1)) = 0.

We are thus reduced to prove that H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ ∧^{i_{1}}W_{n−1}⊗ W(2, …, 2)) = 0 and, using repeatedly the same argument, what we finally have to show is the vanishing of

H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ ∧^{i_{1}}W_{1}⊗ W(2, …, 2)).

This is clear if i_{1} > 1. If i_{1} = 1, we can rewrite this cohomology group as

H^{q}(𝕃𝔽_{n}, K_{𝕃𝔽_{n}}⊗ W(1, 2, …, 2)),

which is zero by the first part of 5.5. □

5.3 Questions and conjectures

Let X = G/P be a complex projective homogeneous space, where G is a complex semisimple Lie group and P is a parabolic subgroup. If P is not maximal and Q is a parabolic subgroup of G containing P, we have a smooth fibration G/P → G/Q. Then we obtain a branched covering f of X by pulling back the one of G/Q. Note that its associated bundle, being a pull-back by f, cannot be ample: it is at best (dim Q/P)-ample in the sense of Sommese [18]. Let us define

k_{X} = max_{Q⊃P} dim Q/P.

If P is defined (up to conjugacy) by a set I of simple roots, one can compute k_{X} as the maximum, as α describes I, of the number of positive roots having positive coefficient on the simple root α, but zero coefficient on the other simple roots in I.

We can then extend the conjecture of Debarre stated in the introduction in the following naive way:

Conjecture. Let f : Y → X = G/P be a branched covering, with Y smooth and connected. Then the associated vector bundle E is k_{X}-ample.

Once again, this would imply a Barth-Lefschetz type theorem: the natural map

f^{∗} : H^{i}(X, ℂ) ⟶ H^{i}(Y, ℂ)

would then be an isomorphism for i ≤ dim X − k_{X} − d + 1, where d is the degree of f. Moreover, one could expect this k_{X}-ampleness to be optimal only when f is obtained by pull-back:

Question. Let f : Y → X = G/P be a branched covering, with a nonsingular connected complex projective variety Y. Suppose that the associated vector bundle E is not (k_{X} − 1)- ample. Then, is f necessarily a pull-back of a covering of some smaller homogeneous space G/Q ?

To attack the previous conjecture by the methods of this paper, we would need efficient Castelnuovo-Mumford criteria. Even when P is maximal, we could not always find good enough such criteria, for example on quadrics, or on spinor varieties, which being hermitian symmetric should be easier to deal with than orthogonal grassmannians.

Question. How to find “good” Castelnuovo-Mumford criteria on homogeneous spaces ?

When the parabolic group P is not maximal, such criteria should certainly involve several line bundles. A special case would be the following:

Question. Let L, M be globally generated line bundles on a projective variety X, with L⊗ M very ample. If F is a coherent sheaf on X, can one give a cohomological criterion for the surjectivity of the mixed evaluation morphism

H^{0}(X, F⊗ L^{−1})⊗ L ⊕ H^{0}(X, F⊗ M^{−1})⊗ M ⟶ F,

that would not necessarily imply that F⊗ L^{−1} or F⊗ M^{−1} is globally generated ?

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Meeyoung Kim  Laurent Manivel
Max-Planck-Institut für Mathematik Institut Fourier, UMR 5582 UJF/CNRS
Gottfried-Claren-Str. 26  Université de Grenoble I, BP 74
D-53225 Bonn, Germany  F-38402 Saint Martin d’Hères, France
kim@mpim-bonn.mpg.de  laurent.manivel@ujf-grenoble.fr