Journal of Lie Theory Volume 14 (2004) 509–522 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Injectivity of the Double Fibration Transform for Cycle Spaces of Flag Domains

Alan T. Huckleberry^{1} and Joseph A. Wolf ^{2}

Communicated by K.-H. Neeb

Abstract. The basic setup consists of a complex flag manifold Z = G/Q where G is a complex semisimple Lie group and Q is a parabolic subgroup, an open orbit D = G_{0}(z) ⊂ Z where G_{0} is a real form of G, and a G_{0} – homogeneous holomorphic vector bundle 𝔼 → D. The topic here is the double fibration transform P : H^{q}(D; O(𝔼)) → H^{0}(M_{D}; O(𝔼′)) where q is given by the geometry of D, M_{D} is the cycle space of D, and 𝔼′ → M_{D} is a certain naturally derived holomorphic vector bundle. Schubert intersection theory is used to show that P is injective whenever 𝔼 is sufficiently negative.

Contents

1. Introduction 509

2. Basics of the Double Fibration Transform 511

3. A Computable Description of M_{D} 514

4. Globalization of the Bundles 516

5. The Method of Schubert Slices 518

6. Canonical Fibrations and DFT–Injectivity 519

1. Introduction

Let G_{0} be a non–compact real form of a complex semisimple Lie group G and consider its action on a compact G–homogeneous projective algebraic manifold Z = G/Q. It is of interest to understand the G_{0}–representation theory associated

1Research partially supported by the DFG Schwerpunkt “Global Methods in Complex Geometry”.

2Research partially supported by DFG Schwerpunkt “Global Methods in Complex Geometry” and by NSF Grant DMS 99–88643.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag to each of its (finitely many) orbits. In this note we restrict to the case of an open orbit D = G_{0}(z_{0}), and we often refer to such open orbits as flag domains.

The simplest example of this situation, where G_{0} = SU(1, 1), G = SL_{2}(ℂ) and Z = ℙ_{1}(ℂ), is at first sight perhaps somewhat misleading. In this case D is either the set of of negative or positive lines and therefore is biholomorphic to the unit disk Δ = {z ∈ ℂ : |z| < 1}. The naturally associated G_{0}–representations can be regarded as being in L^{2}–spaces of holomorphic functions on O(Δ), or their complex conjugates.

Unless G_{0} is of Hermitian type and D is biholomorphically equivalent to the associated bounded symmetric domain as in the above example, the holomorphic functions do not separate the points of D; in fact in most cases O(D) ≅ ℂ. One explanation for this is that D generally contains positive–dimensional compact analytic subsets which are in fact closely related to the representation theory at hand. These arise initially as orbits of maximal compact subgroups K_{0} of G_{0}: There is a unique complex K_{0}–orbit C_{0} in D [20].

Of course C_{0} can be just a point, which is exactly the case when D is a bounded Hermitian symmetric domain, but in general it has positive dimension.

For example, if G_{0} = SL_{3}(ℝ), G = SL_{3}(ℂ) and Z = ℙ_{2}(ℂ), then the closed G_{0}–orbit in Z is the set of real points Z_{ℝ} = ℙ_{2}(ℝ) and its complement D = Z \Z_{ℝ} is the only other orbit. If K_{0} is chosen to be the real orthogonal group K_{0} = SO_{3}(ℝ), then

C_{0} = {[z_{0} : z_{1} : z_{2}] : z_{0}^{2} + z_{1}^{2} + z_{2}^{2} = 0}

is the standard quadric. Here one easily checks that O(D) ≅ ℂ and, in view of basic results of Andreotti and Grauert [2], looks for G_{0}–representations in Dolbeault cohomology H^{1}(D, 𝔼), where 𝔼 → D is a sufficiently negative holomorphic vector bundle.

Cohomology classes, e.g., classes of bundle valued differential forms, are technically and psychologically more difficult to handle than holomorphic functions or sections of vector bundles. Thus, for q := dim_{ℂ} C_{0} one is led to consider the space C^{q}(D) of q–dimensional cycles in D, where C_{0} can be considered as a point.

An element C ∈ C^{q}(D) is a linear combination, C = n_{1}C_{1} +… n_{m}C_{m}, with n_{j} ∈ ℕ^{>0} and where C_{j} is a q–dimensional irreducible compact analytic subset of D. It is also necessary to consider C^{q}(Z), where the C_{j} are not required to be contained in D.

The cycle space C^{q}(X) of a complex space X has a canonical structure of (locally) finite–dimensional complex space (see [4] and, e.g., [14], for this and other basic properties). In the case at hand, where Z is projective algebraic, C^{q}(Z) is often referred to as the Chow variety; in particular, its irreducible components are projective algebraic varieties.

To simplify the notation we redefine C^{q}(Z) to be the topological component containing the base cycle C_{0} . It contains C^{q}(D) as an open semialgebraic subset. This is well defined independent of the choice of the maximal compact subgroup K_{0}.

It is known that the induced action of G on C^{q}(Z) is algebraic (see, e.g.,

[15] for a detailed proof) and therefore the orbit M_{Z} := G.C_{0} is Zariski open in its closure X in C^{q}(Z). It should be noted that M_{Z} is a spherical homogeneous space of the reductive group G and therefore the Luna–Vust theory of compactifications [5] applies. It would be extremely interesting to determine the G–varieties X which occur in this way.

In general it may be difficult to understand the full cycle space C^{q}(D) and therefore one cuts down to a simpler space which is more closely related to the group actions at hand. That simpler space is

M_{D} : topological component of C_{0} in M_{Z} ∩ C^{q}(D).

We have the incidence space

𝔛_{D} = {(z, C) ∈ D × M_{D} : z ∈ C}.

and the projections μ : 𝔛_{D} → D defined by (z, C) ↦ z, and ν : 𝔛_{D} → M_{D} defined by (z, C) ↦ C.

If 𝔼 → D is a holomorphic vector bundle, we may lift it to μ^{∗}𝔼 → 𝔛_{D} consider the associated ν –direct image sheaves on M_{D} . In this way Dolbeault cohomology spaces H^{q}(D, 𝔼) are transformed to the level of sections of holomorphic vector bundles 𝔼′ → M_{D} . This double fibration transform is explained in detail in the next section. Since M_{D} is now known to be a Stein domain ([21], [17]), our somewhat technical initial setting is transformed to one that is more tractable.

In recent work we developed complex geometric methods aimed at describing the cycle spaces M_{D}. For example, for a fixed real form, the space M_{D} is essentially always biholomorphically equivalent to a universal domain U which is defined independent of D and Z ([17], [10]).

On the other hand, here we prove that M_{D} possesses canonically defined holomorphic fibrations which do indeed depend on D and Z and which give it interesting refined structure. As a consequence we in particular show that the double fibration transform is injective, a fact that should be useful for representation theoretic considerations.

The present paper is organized as follows. In §2 we recall the basics of the double fibration transformation. The recently proved results of [17] and [10] are summarized in §3 and in §4 we give details on the method of Schubert slices. In the final section these slices are used to define the fibrations of M_{D} which were mentioned above. The injectivity of the double fibration transform is a consequence of the topological triviality of these bundles.

2. Basics of the Double Fibration Transform

Let D be a complex manifold (later it will be an open orbit of a real reductive group G_{0} on a complex flag manifold Z = G/Q of its complexification ). We suppose that D fits into what we loosely call a holomorphic double fibration. This means that there are complex manifolds M and 𝔛 with maps

[figure: 𝔛; μ; D; ν; M]  (2.1)

where μ is a holomorphic submersion and ν is a proper holomorphic map which is a locally trivial bundle. Given a locally free coherent analytic sheaf E → D we construct a locally free coherent analytic sheaf E′ → M and a transform

P : H^{q}(D; E) → H^{0}(M; E′)  (2.2)

under mild conditions on (2.1). This construction is fairly standard, but we need several results specific to the case of flag domains.

Pull–back.

The first step is to pull cohomology back from D to 𝔛. Let μ^{−1}(E) → 𝔛 denote the inverse image sheaf. For every integer r ≧ 0 there is a natural map

μ^{(r)} : H^{r}(D; E) → H^{r}(𝔛; μ^{−1}(E))  (2.3)

given on the Čech cocycle level by μ^{(r)}(c)(σ) = c(μ(σ)) where c ∈ Z^{r}(D; E) and where σ = (w_{0}, …, w_{r}) is a simplex. For q ≧ 0 we consider the Buchdahl q– condition on the fiber F of μ : 𝔛 → D:

F is connected and H^{r}(F; ℂ) = 0 for 1 ≦ r ≦ q − 1.  (2.4)

Proposition 2.5. (See [6].) Fix q ≧ 0. If (2.4) holds, then (2.3) is an isomorphism for r ≦ q − 1 and is injective for r = q. If the fibers of μ are cohomologically acyclic then (2.3) is an isomorphism for all r.

As usual, O_{X} → X denotes the structure sheaf of a complex manifold X and O(𝔼) → X denotes the sheaf of germs of holomorphic sections of a holomorphic vector bundle 𝔼 → X. Let μ^{∗}(E) := μ^{−1}(E)ˆ⊗_{μ^{−1}(O_{D})}O_{𝔛} → 𝔛 denote the pull–back sheaf. It is a coherent analytic sheaf of O_{𝔛}–modules. Now [σ] ↦ [σ] ⊗ 1 defines a map i : μ^{−1}(E) → μ^{∗}(E) which in turn specifies maps in cohomology, the coefficient morphisms

i_{p} : H^{p}(𝔛; μ^{−1}(E)) → H^{p}(𝔛; μ^{∗}(E)) for p ≧ 0.  (2.6)

Our natural pull–back maps are the compositions j^{(p)} = i_{p} · μ^{(p)} of (2.3) and (2.6):

j^{(p)} : H^{p}(D; E) → H^{p}(𝔛; μ^{∗}(E)) for p ≧ 0.  (2.7)

We have E = O(𝔼) for some holomorphic vector bundle 𝔼 → D, because we assumed E → D locally free. Thus μ^{∗}(E) = O(μ^{∗}(𝔼)) and we realize these sheaf cohomologies as Dolbeault cohomologies. In the context of Dolbeault cohomology, the pull–back maps (2.7) are given by pulling back [ω] ↦ [μ^{∗}(ω)] on the level of differential forms.

Push–down.

In order to push the H^{q}(𝔛; μ^{∗}(E)) down to M we assume that

M is a Stein manifold.  (2.8)

Since ν : 𝔛 → M was assumed proper, we have the Leray direct image sheaves R^{p}(μ^{∗}(E)) → M. Those sheaves are coherent [13]. As M is Stein

H^{q}(M; R^{p}(E)) = 0 for p ≧ 0 and q > 0.  (2.9)

Thus the Leray spectral sequence of ν : 𝔛 → M collapses and gives

H^{p}(𝔛; μ^{∗}(E)) ≅ H^{0}(M; R^{p}(μ^{∗}(E))).  (2.10)

Definition. The double fibration transform for the double fibration (2.1) is the composition

P : H^{p}(D; E) → H^{0}(M; R^{p}(μ^{∗}(E)))  (2.11)

of the maps (2.7) and (2.10).

In order that the double fibration transform (2.11) be useful, one wants two conditions to be satisfied. They are

P : H^{p}(D; E) → H^{0}(M; R^{p}(μ^{∗}(E))) should be injective, and (2.12)
there should be an explicit description of the image of P.  (2.13)

Assuming (2.8), injectivity of P is equivalent to injectivity of j^{(p)} in (2.7). The most general way to approach this is the combination of vanishing and negativity in Theorem 2.14 below, taking the Buchdahl conditions (2.4) into consideration.

Given the setting of (2.8) our attack on the injectivity question uses a spectral sequence argument for the relative de Rham complex of the holomorphic submersion μ : 𝔛 → D. See [23] for the details. The end result is

Theorem 2.14. Let E = O(𝔼) for some holomorphic vector bundle 𝔼 → D. Fix q ≧ 0. Suppose that the fiber F of μ : 𝔛 → D is connected and satisfies (2.4). Assume (2.8) that M is Stein. Suppose that H^{p}(C; Ω_{μ}^{r}(𝔼)|_{C}) = 0 for p < q, and r ≧ 1 for every fiber C of ν : 𝔛 → M where Ω_{μ}^{r}(𝔼) → 𝔛 denotes the sheaf of relative μ^{∗}𝔼–valued holomorphic r–forms on 𝔛 with respect to μ : 𝔛 → D. Then P : H^{q}(D; E) → H^{0}(M; R^{q}(μ^{∗}E)) is injective.

Flag domain case.

In the cases of interest to us, D will be a flag domain, we will have E = O(𝔼) as in Theorem 2.14, and the transform P will have an explicit formula. The Leray derived sheaf will be given by

R^{q}(μ^{∗}(O(𝔼))) = O(𝔼′) where
𝔼′ → M has fiber H^{q}(ν^{−1}(C); O(μ^{∗}(𝔼)|_{ν^{−1}(C)})) at C.

(2.15)

Then P will be given on the level of Dolbeault cohomology, as follows. Let ω be an 𝔼–valued (0, q)–form on D and [ω] ∈ Hq ; \overline{∂}(D, 𝔼) its Dolbeault class. Then

P([ω]) is the holomorphic section of 𝔼′ → M
whose value P([ω])(C) at C ∈ M is [μ^{∗}(ω)|_{ν^{−1}(C)}].

In other words,

P([ω])(C) = [μ^{∗}(ω)|_{ν^{−1}(C)}] ∈ H0 ; \overline{∂}(M; 𝔼′)  (2.16)

This is most conveniently interpreted by viewing P([ω])(C) as the Dolbeault class of ω|_{C}, and by viewing C ↦ [ω|_{C}] as a holomorphic section of the holomorphic vector bundle 𝔼′ → M.

Now let D = G_{0}(z_{0}) be an open orbit in the complex flag manifold Z = G/Q, and M is replaced by the cycle space M_{D}. Our double fibration (2.1) is replaced by

[figure: 𝔛_{D}; μ; D; ν; M_{D}]  (2.17)

where 𝔛_{D} := {(z, C) ∈ D × M_{D} | z ∈ C} is the incidence space. Given a G_{0}– homogeneous holomorphic vector bundle 𝔼 → D, and the number q = dim_{ℂ} C_{0} , we will see in Section 4. that the Leray derived sheaf involved in the double fibration transform satisfies (2.15). Here (2.15) will become a little bit more explicit and take the form

R^{q}(μ^{∗}(O(𝔼))) = O(𝔼′) where
𝔼′ → M_{D} has fiber H^{q}(C; O(𝔼|_{C})) at C ∈ M_{D} .

(2.18)

Evidently, 𝔼′ → M_{D} will be globally G_{0}–homogeneous. It cannot be G– homogeneous unless M_{D} is G–invariant, and that only happens in the degenerate case where G_{0} is transitive on Z. However, in Section 4. we will see that 𝔼′ → M_{D} is the restriction of a G–homogenous holomorphic vector bundle 𝔼̃′ → M_{Z} , and in particular (2.15) is satisfied. In any case, H^{q}(C; O(𝔼|_{C})) can be calculated from the Bott–Borel–Weil Theorem. Thus R^{q}(μ^{∗}(O(𝔼))) will be given explicitly by (2.18) in the flag domain case.

Using methods of complex geometry as described in Section 3. below it was shown ([17] plus [10]) that M_{D} is biholomorphically equivalent to a certain universal domain U or to a bounded symmetric domain. It follows in general that M_{D} is a contractible Stein manifold, so 𝔼′ → M_{D} is holomorphically trivial. We will use those same methods in Section 5. to construct certain holomorphic fibrations of the M_{D} and use those “Schubert fibrations” to show that F satisfies (2.4) for all q. That is how we will prove that the double fibration transforms are injective. Thus, in the flag domain case, we will have a complete answer to (2.12) and some progress toward (2.13).

3. A Computable Description of M_{D}

In order to understand the structure of Z, D and M_{D} we may assume that G_{0} is simple, because G_{0} is local direct product of simple groups, and Z, D and M_{D} break up as global direct products along the local direct product decomposition of G_{0} . From this point on G_{0} is simple unless we say otherwise. We also assume that G_{0} is not compact, and we avoid the two trivial noncompact cases (see [22]), where G_{0} acts transitively on Z so that M_{D} is reduced to a single point.

As above let M_{Z} := G.C_{0}. Since C_{0} is a complex manifold, the isotropy group G_{C_{0}} contains the complexification K of K_{0} , which is a closed complex subgroup of G. If G_{0} is not Hermitian, then 𝔨 is a maximal subalgebra of 𝔤, so G_{C_{0}}/K is finite. In this nonhermitian case, without further discussion, we replace M_{Z} by the finite cover G/K. There is no loss of generality in doing this because, as we will see later, M_{D} pulls back biholomorphically to a domain in G/K under the finite covering G/K → G.C_{0} .

If G_{0} is Hermitian, then the symmetric space G_{0}/K_{0} possesses two invariant complex structures, B and \overline{B}, as bounded Hermitian symmetric domains. These are realized as open orbits D in their compact duals G/P and G/P̄ respectively. In these cases the base cycles are just the K_{0} fixed points and clearly G_{C_{0}} is either P or \overline{P} in such a situation; in particular, as opposed to being the affine homogeneous space G/K, the space M_{Z} is a compact homogeneous manifold.

The above mentioned phenomenon is characterized by G_{C_{0}} being either P or P̄. It occurs in more interesting situations than just that where D itself is a bounded domain, but it should be regarded as an exceptional case which is completely understood [21]. In particular, in any such example the cycle space M_{D} is either B or \overline{B}.

In all other cases it was recently shown that M_{D} is naturally biholomorphic to a universal domain U ⊂ M_{Z} = G/K ([10], [17]). It should be emphasized that U, which we now define, depends only on the real form G_{0}, and not on D.

The domain U can be defined in a number of different ways. We choose the historical starting point which is of a differential geometric nature.

Let M denote the Riemannian symmetric space G_{0}/K_{0} of negative curvature and consider its tangent bundle T M. As usual let θ be a Cartan involution of 𝔤 that commutes with complex conjugation over its real form 𝔤_{0} . θ defines the Cartan decompositions 𝔤 = 𝔤_{u} ⊕ i𝔤_{u} and 𝔤_{0} = 𝔨_{0} ⊕ 𝔰_{0} , where 𝔤_{u} is the compact real form of 𝔤 with 𝔨_{0} = 𝔤_{u} ∩ 𝔤_{0} .

We identify 𝔰_{0} with T M_{x_{0}} , where x_{0} is the base point with G_{0}–isotropy K_{0}, and regard T M as the homogeneous bundle G_{0} ×_{K_{0}} 𝔰_{0}. One defines the polar coordinates mapping by

Π : T M → G/K, ([g_{0}, ξ]) ↦ g_{0} exp(iξ).x_{0}.

Clearly Π is a diffeomorphism in a neighborhood of the 0–section and thus it is of interest to consider the canonically defined domain

Ω_{max} := {v ∈ T M : rank(Π_{∗}(v)) = dim T M}^{0}.

Here the connected component is that which contains the 0–section.

It turns out that Ω_{max} is determined by differential geometric properties of the compact dual N := G_{u}.x_{0} = G_{u}/K_{0}, where G_{u} is the maximal compact subgroup of G defined by 𝔤_{u}. For this it is essential that the geodesics emanating from x_{0} in N are just orbits exp(iξ).x_{0}, ξ ∈ 𝔰_{0}, of 1–parameter groups.

Let \frac{1}{2}N denote the set of points in N which are at most halfway from x_{0} to the cut point locus and define Ω_{C} = G_{0}.\frac{1}{2}N.

Theorem 3.1. (Crittenden [8]) The polar coordinates mapping Π restricts to a diffeomorphism from Ω_{max} to Ω_{C}.

The domain Ω_{C} can be computed in an elementary way. For this regard K_{0} as acting on 𝔰_{0} by the adjoint representation, let 𝔞_{0} be a maximal Abelian subalgebra in 𝔰_{0} and recall that K_{0}.𝔞_{0} = 𝔰_{0}. Thus Ω_{C} is determined by a domain in 𝔞_{0}. This is computed as follows.

Since 𝔞_{0} acts on 𝔤_{0} as a commutative algebra of self–adjoint transformations, the associated joint eigenvalues α ∈ 𝔞_{0}^{∗} are real valued. (The nonzero ones are the (𝔤_{0}, 𝔞_{0})–roots or restricted roots.) Consider the convex polygon

V = \{ξ ∈ 𝔞_{0} | |α(ξ)| < \frac{π}{2} for all restricted roots α\}

and define

U := G_{0}. exp(iV).x_{0}.

Theorem 3.2. ([8], [1]) Ω_{C} = U.

Remark 3.3. The domain U, which is indeed explicitly computable, was first brought to our attention by the work in [1]. As a consequence we originally denoted it by Ω_{AG}. It turns out that it is naturally equivalent to a number of other domains, including the cycle spaces, which are defined from a variety of viewpoints. So now, unless we have a particular construction in mind, we denote it by U to underline its universal character.

Proposition 3.4. The domain U is contractible.

Proof. For this we regard U as being contained in T M. Since V is a (polyhedral) domain in 𝔰_{0} star–shaped from 0, scalar multiplication φ_{t} : T M × T M, v ↦ tv, stabilizes V , and for 0 ≤ t ≤ 1 defines a strong deformation retraction of U to the 0–section M. That 0–section is also contractible.

Complex geometric properties of U are also of importance. These include the fact that U is a Stein domain in M_{Z} ([7]; also see [3], [16], [17], [11], [12]).

For later reference let us state the main points in the context of cycle spaces.

Theorem 3.5. If D is not one of the above exceptions where M_{D} is either a single point or B or B̄, then M_{D} = U. Thus in all cases M_{D} is contractible and Stein.

Corollary 3.6. Let P ⊂ Q be parabolic subgroups of G. Let π denote the natural projection 1P ↦ 1Q of W = G/P onto Z = G/Q. Suppose that the flag domain D ⊂ Z is not one of the above exceptions where M_{D} is either a single point or B or B̄. Let D̃ ⊂ W be a flag domain such that π(D̃) = D. Then π induces a holomorphic diffeomorphism of MD̃ onto M_{D} .

4. Globalization of the Bundles

We will show that various bundles are restrictions of bundles homogeneous under the complex group G, and use that to carry bundles over D to bundles over M_{D} . For that we need an old result on homogeneous holomorphic vector bundles from [19, Section 3].

M = A_{0}/B_{0} be a homogeneous complex manifold. Let p : A_{0} → M denote the natural projection. View the Lie algebra 𝔞_{0} , and thus its complexification 𝔞, as Lie algebras of holomorphic vector fields on M. Let m ∈ M denote the base point 1B_{0} and define 𝔭 = {ξ ∈ 𝔞 | ξ_{m} = 0}. Then 𝔭 is an Ad(B_{0})–stable complex subalgebra of 𝔞 such that 𝔞 = 𝔭 + \overline{𝔭} and 𝔟 = 𝔭 ∩ \overline{𝔭}. Here 𝔟 is the complexification of the Lie algebra 𝔟_{0} of B_{0} .

Let χ be a continuous representation of B_{0} on a finite dimensional complex vector space E = E_{χ} . By extension of χ to 𝔭 we mean a Lie algebra representation λ of 𝔭 on E such that λ|_{𝔟_{0}} = dχ and λ(Ad(b)ξ) = χ(b)λ(ξ)χ(b)^{−1} for all b ∈ B_{0} and ξ ∈ 𝔭. Thus an extension of χ to 𝔭 is a (𝔭, B_{0})–module structure on E_{χ} .

The representation χ defines a real analytic, A_{0}–homogeneous, complex vector bundle 𝔼_{χ} → M = A_{0}/B_{0} , by 𝔼_{χ} = A_{0} ×_{B_{0}} E_{χ} . We identify local sections s : U → 𝔼_{χ} with functions f_{s} : p^{−1}(U) → E_{χ} such that f_{s}(gb) = χ(b)^{−1}f_{s}(b).

The holomorphic vector bundle structures on 𝔼_{χ} → M are given as follows.

Theorem 4.1. [19, Theorem 3.6] The structures of A_{0}–homogeneous holomorphic vector bundle on 𝔼_{χ} → M are on one to one correspondence with the extensions λ of χ from B_{0} to 𝔭. The structure corresponding to λ is the one for which the holomorphic sections s over any open set U ⊂ M are characterized by ξ · f_{s} + λ(ξ)f_{s} = 0 on p^{−1}(U) for all ξ ∈ 𝔭.

Now we return to the flag domain setting and extend the double fibration (2.17) to

[figure: 𝔛_{Z}; μ̃; Z; ν̃; M_{Z}]  (4.2)

where M_{Z} = {gC_{0} | g ∈ G} and 𝔛_{Z} := {(z, C) ∈ Z × M_{Z} | z ∈ C} is the incidence space. Evidently M_{D} is an open submanifold of M_{Z} , and μ and ν are the respective restrictions of μ̃ and ν̃.

Theorem 4.3. Let D be an open G_{0}–orbit on Z, let 𝔼 → D be a G_{0}– homogeneous holomorphic vector bundle, and let q ≧ 0. Suppose G_{0} ⊂ G. Then (1) 𝔼 → D is the restriction of a G–homogeneous holomorphic vector bundle 𝔼̃ → Z, (2) the Leray derived sheaf for ν̃ is given by R^{q}(O(μ̃^{∗}𝔼̃)) = O(𝔼̃′) where 𝔼̃′ → M_{Z} is the G–homogeneous, holomorphic vector bundle with fiber H^{q}(C; O(𝔼̃|_{C})) over C ∈ M_{Z} , and (3) the Leray derived sheaf for ν is given by R^{q}(O(μ^{∗}𝔼)) = O(𝔼′) where 𝔼′ is the restriction of 𝔼̃′ to M_{D} .

Proof. We translate the result of Theorem 4.1 to our situation of the flag domain D = G_{0}(z) ≅ G_{0}/L_{0} where L_{0} = G_{0} ∩ Q_{z} is the isotropy subgroup of G_{0} at z. Here G_{0} replaces A_{0}, L_{0} replaces B_{0} , E = E_{χ} is the fiber of 𝔼 → D over z, the representation χ is the action of L_{0} on E, and 𝔮_{z} replaces 𝔭. The homogeneous holomorphic vector bundle structure on 𝔼 → D comes from an extension λ of χ from L_{0} to 𝔮_{z}. Here λ integrates to L_{0} by construction and then to Q_{z} because G_{0} ⊂ G. Thus we have a holomorphic representation χ̃ of Q_{z} on E that extends χ. That defines the G–homogeneous holomorphic vector bundle 𝔼̃ → G/Q_{z} = Z, and 𝔼 = 𝔼̃|_{D} by construction. Statement (1) is proved.

By G–homogeneity of 𝔼̃ → Z all the 𝔼̃|_{C} → C are holomorphically equivalent. Now (2) follows from the construction of Leray derived sheaves. The same considerations prove (3).

5. The Method of Schubert Slices

Schubert slices play a major role in the classification part of the above theorem (see [10] and [17]). Here we present the mini–version of this theory which is all that is needed for our applications in §6. (see [16]).

Let us begin by recalling that a Borel subgroup B of G has only finitely many orbits in Z. Such an orbit O is called a Schubert cell, O ≅ ℂ^{m(O)}, and its closure S = Ō = O ˙∪Y is referred to as the associated Schubert variety. The set S = {S} of all B –Schubert varieties freely generates the integral homology H_{∗}(Z).

If G_{0} = K_{0}A_{0}N_{0} is an Iwasawa decomposition and B ⊃ A_{0}N_{0}, then we refer to B as an Iwasawa–Borel subgroup. Recalling that a given Borel subgroup has a unique fixed point in Z = G/Q, it follows that the Iwasawa–Borel groups are exactly those which fix a point of the closed G_{0}–orbit in Z.

We now prove several elementary propositions.

Proposition 5.1. Let D be an open G_{0}–orbit in Z, C_{0} the base cycle in D and z ∈ D. Then A_{0}N_{0}.z ∩ C_{0} ≠ Ø. In particular, if B is an Iwasawa–Borel subgroup, then every B –orbit O which has non–empty intersection with D satisfies O ∩ C_{0} ≠ Ø.

Proof. Since G_{0} = A_{0}N_{0}K_{0} and C_{0} is a K_{0}–orbit in D, it is immediate that D = A_{0}N_{0}.C_{0}.

The following holds for similar reasons.

Lemma 5.2. If z ∈ D then T_{z}(K_{0}.z) + T_{z}(A_{0}N_{0}.z) = T_{z}D.

Proposition 5.3. If S is a Schubert variety of an Iwasawa–Borel subgroup B with codim_{ℂ}S > q, then S ∩ D = Ø.

Proof. Let O be the open dense B –orbit in S. If S ∩D ≠ Ø, then O ∩D ≠ Ø as well. Thus, by Proposition 5.1, O ∩ C_{0} ≠ Ø. But if z ∈ O ∩ C_{0} then dim_{ℝ} A_{0}N_{.}z < codim_{ℝ}K_{0}.z would contradict Lemma 5.2.

We now come to a basic fact.

Proposition 5.4. If codim_{ℂ}S = q and S∩D ≠ Ø, then S∩C_{0} = {z_{1}, … z_{d}} is non–empty, finite and contained in the B –orbit O. This intersection is transversal in the sense that

T_{z_{i}}C_{0} ⊕ T_{z_{i}}O = T_{z_{i}}Z

for all i. Furthermore, Σ_{i} := A_{0}N_{0}.z_{i} is open in O and closed in D.

Proof. Proposition 5.1 tells us that S ∩ C_{0} is non–empty. If S ∩ C_{0} is infinite, then it has positive dimension at one or more of its points, contrary to Lemma 5.2. The same argument proves the transversality and the fact that A_{0}N_{0}.z_{i} is open in O. If Σ_{i} were not closed in O, then we would find an A_{0}N_{0}–orbit of smaller dimension on its boundary. By Proposition 5.1 this A_{0}N_{0}–orbit would meet C_{0} , contrary to Lemma 5.2.

An A_{0}N_{0}–orbit Σ as above is called a Schubert slice. Since [C_{0}] is non–zero in H_{∗}(Z; ℤ) and S = {S} generates this homology, every Iwasawa–Borel subgroup B gives us q–codimensional Schubert varieties with non–empty intersection S∩D. In particular, there exist such Schubert slices.

It is known that D is retractable to C_{0} ([20], see also [17]) and therefore π_{1}(D) = 1. Let us translate this into a statement on the isotropy groups along the base cycle.

Lemma 5.5. If Σ is a Schubert slice and z ∈ Σ ∩ C_{0}, then

α : (K_{0})_{z} × (A_{0}N_{0})_{z} → (G_{0})_{z}, (k_{0}, a_{0}n_{0}) ↦ k_{0}a_{0}n_{0},

is bijective.

Proof. Since K_{0} is compact and A_{0}N_{0} is closed in G_{0}, it follows that α is a diffeomorphism onto a closed submanifold of G_{0}. A dimension count shows that it is open in G_{z}. But G_{z} is connected, because π_{1}(D) = 1. Thus the image of α is all of G_{z} .

We now come to the main result of this section.

Theorem 5.6. If Σ is a Schubert slice, then for every C ∈ M_{D} the intersection Σ ∩ C is transversal and consists of exactly one point.

Proof. First, we prove this for C = C_{0}. Let z_{0} ∈ C_{0} ∩ Σ. If z_{1} = a_{0}n_{0}.z_{0} were any other point in this intersection, then z_{1} = k_{0}^{−1}.z_{0} for some k_{0} ∈ K_{0}. But then g_{0} = k_{0}a_{0}n_{0} ∈ (G_{0})_{z_{0}} and consequently by Lemma 5.5 both k_{0} and a_{0}n_{0} fix z_{0}, i.e., z_{1} = z_{0} is the unique point in C_{0} ∩ Σ.

Now let C ∈ M_{D} be an arbitrary element of the cycle space. Denote by O the open B -orbit in the Schubert variety S = O ˙∪Y which contains Σ. Since O ≅ ℂ^{m(O)} is Stein, if C ∩ Σ were positive–dimensional then C ∩ Y ≠ Ø. But codim_{ℂ}Y > q and therefore Y ∩ D = Ø. Thus this would be contrary to C ⊂ D. Now C ∩ D is finite.

If C ∩ Σ is empty we obtain a contradiction as follows. Let C_{t} be a curve from C_{0} to C in M_{D} . Define

t_{0} := sup{t : C_{s} ∩ Σ ≠ Ø for all s < t}.

Then there is a sequence {z_{n}} ⊂ D with z_{n} ∈ C_{t_{n}} corresponding to {t_{n}} such that {t_{n}} → t_{0} and {z_{n}} → z_{0} ∈ bd(Σ) ⊂ bd(D). Therefore C_{t_{0}} ∩ bd(D) ≠ Ø, contrary to C_{t_{0}} ∈ M_{D}.

This argument holds for each of the Σ_{i} . In particular, C ∩ S contains at least d distinct (isolated) points. Since [C].[S] = d, it follows that |C ∩ S| = d and that the intersection at each of these points is transversal.

6. Canonical Fibrations and DFT–Injectivity

Let Σ be a Schubert slice defined by an Iwasawa–Borel subgroup B, denote {z_{0}} = Σ ∩ C_{0} , In the context of the double fibration (2.17), the projection ν carries the fiber F = μ^{−1}(z_{0}) = {(z_{0}, C) : z_{0} ∈ C} biholomorphically onto the analytic subset {C ∈ M_{D} : z_{0} ∈ C} of M_{D} .

Now consider an arbitrary element C ∈ M_{Z} with z_{0} ∈ C. By definition C = g(C_{0}) for some g ∈ G. Since z_{0} ∈ C, by adjusting g by an appropriate element of K_{0} we may assume that g ∈ Q = G_{z_{0}} . Thus F̃ ≅ {C ∈ M_{Z} : z_{0} ∈ C} = Q.C_{0} . F̃ is closed in M_{Z} , for if a net {C_{i}} in F̃ converges to C ∈ M_{Z} then z_{0} ∈ C because z_{0} ∈ C_{i} for each i, so C ∈ F̃.

The Z –analog of the double fibration (2.17) is given by (4.2). There the Q–orbit F̃ ⊂ M_{Z} is identified with the fiber μ̃^{−1}(z_{0}). In particular, its open subset F = F̃ ∩ M_{D} is a closed complex submanifold on M_{D}.

By Theorem 5.6, an A_{0}N_{0}–equivariant map φ : M_{D} → Σ is defined by mapping C to its point of intersection with Σ. The fiber over z_{0} ∈ Σ is of course F.

Let J_{0} := (A_{0}N_{0})_{z_{0}} be the A_{0}N_{0}–isotropy at the base point and note that
A_{0}N_{0} ×_{J_{0}} F → M_{D}, defined by [(a_{0}n_{0}, C)] ↦ a_{0}n_{0}(C),

is well–defined, smooth and bijective. Thus φ : M_{D} → Σ is naturally identified with the smooth A_{0}N_{0}–equivariant bundle

π_{Σ} : A_{0}N_{0} ×_{J_{0}} F → A_{0}N_{0}/J_{0} = Σ.

In this sense, every Schubert slice defines a Schubert fibration of the cycle space M_{D}.

Theorem 6.1. Let B be an Iwasawa–Borel subgroup of G and Σ an associated Schubert slice for the open orbit D. Then the fibration π_{Σ} : M_{D} → Σ is a holomorphic map on to a contractible base Σ and diffeomorphically realizes M_{D} as the product Σ × F.

Proof. Let J := B_{z_{0}} . The inclusions A_{0}N_{0} ↪ B and F ↪ F̃ together define a map A_{0}N_{0} ×_{J_{0}} F ↪ B ×_{J} F. That map realizes A_{0}N_{0} ×_{J_{0}} F ≅ M_{D} as an open subset of B ×_{J} F̃. The latter is fibered over the open A_{0}N_{0}–orbit Σ in O = B.z_{0} by the natural holomorphic projection π : B ×_{J} F̃ → B/J. Since π_{Σ} is the restriction π|_{M_{D}} , it follows that π_{Σ} is holomorphic as well.

The fact that Σ is a cell follows form the simple connectivity of the solvable group A_{0}N_{0} and the fact that it is acting algebraically.

Recall the notation: Ω_{μ}^{r}(𝔼) → 𝔛_{D} is the sheaf of relative μ^{∗}𝔼–valued holomorphic r–forms on 𝔛 with respect to μ : 𝔛_{D} → D.

Corollary 6.2. Suppose that 𝔼 → D a holomorphic G_{0}–homogeneous vector bundle which is sufficiently negative so that H^{p}(C; Ω_{μ}^{r}(𝔼)|_{C}) = 0 for p < q, and r ≧ 1. Then the double fibration transform

P : H^{q}(D, O(𝔼)) → H^{0}(M, O(𝔼′))

is injective.

Proof. M_{D} is contractible by Proposition 3.4. Since Σ is likewise contractible and M_{D} is diffeomorphic to Σ × F, it follows that F is cohomologically trivial. The Buchdahl conditions (2.4) follow. Proposition 2.5 now says that (2.3) is an isomorphism for all r. Composing with coefficient morphisms, the maps (2.6) also are isomorphisms. By Theorem 3.5 we know that the conditions (2.8) are satisfied. The assertion now follows from Theorem 2.14.

With a bit more work one can see that the fiber F of M_{D} → Σ is contractible, not just cohomologically trivial. We thank Peter Michor for showing us the following result for the C^{∞} category, from which contractibility of F is immediate. His argument is based on the existence of a complete Ehresmann connections for smooth fiber bundles.

Proposition 6.3. Let p : M → S be a smooth fiber bundle with fiber F = p^{−1}(s_{0}). If both M and S are contractible then F is contractible.

Proof. Since S is contractible and smooth, approximation gives us a smooth contraction h : [0, 1] × S → S; here h(0, s) = s and h(1, s) = s_{0} . Following [18, §9.9] the bundle p : M → S has a complete Ehresmann connection. Completeness means that every smooth curve in S has horizontal lifts to M. If m ∈ M let t ↦ H(t, m) denote the horizontal lift of t ↦ h(t, p(m)) such that H(0, m) = m. Note H(1, m) ∈ F. Fix a base point m_{0} ∈ M and a smooth contraction I : [0, 1] × M → M of M to m_{0} ; if m ∈ M then I(0, m) = m and I(1, m) = m_{0}. Denote f_{0} = H(1, m_{0}) ∈ F. Define J : [0, 1] × F by J(t, f) = H(t, I(t, f)), so J(0, f) = f and J(1, f) = f_{0} . Thus J is a contraction of F to f_{0} , and so F is contractible.

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Alan T. Huckleberry Fakultät für Mathematik Ruhr–Universität Bochum D–44780 Bochum, Germany ahuck@cplx.ruhr-uni-bochum.de Joseph A. Wolf Department of Mathematics University of California Berkeley, CA 94720–3840, USA jawolf@math.berkeley.edu

Received June 2, 2003 and in final form August 22, 2003