ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Pages 90–92 (August 29, 1997) S 1079-6762(97)00028-0

CLASSIFICATION OF COMPACT COMPLEX HOMOGENEOUS SPACES WITH INVARIANT VOLUMES

DANIEL GUAN

(Communicated by Svetlana Katok)

Abstract. In this note we give a classification of compact complex homogeneous spaces with invariant volume.

1. Introduction

We call a 2n-dimensional manifold M a complex homogeneous space with invariant volume if there is a complex structure and a nonzero 2n-form on M such that a transitive Lie transformation group keeps both the complex structure and the 2n-form invariant. There are many papers published in the direction of classification of such manifolds, e.g., [Bo], [DG1], [DG2], [DG3], [DN], [Gu1], [Gu2], [Ha1], [Hk], [HK], [Kz], [Mt1], [Mt2], [Wa2] and the references there (see also [BR], [Gu1], [Gu2], [Gu4], [Ti], [Wa1] for related topics involving compact complex homogeneous spaces). In this paper we will finish the classification in the compact case.

A major break-through in this direction became possible after the following two results were established. Firstly, the Hano-Kobayashi fibration of a compact complex homogeneous space with invariant volume (we might also call it the Ricci form reduction) is holomorphic and coincides with the anticanonical fibration (see [DG1]). Secondly, one can classify compact complex homogeneous spaces with invariant pseudo-Kähler structure (see [DG1], [Hk] and [Gu1], [Gu2], also [Gu4]).

The proof is much harder than the corresponding proof in the Kähler case in [Mt1]. Namely, in the Kähler case one can choose the transitive group to be compact. Then the isotropy group is a subgroup of an orthogonal group. In particular, both groups are reductive.

In [Hk] Huckleberry observed that one can handle the pseudo-Kähler case using methods from symplectic geometry. In particular, he applied here the construction of the moment map. In [Gu1], [Gu2] we observed that his method actually works for a compact complex homogeneous space with an invariant symplectic structure.

Huckleberry’s method was used in [Gu1], [Gu2] to get a structure theorem for compact homogeneous complex manifolds with 2-cohomology classes ω such that ω^{n} ≠ 0 in the top cohomology. This generalized the result of [BR] for the Kähler case (one does not assume that the Kähler form is invariant).

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors May 30, 1997.

1991 Mathematics Subject Classification. Primary 53C15, 57T15, 53C30, 53C56, 53C50.

Key words and phrases. Invariant volume, homogeneous, product, fiber bundles, complex manifolds, parallelizable manifolds, discrete subgroups, classifications.

Supported by NSF Grant DMS-9401755 and DMS-9627434.

For a general compact complex homogeneous space with invariant volume, the symplectic method does not apply. However, our original method (see [HK], [Mt2] and [DG1]) gives a classification.

Main Theorem 1. Every compact complex homogeneous space with an invariant volume form is a homogeneous complex torus bundle over the product of a projective rational homogeneous space and a parallelizable manifold. Conversely, every complex homogeneous manifold M of this kind admits a transitive real transformation Lie group G, acting on M by holomorphic transforms and preserving a volume form on M.

For more details concerning the structure theorem, one might look at Sections 3, 4 and 5 of [Gu3]. We also note that every compact complex homogeneous space M with a 2-cohomology class such that its top power is nonzero in the top cohomology group, admits a transitive real Lie transformation group G, which acts on M by holomorphic transforms and preserves a volume form.

Our proof is better than the proof of both results in [DG1] and [Gu1].

In [Mt2] Matsushima considered the special case of a semisimple group action. He proved that if G/H is a compact complex homogeneous space with a G-invariant volume and if G is semisimple, then G/H is a holomorphic fiber bundle over a projective rational homogeneous space, the typical fiber being a complex parallelizable homogeneous space of a reductive complex Lie group.

Applying our Main Theorem 1 to this situation, we immediately see that the result of Matsushima can be generalized to the case when G is reductive. Moreover, we have the following stronger result.

Main Theorem 2. Assume that G/H is a compact complex homogeneous space with a G-invariant volume and G is reductive. Then G/H is a holomorphic torus bundle over the product of a projective rational homogeneous space and a complex parallelizable homogeneous space of a semisimple complex Lie group.

Even if we drop here the assumption that G/H has a G-invariant volume form, we still have a holomorphic fibration of G/H over a projective rational base with parallelizable fiber (see [BR], [Ti]). J. Hano [Ha2] proved that the fiber is of the form L/Γ, where Γ is a discrete cocompact subgroup of a reductive complex Lie group L. Our methods show that the converse is also true. Namely, we have the following theorem.

Main Theorem 3. Suppose that a compact complex homogeneous space M admits a holomorphic fibration π : M → D, where D is a projective rational homogeneous space. Assume that the typical fiber of π is of the form F = L/Γ, where L is a connected reductive complex Lie group, and Γ a discrete cocompact subgroup of L. Then any transitive effective complex Lie transformation group G, acting on M by holomorphic transforms, is reductive.

Corollary. Every compact complex homogeneous space is a holomorphic fiber bundle whose base is a compact complex homogeneous space of a reductive Lie group and whose typical fiber is a complex parallelizable homogeneous space of a nilpotent complex Lie group.

Having in mind a classification of all compact complex homogeneous spaces as a goal, we hope to use the Corollary in our future research.

Acknowledgement

I thank the referee for pointing out [Ha2] to me and many invaluable suggestions for the composition of this paper.

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Department of Mathematics, Princeton University, Princeton, NJ 08544

E-mail address: zguan@math.princeton.edu