Toward a Classification of Compact Complex Homogeneous
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 273 (2004) 33–59 www.elsevier.com/locate/jalgebra Toward a classification of compact complex homogeneous spaces Daniel Guan 1 Department of Mathematics, University of California at Riverside, Riverside, CA 92521, USA Received 19 October 2001 Communicated by Robert Kottwitz Abstract In this note, we prove some results on the classification of compact complex homogeneous spaces. We first consider the case of a parallelizable space M = G/Γ ,whereG is a complex connected Lie group and Γ is a discrete cocompact subgroup of G. Using a generalization of results in [M. Otte, J. Potters, Manuscripta Math. 10 (1973) 117–127; D. Guan, Trans. Amer. Math. Soc. 354 (2002) 4493–4504, see also Electron. Res. Announc. Amer. Math. Soc. 3 (1997) 90], it will be shown that, up to a finite covering, G/Γ is a torus bundle over the product of two such quotients, one where G is semisimple, the other where the simple factors of the Levi subgroups of G are all of type Al.Inthe general case of compact complex homogeneous spaces, there is a similar decomposition into three types of building blocks. 2004 Elsevier Inc. All rights reserved. Keywords: Homogeneous spaces; Product; Fiber bundles; Complex manifolds; Parallelizable manifolds; Discrete subgroups; Classifications 1. Introduction In this paper, M is a compact complex homogeneous manifold and G a connected complex Lie group acting almost effectively, holomorphically, and transitively on M.We refer to the literature [5,6,8–12,14,21,22,31,32] quoted at the end of this paper for the classification of complex homogeneous spaces which are pseudo-kählerian, symplectic or admit invariant volume forms.
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