Toward a classification of compact complex homogeneous spaces.
type: Item
instance of: scholarly article
title: Toward a classification of compact complex homogeneous spaces.
zbMATH Open document ID: 1050.22025
published in: Journal of Algebra
publication date: 2004-03-14T00:00:00Z
review text: The author considers: \(M\) a compact complex homogeneous manifold, \(G\) a connected complex Lie group acting almost effectively, holomorphically and transitively on \(M,\Gamma\) a discrete cocompact subgroup of \(G\) such that \(M=G/ \Gamma\), \(R\) the radical of \(G\). The following beautiful and important theorems which give a classification of compact complex homogeneous spaces are proved. Theorem A. If \(S\) is a Levi subgroup of \(G\), then any simple factor of \(S\) which acts nontrivially on \(R\) is of type \(A_\ell\). Theorem B. Let \(G,\Gamma, S\), and \(R\) be as in Theorem A. \(S_1\) (respectively \(S_2)\) be the normal subgroup of \(S\) such that each simple factor of \(S_1\) (respectively of \(S_2)\) acts trivially (respectively nontrivially) on \(R\). Then, up to a finite covering, \(G/ \Gamma\) is a torus fiber bundle over a product \(S_1/ \Gamma_1\times S_2R/ \Gamma_2C^0\), where \(C_0\) is the identity component of the center \(C\) of \(G\) and \(\Gamma_1\) (respectively \(\Gamma_2)\) is a discrete subgroup of \(S_1\) (respectively \(S_2 R/C^0)\). In particular, \(S_2\) has only factors of type \(A_\ell\). Theorem C. Let \(M\) be a compact complex homogeneous space. Then \(M\), up to a finite covering, is a torus bundle over \(S_1/H_1\times S_2 R/H_2\) with \(S_1\), \(S_2\) semisimple and \(R\) being the radical of \(S_2 R\) such that each factor of \(S_2\) acts nontrivially on \(R\). If \(J_1= N_{S_1} (H_1^0)\), \(J_2=N_{S_2R} (H_2^0)\), then \(J_2/H_1^0\) is semisimple and \(H_2^0\cap S_2\) is nilpotent, \(J_2\) has only simple factors of type \(A_\ell\), which are not in \(H_2\). Moreover, each simple factor of \(S_2\) is a classical Lie group and each simple factor of \(J_2\) acts nontrivially on \(R/R\cap H\). A classification of the data, for each type of classical Lie algebra, is given in the lists from Theorem D. Theorem E. The reduced primitive compact complex homogeneous spaces, up to actions of some torus, are isogenous on the Lie alg…
Mathematics Subject Classification ID: 22F30
Mathematics Subject Classification ID: 57S15
zbMATH DE Number: 2055657
zbMATH Keywords: Homogeneous spaces
zbMATH Keywords: Product
zbMATH Keywords: Fiber bundles
zbMATH Keywords: Complex manifolds
zbMATH Keywords: Parallelizable manifolds
zbMATH Keywords: Discrete subgroups
zbMATH Keywords: Classifications
author: Daniel Guan
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.11.007
OpenAlex ID: W2098021439
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DOI: 10.1016/J.JALGEBRA.2003.11.007
description: scientific article; zbMATH DE number 2055657