Dolbeault cohomology of compact complex homogeneous manifolds
type: Item
instance of: scholarly article
title: Dolbeault cohomology of compact complex homogeneous manifolds
zbMATH Open document ID: 0935.32017
DOI: 10.1007/BF02837763
author: Vimala Ramani
author: Parameswaran Sankaran
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1999-12-02T00:00:00Z
review text: If a complex transformation group \(G\) acts on a complex manifold \(M\), there is an induced action of \(G\) on the Dolbeault cohomology of \(M\). This action must be trivial if \(M\) is compact Kähler, but may be non-trivial in general (as was shown by examples of Kodaira and Lescure). The authors prove the following result: Let \(M\) be a compact complex homogeneous manifold which can be realized as a fiber bundle with a flag variety as base and a torus \(T\) as fiber. Then the connected component \(\Aut_1(M)\) of the automorphism group of \(M\) acts trivially on the Dolbeault cohomology of \(M\). For certain classes of these manifolds they compute the Dolbeault cohomology and the Picard group. In remark 2.3 the authors raise the question whether \(\Aut_1(M)\) is algebraic if \(T\) is an abelian variety. The reviewer would like to point out that the answer is positive: There is a short exact sequence \(1 \to T \to \Aut_1(M)\to S\to 1\) where \(S\) is a semisimple Lie group. The universal covering \(\widetilde S\to S\) is finite and the pull-back \(\widetilde G\to\widetilde S\) of the bundle \(G=\Aut_1(M)\to S\) to \(\widetilde S\) yields a short exact sequence \(1\to T\to \widetilde G\to \widetilde S\to 1\) which is split, because \(\widetilde S\) is semi-simple and simply-connected. Therefore \(\widetilde G\simeq\widetilde S\times T\) is an algebraic group as well as \(G=\Aut_1(M)\) which is a quotient of \(\widetilde G\) by a finite subgroup.
reviewed by: Jörg Winkelmann
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 1340031
zbMATH Keywords: homogeneous complex manifolds
zbMATH Keywords: Dolbeault cohomology
zbMATH Keywords: generalized Hopf manifolds
zbMATH Keywords: Stiefel manifolds
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02837763
OpenAlex ID: W2018702721
description: scientific article; zbMATH DE number 1340031