An obstruction to homogeneous manifolds being Kähler.
type: Item
instance of: scholarly article
title: An obstruction to homogeneous manifolds being Kähler.
zbMATH Open document ID: 1070.32017
author: Bruce Gilligan
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116187
review text: Throughout this paper consider the setting where \(\Gamma\) is discrete in the connected complex Lie group \(G\) and there is a homogeneous fibration \(X:G/\Gamma \to G/I:=Y\) where \(I^0\) is an abelian normal subgroup of \(G\) and the fiber \(I^0/(I^0\cap \Gamma)\) of this fibration is an abelian complex Lie group that has no noncompact holomorphic functions. Such complex Lie groups are called Cousin groups. Since \(I^0\) is normal in \(G\) set \(\widehat{G}:= G/I^0\) and \(\widehat{\Gamma}:=I/I^0\). Then \(Y=\widehat{G}/\widehat{I}\) with \(\widehat{G}\) acting almost effectively. If \(X\) is Kähler one would like to determine if \(Y\) is Kähler too. Write \(\widehat{G}=\widehat{S} \ltimes \widehat{R}\), where \(\widehat{R}\) is the radical of \(\widehat{G}\) and \(\widehat{S}\) is a maximal complex semisimple subgroup of \(\widehat{G}\). If \(Y=\widehat{G}/\widehat{\Gamma}\) were Kähler then \(\widehat{S}\cap \widehat{\Gamma}\) would be finite. Actually, the author shows that no element of infinite order in \(\widehat{\Gamma}\) is contained in a semisimple group of \(\widehat{G}\), equivalently \(\widehat{S}\cap\widehat{\Gamma}\) is finite for every semisimple subgroup \(\widehat{S}\) of \(\widehat{G}\). Another result is the following one. Assume \(\widehat{G}\) is a linear algebraic group whose radical \(\widehat{R}\) is a vector group with maximal semisimple subgroup of \(\widehat{G}\) acting on \(\widehat{R}\) by a linear representation that has no nonzero invariant vector. Then \(G/\Gamma\) does not admit a Kähler structure. A particular example of such a linear algebraic group \(\widehat{G}\) is the affine group of \(\mathbb C^n\) for \(n >1\).
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 2162472
zbMATH Keywords: homogeneous complex manifolds
zbMATH Keywords: Kähler manifolds
reviewed by: Gabriela Paola Ovando
MaRDI profile type: Publication
OpenAlex ID: W2490919422
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DOI: 10.5802/AIF.2097
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2162472