Quadratic presentations and nilpotent Kähler groups
type: Item
instance of: scholarly article
title: Quadratic presentations and nilpotent Kähler groups
zbMATH Open document ID: 0835.57014
DOI: 10.1007/BF02921801
author: James A. Carlson
author: Domingo Toledo
published in: The Journal of Geometric Analysis
publication date: 1996-04-16T00:00:00Z
review text: A Kähler group is the fundamental group of a compact Kähler manifold. The Lie algebra of the Mal'tsev completion of the fundamental group is quadratically presented if it is the quotient of the free Lie algebra on its abelianization by an ideal generated in degree two. It is known that the Lie algebras of such groups do not admit quadratic presentations. One purpose of this paper is to obtain an infinite family of examples of quadratically presented three-step nilpotent Lie algebras and to classify quadratically presented complex nilpotent Lie algebras with abelianization of dimension at most five. The second purpose of this article is to prove that if \(M\) is a compact Kähler manifold with nilpotent fundamental group which is not almost abelian and if \(\omega\) is a non-zero integral element of type (1,1) in its characteristic subspace \(\mathcal C\), then the rank of \(\omega\) is at least 8. Several applications of the above results are also given.
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 57R15
zbMATH DE Number: 828775
zbMATH Keywords: Kähler group
zbMATH Keywords: fundamental group of a compact Kähler manifold
zbMATH Keywords: Mal'cev completion
zbMATH Keywords: quadratic presentations
zbMATH Keywords: three-step nilpotent Lie algebras
zbMATH Keywords: abelianization
zbMATH Keywords: nilpotent fundamental group
zbMATH Keywords: characteristic subspace
reviewed by: Bang-Yen Chen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921801
OpenAlex ID: W1985502139
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description: scientific article; zbMATH DE number 828775