Compact locally conformal Kähler nilmanifolds
type: Item
instance of: scholarly article
title: Compact locally conformal Kähler nilmanifolds
zbMATH Open document ID: 0601.53035
DOI: 10.1007/BF00182906
author: Luis A. Cordero
author: Manuel de León
author: Marisa Fernández
published in: Geometriae Dedicata
publication date: 1986-01-01T00:00:00Z
review text: A generalized Hopf manifold is a locally conformal Kähler manifold whose Lee form is parallel but not exact [\textit{I. Vaisman}, Geom. Dedicata 13, 231--255 (1982; Zbl 0506.53032)]. The main non-Kähler example of such a manifold is \(S^1\times S^{2k+1}\), \(k\geq 1\). In this paper, the authors construct a large family of compact generalized Hopf manifolds \(M(r,1)=N(r,1)\times S^1\) of dimension \(2r+2\) not admitting Kähler structures. The manifold \(N(r,1)\) is a compact quotient of the generalized Heisenberg group \(H(r,1)\) [\textit{Y. Haraguchi}, Thèse, Univ. du Haute Alsace, Mulhouse, 1981]. In fact \(M(1,1)\) is the Kodaira-Thurston manifold [cf. \textit{W. P. Thurston}, Proc. Am. Math. Soc. 55, 467--468 (1976; Zbl 0324.53031)]. Nevertheless \(M(1,1)\) is a symplectic manifold, and the authors prove that \(M(r,1)\) can have no symplectic structures for \(r\geq 2\). Another interesting result is proved: the minimal model of \(M(r,1)\) is not formal. This fact enlarges the topological differences between compact generalized Hopf manifolds and compact Kähler manifolds, since, as it is well known, the minimal model of a compact Kähler manifold is formal. The complex analog \(N_{\mathbb{C}}(r,1)\) of \(N(r,1)\) is also considered and the following result is proved: \(N_{\mathbb{C}}(r,1)\) can have no structure of a generalized Hopf manifold for \(r\geq 1\).
reviewed by: P. M. Gadea
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3969574
zbMATH Keywords: generalized Hopf manifold
zbMATH Keywords: Kähler structures
zbMATH Keywords: Heisenberg group
zbMATH Keywords: symplectic structures
zbMATH Keywords: minimal model
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00182906
OpenAlex ID: W2044726311
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description: scientific article; zbMATH DE number 3969574