Locally conformally Kähler metrics on Hopf surfaces
type: Item
instance of: scholarly article
title: Locally conformally Kähler metrics on Hopf surfaces
zbMATH Open document ID: 0917.53025
author: Paul Gauduchon
author: Liviu Ornea
publication date: 1998-11-09T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75311
review text: A Hopf surface \(M\) is a compact complex surface whose universal covering is \(\mathbb{C}^2- \{(0,0)\}\). Primary Hopf surfaces are those \(M\) having their fundamental group isomorphic to \(\mathbb{Z}\), generated by the transformation \(\gamma: (u,v)\to (\alpha u+\lambda v^m,\beta v)\), \((u,v)\in \mathbb{C}^2- \{(0,0)\}\), \(m\) being some integer and \(\alpha, \beta, \lambda\in \mathbb{C}\) such that \(|\alpha|\geq |\beta|> 1\) and \((\alpha- \beta^m)\lambda= 0\). Following \textit{R. Harvey} and \textit{H. B. Lawson jun.} [Invent. Math. 74, 169-198 (1983; Zbl 0553.32008)], the primary Hopf surfaces fall into two disjoint classes: (1) of class 1, those whose fundamental group is generated by a transformation \(\gamma\) as above for which \(\lambda =0\), and (2) of class 0, corresponding to \(\lambda\neq 0\) and \(\alpha= \beta^m\) for some positive integer \(m\). The following theorem is proved in this paper: ``Each primary Hopf surface admits a locally conformal Kähler metric. Moreover, each primary Hopf surface of class 1 admits a locally conformal Kähler metric with parallel (non-zero) Lee form.'' An explicit construction of such metrics, as well as an explicit description of the corresponding Sasakian geometry, are given. (Locally conformal Kähler metrics have been first considered by \textit{I. Vaisman} [Trans. Am. Math. Soc. 259, 439-447 (1980; Zbl 0435.53044); ibid. 262, 533-542 (1980; Zbl 0446.53048)]).
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1220355
zbMATH Keywords: Hopf surface
zbMATH Keywords: locally conformal Kähler metric
zbMATH Keywords: Sasakian structure
zbMATH Keywords: contact form
zbMATH Keywords: Killing vector field
zbMATH Keywords: deformation
reviewed by: Luis A. Cordero
MaRDI profile type: Publication
OpenAlex ID: W2074952544
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DOI: 10.5802/AIF.1651
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1220355