Locally conformally Kähler metrics on Hopf surfaces
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)]).
scientific article; zbMATH DE number 1220355
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