Compact locally conformal Kähler nilmanifolds (Q1081130)

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 \(r\geq 1\).