On the metric structure of some non-K\"ahler complex threefolds

Abstract

We introduce a class of hermitian metrics with {\em Lee potential}, that generalize the notion of l.c.K. metrics with potential introduced in \cite{ov} and show that in the classical examples of Calabi and Eckmann of complex structures on $S^{2p+1}\x S^{2q+1}$, the corresponding hermitian metrics are of this type. These examples satisfy, actually, a stronger differential condition, that we call {\em generalized Calabi-Eckmann}, condition that is satisfied also by the {\em Vaisman} metrics (previously also refered to as {\em generalized Hopf manifolds}). This condition means that, in addition to being with Lee potential, the torsion of the {\em characteristic} (or Bismut) connection is parallel. We give a local geometric characterization of these generalized Calabi-Eckmann metrics, and, in the case of a compact threefold, we give detailed informations about their global structure. More precisely, the cases which can not be reduced to Vaisman structures can be obtained by deformation of locally homogenous hermitian manifolds that can be described explicitly.