LVM Manifolds and lck Metrics

In this paper, we compare two types of complex non-Kähler manifolds: LVM and lck manifolds. First, lck manifolds (for locally conformally Kähler manifolds) admit a metric which is locally conformal to a Kähler metric. On the other side, LVM manifolds (for López de Medrano, Verjovsky and Meersseman) are quotients of an open subset of \({\mathbb {C}}^n\) ${\mathbb{C}}^n$ by an action of \({\mathbb {C}}^*\times {\mathbb {C}}^m\) ${\mathbb{C}}^{\ast }\times {\mathbb{C}}^m$ . LVM and lck manifolds have a fundamental common point: Hopf manifolds which are a specific case of LVM manifolds and which admit also lck metric. Therefore, the question of this paper is:

Are LVM manifolds lck ?

We provide some answers to this question. The results obtained are as follows. In the set of all LVM manifolds, there is a dense subset of LVM manifolds which are not lck. And if we consider lck manifolds with potential (whose metric derives from a potential), the diagonal Hopf manifolds are the only LVM manifolds which admit an lck metric with potential. However, we show that there exists an lck covering with potential (non-compact) of a certain subclass of LVM manifolds. Finally, we present some examples.