LVM manifolds and lck metrics

Abstract In this paper, we compare two type of complex non-Kähler manifolds : LVM and lck manifolds. First, lck manifolds (for locally conformally Kähler manifolds) admit a metric which is locally conforme to a Kähler metric. On the other side, LVM manifolds (for S.Lopez de Medrano, A.Verjovsky and L.Meersseman) are quotient of an open subset of C^n by an action of C*xC^m. LVM and lck manifolds have a fondamental common point : Hopf manifolds which are a specific case of LVM manifolds and which admit also lck metric. So 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 which are not lck. An 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. Based on this observation, we show that there exists an lck covering with potential (non-compact) of a certain subclass of LVM manifolds. Finally, we present some examples. Mathematics Subject Classification (2000). Primary 32J27, 32L05, 32M99, 32Q15, 32Q60, 32T15, 32V99; Secondary 53A30, 53B35, 53C55, 53C56.