International Mathematics Research Notices Advance Access published online on September 25, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp144
Topology of Locally Conformally Kähler Manifolds with Potential
1 Faculty of Mathematics, University of Bucharest, 14 Academiei Street, 70109 Bucharest, Romania and Institute of Mathematics "Simion Stoilow" of the Romanian Academy, 21, Calea Grivitei Street 010702-Bucharest, Romania
2 Theoretical, Mathematical and String Physics lab, Institute of Theoretical and Experimental Physics B. Cheremushkinskaya, 25, Moscow, 117259, Russia
Correspondence: Correspondence to be sent to: verbit{at}mccme.ru
Locally conformally Kähler (LCK) manifolds with potential are those which admit a Kähler covering with a proper, automorphic, global potential. The existence of a potential can be characterized cohomologically as vanishing of a certain cohomology class, called the Bott–Chern class. Compact LCK manifolds with potential are stable at small deformations and admit holomorphic embeddings into Hopf manifolds. This class strictly includes the Vaisman manifolds. We show that every compact LCK manifold with potential can be deformed into a Vaisman manifold. Therefore, every such manifold is diffeomorphic to a smooth elliptic fibration over a Kähler orbifold. We show that the pluricanonical condition on LCK manifolds introduced by G. Kokarev is equivalent to vanishing of the Bott–Chern class. This gives a simple proof of some of the results on topology of pluricanonical LCK manifolds, discovered by Kokarev and Kotschick.