Victor Vuletescu
University of Bucharest, Bucharest, Romania

An LCK (locally conformally Kähler) manifold is a Hermitian manifold which admits a Galois cover which has a Kähler metric such that the deck group is acting by holomorphic homotheties. The product of two LCK manifolds does not have a natural product LCK structure. It is believed that a product of two compact complex manifolds is never LCK.

We classify all known examples of compact LCK manifolds in two classes: manifolds containing a curve and manifolds of Inoue type. In the talk we will outline the description of these classes and the proof that a product of a compact complex manifold and an LCK manifold belonging to one of these classes above does not admit an LCK structure. The talk is based on joined work with L. Ornea and M. Verbitsky.