Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds

Abstract

We study compact complex manifolds bimeromorphic to locally conformally Kahler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kahler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim $$\ge 1$$ ) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds.