A note on locally conformally Kaehler structures and small deformations of Hopf manifolds

Abstract

A Hopf manifold is a compact complex manifold of which the universal covering is C^n\{0}. In this note we show that any Hopf manifold admits a locally conformally Kaehler structure (shortly lcK structure), by constructing a complex analytic family around a Hopf manifold of diagonal type, which admits a lcK potential, and applying a well known fact (due to Ornea and Verbitsky) that the property of lcK potential is preserved under a complex analytic family over a sufficiently small parameter space.