LCK metrics on complex spaces with quotient singularities

Abstract

In this article we introduce a generalization of locally conformally Kahler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kahler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singularities, then it admits a locally conformally Kahler metric if and only if its universal cover admits a Kahler metric such that the deck automorphisms act by homotheties of the Kahler metric. We also prove that the blow-up at a point of an LCK complex space is also LCK.