In this paper, we study the properties of coverings of locally conformally Kähler (LCK) spaces with singularities. We begin by proving that a space is LCK if any only if its universal cover is Kähler, thereby generalizing a result from Ioniţȃ and Preda (Manuscripta Math,

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

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In this paper we present a result for the local Steinness problem: if is a locally Stein open subset of a Stein space X , does it follow that is itself Stein? We will prove that if X is normal, then for every sequence of points (x n ) n which tends to a limit x ∈ ∂ Sing(X ), there exists a holomorphic function f on which is unbounded on (x n ) n . Then, we will use this result to obtain a characterization theorem for a particular case of the Serre problem.