LCK metrics on complex spaces with quotient singularities

Abstract

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.

Coverings of locally conformally Kähler complex spaces

“…2. This generalizes a result in [2]: Theorem 3.10 Let X be a complex space. Then X admits an LCK metric if and only if its universal coveringX admits a Kähler metric such that the deck automorphisms act onX by positive homothethies of the Kähler metric.…”

“…Inspired by the characterization given in Remark 3.2, the following definition for LCK spaces was first introduced in [2], adapting the most well-suited of the equivalent definitions of LCK manifolds. For technical reasons, we introduce at the same time what we call locally conformally preKähler metrics.…”

“…Notable results about the stability of Kähler structures were obtained by [11]. Recently, in [2], the authors gave a natural definition of the same type for LCK spaces and studied the properties of its universal cover, proving, using resolutions of singularities, that, if the space has quotient singularities, the result from the smooth case holds true i.e. the universal cover is Kähler.…”

“…In this paper, we firstly improve the main result from [2] by removing any requirement on the singularity types and in fact finding a way to not mention the singularities at all. Instead, we develop a method for defining closed 1-forms on complex spaces (without first saying what a 1-form is) and integrating them along curves.…”

“…In Sect. 3, we are then able to improve the result from [2] by completely recovering the characterization of LCK spaces as quotients of Kähler spaces by homotheties of its Kähler metric. More precisely, we prove: Theorem 3.10 Let X be a complex space.…”