# LCK metrics on complex spaces with quotient singularities

## TL;DR

This paper extends the concept of locally conformally Kaehler (LCK) metrics to complex spaces with quotient singularities, establishing conditions for their existence and behavior under blow-ups.

## Contribution

It introduces LCK metrics on singular complex spaces, characterizes their existence on quotient singularities, and shows stability under blow-ups.

## Key findings

- LCK metrics exist on quotient singularities if the universal cover admits a compatible Kähler metric.
- Blow-ups at points of LCK complex spaces preserve the LCK property.

## Abstract

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