LCK metrics on complex spaces with quotient singularities
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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.
Mathematics Subject Classification32C15 53C55
Ovidiu Preda is grateful to Professor Liviu Ornea for guiding his learning of geometry which led to the problem studied in this article. Both authors are thankful to Alexandra Otiman for repeatedly taking the time to answer their questions.
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