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.

中文翻译：

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具有商奇点的复杂空间上的 LCK 度量

在本文中，我们介绍了从复杂流形到具有奇点的复杂解析空间的局部共形 Kähler 度量的推广，并研究了局部共形 Kähler 流形的哪些性质在这个新设置中仍然成立。我们证明，如果一个复解析空间只有商奇点，那么当且仅当它的全覆盖允许 Kähler 度量，使得甲板自同构通过 Kähler 度量的同位起作用时，它才允许局部保形 Kähler 度量。我们还证明了 LCK 复空间的某个点的爆炸也是 LCK。