On the construction of a complete Kahler-Einstein metric with negative scalar curvature near an isolated log-canonical singularity

Abstract

In this short note we are concerned with the Kähler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a complete Kähler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold, which provides a local model for the future study of the global Kähler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is a desired one. In the second approach, we obtain a complete Kähler-Einstein metric with negative curvature by using Calabi Ansatz. At last, we show that these two metrics are indeed the same.

Kahler-Einstein metric near an isolated log canonical singularity

“…Remark 5.2. Another interesting example of isolated log canonical singularity unifomized by the Bergman metric is proved in [16]. Let A be an abeliean variety with complex dimension n and N a negative line bundle on A.…”

“…Remark 5.3. The invariant Kähler-Einstein metrics in Lemma 5.1 and [16] have a system of quasi coordinates in a punctured neighborhood of the isolated log canonical singularities. This is the main property we will use in the following proof of Corollary 1.1.…”

“…For example, a metric uniformization is obtained in [23,24] for isolated log canonical singularity in complex dimension 2. Another interesting example of cones over an abelian variety is constructed in [16]. We apply the method of bounded geometry by [9] (see also [23,24,38]) to prove the following rigidity result.…”