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 Kahler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a complete Kahler-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 Kahler-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 Kahler-Einstein metric with negative curvature by using Calabi Ansatz. At last, we show that these two metrics are indeed the same.