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

TL;DR

This paper constructs a complete Kähler-Einstein metric with negative scalar curvature near certain singularities, using two different methods, and shows they are equivalent, aiding the understanding of singular varieties.

Contribution

It introduces two approaches to construct and compare a complete Kähler-Einstein metric near log-canonical singularities, providing a local model for singular variety metrics.

Findings

The singularity is uniformized by a complex ball.

The induced metric from the Bergman metric is a valid Kähler-Einstein metric.

The two constructed metrics are proven to be the same.

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.