Uniqueness of Tangent Cone of Kahler Einstein Metrics on Singular Varieties with Crepant Singularities

TL;DR

This paper proves the uniqueness of tangent cones at any point for Kahler Einstein metrics on singular Calabi-Yau varieties with crepant singularities, extending understanding of metric behavior near singularities.

Contribution

It establishes the uniqueness of tangent cones for Kahler Einstein metrics on singular varieties with crepant singularities, a significant advancement in geometric analysis.

Findings

Tangent cones at any point are unique for these metrics.

The result applies to Ricci flat and negative scalar curvature Kahler Einstein currents.

Enhances understanding of metric singularities in Calabi-Yau varieties.

Abstract

Let $(X, L)$ be a polarized Calabi Yau variety (or canonical polarized variety) with crepant singularity. Suppose $\omega_{KE} \in c_1(L)$ (or $\omega_{KE} \in c_1(K_X)$) is the unique Ricci flat current (or Kahler Einstein current with negative scalar curvature) with local bounded potential constructed in [18], we show that the local tangent at any point $p \in X$ of metric $\omega_{KE}$ is unique