On polynomial convergence to tangent cones for singular K\"ahler-Einstein metrics

TL;DR

This paper establishes a precise criterion for when singular K"ahler-Einstein metrics are conical at a point, linking geometric degeneration theory with curvature growth conditions, and applies this to algebraic singularities.

Contribution

It provides a new characterization of conical singularities in K"ahler-Einstein metrics using Donaldson-Sun's degeneration theory and extends the understanding to algebraic singularities with specific degeneration properties.

Findings

Conicality of metrics is equivalent to a specific degeneration condition.

Curvature growth bounds are crucial for conicality characterization.

Results apply to both metric limits and algebraic singularities.

Abstract

Let $(Z,p)$ be a pointed Gromov-Hausdorff limit of non-collapsing K\"ahler-Einstein metrics with uniformly bounded Ricci curvature. We show that the singular K\"ahler-Einstein metric on $Z$ is conical at $p$ if and only if $\mathcal C=W$ in Donaldson-Sun's two-step degeneration theory, assuming curvature grows at most quadratically near $p$. Let $(X,p)$ be a germ of an isolated log terminal algebraic singularity. Following Hein-Sun's approach, we show that if $\mathcal C=W$ in the two-step stable degeneration of $(X,p)$ and $\mathcal C$ has a smooth link, then every singular K\"ahler-Einstein metric on $X$ with non-positive Ricci curvature and bounded potential is conical at $p$.