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

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 $p$.