Higher regularity for singular K\"ahler-Einstein metrics

TL;DR

This paper investigates the regularity and convergence rates of singular Kähler-Einstein metrics near singularities, establishing polynomial convergence under certain conditions and extending known results to broader classes of tangent cones.

Contribution

It proves polynomial convergence of singular Kähler-Einstein metrics to their tangent cones when the tangent cone matches the singularity germ, generalizing previous results and including cases with non-isomorphic tangent cones.

Findings

Polynomial convergence of metrics to tangent cones when the cone matches the singularity germ.

Extension of convergence results to cases with non-isomorphic tangent cones.

Rigidity result for complete Calabi-Yau metrics with maximal volume growth.

Abstract

We study singular K\"ahler-Einstein metrics that are obtained as non-collapsed limits of polarized K\"ahler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of K\"ahler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein-Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally we prove a rigidity result for complete $\partial\bar\partial$-exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon-Hein, which applies to the case of asymptotically conical manifolds.