Higher regularity for singular Kähler–Einstein metrics

Abstract

We study singular Kähler–Einstein metrics that are obtained as noncollapsed limits of polarized Kähler–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 Kähler 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 and 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 $\partial \overline{\partial }$-exact Calabi–Yau metrics with maximal volume growth. This generalizes a result of Conlon and Hein, which applies to the case of asymptotically conical manifolds.