Nash entropy, Calabi energy and geometric regularization of singular K\"ahler metrics

TL;DR

This paper establishes uniform Sobolev bounds for solutions on K"ahler manifolds with bounded Nash entropy and Calabi energy, linking singular K"ahler metrics to RCD spaces and their geometric regularization.

Contribution

It introduces new uniform Sobolev estimates for solutions on K"ahler manifolds and connects singular K"ahler metrics with RCD space theory, providing examples related to projective varieties.

Findings

Uniform Sobolev bounds for Laplace solutions on K"ahler manifolds.

Connection between singular K"ahler metrics and RCD spaces.

Examples of RCD spaces topologically and holomorphically equivalent to projective varieties.

Abstract

We prove uniform Sobolev bounds for solutions of the Laplace equation on a general family of K\"ahler manifolds with bounded Nash entropy and Calabi energy. These estimates establish a connection to the theory of RCD spaces and provide abundant examples of RCD spaces topologically and holomorphically equivalent to projective varieties. Suppose $X$ is a normal projective variety that admits a resolution of singularities with relative nef or relative effective anti-canonical bundle. Then every admissible singular K\"ahler metric on $X$ with Ricci curvature bounded below induces a non-collapsed RCD space homeomorphic to the projective variety $X$ itself.