Continuity of singular K\"ahler-Einstein potentials

TL;DR

This paper studies the regularity of solutions to degenerate complex Monge-Ampère equations on singular spaces, proving continuity of Kähler-Einstein potentials at singularities and on Calabi-Yau varieties.

Contribution

It establishes the continuity of solutions to DCMAE on singular spaces and extends this to Kähler-Einstein potentials on various singular varieties.

Findings

Kähler-Einstein potentials are continuous at isolated singularities.

Solutions to DCMAE are globally continuous when the reference class is in the Néron-Severi group.

Continuity of Kähler-Einstein potentials on irreducible Calabi-Yau varieties.

Abstract

In this note, we investigate some regularity aspects for solutions of degenerate complex Monge-Amp\`ere equations (DCMAE) on singular spaces. First, we study the Dirichlet problem for DCMAE on singular Stein spaces, showing a general continuity result. A consequence of our results is that K\"ahler-Einstein potentials are continuous at isolated singularities. Next, we establish the global continuity of solutions to DCMAE when the reference class belongs to the real N\'eron-Severi group. This yields in particular the continuity of K\"ahler-Einstein potentials on any irreducible Calabi-Yau variety.