$C^{1,1}$ regularity for degenerate complex Monge-Amp\`ere equations and geodesic rays

TL;DR

This paper establishes a $C^{1,1}$ regularity estimate for solutions to degenerate complex Monge-Ampère equations on compact Kähler manifolds, with applications to geodesic rays and quasi-psh envelopes.

Contribution

It extends previous regularity results by proving a $C^{1,1}$ estimate in degenerate settings and applies it to geodesic rays and envelopes in Kähler geometry.

Findings

Proves $C^{1,1}$ regularity for degenerate complex Monge-Ampère solutions.

Shows local $C^{1,1}$ regularity of geodesic rays in Kähler metrics.

Establishes regularity of quasi-psh envelopes away from the non-Kähler locus.

Abstract

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Amp\`ere equations on compact K\"ahler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of K\"ahler metrics associated to a test configuration, as well as the local $C^{1,1}$ regularity of quasi-psh envelopes in nef and big classes away from the non-K\"ahler locus.