Pluripotential Monge-Amp{\`e}re flows in Big Cohomology Classes

TL;DR

This paper develops a new pluripotential approach to Monge-Ampère flows in big cohomology classes, establishing existence, uniqueness, and regularity of solutions on compact Kähler manifolds, with applications to Kähler-Ricci flows.

Contribution

It introduces a Perron method-based framework for pluripotential Monge-Ampère flows, proving regularity and uniqueness of solutions in big cohomology classes.

Findings

The upper envelope of subsolutions is continuous and semi-concave.

Unique pluripotential solutions with regularity are obtained.

Applications to Kähler-Ricci flows on complex varieties.

Abstract

We study pluripotential complex Monge-Amp\`ere flows in big cohomology classes on compact K{\"a}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{\"a}hler-Ricci flows on compact K{\"a}hler manifolds of general type as well as on stable varieties.