Pluripotential Chern-Ricci Flows

TL;DR

This paper develops a pluripotential approach to degenerate parabolic complex Monge-Ampère equations on compact Hermitian manifolds, establishing existence, uniqueness, and regularity of solutions, and applying these results to the weak Chern-Ricci flow on singular varieties.

Contribution

It extends pluripotential theory to Hermitian manifolds and proves existence, uniqueness, and regularity results for the weak Chern-Ricci flow on complex varieties with singularities.

Findings

Existence and uniqueness of pluripotential solutions to the equations.

Solutions are semi-concave in time and continuous in space.

Application to the weak Chern-Ricci flow on varieties with log terminal singularities.

Abstract

Extending a recent theory developed on compact K\"ahler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds. Under natural assumptions on the Cauchy boundary data, we show that the pluripotential solution is semi-concave in time and continuous in space and that such a solution is unique. We also establish a partial regularity of such solutions under some extra assumptions of the densities and apply it to prove the existence and uniqueness of the weak Chern-Ricci flow on complex compact varieties with log terminal singularities.