Log-concavity of volume and complex Monge-Amp\`ere equations with prescribed singularity

TL;DR

This paper proves existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularities on compact Kähler manifolds, confirming a conjecture on log-concavity of volume and linking it to convex geometry.

Contribution

It extends previous results by removing the small unbounded locus assumption and works with general model type singularities in big cohomology classes.

Findings

Confirmed a conjecture on log-concavity of volume of positive currents.

Established a connection between complex Monge-Ampère equations and convex geometry.

Extended existence and uniqueness results to broader singularity types.

Abstract

Let $(X,\omega)$ be a compact K\"ahler manifold. We prove the existence and uniqueness of solutions to complex Monge-Amp\`ere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the K\"ahler case as well. As an application we confirm a conjecture by Boucksom-Eyssidieux-Guedj-Zeriahi concerning log-concavity of the volume of closed positive $(1,1)$-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn-Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and $P$-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.