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

Abstract

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