Monge-Amp\`ere operators on compact K\"ahler manifolds

TL;DR

This paper investigates the complex Monge-Ampère operator on compact Kähler manifolds, providing a comprehensive description of its range on specific classes of functions, with applications to complex dynamics and Kähler-Einstein metrics.

Contribution

It generalizes local pluripotential theory results to the compact setting, describing the operator's range on functions with $L^2$ gradient and finite energy.

Findings

Complete description of the Monge-Ampère operator's range on certain functions.

Applications to complex dynamics.

Results relevant to Kähler-Einstein metrics on singular manifolds.

Abstract

We study the complex Monge-Amp\` ere operator on compact K\"ahler manifolds. We give a complete description of its range on the set of $\omega-$plurisubharmonic functions with $L^2$ gradient and finite self energy, generalizing to this compact setting results of U.Cegrell from the local pluripoltential theory. We give some applications to complex dynamics and to the existence of K\"ahler-Einstein metrics on singular manifolds.