Monge-Ampere measures on pluripolar sets

TL;DR

This paper extends the theory of complex Monge-Ampere equations to measures with highly singular parts on pluripolar sets, broadening classical results and generalizing Kolodziej's subsolution theorem.

Contribution

It introduces a method to solve the complex Monge-Ampere equation for measures with large singular parts, generalizing previous results on discrete singularities.

Findings

Solved Monge-Ampere equations for measures with large singular parts

Generalized classical results by Demailly, Lelong, Lempert

Extended Kolodziej's subsolution theorem

Abstract

In this article we solve the complex Monge-Ampere equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Kolodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge-Ampere measure, then it is a complex Monge-Ampere measure.