Weak convergence of Monge-Amp\`ere measures on compact Hermitian manifolds

TL;DR

This paper establishes a sufficient condition for the weak convergence of Monge-Ampère measures associated with bounded -plurisubharmonic functions on compact Hermitian manifolds, aiding in solving the Monge-Ampère equation.

Contribution

It provides a new criterion for the weak convergence of Monge-Ampère measures on Hermitian manifolds, facilitating the construction of bounded solutions to the Monge-Ampère equation.

Findings

Established a sufficient condition for weak convergence of measures.

Applied the criterion to obtain bounded solutions to the Monge-Ampère equation.

Enhanced understanding of -plurisubharmonic functions on Hermitian manifolds.

Abstract

We give a sufficient condition on a sequence of uniformly bounded $\omega$-plurisubharmonic functions, $\omega$ being a Hermitian metric, for which the sequence of associated Monge-Amp\`ere measures converges weakly. This criterion can be used to obtained a bounded $\omega$-plurisubharmonic solution to the Monge-Amp\`ere equation.