Continuity of the complex Monge-Amp\`ere operator on compact Hermitian manifolds

TL;DR

This paper investigates the continuity properties of the complex Monge-Ampère operator on compact Hermitian manifolds and constructs weak solutions to the associated equations under specific conditions.

Contribution

It establishes new results on the operator's continuity and demonstrates the existence of weak solutions assuming a smooth subsolution exists.

Findings

Proved continuity (or weak convergence) of the complex Monge-Ampère operator on compact Hermitian manifolds.

Constructed weak solutions to the complex Monge-Ampère equation under the assumption of a smooth subsolution.

Abstract

In this note, we establish several results concerning the continuity (or weak convergence) of the complex Monge-Amp\`ere operator on compact Hermitian manifolds. At the end of this note, we find a weak solution of the complex Monge-Amp\`ere equation on a compact Hermitian manifold under the assumption of the existence of a smooth subsolution.