Mixed Hessian inequalities on Hermitian manifolds and applications

TL;DR

This paper establishes a weak convergence theorem for complex Hessian operators on Hermitian manifolds, proving a general mixed Hessian inequality and demonstrating the existence of bounded solutions to complex Hessian equations with controlled measures.

Contribution

It introduces a Ko extl{}odziej-Nguyen type convergence theorem and a general mixed Hessian inequality on Hermitian manifolds, extending previous results to broader settings.

Findings

Proved a weak convergence theorem for complex Hessian operators.

Established a general mixed Hessian inequality on Hermitian manifolds.

Demonstrated the existence of bounded solutions to complex Hessian equations under capacity domination.

Abstract

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. In this paper we establish a Ko\l odziej-Nguyen type weak convergence theorem of complex Hessian operators. Utilizing this result, we prove a general mixed Hessian inequality with respect to a background Hermitian metric, covering both local and global case. As an application, we prove the existence of bounded solutions of complex Hessian equations where the right-hand side measure is well dominated by capacities.