Mixed Hessian inequalities and uniqueness in the class $\mathcal{E}(X,\omega,m)$

TL;DR

This paper establishes a general inequality for mixed Hessian measures, simplifies the complex Monge-Ampère case, and proves the uniqueness of solutions to the complex Hessian equation on compact Kähler manifolds under certain measure conditions.

Contribution

It introduces a new inequality for mixed Hessian measures, simplifies existing methods for the complex Monge-Ampère equation, and proves uniqueness of solutions in specific measure scenarios.

Findings

Proved a general inequality for mixed Hessian measures.

Simplified the approach for the complex Monge-Ampère equation.

Established uniqueness of solutions for the complex Hessian equation with measures vanishing on m-polar sets.

Abstract

We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Amp\`ere equation. Exploiting this and using Ko{\l}odziej's mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact K\"ahler manifolds in the case of probability measures vanishing on $m$-polar sets.