Complex Hessian measures with respect to a background Hermitian form

TL;DR

This paper develops a potential theory for m-subharmonic functions on Hermitian manifolds, extending complex Hessian equations and establishing foundational properties like capacity and quasi-continuity.

Contribution

It introduces a well-defined complex Hessian operator for bounded functions and extends pluripotential theory results to Hermitian manifolds.

Findings

Defined the complex Hessian operator for bounded functions.

Established the quasi-continuity of m-subharmonic functions.

Extended results on complex Hessian equations to Hermitian manifolds with boundary.

Abstract

We develop potential theory for $m$-subharmonic functions with respect to a Hermitian metric on a Hermitian manifold. First, we show that the complex Hessian operator is well-defined for bounded functions in this class. This allows to define the $m$-capacity and then showing the quasi-continuity of $m$-subharmonic functions. Thanks to this we derive other results parallel to those in pluripotential theory such as the equivalence between polar sets and negligible sets. The theory is then used to study the complex Hessian equation on compact Hermitian manifold with boundary, with the right hand side of the equation admitting a bounded subsolution. This is an extension of a recent result of Collins and Picard dealing with classical solutions.