Complex Hessian measures with respect to a background Hermitian form
Slawomir Kolodziej, Ngoc Cuong Nguyen
TL;DR
The paper develops a comprehensive potential theory for $m$-subharmonic functions with respect to a background Hermitian form on Hermitian manifolds, aiming to handle Hessian-type equations beyond the Kähler setting.It defines Hessian measures for bounded $m- ext{ω}$-subharmonic functions through an inductive, order-zero current limit, and builds a robust capacity theory with quasi-continuity and convergence results for wedge products.Key contributions include Cauchy–Schwarz-type inequalities in the Hermitian context, a well-posed wedge-product framework for bounded functions, a weighted Sobolev structure, a comparison principle, and a full Dirichlet problem theory for complex Hessian equations with bounded subsolutions, extending known results to non-Kähler manifolds.These developments enable weak solutions under minimal regularity assumptions and provide a blueprint for solving Hessian equations on compact Hermitian manifolds with boundary, with potential impact on geometric PDE and complex geometry in non-Kähler settings.
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.