Complex Hessian measures with respect to a background Hermitian form

2023/08/21 by Sławomir Kołodziej, Ngoc Cuong Nguyen, Kolodziej, Slawomir +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2308.10405

openalex publication_date 2023/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

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.

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• Continuity of solutions to complex Hessian equations on compact Hermitian manifolds

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