Complex Hessian measures with respect to a background Hermitian form

Kołodziej, Sławomir; Nguyen, Ngoc Cuong

doi:10.2140/apde.2026.19.107

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Complex Hessian measures with respect to a background Hermitian form

Sławomir Kołodziej and Ngoc Cuong Nguyen

Vol. 19 (2026), No. 1, 107–166

DOI: 10.2140/apde.2026.19.107

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 us to define the $m$ -capacity and then show the quasicontinuity 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.

To the memory of Jean-Pierre Demailly

Keywords

complex Hessian measure, weak solutions, Hermitian manifolds, capacity

Mathematical Subject Classification

Primary: 32U20, 32W50, 53C55, 58J05

Milestones

Received: 3 September 2023

Revised: 28 November 2024

Accepted: 2 February 2025

Published: 26 November 2025

Authors

Sławomir Kołodziej
Faculty of Mathematics and Computer Science Jagiellonian University Krakow Poland

Ngoc Cuong Nguyen
Department of Mathematical Sciences KAIST Daejeon South Korea

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Vol. 19, No. 1, 2026