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Polar sets for $m$-subharmonic functions on compact Hermitian manifolds

Slawomir Kolodziej, Ngoc Cuong Nguyen

TL;DR

This work extends global pluripotential theory to $(oldsymbol{ heta},m)$-subharmonic functions on compact Hermitian manifolds by introducing the global $m$-capacity $cap_m(E)$ and the Hessian measure $H_m(u)=(oldsymbol{ heta}+dd^c u)^m\wedge oldsymbol{ heta}^{n-m}$. It proves a sharp decay for sublevel sets, establishes uniform integrability of normalized $(oldsymbol{ heta},m)$-subharmonic functions, and develops a robust wedge-product calculus including weak convergence and comparison principles. The paper then shows that cap_m is comparable to a family of local capacities cap'_m, derives quasi-continuity with respect to cap_m, and provides a complete characterization of polar sets via extremal functions $h_E^*$ and $V_E^*$, generalizing Josefson-type results to the Hessian setting on Hermitian manifolds. The results offer a comprehensive framework for global Hessian equations on non-Kähler backgrounds, with implications for complex geometry and fully nonlinear PDEs in Hermitian contexts.

Abstract

We prove a sharp decay of capacity of sublevel sets of a $(ω,m)$-subharmonic functions on a $n$-dimensional compact Hermitian manifold $(X,ω)$ which generalizes the case $m=n$ as well as the case $1\leq m\leq n$ on a compact Kähler manifold. We also obtain the full characterizations of polar sets of such functions in terms of the corresponding local and global capacities, and of the extremal functions.

Polar sets for $m$-subharmonic functions on compact Hermitian manifolds

TL;DR

This work extends global pluripotential theory to -subharmonic functions on compact Hermitian manifolds by introducing the global -capacity and the Hessian measure . It proves a sharp decay for sublevel sets, establishes uniform integrability of normalized -subharmonic functions, and develops a robust wedge-product calculus including weak convergence and comparison principles. The paper then shows that cap_m is comparable to a family of local capacities cap'_m, derives quasi-continuity with respect to cap_m, and provides a complete characterization of polar sets via extremal functions and , generalizing Josefson-type results to the Hessian setting on Hermitian manifolds. The results offer a comprehensive framework for global Hessian equations on non-Kähler backgrounds, with implications for complex geometry and fully nonlinear PDEs in Hermitian contexts.

Abstract

We prove a sharp decay of capacity of sublevel sets of a -subharmonic functions on a -dimensional compact Hermitian manifold which generalizes the case as well as the case on a compact Kähler manifold. We also obtain the full characterizations of polar sets of such functions in terms of the corresponding local and global capacities, and of the extremal functions.

Paper Structure

This paper contains 12 sections, 33 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Hessian measures for bounded functions
  3. $(\omega,m)$-subharmonic functions
  4. Wedge product of bounded $(\omega,m)$-subharmonic functions
  5. Weak convergence
  6. Weak comparison principles
  7. Cauchy-Schwarz's inequality
  8. Uniform constants and integral symbols
  9. Basic integral estimates
  10. Uniform integrability
  11. capacity
  12. Characterization of polar sets

Key Result

Theorem 1.1

Let $v_0\in SH_m(X,\omega)$ be such that $\sup_X v_0 =0$. There exists a uniform constant $A$ depending only on $\omega, m,n$ such that In particular, $cap_m(P) =0$ if $P$ is a globally $m$-polar set.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: KN3
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4: weak comparison principle
  • Corollary 2.5
  • Corollary 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 50 more