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

TL;DR

This paper establishes precise decay rates for the capacity of sublevel sets of $( heta,m)$-subharmonic functions on compact Hermitian manifolds, extending known results from Kähler to Hermitian settings and characterizing polar sets via capacities.

Contribution

It generalizes decay estimates and polar set characterizations for subharmonic functions from Kähler to Hermitian manifolds, covering the full range of $m$.

Findings

Decay rates of capacity for sublevel sets are sharp and explicitly characterized.

Full characterization of polar sets in terms of capacities and extremal functions.

Extension of results from Kähler to Hermitian manifolds for all $m$.

Abstract

We prove a sharp decay of capacity of sublevel sets of a $(\omega,m)$-subharmonic functions on a $n$-dimensional compact Hermitian manifold $(X,\omega)$ which generalizes the case $m=n$ as well as the case $1\leq m\leq n$ on a compact K\"ahler 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.