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

Abstract

We prove a sharp decay of capacity of sublevel sets of a $(\ddot{I}‰,m)$-subharmonic functions on a $n$-dimensional compact Hermitian manifold $(X,\ddot{I}‰)$ which generalizes the case $m=n$ as well as the case $1\le m\le n$ 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.

31 pages