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

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 $1\le m\le n$ 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.