Integrability of $(ω,m)$-subharmonic functions on compact Hermitian manifolds
Yuetong Fang
TL;DR
This work proves that on a compact Hermitian manifold $(X,\omega)$, every $(\omega,m)$-subharmonic function belongs to $L^p(X,\omega^n)$ for all $p<\frac{n}{n-m}$. The authors adapt the Hessian potential theory to the Hermitian setting by developing a variant volume-capacity framework based on $\widetilde{\mathrm{Cap}}_{\omega,m}$, addressing torsion and non-positivity issues that arise when $m<n$. The key contribution is the derivation of a volume-capacity inequality and a decay estimate for sublevel-set capacities, which via a layer-cake argument yield the desired $L^p$ integrability. This extends the Dinew–Kołodziej approach beyond the Kähler case and sharpens the understanding of Hessian equations on Hermitian manifolds, recovering known results in the Kähler limit.
Abstract
Let $(X,ω)$ be a compact Hermitian manifold of dimension $n$. We show that all $(ω,m)$-subharmonic functions are $L^p$ integrable on $X$, for any $p < \frac{n}{n-m}$.