Aeppli cohomology and Gauduchon metrics

TL;DR

This paper proves vanishing results for certain cohomology groups on Hermitian manifolds under conditions involving boundedness of powers of the metric form, with implications for Gauduchon metrics and Aeppli cohomology.

Contribution

It establishes new vanishing theorems for Bott-Chern cohomology on Hermitian manifolds with bounded Aeppli classes of metric powers.

Findings

Vanishing of $H^{p,0}_{BC}(M)$ under boundedness conditions.

Aeppli class of $ ext{ω}^{n-p}$ vanishing implies cohomology vanishing.

Results apply to compact Hermitian manifolds with Gauduchon metrics.

Abstract

Let $(M,J,g,\omega)$ be a complete Hermitian manifold of complex dimension $n\ge2$. Let $1\le p\le n-1$ and assume that $\omega^{n-p}$ is $(\partial+\overline{\partial})$-bounded. We prove that, if $\psi$ is an $L^2$ and $d$-closed $(p,0)$-form on $M$, then $\psi=0$. In particular, if $M$ is compact, we derive that if the Aeppli class of $\omega^{n-p}$ vanishes, then $H^{p,0}_{BC}(M)=0$. As a special case, if $M$ admits a Gauduchon metric $\omega$ such that the Aeppli class of $\omega^{n-1}$ vanishes, then $H^{1,0}_{BC}(M)=0$.