Aeppli cohomology and Gauduchon metrics

Abstract

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