Aeppli cohomology and Gauduchon metrics

Abstract

Let $(M,J,g,\omega )$ be a complete Hermitian manifold of complex dimension $n\ge 2$. 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 $H_{BC}^{p,0}(M)=0$. As a special case, if $M$ admits a Gauduchon metric $\omega$ such that the Aeppli class of ${\omega }^{n-1}$ vanishes, then $H_{BC}^{1,0}(M)=0$.

10 pages