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 $$H^{p,0}_{BC}(M)=0$$. As a special case, if M admits a Gauduchon metric $$\omega $$ such that the Aeppli class of $$\omega ^{n-1}$$ vanishes, then $$H^{1,0}_{BC}(M)=0$$.