Aeppli cohomology and Gauduchon metrics

Abstract

Let (M, J, g, ω) be a complete Hermitian manifold of complex dimension n ≥ 2. Let 1 ≤ p ≤ n − 1 and assume that ω n−p is (∂ + ∂)-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.