Formality of the Dolbeault complex and deformations of holomorphic Poisson manifolds

Abstract

The purpose of this paper is to study the properties of holomorphic Poisson manifolds (M,π) under the assumption of ∂∂¯–lemma or ∂π∂¯–lemma. Under these assumptions, we show that the Koszul–Brylinski homology can be recovered by the Dolbeault cohomology, and prove that the DGLA (AM•,•,∂¯,[−,−]∂π) is formal. Furthermore, we discuss the Maurer–Cartan elements of (AM•,•[[t]],∂¯,[−,−]∂π) which induce the deformations of complex structure of M.