Formality of the Dolbeault complex and deformations of holomorphic Poisson manifolds

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.