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, pi) under the assumption of partial differential partial differential over line -lemma or partial differential pi partial differential over line -lemma. Under these assumptions, we show that the Koszul-Brylinski homology can be recovered by the Dolbeault cohomology, and prove that the DGLA (A center dot,center dot M , partial differential over line , [-, -] partial differential pi) is formal. Furthermore, we discuss the Maurer- Cartan elements of (A center dot,center dot M [[t]], partial differential over line , [-, -] partial differential pi) which induce the deformations of complex structure of M.(c) 2022 Elsevier B.V. All rights reserved.