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 ${\mathrm{\partial }}_{}\bar{\mathrm{\partial }}$–lemma or ${\mathrm{\partial }}_{\pi }\bar{\mathrm{\partial }}$–lemma. Under these assumptions,we show that the Koszul–Brylinski homology can be recovered by the Dolbeault cohomology, and prove that the DGLA $(A_M^{\bullet ,\bullet },\bar{\mathrm{\partial }},[-,-{]}_{{\mathrm{\partial }}_{\pi }})$ is formal.Furthermore,we discuss the Maurer–Cartan elements of $(A_M^{\bullet ,\bullet }[[t]],\bar{\mathrm{\partial }},[-,-{]}_{{\mathrm{\partial }}_{\pi }})$ which induce the deformations of complex structure of $M$.