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

18 pages. Comments are welcome! arXiv admin note: text overlap with arXiv:2202.09764

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