Formality of the Dolbeault complex and deformations of holomorphic Poisson manifolds

TL;DR

This paper investigates the properties of holomorphic Poisson manifolds under certain lemmas, demonstrating formality of associated DGLA and exploring deformations of complex structures via Maurer--Cartan elements.

Contribution

It establishes the formality of a specific DGLA for holomorphic Poisson manifolds and links Koszul--Brylinski homology to Dolbeault cohomology under certain conditions.

Findings

Koszul--Brylinski homology recovered by Dolbeault cohomology

DGLA $(A_M^{ullet,ullet},ar{ abla},[-,-]_{ abla})$ is formal

Maurer--Cartan elements induce deformations of complex structure

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 $M$.