Formality of Koszul brackets and deformations of holomorphic Poisson manifolds

TL;DR

This paper proves that certain algebraic conditions imply the formality of Koszul brackets, with applications to deformations of holomorphic Poisson manifolds, generalizing recent results in the field.

Contribution

It establishes a criterion for the formality of Koszul brackets based on Cartan-type identities, extending to geometric contexts like Poisson and symplectic manifolds.

Findings

Lie bracket is formal under specific algebraic conditions

Generalization of Hitchin's result on holomorphic Poisson deformations

Applications to shifted de Rham complexes and Lagrangian submanifolds

Abstract

We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrangian submanifold, endowed with the Koszul bracket. As a corollary we generalize a recent result by Hitchin on deformations of holomorphic Poisson manifolds.