Formality of Koszul brackets and deformations of holomorphic Poisson manifolds

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.

V2 exposition improved; new examples included; references added. We learned from Florian Schaetz that Theorem 1.2 is originally due to Sharygin and Talalaev arXiv:math/0503635. V3 misprints corrected and references added