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

Cited by in corpus (9)

• Shifted Poisson Structures and Deformation Quantization
• Smoothing pairs over degenerate Calabi-Yau varieties
• Poisson quasi-Nijenhuis manifolds and the Toda system
• Formality of the Dolbeault complex and deformations of holomorphic Poisson manifolds
• Endomorphisms of Koszul complexes: formality and application to deformation theory
• Uniqueness and intrinsic properties of non-commutative Koszul brackets
• Poisson structures on twistor spaces of hyperkaehler and HKT manifolds
• An operadic proof of the BTT Theorem
• Logarithmic $\partial \bar{\partial }$-lemma and several geometric applications (with an Appendix joint with Sheng Rao)