Holonomic Poisson manifolds and deformations of elliptic algebras

TL;DR

This paper introduces holonomic Poisson manifolds with a nondegeneracy condition, showing their deformation spaces are finite-dimensional and applying this to elliptic algebras related to Feigin and Odesskii's work.

Contribution

It defines holonomic Poisson structures, analyzes their properties, and proves deformation-invariance of certain elliptic algebra families.

Findings

Holonomic Poisson manifolds have finite-dimensional deformation spaces.

Deformation-invariance established for families of elliptic algebras.

Structural role of divergence of Hamiltonian vector fields highlighted.

Abstract

We introduce a natural nondegeneracy condition for Poisson structures, called holonomicity, which is closely related to the notion of a log symplectic form. Holonomic Poisson manifolds are privileged by the fact that their deformation spaces are as finite-dimensional as one could ever hope: the corresponding derived deformation complex is a perverse sheaf. We develop some basic structural features of these manifolds, highlighting the role played by the divergence of Hamiltonian vector fields. As an application, we establish the deformation-invariance of certain families of Poisson manifolds defined by Feigin and Odesskii, along with the "elliptic algebras" that quantize them.