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Mathematics, Algebraic Geometry, Mathematical Physics, Quantum Algebra, Symplectic Geometry
Abstract:
We introduce a natural non-degeneracy 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 modular vector field, which measures the failure of volume forms to be preserved by Hamiltonian flows . 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.
