Holonomic Poisson manifolds and deformations of elliptic algebras

Brent Pym, Travis Schedler

Research output: Chapter in Book/Report/Conference proceedingChapter (peer-reviewed)peer-review


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 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.
Original languageEnglish
Title of host publicationGeometry and Physics: Volume 2
Subtitle of host publicationA Festschrift in honour of Nigel Hitchin
EditorsAndrew Dancer, Jørgen Ellegaard Andersen, Oscar García-Prada
PublisherOxford University Press
Number of pages23
ISBN (Print)9780198802020
Publication statusPublished - 1 Nov 2018


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