\(D\)-modules, Poisson homology, and symplectic resolutions

Travis Schedler (Imperial)

Frank Adams 2,

Given a Lie algebra of vector fields, one can ask: Is the coinvariant space of functions (or its dual, the invariant distributions) finite-dimensional? I will explain a simple construction, joint with Etingof, using \(D\)-modules, which implies that the result is finite-dimensional if there are finitely many leaves under the flow. In the case that the variety is Poisson, this recovers the zeroth Poisson homology. In the case the variety is singular and admits a symplectic resolution, this homology conjecturally agrees with the top cohomology of the resolution. To obtain all cohomology, one defines a new theory, the Poisson-de Rham homology, on the singularity. As time permits, I will explain various generalisations, such as recent work with Brent Pym concerning smooth Poisson varieties under the modular Hamiltonian flow, and with Thomas Bitoun on the lengths of \(D\)-modules related to the \(b\)-function.

Import this event to your Outlook calendar
▲ Up to the top