I will give an overview of joint work with T. Schedler, in which we introduce the notion of a "holonomic" Poisson manifold---one whose local derived deformation spaces are finite dimensional. Using D-modules, we give a checkable geometric criterion for holonomicity in terms of the divergence of Hamiltonian vector fields. As an application, we establish the deformation-invariance of certain families of Poisson structures on moduli spaces of bundles over elliptic curves, defined by Feigin and Odesskii.