Poisson traces, D-modules, and symplectic resolutions

Abstract

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein–Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.

Untitled

“…Let ρ : X → X be a symplectic resolution with X affine, then HP 0 (O(X)) = H dimX ( X). Conjecture 2.27 holds in many cases (see Examples 6.4 − 6.7 in [13] for details):…”

“…On the other hand, if the equivalence does not hold, there exists a nontrivial bimodule M in the kernel or cokernel of one of the maps in (13)…”

Tamarkin–tsygan Calculus and Chiral Poisson Cohomology

“…In general, Poisson (co)homology is somewhat difficult to compute; it is not even known when it is finite (cf. [4] for results in this direction); nonetheless, one can compute it in the case that π is nondegenerate; and in all cases, one can compute the totalization H π * (M ) S 1 .…”

On the derived ring of differential operators on a singularity

“…In [HK84] and later in [ES09] (see also [ES18]) a certain quotient M (X) of our compact generator D X was considered which governs the invariants under Hamiltonian flows. It was used to define a new homology theory which fuses Poisson homology with the de Rham cohomology, which is particularly nice in the case of symplectic singularities.…”