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.
Highlights
We show that M(X ) ∼= δμ−g ⊕ M(X )ind, where M(X )ind is an indecomposable D-module containing δ2g, such that M(X )ind/δ2g is an indecomposable extension of δg by the intersection cohomology D-module IC(X ). (Here δ is the δ-function D-module supported at the vertex of X .)
TX is restricted to the case that X is smooth, which implies that TX is a vector bundle, but in general TX need not be locally free; see, e.g., [54, pp. 88–89] for a reference for TX in general.) Let Vect(X ) := (X, TX ), which is a Lie algebra whose elements are called global vector fields on X, and which is a module over the ring O(X ) of global functions
If X admits a symplectic resolution ρ : X → X, by [37, Theorem 2.5], it has finitely many symplectic leaves: for every closed irreducible subvariety Y ⊆ X which is invariant under Hamiltonian flow, if U ⊆ Y is the open dense subset such that the map ρ|ρ−1(U) : ρ−1(U ) → U is generically smooth on every fiber ρ−1(u), u ∈ U, U is an open subset of a leaf
Summary
The proof of this more general result is similar: we define a canonical D-module M(X, g), obtained by dividing D(V ) by the equations of X and by g, and show that its underived direct image H 0π∗ M(X, g) to a point is O(X )/{g, O(X )} and that it is holonomic if X has finitely many g-leaves In this setting, the result is well known if the action of g integrates to an action of a connected algebraic group G with LieG = g, and in this case M(X, g) is regular (see, e.g., [57, Lemma 1], [40, Theorem 4.1.1], [35, Section 5]); cf Remark 6.10 below. In the appendix we review background on D-modules used in the body of the paper
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