Poisson traces, D-modules, and symplectic resolutions

作     者：Etingof, Pavel Schedler, Travis 

作者机构：MIT 77 Massachusetts Ave Cambridge MA 02139 USA Imperial Coll London London England 

出 版 物：《LETTERS IN MATHEMATICAL PHYSICS》 (数理物理学快报)

年 卷 期：2018年第108卷第3期

页      面：633-678页

核心收录：

学科分类：07[理学] 070201[理学-理论物理] 0702[理学-物理学] 

基　　金：NSF [DMS-1502244  DMS-1406553] 

主　　题：Hamiltonian flow Complete intersections Milnor number D-modules Poisson homology Poisson varieties Poisson homology Poisson traces Milnor fibration Calabi-Yau varieties Deformation quantization Kostka polynomials Symplectic resolutions Twistor deformations 

摘      要：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.