Poisson traces, D-modules, and symplectic resolutions
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 sym...
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| Format: | Article |
| Language: | English |
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Springer Netherlands
2018
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| Online Access: | http://hdl.handle.net/1721.1/114161 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1826203774583046144 |
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| author | Schedler, Travis Etingof, Pavel I |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Schedler, Travis Etingof, Pavel I |
| author_sort | Schedler, Travis |
| collection | MIT |
| description | 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. Keywords:
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 |
| first_indexed | 2024-09-23T12:42:50Z |
| format | Article |
| id | mit-1721.1/114161 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:42:50Z |
| publishDate | 2018 |
| publisher | Springer Netherlands |
| record_format | dspace |
| spelling | mit-1721.1/1141612022-10-01T10:41:16Z Poisson traces, D-modules, and symplectic resolutions Schedler, Travis Etingof, Pavel I Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I 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. Keywords: 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 National Science Foundation (U.S.) (Grant DMS-1502244) 2018-03-14T20:05:37Z 2018-03-14T20:05:37Z 2017-12 2017-10 2018-02-20T05:31:50Z Article http://purl.org/eprint/type/JournalArticle 0377-9017 1573-0530 http://hdl.handle.net/1721.1/114161 Etingof, Pavel, and Travis Schedler. “Poisson Traces, D-Modules, and Symplectic Resolutions.” Letters in Mathematical Physics, vol. 108, no. 3, Mar. 2018, pp. 633–78. https://orcid.org/0000-0002-0710-1416 en http://dx.doi.org/10.1007/s11005-017-1024-1 Letters in Mathematical Physics Creative Commons Attribution http://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Netherlands Springer Netherlands |
| spellingShingle | Schedler, Travis Etingof, Pavel I Poisson traces, D-modules, and symplectic resolutions |
| title | Poisson traces, D-modules, and symplectic resolutions |
| title_full | Poisson traces, D-modules, and symplectic resolutions |
| title_fullStr | Poisson traces, D-modules, and symplectic resolutions |
| title_full_unstemmed | Poisson traces, D-modules, and symplectic resolutions |
| title_short | Poisson traces, D-modules, and symplectic resolutions |
| title_sort | poisson traces d modules and symplectic resolutions |
| url | http://hdl.handle.net/1721.1/114161 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT schedlertravis poissontracesdmodulesandsymplecticresolutions AT etingofpaveli poissontracesdmodulesandsymplecticresolutions |