Poisson Traces for Symmetric Powers of Symplectic Varieties

Etingof, Pavel; Schedler, Travis

doi:10.1093/imrn/rnt031

Cited by 8 publications

(19 citation statements)

References 27 publications

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“…In this case one has the resolution . In this case, Conjecture 6.1 (c) (and hence the entire conjecture) follows from [ 24 , Theorem 1.17], which gives a direct computation of M ( X ) in this case; see Sect. 7 for more details.…”

Section: Conjectures On Symplectic Resolutionsmentioning

confidence: 98%

“…We obtain from the previous example the resolution , and we can compose this with to obtain the resolution . In this case, Conjecture 6.1 is proved in [ 24 , Sect. 1.3], using the main result of [ 22 ] together with [ 32 , Theorem 3]: see Sect.…”

Section: Conjectures On Symplectic Resolutionsmentioning

confidence: 99%

“…In this section we would like to discuss results from [ 24 ] on the zeroth Poisson homology of symmetric powers. We continue to set .…”

Section: Symmetric Powers and Hilbert Schemesmentioning

confidence: 99%

“…[ 24 , Theorem 1.1] Let Y be an affine symplectic variety. Then, there is a canonical isomorphism of graded algebras, where the grading is given by (on both sides of the isomorphism), and .…”

Section: Symmetric Powers and Hilbert Schemesmentioning

confidence: 99%

“…By a well-known result from D-module theory ([ 44 , Theorem 1.2.2], see also [ 45 , 1.4]), this implies that . (For a slightly different argument avoiding [ 44 , Theorem 1.2.2], see [ 24 ], Lemma 2.1).…”

Section: Symmetric Powers and Hilbert Schemesmentioning

confidence: 99%

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Poisson traces, D-modules, and symplectic resolutions

Etingof

Schedler

2017

Lett Math Phys

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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.

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Section: Conjectures On Symplectic Resolutionsmentioning

confidence: 98%

Section: Conjectures On Symplectic Resolutionsmentioning

confidence: 99%

“…In this section we would like to discuss results from [ 24 ] on the zeroth Poisson homology of symmetric powers. We continue to set .…”

Section: Symmetric Powers and Hilbert Schemesmentioning

confidence: 99%

Section: Symmetric Powers and Hilbert Schemesmentioning

confidence: 99%

Section: Symmetric Powers and Hilbert Schemesmentioning

confidence: 99%

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Poisson traces, D-modules, and symplectic resolutions

Etingof

Schedler

2017

Lett Math Phys

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Poisson–de Rham homology of hypertoric varieties and nilpotent cones

Proudfoot

Schedler

2016

Sel. Math. New Ser.

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We prove a conjecture of Etingof and the second author for hypertoric varieties that the Poisson–de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson–de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson–de Rham–Poincaré polynomial and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham (J Algebra 242(1):160–175, 2001). We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2

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Automorphism Groups of Deformations and Quantizations of Kleinian Singularities

Castellan

2024

Transformation Groups

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It is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $$\textbf{A}$$ A or $$\textbf{D}$$ D , isomorphisms between the quantizations are essentially the same as Poisson isomorphisms between deformations. In particular, the group of automorphisms of the deformation and the quantization corresponding to the same deformation parameter are isomorphic. We additionally describe the groups of automorphisms as abstract groups: for type $$\textbf{A}$$ A they have an amalgamated free product structure, for type $$\textbf{D}$$ D they are subgroups of the groups of Dynkin diagram automorphisms. For type $$\textbf{D}$$ D we also compute all the possible affine isomorphisms between deformations; this was not known before.

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Poisson Traces for Symmetric Powers of Symplectic Varieties

Cited by 8 publications

References 27 publications

Poisson traces, D-modules, and symplectic resolutions

Poisson traces, D-modules, and symplectic resolutions

Poisson–de Rham homology of hypertoric varieties and nilpotent cones

Automorphism Groups of Deformations and Quantizations of Kleinian Singularities

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