On the (non)existence of symplectic resolutions of linear quotients

Bellamy, Gwyn; Schedler, Travis

doi:10.4310/mrl.2016.v23.n6.a1

Cited by 15 publications

(17 citation statements)

References 15 publications

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“…In this paper, we show that these questions have affirmative answers for linear quotient singularities (which rarely admit symplectic resolutions, as explained in [BS13]) and that (1) and (2) have affirmative answers for hypertoric varieties, and we explicitly compute the decompositions. For the case where X is the nilpotent cone of a semisimple Lie algebra (or its Kostant-Slodowy slices), see Remark 11 for a discussion.…”

Section: Introductionmentioning

confidence: 76%

Equivariant Slices for Symplectic Cones

Schedler

2016

Int Math Res Notices

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The Darboux-Weinstein decomposition is a central result in the theory of Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the point and a formal slice.Recently, conical symplectic resolutions, and more generally, Poisson cones, have been very actively studied in representation theory and algebraic geometry. This motivates asking for a C × -equivariant version of the Darboux-Weinstein decomposition.In this paper, we develop such a theory, prove basic results on their existence and uniqueness, study examples (quotient singularities and hypertoric varieties), and applications to noncommutative algebra (their quantization). We also pose some natural questions on existence and quantization of C × -actions on slices to conical symplectic leaves.

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Section: Introductionmentioning

confidence: 76%

Equivariant Slices for Symplectic Cones

Schedler

2016

Int Math Res Notices

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“…As noted by the referee, choosing a suitable normal subgroup K of G may allow one to apply the geometric techniques of [5] in order to show that some of the remaining open cases of symplectically primitive but complex imprimitive groups do not give rise to quotient singularities admitting a symplectic resolution. Specifically, if there exists a symplectic resolution Y → V /K such that the action of G/K on V /K lifts to Y , then the existence of particularly bad singularities on Y /(G/K ) implies that V /G cannot admit a symplectic resolution.…”

Section: Remark 15mentioning

confidence: 99%

“…In this section we show that for the group W (S 2 ) of [11, Table III] there is no symplectic resolution of the corresponding linear quotient. This group is one of the few symplectically and complex primitive groups; we follow the same strategy used in [5] to treat these groups. Namely, we are going to show, or rather compute that there is a subgroup of W (S 2 ), say H , which is the stabilizer of a vector.…”

Section: Theorem 45 If There Existsmentioning

confidence: 99%

“…These groups split into infinite families given in [11,Theorem 2.6] (in dimension 4) and [11,Theorem 2.9] (in dimension greater than 4). Linear quotients of these groups are treated in [5], where the above question is answered for almost all cases. The remaining ones are covered by [29].…”

Section: Introduction To the Problem And Current Statusmentioning

confidence: 99%

“…The complex primitive groups are given in [11,Table III]. In [5], the authors prove that for three of them (W (Q), W (S 3 ), W (T )) no symplectic resolution exists. Using the same strategy, we prove in Sect.…”

Section: Introduction To the Problem And Current Statusmentioning

confidence: 99%

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Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution

2021

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Over the past 2 decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4—the symplectically primitive but complex imprimitive groups—and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving $$39+9=48$$ 39 + 9 = 48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.

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Symplectic resolutions of quiver varieties

Bellamy

Schedler

2021

Sel. Math. New Ser.

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In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic resolutions. Moreover we show that the smooth locus coincides with the locus of canonically $$\theta $$ θ -polystable points, generalizing a result of Le Bruyn; we study their étale local structure and find their symplectic leaves. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.

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On the (non)existence of symplectic resolutions of linear quotients

Cited by 15 publications

References 15 publications

Equivariant Slices for Symplectic Cones

Equivariant Slices for Symplectic Cones

Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution

Symplectic resolutions of quiver varieties

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