On the normality of the null-fiber of the moment map for θ - and tori representations

Bulois, Michael

doi:10.1016/j.jalgebra.2018.04.011

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…Reducing to the case of (V, G) stable and faithful, we can now demonstrate the following. See also [Bul18,Theorem 2.2], where it was demonstrated independently and simultaneously without the hypothesis of stable that N is normal if and only if it is irreducible. Lemma 5.2.…”

Section: The Case Of G 0 a Torusmentioning

confidence: 99%

“…Note that in the case that (V, K) is unimodular, meaning that all maximal minors of the weight matrix of the action are ±1 or 0, Theorem 1.3 was established in [MN14,Lemma 3.8]. See also [Bul18,Section 3], where it is demonstrated that when G is a torus, the alternate definition given in Equation (2.5) yields complex symplectic quotients that are in general reduced and normal.…”

Section: Introductionmentioning

confidence: 99%

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Symplectic quotients have symplectic singularities

Herbig¹,

Schwarz²,

Seaton³

2020

Compositio Math.

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Let K be a compact Lie group with complexification G, and let V be a unitary K-module. We consider the real symplectic quotient M 0 at level 0 of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of M 0 . We show that if (V, G) is 3-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This in particular implies that the real symplectic quotient is graded Gorenstein. In the case that K is a torus or SU 2 , we show that these results hold without the hypothesis that (V, G) is 3-large.2010 Mathematics Subject Classification. Primary 53D20, 13A50; Secondary 20G20, 57S15, 13H10.

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Section: The Case Of G 0 a Torusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symplectic quotients have symplectic singularities

Herbig¹,

Schwarz²,

Seaton³

2020

Compositio Math.

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Constructing symplectomorphisms between symplectic torus quotients

Herbig¹,

Lawler

Seaton

2020

Beitr Algebra Geom

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We identify a family of torus representations such that the corresponding singular symplectic quotients at the 0-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic.

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Stratification of singular hyperkähler quotients

Mayrand

2022

Complex Manifolds

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Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

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On the normality of the null-fiber of the moment map for θ - and tori representations

Cited by 3 publications

References 12 publications

Symplectic quotients have symplectic singularities

Symplectic quotients have symplectic singularities

Constructing symplectomorphisms between symplectic torus quotients

Stratification of singular hyperkähler quotients

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