The equivariant cohomology of hypertoric varieties and their real loci

Harada, Megumi; Holm, Tara S.

doi:10.4310/cag.2005.v13.n3.a3

Cited by 18 publications

(19 citation statements)

References 12 publications

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“…First, we recall the tangent spaces of fixed points of toric hyperkähler manifolds (see [9,Sect. 3] for details).…”

Section: Hyperplane Arrangements Induced By the Equivariant Cohomologymentioning

confidence: 99%

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Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Kuroki

2011

Proc. Steklov Inst. Math.

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Toric hyperkähler manifolds are the hyperkähler analogue of symplectic toric manifolds. The theory of Bielawski and Dancer tells us that, while a symplectic toric manifold is determined by a Delzant polytope, a toric hyperkähler manifold is determined by a smooth hyperplane arrangement. The purpose of this paper is to show that a toric hyperkähler manifold up to weak hyperhamiltonian T -isometry is determined not only by a smooth hyperplane arrangement up to weak linear equivalence but also by its equivariant cohomology H * T (M ; Z) with a point a in H 2 (M ; R) \ {0} up to weak H * (BT ; Z)-algebra isomorphism preserving a.

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“…First, we recall the tangent spaces of fixed points of toric hyperkähler manifolds (see [9,Sect. 3] for details).…”

Section: Hyperplane Arrangements Induced By the Equivariant Cohomologymentioning

confidence: 99%

“…Therefore, one can easily show that r τ i | p = x i for all p ∈ M T α (e.g., by using the localization theorem; see also [9,11]). Because x i is a primitive vector in H 2 (BT ), we have r = ±1.…”

Section: Proceedings Of the Steklov Institute Of Mathematicsmentioning

confidence: 99%

Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Kuroki

2011

Proc. Steklov Inst. Math.

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“…4 There are many variants on the definition of GKM actions (see e.g. [20][21][22][23]). In particular, in less restrictive versions,…”

Section: Gkm Spacesmentioning

confidence: 99%

Torsion in the full orbifold K-theory of abelian symplectic quotients

2011

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Let (M, ω, ) be a Hamiltonian T -space and let H ⊆ T be a closed Lie subtorus. Under some technical hypotheses on the moment map , we prove that there is no additive torsion in the integral full orbifold K -theory of the orbifold symplectic quotient [M//H ]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Hamiltonian T -space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S 1 -quotients of GKM spaces, have integral full orbifold K -theory that is free of additive torsion. Finally, we introduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses of the main theorem hold. We illustrate our results using low-rank coadjoint orbits of type A and B.

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“…(In particular, the action of C × is not hamiltonian.) This extra C × -action was studied in [HP1], and the equivariant cohomology ring H * C × (M α,0 (T )) was given an explicit combinatorial description in [HP1,4.5] and [HH,3.5]. Thus, modulo surjectivity of the Kirwan map, Theorem 3.4.1 tells us how to compute the cohomology ring of arbitrary symplectic quotients constructed in the manner of Section 1.1.…”

Section: Abelianizationmentioning

confidence: 99%

A survey of hypertoric geometry and topology

Proudfoot¹

2008

Toric Topology

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Abstract. Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkähler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject.Given a linear representation of a reductive complex algebraic group G, there are two natural quotient constructions. First, one can take a geometric invariant theory (GIT) quotient, which may also be interpreted as a Kähler quotient by a maximal compact subgroup of G. Examples of this sort include toric varieties (when G is abelian), moduli spaces of spacial polygons, and, more generally, moduli spaces of semistable representations of quivers. A second construction involves taking an algebraic symplectic quotient of the cotangent bundle of V , which may also be interpreted as a hyperkähler quotient. The analogous examples of the second type are hypertoric varieties, hyperpolygon spaces, and Nakajima quiver varieties.The subject of this survey will be hypertoric varieties, which are by definition the varieties obtained from the second construction when G is abelian. Just as the geometry and topology of toric varieties is deeply connected to the combinatorics of polytopes, hypertoric varieties interact richly with the combinatorics of hyperplane arrangements and matroids. Furthermore, just as in the toric case, the flow of information goes in both directions.On one hand, Betti numbers of hypertoric varieties have a combinatorial interpretation, and the geometry of the varieties can be used to prove combinatorial results. Many purely algebraic constructions involving matroids acquire geometric meaning via hypertoric varieties, and this has led to geometric proofs of special cases of the g-theorem for matroids [HSt, 7.4] and the Kook-Reiner-Stanton convolution formula [PW, 5.4]. Future plans include a geometric interpretation of the Tutte polynomial and of the phenomenon of Gale duality of matroids [BLP].On the other hand, hypertoric varieties are important to geometers with no interest in combinatorics simply because they are among the most explicitly understood examples of algebraic symplectic or hyperkähler varieties, which are becoming increasingly prevalent in many areas of mathematics. For example, Nakajima's quiver varieties include resolutions of Slodowy slices and Hilbert schemes of points on ALE spaces, both of which play major roles in modern representation theory. Moduli spaces of Higgs bundles are currently receiving a lot of attention in string theory, and character varieties of fundamental groups of surfaces and 3-manifolds have become an important tool in low-dimensional topology. Hypertoric

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The equivariant cohomology of hypertoric varieties and their real loci

Cited by 18 publications

References 12 publications

Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Torsion in the full orbifold K-theory of abelian symplectic quotients

A survey of hypertoric geometry and topology

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