Centered complexity one Hamiltonian torus actions

Karshon, Yael; Tolman, Susan

doi:10.1090/s0002-9947-01-02799-4

Cited by 38 publications

(47 citation statements)

References 39 publications

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“…On the other hand the completely integrable systems with local torus actions of Kogan [30] form a relatively close generalization of torus actions with Lagrangian principal orbits. The classification of Hamiltonian circle actions on compact connected four-dimensional manifolds in Karshon [27], and of centered complexity one Hamiltonian torus actions in arbitrary dimensions in Karshon and Tolman [28], are also much richer than our classification in the case that n − d h ≤ 1. McDuff [37] and McDuff and Salamon [38] studied non--Hamiltonian circle actions, and Ginzburg [18] non-Hamiltonian symplectic actions of compact groups under the assumption of a "Lefschetz condition".…”

Section: Introductionmentioning

confidence: 81%

Symplectic torus actions with coisotropic principal orbits

Duistermaat¹,

Pelayo²

2007

Ann. inst. Fourier

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In this paper we completely classify symplectic actions of a torus T on a compact connected symplectic manifold (M, σ) when some, hence every, principal orbit is a coisotropic submanifold of (M, σ). That is, we construct an explicit model, defined in terms of certain invariants, of the manifold, the torus action and the symplectic form. The invariants are invariants of the topology of the manifold, of the torus action, or of the symplectic form.In order to deal with symplectic actions which are not Hamiltonian, we develop new techniques, extending the theory of Atiyah, Guillemin-Sternberg, Delzant, and Benoist. More specifically, we prove that there is a well-defined notion of constant vector fields on the orbit space M/T . Using a generalization of the Tietze-Nakajima theorem to what we call V -parallel spaces, we obtain that M/T is isomorphic to the Cartesian product of a Delzant polytope with a torus.We then construct special lifts of the constant vector fields on M/T , in terms of which the model of the symplectic manifold with the torus action is defined. * Research stimulated by a KNAW professorship If the effective action of T on (M, σ) is Hamiltonian, then d T = n and the principal orbits are Lagrange submanifolds. Moreover, the image of the momentum mapping is a convex polytope ∆ in the dual space t * of t, where t denotes the Lie algebra of T . ∆ has the special property that at each vertex of ∆ there are precisely n codimension one faces with normals which form a Z-basis of the integral lattice T Z in t, where T Z is defined as the kernel of the exponential mapping from t to T . The classification of Delzant [11] says that for each such polytope ∆ there is a compact connected symplectic manifold with Hamiltonian torus action having ∆ as image of the momentum mapping, and the symplectic manifold with torus action is unique up to equivariant symplectomorphisms. Such polytopes ∆ and corresponding symplectic T -manifolds (M, σ, T ) are called Delzant polytopes and Delzant manifolds in the exposition of this subject by Guillemin [24], after Delzant [11]. Each Delzant manifold has a T -invariant Kähler structure such that the Kähler form is equal to σ.Because critical points of the Hamiltonian function correspond to zeros of the Hamiltonian vector field, a Hamiltonian action on a compact manifold always has fixed points. Therefore the other extreme case of a symplectic torus action with coisotropic principal orbits occurs if the action is free. In this case, M is a principal torus bundle over a torus, hence a nilmanifold for a two-step nilpotent Lie group as described in Palais and Stewart [44]. If the nilpotent Lie group is not commutative, then M does not admit a Kähler structure, cf. Benson and Gordon [5]. For four-dimensional manifolds M, these were the first examples of compact symplectic manifolds without Kähler structure, introduced by Thurston [49]. See the end of Remark 7.6.The general case is a combination of the Hamiltonian case and the free case, in the sense that

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Section: Introductionmentioning

confidence: 81%

Symplectic torus actions with coisotropic principal orbits

Duistermaat¹,

Pelayo²

2007

Ann. inst. Fourier

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“…The terminology semi-toric may be confusing, with the risk of being mistaken for almost-toric. A more precise phrase would be "almost-toric with deficiency index one" or "almost-toric with complexity one" [11]. We shall keep semi-toric for its shortness.…”

Section: Definition 31mentioning

confidence: 99%

Moment polytopes for symplectic manifolds with monodromy

Ngọc¹

2007

Advances in Mathematics

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A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such "almost-toric 4-manifolds" which admits a Hamiltonian S 1-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application, we derive a Duistermaat-Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system.

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“…This paper is a byproduct of our work on the classification of complexity one Hamiltonian torus actions [18,20,21,22,23], but, in fact, it relies only on elementary aspects of such actions. It is motivated by a number of recent works by toric topologists (specifically, the papers [7,9,8,3] by Buchstaber and Terzic and by Ayzenberg) that explore the topology of the geometric quotients of manifolds with certain torus actions.…”

Section: Introductionmentioning

confidence: 99%

Topology of complexity one quotients

Karshon,

Tolman

2018

Preprint

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Centered complexity one Hamiltonian torus actions

Cited by 38 publications

References 39 publications

Symplectic torus actions with coisotropic principal orbits

Symplectic torus actions with coisotropic principal orbits

Moment polytopes for symplectic manifolds with monodromy

Topology of complexity one quotients

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