Basic Kirwan surjectivity for K-contact manifolds

Casselmann, Lana

doi:10.1007/s10455-017-9552-6

Cited by 3 publications

(5 citation statements)

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“…where F P = F P | P ′ /O h is the lift of the foliation F to P . As in (7), N F P ≃ N F P ⊕ h P , where h P denotes the subbundle of orbit directions for the lifted h action. Thus the foliation ( P , F P ) has a transverse parallelism, obtained by combining the frame V 1 , .…”

Section: 4mentioning

confidence: 98%

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Riemannian foliations and geometric quantization

Lin¹,

Loizides²,

Sjamaar³

et al. 2022

Preprint

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We introduce geometric quantization for constant rank presymplectic structures with Riemannian null foliation and compact leaf closure space. We prove a quantizationcommutes-with-reduction theorem in this context. Examples related to symplectic toric quasifolds, suspensions of isometric actions of discrete groups, and K-contact manifolds are discussed. YI LIN, YIANNIS LOIZIDES, REYER SJAMAAR, AND YANLI SONG 6.4. Haefliger suspensions 36 Appendix A. Transverse metrics, completeness and the Molino package 36 Appendix B. Reduction theory for transverse symplectic structures 39 B.1. Transverse Hamiltonian Lie algebra actions 40 B.2. Reduction at a value 42 B.3. Reduction at a coadjoint orbit 43 B.4. Additional properties 45 References 46

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Section: 4mentioning

confidence: 98%

“…[24] for this point of view. Analogues of many familiar theorems from symplectic geometry have been studied by various authors in this context, including [28,44,14,29,15,7,30].…”

Section: Introductionmentioning

confidence: 99%

Riemannian foliations and geometric quantization

Lin¹,

Loizides²,

Sjamaar³

et al. 2022

Preprint

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“…We now define equivariant basic cohomology and give its basic properties. See [GNT12,Töb14,GNT17,Cas17] for related constructions. We suppose now that a torus G acts on M, preserving the contact form.…”

Section: Equivariant Basic Cohomologymentioning

confidence: 99%

“…Remark 1.6. The first named author has obtained a surjectivity result for the basic contact Kirwan map [Cas17]. Since basic cohomology satisfies Poincaré duality (see Lemma 2.3), Theorem 1.4 in principle provides a method to compute the kernel of the basic Kirwan map, and therefore allows one to compute the basic cohomology ring of the quotient.…”

Section: Introductionmentioning

confidence: 99%

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Localization for $K$-contact manifolds

Casselmann

Fisher²

2019

Journal of Symplectic Geometry

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We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of K-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact manifold M to basic integrals on the contact quotient M 0 := µ −1 (0)/G, where µ denotes the contact moment map for the action of a torus G. In the special case that M → N is an equivariant Boothby-Wang fibration, our formulae reduce to the usual ones for the symplectic manifold N . ContentsRemark 1.5. In §5.1 we explain in detail how Theorems 1.1 and 1.4 may be used to deduce the analogous theorems for symplectic manifolds that occur as M/F in the case that R induces a free S 1 -action. In this sense, these theorems provide a strict generalization of their symplectic analogues, at least in the case of an integral symplectic form and a Hamiltonian group action that lifts to the S 1 -bundle in the Boothby-Wang fibration [BW58].Remark 1.6. The first named author has obtained a surjectivity result for the basic contact Kirwan map [Cas17]. Since basic cohomology satisfies Poincaré duality (see Lemma 2.3), Theorem 1.4 in principle provides a method to compute the kernel of theProposition 4.3. The distribution Q η (y) may be represented by a piecewise polynomial function.As in the proof of Lemma 4.2, the previous equation becomesRecall that the local normal form of the moment map is given by µ(p, z) = z. Then, for sufficiently small y, δ(−µ − y) is supported away from S h and it follows that ∆ = 0. This means that, for sufficiently small y, Q η (y) = 1 (2π) s/2 g µ −1 (0)×B h α ∧ π * q * η 0 ∧ e idα+i −µ−y,φ dφ = (2π) s/2 µ −1 (0)×B h α ∧ π * q * η 0 ∧ e idα δ(−µ − y)

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