On the filtered symplectic homology of prequantization bundles

Ginzburg, Viktor L.; Shon, Jeongmin

doi:10.1142/s0129167x18500714

Cited by 13 publications

(8 citation statements)

References 56 publications

(120 reference statements)

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“…When ω| π 2 pBq " 0 and B satisfies some extra conditions (for instance, when π i pBq " 0 for every i ě 2) it is proved in [23] (c.f. [26]) that every Reeb flow on M (possibly degenerate) carries infinitely many simple closed orbits.…”

Section: Prequantization Circle Bundles Over Symplectic Manifoldsmentioning

confidence: 99%

Two closed orbits for non-degenerate Reeb flows

Abreu¹,

Gutt²,

Kang³

et al. 2020

Math. Proc. Camb. Phil. Soc.

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We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M . Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantization circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik-Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.Remark 1.20. It follows from the Gysin exact sequence that H 1 pM, Rq " 0 whenever H 1 pB; Qq " 0.Remark 1.21. When ω| π 2 pBq " 0 and B satisfies some extra conditions (for instance, when π i pBq " 0 for every i ě 2) it is proved in [23] (c.f. [26]) that every Reeb flow on M (possibly degenerate) carries infinitely many simple closed orbits.Remark 1.22. We have that H˚pB; Qq vanishes in odd degrees and c 1 pT Bq| π 2 pBq ‰ 0 whenever B admits a Hamiltonian circle action with isolated fixed points.

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Section: Prequantization Circle Bundles Over Symplectic Manifoldsmentioning

confidence: 99%

Two closed orbits for non-degenerate Reeb flows

Abreu¹,

Gutt²,

Kang³

et al. 2020

Math. Proc. Camb. Phil. Soc.

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“…The advantage of the latter is that it produces a homology which has exactly one generator per simple orbit, which is discussed in [GG10]. This is helpful when one's goal is counting Reeb orbits via these homology groups, as is the case when proving results about the multiplicity of Reeb orbits, as is the case in [AGKM, GM,GS18,GG20].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…This definition is sometimes called being a contact SDM as given in [GHHM13]. This definition, as well as its iterated analogue, are discussed at some length in [GS18].…”

Section: What This Results Tells Us About Symplectically Degenerate M...mentioning

confidence: 99%

Local Symplectic Homology of Reeb Orbits

Fender

2020

Preprint

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In this paper we prove two isomorphisms in the local symplectic homology of a non iterated, isolated Reeb orbit. The isomorphisms are in S 1equivariant and nonequivariant symplectic homology relating the local Floer homology group of the orbit to that of the return map. The S 1 -equivariant symplectic homology isomorphism can be stated succinctly as the local S 1equivariant symplectic homology of an uniterated isolated Reeb orbit is isomorphic to the local Hamiltonian Floer homology of the return map. An application of this result gives the equivalence of two different definitions of a Reeb orbit being a symplectically degenerate maximum.

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“…Moreover, when K is smooth, all these capacities are given by the minimal action among all the closed characteristics on the boundary ∂K * . The above claims follow from a combination of results from [1,11,12,16,18]. In what follows, for a convex domain K in R 2n we shall denote the above mentioned coinciding capacities by c EHZ (K).…”

Section: The Systolic Ratio Of Symplectic P-productsmentioning

confidence: 96%

Remarks on symplectic capacities of $p$-products

Haim-Kislev¹,

Ostrover²

2021

Preprint

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In this note we study the behavior of symplectic capacities of convex domains in the classical phase space with respect to symplectic p-products. As an application, by using a "tensor power trick", we show that it is enough to prove the weak version of Viterbo's volume-capacity conjecture in the asymptotic regime, i.e., when the dimension is sent to infinity. In addition, we introduce a conjecture about higher-order capacities of p-products, and show that if it holds, then there are no non-trivial pdecompositions of the symplectic ball.of K is defined as the ratio of this capacity of K to the normalized ω-volume of K. Note that sys n (B) = 1, for any Euclidean ball B in R 2n .Recall the following weak version of Viterbo's volume-capacity conjecture [24].Our first result concerns the systolic ratio of symplectic p-products. We show that if two convex bodies K ⊂ R 2n and T ⊂ R 2m fulfill Conjecture 1.1, then the same is true for the p-product of K and T . More precisely, * If the boundary of K is not smooth, the above capacities coincide with the minimal action among "generalized closed characteristics", as explained, e.g., in [2].

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On the filtered symplectic homology of prequantization bundles

Cited by 13 publications

References 56 publications

Two closed orbits for non-degenerate Reeb flows

Two closed orbits for non-degenerate Reeb flows

Local Symplectic Homology of Reeb Orbits

Remarks on symplectic capacities of $p$-products

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