Poisson structures on wrinkled fibrations

Suárez–Serrato, Pablo; Orozco, Jonatán Torres

doi:10.1007/s40590-015-0072-8

Cited by 6 publications

(9 citation statements)

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“…This in turn implies that on any homotopy class of maps from a 4-manifold to S 2 there is such a singular Poisson structure. Similar structures also appear on wrinkled fibrations, where the local models of the Poisson bivectors and induced symplectic forms have been explicitly constructed [17].…”

Section: Introductionmentioning

confidence: 78%

“…Example 2.19. The deformations of wrinkled fibrations introduced by Lekili [14] have been used to describe Poisson structures on the associated fibrations [17]. In a similar way to the previous example assume that M 0 is a closed smooth oriented 4-manifold with a wrinkled fibration whose fibres do not contain 2-spheres.…”

Section: Morita Inequivalent Structuresmentioning

confidence: 99%

“…In a similar way to the previous example assume that M 0 is a closed smooth oriented 4-manifold with a wrinkled fibration whose fibres do not contain 2-spheres. Then the associated Poisson structure Π 0 is integrable [17]. Perform one of Lekili's deformations on (M 0 , Π 0 ) which increases the fibre genus, then the resulting manifold (M 1 , Π 1 ) is Poisson [17].…”

Section: Morita Inequivalent Structuresmentioning

confidence: 99%

“…Then the associated Poisson structure Π 0 is integrable [17]. Perform one of Lekili's deformations on (M 0 , Π 0 ) which increases the fibre genus, then the resulting manifold (M 1 , Π 1 ) is Poisson [17]. Then Lemma 2.17 implies (M 0 , Π 0 ) and (M 1 , Π 1 ) are not Morita equivalent.…”

Section: Morita Inequivalent Structuresmentioning

confidence: 99%

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Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

2019

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We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinite singularities we can provide these wrinkled fibrations in dimension 6 with near-symplectic structures.2010 Mathematics Subject Classification. 57R17, 53D17. after combining the definition of generalized bLf [19] together with the results of [6] and [10] on Poisson structures. Theorem 1.1. Let M and X be closed oriented smooth manifolds with dim(M ) = 2n, dim(X) = 2n − 2, and f : M → X a generalized broken Lefschetz fibration. Then there is a complete singular Poisson structure of rank 2 whose associated bivector vanishes on the singularity set of f . If none of its symplectic leaves are, or contain, 2-spheres, then this Poisson structure is integrable.The proof of this theorem is a direct application of Theorem 2.11 and the definition of completeness. The integrability condition is verified in the relevant cases, as explained in 2.4.In section 3 we focus on the Poisson structure on the total space of a generalized wrinkled fibration in dimension 6. We give the general steps for buidling Poisson bivectors around all types of singularities of corank 1, which can be applied on any given dimension. This idea allows us to show the following. Theorem 1.2. Let M be a closed, orientable, smooth 6-manifold equipped with a generalized wrinkled fibration f : M → X over a smooth 4-manifold X. Then there exists a complete Poisson structure whose symplectic leaves correspond to the fibres of the given fibration structure, and the singularities of both the fibration and the Poisson structures coincide. Moreover, for each singularity, the Poisson bivector and induced symplectic form on the leaves are given by the following equations:Folds: 13) If none of its symplectic leaves are, or contain, 2-spheres, then this Poisson structure is integrable.The existence of a Poisson structure with the stated properties follows from Theorem 2.11, previously shown by the first and third named authors together with García-Naranjo [10]. The proof of this theorem follows from an application of Theorem 2.11 and the definition of completeness.These results allow us to present in section 2.5.1 countably many examples of Poisson structures on the same underlying smooth manifold that are Morita inequivalent. In our examples the leaves of the symplectic foliations change topology, as the fibrations involved undergo deformations.In section 3, the local models for the bivectors are shown to hold true. Then in section 4 we prove that the local models for the symplectic forms on the leaves are also the claimed ones.Finally, in section 5 we turn to near-symplectic geometry. Explicit models of near-symplectic forms have appeared in previous work [11,1,14,19]. Here we show a further construction of local near-symplectic forms that follows a...

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Section: Introductionmentioning

confidence: 78%

Section: Morita Inequivalent Structuresmentioning

confidence: 99%

Section: Morita Inequivalent Structuresmentioning

confidence: 99%

Section: Morita Inequivalent Structuresmentioning

confidence: 99%

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Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

2019

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“…The function flaschka ratiu bivector computes the Flaschka-Ratiu bivector field, with respect to Ω 0 , and the corresponding symplectic form of a 'maximal' set of scalar functions [12,18,35,15]. differential Ki ← an array encoding the differential dK i of K i…”

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confidence: 99%

On computational Poisson geometry I: Symbolic foundations

Evangelista-Alvarado¹,

Ruíz-Pantaleón²,

Suárez–Serrato³

2021

JGM

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<p style='text-indent:20px;'>We present a computational toolkit for (local) Poisson-Nijenhuis calculus on manifolds. Our Python module $\textsf{PoissonGeometry}$ implements our algorithms and accompanies this paper. Examples of how our methods can be used are explained, including gauge transformations of Poisson bivector in dimension 3, parametric Poisson bivector fields in dimension 4, and Hamiltonian vector fields of parametric families of Poisson bivectors in dimension 6.</p>

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