Augmentations, Fillings, and Clusters

Gao, Honghao; Shen, Linhui; Weng, Daping

doi:10.48550/arxiv.2008.10793

Cited by 20 publications

(43 citation statements)

References 43 publications

(51 reference statements)

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“…and with augmentation varieties of Legendrian links by H. Gao, L. Shen and D. Weng in [9]. We also point out the recent work [1] by D. Alvarez, which contains a construction of a Poisson groupoid over F n as the moduli space of flat G-bundles over the disk with decorated boundary.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 87%

“…Thirdly, in their work [9] on Lagrangian fillings of Legendrian links, H. Gao, L. Shen, and D. Weng show that varieties of the form O w e for G = SL n are isomorphic to augmentation varieties of certain positive braid Legendrian links. It would be very interesting to explore connections between the Poisson groupoid structure on O (u,u −1 ) e × T in this paper and the results in [9]. 1.4.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 98%

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Configuration Poisson groupoids of flags

Lu¹,

Mouquin²,

Yu³

2021

Preprint

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Let G be a connected complex semi-simple Lie group and B its flag variety. For every positive integer n, we introduce a Poisson groupoid over B n , called the nth total configuration Poisson groupoid of flags of G, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in B n . Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to G. Contents 1. Introduction and statements of results 1 2. A series of Poisson groupoids associated to Poisson Lie groups 9 3. The total configuration Poisson groupoids of flags and isomorphic models 13 4. Special configuration Poisson groupoids of flags and isomorphic models 17 5. Configuration symplectic groupoids of flags 22 Appendix A. Mixed product Poisson structures 25 Appendix B. The Poisson isomorphism J n 29 Appendix C. Generalized double Bruhat cells 32 Appendix D. Symplectic leaves of (O w e × T, π n ⊲⊳ 0) 40 References 55

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Section: Introduction and Statements Of Resultsmentioning

confidence: 87%

Section: Introduction and Statements Of Resultsmentioning

confidence: 98%

Configuration Poisson groupoids of flags

Lu¹,

Mouquin²,

Yu³

2021

Preprint

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“…The first examples of Legendrian links with infinitely many distinct exact Lagrangian fillings up to Hamiltonian isotopy were given by Casals and Gao [6] who used the theory of microlocal sheaves to distinguish the fillings. That same year Gao, Shen, and Weng [22,23], Casals and Zaslow [9], Casals and Ng [8] found more such examples. The techniques from cluster algebras and microlocal theory of sheaves used to differentiate exact Lagrangian fillings in [6,22,23,9] are constrained to studying rainbow closures of positive braids.…”

Section: Introductionmentioning

confidence: 99%

“…That same year Gao, Shen, and Weng [22,23], Casals and Zaslow [9], Casals and Ng [8] found more such examples. The techniques from cluster algebras and microlocal theory of sheaves used to differentiate exact Lagrangian fillings in [6,22,23,9] are constrained to studying rainbow closures of positive braids. In contrast, the Floer theoretic methods developed by Casals and Ng [8] are not constrained to this set of Legendrian links.…”

Section: Introductionmentioning

confidence: 99%

Infinitely many planar fillings and symplectic Milnor fibers

Capovilla-Searle¹

2022

Preprint

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We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of Legendrian links with infinitely many distinct planar exact Lagrangian fillings, which can be viewed as the smallest Legendrian links currently known to have infinitely many distinct exact Lagrangian fillings. As an application we find new examples of infinitely many exact Lagrangian spheres and tori 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality. We also provide a generalization of a criterion for when the symplectic homology of a Weinstein 4-manifold is non-vanishing given an explicit Weinstein handlebody diagram.

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“…First it has been positively answered by Casals and Gao [4]. Later the works of An-Bae-Lee [1,2], Casals-Zaslow [6], and Gao-Shen-Weng [14,15] have continued to develop various cluster and sheaf-theoretic methods to detect infinitely many exact Lagrangian fillings for Legendrian links in the standard 3-dimensional contact vector space.…”

Section: Introductionmentioning

confidence: 99%