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Symmetry, Integrability and Geometry: Methods and Applications, 2015, Volume 11, 055, 37 pp.
DOI: https://doi.org/10.3842/SIGMA.2015.055
(Mi sigma1036)
 

This article is cited in 22 scientific papers (total in 22 papers)

Non-Compact Symplectic Toric Manifolds

Yael Karshona, Eugene Lermanb

a Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
b Department of Mathematics, The University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA
References:
Abstract: A key result in equivariant symplectic geometry is Delzant's classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular (“Delzant”) polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant's classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map on the quotient need not be an embedding. Moreover, even when the map on the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class on the quotient makes an appearance. Nevertheless, the quotient is a manifold with corners, and the induced map from the quotient to the dual of the Lie algebra is what we call a unimodular local embedding. We classify non-compact symplectic toric manifolds in terms of manifolds with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the Lie algebra of the corresponding torus. The main new ingredient is the construction of a symplectic toric manifold from such data. The proof passes through an equivalence of categories between symplectic toric manifolds and symplectic toric bundles over a fixed unimodular local embedding. This equivalence also gives a geometric interpretation of the degree two cohomology class.
Keywords: Delzant theorem; symplectic toric manifold; Hamiltonian torus action; completely integrable systems.
Received: August 15, 2014; in final form July 10, 2015; Published online July 22, 2015
Bibliographic databases:
Document Type: Article
Language: English
Citation: Yael Karshon, Eugene Lerman, “Non-Compact Symplectic Toric Manifolds”, SIGMA, 11 (2015), 055, 37 pp.
Citation in format AMSBIB
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\by Yael~Karshon, Eugene~Lerman
\paper Non-Compact Symplectic Toric Manifolds
\jour SIGMA
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\papernumber 055
\totalpages 37
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  • https://www.mathnet.ru/eng/sigma1036
  • https://www.mathnet.ru/eng/sigma/v11/p55
  • This publication is cited in the following 22 articles:
    1. Charles Cifarelli, “Weighted K-Stability for a Class of Non-compact Toric Fibrations”, J Geom Anal, 34:5 (2024)  crossref
    2. Yohann Le Floch, Joseph Palmer, “Semitoric Families”, Memoirs of the AMS, 302:1514 (2024)  crossref
    3. Yu Du, Gabriel Kosmacher, Yichen Liu, Jeff Massman, Joseph Palmer, Timothy Thieme, Jerry Wu, Zheyu Zhang, “Packing Densities of Delzant and Semitoric Polygons”, SIGMA, 19 (2023), 081, 42 pp.  mathnet  crossref
    4. Hajime Fujita, “The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry”, Info. Geo., 2023  crossref
    5. Nikita Nekrasov, Nicolo Piazzalunga, Maxim Zabzine, “Shifts of prepotentials (with an appendix by Michele Vergne)”, SciPost Phys., 12:5 (2022)  crossref
    6. Oliveira G., Sena-Dias R., “Minimal Lagrangian Tori and Action-Angle Coordinates”, Trans. Am. Math. Soc., 374:11 (2021), 7715–7742  crossref  mathscinet  isi
    7. Pelayo A., “Symplectic Invariants of Semitoric Systems and the Inverse Problem For Quantum Systems”, Indag. Math.-New Ser., 32:1 (2021), 246–274  crossref  mathscinet  isi
    8. Dixon K., “Regular Ambitoric 4-Manifolds: From Riemannian Kerr to a Complete Classification”, Commun. Anal. Geom., 29:3 (2021), 629–679  crossref  mathscinet  isi
    9. Haniya Azam, Catherine Cannizzo, Heather Lee, Association for Women in Mathematics Series, 27, Research Directions in Symplectic and Contact Geometry and Topology, 2021, 159  crossref
    10. S. Wolbert, “Symplectic toric stratified spaces with isolated singularities”, J. Symplectic Geom., 18:5 (2020), 1391–1473  crossref  mathscinet  isi
    11. X. Tang, “Removing a ray from a noncompact symplectic manifold”, Int. Math. Res. Notices, 2020:22 (2020), 8878–8895  crossref  mathscinet  isi
    12. N. Istrati, “A characterisation of toric locally conformally Kahler manifolds”, J. Symplectic Geom., 17:5 (2019), 1297–1316  crossref  mathscinet  isi  scopus
    13. V. G. B. Ramos, D. Sepe, “On the rigidity of Lagrangian products”, J. Symplectic Geom., 17:5 (2019), 1447–1478  crossref  mathscinet  isi
    14. T. Ratiu, Nguyen Tien Zung, “Presymplectic convexity and (ir)rational polytopes”, J. Symplectic Geom., 17:5 (2019), 1479–1511  crossref  mathscinet  isi  scopus
    15. T. B. Madsen, A. Swann, “Toric geometry of g(2)-manifolds”, Geom. Topol., 23:7 (2019), 3459–3500  crossref  mathscinet  isi  scopus
    16. Ya. Karshon, F. Ziltener, “Hamiltonian group actions on exact symplectic manifolds with proper momentum maps are standard”, Trans. Am. Math. Soc., 370:2 (2018), 1409–1428  crossref  mathscinet  zmath  isi
    17. Sonja Hohloch, Silvia Sabatini, Daniele Sepe, Margaret Symington, “Faithful Semitoric Systems”, SIGMA, 14 (2018), 084, 66 pp.  mathnet  crossref
    18. J. LANE, “CONVEXITY AND THIMM'S TRICK”, Transformation Groups, 23:4 (2018), 963  crossref
    19. J. Palmer, “Moduli spaces of semitoric systems”, J. Geom. Phys., 115 (2017), 191–217  crossref  mathscinet  zmath  isi  scopus
    20. M. Gualtieri, S. Li, A. Pelayo, T. S. Ratiu, “The tropical momentum map: a classification of toric log symplectic manifolds”, Math. Ann., 367:3-4 (2017), 1217–1258  crossref  mathscinet  zmath  isi  scopus
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    Symmetry, Integrability and Geometry: Methods and Applications
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