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

  • P. Suárez-Serrato
  • J. Torres OrozcoEmail author
  • R. Vera
Article

Abstract

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.

Keywords

Singular Poisson Near-symplectic Broken Lefschetz fibrations Wrinkled Singularity theory Stable maps Fold Cusp Swallowtail Butterfly 

Mathematics Subject Classification

MSC 57R17 MSC 53D17 

Notes

Acknowledgements

We warmly thank Yankı Lekili for answering detailed questions about his paper. His explanations allowed us to complete our computations for the near-symplectic forms. We also thank Alan Weinstein for commenting on an early version of this paper. PSS thanks DGAPA PAPIIT-UNAM IN102716 and The University of California Institute for Mexico and the United States (UC MEXUS) Grant CN-16-43, the organizers of the meeting ’Gone fishing 2016’ in Boulder, and IPAM in UCLA where some of this work was done. RV thanks UNAM-DGAPA and the partial support by the FWO under EOS project G0H4518N. JTO thanks support from CONACyT Project CB2016/283960.

References

  1. 1.
    Akhmedov, A.: Symplectic Calabi–Yau 6-manifolds. Adv. Math. 262, 115–125 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auroux, D., Donaldson, S.K., Katzarkov, L.: Singular Lefschetz pencils. Geom. Topol. 9, 1043–1114 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avendaño-Camacho, M., Vorobiev, Y.: Deformations of Poisson structures on fibered manifolds and adiabatic slow–fast systems. Int. J. Geom. Methods Mod. Phys. 14(6), 1750086 (2017). 15 ppMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brahic, O., Fernandes, R.L.: Poisson geometry in mathematics and physics. Contemp. Math. 450, 41–59 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bursztyn, H., Weinstein, A.: Poisson geometry and Morita equivalence, Poisson geometry, deformation quantisation and group representations. Lond. Math. Soc. Lecture Note Ser., vol. 323, pp. 1–78. Cambridge Univ. Press, Cambridge (2005)zbMATHGoogle Scholar
  6. 6.
    Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. Math. (2) 157(2), 575–620 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Crainic, M., Fernandes, R.L.: Integrability of Poisson brackets. J. Differ. Geom. 66(1), 71–137 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Damianou, P.A.: Nonlinear Poisson Brackets. Ph.D. Dissertation, University of Arizona (1989)Google Scholar
  9. 9.
    Damianou, P.A., Petalidou, F.: Poisson brackets with prescribed casimirs. Can. J. Math. 64(5), 991–1018 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Del Zotto, M., Heckman, J.J., Morrison, D.R.: \(6D\) SCFTs and phases of \(5D\) theories. J. High Energy Phys. 5(9), 147 (2017). front matter+37 ppMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Donaldson, S.K.: Lefschetz pencils on symplectic manifolds. J. Differ. Geom. 53(2), 205–236 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dufour, J.-P., Zung, N.T.: Poisson Structures and Their Normal Forms. Progress in Mathematics, vol. 242. Birkhäuser, Basel (2005)zbMATHGoogle Scholar
  13. 13.
    García-Naranjo, L., Suárez-Serrato, P., Vera, R.: Poisson structures on smooth 4-manifolds. Lett. Math. Phys. 105(11), 1533–1550 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Honda, K.: Local properties of self-dual harmonic 2-forms on a 4-manifold. J. Reine Angew. Math. 577, 105–116 (2004)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York, Heidelberg (1973)zbMATHGoogle Scholar
  16. 16.
    Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson Structures. Grundlehren der Mathematischen Wissenschaften, vol. 347. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  17. 17.
    Lekili, Y.: Wrinkled fibrations on near-symplectic manifolds. Geom. Topol. 13, 277–318 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mather, J.: Stability of \(C^{\infty }\) mappings VI: the nice dimensions, from. In: Proceedings of Liverpool Singularities Symposium I (1969–70). Lecture Notes in Mathematics, vol. 192, pp. 207–253. Springer, Berlin (1971)Google Scholar
  19. 19.
    Perutz, T.: Zero-sets of near-symplectic forms. J. Symp. Geom. 4(3), 237–257 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Suárez-Serrato, P., Torres Orozco, J.: Poisson structures on wrinkled fibrations. Bol. Soc. Mat. Mex. (3) 22(1), 263–280 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Thurston, W.P.: Some simple examples of symplectic manifolds. Proc. Am. Math. Soc. 55, 467–468 (1976)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Birkhäuser, Basel (1994)CrossRefzbMATHGoogle Scholar
  23. 23.
    Vaisman, I.: Foliation-coupling Dirac structures. J. Geom. Phys. 56(6), 917–938 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Vallejo, J.A., Vorobiev, Y.: G-invariant deformations of almost-coupling Poisson structures. SIGMA Symmetry Integrability Geom. Methods Appl. 13, Paper No. 022 (2017)Google Scholar
  25. 25.
    Vera, R.: Near-symplectic \(2n\)-manifolds. Alg. Geom. Topol. 16(3), 1403–1426 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vorobjev, Y.: Coupling tensors and Poisson geometry near a single symplectic leaf. In: Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000), vol. 54, pp. 249–274. Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw (2001)Google Scholar
  27. 27.
    Wade, A.: Poisson fiber bundles and coupling Dirac structures. Ann. Glob. Anal. Geom. 33(3), 207–217 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Weinstein, A.: The local structure of Poisson manifolds. J. Differ. Geom. 18(3), 523–557 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Am. Math. Soc. (N.S.) 16(1), 101–104 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Xu, P.: Morita equivalence of Poisson manifolds. Commun. Math. Phys. 142(3), 493–509 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Xu, P.: Morita equivalence and symplectic realizations of Poisson manifolds. Ann. Sci. École Norm. Sup. (4) 25(3), 307–333 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMexico CityMexico
  2. 2.Department of MathematicsUniversity of California Santa BarbaraGoletaUSA
  3. 3.Centro de Ciencias MatemáticasUniversidad Nacional Autónoma de MéxicoMoreliaMexico
  4. 4.Department of MathematicsKU LeuvenLeuven (Heverlee)Belgium

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