Skip to main content
SpringerLink
Log in

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

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akhmedov, A.: Symplectic Calabi–Yau 6-manifolds. Adv. Math. 262, 115–125 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auroux, D., Donaldson, S.K., Katzarkov, L.: Singular Lefschetz pencils. Geom. Topol. 9, 1043–1114 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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 pp

    Article  MathSciNet  MATH  Google Scholar 

  4. Brahic, O., Fernandes, R.L.: Poisson geometry in mathematics and physics. Contemp. Math. 450, 41–59 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  6. Crainic, M., Fernandes, R.L.: Integrability of Lie brackets. Ann. Math. (2) 157(2), 575–620 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crainic, M., Fernandes, R.L.: Integrability of Poisson brackets. J. Differ. Geom. 66(1), 71–137 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Damianou, P.A.: Nonlinear Poisson Brackets. Ph.D. Dissertation, University of Arizona (1989)

  9. Damianou, P.A., Petalidou, F.: Poisson brackets with prescribed casimirs. Can. J. Math. 64(5), 991–1018 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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 pp

    Article  MathSciNet  MATH  Google Scholar 

  11. Donaldson, S.K.: Lefschetz pencils on symplectic manifolds. J. Differ. Geom. 53(2), 205–236 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dufour, J.-P., Zung, N.T.: Poisson Structures and Their Normal Forms. Progress in Mathematics, vol. 242. Birkhäuser, Basel (2005)

    MATH  Google Scholar 

  13. García-Naranjo, L., Suárez-Serrato, P., Vera, R.: Poisson structures on smooth 4-manifolds. Lett. Math. Phys. 105(11), 1533–1550 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Honda, K.: Local properties of self-dual harmonic 2-forms on a 4-manifold. J. Reine Angew. Math. 577, 105–116 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York, Heidelberg (1973)

    MATH  Google Scholar 

  16. Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson Structures. Grundlehren der Mathematischen Wissenschaften, vol. 347. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  17. Lekili, Y.: Wrinkled fibrations on near-symplectic manifolds. Geom. Topol. 13, 277–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  19. Perutz, T.: Zero-sets of near-symplectic forms. J. Symp. Geom. 4(3), 237–257 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Suárez-Serrato, P., Torres Orozco, J.: Poisson structures on wrinkled fibrations. Bol. Soc. Mat. Mex. (3) 22(1), 263–280 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Thurston, W.P.: Some simple examples of symplectic manifolds. Proc. Am. Math. Soc. 55, 467–468 (1976)

    MathSciNet  MATH  Google Scholar 

  22. Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Birkhäuser, Basel (1994)

    Book  MATH  Google Scholar 

  23. Vaisman, I.: Foliation-coupling Dirac structures. J. Geom. Phys. 56(6), 917–938 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  25. Vera, R.: Near-symplectic \(2n\)-manifolds. Alg. Geom. Topol. 16(3), 1403–1426 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  27. Wade, A.: Poisson fiber bundles and coupling Dirac structures. Ann. Glob. Anal. Geom. 33(3), 207–217 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Weinstein, A.: The local structure of Poisson manifolds. J. Differ. Geom. 18(3), 523–557 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  29. Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Am. Math. Soc. (N.S.) 16(1), 101–104 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, P.: Morita equivalence of Poisson manifolds. Commun. Math. Phys. 142(3), 493–509 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, P.: Morita equivalence and symplectic realizations of Poisson manifolds. Ann. Sci. École Norm. Sup. (4) 25(3), 307–333 (1992)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Coyoacán, 04510, Mexico City, Mexico

    P. Suárez-Serrato

  2. Department of Mathematics, University of California Santa Barbara, Goleta, CA, 93106, USA

    P. Suárez-Serrato

  3. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Morelia, Michoacán, Mexico

    J. Torres Orozco

  4. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001, Leuven (Heverlee), Belgium

    R. Vera

Authors
  1. P. Suárez-Serrato
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Torres Orozco
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Vera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Torres Orozco.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suárez-Serrato, P., Torres Orozco, J. & Vera, R. Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6. Ann Glob Anal Geom 55, 777–804 (2019). https://doi.org/10.1007/s10455-019-09651-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-019-09651-2

Keywords

Mathematics Subject Classification

Search

Navigation