Skip to main content
SpringerLink
Log in

Manifolds Which Are Complex and Symplectic But Not Kähler

  • Chapter
  • First Online:
Essays in Mathematics and its Applications

Abstract

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a Kähler structure is the renowned Kodaira–Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply connected eight-dimensional compact manifold of Fernández and Muñoz (Ann Math (2), 167(3):1045–1054, 2008) admits both symplectic and complex structures but does not carry Kähler metrics.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Amorós, J., Burger, M., Corlette, K., Kotschick, D., Toledo, D.: Fundamental Groups of Compact Kähler Manifolds. Mathematical Surveys and Monographs, vol. 44. American Mathematical Society, Providence, RI (1996)

    Google Scholar 

  2. Babenko, I.K., Taǐmanov, I.A.: On nonformal simply-connected symplectic manifolds. Sib. Math. J. 41 (2), 204–217 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babenko, I.K., Taǐmanov, I.A.: Massey products in symplectic manifolds. Sb. Math. 191, 1107–1146 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  5. Bazzoni, G.: Homotopy theory and geometric structures. Ph.D. thesis (2013) eprints.ucm.es/20065

    Google Scholar 

  6. Bazzoni, G., Muñoz, V.: Classification of minimal algebras over any field up to dimension 6. Trans. Am. Math. Soc. 364 (2), 1007–1028 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benson, C., Gordon, C.S.: Kähler and symplectic structures on nilmanifolds. Topology 27, 513–518 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Biswas, I., Fernández, M., Muñoz, V., Tralle, A.: On formality of Sasakian manifolds. J. Topology 9 (1), 161–180 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bock, C.: Geography of non-formal symplectic and contact manifolds. Forum Math. 23 (4), 713–727 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cavalcanti, G.R.: The Lefschetz property, formality and blowing up in symplectic geometry. Trans. Am. Math. Soc. 359 (1), 333–348 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cavalcanti, G.R., Fernández, M., Muñoz, V.: Symplectic resolutions, Lefschetz property and formality. Adv. Math. 218 (2), 576–599 (2008)

    MATH  Google Scholar 

  12. Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 29, 245–274 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Enrietti, N., Fino, A., Vezzoni, L.: Tamed symplectic forms and strong Kähler with torsion metrics. J. Symplectic Geom. 10 (2), 203–223 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Félix, Y., Halperin, S., Thomas, J.-C.: Rational Homotopy Theory. Graduate Texts in Mathematics, vol. 205. Springer, New York (2001)

    Google Scholar 

  15. Félix, Y., Oprea, J., Tanré, D.: Algebraic Models in Geometry. Oxford Graduate Texts in Mathematics, vol. 17. Oxford University Press, Oxford (2008)

    Google Scholar 

  16. Fernández, M., Muñoz, V.: Formality of Donaldson submanifolds. Math. Zeit. 250, 149–175 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fernández, M., Muñoz, V.: An 8-dimensional non-formal simply connected symplectic manifold. Ann. Math. (2) 167 (3), 1045–1054 (2008)

    Google Scholar 

  18. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1994)

    Book  MATH  Google Scholar 

  19. Griffiths, P., Morgan, J.: Rational Homotopy Theory and Differential Forms, 2nd edn. Progress in Mathematics, vol. 16. Birkhäuser, Basel (2013)

    Google Scholar 

  20. Guan, Z.-D.: Toward a classification of compact nilmanifolds with symplectic structures. Int. Math. Res. Not. 2010 (22), 4377–4384 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Guggenheimer, H.: Sur les variétés qui possèdent une forme extérieure quadratique fermée. C. R. Acad. Sci. Paris 232, 470–472 (1951)

    MathSciNet  MATH  Google Scholar 

  22. Hasegawa, K.: Minimal models of nilmanifolds. Proc. Am. Math. Soc. 106 (1), 65–71 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kodaira, K.: On the structure of compact complex analytic surfaces. I. Am. J. Math. 86, 751–798 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kraines, D.: Massey higher products. Trans. Am. Math. Soc. 124, 431–449 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lerman, E.: Symplectic cuts. Math. Res. Lett. 167 (3), 247–258 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lupton, G., Oprea, J.: Symplectic manifolds and formality. J. Pure Appl. Algebra 91, 193–207 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Massey, W.S.: Some higher order cohomology operations. In: International Symposium on Algebraic Topology, Mexico, pp. 145–154 (1958)

    Google Scholar 

  28. McDuff, D.: Examples of symplectic simply connected manifolds with no Kähler structure. J. Diff. Geom. 20, 267–277 (1984)

    MathSciNet  MATH  Google Scholar 

  29. Miller, J.T.: On the formality of (k − 1)−connected compact manifolds of dimension less than or equal to (4k − 2). Ill. J. Math. 23, 253–258 (1979)

    MathSciNet  MATH  Google Scholar 

  30. Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. (2) 65, 391–404 (1957)

    Google Scholar 

  31. Nomizu, K.: On the cohomology of compact homogeneous space of nilpotent Lie group, Ann. Math. (2), 59, 531–538 (1954)

    Google Scholar 

  32. Oprea, J., Tralle, A.: Symplectic Manifolds with No Kähler Structure. Lecture Notes in Mathematics, vol. 1661. Springer, Berlin (1997)

    Google Scholar 

  33. Quillen, D.: Rational homotopy theory. Ann. Math. (2) 90, 205–295 (1969)

    Google Scholar 

  34. Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Erg. der Math. vol. 68. Springer, Berlin (1972)

    Google Scholar 

  35. Rollenske, S.: The Frölicher spectral sequence can be arbitrarily non-degenerate. Math. Ann. 341 (2), 623–628 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Salamon, S.: Complex structures on nilpotent Lie algebras. J. Pure Appl. Algebra 157, 311–333 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sullivan, D.: Infinitesimal computations in topology. Publ. Math. I.H.É.S. 47, 269–331 (1978); 311–333 (2001)

    Google Scholar 

  38. Thurston, W.: Some simple examples of symplectic manifolds. Proc. Am. Math. Soc. 55 (2), 467–468 (1976)

    MathSciNet  MATH  Google Scholar 

  39. Wells, R.O.: Differential Analysis on Complex Manifolds, 3rd edn. Graduate Texts in Mathematics, vol. 65. Springer, Berlin (2008)

    Google Scholar 

Download references

Acknowledgements

The main result of this paper has been taken from Chap. 1 of the Ph.D. thesis [5] of the first author. We are grateful to Jesús Ruiz who suggested us this question. Partially supported by (Spain) MICINN grant MTM2010-17389.

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100301, Bielefeld, 33501, Germany

    Giovanni Bazzoni

  2. Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Vicente Muñoz

Authors
  1. Giovanni Bazzoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vicente Muñoz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vicente Muñoz .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical Unversity of Athens, Athens, Greece

    Themistocles M. Rassias

  2. University of Florida, GAINESVILLE, Florida, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bazzoni, G., Muñoz, V. (2016). Manifolds Which Are Complex and Symplectic But Not Kähler. In: Rassias, T., Pardalos, P. (eds) Essays in Mathematics and its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-31338-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation