Skip to main content
Log in

Topological and geometric aspects of almost Kähler manifolds via harmonic theory

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of d-harmonic forms of pure bidegree. There is also a generalization of Hodge Index Theorem for compact almost Kähler 4-manifolds. In particular, these provide topological bounds on the dimension of the space of d-harmonic forms of pure bidegree, as well as several new obstructions to the existence of a symplectic form compatible with a given almost complex structure.

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

Access this article

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. Alvarez-Gaumé, L., Freedman, D.Z.: Geometrical structure and ultraviolet finiteness in the supersymmetric \(\sigma \)-model. Commun. Math. Phys. 80(3), 443–451 (1981)

    Article  MathSciNet  Google Scholar 

  2. Bär, Christian: On nodal sets for Dirac and Laplace operators. Commun. Math. Phys. 188(3), 709–721 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benson, C., Gordon, C.S.: Kähler and symplectic structures on nilmanifolds. Topology 27(4), 513–518 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bazzoni, G., Muñoz, V.: Manifolds which are complex and symplectic but not Kähler. In: Rassias, Th.M., Pardalos, P.M. (eds.) Essays in Mathematics and Its Applications, pp. 49–69. Springer, Cham (2016)

  5. Brylinski, J.-L.: A differential complex for Poisson manifolds. J. Differ. Geom. 28(1), 93–114 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cordero, L.A., Fernández, M., Gray, A., Ugarte, L.: Compact nilmanifolds with nilpotent complex structures: dolbeault cohomology. Trans. Am. Math. Soc. 352(12), 5405–5433 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, K.: Algebras of iterated path integrals and fundamental groups. Trans. Am. Math. Soc. 156, 359–379 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, K.: Differential forms and homotopy groups. J. Differ. Geom. 6, 231–246 (1971/72)

  9. Cirici, J., Wilson, S.O.: Dolbeault Cohomology for Almost Complex Manifolds (Preprint). arxiv:1809.1416 (2018)

  10. de Bartolomeis, P., Tomassini, A.: On formality of some symplectic manifolds. Int. Math. Res. Not. 24, 1287–1314 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Draghici, T., Li, T.-J., Zhang, W.: Symplectic forms and cohomology decomposition of almost complex four-manifolds. Int. Math. Res. Not. IMRN 1, 1–17 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Donaldson, S.K.: Yang–Mills invariants of four-manifolds. In: Donaldson, S.K., Thomas, C.B. (eds.) Geometry of Low-Dimensional Manifolds. 1 (Durham, 1989), volume 150 of London Mathematical Society Lecture Note Series, pp. 5–40. Cambridge University Press, Cambridge (1990)

  13. Fröhlich, J., Grandjean, O., Recknagel, A.: Supersymmetric quantum theory and differential geometry. Commun. Math. Phys. 193(3), 527–594 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fino, A., Tomassini, A.: On some cohomological properties of almost complex manifolds. J. Geom. Anal. 20(1), 107–131 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994). (Reprint of the 1978 original)

    Book  Google Scholar 

  16. Gompf, R.E.: A new construction of symplectic manifolds. Ann. Math. 142, 527–595 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gompf, R.E.: The topology of symplectic manifolds. Turk. J. Math. 25(1), 43–59 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Hirzebruch, F.: Some problems on differentiable and complex manifolds. Ann. Math. 2(60), 213–236 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hitchin, N.J., Karlhede, A., Lindström, U., Roček, M.: Hyper–Kähler metrics and supersymmetry. Commun. Math. Phys. 108(4), 535–589 (1987)

    Article  MATH  Google Scholar 

  20. Huybrechts, D.: Complex Geometry. Universitext. Springer, Berlin (2005). (An introduction)

    MATH  Google Scholar 

  21. Holt, T., Zhang, W.: Harmonic Forms on the Kodaira–Thurston Manifold (Preprint). arxiv:2001.10962 (2020)

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Kotschick, D.: The Seiberg-Witten invariants of symplectic four-manifolds (after C. H. Taubes). Astérisque, (241):Exp. No. 812, 4, 195–220 (1997) Séminaire Bourbaki, Vol. 1995/96

  24. Li, T.-J., Zhang, W.: Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds. Commun. Anal. Geom. 17(4), 651–683 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mathieu, O.: Harmonic cohomology classes of symplectic manifolds. Comment. Math. Helv. 70(1), 1–9 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. Math. 2(59), 531–538 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  27. Taubes, C.H.: The Seiberg–Witten invariants and symplectic forms. Math. Res. Lett. 1(6), 809–822 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Taubes, C.H.: More constraints on symplectic forms from Seiberg–Witten invariants. Math. Res. Lett. 2(1), 9–13 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Tardini, N., Tomassini, A.: Differential Operators on Almost-Hermitian Manifolds and Harmonic Forms. To appear in Complex Manifolds (preprint). arXiv:1909.06569 (2019)

  31. Tseng, L.-S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: I. J. Differ. Geom. 91(3), 383–416 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tseng, L.-S., Yau, S.-T.: Cohomology and Hodge theory on symplectic manifolds: II. J. Differ. Geom. 91(3), 417–443 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Verbitsky, M.: Hodge theory on nearly Kähler manifolds. Geom. Topol. 15(4), 2111–2133 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Voisin, C.: Hodge Theory and Complex Algebraic Geometry. I, Volume 76 of Cambridge Studies in Advanced Mathematics, english edn. Cambridge University Press, Cambridge (2007). (Translated from the French by Leila Schneps)

    Google Scholar 

  35. Weil, A.: Introduction à l’étude des variétés kählériennes. Publications de l’Institut de Mathématique de l’Université de Nancago, VI. Actualités Sci. Ind. no. 1267. Hermann, Paris (1958)

  36. Yan, D.: Hodge structure on symplectic manifolds. Adv. Math. 120(1), 143–154 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zumino, B.: Supersymmetry and Kähler manifolds. Phys. Lett. 87B, 203 (1979)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank Thomas Holt for pointing out an important sign error that appeared in a preprint of this paper. We would also like to thank the anonymous referee for his suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Scott O. Wilson.

Additional information

Publisher's Note

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

Joana Cirici would like to acknowledge partial support from the AEI/FEDER, UE (MTM2016-76453-C2-2-P) and the Serra Húnter Program. Scott O. Wilson acknowledges support provided by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cirici, J., Wilson, S.O. Topological and geometric aspects of almost Kähler manifolds via harmonic theory. Sel. Math. New Ser. 26, 35 (2020). https://doi.org/10.1007/s00029-020-00568-4

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s00029-020-00568-4

Keywords

Mathematics Subject Classification

Navigation