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Cohomological Aspects on Complex and Symplectic Manifolds

  • Nicoletta Tardini
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 21)

Abstract

We discuss how quantitative cohomological informations could provide qualitative properties on complex and symplectic manifolds. In particular we focus on the Bott-Chern and the Aeppli cohomology groups in both cases, since they represent useful tools in studying non Kähler geometry. We give an overview on the comparisons among the dimensions of the cohomology groups that can be defined and we show how we reach the \(\partial \overline{\partial }\)-lemma in complex geometry and the Hard-Lefschetz condition in symplectic geometry. For more details we refer to Angella and Tardini (Proc Am Math Soc 145(1):273–285, 2017) and Tardini and Tomassini (Int J Math 27(12), 1650103 (20 pp.), 2016).

Notes

Acknowledgements

The author would like to thank the organizers Daniele Angella, Paolo de Bartolomeis, Costantino Medori and Adriano Tomassini for the kind invitation to the INdAM Meeting Complex and Symplectic Geometry held at Palazzone of Cortona. Many thanks to all the participants at the conference who contributed to create such a nice environment. Special thanks also to Adriano Tomassini and Daniele Angella for their constant support, encouragement, for many useful discussions on the subject and for their contribution to the results obtained jointly with the author. This proceeding is dedicated to the memory of the very kind person and excellent mathematician Paolo de Bartolomeis.

The project is partially supported by the GNSAGA of INdAM.

References

  1. 1.
    A. Aeppli, On the cohomology structure of Stein manifolds, in Proceedings of the Conference on Complex Analysis (Minneapolis, MN, 1964) (Springer, Berlin, 1965), pp. 58–70Google Scholar
  2. 2.
    D. Angella, The cohomologies of the Iwasawa manifold and of its small deformations. J. Geom. Anal. 23(3), 1355–1378 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Angella, On the Bott-Chern and Aeppli cohomology, in Bielefeld Geometry & Topology Days (2015). Available via https://www.math.uni-bielefeld.de/sfb701/suppls/ssfb15001.pdf
  4. 4.
    D. Angella, H. Kasuya, Bott-Chern cohomology of solvmanifolds (2016). arXiv:1212.5708 [math.DG]Google Scholar
  5. 5.
    D. Angella, H. Kasuya, Symplectic Bott-Chern cohomology of solvmanifolds (2016). arXiv:1308.4258 [math.SG]Google Scholar
  6. 6.
    D. Angella, N. Tardini, Quantitative and qualitative cohomological properties for non-Kähler manifolds. Proc. Am. Math. Soc. 145(1), 273–285 (2017)CrossRefzbMATHGoogle Scholar
  7. 7.
    D. Angella, A. Tomassini, On the \(\partial \overline{\partial }\)-lemma and Bott-Chern cohomology. Invent. Math. 192(1), 71–81 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D. Angella, A. Tomassini, Inequalities à la Frölicher and cohomological decompositions. J. Noncommut. Geom. 9(2), 505–542 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. Angella, A. Tomassini, On Bott-Chern cohomology and formality. J. Geom. Phys. 93, 52–61 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Angella, G. Dloussky, A. Tomassini, On Bott-Chern cohomology of compact complex surfaces. Ann. Mat. Pura Appl. 195(1), 199–217 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Angella, A. Tomassini, M. Verbitsky, On non-Kähler degrees of complex manifolds (2016). arXiv:1605.03368Google Scholar
  12. 12.
    D. Angella, A. Otiman, N. Tardini, Cohomologies of locally conformally symplectic manifolds and solvmanifolds, to appear in Ann. Global Anal. Geom., DOI: 10.1007/s10455-017-9568-yGoogle Scholar
  13. 13.
    J.-M. Bismut, A local index theorem for non-Kähler manifolds. Math. Ann. 284, 681–699 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    R. Bott, S.S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114(1), 71–112 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J.-L. Brylinski, A differential complex for Poisson manifolds. J. Differ. Geom. 28(1), 93–114 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    G.R. Cavalcanti, New aspects of the dd c-lemma, Oxford University. D. Phil. thesis, 2005. arXiv:math/0501406v1 [math.DG]Google Scholar
  17. 17.
    K. Chan, Y.-H. Suen, A Frölicher-type inequality for generalized complex manifolds. Ann. Global Anal. Geom. 47(2), 135–145 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    P. Deligne, Ph.A. Griffiths, J. Morgan, D.P. Sullivan, Real homotopy theory of Kähler manifolds. Invent. Math. 29(3), 245–274 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants. Proc. Natl. Acad. Sci. USA 41(9), 641–644 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    K. Kodaira, On the structure of compact complex analytic surfaces. I. Am. J. Math. 86, 751–798 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    K. Kodaira, D.C. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures. Ann. Math. (2) 71, 43–76 (1960)Google Scholar
  22. 22.
    O. Mathieu, Harmonic cohomology classes of symplectic manifolds. Comment. Math. Helv. 70(1), 1–9 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    S.A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds. Int. Math. Res. Not. 1998(14), 727–733 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    M.L. Michelsohn, On the existence of special metrics in complex geometry. Acta Math. 143, 261–295 (1983)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Y. Miyaoka, Kähler metrics on elliptic surfaces. Proc. Jpn. Acad. 50(8), 533–536 (1974)CrossRefzbMATHGoogle Scholar
  26. 26.
    M. Schweitzer, Autour de la cohomologie de Bott-Chern, Prépublication de l’Institut Fourier no. 703 (2007). arXiv:0709.3528Google Scholar
  27. 27.
    Y.T. Siu, Every K3 surface is Kähler. Invent. Math. 73(1), 139–150 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    N. Tardini, A. Tomassini, On the cohomology of almost-complex and symplectic manifolds and proper surjective maps. Int. J. Math. 27(12), 1650103 (20 pp.) (2016)Google Scholar
  29. 29.
    N. Tardini, A. Tomassini, On geometric Bott-Chern formality and deformations. Ann. Mat. Pura Appl. (4) 196 (1), 349–362 (2017)Google Scholar
  30. 30.
    A. Teleman, The pseudo-effective cone of a non-Kählerian surface and applications. Math. Ann. 335, 965–989 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    L.-S. Tseng, S.-T. Yau, Cohomology and Hodge theory on symplectic manifolds: I. J. Differ. Geom. 91(3), 383–416 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    J. Varouchas, Proprietés cohomologiques d’une classe de variétés analytiques complexes compactes, in Séminaire d’analyse P. Lelong-P. Dolbeault-H. Skoda, années 1983/1984. Lecture Notes in Mathematics, vol. 1198 (Springer, Berlin, 1986), pp. 233–243Google Scholar
  33. 33.
    C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés, 10 (Société Mathématique de France, Paris, 2002)zbMATHGoogle Scholar
  34. 34.
    C.-C. Wu, On the geometry of superstrings with torsion. Thesis (Ph.D.), Harvard University, Proquest LLC, Ann Arbor, MI, 2006Google Scholar
  35. 35.
    D. Yan, Hodge structure on symplectic manifolds. Adv. Math.120(1), 143–154 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, a part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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