Advertisement

Vertical–horizontal decomposition of Laplacians and cohomologies of manifolds with trivial tangent bundles

  • Sönke Rollenske
  • Adriano TomassiniEmail author
  • Xu Wang
Article
  • 36 Downloads

Abstract

In this paper, we obtain a vertical–horizontal decomposition formula of Laplacians on manifolds with a special foliation structure. Two Nomizu-type theorems for cohomologies of nilmanifolds follow as applications.

Keywords

Künneth formula Horizontal lift Vertical form Laplacian Nilpotent foliation Torus fibration Nomizu-type theorem Nilpotent group 

Mathematics Subject Classification

53C25 53C55 

Notes

Acknowledgements

Xu Wang would like to thank B. Berndtsson, J.P. Demailly and N. Mok for several useful discussions about the topics of this paper. Sönke Rollenske is grateful to the other authors for the invitation to join the project at a relatively late stage. He is also grateful to A. Fino and J. Ruppenthal for many discussions about the Dolbeault cohomology of nilmanifolds that culminated in the paper [13]. We are also pleased to thank the anonymous referee for valuable remarks and suggestions for a better presentation of our results.

References

  1. 1.
    Alvarez López, J.A.: A finiteness theorem for the spectral sequence of a Riemannian foliation. Ill. J. Math. 33, 79–92 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alvarez López, J.A.: A decomposition theorem for the spectral sequence of Lie foliations. Trans. Am. Math. Soc. 329, 173–184 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alvarez López, J.A., Tondeur, P.: Hodge decomposition along the leaves of a Riemannian foliation. J. Funct. Anal. 99, 443–458 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berndtsson, B., Păun, M., Wang, X.: Algebraic fiber spaces and curvature of higher direct images. arXiv:1704.02279
  5. 5.
    Blanchard, A.: Sur les variétés analytiques complexes. Ann. Sci. Ecole Norm. Sup. 73, 157–202 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Console, S., Fino, A.: On the de Rham cohomology of solvmanifolds. Ann. Scuola Norm. Sup. 10, 801–818 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Console, S., Fino, A., Kasuya, H.: On de Rham and Dolbeault cohomology of solvmanifolds. Transf. Groups 21, 653–680 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cordero, L.A., Fernández, M., Gray, A., Ugarte, L.: Nilpotent complex structures on compact nilmanifolds. Rend. Circ. Mat. Palermo 49, 83–100 (1997)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cordero, L.A., Fernández, M., Gray, A., Ugarte, L.: Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology. Trans. Am. Math. Soc. 352, 5405–5433 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Corwin, L.J., Greenleaf, F.P.: Representations of nilpotent Lie groups and their applications. Part I. In: Cambridge Studies in Advanced Mathematics, vol. 18, Cambridge University Press, Cambridge (1990)Google Scholar
  11. 11.
    Deligne, P.: Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. l’Inst. Ht. Ét. Sci. 35, 107–126 (1968)CrossRefzbMATHGoogle Scholar
  12. 12.
    Demailly, J.P.: Complex analytic and differential geometry. Université de Grenoble I Institut Fourier, Demailly’s homepageGoogle Scholar
  13. 13.
    Fino, A., Rollenske, S., Ruppenthal, J.: Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups. arXiv:1808.08090 [math.DG]
  14. 14.
    Frölicher, A.: Relations between the cohomology groups of Dolbeault and topological invariants. Proc. Natl. Acad. Sci. USA 41, 641–644 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Griffiths, P.A.: The extension problem in complex analysis II; embeddings with positive normal bundle. Am. J. Math. 88, 366–446 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kasuya, H.: An extention of Nomizu’s theorem—a user’s guide. In: Complex Manifolds, pp. 231–238 (2016)Google Scholar
  17. 17.
    Kodaira, K.: On the structure of compact complex analytic surfaces. I. Am. J. Math. 86, 751–798 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kordyukov, Y.A.: Adiabatic limits and spectral geometry of foliations. Math. Ann. 313, 763–783 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kamber, F.W., Tondeur, P.: de Rham-Hodge theory for Riemannian foliations. Math. Ann. 277, 415–431 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. Math. 59, 531–538 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Reinhart, B.L.: Harmonic integrals on foliated manifolds. Am. J. Math. 81, 529–536 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rollenske, S.: Geometry of nilmanifolds with left-invariant complex structure and deformations in the large. Proc. Lond. Math. Soc. 99(2), 425–460 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rollenske, S.: Dolbeault cohomology of nilmanifolds with left-invariant complex structure. In: Complex and Differential Geometry, Springer Proceedings in Mathematics, vol. 8, Springer, Heidelberg, pp. 369–392 (2011)Google Scholar
  24. 24.
    Sakane, Y.: On compact complex parallelisable solvmanifolds. Osaka J. Math. 13, 187–212 (1976)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Salamon, S.M.: Complex structures on nilpotent Lie algebras. J. Pure Appl. Algebra 157(2–3), 311–333 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Takagi, R., Yorozu, S.: Notes on the Laplace-Beltrami operator on a foliated Riemannian manifold with a bundle-like metric. Nih. Math. J. 1, 89–106 (1990)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Tomassini, A., Wang, X.: Some results on the Hard Lefschetz condition. Int. J. Math. 29, 1850095 (2018). (30 pp.)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Thurston, W.P.: Some simple examples of symplectic manifolds. Proc. Am. Math. Soc. 55, 467–468 (1976)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ugarte, L.: Hermitian structures on six-dimensional nilmanifolds. Transf. Groups 12, 175–202 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.FB 12/Mathematik und InformatikPhilipps–Universität MarburgMarburgGermany
  2. 2.Dipartimento di Scienze Matematiche, Fisiche e InformaticheUnità di Matematica e Informatica, Universitá degli Studi di ParmaParmaItaly
  3. 3.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations