Advertisement

Vanishing Theorems on Holomorphic Lie Algebroids

  • Alexandru IonescuEmail author
Article
  • 26 Downloads

Abstract

The paper describes a Bochner-type study for holomorphic horizontal vector fields defined on a holomorphic Finsler algebroid E. We obtain in this setting a vanishing theorem for horizontal fields with compact support on E.

Keywords

Holomorphic Lie algebroid Laplacian for functions Bochner technique Vanishing theorem Prolongation 

Mathematics Subject Classification

Primary 17B66 53B40 Secondary 53B35 

Notes

Acknowledgements

The author would like to thank the anonymous referee for the suggestions and comments that helped him improve this article.

References

  1. 1.
    Aikou, T.: Finsler geometry on complex vector bundles. Riemann Finsler Geom 50, 85–107 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bao, D., Lackey, B.: A Hodge decomposition theorem for Finsler spaces, C.R. Acad. Sci. Paris, vol. 323, Serie 1, p. 51–56 (1996)Google Scholar
  3. 3.
    Bochner, S.: Vector fields and Ricci curvature. Bull. Am. Math. Soc. 52, 776–797 (1946)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bochner, S.: Curvature in Hermitian metric. Bull. Am. Math. Soc. 53, 179–195 (1947)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. In: Pure and Applied Mathematics. Wiley-Interscience, New York (1978)Google Scholar
  6. 6.
    Ida, C.: A vanishing theorem for vertical tensor fields on complex Finsler bundles. Analele Şt. ale Univ. “Al. I. Cuza” din Iaşi, Mat., Tomul LVII, Suppl., p. 103–112 (2011)Google Scholar
  7. 7.
    Ida, C., Popescu, P.: On Almost complex Lie algebroids. Mediterr. J. Math. 13(2), 803–824 (2016). arXiv:1341.32021 (Zbl)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ionescu, A.: On holomorphic Lie algebroids. Bull. Transilv. Univ. 58(1), 53–66 (2016)Google Scholar
  9. 9.
    Ionescu, A., Munteanu, G.: Connections in holomorphic Lie algebroids. Mediterr. J. Math. 14, 163 (2017).  https://doi.org/10.1007/s00009-017-0960-4 CrossRefzbMATHGoogle Scholar
  10. 10.
    Ionescu, A.: Finsler structures on holomorphic Lie algebroids. Novi Sad J. Math. 47(2), 117–132 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ionescu, A.: Laplace operators on holomorhphic Lie algebroids. An. Şt. Univ. Ovidius Constanţa 26(1), 141–158 (2018)Google Scholar
  12. 12.
    Kobayashi, S.: Negative vector bundles and complex Finsler structures. Nagoya Math. J. 57, 153–166 (1975)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Laurent-Gengoux, C., Stiénon, M., Xu, P.: Holomorphic Poisson manifolds and holomorphic Lie algebroids. Int. Math. Res. Not. IMRN rnn88, 46 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Marle, C.-M.: Calculus on Lie algebroids, Lie groupoids and Poisson manifolds, Dissertationes Mathematicae. Inst. Math. Polish Acad. Sci. 457, 57 (2008)zbMATHGoogle Scholar
  15. 15.
    Martinez, E.: Geometric formulation of mechanics on Lie algebroids. In: Proc. of the VIIIth Workshop on Geometry and Physics (Medina del Campo, 1999), vol. 2. Publ. R. Soc. Mat. Esp., p. 209–222 (2001)Google Scholar
  16. 16.
    Morrow, J., Kodaira, K.: Complex Manifold. Holt, Rinehart and Winston Inc, New York (1971)zbMATHGoogle Scholar
  17. 17.
    Munteanu, G.: Complex Spaces in Finsler, Lagrange and Hamilton Geometries. Kluwer Academic Publishers, Dordrecht (2004)CrossRefGoogle Scholar
  18. 18.
    Wu, H.: The Bochner technique in differential geometry. Math. Rep. 3, 289–538 (1988)MathSciNetGoogle Scholar
  19. 19.
    Xiao, J., Zhong, T., Qiu, C.: Bochner technique on strong Kähler–Finsler manifolds. Acta Math. Sci. Ser. B Engl. Ed. 30, 89–106 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zhong, C.: A vanishing theorem on Kähler Finsler manifolds. Differ. Geom. Appl. 27, 551–565 (2009)CrossRefGoogle Scholar
  21. 21.
    Zhong, C., Zhong, T.: Hodge decomposition theorem on strongly Kähler Finsler manifolds. Sci. China Ser. A Math. 49(11), 1696–1714 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceTransilvania University of BraşovBraşovRomania

Personalised recommendations