Skip to main content
SpringerLink
Log in

Dolbeault cohomology of compact complex homogeneous manifolds

  • Published:
Proceedings of the Indian Academy of Sciences - Mathematical Sciences Aims and scope Submit manuscript

Abstract

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S1 ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.

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

Access this article

Log in via an institution

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. Akhiezer D, Sur les représentations de groupes de Lie dans les espaces de cohomologie de Dolbeault,C R Acad. Sci. Paris 321 (1995) 1583–1586

    MATH  MathSciNet  Google Scholar 

  2. Akhiezer D, Group actions on the Dolbeault cohomology of homogeneous manifolds,Math. Zeit. 226 (1997) 607–621

    Article  MATH  MathSciNet  Google Scholar 

  3. Akhiezer D, Lie group actions in complex analysis,Aspects Math. E 27 (Vieveg) (1995)

  4. Akhiezer D, Homogeneous complex manifolds, inSeveral Complex Variables-IV, Encycl. Math. Sci. 10 Springer-Verlag, 1990

  5. Atiyah M F, Complex analytic connections in fibre bundles,Trans. Am. Math. Soc. 85 (1957) 181–207

    Article  MATH  MathSciNet  Google Scholar 

  6. Blanchard A, Automorphismes des variété fibrée analytiques complexes,C R Acad. Sci. Paris 233 (1951) 1337–1339

    MATH  MathSciNet  Google Scholar 

  7. Bochner S and Montgomery D, Groups on analytic manifolds,Ann. Math. 48 (1947) 659–669

    Article  MathSciNet  Google Scholar 

  8. Borel A, A spectral sequence for complex analytic bundles, Appendix-II to F. Hirzebruch,Topological methods in algebraic geometry, Grundlehren131 (Third Edition), Springer-Verlag, 1976

  9. Bourbaki N,Groupes et algèbre de Lie, (Ch. IV-VI) Hermann, Paris, 1968

    Google Scholar 

  10. Gorbatsevich V V and Onishchik A L, Lie transformation groups, in Lie groups and Lie algebras-I,Encycl. Math. Sci. 20, Springer-Verlag, 1993

  11. Gugenheim V K A M and Spencer D C, Chain homotopy and the de Rham theory,Proc. Am. Math. Soc. 7 (1956) 144–152

    Article  MATH  MathSciNet  Google Scholar 

  12. Hochschild G P,Basic theory of algebraic groups and Lie algebra, GTM-75, Springer-Verlag, 1981

  13. Ise M, Some properties of complex analytic vector bundles over compact complex homogeneous spaces,Osaka Math. J. 12 (1960) 217–252

    MATH  MathSciNet  Google Scholar 

  14. Ise M, On the geometry of Hopf manifolds,Osaka Math. J. 12 (1960) 387–402

    MATH  MathSciNet  Google Scholar 

  15. Lescure F, Action nontriviale sur le premier groupe de cohomologie de Dolbeault,C R Acad. Sci. Paris 316 (1993) 823–825

    MATH  MathSciNet  Google Scholar 

  16. Lescure F, Example d’actions induites non résolubles sur la cohomologie de Dolbeault,Topology 35 (1996) 561–581

    Article  MATH  MathSciNet  Google Scholar 

  17. Montgomery D, Simply connected homogeneous spaces,Proc. Am. Math. Soc. 1 (1950) 467–469

    Article  MATH  Google Scholar 

  18. Mumford D,Abelian Varieties, Oxford University Press, Bombay, 1974

    MATH  Google Scholar 

  19. Wang H.-C, Closed manifolds with homogeneous complex structures,Am. J. Math. 76 (1954) 1–32

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SPIC Mathematical Institute, 92 G N Chetty Road, 600017, Chennai, India

    Vimala Ramani & Parameswaran Sankaran

Authors
  1. Vimala Ramani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Parameswaran Sankaran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vimala Ramani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ramani, V., Sankaran, P. Dolbeault cohomology of compact complex homogeneous manifolds. Proc. Indian Acad. Sci. (Math. Sci.) 109, 11–21 (1999). https://doi.org/10.1007/BF02837763

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02837763

Keywords

Search

Navigation