Skip to main content
SpringerLink
Log in

A rigidity theorem for ℙ3(ℂ)

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this paper the following theorem is proved. Let X be a compact complex manifold of dimension 3 which carries 3 algebraically independent meromorphic functions. Assume that X is topologically isomorphic to the 3-dimensional projective space ℙ3. Then X is biholomorphically equivalent to ℙ3. In particular any deformation of ℙ3 is ℙ3.

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. Grauert,H., Riemenschneider,O.: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Inv. Math. 11, 263–292 (1970)

    Google Scholar 

  2. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Math. 52, Springer, Berlin-Heidelberg-New York 1977

    Google Scholar 

  3. Hartshorne,R.: Ample vector bundles. Publ.Math.Inst.Hautes Etud. Sci. 29, 63–94 (1966)

    Google Scholar 

  4. Hirzebruch,F., Kodaira,K.: On the complex projective spaces. J. Math. Pures Appl. 36, 201–216 (1957)

    Google Scholar 

  5. Kawamata,Y.: A generalization of Kodaira-Ramanujam's vanishing theorem. Math. Ann. 261, 43–46 (1982)

    Google Scholar 

  6. Kawamata,Y.: On the finiteness of generators of a pluri-canonical ring for a 3-fold of general type. Amer.J. Math. 106, 1503–1512 (1984)

    Google Scholar 

  7. Kawamata,Y.: Elementary contractions of algebraic 3-folds. Ann. Math. 119, 95–110 (1984)

    Google Scholar 

  8. Kawamata,Y.: The cone of curves of algebraic varieties. Ann. Math. 119, 603–633 (1984)

    Google Scholar 

  9. Laufer, H.: On CP1 as exceptional set. Ann. Math. Studies vol. 100, 261–275. Princeton 1981

    Google Scholar 

  10. Miyanishi,M.: Algebraic methods in the theory of algebraic 3-folds. Adv. Studies in Pure Math. 1, 69–99. Tokyo 1981

    Google Scholar 

  11. Moišezon,B.G.: On n-dimensional varieties with n algebraically independent meromorphic functions. Amer. Math. Soc. Transl. 51–177 (1967)

  12. Mori,S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. 116, 133–176 (1982)

    Google Scholar 

  13. Nagata,M.: On self intersection number of a section on a ruled surface. Nagoya Math. J. 37, 191–196 (1870)

    Google Scholar 

  14. Peternell,T.: Algebraicity criteria for compact complex manifolds. Preprint 1984

  15. Ueno,K.: Classification theory of algebraic varieties and compact complex spaces. Lecture Notes in Math. 439, Springer, Berlin-Heidelberg-New York 1975

    Google Scholar 

  16. Viehweg,E.: Vanishing theorems. J. reine u. angew. Math. 335, 1–8 (1982)

    Google Scholar 

  17. Yau,S.T.: Calabi's conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA 74, 1789–1799 (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut der Universität, Einsteinstr. 64, 4400, Münster, Bundesrepublik Deutschland

    Thomas Peternell

Authors
  1. Thomas Peternell
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Karl Stein

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peternell, T. A rigidity theorem for ℙ3(ℂ). Manuscripta Math 50, 397–428 (1985). https://doi.org/10.1007/BF01168838

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01168838

Keywords

Search

Navigation