Skip to main content
SpringerLink
Log in

On fixed points of algebraic actions on ℂn

  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

It is shown that the action regarded for a rather long time by experts as a possible example disproving the conjecture on the existence of fixed points for reductive algebraic group actions on affine spaces is not an action on an affine variety, and therefore provides no example of this kind. Moreover, it is shown that the actions naturally related to the original one provide no examples of this kind as well.

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. E. B. Elliot, An Introduction to the Algebra of Quantics, 2nd ed., Oxford Univ Press, Oxford, 1913.

    Google Scholar 

  2. M. Fankhauser, “Fixed points for reductive group actions on acyclic varieties,” Ann. Inst. Fourier,45, No. 5, 1249–1281 (1995).

    MATH  MathSciNet  Google Scholar 

  3. R. Fossum and B. Iversen, “On Picard groups of algebraic fibre spaces,” J. Pure Appl. Algebra,3, 269–280 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  4. I. M. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston-Basel-Berlin, 1994.

    MATH  Google Scholar 

  5. H. Kraft, “Challenging problems on affinen-space,” Séminaire Bourbaki, 47ème année,802, 1–19 (1994–95).

    Google Scholar 

  6. R. Oliver, “Weight systems for SO(3)-actions,” Ann. of Math.,103, 637–644 (1979).

    Article  MathSciNet  Google Scholar 

  7. V. L. Popov, “The Picard groups of homogeneous spaces of linear algebraic groups, and one-dimensional homogeneous vector bundles,” Izv. Akad. Nauk SSSR, Ser. Mat.,38, 294–322 (1974).

    MATH  MathSciNet  Google Scholar 

  8. M. Rosenlicht, “Toroidal algebraic groups,” Proc. Amer. Math. Soc.,12, 984–988 (1961).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. P. I. Katsylo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. L. Popov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by CRDF grant RM1-206 and INTAS grant INTAS-OPEN-97-1570.

Moscow Independent University. Moscow Institute of Electronics and Mathematics. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 41–50, January–March, 2000

Translated by V. L. Popov

Rights and permissions

Reprints and permissions

About this article

Cite this article

Katsylo, P.I., Popov, V.L. On fixed points of algebraic actions on ℂn . Funct Anal Its Appl 34, 33–40 (2000). https://doi.org/10.1007/BF02467065

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02467065

Keywords

Search

Navigation