Transformation Groups

, Volume 1, Issue 3, pp 249–258 | Cite as

Toute variété magnifique est sphérique



LetG be a (connected) reductive group (over C). An algebraicG-varietyX is called “wonderful”, if the following conditions are satisfied:X is (connected) smooth and complete;X containsr irreducible smoothG-invariant divisors having a non void transversal intersection;G has 2 r orbits inX. We show that wonderful varieties are necessarily spherical (i.e., they are almost homogeneous under any Borel subgroup ofG).


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  1. [A]
    D. Akhiezer,Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Glob. Analysis and Geometry1 (1983), 49–78.CrossRefMathSciNetzbMATHGoogle Scholar
  2. [B-B]
    A. Bialynicki-Birula,Some theorems on actions of algebraic groups, Ann. of Math.98 (1980), 480–497.CrossRefMathSciNetGoogle Scholar
  3. [B1]
    M. Brion,On spherical varieties of rank one, CMS Conf. Proceedings10 (1989), 31–41.MathSciNetGoogle Scholar
  4. [B2]
    —,Vers une généralisation des espaces symétriques, J. Algebra134 (1990), 115–143.CrossRefMathSciNetzbMATHGoogle Scholar
  5. [BLV]
    M. Brion, D. Luna, and Th. Vust,Espaces homogènes sphériques, Inventiones Math.84 (1986), 617–632.CrossRefMathSciNetzbMATHGoogle Scholar
  6. [BP]
    M. Brion and F. Pauer,Valuations des espaces homogènes sphériques, Comment. Math. Helvetici62 (1987), 265–285.MathSciNetzbMATHGoogle Scholar
  7. [Bou]
    N. Bourbaki,Groupes et Algébres de Lie, Hemann, Paris (1968).zbMATHGoogle Scholar
  8. [DeC-P]
    C. De Concini and C. Procesi,Complete symmetric varieties, Springer Lect. Notes in Math.996 (1983), 1–44.CrossRefGoogle Scholar
  9. [K1]
    F. Knop,The Luna-Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups, Manoj Prakashan, Madras (1991), 225–249.Google Scholar
  10. [K2]
    —,Automorphismes, root systems and compactifications of homogeneous varieties, J. Amer. Math. Soc.9 (1996), no. 1, 153–174.CrossRefMathSciNetzbMATHGoogle Scholar
  11. [L1]
    D. Luna,Slices étales, Bull. Soc. Math. France, Mémoire33 (1973), 81–105.zbMATHGoogle Scholar
  12. [L2]
    D. Luna,Grosses cellules pour les variétés sphériques, à paraître dans Algebraic groups and related subjects, a volume in honor of R. W. Richardson (1996), Australian Math. Soc. Lecture Series, G. I. Lehrer et al. (eds.), Cambridge University Press.Google Scholar
  13. [LV]
    D. Luna and Th. Vust,Plongements d’espaces homogènes, Comment. Math. Helvetici58 (1983), 186–245.MathSciNetzbMATHGoogle Scholar
  14. [M-J]
    L. Moser-Jauslin,Some almost homogeneous group actions on smooth complete rational surfaces, L’Enseignement Mathématique34 (1988), 313–332.MathSciNetzbMATHGoogle Scholar
  15. [W]
    B. Wasserman,Wonderful varieties of rank 2,à paraître.Google Scholar

Copyright information

© Birkhäuser 1996

Authors and Affiliations

  • D. Luna
    • 1
  1. 1.Institut FourierUniversité de Grenoble ISaint Martin d’HèresFrance

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