Transformation Groups

, Volume 4, Issue 2–3, pp 273–300

Compactification of symmetric varieties

  • C. De Concini
  • T. A. Springer


The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic ≠ 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. In the casek=C, De Concini and Procesi (1983) constructed a “wonderful” compactification ofG/H. We prove the existence of such a compactification for arbitraryk. We also prove cohomology vanishing results for line bundles on the compactification.


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  1. [B] A. Borel et al.,Seminar on Algebraic Groups and Related Finite Groups, Lect. Notes in Math., vol. 131, Springer, second printing, 1986.Google Scholar
  2. [BB] A. Bialynicki-BirulaSome theorems on actions of algebraic groups, Ann. of Math.98 (1973), 480–497.Google Scholar
  3. [BI] M. Brion and S. P. Inamdar,Frobenius splitting of spherical varieties, In: Algebraic Groups and Their Generalizations: Classical Methods (University Park, PA, 1991), 207–218, Proc. Sympos. Pure Math., vol. 56, part 1, Amer. Math. Soc., Providence, RI, 1994.Google Scholar
  4. [CP] C. De Concini and C. Procesi,Complete symmetric varieties, In: Invariant Theory, pp. 1–44, Lect. Notes in Math., vol. 996, Springer, 1983.Google Scholar
  5. [CS] C. De Concini and T. A. Springer,Betti numbers of complete symmetric varieties, In: Geometry Today, pp. 87–107, Birkhäuser, 1985.Google Scholar
  6. [D] S. Donkin,On tilting modules for algebraic groups, Math. Z.212 (1993), no. 1, 39–60.Google Scholar
  7. [F] G. Faltings,Explicit resolution of local singularities of moduli-spaces, J. Reine Angew. Math.483 (1997), 183–196.Google Scholar
  8. [EGA] A. Grothendieck and J. Dieudonné,Eléments de Géométrie Algèbrique I, Publ. Math. I.H.E.S.4 (1960).Google Scholar
  9. [J] J. C. Jantzen,Representations of Algebraic Groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987.Google Scholar
  10. [K] F. Knop,The Luna-Vust theory of spherical embeddings, In: Proceedings of the Hyderabad Conference on Algebraic Groups, pp. 226–249, Madras, 1991.Google Scholar
  11. [MR] V. B. Mehta and A. Ramanathan,Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2)122 (1985), no. 1, 27–40.Google Scholar
  12. [R] R. W. Richardson,Orbits, invariants and representations associated to involutions of reductive groups, Inv. Math.71 (1983), 287–312.Google Scholar
  13. [RS] R. W. Richardson and T. A. Springer,The Bruhat order on symmetric varieties, Geom. Dedic.35 (1990), 389–436.Google Scholar
  14. [Se] J.-P. Serre,Cohomologie Galoisienne, fifth ed., Lect. Notes in Math., vol. 5, Springer, 1994.Google Scholar
  15. [Sp1] T. A. Springer,Some results on algebraic groups with involutions, In: Algebraic Groups and Related Topics, Kinokuniya/North-Holland, 1985, pp. 525–543.Google Scholar
  16. [Sp2] T. A. Springer,The classification of involutions of simple algebraic groups, Journal Fac. Sci. Tokyo University,34 (1987), 655–670.Google Scholar
  17. [Sp3] T. A. Springer,Linear Algebraic Groups, second ed., Birkhäuser, 1998.Google Scholar
  18. [St] E. Strickland,A vanishing theorem for group compactifications, Math. Ann.277 (1987), no. 1, 165–171.Google Scholar
  19. [T] J. Tits,Reductive groups over local fields, in: Automorphic Forms, Representations andL-Groups, vol. I, pp. 29–69, Amer. Math. Soc., 1979.Google Scholar
  20. [U] T. Uzawa,On equivariant completions of algebraic symmetric spaces, In: Algebraic and Topological Theories, Kinokuniya/North-Holland, 1985, pp. 569–577.Google Scholar
  21. [W] A. Weil,The field of definition of a variety, Amer. J. Math.78 (1956), 509–524.Google Scholar

Copyright information

© Birkhäuser 1999

Authors and Affiliations

  • C. De Concini
    • 1
  • T. A. Springer
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomeItaly
  2. 2.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands

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