Transformation Groups

, Volume 2, Issue 3, pp 225–267

Equivariant Chow groups for torus actions

  • M. Brion
Article

Abstract

We study Edidin and Graham's equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Arabia,Cycles de Schubert et cohomologie équivariante de K/T, Invent. Math.85 (1986), 39–52.Google Scholar
  2. [2]
    qI. Н. Бернштейн, И. М. Гелъфанд, С. И. Гелъфанд, Кпетки Шуберта и когомопогии пространств G/P,YMH XXVIII 3 (171) (1973), 3–26. English translation: I. N. Bernstein, I. M. Gelfand and S. I. Gelfand,Schubert cells and cohomology of the spaces G/P, Russian Math. Surveys,28 (1973), 1–26.Google Scholar
  3. [3]
    A. Bialynicki-Birula,Some theorems on actions of algebraic groups, Ann. Math.98 (1973), 480–497.Google Scholar
  4. [4]
    A. Bialynicki-Birula,On fixed points of torus actions on projective varieties, Bull. Acad. Polon. Sci. Séri. Sci. Math. Astronom. Phys.22 (1974), 1097–1101.Google Scholar
  5. [5]
    A. Bialynicki-Birula,Some properties of the decomposition of algebraic varieties determined by the action of a torus, Bull. Acad. Polon. Sci. Séri. Sci. Math. Astronom. Phys.24, (1976), 667–674.Google Scholar
  6. [6]
    E. Bifet, C. De Concini and C. Procesi,Cohomology of regular embeddings, Adv. Math.82 (1990), 1–34.Google Scholar
  7. [7]
    A. Borel,Linear Algebraic Groups Springer-Verlag, New York, 1991. Russian translation: A. Борелъ, Линейные агебраические руппы, Москва, Мир, 1972.Google Scholar
  8. [8]
    W. Borho, J-L. Brylinski and R. MacPherson,Nilpotent Orbits, Primitive Ideals, and Characteristic Classes, Birkhäuser, Boston, 1989.Google Scholar
  9. [9]
    P. Bressler and S. Evens,The Schubert calculus, braid relations, and generalized cohomology, Trans. AMS317 (1990), 799–811.Google Scholar
  10. [10]
    M. Brion,Piecewise polynomial functions, convex polytopes and enumerative geometry, Parameter spaces (P. Pragacz, ed.), Banach Center Publications, 1996, pp. 25–44.Google Scholar
  11. [11]
    M. Brion and M. Vergne,An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Crelle482 (1997), 67–92.Google Scholar
  12. [12]
    J. B. Carrell,The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, Proc. Symp. in Pure Math., 1994, pp. 53–61.Google Scholar
  13. [13]
    M. Demazure,Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math.21 (1973), 287–301.Google Scholar
  14. [14]
    M. Demazure,Désingularisation des variétés de Schubert généralisées, Ann. Scient. Éc. Norm. Sup.7 (1974), 53–88.Google Scholar
  15. [15]
    D. Edidin and W. Graham,Equivariant intersection theory, preprint 1996.Google Scholar
  16. [16]
    D. Edidin and W. Graham,Localization in equivariant intersection theory and the Bott residue formula, preprint 1996.Google Scholar
  17. [17]
    W. Fulton,Intersection Theory, Springer-Verlag, New York, 1984. Russian translation: У, Фултон, Теория пересечений, Москва, Мир, 1989.Google Scholar
  18. [18]
    W. Fulton,Introduction to Toric Varieties, Princeton University Press, Princeton, 1993.Google Scholar
  19. [19]
    W. Fulton,Flags, Schubert polynomials, degeneracy loci and determinantal formulas, Duke Math. J.65 (1992), 381–420.Google Scholar
  20. [20]
    W. Fulton,Schubert varieties in flag bundles for classical groups, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, Bar-Ilan University, 1996, pp. 241–262.Google Scholar
  21. [21]
    W. Fulton, R. MacPherson, F. Sottile and B. Sturmfels,Intersection theory on spherical varieties, J. Alg. Geom.4 (1995), 181–193.Google Scholar
  22. [22]
    H. Gillet,Riemann-Roch theorems for higher algebraic K-theory, Adv. Math.40 (1981), 203–289.Google Scholar
  23. [23]
    M. Goresky, R. Kottwitz and R. MacPherson,Equivariant cohomology, Koszul duality, and the localization theorem, preprint 1996.Google Scholar
  24. [24]
    W. Graham,The class of the diagonal in flag bundles, preprint 1996.Google Scholar
  25. [25]
    A. Joseph,On the variety of a highest weight module, J. Algebra88 (1984), 238–278.Google Scholar
  26. [26]
    F. Kirwan,Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press, Princeton, 1984.Google Scholar
  27. [27]
    F. Knop,On the set of orbits for a Borel subgroup, Comment. Math. Helv.70 (1995), 285–309.Google Scholar
  28. [28]
    B. Kostant and S. Kumar,The nil Hecke ring and cohomology of G/P for a Kac-Moody group G. Adv. Math.,62 (1986), 187–237.Google Scholar
  29. [29]
    B. Kostant and S. Kumar,T-equivariant K-theory of generalized flag varieties, J. Differ. Geom.32 (1990), 549–603.Google Scholar
  30. [30]
    S. Kumar,The nil Hecke ring and singularity of Schubert varieties, Invent. math.123 (1996), 471–506.Google Scholar
  31. [31]
    P. Littelmann and C. Procesi,Equivariant cohomology of wonderful compactifications, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Birkhäuser, Basel, 1990.Google Scholar
  32. [32]
    M. Nyenhuis,Equivariant Chow groups and multiplicities, preprint 1996.Google Scholar
  33. [33]
    M. Nyenhuis,Equivariant Chow groups and equivariant Chern classes, preprint 1996.Google Scholar
  34. [34]
    P. Polo,On Zariski tangent spaces of Schubert varieties, and a proof of a conjecture of Deodhar, Indag. Math.5 (1994), 483–493.Google Scholar
  35. [35]
    P. Pragacz,Symmetric polynomials and divided differences in formulas of intersection theory, Parameter Spaces, Banach Center Publications, 1996, pp. 125–177.Google Scholar
  36. [36]
    P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci: the\(\bar Q\)-polynomials, Compositio Math., to appear.Google Scholar
  37. [37]
    R. W. Richardson and T. A. Springer,The Bruhat order on symmetric varieties, Geometriae Dedicata35, (1990), 389–436.Google Scholar
  38. [38]
    W. Rossmann,Equivariant multiplicities on complex varieties, Astérisque173–174 (1989), 313–330.Google Scholar
  39. [39]
    W. Smoke,Dimension and multiplicity for graded algebras, J. Algebra21 (1972), 149–173.Google Scholar
  40. [40]
    E. Strickland,A vanishing theorem for group compactifications, Math. Ann.277 (1987), 165–171.Google Scholar
  41. [41]
    A. Vistoli,Characteristic classes of principal bundles in algebraic intersection theory, Duke Math. J.58 (1989), 299–315.Google Scholar

Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • M. Brion
    • 1
  1. 1.Institut FourierSaint-Martin d'Hères CedexFrance

Personalised recommendations