Equivariant Chow groups for torus actions

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.

[1]  H. Gillet.  Riemann-Roch theorems for higher algebraic K-theory , 1981 .

[2]  C. Procesi,et al.  Cohomology of regular embeddings , 1990 .

[3]  P. Pragacz,et al.  Formulas for Lagranigian and orthogonal degeneracy loci; $$\widetilde Q$$ -polynomial approach , 1997, Compositio Mathematica.

[4]  T. A. Springer,et al.  The Bruhat order on symmetric varieties , 1990 .

[5]  P. Polo.  On Zariski tangent spaces of Schubert varieties, and a proof of a conjecture of Deodhar , 1994 .

[6]  Michel Demazure,et al.  Désingularisation des variétés de Schubert généralisées , 1974 .

[7]  M. Goresky,et al.  Equivariant cohomology, Koszul duality, and the localization theorem , 1997 .

[8]  William Fulton,et al.  Flags, Schubert polynomials, degeneracy loci, and determinantal formulas , 1992 .

[9]  Symmetric polynomials and divided differences in formulas of intersection theory , 1996, alg-geom/9605014.

[10]  W. Fulton.  Introduction to Toric Varieties. , 1993 .

[11]  William Smoke,et al.  Dimension and multiplicity for graded algebras , 1970 .

[12]  Elisabetta Stricland.  A vanishing theorem for group compactifications , 1987 .

[13]  A. Białynicki-Birula.  Some theorems on actions of algebraic groups , 1973 .

[14]  Angelo Vistoli.  Characteristic classes of principal bundles in algebraic intersection theory , 1989 .

[15]  P. Pragacz,et al.  Formulas for Lagrangian and orthogonal degeneracy loci; the Q-polynomials approach , 1996, alg-geom/9602019.

[16]  W. Borho.  Nilpotent orbits, primitive ideals, and characteristic classes , 1989 .

[17]  A. Joseph.  On the variety of a highest weight module , 1984 .

[18]  A. Białynicki-Birula.  On fixed points of torus actions on projective varieties , 1974 .

[19]  Santhosh K. P. Kumar,et al.  T-equivariant K-theory of generalized flag varieties. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[20]  The nil Hecke ring and singularity of Schubert varieties , 1995, alg-geom/9503015.

[21]  Armand Borel.  Linear Algebraic Groups , 1991 .

[22]  W. Graham,et al.  Localization in equivariant intersection theory and the Bott residue formula , 1995, alg-geom/9508001.

[23]  F. Kirwan.  Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31 , 1984 .

[24]  B Kostant,et al.  The nil Hecke ring and cohomology of G/P for a Kac-Moody group G. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[25]  W. Graham.  The class of the diagonal in flag bundles , 1997 .

[26]  F. Knop.  On the set of orbits for a Borel subgroup , 1995 .

[27]  A. Białynicki-Birula.  Some properties of the decompositions of algebraic varieties determined by actions of a torus , 1976 .

[28]  M. Brion.  Piecewise polynomial functions, convex polytopes and enumerative geometry , 1996 .

[29]  S. Evens,et al.  The Schubert calculus, braid relations, and generalized cohomology , 1990 .

[30]  William Fulton,et al.  Introduction to Toric Varieties. (AM-131) , 1993 .

[31]  Michel Demazure,et al.  Invariants symétriques entiers des groupes de Weyl et torsion , 1973 .

[32]  A. Arabia.  Cycles de Schubert et cohomologie équivariante de K/T , 1986 .

[33]  Michèle Vergne,et al.  An equivariant Riemann-Roch theorem for complete, simplicial toric varieties. , 1997 .