Transformation Groups

, Volume 2, Issue 3, pp 225–267

Equivariant Chow groups for torus actions


  • M. Brion
    • Institut Fourier

DOI: 10.1007/BF01234659

Cite this article as:
Brion, M. Transformation Groups (1997) 2: 225. doi:10.1007/BF01234659


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.

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© Birkhäuser Boston 1997