Localization in Equivariant Intersection Theory and the Bott Residue Formula

The purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections (cf. [BFQ]) and Schubert varieties. Let T be a split torus acting on a scheme X. The T -equivariant Chow groups of X are a module over RT = Sym(T̂ ), where T̂ is the character group of T . The localization theorem states that up to RT -torsion, the equivariant Chow groups of the fixed locus X are isomorphic to those of X. Such a theorem is a hallmark of any equivariant theory. The earliest version (for equivariant cohomology) is due to Borel [Bo]. Subsequently K-theory versions were proved by Segal [Se] (in topological K-theory), Quart [Qu] (for actions of a cyclic group), and Thomason [Th] (for algebraic K-theory [Th]). For equivariant Chow groups, the localization isomorphism is given by the equivariant pushforward i∗ induced by the inclusion of X T to X. An interesting aspect of this theory is that the push-forward is naturally defined on the level of cycles, even in the singular case. The closest topological analogue of this is equivariant Borel-Moore homology (see [E-G] for a definition), and a similar proof establishes localization in that theory. For smooth spaces, the inverse to the equivariant push-forward can be written explicitly. It was realized independently by several authors ([I-N], [A-B], [B-V]) that for compact spaces, the formula for the inverse implies the Bott residue formula. In this paper, we prove the Bott