Localization in equivariant intersection theory and the Bott residue formula

Abstract

We 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 and Schubert varieties.

A mathematical theory of the topological vertex

“…1 /z ˆ ; . 2 / and (2-10)ˆ C ; .0/ D ı C ; z C : Equation (2-9) is a sum formula for double Hurwitz numbers; Equation (2)(3)(4)(5)(6)(7)(8)(9)(10) gives the initial values for the double Hurwitz numbers.…”

On the degree of Fano schemes of linear subspaces on hypersurfaces

“…Edidin and Graham [6,7] gave an algebraic construction to equivariant intersection theory. In this section, we review the basic notions and results of this theory.…”

“…The localization theorem states that up to R T -torsion, the T -equivariant Chow group of the fixed points locus X T is isomorphic to that of X. Moreover, the localization isomorphism is given by the equivariant push-forward induced by the inclusion of X T to X (see [7,Theorem 1]). For smooth varieties, the inverse to the equivariant push-forward can be written explicitly (see [7,Theorem 2]).…”

“…Moreover, the localization isomorphism is given by the equivariant push-forward induced by the inclusion of X T to X (see [7,Theorem 1]). For smooth varieties, the inverse to the equivariant push-forward can be written explicitly (see [7,Theorem 2]). Using these results, Edidin and Graham gave an algebraic proof of Bott's residue formula for Chern numbers of vector bundles on smooth complete varieties (see [7,Theorem 3]).…”

“…For smooth varieties, the inverse to the equivariant push-forward can be written explicitly (see [7,Theorem 2]). Using these results, Edidin and Graham gave an algebraic proof of Bott's residue formula for Chern numbers of vector bundles on smooth complete varieties (see [7,Theorem 3]). Bott's residue formula shows that we can compute the degree of a zero-dimensional cycle class on a smooth complete variety X in terms of local contributions supported on the components of the fixed point locus of a torus action on X.…”