Localization in equivariant intersection theory and the Bott residue formula. (English) Zbl 0980.14004

Let \(T\) be a split torus acting on a scheme, \(X\). The localisation theorem is an isomorphism between localised higher Chow groups of the form \[ i_{*} : A_{*}(X^{T}) \otimes {\mathbb Q} @>\cong>>A_{*}^{T}(X) \otimes {\mathbb Q} . \] Here \(A_{*}^{T}\) is the authors’ equivariant Chow group [D. Edidin and W. Graham, Invent. Math. 131, No. 3, 595-634 (1998; Zbl 0940.14003)]. When \(X\) is smooth the authors give an explicit formula for \((i_{*})^{-1}\) from which, as is well-known, one may obtain Bott’s residue formula for Chern numbers of bundles on smooth, complete varieties. The authors also manage to extend their method to some classes of singular \(X\).


14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
55N91 Equivariant homology and cohomology in algebraic topology


Zbl 0940.14003
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