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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 \(X\).

MSC:

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

Citations:

Zbl 0940.14003
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