Atiyah–Bott Localization in Equivariant Witt Cohomology

doi:10.1017/9781009497206.010

9 - Atiyah–Bott Localization in Equivariant Witt Cohomology

Published online by Cambridge University Press: 31 October 2025

Marc Levine

Edited by

Pedro L. del Ángel R. ,

Frank Neumann and

Alexander H. W. Schmitt

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Pedro L. del Ángel R.: Affiliation:
Centro de Investigación en Matemáticas
Frank Neumann: Affiliation:
Università di Pavia
Alexander H. W. Schmitt: Affiliation:
Freie Universität Berlin

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Summary

Let N be the normalizer of the diagonal torus T1≅Gm in SL2. We prove localization theorems for SL2n and Nn for equivariant cohomology with coefficients in the (twisted) Witt sheaf, along the lines of the classical localization theorems for equivariant cohomology for a torus action. We also have an analog of the Bott residue formula for SL2n and N. In the case of an SL2n-action, there is a rather serious restriction on the orbit type. For an N-action, there is no restriction for the localization result, but for the Bott residue theorem, one requires a certain type of decomposition of the fixed points for the T1-action, which is always available if the subscheme of T1 fixed points has dimension zero.

Keywords

Atiyah-Bott localization equivariant cohomology Grothendieck-Witt ring motivic cohomology

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Type: Chapter
Information: Moduli, Motives and Bundles
New Trends in Algebraic Geometry
, pp. 265 - 405

DOI: https://doi.org/10.1017/9781009497206.010 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2025

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