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Atiyah-Bott localization in equivariant Witt cohomology

Marc Levine

TL;DR

This work develops Atiyah-Bott localization in equivariant Witt cohomology for actions of the product group $\operatorname{SL}_2^n$ and its normalizer $N^n$. It proves localization theorems after inverting explicit Euler-type polynomials $e_*$ (and in some $N^n$-cases a polynomial $P_X$), and establishes Bott residue-type formulas under char-zero and smooth/regular-embedding assumptions, while carefully handling orbit-type restrictions for $\operatorname{SL}_2^n$ actions and semi-strict vs strict $N$-actions. The paper also provides detailed Witt-sheaf cohomology computations for classifying spaces $BN$ and $\operatorname{BSL}_2$ and constructs equivariant Witt Borel-Moore homology within the motivic framework, including Rost-Schmidt complexes and purity/theory of Euler classes. Collectively, these results extend localization techniques to quadratic refinements of motivic invariants, enabling precise fixed-point computations in Grothendieck-Witt groups with potential applications to motivic and arithmetic geometry. The methods yield concrete formulas for fixed-point contributions and demonstrate the interplay between representation theory, equivariant motivic homotopy theory, and Witt-theoretic invariants, thereby enhancing computational tools for refined characteristic classes.

Abstract

Let $N$ be a normalizer of the diagonal torus $T_1\cong \mathbb{G}_m$ in $\text{SL}_2$. We prove localization theorems for $\text{SL}_2^n$ and $N^n$ for equivariant cohomology with coefficients in the (twisted) Witt sheaf, along the lines of the classical Atiyah-Bott localization theorems for equivariant cohomology for a torus action. We also have an analog of the Bott residue formula for $\text{SL}_2^n$ and $N$. In the case of an $\text{SL}_2^n$-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 $T_1$-action.

Atiyah-Bott localization in equivariant Witt cohomology

TL;DR

This work develops Atiyah-Bott localization in equivariant Witt cohomology for actions of the product group and its normalizer . It proves localization theorems after inverting explicit Euler-type polynomials (and in some -cases a polynomial ), and establishes Bott residue-type formulas under char-zero and smooth/regular-embedding assumptions, while carefully handling orbit-type restrictions for actions and semi-strict vs strict -actions. The paper also provides detailed Witt-sheaf cohomology computations for classifying spaces and and constructs equivariant Witt Borel-Moore homology within the motivic framework, including Rost-Schmidt complexes and purity/theory of Euler classes. Collectively, these results extend localization techniques to quadratic refinements of motivic invariants, enabling precise fixed-point computations in Grothendieck-Witt groups with potential applications to motivic and arithmetic geometry. The methods yield concrete formulas for fixed-point contributions and demonstrate the interplay between representation theory, equivariant motivic homotopy theory, and Witt-theoretic invariants, thereby enhancing computational tools for refined characteristic classes.

Abstract

Let be a normalizer of the diagonal torus in . We prove localization theorems for and for equivariant cohomology with coefficients in the (twisted) Witt sheaf, along the lines of the classical Atiyah-Bott localization theorems for equivariant cohomology for a torus action. We also have an analog of the Bott residue formula for and . In the case of an -action, there is a rather serious restriction on the orbit type. For an -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 -action.
Paper Structure (21 sections, 67 theorems, 620 equations)

This paper contains 21 sections, 67 theorems, 620 equations.

Table of Contents

  1. Quotients and fixed points
  2. Some representation theory
  3. Cohomology and Borel-Moore homology
  4. ${\mathcal{E}}$-cohomology, Borel-Moore homology, products
  5. Purity isomorphisms
  6. Thom isomorphisms, oriented spectra
  7. Functorialities
  8. ${\mathcal{E}}$-cohomology with supports
  9. Localization in Borel-Moore homology
  10. Euler class
  11. Refined Gysin morphism
  12. Compatibilities
  13. Rost-Schmid complexes
  14. Equivariant Borel-Moore Witt homology
  15. Witt-sheaf cohomology of $BN$ and $\operatorname{BSL}_2$
  16. ...and 6 more sections

Key Result

Theorem 2

Let $k$ be a field of characteristic $\neq 2$ and let $X$ be in ${\operatorname{\mathbf{Sch}}}^{\operatorname{SL}_2^n}/k$. Let ${\mathcal{L}}$ be a $\operatorname{SL}_2^n$-linearized invertible sheaf on $X$ and let $i{\colon}X^{\operatorname{SL}_2^n}\to X$ be the inclusion of the $\operatorname{SL}_ Then the push-forward map is an isomorphism. Moreover

Theorems & Definitions (186)

  • Definition 1
  • Theorem 2: Localization for $\operatorname{SL}_2^n$: Theorem \ref{['thm:MainSLLoc']}
  • Example 3
  • Theorem 4: Localization for $N^n$: Theorem \ref{['thm:NnLocalization']}
  • Definition 5
  • Example 6
  • Theorem 7: Localization for an $N$-action: Theorem \ref{['thm:AtiyahBottLocalizationN']}
  • Remark 8
  • Theorem 9: Bott Residue Theorem for $\operatorname{SL}_2^n$: Theorem \ref{['thm:BottResidue']}
  • Theorem 10: Bott Residue Theorem for $N$: Theorem \ref{['thm:BottResidue']}
  • ...and 176 more