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Virtual Localisation Formula for $ SL_η $-Oriented Theories

Alessandro D'Angelo

Abstract

In this paper, we extend the Virtual Localization Formula of Levine to a wide class of motivic ring spectra, obtaining in particular a localization formula for virtual fundamental classes in Witt theory $ \mathrm{KW} $. Applying standard tools of $\mathbb A^1$-intersection theory to any $ SL_η $-oriented spectra $ \mathrm A $, we obtain an additive presentation of $ \mathrm A(BN) $, for $ N $ the normaliser of the torus in $ SL_2 $. Then we establish an equivariant Atiyah-Bott localization theorem for $ \mathrm A_N^{\mathrm{BM}}(X) $ and we conclude with the $N$-equivariant virtual localisation formula. Of independent interest, we also describe the ring structure of $ \mathrm{KW}(BN) $.

Virtual Localisation Formula for $ SL_η $-Oriented Theories

Abstract

In this paper, we extend the Virtual Localization Formula of Levine to a wide class of motivic ring spectra, obtaining in particular a localization formula for virtual fundamental classes in Witt theory . Applying standard tools of -intersection theory to any -oriented spectra , we obtain an additive presentation of , for the normaliser of the torus in . Then we establish an equivariant Atiyah-Bott localization theorem for and we conclude with the -equivariant virtual localisation formula. Of independent interest, we also describe the ring structure of .

Paper Structure

This paper contains 3 sections, 3 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notations and Conventions

Key Result

Proposition 1

For any $SL_{\eta}$-oriented ring spectrum, we get the following isomorphisms of graded $\mathrm A^{\bullet}(S)$-modules: where $\mathcal{T}$ is the tangent bundle of $\left[ {\raisebox{.2em}{$\mathbb{P}(\mathop{\mathrm{Sym}}\nolimits^2(F))$}\!\!\,\left/\!\raisebox{-.2em}{$SL_2$}\right.} \right]$ over $\mathcal{B}SL_2$ and $\gamma_N$ is the generator of $\mathrm{Pic}(\mathcal{B}N)$.

Theorems & Definitions (3)

  • Proposition 1: Proposition\ref{['ch2:_5.3_MEC']}
  • Theorem 2: Theorem \ref{['ch4:_ABL_8.6']}
  • Theorem 3: Theorem \ref{['ch4:_VLF_for_general_A']}