Virtual Localisation Formula for SL_η -Oriented Theories

2024/11/10 by Alessandro D'Angelo, D'Angelo, Alessandro
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2411.06570

openalex publication_date 2024/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

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 KW . Applying standard tools of \mathbb A¹-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 SL₂ . Then we establish an equivariant Atiyah-Bott localization theorem for \mathrm A_N^BM(X) and we conclude with the N-equivariant virtual localisation formula. Of independent interest, we also describe the ring structure of KW(BN) .

Related

• Virtual Localization in equivariant Witt cohomology
• Localization of virtual classes
• Virtual intersection theories
• Remarks on chromatically localised hermitian K-theory
• Virtual intersections on the Quot-scheme and Vafa-Intriligator formulas