Cohomological and categorical concentration

TL;DR

This paper extends the classical concentration results of equivariant cohomology from topological spaces to algebraic varieties over any field, using advanced categorical and motivic homotopy techniques.

Contribution

It introduces a categorification approach at the level of equivariant derived categories and motivic homotopy categories, generalizing the classical concentration theorem.

Findings

Concentration results hold for Voevodsky motives.

Concentration applies to homotopy K-theory.

Framework extends classical topological results to algebraic geometry.

Abstract

Given a torus action on a compact space X, a fundamental result of Borel and Atiyah-Segal asserts that the equivariant cohomology of X is concentrated in the fixed locus X^T, up to inverting enough Chern classes. We prove an analogue for algebraic varieties over an arbitrary field. In fact, we deduce this from a categorification at the level of equivariant derived categories and even equivariant stable motivic homotopy categories, which also gives concentration at the level of Voevodsky motives and for homotopy K-theory.