The stacky concentration theorem

TL;DR

This paper generalizes the torus fixed-point localization theorem to algebraic stacks and extends localization results to actions of any algebraic group, broadening the scope of equivariant intersection theory.

Contribution

It provides a sufficient criterion for localizing Chow and algebraic bordism groups of algebraic stacks, generalizing classical localization theorems to a broader stack context.

Findings

Generalization of torus localization to algebraic stacks

Localization theorem for any algebraic group actions

Application to Chow and algebraic bordism groups

Abstract

We give a sufficient criterion for the Chow or algebraic bordism groups of an algebraic stack, localized at a set of Chern classes of line bundles, to be concentrated in some closed substack. This is a vast generalization of the torus fixed-point localization theorem in equivariant intersection theory, which is the special case of the stack quotient of a scheme $X$ by an action of a torus $T$. Taking on the one hand an algebraic stack in place of $X$, we deduce a generalization of torus localization to algebraic stacks. Taking on the other hand any algebraic group $G$ instead of $T$, we obtain a localization theorem in $G$-equivariant intersection theory.