Motivic concentration theorem

TL;DR

This paper proves a concentration theorem for additive invariants of quotient stacks under diagonalizable group actions, showing they are determined by fixed points after certain localizations, and applies this to compute invariants and establish a Riemann-Roch formula.

Contribution

It introduces a motivic concentration theorem for quotient stacks with diagonalizable group actions, linking invariants to fixed points and roots of unity, and derives a Lefschetz-Riemann-Roch formula.

Findings

Additive invariants of [X/G] are concentrated at fixed points after localization.

In the connected case with finite stabilizers, invariants are computed using roots of unity.

A Lefschetz-Riemann-Roch formula for proper push-forwards is established.

Abstract

In this short article, given a smooth diagonalizable group scheme G of finite type acting on a smooth quasi-compact quasi-separated scheme X, we prove that (after inverting some elements of representation ring of G) all the information concerning the additive invariants of the quotient stack [X/G] is "concentrated" in the subscheme of G-fixed points X^G. Moreover, in the particular case where G is connected and the action has finite stabilizers, we compute the additive invariants of [X/G] using solely the subgroups of roots of unity of G. As an application, we establish a Lefschtez-Riemann-Roch formula for the computation of the additive invariants of proper push-forwards.