Mathematical Research Letters

Volume 27 (2020)

Number 2

Motivic concentration theorem

Pages: 565 – 589

DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n2.a10

Authors

Gonçalo Tabuada (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Ma., U.S.A.; Departamento de Matemática and Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidad Nova de Lisboa, Portugal)

Michel Van Den Bergh (Department of Mathematics, Universiteit Hasselt, Diepenbeek, Belgium)

Abstract

In this short article, given a smooth diagonalizable group scheme $G$ of finite type acting on a smooth quasi-compact 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.

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G. Tabuada was partially supported by a NSF CAREER Award #1350472, and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matemática e Aplicações).

M. Van den Bergh is a senior researcher at the Research Foundation – Flanders.

Received 22 June 2018

Accepted 15 April 2019

Published 8 June 2020