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Mathematical Research Letters
Volume 27 (2020)
Motivic concentration theorem
Pages: 565 – 589
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.
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