Motivic concentration theorem

2018/06/18 by Tabuada, Goncalo, Bergh, Michel Van den
#14A20 #14A22 #14F30 #14F40 #18D20 #19D25 #19E08 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1806.06846

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.

Related

• The geometric concentration theorem
• Equivariant motivic homotopy theory
• Motivic (Representation) Stability of Representation Varieties and Character Stacks
• Additivity for the Motivic Trace and the Motivic Euler Characteristic
• Stratifying quotient stacks and moduli stacks