Mixed motives and quotient stacks: Abelian varieties

TL;DR

This paper characterizes the category of mixed motives generated by abelian varieties as modules over a derived quotient stack and analyzes the structure of their motivic Galois groups.

Contribution

It provides a new description of the mixed motives of abelian varieties as modules over a derived quotient stack and studies their motivic Galois groups' structure.

Findings

The category of mixed motives generated by an abelian variety is equivalent to modules over a derived quotient stack.

The motivic Galois group decomposes into a unipotent part and a reductive quotient.

The reductive quotient corresponds to the Tannaka dual of Grothendieck numerical motives.

Abstract

We prove that the symmetric monoidal category of mixed motives generated by an abelian variety (more generally, an abelian scheme) can be described as a certain module category. More precisely, we describe it as the category of quasi-coherent complexes over a derived quotient stack constructed from a motivic algebra of the abelian variety. We then study the structure of the motivic Galois groups of their mixed motives. We prove that the motivic Galois group is decomposed into a unipotent part constructed from the motivic algebra, and the reductive quotient which is the Tannaka dual of Grothendieck numerical motives.