A note on 1-motives

TL;DR

This paper establishes that for 1-motives over an algebraically closed subfield of complex numbers, the motivic Galois group coincides with the Mumford-Tate group, confirming the full faithfulness of the Hodge realization.

Contribution

It proves the equivalence of motivic Galois and Mumford-Tate groups for 1-motives and shows the Hodge realization is fully faithful in this context.

Findings

Motivic Galois group equals Mumford-Tate group for 1-motives.

Hodge realization of the category generated by 1-motives is fully faithful.

Results hold over algebraically closed subfields of complex numbers.

Abstract

We prove that for $1$-motives defined over an algebraically closed subfield of $\C$, viewed as Nori motives, the motivic Galois group is the Mumford-Tate group. In particular, the Hodge realization of the tannakian category of (Nori) motives generated by $1$-motives is fully faithful.