A Note on 1-Motives

Abstract

Abstract We prove that for $1$-motives defined over an algebraically closed subfield of $\mathbf{C}$, viewed as Nori motives, the motivic Galois group coincides with the Mumford–Tate group. In particular, the Hodge realization of the Tannakian category of Nori motives generated by $1$-motives is fully faithful. This result extends an earlier result by the author, according to which Hodge cycles on abelian varieties are motivated (a weak form of the Hodge conjecture).