Nori 1-motives

TL;DR

This paper establishes an equivalence between Nori's category of effective homological motives generated by low-degree homology and Deligne's category of 1-motives with torsion, providing new insights and proofs in the theory of motives.

Contribution

It proves that the subcategory generated by low-degree homology is equivalent to Deligne's 1-motives, connecting Nori's formalism with classical motive theory.

Findings

EHM_1 is equivalent to M_1 of Deligne 1-motives with torsion.

Realization of M_1 from a diagram of curves using Nori's formalism.

Proof of Deligne's conjecture on extensions of 1-motives in mixed realizations.

Abstract

Let EHM be Nori's category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM_1 generated by the i-th relative homology of pairs of varieties for i = 0,1. We show that EHM_1 is naturally equivalent to the abelian category M_1 of Deligne 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize M_1 as the universal abelian category obtained, using Nori's formalism, from the Betti representation of an explicit diagram of curves. Secondly, we obtain a conceptual proof of a theorem of Vologodsky on realizations of 1-motives. Thirdly, we verify a conjecture of Deligne on extensions of 1-motives in the category of mixed realizations for those extensions that are effective in Nori's sense.