Hodge realizations of 1-motives and the derived Albanese

TL;DR

This paper demonstrates that the embedding of 1-motives into Voevodsky motives and its left adjoint commute with the Hodge realization, establishing compatibility between these categories and Hodge structures.

Contribution

It proves the compatibility of the derived category embedding of 1-motives and its left adjoint with the Hodge realization functor.

Findings

The embedding of 1-motives into Voevodsky motives commutes with Hodge realization.

The left adjoint functor LAlb also commutes with Hodge realization.

This compatibility enhances understanding of the relationship between motives and Hodge structures.

Abstract

We prove that the embedding of the derived category of 1-motives into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor $LAlb$, commute with the Hodge realization.