Realizations of 1-motives over a scheme of characteristic 0

TL;DR

This paper constructs and proves an equivalence of categories between 1-motives over a complex scheme and certain variations of mixed Hodge structures, extending Deligne's results to the relative case.

Contribution

It extends Deligne's equivalence between 1-motives and mixed Hodge structures to schemes over complex numbers, addressing a question by André.

Findings

Constructed the Hodge realization functor for 1-motives over schemes of characteristic 0.

Proved the equivalence of categories between 1-motives and admissible variations of mixed Hodge structures.

Described the l-adic and de Rham realizations within Deligne's framework.

Abstract

Let S be a connected and smooth scheme of finite type over the complex numbers. We construct functorially the Hodge realization of a 1-motive over S as a torsion-free, polarizable and admissible variation of mixed Hodge structures of type (0,0),(-1,0),(0,-1),(-1,-1). We prove that this construction yields an equivalence between the category of 1-motives over S and the category of such variations of mixed Hodge structures, thereby extending Deligne's equivalence over the complex numbers to the relative case and providing a positive answer to a question of Andr\'e concerning the geometric origin of admissible variations of mixed Hodge structures of the above type. We also describe the l-adic and de Rham realizations of 1-motives and show that these realizations fit naturally into Deligne's framework of smooth mixed realizations.