1-motives and admissible variations of mixed Hodge structures

Let S be a connected scheme smooth and of finite type over the field of complex numbers. To every 1-motive over S, Andr\'e associated the enriched Hodge realization given by a torsion-free, graded-polarizable and admissible variation of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1) over the associated complex analytic space. In this paper, we prove that every admissible variation of mixed Hodge structures of the above type arises, up to isogeny, from a 1-motive over S, thereby providing a positive answer to a question of Andr\'e concerning the geometric origin of such variations. More precisely, we establish a Hodge-theoretic interpretation of sections of semi-abelian varieties by combining Andr\'e's description of the abelian case with a new analysis of the toric part. As a consequence, we prove a relative analogue of Deligne's equivalence over the field of complex numbers. Namely, under suitable assumptions on S and on the lattices and the tori underlying 1-motives, the enriched Hodge realization functor induces an equivalence between the category of 1-motives over S and the category of torsion-free, graded-polarizable and admissible variations of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1). In general, the corresponding statement holds only up to isogeny. Finally, we introduce the global Mumford--Tate group of a 1-motive over S and show that its neutral connected component identifies with the Mumford-Tate group of the generic fiber.