1-motives and admissible variations of mixed Hodge structures
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910262710239232 |
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| author | Bertolin, Cristiana |
| author_facet | Bertolin, Cristiana |
| contents | Let S be a connected scheme smooth and of finite type over the field of complex numbers. To every 1-motive over S, André 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 André concerning the geometric origin of such variations. More precisely, we establish a Hodge-theoretic interpretation of sections of semi-abelian varieties by combining André'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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16545 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 1-motives and admissible variations of mixed Hodge structures Bertolin, Cristiana Algebraic Geometry Number Theory Let S be a connected scheme smooth and of finite type over the field of complex numbers. To every 1-motive over S, André 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 André concerning the geometric origin of such variations. More precisely, we establish a Hodge-theoretic interpretation of sections of semi-abelian varieties by combining André'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. |
| title | 1-motives and admissible variations of mixed Hodge structures |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2603.16545 |