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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2205.13073 |
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| _version_ | 1866917119775473664 |
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| author | Jacobsen, Emil |
| author_facet | Jacobsen, Emil |
| contents | We prove that, on a smooth, connected variety in characteristic zero admitting a rational point, local systems of geometric origin are stable under extension in the category of all local systems. As a consequence of this, we obtain a (Nori) motivic strengthening of Hain's theorem on Malcev completions of monodromy representations.
Our methods are Tannakian, and rely on an abstract criterion for ``Malcev completeness'', which is proved in the first part of the paper. A couple of secondary applications of this criterion are given: an alternative proof of D'Addezio--Esnault's theorem, which says that local systems of Hodge origin are stable under extension in the category of all local systems; a generalisation of the theorem of Hain, mentioned above, which also affirms a conjecture of Arapura; and an alternative proof of a theorem of Lazda, which under suitable assumptions gives an isomorphism between the relative unipotent de Rham fundamental group and the unipotent de Rham fundamental group of the special fibre. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_13073 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Malcev Completions, Hodge Theory, and Motives Jacobsen, Emil Algebraic Geometry Representation Theory We prove that, on a smooth, connected variety in characteristic zero admitting a rational point, local systems of geometric origin are stable under extension in the category of all local systems. As a consequence of this, we obtain a (Nori) motivic strengthening of Hain's theorem on Malcev completions of monodromy representations. Our methods are Tannakian, and rely on an abstract criterion for ``Malcev completeness'', which is proved in the first part of the paper. A couple of secondary applications of this criterion are given: an alternative proof of D'Addezio--Esnault's theorem, which says that local systems of Hodge origin are stable under extension in the category of all local systems; a generalisation of the theorem of Hain, mentioned above, which also affirms a conjecture of Arapura; and an alternative proof of a theorem of Lazda, which under suitable assumptions gives an isomorphism between the relative unipotent de Rham fundamental group and the unipotent de Rham fundamental group of the special fibre. |
| title | Malcev Completions, Hodge Theory, and Motives |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2205.13073 |