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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.03758 |
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| _version_ | 1866910149568888832 |
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| author | Hai, Phùng Hô Bao, Võ Quôc Bao, Trân Phan Quôc |
| author_facet | Hai, Phùng Hô Bao, Võ Quôc Bao, Trân Phan Quôc |
| contents | Let $X/S$ be a smooth family of smooth projective varieties, where $S$ is a smooth affine curve over a field $k$ of characteristic $0.$ We relate the differential fundamental groupoid scheme of $X/k$ with the differential fundamental groupoid scheme of $S/k$ and the relative differential fundamental group of $X/S$ in a short exact sequence. This yields natural maps from the group cohomology of the geometric relative fundamental group to the Gauss-Manin connections. For families of curves of genus at least $1,$ we prove that these maps are isomorphisms. This gives an interpretation of the Gauss-Manin connection in terms of cohomology of the differential fundamental group. As a consequence we can shrink $X$ (as a family on $S$) to obtain a de Rham $K(π,1)$ surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03758 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tannakian duality and Gauss-Manin connections for a family of curves Hai, Phùng Hô Bao, Võ Quôc Bao, Trân Phan Quôc Algebraic Geometry 14F40, 14F43, 14L15, 14L30, 18G15, 18G40, 18M25 Let $X/S$ be a smooth family of smooth projective varieties, where $S$ is a smooth affine curve over a field $k$ of characteristic $0.$ We relate the differential fundamental groupoid scheme of $X/k$ with the differential fundamental groupoid scheme of $S/k$ and the relative differential fundamental group of $X/S$ in a short exact sequence. This yields natural maps from the group cohomology of the geometric relative fundamental group to the Gauss-Manin connections. For families of curves of genus at least $1,$ we prove that these maps are isomorphisms. This gives an interpretation of the Gauss-Manin connection in terms of cohomology of the differential fundamental group. As a consequence we can shrink $X$ (as a family on $S$) to obtain a de Rham $K(π,1)$ surface. |
| title | Tannakian duality and Gauss-Manin connections for a family of curves |
| topic | Algebraic Geometry 14F40, 14F43, 14L15, 14L30, 18G15, 18G40, 18M25 |
| url | https://arxiv.org/abs/2603.03758 |