A necessary condition for liftings of positive characteristic varieties with finite fundamental groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914043805040640 |
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| author | Di, Ruida Hu, Runjie Zhang, Siqing |
| author_facet | Di, Ruida Hu, Runjie Zhang, Siqing |
| contents | In this paper, we introduce a necessary condition for the existence of characteristic zero liftings of certain smooth, proper varieties in positive characteristic, using etale homotopy theory and Wall's finiteness obstruction. For a variety with finite etale fundamental group pi, we define a notion of mod-l finite dominatedness based on the F_l-chain complex of the universal cover of its l-profinite etale homotopy type. We prove that such a variety X can be lifted to characteristic zero only if the above chain complex of X is quasi-isomorphic to a bounded complex of finitely generated projective F_l[pi]-modules. To prove this result, we extend Wall's discussions of finiteness obstructions to l-profinite complete spaces with finite fundamental group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07914 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A necessary condition for liftings of positive characteristic varieties with finite fundamental groups Di, Ruida Hu, Runjie Zhang, Siqing Algebraic Topology Algebraic Geometry In this paper, we introduce a necessary condition for the existence of characteristic zero liftings of certain smooth, proper varieties in positive characteristic, using etale homotopy theory and Wall's finiteness obstruction. For a variety with finite etale fundamental group pi, we define a notion of mod-l finite dominatedness based on the F_l-chain complex of the universal cover of its l-profinite etale homotopy type. We prove that such a variety X can be lifted to characteristic zero only if the above chain complex of X is quasi-isomorphic to a bounded complex of finitely generated projective F_l[pi]-modules. To prove this result, we extend Wall's discussions of finiteness obstructions to l-profinite complete spaces with finite fundamental group. |
| title | A necessary condition for liftings of positive characteristic varieties with finite fundamental groups |
| topic | Algebraic Topology Algebraic Geometry |
| url | https://arxiv.org/abs/2506.07914 |