Symmetry and parity in Frobenius action on cohomology
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909332849819648 |
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| author | Suh, Junecue |
| author_facet | Suh, Junecue |
| contents | We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01184 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetry and parity in Frobenius action on cohomology Suh, Junecue Algebraic Geometry Number Theory 11G25 (Primary) 14F20, 14F30 (Secondary) We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz. |
| title | Symmetry and parity in Frobenius action on cohomology |
| topic | Algebraic Geometry Number Theory 11G25 (Primary) 14F20, 14F30 (Secondary) |
| url | https://arxiv.org/abs/2410.01184 |