The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914316692750336 |
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| author | Floccari, Salvatore Fu, Lie |
| author_facet | Floccari, Salvatore Fu, Lie |
| contents | We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on hyper-Kähler varieties of generalized Kummer type, and by constructing semiregular sheaves on abelian varieties. Our proof instead relies on a direct geometric relation between abelian fourfolds of Weil type with discriminant 1 and the six-dimensional hyper-Kähler varieties $\widetilde{K}$ of O'Grady type arising as crepant resolutions $\widetilde{K}\to K$ of a locally trivial deformation of a singular moduli space of sheaves on an abelian surface. As applications, we establish the Hodge conjecture and the Tate conjecture for any variety $\widetilde{K}$ of OG6-type as above, and all of its powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties Floccari, Salvatore Fu, Lie Algebraic Geometry We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on hyper-Kähler varieties of generalized Kummer type, and by constructing semiregular sheaves on abelian varieties. Our proof instead relies on a direct geometric relation between abelian fourfolds of Weil type with discriminant 1 and the six-dimensional hyper-Kähler varieties $\widetilde{K}$ of O'Grady type arising as crepant resolutions $\widetilde{K}\to K$ of a locally trivial deformation of a singular moduli space of sheaves on an abelian surface. As applications, we establish the Hodge conjecture and the Tate conjecture for any variety $\widetilde{K}$ of OG6-type as above, and all of its powers. |
| title | The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2504.13607 |