Matroids and the integral Hodge conjecture for abelian varieties
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911549294116864 |
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| author | Engel, Philip Fortman, Olivier de Gaay Schreieder, Stefan |
| author_facet | Engel, Philip Fortman, Olivier de Gaay Schreieder, Stefan |
| contents | We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. Our proof is motivated by tropical geometry; it relies on multivariable Mumford constructions, monodromy considerations, and the combinatorial theory of matroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matroids and the integral Hodge conjecture for abelian varieties Engel, Philip Fortman, Olivier de Gaay Schreieder, Stefan Algebraic Geometry Combinatorics 05B35, 14C25, 14C30, 14E08 We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. Our proof is motivated by tropical geometry; it relies on multivariable Mumford constructions, monodromy considerations, and the combinatorial theory of matroids. |
| title | Matroids and the integral Hodge conjecture for abelian varieties |
| topic | Algebraic Geometry Combinatorics 05B35, 14C25, 14C30, 14E08 |
| url | https://arxiv.org/abs/2507.15704 |