Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918033878941696 |
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| author | Hu, Jiajun Xiao, Jian |
| author_facet | Hu, Jiajun Xiao, Jian |
| contents | For $(n-2)$ free divisor classes on a smooth projective variety of dimension $n$, the product of these free divisor classes induces a Lefschetz type operator acting on the Néron-Severi space or the cohomology group of $(1,1)$ classes. We give a characterization of this kernel space, when the collection of these free divisor classes is supercritical. This resolves Shenfeld-van Handel's open problem in this setting. As consequences, we provide an algebro-geometric proof of the characterization of the extremals of the Alexandrov-Fenchel inequality for a supercritical collection of rational convex polytopes; we also give a characterization of the extremals of the Khovanskii-Teissier inequality given by the intersection numbers of two arbitrary free divisor classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_18729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes Hu, Jiajun Xiao, Jian Algebraic Geometry Metric Geometry For $(n-2)$ free divisor classes on a smooth projective variety of dimension $n$, the product of these free divisor classes induces a Lefschetz type operator acting on the Néron-Severi space or the cohomology group of $(1,1)$ classes. We give a characterization of this kernel space, when the collection of these free divisor classes is supercritical. This resolves Shenfeld-van Handel's open problem in this setting. As consequences, we provide an algebro-geometric proof of the characterization of the extremals of the Alexandrov-Fenchel inequality for a supercritical collection of rational convex polytopes; we also give a characterization of the extremals of the Khovanskii-Teissier inequality given by the intersection numbers of two arbitrary free divisor classes. |
| title | Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes |
| topic | Algebraic Geometry Metric Geometry |
| url | https://arxiv.org/abs/2505.18729 |