On the Complexity of Atypical Special Points
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915653312577536 |
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| author | Urbanik, David |
| author_facet | Urbanik, David |
| contents | Given an integral variation of Hodge structure $\mathbb{V}$ on a complex algebraic variety $S$, polarized by some bilinear form $Q : \mathbb{V} \otimes \mathbb{V} \to \mathbb{Z}$, it is believed that the set $\mathcal{A}^{\textrm{iso}}_{0} \subset S(\mathbb{C})$ of isolated atypical special points associated to $(\mathbb{V}, Q)$ forms a finite set. Here we show that the number of such points $s$ is $O(Q(t_{s}, t_{s})^{\varepsilon})$ for any $\varepsilon > 0$, where $t_{s}$ is a minimal integral Hodge tensor defining $s$ (in an appropriate sense). This resolves a conjecture of Grimm and Monnee. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04491 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Complexity of Atypical Special Points Urbanik, David Algebraic Geometry Given an integral variation of Hodge structure $\mathbb{V}$ on a complex algebraic variety $S$, polarized by some bilinear form $Q : \mathbb{V} \otimes \mathbb{V} \to \mathbb{Z}$, it is believed that the set $\mathcal{A}^{\textrm{iso}}_{0} \subset S(\mathbb{C})$ of isolated atypical special points associated to $(\mathbb{V}, Q)$ forms a finite set. Here we show that the number of such points $s$ is $O(Q(t_{s}, t_{s})^{\varepsilon})$ for any $\varepsilon > 0$, where $t_{s}$ is a minimal integral Hodge tensor defining $s$ (in an appropriate sense). This resolves a conjecture of Grimm and Monnee. |
| title | On the Complexity of Atypical Special Points |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2512.04491 |