Volumes of Subvarieties of Complex Ball Quotients and Sparsity of Rational Points
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909870200979456 |
|---|---|
| author | Memariansorkhabi, Soheil |
| author_facet | Memariansorkhabi, Soheil |
| contents | Let $X=Γ\backslash \mathbb{B}^{n} $ be an $n$-dimensional complex ball quotient by a torsion-free non-uniform lattice $Γ$ whose parabolic subgroups are unipotent. We prove that the volumes of subvarieties of $X$ are controlled by the systole of $X,$ which is the length of the shortest closed geodesic of $X$.
There are a number of arithmetic and geometric consequences: the systole of $X$ controls the growth rate of rational points on $X,$ uniformly in the field of definition. Also, we obtain effective global generation and very ampleness results for multiples of the canonical bundle $K_{\overline{X}},$ where $\overline{X}$ is the toroidal compactification of $X.$ These results follow from the bound we find for the Seshadri constant of $K_{\overline{X}}$ in terms of the systole. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08060 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Volumes of Subvarieties of Complex Ball Quotients and Sparsity of Rational Points Memariansorkhabi, Soheil Algebraic Geometry Complex Variables 32Q45 (primary), 14G05, 14E25, 32M15, 32Q40 (secondary) Let $X=Γ\backslash \mathbb{B}^{n} $ be an $n$-dimensional complex ball quotient by a torsion-free non-uniform lattice $Γ$ whose parabolic subgroups are unipotent. We prove that the volumes of subvarieties of $X$ are controlled by the systole of $X,$ which is the length of the shortest closed geodesic of $X$. There are a number of arithmetic and geometric consequences: the systole of $X$ controls the growth rate of rational points on $X,$ uniformly in the field of definition. Also, we obtain effective global generation and very ampleness results for multiples of the canonical bundle $K_{\overline{X}},$ where $\overline{X}$ is the toroidal compactification of $X.$ These results follow from the bound we find for the Seshadri constant of $K_{\overline{X}}$ in terms of the systole. |
| title | Volumes of Subvarieties of Complex Ball Quotients and Sparsity of Rational Points |
| topic | Algebraic Geometry Complex Variables 32Q45 (primary), 14G05, 14E25, 32M15, 32Q40 (secondary) |
| url | https://arxiv.org/abs/2310.08060 |