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過去の記録 ～12/07｜次回の予定｜今後の予定 12/08～
| 開催情報 | 月曜日 10:30～12:00 数理科学研究科棟(駒場) 128号室 |
|---|---|
| 担当者 | 平地 健吾, 高山 茂晴 |
2024年12月23日(月)
10:30-12:00 数理科学研究科棟(駒場) 128号室
野口 潤次郎 氏 (東京大学)
Hyperbolicity and sections in a ramified cover over abelian varieties
with trace zero (Japanese)
https://forms.gle/gTP8qNZwPyQyxjTj8
野口 潤次郎 氏 (東京大学)
Hyperbolicity and sections in a ramified cover over abelian varieties
with trace zero (Japanese)
[ 講演概要 ]
We discuss a higher dimensional generalization of the Manin-Grauert Theorem ('63/'65) in relation with the function field analogue of Lang's conjecture on the finiteness of rational points in a Kobayashi hyperbolic algebraic variety over a number field. Let $B$ be a possibly open algebraic curve over $\mathbf{C}$, and let $\pi:X \to B$ be a smooth or normal projective fiber space. In '81 I proved such theorems for $\dim \geq 1$, assuming the ampleness of the cotangent bundle $T^*(X_t)$, and in '85 the Kobayashi hyperbolicity of $X_t$ with some boundary condition (BC) (hyperbolic embedding condition relative over $\bar{B}$).
It is interesting to study if (BC) is really necessary or not. If $\dim X_t=1$, (BC) is automatically satisfied, and if $T^*(X_t)$ is ample, (BC) is not necessary; thus in those cases, (BC) is unnecessary. Lately, Xie-Yuan in arXiv '23 obtained such a result without (BC) for $X$ which is a hyperbolic finite cover of an abelian variety $A/B$.
The aim of this talk is to present a simplified treatment of the Xie-Yuan theorem from the viewpoint of Kobayashi hyperbolic geometry. In particular, if the $K/\mathbf{C}$-trace $Tr(A/B)=0$ with $K=\mathbf{C}(B)$, there are only finitely many $X(K)$-points or sections in $X \to B$. In this case, Bartsch-Javanpeykar in arXiv '24 gave another proof based on Parshin's topological rigidity theorem ('90). We will discuss the proof which is based on the Kobayashi hyperbolicity.
[ 参考URL ]We discuss a higher dimensional generalization of the Manin-Grauert Theorem ('63/'65) in relation with the function field analogue of Lang's conjecture on the finiteness of rational points in a Kobayashi hyperbolic algebraic variety over a number field. Let $B$ be a possibly open algebraic curve over $\mathbf{C}$, and let $\pi:X \to B$ be a smooth or normal projective fiber space. In '81 I proved such theorems for $\dim \geq 1$, assuming the ampleness of the cotangent bundle $T^*(X_t)$, and in '85 the Kobayashi hyperbolicity of $X_t$ with some boundary condition (BC) (hyperbolic embedding condition relative over $\bar{B}$).
It is interesting to study if (BC) is really necessary or not. If $\dim X_t=1$, (BC) is automatically satisfied, and if $T^*(X_t)$ is ample, (BC) is not necessary; thus in those cases, (BC) is unnecessary. Lately, Xie-Yuan in arXiv '23 obtained such a result without (BC) for $X$ which is a hyperbolic finite cover of an abelian variety $A/B$.
The aim of this talk is to present a simplified treatment of the Xie-Yuan theorem from the viewpoint of Kobayashi hyperbolic geometry. In particular, if the $K/\mathbf{C}$-trace $Tr(A/B)=0$ with $K=\mathbf{C}(B)$, there are only finitely many $X(K)$-points or sections in $X \to B$. In this case, Bartsch-Javanpeykar in arXiv '24 gave another proof based on Parshin's topological rigidity theorem ('90). We will discuss the proof which is based on the Kobayashi hyperbolicity.
https://forms.gle/gTP8qNZwPyQyxjTj8
