[geometry-ml:06063] 東大数理・複素解析幾何セミナー 12/23

[shigeharu takayama]

皆様、

東大数理・複素解析幾何セミナーのお知らせです。

2024年12月23日(月) 10:30-12:00
数理科学研究科棟(駒場) 128号室
講演者：野口 潤次郎 氏 (東京大学)
講演題目：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.

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今後の予定はこちら
https://www.ms.u-tokyo.ac.jp/seminar/geocomp/future.html

世話人
高山 茂晴、平地 健吾