複素解析幾何セミナー

過去の記録 ~02/07次回の予定今後の予定 02/08~

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, 野村 亮介

2022年01月24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)
Analytic Ax-Schanuel Theorem for semi-abelian varieties and Nevanlinna theory (Japanese)
[ 講演概要 ]
The present study is motivated by $\textit{Schanuel Conjecture}$, which in particular implies the algebraic independence of $e$ and $\pi$. Our aim is to explore, as a transcendental functional analogue of Schanuel Conjecture, the value distribution theory (Nevanlinna theory) of the entire curve $\widehat{\mathrm{ex}}_A f:=(\exp_Af,f):\mathbf{C} \to A \times \mathrm{Lie}(A)$ associated with an entire curve $f: \mathbf{C} \to \mathrm{Lie}(A)$, where $\exp_A:\mathrm{Lie}(A)\to A$ is an exponential map of a semi-abelian variety $A$.

We firstly give a Nevanlinna theoretic proof to the $\textit{analytic Ax-Schanuel Theorem}$ for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series $\mathbf{C}[[t]]$ (Ax-Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that in the case of $A=(\mathbf{C}^*)^n$ and $\mathrm{Lie}((\mathbf{C}^*)^n)=\mathbf{C}^n$, the elements of the vector-valued function $f(z)-f(0)$ are $\mathbf{Q}$-linearly independent. Then by the method of Nevanlinna theory (the Log Bloch-Ochiai Theorem), we prove that $\mathrm{tr.deg}_\mathbf{C}\, \widehat{\mathrm{ex}}_A f \geq n+ 1.$

Secondly, we prove a $\textit{Second Main Theorem}$ for $\widehat{\mathrm{ex}}_A f$ and an algebraic divisor $D$ on $A \times \mathrm{Lie}(A)$ with compactifications $\bar D \subset \bar A \times \overline{\mathrm{Lie}(A)}$ such that
\[
T_{\widehat{\mathrm{ex}}_Af}(r, L({\bar D})) \leq N_1 (r,
(\widehat{\mathrm{ex}}_A f)^* D)+
\varepsilon T_{\exp_Af}(r)+O(\log r) ~~ ||_\varepsilon.
\]
We will also deal with the intersections of $\widehat{\mathrm{ex}}_Af$ with higher codimensional algebraic cycles of $A \times \mathrm{Lie}(A)$ as well as the case of higher jets.
[ 参考URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB