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複素解析幾何セミナー
過去の記録 ～06/24｜次回の予定｜今後の予定 06/25～
| 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 |
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)
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB
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 Schanuel Conjecture, which in particular implies the algebraic independence of e and π. Our aim is to explore, as a transcendental functional analogue of Schanuel Conjecture, the value distribution theory (Nevanlinna theory) of the entire curve ^exAf:=(expAf,f):C→A×Lie(A) associated with an entire curve f:C→Lie(A), where expA:Lie(A)→A is an exponential map of a semi-abelian variety A.
We firstly give a Nevanlinna theoretic proof to the analytic Ax-Schanuel Theorem for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series C[[t]] (Ax-Schanuel Theorem). We assume some non-degeneracy condition for f such that in the case of A=(C∗)n and Lie((C∗)n)=Cn, the elements of the vector-valued function f(z)−f(0) are Q-linearly independent. Then by the method of Nevanlinna theory (the Log Bloch-Ochiai Theorem), we prove that tr.degC^exAf≥n+1.
Secondly, we prove a Second Main Theorem for ^exAf and an algebraic divisor D on A×Lie(A) with compactifications ˉD⊂ˉAׯLie(A) such that
T^exAf(r,L(ˉD))≤N1(r,(^exAf)∗D)+εTexpAf(r)+O(logr) ||ε.
We will also deal with the intersections of ^exAf with higher codimensional algebraic cycles of A×Lie(A) as well as the case of higher jets.
[ 参考URL ]The present study is motivated by Schanuel Conjecture, which in particular implies the algebraic independence of e and π. Our aim is to explore, as a transcendental functional analogue of Schanuel Conjecture, the value distribution theory (Nevanlinna theory) of the entire curve ^exAf:=(expAf,f):C→A×Lie(A) associated with an entire curve f:C→Lie(A), where expA:Lie(A)→A is an exponential map of a semi-abelian variety A.
We firstly give a Nevanlinna theoretic proof to the analytic Ax-Schanuel Theorem for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series C[[t]] (Ax-Schanuel Theorem). We assume some non-degeneracy condition for f such that in the case of A=(C∗)n and Lie((C∗)n)=Cn, the elements of the vector-valued function f(z)−f(0) are Q-linearly independent. Then by the method of Nevanlinna theory (the Log Bloch-Ochiai Theorem), we prove that tr.degC^exAf≥n+1.
Secondly, we prove a Second Main Theorem for ^exAf and an algebraic divisor D on A×Lie(A) with compactifications ˉD⊂ˉAׯLie(A) such that
T^exAf(r,L(ˉD))≤N1(r,(^exAf)∗D)+εTexpAf(r)+O(logr) ||ε.
We will also deal with the intersections of ^exAf with higher codimensional algebraic cycles of A×Lie(A) as well as the case of higher jets.
https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB
