New developments of higher dimensional value distribution theory and the fundamentals of complex analysis in several variables

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03511
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: The University of Tokyo
• Principal Investigator: Noguchi Junjiro 東京大学, 大学院数理科学研究科, 名誉教授 (20033920)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 解析学 / 高次元値分布理論 / Nevanlinna理論 / 多変数関数論 / 岡理論

Outline of Final Research Achievements

(1) Based on the results of the previous project ``15K04917'', I proceeded with research from the viewpoint of higher dimensional value distribution theory-Diophantine geometry-the o-minimal theory. In collaboration with P. Corvaja (Italy) and U. Zannier (ibid.), I studied the the removability of singularities of holomorphic sections in families of elliptic curves and semi-abelian vareities over function fields. In view of transcendental number theory, I formulated and proved the Ax-Schanuel type theorem by making use of the value distribution theory, that provides the basis for future researches of the theory and the o-minimal theory.
(2) Regarding the basics of complex analysis in several variables, Istudied Kiyoshi Oka's unpublished papers, revealed new knowledges which had been over looked. Based on the results, I investigated a new easier appraoch to develop the theory of several complex varaibles as the Oka theory; in practice, I published such four textbooks.

Free Research Field

基礎解析学関連

Academic Significance and Societal Importance of the Research Achievements

前研究課題「15K04917」で端緒が見出された高次元値分布理論-ディオファントス幾何-o-minimal理論の関係について研究が進んだ。超越数論におけるAx-Schanuelの定理にはo-minimal理論を含めた幾つかの証明が知られていたが、値分布理論を用いた証明が得られたことは興味深い。未だo-minimal理論との関係は未開発で今後の研究に待つ。
多変数複素解析の基礎をなす岡潔の成果は、これまで岡-Cartan理論として認識され紹介されてきた。入門理論としてはレベルが高く講義も易しくはない。本研究によりむしろ元の岡理論として入門的な方法が開発されたのは社会的に意義深いと考える。