Fundamental groups and moduli spaces of curves in positive characteristic

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14283
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kyoto University
• Principal Investigator: YANG YU 京都大学, 数理解析研究所, 特定助教 (30838131)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: curve / moduli space / fundamental group / anabelian geometry / positive characteristic

Outline of Final Research Achievements

I introduced a new topological space called the moduli space of fundamental groups and formulated the conjecture of topological isomorphism. I also proved that this topological isomorphism conjecture holds for one-dimensional moduli spaces. By using the moduli space of fundamental groups, I posed a new anabelian philosophy for curves over algebraically closed fields of positive characteristic. Furthermore, based on this philosophy, I formulated numerous new conjectures. As an application of the philosophy, I successfully provided a new proof for the famous theorem of Mochizuki concerning the Isom version of the Grothendieck conjecture for curves over sub-p-adic fields.

Free Research Field

代数幾何

Academic Significance and Societal Importance of the Research Achievements

私の研究では、基本群のモジュライ空間という新たな理論を提唱し、いくつかの基本的な予想を定式化し、その予想が成り立つ証拠も提供した。基本群のモジュライ空間理論は、正標数の代数閉体上の曲線の遠アーベル幾何学に対する一般的な哲学を提供し、特にその主予想である位相同型予想は、今後の研究の方向性を指し示している。