The McKay correspondence over number fields

2020 Fiscal Year Final Research Report

• Project/Area Number: 18K18710
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 11:Algebra, geometry, and related fields
• Research Institution: Osaka University (2020); Tohoku University (2018-2019)
• Principal Investigator: Yasuda Takehiko 大阪大学, 理学研究科, 教授 (30507166)
• Project Period (FY): 2018-06-29 – 2021-03-31
• Keywords: マッカイ対応 / 非線形作用 / 局所体 / KLT特異点 / p冪位数巡回群 / モジュライ空間

Outline of Final Research Achievements

This research was aimed at studying the McKay correspondence over global fields, that is, number fields and functions fields. Obtained results are mainly, not the McKay correspondence over global fields itself, but deepening our understanding of the McKay correspondence over local fields, which would play an important role to study the case of global fields. We obtained results about the McKay correspondence for cyclic group of prime power order and the McKay correspondence for non-linear actions as well as about the moduli space of Galois extensions of a power series field. By closely related methods, we obtained also the finiteness of local fundamental group of 2-dimensional KLT singularities in arbitrary characteristic. We held two international workshops and promoted sharing of research information.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

大域体や局所体のような数論的体上のMcKay対応を研究することで、整数論と特異点論を結ぶ新しい橋をかけることが期待できる。整数論と特異点論は、それぞれ整数と特異点という非常に基本的な研究対象を扱うため、様々な研究分野と関連する重要な分野である。本研究課題は、この２つの研究領域の融合分野に関するものだったが、得られた成果により両分野の結びつきをより強くすることができた。