Existence of higher dimensional crepant resolutions and generlization of the McKay correspondence

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03209
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: The University of Tokyo (2019-2023); Nagoya University (2018)
• Principal Investigator: ITO YUKARI 東京大学, カブリ数物連携宇宙研究機構, 教授 (70285089)
• Co-Investigator(Kenkyū-buntansha): 石井 亮 名古屋大学, 多元数理科学研究科, 教授 (10252420) ; 伊山 修 東京大学, 大学院数理科学研究科, 教授 (70347532)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: crepant resolution / McKay correspondence / tilting theory / Austanfer-Reiten theory / dimar model / exceptional correction

Outline of Final Research Achievements

Ito characterizes the exceptional set of resolutions of quotient singularities of a finite group G and their corresponding irreducible representations when resolutions of singularities are obtained. Ishii investigates the Dimer models of quotient singularities and exceptional corrections on Hirzebruch surfaces. Iyama gave an invited lecture at the International Congress of Mathematicians (ICM) in 2018 and conducted research on triangulated categories, cluster categories, and AR theory. Additionally, international research meetings were held in 2018, 2020, and 2023, and in April 2023, a collection of papers related to this research project titled "McKay correspondence, tilting theory and related topics" was published as Advanced Studies in Pure Mathematics 88.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究課題の成果のうち、伊藤と石井は、2次元で知られているMcKay対応の3次元への一般化について、導来圏や新たに定義したessential representation、ダイマー模型を用いて研究をし、伊山は多元環の表現論の研究を発展させた点が数学の代数幾何学における学術的意義である。また上記の出版論文集には、サーベイも含まれ、本研究課題周辺を新たに勉強したい学生や研究者の教科書ともなる有意義な一冊である。