Singularities and derived catetories

• Project/Area Number: 15K04819
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nagoya University (2018); Hiroshima University (2015-2017)
• Principal Investigator: Ishii Akira 名古屋大学, 多元数理科学研究科, 教授 (10252420)
• Research Collaborator: Nakamura Iku ; Ueda Kazushi ; Álvaro Nolla
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: McKay対応 / モジュライ空間 / ダイマー模型 / 非可換クレパント解消 / トーリック多様体 / McKay 対応 / 導来圏 / 最大特異点解消 / 商特異点

Outline of Final Research Achievements

(1) We studied some globalisation of the so-called special McKay correspondence for a two-dimensional quotient singularity with Iku Nakamura and obtained a description of it.
(2) For a two-dimensional quotient singularity, we characterized those resolutions dominated by the so-called maximal resolution as the moduli spaces of G-constellations.
(3) We studied dimer models with group actions with Alvaro Nolla and Kazushi Ueda and constructed non-commutative crepant resolutions of the quotient of an affine topic variety by the symmetry of a lattice polygon.

Academic Significance and Societal Importance of the Research Achievements

(1) これまで知られていたことに対し，より深い洞察を与えることができた．
(2) これまで違う文脈で出てきた二つの概念を結びつけるとともに，導来圏に関する一般的な予想とよく適合する結果である．
(3) 群の作用を考えることで，これまでの結果を一般化するとともに，新しい例の構成に成功した．

Research Products

• [Int'l Joint Research] Universidad Autonoma de Madrid(スペイン)
• [Int'l Joint Research] Universidad Autonoma de Madrid(スペイン)
• [Journal Article] Extended McKay correspondence for quotient surface singularities (2018)
  • Author(s): Akira Ishii and Iku Nakamura
  • Journal Title: The Quarterly Journal of Mathematics Volume: - Issue: 2 Pages: 00-00
  • DOI: 10.1093/qmath/hay047
  • NAID: 120006767378
  • Peer Reviewed / Open Access
• [Journal Article] Dimer models and crepant resolutions (2016)
  • Author(s): Akira Ishii and Kazushi Ueda
  • Journal Title: Hokkaido Math. J. Volume: 45 Pages: 1-42
  • Peer Reviewed
• [Journal Article] Dimer models and the special McKay correspondence (2015)
  • Author(s): Akira Ishii and Kazushi Ueda
  • Journal Title: Geometry and Topology Volume: 19 Issue: 6 Pages: 3405-3466
  • DOI: 10.2140/gt.2015.19.3405
  • Peer Reviewed / Open Access
• [Journal Article] The special McKay correspondence and exceptional collections (2015)
  • Author(s): Akira Ishii and Kazushi Ueda
  • Journal Title: Tohoku Math. J. Volume: 67 Issue: 4 Pages: 585-609
  • DOI: 10.2748/tmj/1450798075
  • NAID: 130007961945
  • Peer Reviewed
• [Presentation] G-constellations and the maximal resolution of a quotient surface singularity (2018)
  • Author(s): Akira Ishii
  • Organizer: Higher dimensional algebraic geometry
  • Int'l Joint Research / Invited
• [Presentation] Dimer models with group actions (2018)
  • Author(s): Akira Ishii
  • Organizer: Differential, Algebraic and Topological Methods in Complex Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] G-constellations and the maximal resolution of a quotient surface singularity (2017)
  • Author(s): Akira Ishii
  • Organizer: 玉原代数幾何サマー スクール 2017
  • Int'l Joint Research / Invited
• [Presentation] G-constellations and the maximal resolution of a quotient surface singularity (2017)
  • Author(s): Akira Ishii
  • Organizer: Classification and Moduli Theory of Algebraic Varieties
  • Int'l Joint Research / Invited
• [Presentation] G-constellations and the maximal resolution of a quotient surface singularity (2017)
  • Author(s): Akira Ishii
  • Organizer: Categorical and Analytic Invariants in Algebraic Geometry V
  • Int'l Joint Research / Invited
• [Presentation] G-constellations and the maximal resolution of a quotient surface singularity (2017)
  • Author(s): Akira Ishii
  • Organizer: Moduli spaces of sheaves and related topics
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Introduction to the McKay correspondence and Artin-Verdier theory (2016)
  • Author(s): Akira Ishii
  • Organizer: Non-commutative crepant resolutions, Ulrich modules and generalizations of the McKay correspondence
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Dimer models with group actions (2016)
  • Author(s): Akira Ishii
  • Organizer: Categorical and analytic invariants in Algebraic geometry 3
  • Place of Presentation: Higher School of Economics, Moscow(Russia)
  • Invited
• [Presentation] On the special McKay correspondence (2015)
  • Author(s): Akira Ishii
  • Organizer: Categorical and analytic invariants in Algebraic geometry 1
  • Place of Presentation: Steklov Math Inst, Moscow (Russia)
  • Year and Date: 2015-09-17
  • Int'l Joint Research / Invited
• [Presentation] On the special McKay correspondence (2015)
  • Author(s): 石井亮
  • Organizer: Symplectic Geometry and Bridgeland stability
  • Place of Presentation: 北海道大学理学部三号館 3-210
  • Year and Date: 2015-08-24
  • Invited