Orbifold cohomology and generalization of the McKay correspondence

• Project/Area Number: 19540022
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: ITO Yukari Nagoya University, 多元数理科学研究科, 准教授 (70285089)
• Co-Investigator(Kenkyū-buntansha): GUEST Martin 首都大学東京, 大学院・理工学研究科, 教授 (10295470) ; MAENO Toshiaki 京都大学, 大学院・工学研究科, 講師 (60291423) ; IRITANI Hiroshi 京都大学, 大学院・理学研究科, 准教授 (20448400)
• Co-Investigator(Renkei-kenkyūsha): SATO Fumitoshi 香川高等専門学校, 講師 (20548309)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: トポロジー / マッカイ対応 / モジュライ空間 / 代数学 / 商特異点 / 幾何学 / 数理物理 / 特異点解消 / 特異点 / 有限群

Research Abstract

We studied on the ring structure and several properties of Orbifold cohomology and constructed crepant resolutions of Gorenstein singularities as moduli spaces of the quivers which are given by the representations of finite groups to see the generalized McKay correspondence.

Research Products

• [Journal Article] 「鏡の国へ行ってみよう!」～対称性のはなし～ (2011)
  • Author(s): Yukari Ito
  • Journal Title: 日本数学会「数学通信」 15巻4号 Pages: 20-30
  • NAID: 40018733087
• [Journal Article] 「鏡の国へ行ってみよう!」～対称性のはなし～ (2011)
  • Author(s): 伊藤由佳理
  • Journal Title: 日本数学会「数学通信」 Volume: 15巻4号 Pages: 20-30
  • NAID: 40018733087
• [Journal Article] Ruan's conjecture and integral structures in quantum cohomology. New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry(RIMS, Kyoto, 2008) (2010)
  • Author(s): Hiroshi Iritani
  • Journal Title: Adv.Stud.Pure Math. 59 Pages: 111-166
  • Peer Reviewed
• [Journal Article] The tt* Structure of the Quantum Cohomology of CP1 from the Viewpoint of Differential Geometry (2010)
  • Author(s): J.Dorfmeister, M.Guest, W.Rossman
  • Journal Title: Asian Jour.Math. 14 Pages: 417-438
  • Peer Reviewed
• [Journal Article] Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials (2009)
  • Author(s): Toshiaki Maeno, Junzo Watanabe
  • Journal Title: Illinois J.Math. 53 Pages: 591-603
  • Peer Reviewed
• [Journal Article] Wall-crossings in toric Gromov-Witten theories I, Crepant examples. (2009)
  • Author(s): Tom Coates, Hiroshi Iritani, Hsian-Hua Tseng
  • Journal Title: Geometry and Topology 13, no.5 Pages: 2675-2744
  • Peer Reviewed
• [Journal Article] Computing genus zero twisted Gromov-Witten invariants. (2009)
  • Author(s): Tom Coates, Alessio Corti, Hiroshi Iritani, Hsian-Hua Tseng
  • Journal Title: Duke Math.J. 147, no.3 Pages: 377-438
  • Peer Reviewed
• [Journal Article] An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. (2009)
  • Author(s): Hiroshi Iritani
  • Journal Title: Adv.Math. 222, no.3 Pages: 1016-1079
  • Peer Reviewed
• [Journal Article] Recursive formula for $\psi^g-\lambda_1 \psi^{g-1}+\cdots+(-1)^g\lambda_g$ in $\overline{M}_{g,1}$'' (2009)
  • Author(s): Arcara, D., Sato, F.
  • Journal Title: Proc.of Amer.Math.Soc. vol.137, No.12 Pages: 4077-4081
  • Peer Reviewed
• [Journal Article] Schur-Weyl duality and a q-deformation of Specht polynomial (2007)
  • Author(s): Toshiaki Maeno
  • Journal Title: Comm.Alg 35 Pages: 1307-1321
  • Peer Reviewed
• [Journal Article] Convergence of quantum cohomology by quantum Lefschetz (2007)
  • Author(s): Hiroshi Iritani
  • Journal Title: J.Reine.Angew.Math 610 Pages: 29-69
  • Peer Reviewed
• [Journal Article] Schur-Weyl duality and a q-deformation of Specht polynomial (2007)
  • Author(s): Toshiaki Maeno
  • Journal Title: Comm. Alg 35 Pages: 1307-1321
  • Peer Reviewed
• [Journal Article] Convergence of quantum cohomology by quantum Lefschetz (2007)
  • Author(s): Hiroshi Iritani
  • Journal Title: J. Reine. Angew. Math. 610 Pages: 29-69
  • Peer Reviewed
• [Presentation] Alcove pathとaffine Weyl群上の非可換微分構造 (2011)
  • Author(s): Toshiaki Maeno
  • Organizer: 日本数学会年会の中止に伴う臨時講演会
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] マッカイ対応入門 (2010)
  • Author(s): Yukari Ito
  • Organizer: 情報数理談話会
  • Place of Presentation: 大阪府立大学
  • Year and Date: 2010-12-21
• [Presentation] 「鏡の国へ行ってみよう!」～対称性のはなし～ (2010)
  • Author(s): Yukari Ito
  • Organizer: 日本数学会市民講演会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-26
• [Presentation] Troidal resolution of non-abelian quotient singularities (2010)
  • Author(s): Yukari Ito
  • Organizer: Geometry and Algebra of Orbifolds and the McKay Correspondence
  • Place of Presentation: ウォーリック大学(イギリス)
  • Year and Date: 2010-08-12
• [Presentation] McKay対応入門I, II (2010)
  • Author(s): Yukari Ito
  • Organizer: 静岡大学代数学セミナー
  • Place of Presentation: 静岡大学
  • Year and Date: 2010-07-02
• [Presentation] McKay対応入門I, II (2010)
  • Author(s): 伊藤由佳理
  • Organizer: 第7回静岡代数学セミナー
  • Place of Presentation: 静岡大学
  • Year and Date: 2010-07-02
• [Presentation] On Hilb of Hilb (2010)
  • Organizer: 研究集会 "Seminar on McKay correspondence"(Join work with Alvaro Nolla de Celis)
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-03-09
• [Presentation] Hilb of Hilb (2010)
  • Author(s): 伊藤由佳理
  • Organizer: Seminar on McKay correspondence
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-03-09
• [Presentation] Existence of crepant resolutions (2009)
  • Author(s): Yukari Ito
  • Place of Presentation: ソガン大学(韓国)
  • Year and Date: 2009-08-07
• [Presentation] Existence of crepant resolutions (2009)
  • Organizer: 代数幾何学セミナー
  • Place of Presentation: 韓国・ソガン大学
  • Year and Date: 2009-08-07
• [Presentation] Exsistence of crepant resolutions (2009)
  • Author(s): 伊藤由佳理
  • Organizer: 代数幾何学セミナー
  • Place of Presentation: ソガン大学(韓国/ソウル)
  • Year and Date: 2009-08-07
• [Presentation] Quantum cohomology D-modules of toric stacks (2009)
  • Author(s): Hiroshi Iritani
  • Organizer: Equivariant Gromov-Witten theory and symplectic vortices
  • Place of Presentation: C.I.R.M. Marseille-Luminy
  • Year and Date: 2009-07-07
• [Presentation] Existenoe of crepant resolutions (2008)
  • Author(s): Yukari Ito
  • Organizer: COE COW Tokyo.
  • Place of Presentation: 東京大学
  • Year and Date: 2008-12-18
• [Presentation] Existence of crepant resolutions (2008)
  • Organizer: COE COW Tokyo
  • Place of Presentation: 東京大学
  • Year and Date: 2008-12-18
• [Presentation] Existenoe of crepant resolutions (2008)
  • Author(s): 伊藤由佳理
  • Organizer: COE COW Tokyo
  • Place of Presentation: 東京大学
  • Year and Date: 2008-12-18
• [Presentation] Existence of crepant resolutions (2008)
  • Author(s): Yukari Ito
  • Organizer: WAG Concluding Conference.
  • Place of Presentation: ウォーリック大学(イギリス)
  • Year and Date: 2008-07-23
• [Presentation] Existence of crepant resolutions (2008)
  • Organizer: Warwick Algebraic Geometry Concluding Conference
  • Place of Presentation: イギリス・ウォーリック大学
  • Year and Date: 2008-07-23
• [Presentation] Existence of crepant resolutions (2008)
  • Author(s): 伊藤由佳理
  • Organizer: WAG Concluding Conference
  • Place of Presentation: ウォーリック大学(イギリス)
  • Year and Date: 2008-07-23
• [Presentation] The existence of crepant resolutions (2008)
  • Author(s): Yukari Ito
  • Organizer: Mathematics and Physics seminar.
  • Place of Presentation: ペンシルバニア大学(アメリカ合衆国)
  • Year and Date: 2008-02-29
• [Presentation] The existence of crepant resolutions (2008)
  • Organizer: Mathematics and Physics Seminar
  • Place of Presentation: アメリカ・ペンシルバニア大学
  • Year and Date: 2008-02-29
• [Presentation] The existence of crepant resolutions (2008)
  • Author(s): Yukari Ito
  • Organizer: Mathematics and Physics seminar
  • Place of Presentation: ペンシルバニア大学
  • Year and Date: 2008-02-29
• [Presentation] tt*-geometry
  • Author(s): Martin Guest
  • Organizer: Progress on Surface Theory
  • Place of Presentation: MFI Oberwolfach(ドイツ)
• [Book] Special McKay correspondence「Singularity Seminar 2008」多元数理講義録 (2009)
  • Author(s): Yukari Ito
• [Book] この定理が美しい (2009)
  • Author(s): 伊藤由佳理, 他19名
  • Total Pages: 200
  • Publisher: 数学書房
• [Book] From Quantum cohomology to Integrable Systems (2008)
  • Author(s): Martin Guest
  • Total Pages: 305
  • Publisher: Oxford Univ.(Press.)
• [Book] From Quantum cohomology to Integrable Systems (2008)
  • Author(s): Martin Guest
  • Total Pages: 305
  • Publisher: Oxford Univ. Press
• [Book] この定理が美しい(共著)
  • Author(s): 伊藤由佳理
  • Publisher: 数学書房