Moduli spaces and derived categories

• Project/Area Number: 21540039
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Hiroshima University
• Principal Investigator: ISHII Akira 広島大学, 大学院・理学研究科, 准教授 (10252420)
• Co-Investigator(Kenkyū-buntansha): SHIMADA Ichiro 広島大学, 大学院・理学研究科, 教授 (10235616) ; SHUN-ICHI Kimura 広島大学, 大学院・理学研究科, 教授 (10284150) ; HIDEYASU Sumihiro 広島大学, 大学院・理学研究科, 名誉教授 (60068129)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 導来圏 / McKay対応 / ダイマー模型 / ミラー対称性 / ヒルベルトスキーム / 代数幾何学 / マッカイ対応 / モジュライ

Research Abstract

We determined the consistency condition for dimer models which ensures the derived equivalence between the quiver with relations and the 3-dimensional Gorenstein affine toric variety associated to it. We constructed a full strong exceptional collection consisting of line bundles on a 2-dimensional toric weak Fano stack. We proved that the remaining component in the special McKay correspondence is generated by an exceptional collection. We obtained a certain kind of description of the derived category of a Fermat variety. We obtained some results on iterated G-Hilbert schemes.

Research Products

• [Journal Article] A note on derived categories of Fermat varieties (2012)
  • Author(s): Akira Ishii and Kazushi Ueda
  • Journal Title: RIMS Kokyuroku Bessatsu
  • Peer Reviewed
• [Journal Article] A note on derived categories of Fermat varieties (2012)
  • Author(s): Akira Ishii, Kazushi Ueda
  • Journal Title: Proceedings of the conference "Dereived Categories Tokyo 2011", EMS series of Congress Reprots Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] A note on consistency conditions on dimer models (2011)
  • Author(s): Akira Ishii and Kazushi Ueda
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 24 Pages: 143-164
  • Peer Reviewed
• [Journal Article] Finite dimensional morphisms in a tensor category (2011)
  • Author(s): Perdo Luis del Angel and Shun-ichi Kimura
  • Journal Title: J. Reine Angew. Math Volume: 656 Pages: 213-222
  • Peer Reviewed
• [Journal Article] A note on consistency conditions on dimer models (2011)
  • Author(s): Akira Ishii, Kazushi Ueda
  • Journal Title: RIMS Kokyyroku Bessatsu Volume: B24 Pages: 143-164
  • Peer Reviewed
• [Journal Article] Finite dimensional morphisms in a tensor cateogry (2011)
  • Author(s): P.L.del Angel, S.Kimura
  • Journal Title: Journal fuer die reine und angewandte Math Volume: 656 Issue: 656 Pages: 213-222
  • DOI: 10.1515/crelle.2011.051
  • Peer Reviewed
• [Journal Article] A note on consistency conditions on dimer models (2011)
  • Author(s): Akira Ishii, Kazushi Ueda
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B24(掲載決定)
  • Peer Reviewed
• [Journal Article] Stability conditions on A_n-singularities (2010)
  • Author(s): Akira Ishii, Kazushi Ueda and Hokuto Uehara
  • Journal Title: J. Differential Geom Volume: 84 Pages: 87-126
  • Peer Reviewed
• [Journal Article] Generalized Zariski-van Kampen theorem and its application to Grassmannian dual varieties, Internat (2010)
  • Author(s): Ichiro Shimada
  • Journal Title: Internat. J. Math Volume: 21 Pages: 591-637
  • Peer Reviewed
• [Journal Article] Stability conditions on A_n-singularities (2010)
  • Author(s): Akira Ishii, Kazushi Ueda, Hokuto Uehara
  • Journal Title: J.Differential Geom. Volume: 84 Pages: 87-126
  • Peer Reviewed
• [Journal Article] Generalized Zariski-van Kampen theorem and its application to Grassm annian dual varieties (2010)
  • Author(s): Ichiro Shimada
  • Journal Title: Internat.J.Math. Volume: 21 Pages: 591-637
  • NAID: 120002598782
  • Peer Reviewed
• [Journal Article] Zariski-van Kampen method and transcendental lattices of certain singular K3 surfaces (2009)
  • Author(s): Ken-ichiro Arima and Ichiro Shimada
  • Journal Title: Tokyo J. Math Volume: 32-1 Pages: 201-227
  • Peer Reviewed
• [Journal Article] Irrationality of motivic series of Chow varieties (2009)
  • Author(s): E. Javier Elizondo and Shun-Ichi Kimura
  • Journal Title: Math. Z Volume: 263-1 Pages: 27-32
  • Peer Reviewed
• [Journal Article] Zariski-van Kampen method and transcendental lattices of certain singular K3 surfaces (2009)
  • Author(s): Ken-ichiro Arima, Ichiro Shimada
  • Journal Title: Tokyo J.Math. 32・1 Pages: 201-227
  • Peer Reviewed
• [Journal Article] Irrationality of motivic series of Chow varieties (2009)
  • Author(s): E.Javier Elizondo, Shun-Ichi Kimura
  • Journal Title: Math.Z. 263・1 Pages: 27-32
  • Peer Reviewed
• [Journal Article] Stability conditions on A_n-singularities
  • Author(s): Akira Ishii, Kazushi Ueda, Hokuto Uehara
  • Journal Title: J.Differential Geom. (掲載決定)
  • Peer Reviewed
• [Presentation] On iterated G-Hilbert schemes,高次元代数多様体とベクトル束の代数幾何学 (2012)
  • Author(s): 石井亮
  • Organizer: 高次元代数多様体とベクトル束の代数幾何学
  • Place of Presentation: 九州大学数理学研究院
  • Year and Date: 2012-03-16
• [Presentation] On iterated G-Hilbert schemes (2012)
  • Author(s): 石井亮
  • Organizer: 高次元代数多様体とベクトル束の代数幾何学
  • Place of Presentation: 九州大学数理学研究院(招待講演)
  • Year and Date: 2012-03-16
• [Presentation] McKay correspondence and dimer models (2012)
  • Author(s): Akira Ishii
  • Organizer: Mini-workshop of Algebra
  • Place of Presentation: 国立台湾大学,台湾
• [Presentation] McKay correspondence and dimer models (2012)
  • Author(s): Akira Ishii
  • Organizer: Mini-workshop of Algebra
  • Place of Presentation: 国立台湾大学(招待講演)
• [Presentation] Dimer models and crepant resolutions (2011)
  • Author(s): Akira Ishii
  • Organizer: Noncommutative Algebraic Geometry and D-branes
  • Place of Presentation: USA
  • Year and Date: 2011-12-13
• [Presentation] Dimer models and crepant resolutions (2011)
  • Author(s): Akira Ishii
  • Organizer: Noncommutative Algebraic Geometry and D-branes
  • Place of Presentation: Simons Center for Geometry and Physics, Stony Brook University, USA(招待講演)
  • Year and Date: 2011-12-13
• [Presentation] Special McKay correspondence and exceptional collections (2011)
  • Author(s): Akira Ishii
  • Organizer: Complex Algebraic Geometry
  • Place of Presentation: Germany
  • Year and Date: 2011-09-26
• [Presentation] Special McKay correspondence and exceptional collections (2011)
  • Author(s): Akira Ishii
  • Organizer: Complex Algebraic Geometry
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach, Germany(招待講演)
  • Year and Date: 2011-09-26
• [Presentation] Special McKay correspondence and exceptional collections (2011)
  • Author(s): 石井亮
  • Organizer: 特異点論とそのひろがり
  • Place of Presentation: 京都大学理学部
  • Year and Date: 2011-08-25
• [Presentation] A remark on invertible polynomials and exceptional collections (2011)
  • Author(s): Akira Ishii
  • Organizer: WORKSHOP ON NON-COMMUTATIVE GEOMETRY AND THE McKAY CORRESPONDENCE
  • Place of Presentation: 名古屋大学多元数理科学研究科
  • Year and Date: 2011-03-14
• [Presentation] A remazk on invertible polynomials and exceptional collections (2011)
  • Author(s): Akira Ishii
  • Organizer: WORKSHOP ON NON-COMMUTATIVE GEOMETRY AND THE McKAY CORRESPOND ENCE
  • Place of Presentation: 名古屋大学多元数理科学研究科(招待講演)
  • Year and Date: 2011-03-14
• [Presentation] Special McKay correspondence and universal localization (2010)
  • Author(s): Akira Ishii
  • Organizer: Geometry and Algebra of Orbifolds and the McKay Correspondence
  • Place of Presentation: Warwick University, U. K
  • Year and Date: 2010-08-12
• [Presentation] Special McKay correspondence and universal localisation (2010)
  • Author(s): Akira Ishii
  • Organizer: Geometry and Algebra of Orbifolds and the McKay Correspondence
  • Place of Presentation: Warwick University, U.K.(招待講演)
  • Year and Date: 2010-08-12
• [Presentation] Dimer models and exceptional collections (2010)
  • Author(s): Akira Ishii
  • Organizer: DMM seminar
  • Place of Presentation: 東京大学IPMU
  • Year and Date: 2010-04-26
• [Presentation] Dimer models and exceptional collections (2010)
  • Author(s): Akira Ishii
  • Organizer: DMM seminar
  • Place of Presentation: 東京大学IPMU(招待講演)
  • Year and Date: 2010-04-26
• [Presentation] Dimer models and exceptional collections (2009)
  • Author(s): Akira Ishii
  • Organizer: Mirror Symmetry and Gromov-Witten Invariants
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2009-12-07
• [Presentation] Dimer models and tilting bundles (2009)
  • Author(s): Akira Ishii
  • Organizer: Complex Algebraic Geometry
  • Place of Presentation: MathematischesForschungsinstitut Oberwolfach, Germany
  • Year and Date: 2009-10-01
• [Presentation] Dimer models and tilting bundles (2009)
  • Author(s): Akira Ishii
  • Organizer: Complex Algebraic Geometry
  • Place of Presentation: Oberwolfach, Germany
  • Year and Date: 2009-10-01
• [Presentation] Dimer models and the special McKay correspondence (2009)
  • Author(s): Akira Ishii
  • Organizer: Algebraic Triangulated Categories and Related Topics
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-07-24