Geometry of quiver varieties and representation theory

• Project/Area Number: 19340006
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: NAKAJIMA Hiraku Kyoto University, 数理解析研究所, 教授 (00201666)
• Co-Investigator(Kenkyū-buntansha): ISHII Akira 広島大学, 大学院・理学研究科, 准教 (10252420) ; YOSHIOKA Kota 神戸大学, 大学院・理学研究科, 教授 (40274047)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥8,190,000 (Direct Cost: ¥6,300,000、Indirect Cost: ¥1,890,000); Fiscal Year 2010: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2009: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2008: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2007: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: インスタントン / ドナルドソン不変量 / 偏屈連接層 / 壁越え公式 / 量子アファイン展開環 / 箙多様体 / クラスター代数 / インスタントンの数え上げ / 壁越え / モジュライ空間

Research Abstract

Consider the blowup of a complex algebraic surface at a point. Together with Yoshioka, I introduced an abelian category in the derived category of coherent sheaves, called the category of perverse coherent sheaves, and study its moduli spaces. As an application, further with Gottsche, I proved Witten's conjecture equating Donaldson invariants and Seiberg-Witten invariants for complex surfaces.

Research Products

• [Journal Article] Perverse coherent sheaves on blow-up.II, wall-crossing and Betti numbers formula (2011)
  • Author(s): Hiraku Nakajima, Kota Yoshioka
  • Journal Title: J.Algebraic Geom 20 Pages: 47-100
  • Peer Reviewed
• [Journal Article] Quiver varieties and cluster algebras, Kyoto (2011)
  • Author(s): Hiraku Nakajima
  • Journal Title: J.Math. Volume 51, Number 1 Pages: 71-126
  • Peer Reviewed
• [Journal Article] Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting (2011)
  • Author(s): Lothar Gottsche, Hiraku Nakajima, Kota Yoshioka
  • Journal Title: Publ.of RIMS 47 No.1 Pages: 307-359
  • NAID: 110008425903
  • Peer Reviewed
• [Journal Article] Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula (2011)
  • Author(s): Hiraku Nakajima, Kota Yoshioka
  • Journal Title: J.of Algebraic Geometry Volume: 20 Pages: 47-100
  • NAID: 120005593584
  • Peer Reviewed
• [Journal Article] Quiver varieties and cluster algebras (2011)
  • Author(s): Hiraku Nakajima
  • Journal Title: Kyoto J.of Math. Volume: 51(1) Pages: 71-126
  • NAID: 120005593585
  • Peer Reviewed
• [Journal Article] Perverse coherent sheaves on blow-up. III. Blow-up formula from wall-crossing (2011)
  • Author(s): Hiraku Nakajima, Kota Yoshioka
  • Journal Title: Kyoto J.of Math Volume: (掲載確定)
  • NAID: 120005593586
  • Peer Reviewed
• [Journal Article] Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting (2011)
  • Author(s): Lothar Gottsche, Hiraku Nakajima, Kota Yoshioka
  • Journal Title: Publ.of RIMS Volume: (掲載確定)
  • NAID: 110008425903
  • Peer Reviewed
• [Journal Article] PERVERSE COHERENT SHEAVES ON BLOW-UP. II. WALL-CROSSING AND BETTI NUMBERS FORMULA (2010)
  • Author(s): Hiraku Nakajima, Kota Yoshioka
  • Journal Title: Journal of Algebraic Geometry (掲載予定, 現在 online で発行)
  • NAID: 120005593584
  • Peer Reviewed
• [Journal Article] K-theoretic Donaldson invariants via instanton counting (2009)
  • Author(s): Lothar Gottsche, Hiraku Nakajima, Kota Yoshioka
  • Journal Title: Pure and Appl.Math.Quaterly 5, No.3 Pages: 1029-1111
  • Peer Reviewed
• [Journal Article] Quiver varieties and branching (2009)
  • Author(s): Hiraku Nakajima
  • Journal Title: SIGMA 5 003 Pages: 37-37
  • Peer Reviewed
• [Journal Article] K-theoretic Donaldson invariants via instanton counting (2009)
  • Author(s): Lothar Gottsche, Hiraku Nakajima, Kota Yoshioka
  • Journal Title: Pure and Appl. Math. Quaterly 5 Pages: 1029-1111
  • Peer Reviewed
• [Journal Article] Quiver varieties and branching (2009)
  • Author(s): Hiraku Nakajima
  • Journal Title: SIGMA 5 Pages: 37-37
  • Peer Reviewed
• [Journal Article] Quiver varieties and branching (2009)
  • Author(s): Hiraku Nakajima
  • Journal Title: SIGMA 5(掲載確定) Pages: 37-37
  • Peer Reviewed
• [Journal Article] Donaldson invariants and instanton counting (2008)
  • Author(s): Lothar Gottsche, Hiraku Nakajima, Kota Yoshioka
  • Journal Title: J. of Diff. Geom. 80 Pages: 343-390
  • Peer Reviewed
• [Journal Article] Instanton counting and Donaldson invariants (2008)
  • Author(s): Lothar Gottsche, Hiraku Nakajima, Kota Yoshioka
  • Journal Title: J. of Differential Geometry 80 Pages: 343-390
  • NAID: 10019542192
  • Peer Reviewed
• [Journal Article] Sheaves on ALE spaces and quiver varieties (2007)
  • Author(s): Hiraku Nakajima
  • Journal Title: Moscow Math.Journal 7 No.4 Pages: 699-722
  • Peer Reviewed
• [Journal Article] Sheaves on ALE spaces and quiver varieties (2007)
  • Author(s): Hiraku Nakajima
  • Journal Title: Moscow Math. Journal 7 Pages: 699-722
  • Peer Reviewed
• [Journal Article] Counting invariant of perverse coherent sheaves and its wall-crossing
  • Author(s): 1Kentaro Nagao, Hiraku Nakajima
  • Journal Title: IMRN, to appear
  • Peer Reviewed
• [Journal Article] Perverse coherent sheaves on blow-up II. Wall-crossing and Betti numbers formula
  • Author(s): Hiraku Nakajima, Kota Yoshioka
  • Journal Title: J. of Alg. Geom. (to appear)
  • NAID: 120005593584
  • Peer Reviewed
• [Presentation] インスタントンの数え上げとドナルドソン不変量 (2011)
  • Author(s): Hiraku Nakajima
  • Organizer: 臨時講演会(日本数学会年会の中止に伴う)
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-03-22
• [Presentation] Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting (2011)
  • Author(s): Hiraku Nakajima
  • Organizer: Kaleidoscopic View of Modern Mathematics
  • Place of Presentation: Institute Henri-Poincare Paris(フランス)(招待講演)
  • Year and Date: 2011-01-10
• [Presentation] Cluster algebras and quantum affine algebras (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: Representation Theory of Algebraic Groups and quantum groups
  • Place of Presentation: 名古屋大学(招待講演)
  • Year and Date: 2010-08-03
• [Presentation] Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: The 5th Pacific Rim Complex and Symplectic Geometry Conference 2010
  • Place of Presentation: 名古屋大学(招待講演)
  • Year and Date: 2010-07-26
• [Presentation] Instanton counting and Donaldson invariants (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: London Math.Soc.Hardy lectures
  • Place of Presentation: London
  • Year and Date: 2010-07-02
• [Presentation] Instanton counting and Donaldson invariants (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: London Math.Soc.Hardy lectures
  • Place of Presentation: London Math.Soc.(UK)(招待講演)
  • Year and Date: 2010-07-02
• [Presentation] Quiver varieties and double affine Grassmann (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: London Math.Soc.Hardy lectures
  • Place of Presentation: Oxford, UK
  • Year and Date: 2010-06-28
• [Presentation] Quiver varieties and double affine Grassmann (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: London Math.Soc.Hardy lectures
  • Place of Presentation: Oxford大学(UK)(招待講演)
  • Year and Date: 2010-06-28
• [Presentation] Quiver varieties and cluster algebras (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: London Math.Soc.Hardy lectures
  • Place of Presentation: Leeds, UK
  • Year and Date: 2010-06-22
• [Presentation] Quiver varieties and cluster algebras (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: London Math.Soc.Hardy lectures
  • Place of Presentation: Leeds大学(UK)(招待講演)
  • Year and Date: 2010-06-22
• [Presentation] Quiver varieties and cluster algebras (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: London Math.Soc.Hardy lectures
  • Place of Presentation: Edinburgh大学(UK)(招待講演)
  • Year and Date: 2010-06-18
• [Presentation] Quiver varieties and cluster algebras, I, II, III, Derived categories of Algebro-Geometric origin and integrable systems (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: the 15th Midrasha Mathmaticae
  • Place of Presentation: Jerusalem
• [Presentation] Quiver varieties and crystal bases of quantum affine algebra (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: NSF-CBMS Conferences
  • Place of Presentation: North Carolina State University, USA
• [Presentation] Quiver varieties and crystal bases of quantum affine algebra (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: NSF-CBMS Conferences
  • Place of Presentation: North Carolilna State University (USA)(招待講演)
• [Presentation] Quiver varieties and cluster algebras, I, II, III (2010)
  • Author(s): Hiraku Nakajima
  • Organizer: Derived categories of Algebro-Geometric origin and integrable systems, the 15th Midrasha Mathmaticae
  • Place of Presentation: The Institute for Advanced Studies of Jerusalem(イスラエル)(招待講演)
• [Presentation] Donaldson = Seiberg- Witten from Mochizuki's formula and instanton counting for the theory with a fundamental matter (2009)
  • Author(s): Hiraku Nakajima
  • Organizer: Mirror Symmetry and Gromov-Witten Invariants
  • Place of Presentation: 東京大学
  • Year and Date: 2009-12-07
• [Presentation] Donaldson=Seiberg-Witten from Mochizuki's formula and instanton counting for the theory with a fundamental matter (2009)
  • Author(s): Hiraku Nakajima
  • Organizer: Mirror Symmetry and Gromov-Witten Invariants
  • Place of Presentation: University of Tokyo
  • Year and Date: 2009-12-07
• [Presentation] Perverse Coherent Sheaves on Blow-up (2009)
  • Author(s): Hiraku Nakajima
  • Organizer: Arithmetic Geometry and Moduli Spaces in Algebraic Geometry
  • Place of Presentation: Hang-Zhou, China
  • Year and Date: 2009-08-25
• [Presentation] Quiver varieties and cluster algebras (2009)
  • Author(s): Hiraku Nakajima
  • Organizer: Summer School and Conference in Geometric Representation Theory and Extended Affine Lie Algebras
  • Place of Presentation: Univ.Ottawa, Canada
  • Year and Date: 2009-07-01
• [Presentation] Quiver varieties and cluster algebras (2009)
  • Author(s): Hiraku Nakajima
  • Organizer: Summer School and Conference in Geometric Representation Theory and Extended Affine Lie Algebras
  • Place of Presentation: University of Ottawa, Ontario, Canada
  • Year and Date: 2009-07-01
• [Presentation] Instanton counting and wall-crossing in Donaldson invariants, Quiver varieties (2009)
  • Author(s): Hiraku Nakajima
  • Organizer: Donaldson-Thomas invariants and instantons
  • Place of Presentation: CIRM, Luminy
• [Presentation] Instanton counting and wall-crossing in Donaldson invariants (2009)
  • Author(s): Hiraku Nakajima
  • Organizer: Quiver varieties, Donaldson-Thomas invariants and instantons
  • Place of Presentation: CIRM, Luminy
• [Presentation] Quiver varieties and double affine Grassmannian (2008)
  • Author(s): Hiraku Nakajima
  • Organizer: Bert Kostant's 80th birthday conference
  • Place of Presentation: バンクーバー、カナダ
  • Year and Date: 2008-05-19
• [Presentation] Quiver varieties and double affine Grassmannian (2008)
  • Author(s): Hiraku Nakajima
  • Organizer: Bert Kostant's 80th birtday conference
  • Place of Presentation: ブリテイッシュ・コロンビア大学 (カナダ)
  • Year and Date: 2008-05-19
• [Presentation] Quiver varieties and double affine Grassmannian (2008)
  • Author(s): Hiraku Nakajima
  • Organizer: Bert Kostant's 80th birtday conference
  • Place of Presentation: ブリティッシュ・コロンビア大学(カナダ)
  • Year and Date: 2008-05-19
• [Remarks] ホームページ等
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• [Remarks]
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• [Remarks] ホームページ
  • URL: http://www.kurims.kyoto-u.ac.jp/~nakajima
• [Remarks]
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• [Remarks]
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