Study of Moduli of algebraic vector bundles

• Project/Area Number: 20740009
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: ABE Takeshi Kyoto University, 大学院・自然科学研究科, 准教授 (90362409)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: ベクトル束 / モジュライ

Research Abstract

I studied on the strange duality of generalized theta functions on moduli spaces of vector bundles. I described the behavior of the strange duality map for symplectic bundles as the base curve degenerates to a singular curve, and proved the strange duality conjecture for symplectic bundles. By considering quasi-bundles on a projective plane, I proved a special case of Le Potier's strange duality conjecture. I also computed the dimension of generalized theta functions on the moduli spaces of pure one-dimensional sheaves of degree four.

Research Products

• [Journal Article] Deformation of rank 2 quasi-bundles and some strange dualities for rational surfaces (2010)
  • Author(s): 阿部健
  • Journal Title: Duke Mathematical Journal Vol.155, no.3 Pages: 577-620
  • Peer Reviewed
• [Journal Article] Projective normality of the moduli space of rank $2$ vector bundles on a generic curve. (2010)
  • Author(s): Takeshi Abe
  • Journal Title: Transactions of AMS 362 Pages: 477-490
  • Peer Reviewed
• [Journal Article] Degeneration of the strange duality map for symplectic bundles (2009)
  • Author(s): 阿部健
  • Journal Title: Journal fur die Reine und Angewandte Mathematik Vol.631, no.3 Pages: 181-220
  • Peer Reviewed
• [Journal Article] The moduli stack of rank-two Gieseker bundles with fixed determinant on a nodal curve II. (2009)
  • Author(s): Takeshi Abe
  • Journal Title: International Journal of Mathematics 20 Pages: 859-882
  • Peer Reviewed
• [Journal Article] Degeneration of the strange dulity map for symplectic bundles (2009)
  • Author(s): Takeshi Abe
  • Journal Title: Journal fur die Reine und Angewandte Mathematik 631 Pages: 181-220
  • Peer Reviewed
• [Journal Article] Strange Duality for Parabolic Symplectic Bundles on a Pointed Projective Line (2008)
  • Author(s): 阿部健
  • Journal Title: International Mathematics Research Notices rnn121 Pages: 47-47
  • Peer Reviewed
• [Journal Article] Compactification of the symplectic group via generalized symplectic isomorphisms (2008)
  • Author(s): 阿部健
  • Journal Title: RIMS Kokyuroku Bessatsu B9 Pages: 1-50
  • Peer Reviewed
• [Journal Article] Strange duality for parabolic symplectic bundles on a pointed projective line (2008)
  • Author(s): 阿部健
  • Journal Title: International mathematics research notices Vol.2008
  • Peer Reviewed
• [Presentation] The moduli space of pure one dimensional sheaves with $c_{1}=4$ and $\chi=0$ on $\mathbb{P}^{2}$ (2010)
  • Author(s): 阿部健
  • Organizer: 研究集会「代数幾何の関連する諸分野」
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-08-31
• [Presentation] The moduli space of pure one dimensional sheaves with $c_{1}=4$ and $￥chi=0$ on $￥mathbb{P}^{2}$ (2010)
  • Author(s): 阿部健
  • Organizer: 研究集会「代数幾何の関連する諸分野」
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-08-31
• [Presentation] On strange duality for symplectic bundles (2009)
  • Author(s): 阿部健
  • Organizer: 研究集会「Eigenvalue and Saturation Problems for Reductive Groups」
  • Place of Presentation: North Carolina大学アメリカ
  • Year and Date: 2009-05-26
• [Presentation] On strange duality for symplectic bundles (2009)
  • Author(s): 阿部健
  • Organizer: Eigenvalue and Saturation Problems for Reductive Groups
  • Place of Presentation: Univ.of North Carolina
  • Year and Date: 2009-05-26
• [Presentation] On strange duality for symplectic bundles (2008)
  • Author(s): 阿部健
  • Organizer: 研究集会「Moduli and invariants in complex analysis and algebraic geometry」
  • Place of Presentation: 京都工芸繊維大学
  • Year and Date: 2008-11-26
• [Presentation] On strange duality for symplectic bundles (2008)
  • Author(s): 阿部健
  • Organizer: Moduli and invariants in complex analysis and al-gebraic geometry
  • Place of Presentation: 京都工芸繊維大学
  • Year and Date: 2008-11-26
• [Presentation] Level-rank duality of conformal blocks of affine symplectic Lie algebra and symplectic strange duality (2008)
  • Author(s): 阿部健
  • Organizer: 研究集会「表現論と非可換調和解析における新しい視点」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-09-16
• [Presentation] Level-rank duality of conformal blocks of affirm symplectic Lie algebra and symplectic strange duality (2008)
  • Author(s): 阿部健
  • Organizer: 表現論と非可換調和解析における新しい視点
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2008-09-16
• [Presentation] Strange duality for symplectic bundles (2008)
  • Author(s): 阿部健
  • Organizer: 研究集会「代数幾何の関連する諸分野」
  • Place of Presentation: 北海道大学
  • Year and Date: 2008-07-23
• [Presentation] Strange duality for symplectic bundles (2008)
  • Author(s): 阿部健
  • Organizer: 代数幾何の関連する諸分野
  • Place of Presentation: 北海道大学
  • Year and Date: 2008-07-23
• [Presentation] On strange duality for symplectic bundles (2008)
  • Author(s): 阿部健
  • Organizer: 研究集会「Conference on Moduli spaces」
  • Place of Presentation: Warwick大学イギリス
  • Year and Date: 2008-07-08
• [Presentation] On strange duality for symplectic bundles (2008)
  • Author(s): 阿部健
  • Organizer: Conference on Moduli spaces
  • Place of Presentation: Warwick大学
  • Year and Date: 2008-07-08
• [Presentation] Degeneration of the Strange Duality Map for Symplectic Bundles (2008)
  • Author(s): 阿部健
  • Organizer: 研究集会「moduli 空間と双有理幾何」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-06-12
• [Presentation] Degeneration of the Strange Duality Map for Symplectic Bundles (2008)
  • Author(s): 阿部健
  • Organizer: moduli空間と双有理幾何
  • Place of Presentation: 数理解析研究所
  • Year and Date: 2008-06-12