Explicit study of structures of irreducible symplectic manifolds

2012 Fiscal Year Final Research Report

• Project/Area Number: 22740004
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Waseda University (2011-2012); The University of Tokyo (2010)
• Principal Investigator: NAGAI Yasunari 早稲田大学, 理工学術院, 准教授 (50572525)
• Project Period (FY): 2010 – 2012
• Keywords: 代数多様体 / 双有理写像 / 正則シンプレクティック幾何 / 計算機代数の応用

Research Abstract

O’Grady constructed two sporadic examples of compact irreducible symplectic Kahler manifolds using moduli spaces of rank 2 semi-stable coherent sheaves on a K3 surface or an Abelian surface. I studied the locus of non-locally free sheaves in these moduli spaces and clarified some birational-geometric structures on the O’Grady’s examples. In the course of study, I investigated a method to calculate explicitly some invariant rings associated to the moduli problem using classical invariant theory, with the aid of computational algebra.

Research Products

• [Journal Article] Non-locally-free locus of O'Grady's ten dimensional example (2012)
  • Author(s): Nagai, Yasunrai
  • Journal Title: manuscripta mathematica
  • DOI: DOI:10.1007/s00229-012-0595-5
  • Peer Reviewed
• [Presentation] Computing classical invariant rings with application to moduli spaces (2012)
  • Author(s): 永井 保成
  • Organizer: 第57回代数学シンポジウム
  • Year and Date: 20120820-23
• [Presentation] Abelian fibrations and Lagrangian fibrations (2012)
  • Author(s): Yasunari Nagai
  • Organizer: KIAS Winter School on Algebraic Geometry
  • Place of Presentation: 韓国 Duksan
  • Year and Date: 20120130-0202
• [Presentation] An explicit study of O'Grady's examles (2010)
  • Author(s): Yasunari Nagai
  • Organizer: 都の西北 代数幾何学シンポジウム
  • Place of Presentation: 早稲田大学
  • Year and Date: 20101110-13
• [Presentation] An explicit study of non-locally free locus of O'Grady's example (2010)
  • Author(s): Yasunari Nagai
  • Organizer: Miniworkshop on holomorphic symplectic geometry
  • Place of Presentation: 韓国・高等科学院(KIAS)
  • Year and Date: 20100929-30
• [Remarks] Nagai, Yasunari, Birational geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification, preprint (2010) 投稿中
• [Remarks] 永井 保成, Birational Geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification,東京大学数理科学研究科 代数幾何学セミナー,2011.07.04
• [Remarks] 永井 保成, Birational geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification, 京都大学 代数幾何学セミナー,2011.1.14
• [Remarks] 永井 保成, Non-locally free locus of O'Grady's 10 dimensional example, 首都大学東京・複素幾何セミナー, 2010.1.13.
• [Remarks] 永井 保成, Non-locally free locus of O'Grady's 10 dimensional example, 名古屋大学・代数幾何セミナー, 2009.12.21.