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Study of Lagrangian fibrations via projective embeddings

Research Project

Project/Area Number 19740025
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

NOHARA Yuichi  Tohoku University, 教育学部, 講師 (60447125)

Project Period (FY) 2007 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥3,730,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywordsラグランジアンファイブレーション / 完全可積分系 / ミラー対称性 / トーリック退化 / Grassmann多様体 / Gelfand-Cetlin系 / Gelfand-Cetlin型 / ポテンシャル関数 / 幾何学 / 幾何学的量子化
Research Abstract

I have proved that the Gelfand-Cetlin system, a completely integrable system on a flag manifold of type A, can be deformed into a toric moment map on a toric variety (this is a joint work with T. Nishinou and K. Ueda). As an application we computed the potential functions of the Lagrangian torus fibers of the Gelfand-Cetlin system, and showed that it coincides with the superpotential of the Landau-Ginzburg mirror of the flag manifold.
It is known that toric degenerations of a Grassmannian of two-planes in a complex vector space are classified by certain trees. For each such tree I have constructed a completely integrable system on the Grassmannian, and proved that it can be deformed into a toric moment map under the corresponding toric degeneration. This result implies that completely integrable systems on a polygon space constructed by Kapovich and Millson also admit deformations into toric moment maps. I have also studied a relation between Kapovich-Milson's integrable systems and Goldman's integrable systems on a moduli space of parabolic bundles of rank 2 on a projective line.

Report

(6 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report   Self-evaluation Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (50 results)

All 2010 2009 2008 2007 Other

All Journal Article (12 results) (of which Peer Reviewed: 10 results) Presentation (38 results)

  • [Journal Article] Toric degenerations of Gelfand-Cetlin systems and potential functions2010

    • Author(s)
      T.Nishinou, Y.Nohara, K.Ueda
    • Journal Title

      Advances in Mathematics 224

      Pages: 648-706

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Toric degenerations of Gelfand-Cetlin systems and potential functions2010

    • Author(s)
      T.Nishinou, et al.
    • Journal Title

      Advances in Mathematics

      Volume: 224 Pages: 648-706

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lagrangian fibrations and theta functions, "Noncommutativity and Singularities-Proceedings of French-Japanese symposia held at IHES in 20062009

    • Author(s)
      Y.Nohara
    • Journal Title

      Advanced Studies in Pure Mathematics 55

      Pages: 299-308

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Projective embeddings and Lagrangian fibraions of Kummer varieties2009

    • Author(s)
      Y.Nohara
    • Journal Title

      Internat.J.Math. 20, No.5

      Pages: 557-572

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Projective embeddings and Lagrangian fibrations of Kummer varieties2009

    • Author(s)
      Y. Nohara
    • Journal Title

      Internat. J. Math. 20

      Pages: 557-572

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Projective embeddings and Lagrangian fibraions of Kummer varieties2009

    • Author(s)
      Yuichi Nohara
    • Journal Title

      International Journal of Mathematics 20, No.5

      Pages: 557-572

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      T.Nishinou, Y.Nohara, K.Ueda
    • Journal Title

      Advances in Mathematics 現時点では電子版のみ

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Projective embeddings and Lagrangian fibraions of Kummer varieties2009

    • Author(s)
      Yuichi Nohara
    • Journal Title

      International Journal of Mathematics (印刷中)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lagrangian fibrations and theta functions(summary)2009

    • Author(s)
      Yuichi Nohara
    • Journal Title

      Noncommutativity and Singularities-Proceedings of French-Japanese symposia held at IHES in 2006 (印刷中)

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 旗多様体のトーリック退化とGelfand-Cetlin系, 数理解析研究所講究録16132008

    • Author(s)
      野原雄一
    • Journal Title

      Bergman核と代数幾何への応用

      Pages: 66-85

    • Related Report
      2010 Final Research Report
  • [Journal Article] 旗多様体のトーリック退化とGeHland-Cetlin系2008

    • Author(s)
      野原雄一
    • Journal Title

      数理解析研究所講究録 1613

      Pages: 66-85

    • Related Report
      2008 Annual Research Report
  • [Journal Article] Toric degenerations of Gelfand-Cetlin systems and potential functions

    • Author(s)
      T. Nishinou, Y. Nohara, K. Ueda
    • Journal Title

      in Advances in mathematics (掲載決定)

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Presentation] Toric degenerations of polygon spaces and bending flows2010

    • Author(s)
      野原雄一
    • Organizer
      幾何学シンポジウム
    • Place of Presentation
      神戸大学
    • Year and Date
      2010-08-07
    • Related Report
      2010 Final Research Report
  • [Presentation] Toric degenerations of polygon spaces and bending flows2010

    • Author(s)
      野原雄一
    • Organizer
      幾何学シンポジウム
    • Place of Presentation
      神戸大学(招待講演)
    • Year and Date
      2010-08-07
    • Related Report
      2010 Annual Research Report
  • [Presentation] Toric degeneration of the Gelfand-Cetlin system and its applications2010

    • Author(s)
      野原雄一
    • Organizer
      Workshop : Lefschetz fibration and category theory
    • Place of Presentation
      大阪大学
    • Year and Date
      2010-06-05
    • Related Report
      2010 Final Research Report
  • [Presentation] Toric degeneration of the Gelfand-Cetlin system and its applications2010

    • Author(s)
      野原雄一
    • Organizer
      Workshop : Lefschetz fibration and category theory
    • Place of Presentation
      大阪大学(招待講演)
    • Year and Date
      2010-06-05
    • Related Report
      2010 Annual Research Report
  • [Presentation] Gelfand-Cetlin系のトーリック退化とその応用2010

    • Author(s)
      野原雄一
    • Organizer
      トロピカル幾何と超離散系の新展開
    • Place of Presentation
      京都大学
    • Year and Date
      2010-03-12
    • Related Report
      2010 Final Research Report
  • [Presentation] Gelfand-Cetlin 系のトーリック退化とその応用2010

    • Author(s)
      野原雄一
    • Organizer
      トロピカル幾何と超離散系の新展開
    • Place of Presentation
      京都
    • Year and Date
      2010-03-12
    • Related Report
      2009 Annual Research Report
  • [Presentation] Toric degeneration of Gelfand-Cetlin systems2010

    • Author(s)
      Y.Nohara
    • Organizer
      The 2nd GCOE International Symposium on "Weaving Science Web beyond Particle-Matter Hierarchy
    • Place of Presentation
      東北大学
    • Year and Date
      2010-02-18
    • Related Report
      2010 Final Research Report
  • [Presentation] Toric degeneration of Gelfand-Cetlin systems2010

    • Author(s)
      野原雄一
    • Organizer
      The 2nd GCOE International Symposium
    • Place of Presentation
      仙台
    • Year and Date
      2010-02-18
    • Related Report
      2009 Annual Research Report
  • [Presentation] Toric degenerations of Gelfand-Celtin systems and potential functions2009

    • Author(s)
      Y.Nohara
    • Organizer
      Workshop on Recent Advances in Mathematics at IPMU, IPMU
    • Place of Presentation
      東京大学
    • Year and Date
      2009-11-16
    • Related Report
      2010 Final Research Report
  • [Presentation] Toric degenerations of Gelfand-Celtin systems and potential functions2009

    • Author(s)
      野原雄一
    • Organizer
      Workshop on Recent Advances in Mathematics at IPMU
    • Place of Presentation
      柏
    • Year and Date
      2009-11-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      Y.Nohara
    • Organizer
      第15回複素幾何シンポジウム
    • Place of Presentation
      信州菅平高原
    • Year and Date
      2009-11-02
    • Related Report
      2010 Final Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      野原雄一
    • Organizer
      第15回複素幾何シンポジウム
    • Place of Presentation
      菅平
    • Year and Date
      2009-11-02
    • Related Report
      2009 Annual Research Report
  • [Presentation] 旗多様体のトーリック退化とGelfand-Cetlin系2009

    • Author(s)
      野原雄一
    • Organizer
      2009日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学(特別講演)
    • Year and Date
      2009-09-25
    • Related Report
      2010 Final Research Report
  • [Presentation] 旗多様体のトーリック退化とGelfand-Cetlin系2009

    • Author(s)
      野原雄一
    • Organizer
      2009日本数学会秋季総合分科会(特別講演)
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-25
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] 旗多様体のトーリック退化と Gelfand-Cetlin 系2009

    • Author(s)
      野原雄一
    • Organizer
      2009日本数学会秋季総合分科会
    • Place of Presentation
      大阪
    • Year and Date
      2009-09-25
    • Related Report
      2009 Annual Research Report
  • [Presentation] 旗多様体とGelfand-Cetlin多面体2009

    • Author(s)
      野原雄一
    • Organizer
      第2回シューベルト・カルキュラスとその周辺
    • Place of Presentation
      倉敷シーサイドホテル,岡山県倉敷市
    • Year and Date
      2009-09-23
    • Related Report
      2010 Final Research Report
  • [Presentation] 旗多様体と Gelfand-Cetlin 多面体2009

    • Author(s)
      野原雄一
    • Organizer
      第2回シューベルト・カルキュラスとその周辺
    • Place of Presentation
      倉敷
    • Year and Date
      2009-09-23
    • Related Report
      2009 Annual Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      Y.Nohara
    • Organizer
      Third International Conference on Geometry and Quantization
    • Place of Presentation
      University of Luxembourg
    • Year and Date
      2009-09-10
    • Related Report
      2010 Final Research Report 2009 Self-evaluation Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      野原雄一
    • Organizer
      Third International Conference on Geometry and Quantization
    • Place of Presentation
      University of Luxembourg, ドイツ
    • Year and Date
      2009-09-10
    • Related Report
      2009 Annual Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      Y.Nohara
    • Organizer
      East Asian Symplectic Conference 2009, Academia Sinica
    • Place of Presentation
      台北
    • Year and Date
      2009-05-10
    • Related Report
      2010 Final Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      Y. Nohara
    • Organizer
      East Asian Symplectic Conference 2009
    • Place of Presentation
      Academia Sinica, 台北
    • Year and Date
      2009-05-10
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      野原雄一
    • Organizer
      East Asian Symplectic Conference 2009
    • Place of Presentation
      Academia Sinica, 台湾
    • Year and Date
      2009-05-10
    • Related Report
      2009 Annual Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      野原雄一
    • Organizer
      Fusion of Integrable Systems and Geometry
    • Place of Presentation
      東北大学
    • Year and Date
      2009-04-17
    • Related Report
      2010 Final Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      野原雄一
    • Organizer
      Fusion of Integrable Systems and Geometry
    • Place of Presentation
      仙台
    • Year and Date
      2009-04-17
    • Related Report
      2009 Annual Research Report
  • [Presentation] Lectures on Toric Degenerations2009

    • Author(s)
      西納武男, 野原雄一
    • Organizer
      Potential Functions and Gelfand-Cetlin systems
    • Place of Presentation
      KIAS, ソウル
    • Year and Date
      2009-03-17
    • Related Report
      2010 Final Research Report
  • [Presentation] Lectures on Toric Degenerations, Potential Functions, and Gelfand-Cetlin Systems2009

    • Author(s)
      西納武男、野原雄一
    • Organizer
      連続講義
    • Place of Presentation
      KIAS(韓国)
    • Year and Date
      2009-03-17
    • Related Report
      2008 Annual Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      Y.Nohara
    • Organizer
      The 1st GCOE International Symposium "Weaving Science Web beyond Particle-Matter Hierarchy"
    • Place of Presentation
      東北大学
    • Year and Date
      2009-03-07
    • Related Report
      2010 Final Research Report
  • [Presentation] Toric degenerations of Gelfand-Cetlin systems and potential functions2009

    • Author(s)
      野原雄一
    • Organizer
      The 1st GCOE International Symposium
    • Place of Presentation
      東北大学
    • Year and Date
      2009-03-07
    • Related Report
      2008 Annual Research Report
  • [Presentation] Gelfand-Cetlin系のトーリック退化とポテンシャル関数2008

    • Author(s)
      野原雄一
    • Organizer
      幾何学シンポジウム
    • Place of Presentation
      弘前大学
    • Year and Date
      2008-08-23
    • Related Report
      2010 Final Research Report
  • [Presentation] Gelfand-Cethn系のトーリック退化とポテンシャル関数2008

    • Author(s)
      野原雄一
    • Organizer
      幾何学シンポジウム
    • Place of Presentation
      弘前大学
    • Year and Date
      2008-08-23
    • Related Report
      2008 Annual Research Report
  • [Presentation] 旗多様体のトーリック退化とGelfand-Cetlin系2008

    • Author(s)
      野原雄一
    • Organizer
      Bergman核と代数幾何への応用
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-06-05
    • Related Report
      2010 Final Research Report
  • [Presentation] 旗多様体のトーリック退化とGelfand-Cetlin系2008

    • Author(s)
      野原雄一
    • Organizer
      Berglnan核と代数幾何への応用
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2008-06-05
    • Related Report
      2008 Annual Research Report
  • [Presentation] Toric degeneration of flag manifolds and the Gelfand-Cetlin system2008

    • Author(s)
      Y.Nohara
    • Organizer
      The 3rd Geometry Conference for Friendship of Japan and China
    • Place of Presentation
      名古屋大学
    • Year and Date
      2008-01-28
    • Related Report
      2010 Final Research Report 2007 Annual Research Report
  • [Presentation] 旗多様体のトーリック退化とGelfand-Cetlin系2008

    • Author(s)
      野原雄一
    • Organizer
      研究集会「接触幾何,特異点論およびその周辺」
    • Place of Presentation
      金沢大学サテライト・プラザ
    • Year and Date
      2008-01-24
    • Related Report
      2010 Final Research Report
  • [Presentation] 旗多様体のトーリック退化とGelfand-Cetlin系2008

    • Author(s)
      野原 雄一
    • Organizer
      研究集会「接触幾何, 特異点論およびその周辺」
    • Place of Presentation
      金沢大学サテライト・プラザ
    • Year and Date
      2008-01-24
    • Related Report
      2007 Annual Research Report
  • [Presentation] Projective embeddings and Lagrangian fibraions of Abelian varieties and Kummer varieties2007

    • Author(s)
      Y.Nohara
    • Organizer
      Conference on Symplectic geometry
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-07-24
    • Related Report
      2010 Final Research Report 2007 Annual Research Report
  • [Presentation] Projective embeddings and Lagrangian fibrations of Abelian varieties and Kummer varieties (Lecture course)2007

    • Author(s)
      Y.Nohara
    • Organizer
      International School-Conference GEOMETRY and QUANTIZATION, Steklov Mathematical Institute
    • Place of Presentation
      モスクワ,
    • Related Report
      2010 Final Research Report
  • [Presentation] Projective embeddings and Lagrangian fibraions of Abelian varieties and Kummer varieties (Lecture course)2007

    • Author(s)
      野原 雄一
    • Organizer
      International School-Conference GEOMETRY and QUANTIZATION
    • Place of Presentation
      Steklov Mathematical Institute
    • Related Report
      2007 Annual Research Report

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Published: 2007-04-01   Modified: 2016-04-21  

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