Symplectic Algebraic Geometry

• Project/Area Number: 25287003
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: Namikawa Yoshinori 京都大学, 理学(系)研究科(研究院), 教授 (80228080)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥7,410,000 (Direct Cost: ¥5,700,000、Indirect Cost: ¥1,710,000); Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
• Keywords: シンプレクティック特異点 / ポアソン変形 / 双有理幾何 / ポアソン幾何 / 複素シンプレクティック幾何 / べき零軌道 / 複素接触幾何 / 複素シンプレクティック多様体 / 複素接触多様体

Outline of Final Research Achievements

An algebraic variety often has a singularity. Among them an important role plays a singularity that has a symplectic structure.We call it a symplectic singularity. A symplectic singularity is an important object which is closely related with hyperkahler geometry and geometric representation theory.In most cases a symplectic singularity shows up with a 1-dimensional torus action. Such a singularity is particularly called a conical symplectic singularity. We discovered that there is a close relationship between the universal Poisson deformation of a conical symplectic singularity and the birational geometry of its crepant resolution. We also started classifing conical symplectic singularities. Our main results contain a certain finiteness theorem for symplectic singularities and the characterization of a nilpotent orbit closure of complex semisimple Lie algebra.

Research Products

• [Journal Article] Fundamental groups of symplectic singularities (2017)
  • Author(s): Y. Namikawa
  • Journal Title: Adv. Stud. Pure Math. Volume: 74 Pages: 321-336
  • Peer Reviewed
• [Journal Article] A finiteness theorem on symplectic singularities (2016)
  • Author(s): 並河 良典
  • Journal Title: Compositio Math. Volume: 152 Issue: 6 Pages: 1225-1236
  • DOI: 10.1112/s0010437x16007387
  • Peer Reviewed / Open Access
• [Journal Article] On symplectic hypersurfaces (2016)
  • Author(s): M. Lehn, Y. Namikawa, C. Sorger, D. van Straten
  • Journal Title: Adv. Stud. Pure Math. Volume: 70 Pages: 277-298
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On symplectic hypersurfaces (2016)
  • Author(s): Y. Namikawa (joint work with M. Lehn, C. Sorger, Duco van Straten)
  • Journal Title: To appear in Adv. Stud. Pure Math. Volume: 未定
  • Peer Reviewed
• [Journal Article] Fundamental groups of symplectic singularities (2016)
  • Author(s): Y. Namikawa
  • Journal Title: To appear in Adv. Stud. Pure Math. Volume: 未定
  • Peer Reviewed
• [Journal Article] Poisson deformations and birational geometry (2015)
  • Author(s): Y. Namikawa
  • Journal Title: J. Math. Sci. Univ. Tokyo (小平１００周年記念号) Volume: 22 Pages: 339-359
  • Peer Reviewed
• [Journal Article] Poisson deformations and birational geometry (2015)
  • Author(s): Y. Namikawa
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: 22 Pages: 339-359
  • Peer Reviewed
• [Journal Article] Equivalence of symplectic varieties (2013)
  • Author(s): 並河良典
  • Journal Title: Kyoto Journal of Math. Volume: 53 Issue: 2 Pages: 483-514
  • DOI: 10.1215/21562261-2081270
  • Peer Reviewed
• [Journal Article] Birational geometry for nilpotent orbits (2013)
  • Author(s): Yoshinori Namikawa
  • Journal Title: Advanced Lectures in Mathematics Volume: 26 Pages: 1-38
  • Peer Reviewed
• [Journal Article] On the structure of homogeneous symplectic varieties of complete intersection (2012)
  • Author(s): Namikawa, Y
  • Journal Title: Invent. Math. Volume: to appear Issue: 1 Pages: 159-185
  • DOI: 10.1007/s00222-012-0424-1
  • Peer Reviewed
• [Presentation] A characterization of nilpotent orbit closures among symplectic singularities (2016)
  • Author(s): Y. Namikawa
  • Organizer: Algebraic Geometry and Integrable system
  • Place of Presentation: Kobe University
  • Year and Date: 2016-12-06
  • Int'l Joint Research / Invited
• [Presentation] A characterization of nilpotent cones of complex semisimple Lie algebras (2016)
  • Author(s): Y. Namikawa
  • Organizer: Higher dimensional algebraic geometry and around, Kobe-Kyoto
  • Place of Presentation: 神戸大学理学部
  • Year and Date: 2016-02-03
  • Int'l Joint Research / Invited
• [Presentation] A characterization of nilpotent orbit closures among symplectic singularities (2016)
  • Author(s): Y. Namikawa
  • Organizer: On explicit description of holomorphic synplectic varieties
  • Place of Presentation: Onuma International Seminar House, Japan
  • Int'l Joint Research / Invited
• [Presentation] A characterization of nilpotent orbit closures among symplectic singularities (2016)
  • Author(s): Y. Namikawa
  • Organizer: Singularities
  • Place of Presentation: Oberwolfach Math Inst. Germany
  • Int'l Joint Research / Invited
• [Presentation] A finiteness theorem for symplectic singularities (2015)
  • Author(s): Y. Namikawa
  • Organizer: 特異点と不変量
  • Place of Presentation: 京大数理解析研究所
  • Year and Date: 2015-12-18
  • Int'l Joint Research / Invited
• [Presentation] A finiteness theorem for symplectic singularities (2015)
  • Author(s): Y. Namikawa
  • Organizer: 21-th Symposium on Complex geometry
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2015-10-29
  • Int'l Joint Research / Invited
• [Presentation] A characterization of nilpotent cones of complex semisimple Lie algebras (2015)
  • Author(s): Y. Namikawa
  • Organizer: New techniques in geometric invariant theory
  • Place of Presentation: Freie Univ, Berlin, Germany
  • Year and Date: 2015-09-01
  • Int'l Joint Research / Invited
• [Presentation] A finiteness theorem for symplectic singularities (2015)
  • Author(s): Y. Namikawa
  • Organizer: Algebraic Geometry 2015, AMS summer institute
  • Place of Presentation: Univesity of Utah, Salt Lake City, USA
  • Year and Date: 2015-07-14
  • Int'l Joint Research / Invited
• [Presentation] A finiteness theorem for symplectic singularities (2015)
  • Author(s): Y. Namikawa
  • Organizer: Representation theory and geometry of symplectic resolutions
  • Place of Presentation: Northeastern Univ, Boston, USA
  • Year and Date: 2015-05-20
  • Int'l Joint Research / Invited
• [Presentation] べき零多様体の代数幾何学的特徴つけ (2014)
  • Author(s): 並河 良典
  • Organizer: 表現論シンポジウム
  • Place of Presentation: 夢海遊淡路島, 兵庫県, 洲本市
  • Year and Date: 2014-11-27
  • Invited
• [Presentation] A characterization of nilpotent varieties of complex semisimple Lie algebras (2014)
  • Author(s): Y. Namikawa
  • Organizer: Workshop "Complex geometry and Lie groups"
  • Place of Presentation: University of Torino, Italy
  • Year and Date: 2014-06-19
  • Invited
• [Presentation] Poisson deformations and birational geometry (2014)
  • Author(s): Y. Namikawa
  • Organizer: Workshop on Algebraic Geometry
  • Place of Presentation: Sichuan University, China
  • Year and Date: 2014-05-20
  • Invited
• [Presentation] べき零多様体の代数幾何学的特徴つけ (2014)
  • Author(s): 並河 良典
  • Organizer: 京都大学談話会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2014-05-07
  • Invited
• [Presentation] A characterization of nilpotent varieties of complex semisimple Lie algebras I, II, III, IV (2014)
  • Author(s): Y. Namikawa
  • Organizer: Lecture Series
  • Place of Presentation: KAIST, Daejon, Korea
  • Year and Date: 2014-04-29 – 2014-05-01
  • Invited
• [Presentation] Poisson deformations and Mori dream spaces (2014)
  • Author(s): Yoshinori Namikawa
  • Organizer: Pacific Rim Conference 2013
  • Place of Presentation: Dalian, China
  • Invited
• [Presentation] A characterization of the nilpotent cones of complex semisimple Lie algebras (2014)
  • Author(s): Yoshinori Namikawa
  • Organizer: Symposium on projective algebraic varieties and moduli
  • Place of Presentation: Seoul, Korea
  • Invited
• [Presentation] Poisson deformations and Mori dream spaces (2013)
  • Author(s): Yoshinori Namikawa
  • Organizer: Classifications of algebraic varieties and related topics
  • Place of Presentation: Cetraro, Italy
  • Invited
• [Presentation] symplectic varieties of complete intersection and contact geometry (2013)
  • Author(s): Yoshinori Namikawa
  • Organizer: Algebraic Geometry in East Asia
  • Place of Presentation: Beijing, China
  • Invited