The geometry of complex symplectic varieties

• Project/Area Number: 21340005
• 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: NAMIKAWA Yoshinori 京都大学, 大学院・理学研究科, 教授 (80228080)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥8,840,000 (Direct Cost: ¥6,800,000、Indirect Cost: ¥2,040,000); Fiscal Year 2012: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2011: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2010: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2009: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
• Keywords: 複素シンプレクティック多様体 / ポアソン変形 / 双有理幾何 / べき零多様体 / 接触構造 / シンプレクティック特異点 / べき零軌道 / 接触幾何 / Slodowy切片 / シンプレクティック超曲面 / スロードウィースライス / ポアソン幾何 / 特異点解消

Research Abstract

We have studied affine symplectic varieties and their crepant resolutions from the point of view of birational geometry and Poisson deformations.In particular, we proved that the Poisson deformations of affine symplectic varieties are unobstructed and we furthermore showed that those varieties have crepant resolutions if and only if they can be smoothed by Poisson deformations. We also gave a characterization of the nilpotent varieties of complex semisimple Lie algebras.

Research Products

• [Journal Article] Equivalences of symplectic singularities (2013)
  • Author(s): Namikawa, Y
  • Journal Title: J. Math Volume: Vol. 53, no.2
  • Peer Reviewed
• [Journal Article] Birational geometry for nilpotent orbits, in Handbook of Moduli vol III (2013)
  • Author(s): Namikawa, Y.
  • Journal Title: Advanced Lectures in Math Volume: Vol. 26 Pages: 1-38
  • Peer Reviewed
• [Journal Article] Birational geometry for nilpotent orbits (2013)
  • Author(s): Y. Namikawa
  • Journal Title: Handbook of Moduli, International Press Volume: volume III
  • Peer Reviewed
• [Journal Article] Equivalence of symplectic singularities (2013)
  • Author(s): Y. Namikawa
  • Journal Title: Kyoto J. Math. Volume: to appear
  • Peer Reviewed
• [Journal Article] Symplectic hypersurfaces (2013)
  • Author(s): M. Lehn, Y. Namikawa, C. Sorger, D. van Straten
  • Journal Title: Advanced Studies in Pure Mathematics Volume: to appear
  • 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
• [Journal Article] Slodowy slices and universal Poisson deformations (2012)
  • Author(s): Lehn, M., Namikawa, Y., Sorger, C.
  • Journal Title: Compositio Math Volume: Vol 148 Issue: 1 Pages: 121-144
  • DOI: 10.1112/s0010437x11005550
  • Peer Reviewed
• [Journal Article] Birational geometry and nilpotent orbits (2011)
  • Author(s): Namikawa,Y
  • Journal Title: Sugaku Exposition Volume: Vol. 24 Pages: 239-261
  • Peer Reviewed
• [Journal Article] Poisson deformations of affine symplectic varieties (2011)
  • Author(s): Namikawa, Y
  • Journal Title: Duke Math. J. Volume: Vol. 156 Issue: 1 Pages: 51-85
  • DOI: 10.1215/00127094-2010-066
  • Peer Reviewed
• [Journal Article] Birational algebraic geometry and nilpotent orbits (2011)
  • Author(s): Y.Namikawa
  • Journal Title: Sugaku Exposition Volume: 24 Pages: 239-262
  • Peer Reviewed
• [Journal Article] Poisson deformations of affine symplectic varieties II (2010)
  • Author(s): Namikawa, Y
  • Journal Title: Kyoto J. Math Volume: Vol 50 Issue: 4 Pages: 727-752
  • DOI: 10.1215/0023608x-2010-012
  • Peer Reviewed
• [Journal Article] Poisson deformations of affine symplectic varieties II (2010)
  • Author(s): Y.Namikawa
  • Journal Title: Kyoto Journal of Mathematics Volume: 50 Pages: 727-752
  • Peer Reviewed
• [Journal Article] Poisson deformations of affine symplectic varieties (2010)
  • Author(s): Y.Namikawa
  • Journal Title: Duke Mathematical Journal (掲載決定)
  • Peer Reviewed
• [Journal Article] Poisson deformations of affine symplectic varieties II (2010)
  • Author(s): Y.Namikawa
  • Journal Title: Kyoto Journal of Mathematics 50-(4)(掲載決定)
  • Peer Reviewed
• [Journal Article] Induced nilpotent orbits and birational geometry (2009)
  • Author(s): Namikawa, Y
  • Journal Title: Adv. Math Volume: Vol. 222 Pages: 547-564
  • Peer Reviewed
• [Journal Article] Induced nilpotent orbits and birational geometry (2009)
  • Author(s): Y.Namikawa
  • Journal Title: Advances in Mathematics 222 Pages: 547-564
  • Peer Reviewed
• [Journal Article] On symplectic hypersurfaces, to appear in Adv. Stud
  • Author(s): Lehn, M., Namikawa, Y., Sorger, C., van-Straten, D.
  • Journal Title: Pure Math
  • Peer Reviewed
• [Presentation] Symplectic varieties of complete intersection and contact geometry (2013)
  • Organizer: Workshop: Higher dimensional algebraic geometry
  • Place of Presentation: University of Tokyo, Tokyp, Japan
  • Year and Date: 2013-01-11
• [Presentation] Symplectic varieties of complete intersection and contact geometry (2012)
  • Organizer: Workshop: Hyper-kahler geometry, Simons Center for geometry and physics, Stony Brook
  • Place of Presentation: University, New York, USA
  • Year and Date: 2012-11-02
• [Presentation] Symplectic varieties of complete intersection and contact geometry (2012)
  • Organizer: 城崎代数幾何シンポジウム
  • Place of Presentation: 城崎大会議館, 兵庫県城崎町
  • Year and Date: 2012-10-26
• [Presentation] On the structure of homogeneous symplectic varieties of complete intersection (2012)
  • Organizer: Workshop: Singularities
  • Place of Presentation: Oberwolfach 数学研究所, Germany
  • Year and Date: 2012-09-27
• [Presentation] Symplectic varieties of complete intersection and contact geometry (2012)
  • Organizer: Workshop: Representation theory and symplectic algebraic geometry
  • Place of Presentation: CIRS, Luminy, France
  • Year and Date: 2012-07-09
• [Presentation] Symplectic varieties of complete intersection and contact geometry (2012)
  • Organizer: Workshop: Holomorphic symplectic manifolds and moduli spaces
  • Place of Presentation: Fourier Institute, Grenoble, France
  • Year and Date: 2012-06-12
• [Presentation] Fundamental groups of symplectic singularities (2012)
  • Organizer: Workshop on higher dimensional algebraic geometry
  • Place of Presentation: National Taiwan University, Taipei, Taiwan
  • Year and Date: 2012-03-25
• [Presentation] On the structure of homogeneous symplectic varieties of complete intersection (2012)
  • Organizer: Advances in hyperkaehler and holomorphic symplectic geometry
  • Place of Presentation: Banff, Canada
  • Year and Date: 2012-03-12
• [Presentation] On the structure of homogeneous symplectic varieties of complete intersection (2012)
  • Author(s): Y.Namikawa
  • Organizer: Adv in HK and holomorphic symplectic geometry
  • Place of Presentation: Banff, Canada
  • Year and Date: 2012-03-12
• [Presentation] Poisson deformations of affine symplectic varieties (2011)
  • Organizer: Doctorial Forum of Mathematics between Fudan and Kyoto
  • Place of Presentation: Fudan Univ. China
  • Year and Date: 2011-11-02
• [Presentation] Slodowy slices and universal Poisson deformations (2011)
  • Organizer: 特異点論とそのひろがり
  • Year and Date: 2011-08-25
• [Presentation] Slodowy slices and universal Poisson deformations (2011)
  • Author(s): Y.Namikawa
  • Organizer: 特異点とそのひろがり
  • Place of Presentation: Kyoto Univ.
  • Year and Date: 2011-08-25
• [Presentation] Equivalence of symplectic singularities (2011)
  • Organizer: MMP and extremal rays
  • Place of Presentation: RIMS, Kyoto Univ
  • Year and Date: 2011-06-23
• [Presentation] Equivalence of symplectic singularities (2011)
  • Author(s): Y.Namikawa
  • Organizer: MMP and extremal rays
  • Place of Presentation: RIMS, Kyoto Univ.
  • Year and Date: 2011-06-23
• [Presentation] Slodowy slices and universal Poisson deformations (2011)
  • Author(s): Y.Namikawa
  • Organizer: The 10-th workshop on nilpotent orbits and representation theory
  • Place of Presentation: Kyushu Univ
  • Year and Date: 2011-02-21
• [Presentation] Equivalence problems of symplectic varieties (2011)
  • Organizer: Derived categories 2011, Tokyo
  • Place of Presentation: Univ. of Tokyo
  • Year and Date: 2011-01-25
• [Presentation] Equivalence problems of symplectic varieties (2011)
  • Author(s): Y.Namikawa
  • Organizer: Derived categories 2011, Tokyo
  • Place of Presentation: Univ.Tokyo
  • Year and Date: 2011-01-25
• [Presentation] Slodowy slices and universal Poisson deformations (2011)
  • Author(s): Y.Namikawa
  • Organizer: Workshop on algebraic geometry, complex dynamic and their interaction
  • Place of Presentation: National University of Singapore
  • Year and Date: 2011-01-05
• [Presentation] Universal Poisson deformations and birational geometry (2010)
  • Author(s): Y.Namikawa
  • Organizer: Mini-workshop on holomorphic symplectic geometry
  • Place of Presentation: KIAS, Korea
  • Year and Date: 2010-09-29
• [Presentation] Poisson deformations of symplectic varieties and associated Galois covers (2010)
  • Author(s): Y.Namikawa
  • Organizer: Conferences on derived categories, symplectic geometry, birational geometry and deformation theory
  • Place of Presentation: Poicare Inst.Paris, France
  • Year and Date: 2010-05-20
• [Presentation] Poisson deformations of symplectic varieties (2009)
  • Organizer: workshop "Complex algebraic geometry
  • Place of Presentation: Oberwolfach 数学研究所,Germany
  • Year and Date: 2009-09-28
• [Presentation] Poisson deformations of symplectic varieties (2009)
  • Organizer: Singularities
  • Place of Presentation: Oberwolfach 数学研究所,Germany
  • Year and Date: 2009-09-22
• [Presentation] Poisson deformations and symplectic varieties (2009)
  • Organizer: PRIMA 2009
  • Place of Presentation: Sydney, Australia
  • Year and Date: 2009-07-06
• [Presentation] Symplectic varieties of complete intersection and contact geometry
  • Author(s): Y. Namikawa
  • Organizer: Holomorphic symplectic manifolds and moduli spaces
  • Place of Presentation: Grenoble, France
  • Invited
• [Presentation] Symplectic varieties of complete intersection and contact geometry
  • Author(s): Y. Namikawa
  • Organizer: Representation theory and symplectic algebraic geometry
  • Place of Presentation: Luminy, France
  • Invited
• [Presentation] On the structure of homogeneous symplectic varieties of complete intersection
  • Author(s): Y. Namikawa
  • Organizer: Singularities
  • Place of Presentation: Oberwolfach, Germany
  • Invited
• [Presentation] Symplectic varieties of complete intersection and contact geometry
  • Author(s): Y. Namikawa
  • Organizer: 城崎代数幾何シンポジウム
  • Place of Presentation: 兵庫県城崎町
  • Invited
• [Presentation] Symplectic varieties of complete intersection and contact geometry
  • Author(s): Y. Namikawa
  • Organizer: Hyper-Kahker geometry
  • Place of Presentation: Stony-Brook, USA
  • Invited
• [Presentation] Symplectic varieties of complete intersection and contact geometry
  • Author(s): Y. Namikawa
  • Organizer: Higher dimensional algebraic geometry
  • Place of Presentation: 東京大学
  • Invited
• [Presentation] Fundamental groups of symplectic singularities
  • Author(s): Y. Namikawa
  • Organizer: Workshop on higher dimensional algebraic geometry
  • Place of Presentation: Taipei, Taiwan
  • Invited