The geometry of complex symplectic varieties

2012 Fiscal Year Final Research Report

• 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
• 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

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] On the structure of homogeneous symplectic varieties of complete intersection (2012)
  • Author(s): Namikawa, Y
  • Journal Title: Invent. Math.
  • DOI: 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 Pages: 121-144
  • DOI: 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 Pages: 51-85
  • DOI: DOI:10.1215/00127094-2010-066
  • Peer Reviewed
• [Journal Article] Poisson deformations of affine symplectic varieties II (2010)
  • Author(s): Namikawa, Y
  • Journal Title: Kyoto J. Math Volume: Vol 50 Pages: 727-752
  • DOI: DOI:10.1215/0023608X-2010-012
  • 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
  • DOI: DOI:10.1016/j.aim.2009.05.00
  • 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] 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] Equivalence of symplectic singularities (2011)
  • Organizer: MMP and extremal rays
  • Place of Presentation: RIMS, Kyoto Univ
  • Year and Date: 2011-06-23
• [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] 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