Complex analytic surfaces and hyperkaeher manifolds

2008 Fiscal Year Final Research Report

• Project/Area Number: 18540016
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Keio University
• Principal Investigator: OGUISO Keiji Keio University, 経済学部, 教授 (40224133)
• Project Period (FY): 2006 – 2008
• Keywords: 超ケーラー多様体 / 双有理変換群 / モーデル・ヴェイユ群 / ラグランジアンファイバー空間 / カラビ・ヤウ多様体

Research Abstract

研究成果の概要 : 複素解析曲面と超ケーラー多様体に見られる著しい類似性に着目することにより、ケーラー多様体の双有理型変換群の粗分類定理(Tits型定理)を確立した。また、分類に現れる非可換自由群および最大階数のアーベル群のラグランジアンファイブレーションを用いた構成法を確立した。また、自己同型の複素力学的意味, ラグランジアンファイブレーションの一般特異ファイバーの分類, 超ケーラー多様体の代数次元、特に4次元の場合の完全な記述を与えた。以上の研究に加えて、滑らかな小変形を有せず、互いに双有理であるが射影平坦変形でつながりしかも同相ではない3次元多様体カラビ・ヤウ多様体の存在を発見した。以上の研究が評価され日本数学会代数学賞を受賞した。

Research Products

• [Journal Article] A remark on dynamical degrees of automorphisms of hyperk\"ahler manifold (2009)
  • Author(s): Keiji Oguiso
  • Journal Title: to appear in Manuscripta Math
• [Journal Article] Picard number of the generic fiber of an abelian fibered hyperkaehler manifold to appear in Math (2008)
  • Author(s): Keiji Oguiso
  • Journal Title: Annalen
• [Journal Article] Connecting certain rigid birational non-homeomorphic Calabi-Yau threefolds via Hilbert scheme to appear in Comm (2008)
  • Author(s): Nam-Hoon Lee
  • Journal Title: Analysis and Geometry
• [Journal Article] Conjecture of Tits type for complex varieties and Theorem of Lie-Kolchin type for a cone", to appear in Math (2008)
  • Author(s): JongHae Keum, Keiji Oguiso, De-Qi Zhang
  • Journal Title: Research Letters
• [Journal Article] Mordell-Weil groups of a hyperk\"ahler manifold-a question of F. Campana", Publ (2008)
  • Author(s): K. Oguiso
  • Journal Title: Publ. RIMS(special issue for Professor Heisuke Hironaka's 77-th birthday) {\bf 44} Pages: 95-506
  • Peer Reviewed
• [Journal Article] Bimeromorphic automor phism groups of non-projective hyper k\"ahlar manifolds-a note inspired by C (2008)
  • Author(s): K.Oguiso
  • Journal Title: T. McMullen, J. Differential Geometry 78 Pages: 163-191
  • Peer Reviewed
• [Journal Article] Shioda-Tate formula for an abelian fibered variety and applications", to appear in Korean J (2007)
  • Author(s): K. Oguiso
  • Journal Title: Math
• [Journal Article] Salem多項式と超ケーラー多様体の双有理型変換群 (2007)
  • Author(s): 小木曽啓示
  • Journal Title: 数学 59巻 Pages: 1-23
  • Peer Reviewed
• [Journal Article] Groups of automorphisms of null-entropy of hyper k\"ahler manifolds, Contemp (2007)
  • Author(s): K.Oguiso
  • Journal Title: Math 422 Pages: 173-185
  • Peer Reviewed
• [Journal Article] Tits alternative in hyperk\"ahler manifolds, Math (2006)
  • Author(s): K. Oguiso
  • Journal Title: Research Letters 13 Pages: 307-316
  • Peer Reviewed
• [Presentation] 日本数学会2009年度代数学賞受賞講演 (2009)
  • Author(s): Keiji Oguiso
  • Organizer: Connecting certain rigid birational non-homeomorphic Calabi-Yau threefolds via Hilbert scheme
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-28
• [Presentation] Enriques manifolds (2009)
  • Author(s): Keiji Oguiso
  • Organizer: Arithmetic and Algebraic Geometry Related to Moduli Spaces
  • Place of Presentation: 東京大学
  • Year and Date: 2009-01-22
• [Presentation] Fano manifolds and Lagrangian fibration of symplectic manifolds (2008)
  • Author(s): Keiji Oguiso
  • Organizer: 31st Autumn School in Algebraic Geometry
  • Place of Presentation: Lukecin, Poland(seven 90 minute lectures)
  • Year and Date: 20080831-0906
• [Presentation] Non-algebraic hyperk\"ahler manifold (2008)
  • Author(s): Keiji Oguiso
  • Organizer: 平成20年度多変数関数論冬セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2008-12-26
• [Presentation] Mordell-Weil groups of abelian fibered hyperkaehler manifolds and O'Grady's 10-dimensional examples (2008)
  • Author(s): Keiji Oguiso
  • Organizer: Coe-Cow in Tokyo
  • Place of Presentation: 東京大学
  • Year and Date: 2008-12-18
• [Presentation] Non-algebraic hyperk\"ahler manifolds (2008)
  • Author(s): Keiji Oguiso
  • Organizer: Komplexe Analysis
  • Place of Presentation: Oberwolfach, Germany
  • Year and Date: 2008-08-29
• [Presentation] Algebraic reductions of hyperkaehler manifolds (2008)
  • Author(s): Keiji Oguiso
  • Organizer: 第53回代数学シンポジウム
  • Place of Presentation: 盛岡市
  • Year and Date: 2008-08-06
• [Presentation] KIAS School on Holomorphic Symplectic Geometry (2007)
  • Author(s): Keiji Oguiso
  • Organizer: Phoenix Park(Pyeongchang)
  • Place of Presentation: Korea(five 2-hour lectures)
  • Year and Date: 20070205-09
• [Presentation] Connecting certain rigid birational non-homeomorphic Calabi-Yau threefolds via Hilbert scheme (2007)
  • Author(s): Keiji Oguiso
  • Organizer: Fourth Pacific Rim Conference on Mathematics
  • Place of Presentation: Hong-Kong
  • Year and Date: 2007-12-08
• [Presentation] On non-algebraic hyperk\"ahler manifolds" (2007)
  • Author(s): Keiji Oguiso
  • Organizer: Complex Geometry
  • Place of Presentation: in Osaka
  • Year and Date: 2007-11-14
• [Presentation] Mordell-Weil group of an abelian fibered variety and its application to hyperk\"ahler manifolds (2007)
  • Author(s): Keiji Oguiso
  • Organizer: Algebraic Geometry Symposium
  • Place of Presentation: 城崎
  • Year and Date: 2007-10-23
• [Presentation] Mordell-Weil group of an abelian fibered variety and its application to hyperk\"ahler manifolds (2007)
  • Author(s): Keiji Oguiso
  • Organizer: Komplexe Algebraische Geometrie
  • Place of Presentation: Oberwolfach, Germany
  • Year and Date: 2007-10-01
• [Presentation] General singular fibers of holomorphic Lagrangian fibration-a joint work with Jun-Muk Hwang (2007)
  • Author(s): Keiji Oguiso
  • Organizer: Pacific Rim Complex Geometry Conference
  • Place of Presentation: Kobe, Japan
  • Year and Date: 2007-08-08
• [Presentation] Connecting certain rigid birational non-homeomorphic Calabi-Yau threefolds via Hilbert scheme" (2007)
  • Author(s): Keiji Oguiso
  • Organizer: Hayama Symposium on Complex Analysis in several variables
  • Place of Presentation: Hayama, Japan
  • Year and Date: 2007-07-16
• [Remarks] (1) Algebraic Geometry in East Asia, KIAS Seoul, Korea, November 11-15, 2008(with Shigeyuki Kondo, Kazuhiro Konno, JongHae Keum, Sijong Kwak, Namhoon Lee)
• [Remarks] (2) Kinosaki Symposium on Algebraic Geometry", Kinosaki, Hyogo, October 20-24 2008(with Toshiaki Maeno, Masahiko Yoshinaga)
• [Remarks] (3) Conference on Arithmetic and algebraic geometry", University of Tokyo, July 3-6 2008(with Tomohide Terasoma, Masahiko Saito, Yuji Shimizu)
• [Remarks] (4) UK-Japan Winter Scool 2008-Algebraic and Symplectic Geometry, Mathematics Research Centre, University of Warwick, 7-11 January 2008(with Jurgen Berndt, Jhon Bolton, Martin Guest, Yoshiaki Maeda, Mile Reid)
• [Remarks] (5) Birational Automorphisms of Compact Complex Manifold and Dynamical System, Nagoya, Japan, August 2007 (with Shigeyuki Kondo and Shigeru Mukai)
• [Remarks] (6) Higher dimensional Algebraic Geometry Echigo Yuzawa, Yuzawa, Japan, December 2006(with Yujiro Kawamata and Hiromichi Takagi)
• [Remarks] (7) Workshop on Derived Categories of Coherent Sheaves", KIAS, Korea, May 2006(with Bumsig Kim and Yujiro Kawamata)