広義カラビ・ヤウ多様体と有理多様体の複素力学系的観点からの研究

2010 Fiscal Year Annual Research Report

• Project/Area Number: 22340009
• Research Category: Grant-in-Aid for Scientific Research (B)
• Research Institution: Osaka University
• Principal Investigator: 小木曽 啓示 大阪大学, 理学研究科, 教授 (40224133)
• Co-Investigator(Kenkyū-buntansha): 今野 一宏 大阪大学, 理学研究科, 教授 (10186869) ; 臼井 三平 大阪大学, 理学研究科, 教授 (90117002) ; 藤木 明 大阪大学, 理学研究科, 教授 (80027383) ; 後藤 竜司 大阪大学, 理学研究科, 准教授 (30252571) ; 川口 周 大阪大学, 理学研究科, 准教授 (20324600)
• Keywords: カラビ・ヤウ多様体 / 超ケーラー多様体 / エントロピー / 導来圏 / 超単純多様体 / コホモロジカル関手

Research Abstract

次元が2以上で真の固有解析的部分多様体を有しないコンパクト複素多様体のことを超単純多様体(very simple manifold)と命名し,その連接層の有界導来圏上の有限なコホモロジカル関手には表現可能でないものが必ず存在することを明示的に示した。この結果はBondal-Van den Bergh氏の射影代数多様体の場合の表現可能性と対極的な結果であり,また氏らによる曲面の場合の結果の一般化である。結果は,論文"K.Oguiso,Bounded derived categories of very simple manifolds"(掲載決定済)にまとめ,国際研究集会Complex Geometry,Group actions and Moduli spaces(satellite conference to ICM 2010),GCOE Conference,Derived Categories 2011で招待講演した。
Schroeer氏との共著Enriques manifolds(J.Reine Angew.Math.に掲載確定済)により,エンリケス曲面の点のヒルベルトスキームの普遍被覆多様体はカラビ・ヤウ多様体である。エンリケス曲面が十分に一般であれば,このカラビ・ヤウ多様体の自己同型群は位数2のアーベル群3つの自由積を含み,特にその一般元は正のエントロピーをもつことを発見した。正のエントロピーをもった正則自己同型を許容する任意偶数次元のカラビ・ヤウ多様体の存在を示す初めての例であり,その構成も系統的である。この結果と超ケーラー多様体に対する類似の結果を現在論文にまとめている。これらの結果及び先行する結果については,国際研究集会Workshop on Lattices,Reflection Groups and Algebraic Geometry,Research on Complex Dynamics and Related Fields,Workshop on Algebraic Geometry,Complex Dynamics and their Interaction等でも招待講演し,力学系の研究者からも興味をもたれた。

Research Products

• [Journal Article] Enriques manifolds (2011)
  • Author(s): K.Oguiso, S.Schroeer
  • Journal Title: J.Reine Angew.Math. Volume: 661 Pages: 215-235
  • Peer Reviewed
• [Journal Article] Periods of Enriques Manifolds (2011)
  • Author(s): K.Oguiso, S.Schroeer
  • Journal Title: Pure Appl.Math.Q, special issue dedicated to the memory of Eckart Viehweg Volume: (掲載確定済)
  • Peer Reviewed
• [Journal Article] K3 surfaces with order 11 automorphisms (2011)
  • Author(s): K.Oguiso, D.-Q.Zhang
  • Journal Title: Pure Appl.Math.Q, special issue dedicated to the memory of Eckart Viehweg Volume: (掲載確定済)
  • Peer Reviewed
• [Journal Article] A simple remark on a flat projective morphism with a Calabi-Yau fiber (2011)
  • Author(s): K.Oguiso
  • Journal Title: Science of China Series A : Mathematics, special issue of Professor Catanese's 60th birthday Volume: (掲載確定済)
  • Peer Reviewed
• [Journal Article] Bounded derived categories of very simple manifolds (2011)
  • Author(s): K.Oguiso
  • Journal Title: Science of China Series A : Mathematics, special issue of Professor Catanese's 60th birthday Volume: (掲載確定済)
  • Peer Reviewed
• [Journal Article] The third smallest Salem number in automorphisms of K3 surfaces (2010)
  • Author(s): K.Oguiso
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 60 Pages: 331-360
  • Peer Reviewed
• [Journal Article] Non-algebraic hyperkaehler manifolds (2010)
  • Author(s): F.Campana, K.Oguiso, T.Peternell
  • Journal Title: Journal of Differential Geometry Volume: 85 Pages: 397-424
  • Peer Reviewed
• [Journal Article] On volume preserving complex structures on real tori (2010)
  • Author(s): F.Catanese, K.Oguiso, Th.Peternell
  • Journal Title: Kyoto Journal of Mathematics, memorial issue for the Late Professor Masayoshi Nagata Volume: 50 Pages: 753-775
  • Peer Reviewed
• [Presentation] Classification of general singular fibers of proper holomorphic Lagrangian fibrations via characteristic curves (2011)
  • Author(s): K.Oguiso
  • Organizer: UK-Japan Winter School New Methods in Geometry
  • Place of Presentation: King's College London, UK(招待基調講演)
  • Year and Date: 20110111-20110113
• [Presentation] Bounded derived categories of very simple manifolds (2011)
  • Author(s): K.Oguiso
  • Organizer: GCOE Conference, Derived Categories 2011
  • Place of Presentation: 東京大学(招待講演)
  • Year and Date: 2011-01-25
• [Presentation] On Enriques manifolds (2011)
  • Author(s): K.Oguiso
  • Organizer: Workshop on Algebraic Geometry, Complex Dynamics and their Interaction
  • Place of Presentation: National University of Singapore, Singapore(招待講演)
  • Year and Date: 2011-01-06
• [Presentation] Entropy of automorphisms of compact hyperkaeler manifolds (2010)
  • Author(s): K.Oguiso
  • Organizer: Research on Complex Dynamics and Related Fields (Celebrating Professor Shigehiro Ushiki's 60th birthday)
  • Place of Presentation: 京都大学(招待講演)
  • Year and Date: 2010-12-08
• [Presentation] Classifications and constructions of general singular fibers of proper holomorphic Lagrangian fibrations (2010)
  • Author(s): K.Oguiso
  • Organizer: Miniworkshop on holomorphic symplectic geometry
  • Place of Presentation: KIAS Seoul, Korea(招待講演)
  • Year and Date: 2010-11-26
• [Presentation] Periods of Enriques manifolds (2010)
  • Author(s): K.Oguiso
  • Organizer: Workshop on Lattices, Reflection Groups and Algebraic Geometry
  • Place of Presentation: 名古屋大学(招待講演)
  • Year and Date: 2010-11-25
• [Presentation] Classifications and constructions of general singular fibers of proper holomorphic Lagrangian fibrations (2010)
  • Author(s): K.Oguiso
  • Organizer: Perspectives on Algebraic Varieties (Celebrating Professors Fabrizio Catanese, Ciro Ciliberto and Alessandro Verra on the occasion of their 60th birthday)
  • Place of Presentation: Levico Terme, Trento, Italy.(招待講演)
  • Year and Date: 2010-09-11
• [Presentation] Bounded derived categories of very simple manifolds (2010)
  • Author(s): K.Oguiso
  • Organizer: Complex Geometry, Group actions and Moduli spaces (satellite conference to ICM 2010)
  • Place of Presentation: Hyderabad, India(招待講演)
  • Year and Date: 2010-08-15