2000 Fiscal Year Annual Research Report

退化からみた代数幾何学・Hodge理論・Log幾何学相互の関係の研究

Research Project

Project/Area Number	11304001
Research Institution	Osaka University
Principal Investigator	臼井三平大阪大学, 大学院・理学研究科, 教授 (90117002)
Co-Investigator(Kenkyū-buntansha)	向井茂名古屋大学, 大学院・多元数理学研究科, 教授 (80115641) 加藤和也東京大学, 大学院・数理科学研究科, 教授 (90111450) 中村郁北海道大学, 大学院・理学研究科, 教授 (50022687) 今野一宏大阪大学, 大学院・理学研究科, 助教授 (10186869) 森重文京都大学, 数理解析研究所, 教授 (00093328)
Keywords	Log Hodge構造 / 退化 / 周期写像 / モジュライのコンパクト化 / ベクトル束 / 一般型多様体 / 標準環 / symplectic多様体
Research Abstract	以下の研究集会を開催した: 国際シンポジュウム「Algebraic Geometry 2000」2000年7月20日〜30日,長野県安曇野. 組織委員:M.Green,L.Illusie,K.Kato,E.Looijenga,S.Mukai,S.Saito,S.Usui(Chief). トピック:Interaction of Algebraic Geometry,Hodge Theory and Logarithmic Geometry from the view point of Degenerations.海外からの参加者60名を含む150名の参加者が密度の濃い交流をし,本科研費を申請した第1の目標が達成できた. 研究集会「代数幾何学シンポジウム」2000年10月24日〜27日,兵庫県立城崎大会議館, 世話係:大野浩司,加藤文元.恒例の代数幾何学全分野の集会を成功裏に開催できた. ミニワークショップ「代数幾何学」2001年2月6日〜8日,兵庫県八千代町公民館, 世話係:臼井三平.密度の高い討議をした.また勤務時間外に地元の人達との交流をした. それぞれの研究集会の報告集を作成中である. 個々の研究実績・発表として主なものは次の通りである:臼井と加藤は共同研究により,Mumford等のトロイダルコンパクト化理論,Borel-Serreのコンパクト化理論,Griffithsの周期写像理論を一般化した.この成果は7月に国際シンポジュウム,8月にイタリアのトリエステの国際理論物理学研究センター,9月に韓国のソウルのKン国高等研究所で発表した.今野は非特異代数曲面上の曲線の標準環についての1-2-3定理を証明した.並河は特異symplectic多様体の変形理論を展開した.本科研費の代表者・分担者によるその他の成果については発表論文表参照.

Research Products

(27 results)

All Other

All Publications (27 results)

[Publications] Usui,S.: "Recovery of vanishing cycles by log geometry"Tohoku Math.J.. 53-1. (2001)
[Publications] Usui,S.: "Torelli type problems"Sugaku Expositions,Amer.Math.Soc.. 13. 17-38 (2000)
[Publications] Kato,K.and Usui,S.: "Borel-Serre spaces and spaces of SL (2) -orbits"Advanced Studies in Pure Math.. (under referee).
[Publications] Kato,K.,Matsubara,T.and Nakayama,C.: "Log C^∞-functions and degenerations of Hodge structures"Advanced Studies in Pure Math.. (under referee).
[Publications] 向井茂: "曲線と「曲面」をめぐって(数学の未解決問題17)"数理科学. 1月号. 56-64 (2000)
[Publications] Ishida,M.: "Intersection complexes of fans and toric varieties"math.AG/0102085.
[Publications] Konno,K.: "Certain algebraic surfaces with non-reduced moduli space"Portugal J.Math.. 57. 169-178 (2000)
[Publications] Konno,K.: "1-2-3 theorem for curves on algebraic surfaces"J.reine angew.Math.. (in press).
[Publications] Konno,K.: "Relations in canonical algebras on surfaces"
[Publications] Sato,E.: "Behavior of rational curves with minimal degree in projective space bundle in any characteristic"J.Math.Kyoto Univ.. (to appear).
[Publications] Sato,E.and Zhao,Y.: "Smoth 4-folds which contain a P′- bundle as an ample divisor"Manuscripta Math.. 101. 313-323 (2000)
[Publications] Mori,S.: "On a hyperplane section theorem of Gurjar"Math.Annalen. (to appear).
[Publications] Kollar,J.,Miyaoka,Y,Mori,S.,and Takagi,H.: "Boundedness of canonical Q-Fano 3-folds"Proc.Japan Acad.,Ser.A Math.Sci.. 76. 73-77 (2000)
[Publications] Fujino,O.and Mori,S.: "A canonical bundle formula"J.Diff.Geom.. (accepted).
[Publications] Sumihiro,H.and Tagami,S.: "A splitting theorem for rank two vector bundles on projective spaces in positive characteristic"Hiroshima Math.J.. (to appear).
[Publications] Saito,S.: "Infinitesimal logarithmic Torelli problem for degenerating hypersurfaces in P^n"Advanced Studies in Pure Math.. (under referee).
[Publications] Nakamura,I.: "Hilbert schemes of abelian group orbits"J.Alg.Geom.. (to appear).
[Publications] Gomi,Y.,Nakamura,I.and Shinoda,K.: "Hilbert schemes of G-orbits in dimension three"Asian J.Math.. 4. 1-20 (2000)
[Publications] Namikawa,Y.: "Calabi-Yaus and deformation theory"Sugaku Expositions, Amer.Math.Soc.. (to appear).
[Publications] Namikawa,Y.: "Global smoothing of Calabi-Yau threefolds II"Compsitio Math.. (to appear).
[Publications] Namikawa,Y.: "Deformation theory of singular symplectic n-folds"Math.Annalen. (to appear).
[Publications] Namikawa,Y.: "Extension of 2-forms and symplectic varieties"J.reine angew.Math.. (to appear).
[Publications] Namikawa,Y.: "A note on symplectic singularities"Math.AG/0101028.
[Publications] Namikawa,Y.: "Projectivity criterion of Moishezon spaces and density of projective symplectic varieties"Math.AG/0101019.
[Publications] 向井茂: "モジュライ理論II,現在数学の展開"岩波書店. 291 (2000)
[Publications] 石田正典: "トーリック多様体入門"朝倉書店(出版予定).
[Publications] 石田正典: "代数幾何学の基礎"培風館. (2000)

2000 Fiscal Year Annual Research Report

退化からみた代数幾何学・Hodge理論・Log幾何学相互の関係の研究

Principal Investigator

臼井 三平 大阪大学, 大学院・理学研究科, 教授 (90117002)

Research Products

[Publications] Usui,S.: "Recovery of vanishing cycles by log geometry"Tohoku Math.J.. 53-1. (2001)

[Publications] Usui,S.: "Torelli type problems"Sugaku Expositions,Amer.Math.Soc.. 13. 17-38 (2000)

[Publications] Kato,K.and Usui,S.: "Borel-Serre spaces and spaces of SL (2) -orbits"Advanced Studies in Pure Math.. (under referee).

[Publications] Kato,K.,Matsubara,T.and Nakayama,C.: "Log C^∞-functions and degenerations of Hodge structures"Advanced Studies in Pure Math.. (under referee).

[Publications] 向井茂: "曲線と「曲面」をめぐって(数学の未解決問題17)"数理科学. 1月号. 56-64 (2000)

[Publications] Ishida,M.: "Intersection complexes of fans and toric varieties"math.AG/0102085.

[Publications] Konno,K.: "Certain algebraic surfaces with non-reduced moduli space"Portugal J.Math.. 57. 169-178 (2000)

[Publications] Konno,K.: "1-2-3 theorem for curves on algebraic surfaces"J.reine angew.Math.. (in press).

[Publications] Konno,K.: "Relations in canonical algebras on surfaces"

[Publications] Sato,E.: "Behavior of rational curves with minimal degree in projective space bundle in any characteristic"J.Math.Kyoto Univ.. (to appear).

[Publications] Sato,E.and Zhao,Y.: "Smoth 4-folds which contain a P′- bundle as an ample divisor"Manuscripta Math.. 101. 313-323 (2000)

[Publications] Mori,S.: "On a hyperplane section theorem of Gurjar"Math.Annalen. (to appear).

[Publications] Kollar,J.,Miyaoka,Y,Mori,S.,and Takagi,H.: "Boundedness of canonical Q-Fano 3-folds"Proc.Japan Acad.,Ser.A Math.Sci.. 76. 73-77 (2000)

[Publications] Fujino,O.and Mori,S.: "A canonical bundle formula"J.Diff.Geom.. (accepted).

[Publications] Sumihiro,H.and Tagami,S.: "A splitting theorem for rank two vector bundles on projective spaces in positive characteristic"Hiroshima Math.J.. (to appear).

[Publications] Saito,S.: "Infinitesimal logarithmic Torelli problem for degenerating hypersurfaces in P^n"Advanced Studies in Pure Math.. (under referee).

[Publications] Nakamura,I.: "Hilbert schemes of abelian group orbits"J.Alg.Geom.. (to appear).

[Publications] Gomi,Y.,Nakamura,I.and Shinoda,K.: "Hilbert schemes of G-orbits in dimension three"Asian J.Math.. 4. 1-20 (2000)

[Publications] Namikawa,Y.: "Calabi-Yaus and deformation theory"Sugaku Expositions, Amer.Math.Soc.. (to appear).

[Publications] Namikawa,Y.: "Global smoothing of Calabi-Yau threefolds II"Compsitio Math.. (to appear).

[Publications] Namikawa,Y.: "Deformation theory of singular symplectic n-folds"Math.Annalen. (to appear).

[Publications] Namikawa,Y.: "Extension of 2-forms and symplectic varieties"J.reine angew.Math.. (to appear).

[Publications] Namikawa,Y.: "A note on symplectic singularities"Math.AG/0101028.

[Publications] Namikawa,Y.: "Projectivity criterion of Moishezon spaces and density of projective symplectic varieties"Math.AG/0101019.

[Publications] 向井茂: "モジュライ理論II,現在数学の展開"岩波書店. 291 (2000)

[Publications] 石田正典: "トーリック多様体入門"朝倉書店(出版予定).

[Publications] 石田正典: "代数幾何学の基礎"培風館. (2000)

臼井三平大阪大学, 大学院・理学研究科, 教授 (90117002)