1994 Fiscal Year Annual Research Report

数学の諸分野にあらわれる特異点

Research Project

Project/Area Number	06640114
Research Institution	Tokyo Institute of Technology
Principal Investigator	福田拓生東京工業大学, 理学部, 教授 (00009599)
Co-Investigator(Kenkyū-buntansha)	増田一男東京工業大学, 理学部, 助教授 (20016158) 石井志保子東京工業大学, 理学部, 助教授 (60202933) 井上淳東京工業大学, 理学部, 教授 (40011613) 森田茂之東京工業大学, 理学部, 教授 (70011674) 岡睦雄東京工業大学, 理学部, 教授 (40011697)
Keywords	特異点 / 特異点解消 / 多様体 / Casson不変量 / 多重数種 / 特性類 / モジュライ空間 / 対数微分
Research Abstract	岡睦雄はパリ大学のLe教授との共同研究で、平面曲線の特異点解消グラフから決まるある不変量が特異点の解消に必要な最少のトーリック・モディフィケ-シオンと一致することを発見した。また、これは単独で、複素2次元平面の多項式写像の無限遠点から来る臨界地の解析とその評価式を与えた。石井志保子は、Q-factorial terminal modificationを用いて、非有理特異点を持つ3次元Fano多様体を分類した。またここ数年の、多重数種を用いた特異点の分類理論を紹介するサ-ヴェイを行った。森田茂之は3次元ホモロジー球面に対して定義されるCasson不変量が曲面束の第一特性類に同伴する二次不変量として解釈できることを示し、Casson不変量と曲面の写像類群の構造との間にある深い関係を明らかにした。また曲面の写像類群の構造と代数曲面のモジュライ空間のトポロジーに関するこれまでに知られた結果を概観し、これからの進むべき方向を提示するサ-ヴェイを行った。コンパクトケーラー多様体上の対数微分のd-閉性はP.Deligneにより証明されているが野口潤次郎は調和積分論を用いた簡明な解析的証明を与えた。また野口は値分布論の第二主要定理を関数体上の場合に確立し、応用として、双曲的多様体を定義する或る不定方程式の有理点の有限性を示し、さらに、代数体の場合にも、或る解の有限性を証明した。これはS.Langによる双曲的代数多様体の有理点の有限性予想を支えるものである。複素射影空間の次数の高い超平面およびその補空間は双曲的であろうという小林予想が1970年に出されたが、そのような超曲面の存在について複素射影空間の次元が3次元までしか分かっていなかった。増田一男と野口潤次郎は全ての次元についてそのような超曲面の存在を示した。

Research Products

(19 results)

All Other

All Publications (19 results)

[Publications] Le D.T.and M.Oka: "On resolution complexity of plane curves" Kodai Mathematical Journal. 18. 1-36 (1995)
[Publications] Mutsuo Oka: "Note on estimation of the number of the critical value of a polynomial mapping f:C^2 C^2" Kodai Math J.(Proceedding of Hanoi Workshop. 17. 409-420 (1994)
[Publications] Shihoko Ishii: "A Fano 3-fold with non-rational singularities and a two dimensional basis." Kodai Mathematical Journal. 17. 420-427 (1994)
[Publications] Shihoko Ishii: "On Fano 3-folds with non rational singularities and two dimensional basis" Abh.Math.Seminar Univ.Hamburg. 64. 249-277 (1994)
[Publications] Shihoko Ishii: "The weighted blowing ups of singularities with k=0" to appear in proceeding of the Conference of Singularities,Beijin.
[Publications] Shihoko Ishii: "Plurigenera for isolated singularities" To appear in Proceeding of Conference of Singularities,Trieste.
[Publications] Shigeyuki Morita: "Characteristic classes of surface bundles and Casson invariant" Sugaku Exposition. 7. 59-79 (1994)
[Publications] Shigeyuki Morita: "Problems on the structure of the mapping class group of surfaces and the topology of the moduli space ofcurv" Topology,Geometry and Field theory,World Scientific. 101-110 (1994)
[Publications] Junjiro Noguchi: "Some topics in Nevanlinna theory,hyperbolic manifolds and Diophantine Geometry" Geometry and Analysis on Complex Manifolds World Scientific Publ.140-145 (1994)
[Publications] Junjiro Noguchi: "A short analytic proof of closedness of logarithmic form" to appear in Kodai Math.Journal.
[Publications] Kazuo Masuda and J.Noguchi: "A construction of hyperbolic hypersurfaces of pn(C)" to apppear in Math.Ann.
[Publications] Junjiro Noguchi: "Nevanlinna-Cartan theory over function fields and a Diophantine equation" Selected Papers on Geometric Analysis. 97-119 (1994)
[Publications] Shin Ozawa: "Singular variation of domain and L boundedness of eigenfunction for some semi-linear elliptic equations" Proc.Japan Acad.70. 67-70 (1994)
[Publications] Reiko Miyaoka: "On complete minimal surfaces whose Gauss map misses two directions" Archiv der Mathematik. 63. 565-576 (1994)
[Publications] Takao Fujita: "Approximating Zariski decomposition of big line bundles" Kodai Math.J. 17. 1-3 (1994)
[Publications] Takao Fujita: "On Kodaira energy of polarized log varieties" to appear in J.Math.Soc.Japan.
[Publications] Takehiko Morita: "Periodic orbits of a dynamical system in a compound central field and a perturbed billiards system" Ergod.Th.& Dynam.Sys.,. 14. 599-619 (1994)
[Publications] Takehiko Morita: "One-dimensional markov maps and Selberg zeta functions associated with confine Fuchsian groups" Geometry and Analysis in Dynamical Systems. 161-173 (1994)
[Publications] 丹野修吉: "空間図形の幾何学" 培風館, 187 (1994)

1994 Fiscal Year Annual Research Report

数学の諸分野にあらわれる特異点

Principal Investigator

福田 拓生 東京工業大学, 理学部, 教授 (00009599)

Research Products

[Publications] Le D.T.and M.Oka: "On resolution complexity of plane curves" Kodai Mathematical Journal. 18. 1-36 (1995)

[Publications] Mutsuo Oka: "Note on estimation of the number of the critical value of a polynomial mapping f:C^2 C^2" Kodai Math J.(Proceedding of Hanoi Workshop. 17. 409-420 (1994)

[Publications] Shihoko Ishii: "A Fano 3-fold with non-rational singularities and a two dimensional basis." Kodai Mathematical Journal. 17. 420-427 (1994)

[Publications] Shihoko Ishii: "On Fano 3-folds with non rational singularities and two dimensional basis" Abh.Math.Seminar Univ.Hamburg. 64. 249-277 (1994)

[Publications] Shihoko Ishii: "The weighted blowing ups of singularities with k=0" to appear in proceeding of the Conference of Singularities,Beijin.

[Publications] Shihoko Ishii: "Plurigenera for isolated singularities" To appear in Proceeding of Conference of Singularities,Trieste.

[Publications] Shigeyuki Morita: "Characteristic classes of surface bundles and Casson invariant" Sugaku Exposition. 7. 59-79 (1994)

[Publications] Shigeyuki Morita: "Problems on the structure of the mapping class group of surfaces and the topology of the moduli space ofcurv" Topology,Geometry and Field theory,World Scientific. 101-110 (1994)

[Publications] Junjiro Noguchi: "Some topics in Nevanlinna theory,hyperbolic manifolds and Diophantine Geometry" Geometry and Analysis on Complex Manifolds World Scientific Publ.140-145 (1994)

[Publications] Junjiro Noguchi: "A short analytic proof of closedness of logarithmic form" to appear in Kodai Math.Journal.

[Publications] Kazuo Masuda and J.Noguchi: "A construction of hyperbolic hypersurfaces of pn(C)" to apppear in Math.Ann.

[Publications] Junjiro Noguchi: "Nevanlinna-Cartan theory over function fields and a Diophantine equation" Selected Papers on Geometric Analysis. 97-119 (1994)

[Publications] Shin Ozawa: "Singular variation of domain and L boundedness of eigenfunction for some semi-linear elliptic equations" Proc.Japan Acad.70. 67-70 (1994)

[Publications] Reiko Miyaoka: "On complete minimal surfaces whose Gauss map misses two directions" Archiv der Mathematik. 63. 565-576 (1994)

[Publications] Takao Fujita: "Approximating Zariski decomposition of big line bundles" Kodai Math.J. 17. 1-3 (1994)

[Publications] Takao Fujita: "On Kodaira energy of polarized log varieties" to appear in J.Math.Soc.Japan.

[Publications] Takehiko Morita: "Periodic orbits of a dynamical system in a compound central field and a perturbed billiards system" Ergod.Th.& Dynam.Sys.,. 14. 599-619 (1994)

[Publications] Takehiko Morita: "One-dimensional markov maps and Selberg zeta functions associated with confine Fuchsian groups" Geometry and Analysis in Dynamical Systems. 161-173 (1994)

[Publications] 丹野修吉: "空間図形の幾何学" 培風館, 187 (1994)

福田拓生東京工業大学, 理学部, 教授 (00009599)