• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Radon transformation in Nevanlinina theory and Diophantire approximation

Research Project

Project/Area Number 09304007
Research Category

Grant-in-Aid for Scientific Research (A).

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNagoya University

Principal Investigator

KOBAYASHI Ryoichi  Graduate School of Mathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (20162034)

Co-Investigator(Kenkyū-buntansha) NOGUCHI Junjiro  Graduate School of Mathematical Science, Nagoya University, Professor, 大学院・数理科学研究科, 教授 (20033920)
SATO Takeshi  Graduate School of Mathematics, Nagoya University, Assistant Professor, 大学院・多元数理科学研究科, 助手 (60252219)
OHSAWA Takeo  Graduate School of Mathematics, Nagoya University, Professor, 大学院・多元数理科学研究科, 教授 (30115802)
向井 茂  名古屋大学, 大学院・多元数理科学研究科, 教授 (80115641)
金井 雅彦  名古屋大学, 大学院・多元数理科学研究科, 教授 (70183035)
佐藤 肇  名古屋大学, 大学院・多元数理科学研究科, 教授 (30011612)
翁 林  名古屋大学, 大学院・多元数理科学研究科, 助教授 (60304002)
名和 範人  名古屋大学, 大学院・多元数理科学研究科, 助教授 (90218066)
藤原 一宏  名古屋大学, 大学院・多元数理科学研究科, 助教授 (00229064)
橋本 光靖  名古屋大学, 大学院・多元数理科学研究科, 助教授 (10208465)
内藤 久資  名古屋大学, 大学院・多元数理科学研究科, 助教授 (40211411)
三宅 正武  名古屋大学, 大学院・多元数理科学研究科, 教授 (70019496)
Project Period (FY) 1997 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥28,000,000 (Direct Cost: ¥28,000,000)
Fiscal Year 2000: ¥4,800,000 (Direct Cost: ¥4,800,000)
Fiscal Year 1999: ¥8,100,000 (Direct Cost: ¥8,100,000)
Fiscal Year 1998: ¥5,100,000 (Direct Cost: ¥5,100,000)
Fiscal Year 1997: ¥10,000,000 (Direct Cost: ¥10,000,000)
KeywordsNevanlinna theory / Diophantine geometry / Lemma on logauthmic derivative / integral geometry / Radon transformation / ディオファンタス幾何学 / 値分布論 / チェビシエフ不等式 / リッチフラットケーラー計量 / ソボレフ不等式 / K3曲面 / Ricci-flat K“ahler計量 / hyper-Kahler構造 / 対称性 / Weighted Scholev不等式
Research Abstract

The second main conjective is the ultimate purpose of the Nevanlinna theory. In this research, I proposed several ideas of the "conjectural geometry" unifying Nevanlinna theory and Diophantive approximation through the attempt toward the second main conjecture. The analogy between Castan-Ahlfors-Weyl theory of holomorphic curves in Pn and the sulpace theorem of Schmidt on Diophantino approximation on projective spaces is ohsewed not only in their statements but also in their profs. The essence of the analogy lies in the idea of the "Radon-transformation" in integral geomeyty, In this research, I was able to establish a kind of "Radon-transform" which lies behind both Nevanlinna theory and Diophantine approximation. One advantage of this "Radon-transform" is the following : through this transformation we will be able to separate the part of the Diophantine geometry which is completely analogous to the part of Nevanlinna theory in which the lemma on logarithmic derivative plays an essent … More ial role. This part in Diophantine geometry is most number-theoretic and the theorems of Diophantine approximation of Roth-Schmidt-Faltings type, as well as Mihkowski-Bombieri-Vasler "geometry of numbers" play the most enential role. This is most different from holomorphic case, but never the less the result is completely analogous to the lemnia on loganithmic derivature. On the other hand, there was a big progress on the attempt toward 2nd main conjecture. Namely I discovered the geometric framework of n independent commutative holomorphic vector fields and the singularity caused by introducing such vector fields on "noncommutative" projective varieties. The wronskian formalism is possible in terms of these n commutative vector fields. As a result, the difficulty is localized in "singularities" and I was able to establish a new lemma on loganthmic derivative ("projective version") to analyze holomorphic curves in the presence of "singularities". I hope this will provide a new method in complex algebraic geometry. Less

Report

(5 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (36 results)

All Other

All Publications (36 results)

  • [Publications] R.Kobayashi: "Holomorphic cures in Abelian varieties : the 2^<nd> main theorem and applications;"Japanese J.Math. 26-1. 129-152 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi and Y.Itokawa: "Minimizing currents in open manifolds and the n-1 homology of non-negatively Ricci curved manifolds"Amer.J.Math. 121. 1253-1278 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi and M.Henmi: "Hooke's law in statistical manifolds and divergences"Nagoya Math.J.. 159. 1-24 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi: "Nevanlima's lemma on-logarithmic derivative and integral geometry"Nagoya Math.J.. (発表予定). 10 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi: "Methods of integral geometry in Nevanlima theory, lemma on logarithmic derivative and a program toword the 2^<nd> main theorem"Sugaku Exp.AMS. (発表予定). 50 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 小林亮一: "リッチフラットケーラー計量の幾何学と解析学"培風館(発表予定). 300

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 小林亮一: "ネヴァンリンナ理論における積分幾何的方法と対数微分の補題"森脇淳 編「ロジックと代数幾何」収録. 50 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi and Y.Itobawa: "Minimizing currents in open manifolds and the n-1 homology of non-negutudy Ricci curved manifolds"Amer.J.Math. 121. 1253-1278 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi: "Holomorphic curves in Abelian varieties the 2nd main theorem and applications"Japanese J.Math. 26-1. 129-152 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi and M.Henmi: "Hooke's law in statislical manifolds and divergences"Nagoya Math.J. 159. 1-24 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi: "Nevanlinna's lemma on loganithmic derivative and integral geometry"Nagoya Math.J.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi: "Mehtods of integral geometry in Nevanlinna theory, lemma on logarithmic derivative and a program toward the 2nd main theorem"Sugaku Exp.AMS. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi: "Methods of integral geometry in Nevanlinna theory and lemma on loganithmic derivative"Proc.Symp "Logic and algebraic Geometry"ed. A.Moriwaki. (in Japanese).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi: "Geometry and Analysin of Ricci-flat Kuhlor metrics"Baifukan publ (to appear)(English traslation will appear in vieweg pull.)(in Japanese).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] R.Kobayashi: "Holomorphic curves in Abelian varieties : the 2nd main theorem and applications:"Japanese J.Math. 26-1. 129-152 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Kobayashi and Y,Itokawa: "Minimizing currents in open manifolds and the n-1 homology of non-negatively Ricci curved manifolds"Amer.J.Math. 121. 1253-1278 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Kobayashi and M.Hemmi: "Hooke's law in statistical manifolds and divergences "Nagoya Math.J.. 159. 1-24 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Kobayashi: "Nevanlinna's lemma on logarithmic derivative and integral geometry"Nagoya Math.J.. 発表予定. 10 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Kobayashi: "Methods of integral geametry in Nevanlinna theory, lemma on logarithmic derivative and a program toward the 2nd main theorem"Sugaku Exp.AMS.. (発表予定). 50 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 小林亮一: "リッチフラットケーラー計量の幾何学と解析学"培風館(発表予定). 300

    • Related Report
      2000 Annual Research Report
  • [Publications] 小林亮一: "ネヴァンリンナ理論における積分幾何的方法と対数微分の補題"森脇淳編「ロジックと代数幾何」収録. 50 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y. Itokawa & R. Kobayashi: "Minimizing Currents in Open Manifolds and the n-1 Homology of Nonnegatively Ricci Curved Manifolds"Amer. J. Math. 121. 1253-1278 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] R. Kobayashi: "Holomorphic Curves in Abelian Varieties: the Second Main Theorem and Applications"Japanese J. Math. 26-1. 1-22 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] M. Henmi & R. Kobayashi: "Hooke's Laio in Statistical Manifolds and Divergences"Nagoya Malk. J.. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] H. Azad & R. Kobayashi: "Ricci-Flat Kahler Metrics on Symmetric Varieties"Nagoya Math. J.. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] 小林亮一: "Ricci-Flat Kahler 計量の幾何学と解析学"培風館(出版予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Kobayashi: "Geometry and Analysis of Ricci-Flat Kahler Metrics"Vieweg Verlag(出版予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] 小林 亮一・糸川 銚: "Mimimizing currents in open manifolds and the n-1 homology of non-negatively ricci curved manifolds" American Journal of Mathematics. (発表予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] 小林 亮一: "Holomorphic curves in Abelion varieties" Japanese Journal of Mathematics. (発表予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] 小林亮一・Hassen Azad: "Bifurcation of Kd'hler potentials and Ricci-flat Kd'hler metrics on symmetric varieties" Nagoya Math.Journal. (発表予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] 小林 亮一: "リッチフラットケーラー計量の幾何学と解析学" 培風館(発表予定),

    • Related Report
      1998 Annual Research Report
  • [Publications] 小林亮一: "Blaschke予想への複素解析的アプローチ" 京大数理研請究録. (発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Anattempt towas constracting Rici-flat cndefinite Kahler matrio" Proc.Japan Acad.(発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Weighted Sobolevinequalities and Rcu-flat Kahler metrics on symation" Comm.Geometry and Analysis. (発表予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Qusai-Petentialo and Kahler-Einstein motrics on flay vanete." J.of algebra. 196. 620-629 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 小林亮一: "リッチ平坦多様体の幾何学" 培風館. (出版予定).

    • Related Report
      1997 Annual Research Report

URL: 

Published: 1997-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi