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

Toward discretization of Nevanlinna theory

Research Project

Project/Area Number 13304003
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  Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (20162034)

Co-Investigator(Kenkyū-buntansha) KANAI Masahiko  Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70183035)
KIMURA Yoshifumi  Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (70169944)
NAYATANI Shin  Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (70222180)
谷川 好男  名古屋大学, 大学院・多元数理科学研究科, 助教授 (50109261)
金銅 誠之  名古屋大学, 大学院・多元数理科学研究科, 教授 (50186847)
浪川 幸彦  名古屋大学, 大学院・多元数理科学研究科, 教授 (20022676)
山ノ井 克俊  京都大学, 数理解析研究所, 助手 (40335295)
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)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥44,330,000 (Direct Cost: ¥34,100,000、Indirect Cost: ¥10,230,000)
Fiscal Year 2004: ¥10,140,000 (Direct Cost: ¥7,800,000、Indirect Cost: ¥2,340,000)
Fiscal Year 2003: ¥10,660,000 (Direct Cost: ¥8,200,000、Indirect Cost: ¥2,460,000)
Fiscal Year 2002: ¥10,530,000 (Direct Cost: ¥8,100,000、Indirect Cost: ¥2,430,000)
Fiscal Year 2001: ¥13,000,000 (Direct Cost: ¥10,000,000、Indirect Cost: ¥3,000,000)
KeywordsNevanlinna theory / Diophantine approximation / Diophantine approximation Diophantos / Neyanlinna analogue / ramification counting function / abc conjecture / Voita's dictionary / lemma On derivative / Diophantine analogue of derivative / Diophantros近似論 / 正則曲線 / 交点理論 / Diophantus解析 / 数の幾何学 / 正則写像の値分布論 / Radon変換 / height関数 / 対数微分の補題 / Diophantos類似 / Gauss写像 / 積分幾何学 / ディオファントス幾何学 / 値分布論 / Vojta予想 / 第2主要予想
Research Abstract

Nevanlinna theory is a mathematics which is based on classicical calculus. However, this theory has aspects of those Mathematics such as statistical mechanics or arithmetic geometry and this makes the application of classical Differential geometry a difficult issue. More precisely, one can apply differential geometry only after one is successful in putting our problem in a good form by making best use of its statistical or arithmetic nature. Why does Nevablinna theory have such a nature? This question motivated my research project. I aimed at constructing background geometry explaining the origin of such nature of Nevanlinna theory. The guiding principle of my study came from statistical mechanics and arithmetic geometry (Arakelov geometry). Vojta proposed the so called Vojta's dictionary between Nevanlinna theory and Diophantine approximation. The set of rational points of projective varieties defined over a number field is considered to be the Diophantine analogue of transcendental h … More olomorphic curves into the complex variety consisting of its complex points. In my project I asked the question "What is the Diophantine analogue of differentiation of holomorphic curves?" My answer to this queestio is to interpret the lemma on logarithmic derivative in Nevanlinna theory as a defining equation of the derivative of a holomorphic curve. The corresponding statement in Diophantine setting becomes the defining equation of derivatives of rational points. The Diophantine Analogue of "differentiation" thus defined is not an absolute concept. This is defined becomes meaningful only after the target of approximation is given. In Nevanlinna theory, the absolute differentiation obays the relative law, which is the consequence of Lemma on logarithmic derivative. In Diophantine setting we should take finite places into account. I proposed a definition of ramification counting function in Diophantine approximation by extending the Minkovski/Bombierri-Vaaler geometry of numbers. I then proposed a Schmidt Subspace Theorem with truncated counting function. This version of SST enables us to establish some conjectures which is equivalent to the "abc conjecture" which seems to be quite different from the original conjecture. The definition of derivatives contain the rule of counting roots of equations. In the case of transcendental holomorphic curve or the set of rational points, the rule of counting should be based on some non-trivial statistics. In fact the truncated counting function for holomorphic curves in Abelian varieties should be of level 1 regardless of the target's dimension. Such kinds of question arises if we import the Diophantine definition of derivatives in the opposite way to Nevanlinna theory. We started this direction at the end of this project. Less

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (36 results)

All 2005 2003 2002 2001 Other

All Journal Article (17 results) Book (2 results) Publications (17 results)

  • [Journal Article] 対数微分の補題から見たNevanlinna理論2005

    • Author(s)
      小林亮一
    • Journal Title

      Surveys in Geometry, Special Ed. (掲載予定)

      Pages: 31-31

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Toward SST with truncated counting function2005

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      京大数理研講究録「代数的整数論とその周辺」 (掲載予定)

      Pages: 40-40

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Lemma on log.derivative for the Gauss map of algebraic minimal surfaces2005

    • Author(s)
      Yu Kawakami, R.K., Reiko Miyaoka
    • Journal Title

      Proc.COE conference "Geometry and Viisualization" (掲載予定)

      Pages: 17-17

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Nevanlinna theory from the view points of Lemma on logarithmic derivative2005

    • Author(s)
      Rypochi Kobayashi
    • Journal Title

      Surveys in Geometry Spercial Ed. (to appear)

      Pages: 31-31

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] An attempt toward rDiophantine analogue of ramification counting in Nevanlinna theory : Truncated counting function in Schmidt's Subspace Theorem2005

    • Author(s)
      Rypochi Kobayashi
    • Journal Title

      RIMS kokyuroku "Algebraic Number Theory And Related Topics" (to appear)

      Pages: 40-40

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Lemma on logarithmic Derivative for the Gauss map of algebraic minimal surfaces2005

    • Author(s)
      Yu Kawakami, Ryoichi Kobayashi, Reiko Miyaoka
    • Journal Title

      COE conference "Geometry and Visualization", Kyushu University (to appear)

      Pages: 17-17

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] An attempt toward Diophantine analogue of ramification counting in Nevanlinna theory : Truncated counting function in Schmidt Subspace Theorem2005

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      京大数理研講究録「代数的整数論とその周辺」 (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Lemma on logarithmic derivative for the Gauss map of algebraic minimal surfaces2005

    • Author(s)
      Yu Kawakami, Ryoichi Kobayashi Reikio Miyaoka
    • Journal Title

      Proc. COE conference "Geometry and Viisualization" (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Toward Nevanlinna theory as a geometric model for Diophantine Approximation2003

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      Amer.Math.Soc.Suugaku Exp. 16

      Pages: 39-79

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Toward Nevanlinna theory as a geometric model for Diophantine approximation2003

    • Author(s)
      Rypochi Kobayashi
    • Journal Title

      Amer.Math.Soc.Sugaku Exp. 16

      Pages: 39-79

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Toward Nevanlinna teory as a geometric model for Diophantine Approximation2003

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      Amer.Math.Soc.Suugaku Exp. 16

      Pages: 39-79

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Second Main Conjecture as a variant of the Weitzenboeck formula2002

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      Lect.Notes in Math.Osaka Univ. 7

      Pages: 109-149

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Second Main Conjecture as a variant of Weitzenboeck Formula2002

    • Author(s)
      Rypochi Kobayashi
    • Journal Title

      Lect.Notes in Math.Osaka Univ. 7

      Pages: 109-149

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Second Main Conjecture as a variant of the Weitzenboeck formula2002

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      Lect.Notes in Math. Osaka Univ. 7

      Pages: 109-149

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Meromorphically parameter dependent integral geometry2001

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      Proc.Conf.Geometry, Ibaraki Univ.

      Pages: 301-342

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Meromorphically parameter dependent integral geometry and Lemma on logarithmic derivative in Nevanlinna/Diophantos calculus2001

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      Proc.Geometry, Ibaraki Univ.

      Pages: 301-342

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Meromorphically parameter dependent integral geometry and Lemma on logarithmic derivative in Nmevanlinna/Dionhantos calculus2001

    • Author(s)
      Ryoichi Kobayashi
    • Journal Title

      Proc.Conf.Geometry, Ibaraki Univ.

      Pages: 301-342

    • Related Report
      2004 Annual Research Report
  • [Book] Ricci-Flat Kaehler計量の幾何学と解析学

    • Author(s)
      小林亮一
    • Total Pages
      400
    • Publisher
      培風館(出版予定)
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Book] Geometry and Analysis of Ricci-Flal Kaehler Manifolds

    • Author(s)
      Ryoichi Kobayashi
    • Publisher
      Baifukan(To appear)
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Ryoichi Kobayashi: "Toward Nevanlinna theory as a geometric model for Diophantine approximation"Sugaku Explsitions (Amer.Math.Soc). 16-1. 39-79 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Truncated counting function in the Schmidt Subspace Theorem"Proc.Several Complex Variables Hayama. 1-43 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] 小林亮一: "Ricci-Flat Kaehler多様体の幾何学と解析学"培風館(出版予定). 400

    • Related Report
      2003 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Truncated counting function for holomorphic curves in Abelian varieties"Proc. Complex Geometry Tokyo 2002. (発売予定). 1-17 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Truncated counting function in the Schmidt Subspace Theorem"Proc. Several Complex Variables, Hayama 2002. (発売予定). 1-40 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Toward Nevanlinna theory as a geometric model for Diophantine approximation"Sugaku Exp.(Amer.Math.Soc.). (発売予定). 1-43 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Lemma on logarithmic derivative and integral geometry"Nagoya Math J.. (発売予定). 1-10 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Minimizing currents in open manifolds and the n-1 homology of non-negatively Ricci curved manifolds"Amer.J.Math. 121. 1253-1278 (2000)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ryoichi Kobayashi: "Hooke's law in statistical manifolds and divergences"Nagoya Math.J.. 159. 1-24 (2000)

    • Related Report
      2002 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. 350 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Henmi, R.Kobayashi: "Hooke's Law in statistical Manifelds and Divergences"Nagoya Math. J.. 159. 1-24 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Kobayashi: "Holomorphic Curves in Abelian Varieties : ・・・・・"Japanese J. of Math.. 26-1. 129-152 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Kobayashi: "Methods of Integral Geometry in Nevanlinna Theory, Lemma on Logarithmic Derivative and a Program toward Second Main Theorem"Sugaku Expositions. (in press). 1-42 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Kobayashi: "Nevanlinna's Lemma on Logarithmic Derivative and Integral Geometry"Nagoya Math. J.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] R. Kobayashi: "Meromorphically Parameter Dependent Integral Geometry and Lemma on Log. Der. in Nevanlinna/Diophantus Calculus"2001幾何学シンポジウム,講演予稿集. 1-30 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小 林 亮 一: "変種Weitzenback公式としての第2主要予想"竹内勝先生メモリアル. (阪大数学レクチャーノートから出版予定). 1-40 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小林亮一: "リッチフラット計量の幾何学と解析学"培風館(出版予定). 500

    • Related Report
      2001 Annual Research Report

URL: 

Published: 2001-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi