Uniformization, hyperbolicity, and Nevanlinna theory

• Project/Area Number: 24540069
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Osaka University (2015-2016); Tokyo Institute of Technology (2012-2014)
• Principal Investigator: Yamanoi Katsutoshi 大阪大学, 理学研究科, 教授 (40335295)
• Project Period (FY): 2012-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: ネヴァンリンナ理論 / 擬小林双曲性 / 高次元値分布論 / 小林擬距離 / ブロッホ原理 / 小林双曲性 / 特殊集合 / 高次元ネヴァンリンナ理論 / 高次元Nevanlinna理論

Outline of Final Research Achievements

I proved that subvariety of general type on abelian variety is pseudo Kobayashi hyperbolic. This problem starts from Bloch's classical paper in 1920's. Subsequently Ochiai, Kawamata proved Bloch's conjecture for entire curve. After that, it was discussed by several authors whether this result can be generalized as some kind of normality of families of holomorphic maps from the unit disc. Our result answers this question.

Research Products

• [Journal Article] Holomorphic curves in algebraic varieties of maximal albanese dimension (2015)
  • Author(s): Katsutoshi Yamanoi
  • Journal Title: International Journal of Mathematics Volume: 26 Issue: 06 Pages: 129-174
  • DOI: 10.1142/s0129167x15410062
  • Peer Reviewed
• [Journal Article] Kobayashi Hyperbolicity and Higher-dimensional Nevanlinna Theory (2015)
  • Author(s): K. Yamanoi
  • Journal Title: Progress in Mathematics Volume: 308 Pages: 209-273
  • Peer Reviewed / Open Access
• [Journal Article] A generalization of a completeness lemma in minimal surface theory (2014)
  • Author(s): Y. Okuyama and K. Yamanoi
  • Journal Title: Kodai Mathematical Journal Volume: 37 Pages: 506-517
  • NAID: 130004949007
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] On a reversal of the second main theorem for meromorphic functions of finite order (2013)
  • Author(s): K. Yamanoi
  • Journal Title: The proceedings of the 19th ICFIDCAA "Topics in Finite or Infinite Dimensional Complex Analysis" Volume: 1 Pages: 75-83
  • Peer Reviewed
• [Journal Article] Zeros of higher derivatives of meromorphic functions in the complex plane (2013)
  • Author(s): K. Yamanoi
  • Journal Title: Proc. London Math. Soc. Volume: 106 Pages: 703-780
  • Peer Reviewed
• [Journal Article] Degeneracy of Holomorphic Curves into Algebraic Varieties II (2013)
  • Author(s): J. Noguchi, J. Winkelmann, K. Yamanoi
  • Journal Title: Vietnam Journal of Mathematics Volume: 41 Pages: 519-525
  • Peer Reviewed
• [Journal Article] New examples of Sasaki-Einstein manifolds (2013)
  • Author(s): T. Mabuchi and Y. Nakagawa
  • Journal Title: Proc. London Math. Soc. Volume: 掲載確定 Issue: 4 Pages: 703-780
  • DOI: 10.1112/plms/pds051
  • NAID: 130005562143
  • Peer Reviewed
• [Journal Article] On a reversal of the second main theorem for meromorphic functions of finite order (2013)
  • Author(s): K. Yamanoi
  • Journal Title: Topics in Finite or Infinite Dimensional Complex Analysis Volume: 会議録 Pages: 75-84
  • Peer Reviewed
• [Presentation] Complex hyperbolicity problems related to abelian varieties (2016)
  • Author(s): 山ノ井克俊
  • Organizer: 代数幾何学城崎シンポジウム2016
  • Place of Presentation: 城崎国際アートセンター
  • Year and Date: 2016-10-20
• [Presentation] Pseudo Kobayashi hyperbolicity of subvarieties of general type on abelian varieties (2016)
  • Author(s): 山ノ井克俊
  • Organizer: Hayama Symposium on Complex Analysis in Several Variables XVIII
  • Place of Presentation: 湘南国際村センター
  • Year and Date: 2016-07-16
• [Presentation] Bloch-Ochiaiの定理と小林擬双曲性 (2015)
  • Author(s): Katsutoshi Yamanoi
  • Organizer: The 21th Symposium on Complex Geometry
  • Place of Presentation: 金沢大学サテライトプラザ
  • Year and Date: 2015-10-29
  • Invited
• [Presentation] On a degree of curves in algebraic varieties of maximal albanese dimension (2015)
  • Author(s): Katsutoshi Yamanoi
  • Organizer: Japan-China Geometry Conference
  • Place of Presentation: 京都大学
  • Year and Date: 2015-09-07
  • Int'l Joint Research / Invited
• [Presentation] Nevanlinna theory for holomorphic curves into algebraic varieties of maximal albanese (2014)
  • Author(s): K. Yamanoi
  • Organizer: Perspectives of Modern Complex Analysis
  • Place of Presentation: Bedlewo, ポーランド
  • Year and Date: 2014-07-21
  • Invited
• [Presentation] Kobayashi hyperbolicity and higher dimensional Nevanlinna theory (2013)
  • Author(s): K. Yamanoi
  • Organizer: Geometry and Analysis on Manifolds：A Memorial Symposium for Professor Shoshichi Kobayashi
  • Place of Presentation: 東京大学
  • Invited
• [Presentation] Zeros of higher derivatives of meromorphic functions in the complex plane (2013)
  • Author(s): K. Yamanoi
  • Organizer: The Asian Mathematical Conference 2013
  • Place of Presentation: 韓国
  • Invited
• [Presentation] Holomorphic curves in algebraic varieties of maximal albanese dimension (2013)
  • Author(s): K. Yamanoi
  • Organizer: 多変数関数論葉山シンポジウム
  • Place of Presentation: 湘南国際村センター
  • Invited
• [Presentation] 有理型函数の導関数の値分布 (2013)
  • Author(s): 山ノ井克俊
  • Organizer: 日本数学会秋季総合分科会，2013 年度幾何学賞受賞特別講演
  • Place of Presentation: 愛媛大学
  • Invited
• [Presentation] アルバネーゼ次元最大の複素射影多様体への正則曲線 (2012)
  • Author(s): 山ノ井克俊
  • Organizer: 多変数関数論冬セミナー
  • Place of Presentation: 東北大学理学部・数理科学記念館