Local-global correspondance in birational geometry

• Project/Area Number: 18H01108
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: The University of Tokyo
• Principal Investigator: Gongyo Yoshinori 東京大学, 大学院数理科学研究科, 教授 (70634210)
• Co-Investigator(Kenkyū-buntansha): 中村 勇哉 東京大学, 大学院数理科学研究科, 助教 (20780034)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥17,160,000 (Direct Cost: ¥13,200,000、Indirect Cost: ¥3,960,000); Fiscal Year 2022: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2021: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2020: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2019: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2018: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
• Keywords: MMP / MLD / 極小モデル理論 / 極小対数的食い違い係数 / フリップの停止 / フリップ / 相対標準因子の正値性 / 錐定理 / ファノ多様体 / FLIP / フリップの停止問題 / 食い違い係数

Outline of Final Research Achievements

I have published a paper with Yusuke Nakamura and Wei-chung Chen on the generalized minimal log discrepancies. This paper has been published in JMSJ. A further development is the study of the Mukai-type conjecture from the perspective of local-global correspondence. I have introduced a new invariant of total exponent for Fano manifolds and have studied the characterization of the projective space of the Mukai-type conjecture. I have been able to prove that the conjecture holds if we assume the Ambro-Kawamata effective non-vanishing conjecture holds, so the conjecture itself seems to be correct. The paper including the above results are will be published by Kyoto Math. Journal. I have also investigated the case of curved surfaces in more detail in collaboration with Moraga.

Academic Significance and Societal Importance of the Research Achievements

研究成果の学術的意味は、オリジナルの向井予想への新しいアプローチを考えられたので十分意義があったと思う。さらにその視点からショクロフ学派の数学が有用であることを示唆しているので、その方向からのアプローチがMoragaらを中心に活性化させることができたので十分意味があったと思う。

Research Products

• [Int'l Joint Research] ユタ大学/UCLA(米国)
• [Int'l Joint Research] UCLA(米国)
• [Int'l Joint Research] Utah大学(米国)
• [Int'l Joint Research] Yonsei University(韓国)
• [Int'l Joint Research] University of Utah(米国)
• [Journal Article] On generalized minimal log discrepancy (2024)
  • Author(s): CHEN Weichung、GONGYO Yoshinori、NAKAMURA Yusuke
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 76 Issue: 2 Pages: 393-449
  • DOI: 10.2969/jmsj/90119011
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Nef anti-canonical divisors and rationally connected fibrations 5.発行年 2020年 (2020)
  • Author(s): S. Ejiri, Y. Gongyo
  • Journal Title: Compositio Math. Volume: 印刷中
• [Journal Article] Rational points on log Fano threefolds over a finite field (2019)
  • Author(s): Yoshinori Gongyo, Yusuke Nakamura, Hiromu Tanaka
  • Journal Title: J. Eur. Math. Soc. (JEMS) Volume: 21 Issue: 12 Pages: 3759-3795
  • DOI: 10.4171/jems/913
  • Peer Reviewed
• [Journal Article] Nef anti-canonical divisors and rationally conncected fibrations (2019)
  • Author(s): S. Ejiri, Y. Gongyo
  • Journal Title: Compos. Math. Volume: 155 Issue: 7 Pages: 1444-1456
  • DOI: 10.1112/s0010437x19007383
  • Peer Reviewed
• [Journal Article] Rational points on log Fano threefolds over a finite field (2019)
  • Author(s): Y. Gongyo, Y. Nakamura, H. Tanaka
  • Journal Title: Journal of the EMS Volume: 印刷中
• [Journal Article] Koll\'ar's injectivity theorem for globally F-regular varieties (2019)
  • Author(s): Y. Gongyo, S. Takagi
  • Journal Title: European Journal of Mathematics Volume: 印刷中
• [Journal Article] Koll\'ar’s injectivity theorem for globally F-regular varieties (2018)
  • Author(s): Gongyo Yoshinori、Takagi Shunsuke
  • Journal Title: European Journal of Mathematics Volume: 印刷中 Issue: 3 Pages: 872-880
  • DOI: 10.1007/s40879-018-0230-4
  • Peer Reviewed
• [Presentation] 有効消滅性と向井型予想について (2023)
  • Author(s): 權業善範
  • Organizer: 京都大学代数幾何セミナー
  • Invited
• [Presentation] The Mukai type conjecture (2023)
  • Author(s): 權業善範
  • Organizer: Recent development in Algebraic Geometry, Arithmetic and Dynamics
  • Int'l Joint Research / Invited
• [Presentation] 向井型の射影空間の積の特徴づけの予想 (2023)
  • Author(s): 權業善範
  • Organizer: 代数幾何学城崎シンポジウム
  • Int'l Joint Research / Invited
• [Presentation] Effective non-vanishing and Mukai type conjecture (2023)
  • Author(s): 權業善範
  • Organizer: 新潟代数シンポジウム
  • Invited
• [Presentation] 向井型の射影空間の積の特徴づけの予想 (2023)
  • Author(s): 權業善範
  • Organizer: 湯布院代数幾何学ワークショップ
  • Invited
• [Presentation] 高次元極小モデル理論の構築とその応用, (2023)
  • Author(s): 權業善範
  • Organizer: 2023 年度日本数学会年会代数学賞受賞講演,
  • Invited
• [Presentation] Generalized complexity and the Mukai type conjecture. (2023)
  • Author(s): Yoshinori Gongyo
  • Organizer: Korea-Japan Conference in Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] Generalized complexity and the Mukai type conjecture (2023)
  • Author(s): Yoshinori Gongyo
  • Organizer: The 1st Algebraic geometry Atami symposium, 熱海
  • Int'l Joint Research / Invited
• [Presentation] 一般的複雑性と射影空間の直積の特徴付けに関する向井型予想について (2022)
  • Author(s): 權業善範
  • Organizer: 名古屋代数幾何 学セミナー
  • Invited
• [Presentation] Generalized complexity and Mukai’s type conjecture. (2022)
  • Author(s): Yoshinori Gongyo
  • Organizer: YMSC AG seminar (Zoom)
  • Invited
• [Presentation] Nef anti-canonical divisors and rationally conncected fibrations (2020)
  • Author(s): Yoshinori Gongyo
  • Organizer: AG seminar at UCLA
  • Invited
• [Presentation] On a generalized Batyrev’s cone conjecture (2020)
  • Author(s): Yoshinori Gongyo
  • Organizer: Zoom Algbraic Geometry seminar
  • Int'l Joint Research / Invited
• [Presentation] A generailization of Batyrev’s cone conjecture (2019)
  • Author(s): Yoshinori Gongyo
  • Organizer: GMS seminar at MSRI
  • Int'l Joint Research / Invited
• [Presentation] A generailization of Batyrev’s cone conjecture (2019)
  • Author(s): Yoshinori Gongyo
  • Organizer: Algebraic Geometry International Conference, KIAS
  • Int'l Joint Research / Invited
• [Presentation] Nef anti-canonical divisors and rationally conncected fibrations (2019)
  • Author(s): Yoshinori Gongyo
  • Organizer: Birational Geometry and Fano varieties, Moscow
  • Int'l Joint Research / Invited
• [Presentation] ACC for LCT and Boundedness of Fano va- rieties after Birkar, Hacon, Mckernan, and Xu (2019)
  • Author(s): Yoshinori Gongyo
  • Organizer: 第 21 回多変数関数論葉山シンポジウム
  • Int'l Joint Research / Invited
• [Presentation] A generailization of Batyrev’s cone conjecture (2019)
  • Author(s): Yoshinori Gongyo
  • Organizer: JHU AG seminar,
  • Int'l Joint Research / Invited
• [Presentation] A generailization of Batyrev’s cone conjecture (2019)
  • Author(s): Yoshinori Gongyo
  • Organizer: 成功大学, コロキウム, 台南
  • Int'l Joint Research / Invited
• [Presentation] A generailization of Batyrev’s cone conjecture (2019)
  • Author(s): Yoshinori Gongyo
  • Organizer: 都の西北代数幾何学シンポジウム 2019
  • Invited
• [Presentation] A generailization of Batyrev’s cone conjecture (2019)
  • Author(s): Yoshinori Gongyo
  • Organizer: Algebraic Geometry in Auckland
  • Int'l Joint Research / Invited
• [Presentation] ACC for LCT and Boundedness of Fano varieties after Birkar, Hacon, Mckernan, and Xu (2019)
  • Author(s): 權業善範
  • Organizer: Mini Workshop at Saitama University
  • Int'l Joint Research / Invited
• [Presentation] Nef anti-canonical divisors and rationally conncected fibrations (2018)
  • Author(s): 權業善範
  • Organizer: Birational Geometry and Arithmetic
  • Int'l Joint Research / Invited
• [Presentation] Nef anti-canonical divisors and rationally conncected fibrations (2018)
  • Author(s): 權業善範
  • Organizer: Differential, Algebraic and Topological Methods in Complex Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] Nef anti-canonical divisors and rationally conncected fibrations (2018)
  • Author(s): 權業善範
  • Organizer: Workshop on Birational Geometry and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] On log CY structure of surfaces admitting non-trivial polarized endomorphisms (2018)
  • Author(s): 權業善範
  • Organizer: the Xiamen workshop Algebraic Surfaces and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] My understanding of the proof of BAB conjecture (2018)
  • Author(s): 權業善範
  • Organizer: Higher Dimensional Arithmetic Geometry
  • Int'l Joint Research / Invited
• [Remarks] Y.Gongyo's webpage
  • URL: https://www.ms.u-tokyo.ac.jp/~gongyo/
• [Remarks] Yusuke Nakamura
  • URL: https://sites.google.com/site/ynakamuraagmath/home
• [Remarks] Y. Gongyo's webpage
  • URL: https://www.ms.u-tokyo.ac.jp/~gongyo/
• [Remarks] https://www.ms.u-tokyo.ac.jp/~gongyo/
• [Remarks] Y. Gongyo's webpage
  • URL: http://www.ms.u-tokyo.ac.jp/~gongyo/
• [Remarks] Y. Gongyo's webpage
  • URL: http://www.ms.u-tokyo.ac.jp/~gongyo/index.html
• [Funded Workshop] Birational Geometry and Algebraic Dynamics (2023)
• [Funded Workshop] Cambridge--Tokyo Algebraic Geometry workshop (2018)
• [Funded Workshop] London-Tokyo Workshop In Birational Geometry II, Imperial College London (2018)
• [Funded Workshop] London-Tokyo Workshop In Birational Geometry (2018)