Studies on developments and applications for the toric Mori theory

• Project/Area Number: 23540062
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Fukuoka University (2014-2015); Gifu Shotoku Gakuen University (2011-2013)
• Principal Investigator: Sato Hiroshi 福岡大学, 理学部, 准教授 (20433310)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: トーリック多様体 / ２ファノ多様体 / 森理論 / 変形理論 / 端射的収縮写像 / ファノ多様体 / 2サイクル / 高次サイクル

Outline of Final Research Achievements

In this research, we mainly studied about smooth projective toric varieties whose second Chern characters are non-negative. We studied a new method to calculate two-cycles on toric manifolds, and as a result, we completely determined the geometric structure of such manifolds when they have a Fano contraction. Moreover, we obtained a classification result about toric 2-Fano manifolds for a special case. Also we obtained some results about the ordinary toric Mori theory and the deformation theory of toric varieties.

Research Products

• [Journal Article] トーリック・ファノ多様体の第二チャーン指標 (2015)
  • Author(s): 佐藤 拓
  • Journal Title: 福岡大学微分幾何研究会報告集 Volume: 21 Pages: 1-9
• [Journal Article] トーリック・ファノ多様体の第二チャーン指標 (2014)
  • Author(s): 佐藤 拓
  • Journal Title: 数理解析研究所講究録 Volume: 未定
• [Journal Article] The numerical class of a surface on a toric manifold (2012)
  • Author(s): H. Sato
  • Journal Title: Int. J. Math. Math. Sci Volume: 2012 Pages: 9-9
  • DOI: 10.1155/2012/536475
  • Peer Reviewed
• [Presentation] トーリック・ファノ多様体の変形 (2016)
  • Author(s): 佐藤 拓
  • Organizer: ワークショップ「Algebraic Geometry and Singularities」
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2016-03-22
  • Invited
• [Presentation] トーリック多様体のチャーン指標 (2015)
  • Author(s): 佐藤 拓
  • Organizer: 研究集会「射影多様体の幾何とその周辺2015」
  • Place of Presentation: 高知大学理学部
  • Year and Date: 2015-10-31
  • Invited
• [Presentation] 第二チャーン指標が正のトーリック多様体 (2015)
  • Author(s): 佐藤 拓
  • Organizer: RIMS研究集会「幾何学・組み合わせ論に現れる環と代数構造」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-06-09
  • Invited
• [Presentation] トーリック・ファノ多様体の第二チャーン指標 (2014)
  • Author(s): 佐藤 拓
  • Organizer: 福岡大学微分幾何研究会
  • Place of Presentation: 福岡大学セミナーハウス
  • Year and Date: 2014-10-31 – 2014-11-03
  • Invited
• [Presentation] トーリック・ファノ多様体の第二チャーン指標 (2013)
  • Author(s): 佐藤 拓
  • Organizer: Fano多様体の最近の進展
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] トーリック多様体の weak factorization theorem について (2013)
  • Author(s): 佐藤 拓
  • Organizer: 杜の都代数幾何学研究集会
  • Place of Presentation: 東北大学大学院理学研究科
  • Invited
• [Presentation] ファノ多面体の分類理論 (I), (II) (二回連続講演) (2012)
  • Author(s): 佐藤 拓
  • Organizer: 研究集会「学習理論における組合せ論」
  • Place of Presentation: 九州大学マス・フォア・インダストリ研究所
  • Invited
• [Presentation] On the classification of toric 2-Fano manifolds (2011)
  • Author(s): Hiroshi Sato
  • Organizer: 特異点論とその広がり(招待講演)
  • Place of Presentation: 京都大学
• [Presentation] Toric manifolds whose Chern characters are non-negative (2011)
  • Author(s): Hiroshi Sato
  • Organizer: Toric Topology and Automorphic Functions(招待講演)
  • Place of Presentation: Pacific National Univ.