Characterization of homogeneous varieties by means of tangent bundles

• Project/Area Number: 17K14153
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Chuo University (2020); Saitama University (2017-2019)
• Principal Investigator: Watanabe Kiwamu 中央大学, 理工学部, 准教授 (20638176)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 接束 / ネフ / 等質多様体 / 有理曲線 / ファノ多様体 / 有理連結多様体 / 外積 / 正標数 / ネフ接束 / VMRT / 有理等質多様体 / 対角集合 / 代数学 / 幾何学

Outline of Final Research Achievements

I studied projective varieties from the viewpoint of the positivity of the tangent bundle. I wrote three papers on the Campana-Peternell conjecture "Any Fano manifold with nef tangent bundle is homogeneous". I also studied more wide class of varieties. As a consequence, I obtained a structure theorem of varieties with nef second exterior power of the tangent bundle and a positive characteristic analogue of the Demailly-Peternell-Scheneider theorem.

Academic Significance and Societal Importance of the Research Achievements

４年間の研究期間のうちに，5本の論文を執筆し，うち4本が出版された．とりわけ，研究課題であるカンパーナ・ペターネル予想（以下CP予想）に関連し，F4(4)型の等質多様体の特徴付けを得た．また，余指数3のファノ多様体に対してCP予想を肯定的に解決した．CP予想の解決には届かなかったが，その一方で，接束の二階外積がネフである多様体の構造定理と正標数の体上定義されたネフ接束を持つ射影多様体の構造定理を得た．CP予想より広い枠組みで構造定理を得ることができたことで，今後の研究に広がりができた．

Research Products

• [Int'l Joint Research] University of Trento(イタリア)
• [Int'l Joint Research] Trento University(イタリア)
• [Int'l Joint Research] トレント大学(イタリア)
• [Journal Article] Fano manifolds of coindex three admitting nef tangent bundle (2021)
  • Author(s): Kiwamu Watanabe
  • Journal Title: Geometriae Dedicata Volume: 210 Issue: 1 Pages: 165-178
  • DOI: 10.1007/s10711-020-00538-2
  • Peer Reviewed
• [Journal Article] Varieties with nef diagonal (2020)
  • Author(s): Taku Suzuki and Kiwamu Watanabe
  • Journal Title: International Journal of Mathematics Volume: 31 Issue: 02 Pages: 2050011-2050011
  • DOI: 10.1142/s0129167x20500111
  • Peer Reviewed
• [Journal Article] Characterizing the homogeneous variety F4(4) (2018)
  • Author(s): Occhetta, Gianluca; Sola Conde, Luis E.; Watanabe, Kiwamu
  • Journal Title: v Volume: 291 Issue: 14-15 Pages: 2334-2346
  • DOI: 10.1002/mana.201700261
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Characterizing the homogeneous variety F4(4) (2018)
  • Author(s): Gianluca Occhetta, Luis E. Sola Conde; Conde, Kiwamu Watanabe,
  • Journal Title: Mathematische Nachrichten Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A characterization of symplectic Grassmanianns (2017)
  • Author(s): Occhetta, Gianluca; Sola Conde, Luis E., Watanabe, Kiwamu
  • Journal Title: Mathematische Zeitschrift Volume: 印刷中 Issue: 3-4 Pages: 1421-1433
  • DOI: 10.1007/s00209-016-1807-6
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Fano manifolds whose elementary contractions are smooth P^1-fibrations: a geometric characterization of flag varieties (2017)
  • Author(s): Gianluca Occhetta, Luis E. Sola Conde; Conde, Kiwamu Watanabe, Jaroslaw A. Wisniewski
  • Journal Title: Annali della Scuola Normale Superiore di Pisa Volume: XVII, issue 2 Pages: 573-607
  • DOI: 10.2422/2036-2145.201508_007
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Positivity of the exterior power of the tangent bundles (2021)
  • Author(s): Watanabe, Kiwamu
  • Organizer: Zoom Algebraic Geometry Seminar
  • Int'l Joint Research / Invited
• [Presentation] Fano manifolds of coindex three admitting nef tangent bundle (2020)
  • Author(s): 渡邉究
  • Organizer: 第７回代数幾何学研究集会－宇部－
  • Invited
• [Presentation] Characterizing the homogeneous variety F_4(4) (2019)
  • Author(s): Kiwamu Watanabe
  • Organizer: Algebraic Geometry and Moduli Theory
  • Int'l Joint Research / Invited
• [Presentation] Varieties with nef diagonal (2018)
  • Author(s): 渡邉究
  • Organizer: 日大特異点論月曜セミナー
  • Invited
• [Presentation] Varieties with nef diagonal (2018)
  • Author(s): 渡邉究
  • Organizer: 東大数理代数幾何セミナー
  • Invited
• [Presentation] Varieties with nef diagonal (2018)
  • Author(s): 渡邉究
  • Organizer: 第16回アフィン代数幾何学研究集会（関西学院大学大阪梅田キャンパス）
  • Int'l Joint Research / Invited
• [Presentation] Characterizing the homogeneous variety F4(4) (2017)
  • Author(s): 渡邉究
  • Organizer: 代数幾何学セミナー（大阪大学）
  • Invited
• [Presentation] Geometric characterizations of rational homogeneous manifolds (2017)
  • Author(s): 渡邉究
  • Organizer: 代数学シンポジウム（大阪大学）
  • Invited
• [Presentation] Fano manifolds with nef tangent bundle (2017)
  • Author(s): 渡邉究
  • Organizer: 代数幾何学城崎シンポジウム（城崎温泉）
  • Int'l Joint Research / Invited
• [Remarks]
  • URL: https://sites.google.com/site/kiwamuwatanabeshomepage/home?authuser=0
• [Remarks]
  • URL: https://sites.google.com/site/kiwamuwatanabeshomepage/home
• [Remarks]
  • URL: http://www.rimath.saitama-u.ac.jp/lab.jp/Kiwamu.Watanabe.html
• [Remarks]
  • URL: http://www.rimath.saitama-u.ac.jp/lab.jp/Kiwamu.Watanabe.html