Kahler-Einstein metrics and birational geometry

• Project/Area Number: 25800050
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Fukuoka University (2016); Kumamoto University (2013-2015)
• Principal Investigator: Sano Yuji 福岡大学, 理学部, 教授 (00399792)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: ケーラー・アインシュタイン計量 / K 安定性 / トーリックファノ多様体 / 端的ケーラー計量 / balanced 計量 / ケーラーアインシュタイン計量 / 二木不変量 / 多面体の重心 / バランスド計量

Outline of Final Research Achievements

In this research, we achieved the following two results.
(1) For a given toric Fano manifold, we introduced an invariant of a Fano polytope which is likely to be a polar dual of the barycenter of the image under the moment map on a given toric Fano manifold. Under some assumption, it induces a simpler criterion for the existence of Kahler-Einstein metrics (or K stability) on a toric Fano manifold.
(2) With a differential geometrical approach, we extended the moment map interpretation for balanced metrics to relative balanced metrics. Then, it implies the quantization of the extremal metrics and the extremal vector fields on a polarized manifold.

Research Products

• [Int'l Joint Research] Universite de Bretagne Occidentale(France)
• [Int'l Joint Research] Universite de Bretagne Occidentale(France)
• [Presentation] 端的ケーラー計量と相対的 balanced 計量について (2016)
  • Author(s): 佐野友二
  • Organizer: 福岡大学微分幾何研究集会
  • Place of Presentation: 福岡大学
  • Invited
• [Presentation] 端的ケーラー計量と相対的 balanced 計量について (2016)
  • Author(s): 佐野友二
  • Organizer: 多変数関数論冬セミナー
  • Place of Presentation: 福岡工業大学
  • Invited
• [Presentation] Minkowski Problem and Fano polytopes (2015)
  • Author(s): Yuji Sano
  • Organizer: Moduli spaces and singularities in algebraic and Riemannian geometry
  • Place of Presentation: Simons Center for Geometry and Physics (New York, 米国)
  • Year and Date: 2015-11-05
  • Int'l Joint Research / Invited
• [Presentation] ミンコウスキ問題とファノ多面体 (2015)
  • Author(s): 佐野友二
  • 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: 2015-10-30
  • Invited
• [Presentation] Quantization of extremal metrics and modified K-energy (2015)
  • Author(s): Yuji Sano
  • Organizer: International conference on Geometry and Quantization (GEOQUANT2015)
  • Place of Presentation: ICMAT (Madrid, スペイン)
  • Year and Date: 2015-09-17
  • Int'l Joint Research / Invited
• [Presentation] なめらかなトーリックファノ多様体の重心について (2015)
  • Author(s): 佐野友二
  • Organizer: 大阪大学幾何セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2015-01-19
• [Presentation] On extremal vector fields on smooth toric Fano manifolds (2014)
  • Author(s): Yuji Sano
  • Organizer: The 10 Korean conference several complex variables
  • Place of Presentation: 韓国
  • Year and Date: 2014-08-07 – 2014-08-11
  • Invited
• [Presentation] On extremal vector fields on smooth toric Fano manifolds (2014)
  • Author(s): Yuji Sano
  • Organizer: Trends in Modern Geometry
  • Place of Presentation: 東京大学
  • Year and Date: 2014-07-07 – 2014-07-11
  • Invited
• [Presentation] On computations of a bilinear form on toric Fano manifolds (2014)
  • Author(s): Yuji Sano
  • Organizer: The 5th international workshop on differential geometry and analysis
  • Place of Presentation: 佐賀県（唐津）
  • Year and Date: 2014-05-31 – 2014-06-04
  • Invited
• [Presentation] Extremal metrics and lower bound of the modified K-energy (2013)
  • Author(s): Yuji Sano
  • Organizer: Extremal Kahler Metrics
  • Place of Presentation: Centre de recherches mathematiques, Canada
• [Presentation] Extremal metrics and lower bound of the modified K-energy (2013)
  • Author(s): 佐野友二
  • Organizer: 幾何セミナー
  • Place of Presentation: 名古屋大学
• [Presentation] Tian のアルファ不変量と K 安定性 (2013)
  • Author(s): 佐野友二
  • Organizer: 第60回幾何学シンポジウム
  • Place of Presentation: 東京工業大学
• [Remarks] 佐野友二のホームページ
  • URL: http://www.sci.kumamoto-u.ac.jp/~sano/index.html
• [Remarks] 佐野友二のホームページ
  • URL: http://www.sci.kumamoto-u.ac.jp/~sano/research.html
• [Funded Workshop] The 7th International workshop on Differential Geometry (2017)
  • Place of Presentation: 唐津
  • Year and Date: 2017-03-23