An explicit study for K-stability of Fano manifolds

• Project/Area Number: 16H06885
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: Fujita Kento 京都大学, 数理解析研究所, 助教 (40779146)
• Project Period (FY): 2016-08-26 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: ファノ多様体 / K安定性 / 極小モデル理論 / ケーラー・アインシュタイン計量

Outline of Final Research Achievements

I mainly treated the problem that which Fano manifolds admit Kaehler-Einstein metrics. It has been know that the existence of Kaehler-Einstein metrics is equivalent to K-stability, a purely algebro-geometric condition. The purpose of the project was to simplify the condition for K-stability of Fano manifolds. More precisely, we simplify "a valuative criterion" for K-stability of Fano manifolds, which is equivalent to K-stability.

Research Products

• [Journal Article] Optimal bounds for the volumes of K\"ahler-Einstein Fano manifolds (2018)
  • Author(s): Fujita Kento
  • Journal Title: American Journal of Mathematics Volume: 140 Issue: 2 Pages: 391-414
  • DOI: 10.1353/ajm.2018.0009
  • Peer Reviewed
• [Journal Article] Classification of log del Pezzo surfaces of index three (2017)
  • Author(s): FUJITA Kento, YASUTAKE Kazunori
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 69 Issue: 1 Pages: 163-225
  • DOI: 10.2969/jmsj/06910163
  • NAID: 130005310384
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] On three-dimensional semi-terminal singularities (2017)
  • Author(s): Kento Fujita
  • Journal Title: Bulletin of the Korean Mathematical Society Volume: 54 Pages: 1471-1483
• [Journal Article] Examples of K-unstable Fano manifolds with the Picard number 1 (2017)
  • Author(s): Kento Fujita
  • Journal Title: Proceedings of the Edinburgh Mathematical Society Volume: 60 Issue: 4 Pages: 881-891
  • DOI: 10.1017/s0013091516000432
• [Journal Article] On log K-stability for asymptotically log Fano varieties (2016)
  • Author(s): Kento Fujita
  • Journal Title: Ann. Fac. Sci. Toulouse Math. Volume: 25 Issue: 5 Pages: 1013-1024
  • DOI: 10.5802/afst.1520
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Openness results for uniform K-stability (2018)
  • Author(s): Kento Fujita
  • Organizer: Positivity concepts an holomorphic line bundles and theories on canonical K\"ahler metrics
  • Invited
• [Presentation] On globally embeddedness of simple normal crossing varieties (2018)
  • Author(s): Kento Fujita
  • Organizer: Log geometry, degenerations and related topics
  • Invited
• [Presentation] K-stability of log Fano hyperplane arrangements (2018)
  • Author(s): Kento Fujita
  • Organizer: The 16th affine algebraic geometry meeting
  • Invited
• [Presentation] Problems on K-stability of log Fano pairs (2017)
  • Author(s): Kento Fujita
  • Organizer: Workshop on stability and moduli spaces
  • Int'l Joint Research / Invited
• [Presentation] Uniform K-stability and plt blowups of log Fano pairs (2017)
  • Author(s): Kento Fujita
  • Organizer: Spring Meeting on Algebraic Geometry
  • Int'l Joint Research / Invited
• [Presentation] Uniform K-stability and plt blowups of log Fano pairs (2017)
  • Author(s): Kento Fujita
  • Organizer: Symposium in Geometry and Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] Uniform K-stability and plt blowups of log Fano pairs (2017)
  • Author(s): Kento Fujita
  • Organizer: Moduli of K-stable varieties
  • Int'l Joint Research / Invited
• [Presentation] Openness results for uniform K-stability (2017)
  • Author(s): Kento Fujita
  • Organizer: Boundedness and Fano varieties
  • Int'l Joint Research / Invited
• [Presentation] A valuative criterion for uniform K-stability of log Fano pairs (2017)
  • Author(s): Kento Fujita
  • Organizer: Classification and Moduli theory of algebraic varieties
  • Int'l Joint Research / Invited
• [Presentation] K-stability of log Fano hyperplane arrangements (2017)
  • Author(s): Kento Fujita
  • Organizer: Bi-annual Algebraic and Tropical Meetings of Brown and Yale
  • Int'l Joint Research / Invited
• [Presentation] A valuative criterion for uniform K-stability of Fano manifolds (2016)
  • Author(s): Kento Fujita
  • Organizer: 第22 回複素幾何シンポジウム(金沢)
  • Place of Presentation: しいのき迎賓館
  • Invited
• [Presentation] On the K-stability of Fano varieties and anticanonical divisors (2016)
  • Author(s): Kento Fujita
  • Organizer: 都の西北代数幾何学シンポジウム
  • Place of Presentation: 早稲田大学
  • Invited
• [Presentation] Remarks on uniform K-stability of log Fano pairs (2016)
  • Author(s): Kento Fujita
  • Organizer: Workshop on Algebraic Geometry
  • Place of Presentation: Hanga Roa, Chile
  • Int'l Joint Research / Invited