Kahler-Einstein metrics and stability

• Project/Area Number: 22740041
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Kumamoto University (2011-2012); Kyushu University (2010)
• Principal Investigator: SANO Yuji 熊本大学, 大学院・自然科学研究科, 准教授 (00399792)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: ケーラーアインシュタイン計量 / K-安定性 / 幾何学 / 端的ケーラー計量 / balanced計量 / トーリックファノ多様体 / 乗数イデアル層 / リッチソリトン / K安定性 / 対数的標準閾値

Research Abstract

In this project, we proved the following results. (1) We showedthe algebro-geometric proof for the relation between the alpha invariant and K-stability.(2) We gave the first example of asymptotically Chow unstable Kahler-Einstein manifolds.(3) We calculated the support of the multiplier ideal subschemes induced by Kahler-Ricciflow. (4) We gave the lower diameter bounds for compact shrinking Ricci solitons.

Research Products

• [Journal Article] Lower diameterbounds for compact shrinking Riccisolitons (2013)
  • Author(s): Akito Futaki, Yuji Sano
  • Journal Title: The Asian Journal of Mathematics Volume: Vol. 17 Pages: 17-32
  • Peer Reviewed
• [Journal Article] Lower diameter bounds for compact shrinking Ricci solitons (2013)
  • Author(s): Akito Futaki, Yuji Sano
  • Journal Title: The Asian Journal of Mathematics Volume: 17 Pages: 17-32
  • Peer Reviewed
• [Journal Article] Multiplier ideal sheaves andthe Kahler-Ricci flow on toric Fanomanifolds with largesymmetry (2012)
  • Author(s): Yuji Sano
  • Journal Title: Communications in Analysis andGeometry Volume: Vol. 20 Pages: 341-368
  • Peer Reviewed
• [Journal Article] An example of asymptotically Chow unstablemanifolds with constant scalarcurvature (2012)
  • Author(s): Hajime Ono, Yuji Sano, Naoto Yotsutani
  • Journal Title: Annales de L'Institut Fourier Volume: Vol. 62 Pages: 1265-1287
  • Peer Reviewed
• [Journal Article] Alpha invariantand K-stability of Q-Fano varieties (2012)
  • Author(s): Yuji Odaka, Yuji Sano
  • Journal Title: Advances in Mathematics Volume: Vol.229 Pages: 2818-2834
  • Peer Reviewed
• [Journal Article] Multiplierideal sheaves and geometric problems,London Mathematical Society Lecture NoteSeries (2012)
  • Author(s): Akito Futaki, Yuji Sano
  • Journal Title: CambridgeUniversity Press, Cambridge Volume: 394 Pages: 68-93
  • Peer Reviewed
• [Journal Article] Multiplier ideal sheaves and the Kahler-Ricci flow on toric Fano manifolds with large symmetry (2012)
  • Author(s): Yuji Sano
  • Journal Title: Communications in Analysis and Geometry Volume: 20 Pages: 341-368
  • Peer Reviewed
• [Journal Article] An example of asymptotically Chow unstable manifolds with constant scalar curvature (2012)
  • Author(s): Hajime Ono, Yuji Sano, Naoto Yotsutani
  • Journal Title: Annales de L'Institut Fourier Volume: 62 Issue: 4 Pages: 1265-1287
  • DOI: 10.5802/aif.2722
  • Peer Reviewed
• [Journal Article] Alpha invariants and K-stability of Q-Fano varieties (2012)
  • Author(s): Yuji Odaka, Yuji Sano
  • Journal Title: Advances in Mathematics Volume: 229 Issue: 5 Pages: 2818-2834
  • DOI: 10.1016/j.aim.2012.01.017
  • Peer Reviewed
• [Journal Article] Lower diameter bounds for compact shrinking Ricci solitons (2012)
  • Author(s): Akito Futaki, Yuji Sano
  • Journal Title: The Asian Journal of Mathematics Volume: (掲載確定)
  • Peer Reviewed
• [Journal Article] Hilbert series and obstructions toasymptotic semistability (2011)
  • Author(s): Akito Futaki, Ono Hajime, Yuji Sano
  • Journal Title: Advances inMathematics Volume: Vol.226 Pages: 254-284
  • Peer Reviewed
• [Journal Article] ケーラー・アインシュタイン計量の存在への障害的因子について (2011)
  • Author(s): 佐野友二
  • Journal Title: RIMS研究集会報告集(部分多様体の微分幾何学的研究) Volume: 1775 Pages: 45-51
• [Journal Article] Multiplier ideal sheaves and integral invariants on toric Fano manifolds (2011)
  • Author(s): Akito Futaki, Yuji Sano
  • Journal Title: Mathematische Annalen Volume: (In press)
  • Peer Reviewed
• [Journal Article] Hilbert series and obstructions to asymptotic semistability (2011)
  • Author(s): Akito Futaki, Hajime Ono, Yuji Sano
  • Journal Title: Advances in Mathematics Volume: 226 Pages: 254-284
  • Peer Reviewed
• [Journal Article] An example of asymptotically Chow unstable manifolds with constant scalar curvature (2011)
  • Author(s): Hajime Ono, Yuji Sano, Naoto Yotsutani
  • Journal Title: Annales de L'Institut Fourier Volume: (In press)
  • Peer Reviewed
• [Journal Article] Multiplier ideal sheaves and geometric problems (2011)
  • Author(s): Akito Futaki, Yuji Sano
  • Journal Title: London Mathematical Society Lecture Note Series Volume: (In press)
  • Peer Reviewed
• [Presentation] Extremal metrics and lowerbound of the modified K-energy (2012)
  • Author(s): Yuji Sano
  • Organizer: RecentDevelopments in Kahler Geometry
  • Place of Presentation: Institut Henri Poincare (フランス)
  • Year and Date: 2012-12-10
• [Presentation] A dynamical approach tobalanced metrics with automorphism group (2012)
  • Author(s): 佐野友二
  • Organizer: Symposium on ARITHMETIC & GEOMETRY
  • Place of Presentation: 九州大学(福岡)
  • Year and Date: 2012-06-02
• [Presentation] Alpha invariant andK-stability of Fano varieties (joint withY. Odaka) (2012)
  • Author(s): 佐野友二
  • Organizer: 第7回代数・解析・幾何セミナー
  • Place of Presentation: 鹿児島大学(鹿児島)
  • Year and Date: 2012-02-14
• [Presentation] Alpha invariant and K-stability of Fano varieties (joint with Y.Odaka) (2012)
  • Author(s): Yuji Sano
  • Organizer: 第7回代数・解析・幾何セミナー
  • Place of Presentation: 鹿児島大学(鹿児島県)
  • Year and Date: 2012-02-14
• [Presentation] A dynamical approach tobalanced metrics with automorphism group (2012)
  • Author(s): Yuji Sano
  • Organizer: 第7回日中友好幾何学研究集会
  • Place of Presentation: ホテルレジーナ河口湖(山梨県)
  • Year and Date: 2012-01-14
• [Presentation] A dynamical approach to balanced metrics with automorphism group (2012)
  • Author(s): Yuji Sano
  • Organizer: 第7回日中友好幾何学研究集会
  • Place of Presentation: ホテルレジーナ河口湖(山梨県)
  • Year and Date: 2012-01-14
• [Presentation] A dynamical approach to balanced metrics with automorphism group (2012)
  • Author(s): Yuji Sano
  • Organizer: Symposium on ARITHMETIC & GEOMETRY
  • Place of Presentation: 九州大学
  • Invited
• [Presentation] Extremal metrics and lower bound of the modified K-energy (2012)
  • Author(s): Yuji Sano
  • Organizer: Recent Developments in Kahler Geometry
  • Place of Presentation: Institut Henri Poincare（フランス，パリ）
  • Invited
• [Presentation] A dynamical approach tobalanced metrics with automorphism group (2011)
  • Author(s): Yuji Sano
  • Organizer: 2011 Complex Geometry and SymplecticGeometry Conference
  • Place of Presentation: 中国科学技術大学(中国)
  • Year and Date: 2011-08-16
• [Presentation] A dynamical approach to balanced metrics with automorphism group (2011)
  • Author(s): Yuji Sano
  • Organizer: 2011 Complex Geometry and Symplectic Geometry Conference
  • Place of Presentation: 中国科学技術大学(中国)
  • Year and Date: 2011-08-16
• [Presentation] ケーラー・アインシュタイン計量の存在への障害的因子について (2011)
  • Author(s): 佐野友二
  • Organizer: 部分多様体の微分幾何学的研究
  • Place of Presentation: 京都大学数理解析研究所(京都)
  • Year and Date: 2011-06-28
• [Presentation] ケーラー・アインシュタイン計量の存在への障害的因子について (2011)
  • Author(s): 佐野友二
  • Organizer: RIMS研究集会『部分多様体の微分幾何学的研究』
  • Place of Presentation: 京都大学RIMS(京都府)
  • Year and Date: 2011-06-28
• [Presentation] Alpha invariant andK-stability of Fano varietes (joint workwith Yuji Odaka) (2011)
  • Author(s): 佐 野 友 二
  • Organizer: 代数幾何学研究集会 -ファノ多様体と正標数上の話題を中心として-
  • Place of Presentation: 九州大学(福岡)
  • Year and Date: 2011-02-21
• [Presentation] Alpha invariant and K-stability of Fano varietes (joint work with Yuji Odaka) (2011)
  • Author(s): 佐野友二
  • Organizer: 代数幾何学研究集会-ファノ多様体と正標数上の話題を中心として-
  • Place of Presentation: 九州大学(福岡)
  • Year and Date: 2011-02-21
• [Presentation] Alpha invariant andK-stability of Fano varieties (2011)
  • Author(s): Yuji Sano
  • Organizer: Extremalmetrics:evolution equations and stability(Complex and Riemannian Geometry Week2)
  • Place of Presentation: CIRM(フランス)
  • Year and Date: 2011-02-08
• [Presentation] Alpha invariant and K-stability of Fano varietes (2011)
  • Author(s): Yuji Sano
  • Organizer: Extremal metrics : evolution equations and stability in "Complex and Riemannian Geometry"
  • Place of Presentation: Marseille(フランス)
  • Year and Date: 2011-02-08
• [Presentation] Alpha invariant andK-stability of Fano varieties (2010)
  • Author(s): Yuji Sano
  • Organizer: BirationalGeometry in honor of V.V. Shokurov's 60thBirthday
  • Place of Presentation: ICMS Edinburgh(エジンバラ)
  • Year and Date: 2010-12-10
• [Presentation] Alpha invariant and K-stability of Fano varietes (2010)
  • Author(s): Yuji Sano
  • Organizer: Birational Geometry in honor of V.V.Shokurov's 60th Birthday
  • Place of Presentation: Edingburgh(イギリス)
  • Year and Date: 2010-12-10
• [Presentation] Alpha invariant andK-stability (2010)
  • Author(s): Yuji Sano
  • Organizer: 第16回複素幾何シンポジウム
  • Place of Presentation: 信州菅平高原 プチホテル ゾンタック(長野県)
  • Year and Date: 2010-10-23
• [Presentation] Alpha invariant and K-stability (2010)
  • Author(s): Yuji Sano
  • Organizer: 第16回複素幾何シンポジウム
  • Place of Presentation: 菅平(長野)
  • Year and Date: 2010-10-23
• [Presentation] On multiplier ideal sheavesand test configurations on toric Fanomanifolds (Obstructive divisors toKahler-Einstein metrics) (2010)
  • Author(s): Yuji Sano
  • Organizer: KomplexeAnalysis
  • Place of Presentation: Oberwolfach(ドイツ)
  • Year and Date: 2010-09-01
• [Presentation] On multiplier ideal sheaves and test configurations on toric Fano manifolds (Obstructive divisors to Kahler-Einstein metrics) (2010)
  • Author(s): Yuji Sano
  • Organizer: Komplexe Analysis
  • Place of Presentation: Oberwolfach(ドイツ)
  • Year and Date: 2010-09-01
• [Presentation] On multiplier ideal sheavesand test configurations on toric Fanomanifolds (2010)
  • Author(s): Yuji Sano
  • Organizer: Workshop on Moduli andBirational Geometry
  • Place of Presentation: POSTECH (韓国)
  • Year and Date: 2010-08-06
• [Presentation] On multiplier ideal sheaves and test configurations on toric Fano manifolds (2010)
  • Author(s): Yuji Sano
  • Organizer: Workshop on Moduli and Birational Geometry
  • Place of Presentation: Pohang(韓国)
  • Year and Date: 2010-08-06
• [Presentation] Obstructive divisors to theexistence of Kahler-Einstein metrics (2010)
  • Author(s): Yuji Sano
  • Organizer: The 5th Pacific Rim Complex and SymplecticGeometry Conference 2010
  • Place of Presentation: 伊勢志摩ロイヤルホテル(三重県)
  • Year and Date: 2010-07-30
• [Presentation] Obstructive divisors to the existence of Kahler-Einstein metrics (2010)
  • Author(s): Yuji Sano
  • Organizer: The 5th Pacific Rim Complex and Symplectic Geometry Conference 2010
  • Place of Presentation: 伊勢志摩ロイヤルホテル(三重)
  • Year and Date: 2010-07-30
• [Remarks]
  • URL: http://www.sci.kumamoto-u.ac.jp/~sano/research.html
• [Remarks]
  • URL: http://researchmap.jp/sanoyuji/
• [Remarks]
  • URL: http://www.sci.kumamoto-u.ac.jp/~sano/research.html
• [Remarks]
  • URL: http://www2.math.kyushu-u.ac.jp/sano/index.html