2013 Fiscal Year Final Research Report
Geometry of Ricci solitons on complex manifolds
| Project/Area Number |
20244005
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Osaka University |
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Principal Investigator |
MABUCHI Toshiki 大阪大学, 理学(系)研究科(研究院), 教授 (80116102)
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| Co-Investigator(Kenkyū-buntansha) |
GOTO Ryushi 大阪大学, 大学院理学研究科, 教授 (30252571)
UMEHARA Masaaki 東京工業大学, 大学院情報理工学研究科, 教授 (90193945)
SASAKI Takeshi 神戸大学, 大学院理学研究科, 名誉教授 (00022682)
NAKAGAWA Yasuhiro 佐賀大学, 大学院工学系研究科, 教授 (90250662)
HASEGAWA Keizo 新潟大学, 人文社会教育科学系, 教授 (00208480)
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| Co-Investigator(Renkei-kenkyūsha) |
NAKAJIMA Hiraku 京都大学, 数理科学研究科, 教授 (00201666)
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| Project Period (FY) |
2008-04-08 – 2013-03-31
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| Keywords | 複素多様体 / Ricci soliton / K-安定性 / Donaldson-Tian-Yau予想 / テスト配位 / 佐々木アインシュタイン計量 / 偏極射影代数多様体 / トーリック微分幾何 |
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| Research Abstract |
(1) In a joint work (Tohoku Math. J., 65, 2013, 243-252) with Y. Nakagawa, we generalized Sakane-Koiso's construction of Kaehler-Einstein metrics to the Kaehler-Ricci soliton case where the Futaki invariant is non-vanishing. In this case, we obtain Sasaki-Einstein metrics in place of Kaehler-Einstein metrics.
(2) For the Kaehler-Einstein metric on the blowing-up of the complex projective plane at 3 non-colinear points, its detailed description was obtained by asymptotic expansion of the solution of a hyperbolic affine sphere equation on a bounded domain in the real 2-plane (AMS/IP Stud. Adv. Math. 48, 219-229).
(3) As to the Donaldson-Tian-Yau Conjecture, we proved: i) Asymptotic relative Chow stability implies the existence of a sequence of polybalanced metrics (Osaka J. Math. 48, 2011, 845-856); ii) strong relative K-stability implies asymptotic relative Chow stability (joint work with Y.Nitta).
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