2000 Fiscal Year Final Research Report Summary
Symplectic Structures and Geometry of Canonical Bundle
| Project/Area Number |
10440021
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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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 Graduate School of Science Osaka University, Prof., 大学院・理学研究科, 教授 (80116102)
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| Co-Investigator(Kenkyū-buntansha) |
WENG Lin Nagoya Univ., Math, Dept. Osaka University, Assoc.Prof., 多元数理科学研究科, 助教授 (60304002)
KOBAYASHI Ryoichi Nagoya Univ., Math.Dept. Osaka University, Prof., 多元数理科学研究科, 教授 (20162034)
FUJIKI Akira Graduate School of Science Osaka University, Prof., 大学院・理学研究科, 教授 (80027383)
SAKUMA Makoto Graduate School of Science Osaka University, Assoc.Prof., 大学院・理学研究科, 助教授 (30178602)
KONNO Kazuhiro Graduate School of Science Osaka University, Assoc.Prof., 大学院・理学研究科, 助教授 (10186869)
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| Project Period (FY) |
1998 – 2000
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| Keywords | symplactic structure / Kahler structure / canonical bundle / anticanonical bundle / Kahler-Einstein metric / Hitchin-Kobayashi correspondence / Futaki choracter / stability |
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| Research Abstract |
We focussed our study on the "Hitchin-Kobayashi correspondence for manifolds" which is one of the most interesting topics in our project. For a Fano manifold, such a correspondence is supposed to relate the Chow-Mumford stability of the manifold with the existence of Kahler-Einstein metrics. As a first step, we obtained :
(1) For a certain generalization of Kahler-Einstein metrics, we showed the uniqueness of such metrics modulo holomorphic automorphisms on a given Fano manifold. Hence, even in this generalized context, the above correspondence is one-to-one. (In the case of Kahler-Ricci solitons, a similar result was obtained also by Tian and Zhu.)
Recently, we also saw the following :
(2) In a joint work with H.Nakagawa, we succeeded in characterizing the Futaki character of a Fano manifold as an obstruction to Chow-Mumford semistability of the manifold.
(3) In complex analytic studies of a polarized manifold, the asymptotic stability of the manifold has a strong relationship, via the asymptotic behavior of the Bergman metrics, with the existence of metrics of constant scalar curvature in the polarized Kahler class.
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