2002 Fiscal Year Final Research Report Summary
Homogeneous complex manifolds and related problems
| Project/Area Number |
13640091
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Ritsumeikan University |
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Principal Investigator |
NAKAJIMA Kazufumi Ritsumeikan Univ., Fac. Science and Engineering, Professor, 理工学部, 教授 (10025489)
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| Co-Investigator(Kenkyū-buntansha) |
SHIN'YA Hitoshi Ritsumeikan Univ., Fac. Science and Engineering, Professor, 理工学部, 教授 (70036416)
NARUKI Isao Ritsumeikan Univ., Fac. Science and Engineering, Professor, 理工学部, 教授 (90027376)
FUJIMURA Shigeyoshi Ritsumeikan Univ., Fac. Science and Engineering, Professor, 理工学部, 教授 (30066724)
KAGAWA Takaaki Ritsumeikan Univ., Fac. Science and Engineering, Associate Professor, 理工学部, 助教授 (90298175)
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| Project Period (FY) |
2001 – 2002
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| Keywords | complex manifolds / Kahler manifolds / homogeneous spaces / semi-simple Lie groups |
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| Research Abstract |
By Borel or Koszul, every homogeneous Kahler manifold of a semi-simple Lie group G is a coset space of G by C(Z), the centralizer of an element Z of the Lie algebra g of G. Conversely, let G be a semi-simple Lie group and Z an element of g and consider the factor space G/C(Z). We have obtained
1. a necessary and sufficient condition of Z for the homogeneous space G/C(Z) admits a G-invariant Kahler structure.
Secondly, under the condition that G is simple and compact
2. the maximality of C(Z) implies dimC(Z) = 1 and in this case, the Kahler structure of G/C(Z) is unique up to trivial changes of complex structures and Kahler metrics.
In non-compact case, dimC(Z) = 1 if and only if the homogeneous Kahler manifold is symmetric. The rerult 2 imples the existence of non-symmetric homogeneous Kahler manifolds G/C(Z) with dimC(Z) = 1.
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