2001 Fiscal Year Final Research Report Summary
| Project/Area Number |
11640076
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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 | Osaka University |
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Principal Investigator |
GOTO Ryushi Osaka University, Graduate School of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (30252571)
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| Co-Investigator(Kenkyū-buntansha) |
NAMIKAWA Yoshinori Osaka University, Graduate School of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (80228080)
MABUCHI Toshiki Osaka University, Graduate School of Mathematics, Professor, 大学院・理学研究科, 教授 (80116102)
FUJIKI Akira Osaka University, Graduate School of Mathematics, Professor, 大学院・理学研究科, 教授 (80027383)
OHYAMA Yosuke Osaka University, Graduate School of Mathematics, Lecturer, 大学院・理学研究科, 講師 (10221839)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Hyper Kahler manifolds / Calabi-Yau manifolds / G_2 manifolds / Spin(7) manifolds |
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| Research Abstract |
Let X be a compact Riemannian manifold with vanishing Ricci curvature. Then the list of holonomy group of X includes four interesting classes of the holonomy groups: SU(n), Sp(m), G_2 and Spin(7). The Lie group SU(n) arises as the holonomy group of Calabi-Yau manifolds and Sp(m) is the holonomy group of hyper Kahler manifolds. G_2 and Spin(7) occur as the holonomy groups of 7 and 8 dimensional manifolds respectively. There are many intriguing common properties between these four geometries. The author's research is based on the study of hyper Kahler manifolds, from which he obtains some ideas with the potential to unify moduli spaces results of these different kinds of geometric structures. His main results are the followings:
(1) One of the most remarkable common property is smoothness of the deformation spaces of these geometric structures. The author show that these deformations can be regarded as deformations of special kind of differential forms and he constructs a new kind of deformations theory of these differential forms. Then an obstruction of the deformation is given by the certain exact forms. Hence he shows that the obstruction vanishes in terms of cohomological (topological) argument. This result can be considered as a natural generalization of Kodaira-Spencer theory.
(2) He also constructs the moduli space and shows that the local Torelli's type theorem holds in these cases.
(3) As an application he obtain the smooth moduli space of Calabi-Yau structures and Special Lagrangian submanifolds.
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