Unified approach of Ricci-flat manifolds
Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants |
|Research Institution||Osaka University |
GOTO Ryushi Osaka University, Graduate School of Science, Associate professor, 大学院・理学研究科, 助教授 (30252571)
FUJIKI A. Osaka University, Graduate School of Science, professor, 大学院・理学研究科, 教授 (80027383)
MABUCHI T. Osaka University, Graduate School of Science, professor, 大学院・理学研究科, 教授 (80116102)
NAMIKAWA Y. Osaka University, Graduate School of Science, professor, 大学院・理学研究科, 教授 (80228080)
FUKAYA K. Kyoto University, Department of mathematics, professor, 大学院・理学研究科, 教授 (30165261)
ONO K. Hokkaido University, Department of Mathematics, professor, 大学院・理学研究科, 教授 (20204232)
|Project Period (FY)
2003 – 2005
Completed(Fiscal Year 2005)
|Budget Amount *help
¥3,800,000 (Direct Cost : ¥3,800,000)
Fiscal Year 2005 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 2004 : ¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 2003 : ¥1,600,000 (Direct Cost : ¥1,600,000)
|Keywords||Calabi-Yau structure / hyperK"ahler structure / G_2 structure / Spin(7) structure / geometric structure / Deformations / Moduli spaces / generalized structure / カラビーヤオ構造 / カラビ・ヤオ多様体 / 超ケーラー多様体 / G_2多様体 / シンプレクティク多様体 / Spin(7)多様体|
The list of possible Lie groups arising as holonomy groups of Ricci-flat Riemannian manfolds implies that there are four interesting classes of Lie groups : SU(n), Sp(m), G_2 and Spin(7).
The special unitary group SU(n) arises as the holonomy group of Calabi-Yau manifolds and Sp(m) is the holonomy group of hyperK"ahler manifolds. The exceptional Lie group G_2 and Spin(7) are respectively holonomy groups of 7 and 8 dimensional manifolds, which are called G_2 and Spin(7) manifolds.
There are superficial differences between these four classes of Riemannian manifolds, however the author shows that these four structures are regarded as geometric structures defined by special closed differential forms. He obtains a new approach of deformation problems of these structures. He shows that under certain cohomological condition, deformation space becomes a smooth manifolds of finite dimension. Hence he obtains a unified construction of moduli spaces of these four structures.
This approach is quite g
eneral and he expects that there should exist many geometric structures on which his approach can be applied effectively. In fact, he develops deformation problems of (1) holomorphci symplectic structures and (2) generalized geometric structures : (CONTINUE TO NEXT PAGE)
(1)holomorphic symplectic structures
The author studies holomorphic symplectic structures which are not necessary K"ahlerian. He obtains a new criterion of unobstructed deformations and local Torelli type theorem. He also shows that the criterion holds on complex Nilmanifolds and further constructs an example of compact holomorphic symplectic manifold which has just obstructed deformations.
(2)generalized geometric structures
A notion of generlized geometric structures, which is recently introduced by Hitchin
Is based on an idea replacing the tangent bundle with the direct sum of the tangent and cotangent bundle on a manifold. Then complex structures and real symplectic structures are regarded as special cases of generalized complex structures.
The author focuses on the Clifford algebra and shows that generalized structures can be suitably understood as structures defined by the action under the conformal pin group.
Then he obtains a natural notion of generalized Calabi-Yau, hyperK"ahler G_2 and Spin(7) structures and establishes a deformation theory of generalized structures.
In particular, he has unobstructed deformations of generalized Calabi-Yau and Spin(7) structures. Less
Research Products (18results)