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Unified approach of Ricci-flat manifolds

Ricci平坦流形的统一方法

基本信息

批准号：
15540070
负责人：
GOTO Ryushi
金额：
$ 2.43万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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 … More 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 structuresThe 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 structuresA notion of generlized geometric structures, which is recently introduced by HitchinIs 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

由ricci -平坦黎曼倒褶的完整群所产生的可能李群的列表表明，有四类有趣的李群：SU(n)、Sp(m)、G_2和Spin(7)。特殊酉群SU(n)是Calabi-Yau流形的完整群，Sp(m)是hyperK"ahler流形的完整群。特殊李群G_2和Spin(7)分别是7维和8维流形的完整群，分别称为G_2和Spin(7)流形。这四类黎曼流形表面上有区别，但作者认为这四种结构都是由特殊的闭微分形式定义的几何结构。他提出了一种新的方法来研究这些结构的变形问题。他证明了在一定的上同调条件下，变形空间成为有限维的光滑流形。由此得到了这四种结构的模空间的统一构造。这种方法是非常普遍的，他希望存在许多几何结构，他的方法可以有效地应用。事实上，他发展了(1)全纯辛结构和(2)广义几何结构的变形问题：（继续到下一页）(1)全纯辛结构作者研究了不一定是K ' ahlerian的全纯辛结构。他得到了一个新的无阻碍变形判据和局部Torelli型定理。他还证明了该准则在复零流形上成立，并进一步构造了一个刚好有阻塞变形的紧致全纯辛流形的例子。(2)广义几何结构（generalized geometric structures）广义几何结构的概念是由hitchini最近提出的，它基于一个思想，即用流形上的正切束和余切束的直接和来代替正切束。然后将复杂结构和实辛结构作为广义复杂结构的特例。作者着重讨论了Clifford代数，并证明了广义结构可以恰当地理解为由共形针群下的作用所定义的结构。得到了广义Calabi-Yau、hyperK"ahler G_2和Spin(7)结构的自然概念，建立了广义结构的变形理论。特别是，他研究了广义Calabi-Yau和Spin(7)结构的无阻碍变形。少

项目成果

期刊论文数量（36）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

An energy theoretic approach to the Kobayashi-Hitchin correspondence for manifold I

流形 I 的 Kobayashi-Hitchin 对应关系的能量理论方法

DOI：
发表时间：
2004
期刊：
Invent.Math 159
影响因子：
0
作者：
H.Ohta;K.Ono;R.Goto;T.Mabuchi
通讯作者：
T.Mabuchi

Moduli spaces of topological calibrations, Calabi-Yau, hyperK"abler, G_2 and Spin(7) structures

拓扑校准的模空间、Calabi-Yau、hyperK"abler、G_2 和 Spin(7) 结构

DOI：
发表时间：
2004
期刊：
International Journal of Math. 69, No.3
影响因子：
0
作者：
H.Ohta;K.Ono;R.Goto
通讯作者：
R.Goto

Twistor spaces of algebraic dimension two, associated to a connected sum of projective planes

代数二维扭量空间，与射影平面的连通和相关

DOI：
发表时间：
2004
期刊：
Compositio Math. 140
影响因子：
0
作者：
H.Ohta;K.Ono;R.Goto;T.Mabuchi;T.Mabuchi;A.Fujiki;R.goto;T.Mabuchi;T.Mabuchi;A.Fujiki
通讯作者：
A.Fujiki

Moduli spaces of topological calibrations, Calabi-Yau, hyperKahler, G_2 and Spin(7) structures

拓扑校准的模空间、Calabi-Yau、hyperKahler、G_2 和 Spin(7) 结构

DOI：
发表时间：
2004
期刊：
International J. of Math. 69
影响因子：
0
作者：
M.Miyanishi;K.Masuda;Y.Hara;A.Fujiki;Y.Namikawa;T.Mabuchi;R.Goto;A.Fujiki;Y.Namikawa;T.Mabuchi;R.Goto
通讯作者：
R.Goto

Simple Singularities and Symplectic fillings.

简单奇点和辛填充。