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

作为 Ricci 平黎曼流形的完整群而出现的可能李群列表意味着存在四类有趣的李群：SU(n)、Sp(m)、G_2 和 Spin(7)。特殊酉群 SU( n) 作为 Calabi-Yau 流形的完整群而出现，Sp(m) 是 hyperK"ahler 流形的完整群。例外的李群 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。这四类黎曼流形之间存在表面差异，但作者表明这四种结构被视为几何结构他得到了这些结构变形问题的新方法，证明在一定的上同调条件下，变形空间成为有限维的光滑流形。因此，他获得了这四种结构的模空间的统一构造。这种方法非常通用，他期望应该存在许多可以有效应用他的方法的几何结构。事实上，他开发了（的变形问题。 1）全纯辛结构和（2）广义几何结构：（继续下一页）（1）全纯辛结构作者研究了非必要的全纯辛结构K"ahlerian。他得到了新的无阻碍变形判据和局域Torelli型定理。他还证明了该判据在复尼尔曼流形上成立，并进一步构造了一个仅阻碍变形的紧全纯辛流形的例子。(2)广义几何结构A广义几何结构的概念，这是希钦最近提出的，基于用正切和的直和代替切丛的思想然后复数结构和实辛结构被视为广义复数结构的特例。作者重点研究了 Clifford 代数，并表明广义结构可以适当地理解为由共形 pin 群下的作用定义的结构。然后他得到了广义Calabi-Yau、hyperK"ahler G_2和Spin(7)结构的自然概念，并建立了广义结构的变形理论。特别是，他有广义 Calabi-Yau 和 Spin(7) 结构的无阻碍变形。