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-Flat Riemannian Manfolds的自治组的可能谎言群体的清单意味着有四个有趣的谎言组：SU（N），SP（N），SP（M），G_2和Spin（7）。特别单位组SU（N）是calabi-yau歧管和SP（M）的整体群体，是exply spectiation speriate speriate speriate speriate sperialtiation speriate sperialt sperialt sperialtiation spectiate and offords speriorts。自旋（7）分别是7和8维歧管的整体组，称为g_2和自旋（7）歧管，这四个类别的riemannian歧管之间存在浅表差异，但是作者表明，这些四个结构被认为是在特殊差异的情况下所构成的，他认为这些结构被认为是一个新的构造，他将其视为均匀的范围。因此，有限的维度。 In fact, he develops deformation problems of (1) holomorphic symmetry structures and (2) generalized geometric structures: (CONTINUE TO NEXT PAGE)(1) holomorphic symmetry structuresThe author studies holomorphic symplexic 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复杂的Nilmanifold和进一步构建了一个紧凑的圆锥体相称的歧管的例子，这只是妨碍了变形。（2）广义的几何结构的概念是通用的几何结构的概念，最近，Hitchinis引入了Hitchinis，这是基于一个思想，基于一个构想的构造和cot bunstral complenter complent complent and Complent and Complent and Complent and Complent and Comperty and Complent and Complent and Complent and Comperty complent and Comperty complent and Comperty complent and Comperty and Complent and Comperty and Comperty。作者的复杂结构。作者的重点是Clifford代数，并表明可以将广义结构适当地理解为结构所定义的结构，该结构由整形销钉组下的作用定义。自旋（7）结构。较少的