Noncommutative Geometry and equivariant index theorem for twisted group actions

扭曲群作用的非交换几何和等变指数定理

• 批准号: 19540099
• 负责人: MORIYOSHI Hitoshi
• 金额: $ 2.91万
• 依托单位: Keio University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2009
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19540099/
• 关键词: 非可換幾何学; 指数定理; K理論; エータ不変量; 葉層構造; 巡回コホモロジー; 幾何学; 位相幾何学; Godbillon-Vey不変量; 国際研究者交流; イタリア:中国:アメリカ; 接触構造; 佐々木多様体

The objective in this project of research is a generalization of the Atiyah-Singer Index Theorem from the viewpoint of Noncommutative Geometry, in situations such as foliated manifolds and manifolds with boundary. We focused on tow notions, the twisted K-theory and the group C*-algebras twisted by cocycles, and sought for Atiyah-Singer type index theorems related to them. We first established a Atiyah-Singer type index theorem for a twisted group C*-algebras of the fundamental group on K-aspherical kaehler manifolds, and obtained certain inequalities on the arithmetic genera of complex manifolds, which is related to the vanishing theorem due to Green and Lazarsfeld. Second we formulated the Atiyha-Patodi-Singer index theorem in the framework of Noncommutative Geometry and extended the theorem in the cases of covering space with infinite degree and foliated manifold with boundary. This formulation provide us with an interpretation of the eta invariant including the higher one, as a pairing of relative cyclic cohomology and relative K-theory, which makes even clearer the geometric significance of eta invariants.

本课题的研究目标是从非交换几何的角度对叶状流形和带边界流形的Atiyah-Singer指数定理进行推广。重点讨论了两个概念，即扭曲k理论和被环扭曲的群C*-代数，并寻求了与它们相关的Atiyah-Singer型指标定理。我们首先建立了k -非球kaehler流形上基本群的扭曲群C*-代数的Atiyah-Singer型指标定理，并得到了复流形算术属上的若干不等式，这些不等式与Green和Lazarsfeld的消失定理有关。其次，在非交换几何的框架下建立了Atiyha-Patodi-Singer指标定理，并在无限次覆盖空间和有边界的叶状流形的情况下推广了该定理。这个公式为我们提供了对不变量的解释，包括更高的不变量，作为相对循环上同调和相对k理论的配对，这使得不变量的几何意义更加清晰。

项目成果

Twisted Riemann-Roch theorem on K-aspherical manifold

K-非球面流形上的扭曲黎曼-罗赫定理

• 发表时间: 2009
• 作者: Y. Maeda;et. al.;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;H.Moriyoshi;H.Moriyoshi;H.Moriyoshi
• 通讯作者: H.Moriyoshi

Twisted Riemann-Roch theorem on K-aspherical manifold, The godobillon-Vey eta cocycle on foliated manifolds with boundary

K-非球面流形上的扭曲黎曼-罗赫定理，有边界的叶流形上的 godobillon-Vey eta 循环

• 发表时间: 2009
• 作者: Y. Maeda;et. al.;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;H.Moriyoshi;H.Moriyoshi;H.Moriyoshi;森吉仁志;H. Moriyoshi;森吉仁志
• 通讯作者: 森吉仁志

Kahler hyperbolicity and Twisted Index Theorem

卡勒双曲性和扭曲指数定理

• 发表时间: 2007
• 期刊: Geometry and Something, Fukuoka
• 作者: Y. Kamiyama;S. Tsukuda;S. Tsukuda;S. Tsukuda;S. Tsukuda;H.Moriyoshi;H.Moriyoshi;H.Moriyoshi
• 通讯作者: H.Moriyoshi

葉層多様体上の指数定理と非可換幾何学

叶流形上的指数定理和非交换几何

• 发表时间: 2008
• 作者: Y. Maeda;et. al.;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;H.Moriyoshi;H.Moriyoshi;H.Moriyoshi;森吉仁志;H. Moriyoshi;森吉仁志;森吉仁志;H. Moriyoshi;森吉仁志
• 通讯作者: 森吉仁志

wisted Index Theorem and type III factors

扭曲指数定理和第三类因子

• 发表时间: 2007
• 作者: Y. Maeda;et. al.;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;森吉仁志;H.Moriyoshi;H.Moriyoshi;H.Moriyoshi;森吉仁志;H. Moriyoshi;森吉仁志;森吉仁志;H. Moriyoshi;森吉仁志;森吉仁志;森吉仁志;H. Moriyoshi;H. Moriyoshi;森吉仁志
• 通讯作者: 森吉仁志