2009 Fiscal Year Final Research Report
Noncommutative Geometry and equivariant index theorem for twisted group actions
| Project/Area Number |
19540099
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Keio University |
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Principal Investigator |
MORIYOSHI Hitoshi Keio University, 大学院・多元数理科学研究科, 教授 (00239708)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Yoshiaki 慶應義塾大学, 理工学部, 教授 (40101076)
KAMETANI Yukio 慶應義塾大学, 理工学部, 准教授 (70253581)
MIYAZAKI Takuya 慶應義塾大学, 理工学部, 准教授 (10301409)
TOSE Nobuyuki 慶應義塾大学, 経済学部, 教授 (00183492)
MIYAZAKI Naoya 慶應義塾大学, 経済学部, 教授 (50315826)
KATO Tsuyoshi 京都大学, 大学院・理学研究科, 教授 (20273427)
NATSUME Toshikazu 名古屋工業大学, 工学系研究科, 教授 (00125890)
MITSUMATSU Yoshihiko 中央大学, 理工学部, 教授 (70190725)
ONO Kaoru 北海道大学, 大学院・理学研究院, 教授 (20204232)
WATAMUAR Satoshi 東北大学, 大学院・理学研究科, 准教授 (00201252)
KATSURA Takeshi 慶応義塾大学, 理工学部, 講師 (50513298)
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| Project Period (FY) |
2007 – 2009
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| Keywords | 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 |
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| Research Abstract |
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.
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