1998 Fiscal Year Final Research Report Summary
Study of group actions on complex manifolds
| Project/Area Number |
09640145
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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 |
解析学
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| Research Institution | Tohoku University |
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Principal Investigator |
SHIMIZU Satoru Graduate School of Science, Tohoku University Associate Professor, 大学院・理学研究科, 助教授 (90178971)
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| Co-Investigator(Kenkyū-buntansha) |
KODAMA Akio Kanazawa University, Faculty of Science, Professor, 理学部, 教授 (20111320)
TAKEUCHI Shigeru Gifu University, Faculty of Education, Professor, 教育学部, 教授 (30021330)
NAKAGAWA Yasuhiro Graduate School of Science, Tohoku University Research Associate, 大学院・理学研究科, 助手 (90250662)
OGATA Shoetsu Graduate School of Science, Tohoku University Associate Professor, 大学院・理学研究科, 助教授 (90177113)
ARAI Hitoshi Graduate School of Science, Tohoku University Professor, 大学院・理学研究科, 教授 (10175953)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Complex bounded domain / Holomorphic equivalence problem / Holomorphic automorphism group / Reinhardt domain / Torus action / CR-structure / Futaki character / Tangential Cauchy-Riemann equation |
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| Research Abstract |
In this research, we have obtained the following results related to group actions on complex manifolds.
1. Concerning group actions on complex manifolds, we have obtained the fundamental result on the normalization of torus actions on unbounded Reinhardt domains. More precisely speaking, we have given an answer to the holomorphic equivalence problem for a basic class of unbounded Reinhardt domains and, as an application, we have shown the conjugacy of torus actions on such a class of Reinhardt domains.
2. Related to the study of group actions on complex domains, we have given a characterization of generalized complex ellipsoids from the viewpoint of biholomorphic automorphism groups by making use of well-known extension theorems on holomorphic mappings and CR-mappings and applying Webster's CR-invariant metrics.
3. Related to the study of group actions on the boundaries of complex domains, we have investigated CR structures. In particular, we have shown that it is possible to formulate the concepts of dual spaces and tensor products in the category of CR vector spaces under some additional conditions.
4. As a study of torus actions, we have investigated the Bando-Calabi-Futaki character that is a generalization of the Futaki character. Also, we have calculated signature defects of degenerations-of abelian varieties
5. Related to the analysis of the boundaries of complex domains, we have proved new Morrey-Holder estimates for the Cauchy-Riemann equations on strongly pseudoconvex CR manifolds. Also, to study group actions in a wider mathematical framework, we have developed the investigation of equivariant harmonic mappings.
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