2005 Fiscal Year Final Research Report Summary
Lie group-theoretic approach to the holomorphic equivalence problem and related various problems in several complex variables
| Project/Area Number |
14540149
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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 |
Basic analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
SHIMIZU Satoru Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90178971)
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| Co-Investigator(Kenkyū-buntansha) |
KENMOTSU Katsuei Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004404)
TAKEUCHI Shigeru Gifu University, Faculty of Education, Professor, 教育学部, 教授 (30021330)
KODAMA Akio Kanazawa University, Graduate School of Natural Science and Technology, Professor, 大学院・自然科学研究科, 教授 (20111320)
NAKAGAWA Yasuhiro Kanazawa University, Graduate School of Natural Science and Technology, Associate Professor, 大学院・自然科学研究科, 助教授 (90250662)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Holomorphic equivalence problem / Holomorphic automorphism group / Lie group / Reinhardt domain / Riemann mapping theorem / Torus action / Bando-Calabi-Futaki character / Quasi-circular domain |
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| Research Abstract |
In this research, centering our study in the holomorphic equivalence problem, holomorphic automorphism groups, Reinhardt domains, and torus actions, we have obtained the following results.
1.Recently, an answer was given to the holomorphic equivalence problem for certain unbounded Reinhardt domains called elementary Reinhardt domains, whose method uses pluricomplex Green functions. We have given another approach to this problem that uses the theory of holomorphic automorphism groups. Also, related to the study of Reinhardt domains, we have developed the study of a group-theoretic characterization of the space X obtained by omitting the coordinate hyperplanes from the complex Euclidean space, and succeeded in characterizing X in the category of manifolds that are not necessarily Stein. As a by-product of this study, we have obtained a result on the standardization of rank n compact Lie group actions on n-dimensional complex manifolds, and moreover, as an application, we have succeeded in
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characterizing the space given as the direct product of the ball and the complex Euclidean space by its holomorphic automorphism group as well.
2.As part of the study of the holomorphic equivalence problem and holomorphic automorphism groups, we have made one formulation to generalize the Riemann mapping theorem to the higher-dimensional case by utilizing holomorphic automorphism groups. According to this formulation, we have characterized the direct of balls by its holomorphic automorphism group.
3.As part of the study of torus actions, we have studied the Bando-Calabi-Futaki character. In particular, we have made a detailed ovservation of the connection between the stability of polarized algebraic varieties and the existence of Kaehler metrics of constant scalar curvature (what is called Hitchin-Kobayashi correspondence for manifolds). Also, related to the study of one-dimensional torus actions on two-dimensional complex manifolds, we have obtained a new knowledge of the holomorphic automorphism groups of two-dimensional quasi-circular domains. Less
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