2001 Fiscal Year Final Research Report Summary
Development of Geometric Complex Analysis
| Project/Area Number |
12440035
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Nagoya University |
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Principal Investigator |
OHSAWA Takeo Nagoya University, Math. Professor, 大学院・多元数理科学研究科, 教授 (30115802)
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| Co-Investigator(Kenkyū-buntansha) |
SUZUKI Noriaki Nagoya University, Math. Associate Prof., 大学院・多元数理科学研究科, 助教授 (50154563)
NAKANISHI Toshihiro Nagoya University, Math. Associate Prof., 大学院・多元数理科学研究科, 助教授 (00172354)
KAZAMA Hideaki Kyusyu Univ. Math. Professor, 大学院・数理学研究科, 教授 (10037252)
YAMAGUCHI Hiroshi Nara Woman's Univ. Math. Professor, 理学部, 教授 (20025406)
NOGUCHI Junjirou Tokyo Univ. Math. Professor, 大学院・数理科学研究科, 教授 (20033920)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Pseudoconvox domain / ∂^^- equation / L^2 holomorphic function / Levi flatness / L^2 division theorem / Adherent complex curve / Bergman kernel / Hefer decomjposition |
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| Research Abstract |
In spite of a lot of work in complex analysis and conplex analytic geometry in the last century, there seem to remain unnoticed important questions in the basic theory of several complex variables. Relation between the extension and the sivision problems is very likely one of them. The purpose of this research project was to get a new viewpoint of relating extension and division problems on complex manifolds after the author's previous works on the L^2extention theorems for holomorphic functions. As a result we succeeded in improving a well known L^2division theory of H.Skoda. On the other hand, there was a progress concering Levi flat hypersurfaces : Let X be a complex manifold of dimension n and let M be a real hypersurface of X. M is called Levi flat if it locally separates X into two Stein domains i.e.if M is locally psendoconvex from both sides. In recent works of Lins-Nets and Ohsawa, it was proved that complex projective space of dimension n contains no compact real analytic Levi flat hypersurfaces if n 【greater than or equal】 2. It was extended in a joint work with K.Matsumoto by studying the geometry of Levi flat hypersurfaces in complex tori. Unlike the case of projective spaces, tori contain infinitely many compact Levi flat hypersurfaces. We determined all such jypersurfaces under the assumption of real analyticity when the dimension of tori is 2.
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