| Project/Area Number |
18540188
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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 | Kumamoto University |
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Principal Investigator |
ABE Makoto Kumamoto University, School of Health Sciences, Associate Professor (90159442)
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| Co-Investigator(Kenkyū-buntansha) |
SHIMA Tadashi Hiroshima University, Department of Information Engineering, Associate Professor (30226196)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥2,170,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | complex space / Stein space / Stein manifold / meromorphic convexity / rational convexity / meromorphic approximation theorem / Cartier divisor / Picard group |
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| Research Abstract |
Connected to the meromorphic convexity in Stein spaces, Makoto Abe obtained the following results.
(1) It is proved that there exists an open set D of C^n which is not polynomially convex and satisfies the strong disk property in C^n if n≧2. This also gives a negative answer to the problem of Bremermann(1958).
(2) By considering the local Steinness for an open set of possibly non-reduced Cohen-Macaulay Stein space, it is proved that the Steinness of an open set D of a Stein manifold X of dimension n satisfying the condition that dim H^k(D, O)< +∞ (2≦k ≦n-1) by the surjectivity of the canonical homomorphism Div(D)→Pic(D).
(3) By giving examples, it is proved that a rationally convex open set D of C^n, where n≧2, does not satisfy in general the strong meromorphic approximation property in C^n. It is also proved that every open set D of a reduced Stein space X of dimension 1 satisfies the strong meromorphic approximation property in X.
(4) A generalized meromorphic approximation theorem in a reduced Stein space X with respect to a subsemigroup G of the Picard group Pic(X)is obtained, which includes both two different meromorphic approximation theorems due to Hirschowitz(1971) and due to Abe (2005).
Tadashi Shima(with Mikio Furushima and Yasuhiro Ohshima) obtained the following result.
(5) Let G be a small finite subgroup of general linear group GL(2, C). Then the classification of the dual graphs of the boundaries in the minimal normal analytic compactifications of C^2/G is completed, which is a continuation of the earlier works due to Abe-Furushima-Yamasaki (2000) and Abe-Furushima-Shima (2004).
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