| Project/Area Number |
13640092
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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 | Ryukoku University |
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Principal Investigator |
ITO Toshikazu Ryukoku Univ., Faculty of Economics, Professor, 経済学部, 教授 (60110178)
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| Co-Investigator(Kenkyū-buntansha) |
KOKUBO Hiroe (OKA Hiroe) Ryukoku Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (20215221)
YOTSUTANI Shoji Ryukoku Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (60128361)
MATSUMOTO Waichiro Ryukoku Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (40093314)
YAMAGISHI Yoshikazu Ryukoku Univ., Faculty of Sci.and Tech., Assistant, 理工学部, 助手 (40247820)
MORITA Yoshihisa Ryukoku Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (10192783)
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| Project Period (FY) |
2001 – 2003
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| Keywords | holomorphic one form / holomorphic foliation / Poincare-Bendixson theorem / Poincare-Hopf theorem / holomorphic vector field / singularity / transversality |
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| Research Abstract |
The aim of this project is an extension of "Poincare-Bendixson type theorem for holomorphic vector field" proved by A.Douady and T.Ito to a holomorphic foliation of codimension one. We will investigate the following question :
Question Let F be a holomorphic foliatiop of codimension one defined in a neighborhood of a closed disc <B^<2n>(R)>^^^-={Z∈C^n||Z|【less than or equal】R}⊂C^n. If F is transverse to S^<2n-1>(R)=∂<B^<2n>(R)>^^^-, then what can be said about F?
We got the following results :
(1)There is no holomorphic foliation F such that F is transverse to S^<4m+1>(R)⊂C^<2m+1>.
(2)If F is defined by a homogeneous integrable one form ω, F is not transverse to S^<2n-1>(R), n【greater than or equal】3.
(3)Let F be defined in a neighborhood U of a closed polydisk <Δ^n(1)>^^^-⊂C^n. If F is transverse a boundary of <Δ^n(1)>^^^-, then F is ψ^*(L), where L is a hyperbolic linear logarithmic foliation on C^n and a map ψ:U→C^n is holomorphic.
(4)Let ω be a holomorphic one form defined in a neighborhood of <B^<2n>(R)>^^^-⊂C^n. If Ker(ω) is transverse to S^<2n-1>(R), then we have a Poincare-Hopf type theorem for ω.
(5)If Ker(ω) is transverse to S^<2n-1>(R), then there is only one singular point of ω inside B^<2n>(R) and this point is simple.
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