2006 Fiscal Year Final Research Report Summary
A Poincare-Hopf type theorem for holomorphic one forms
| Project/Area Number |
16540086
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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) |
MATSUMOTO Waichiro Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (40093314)
YOTSUTANI Shoji Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (60128361)
KOKUBU Hiroe (OKA Hiroe) Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (20215221)
NINOMIYA Hirokazu Ryukoku Univ., Faculty of Sci. and Tech., Assistant Professor, 理工学部, 助教授 (90251610)
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| Project Period (FY) |
2004 – 2006
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| Keywords | holomorphic one form / Poincare-Hopf type theorem / holomorphic foliation / separatrix / hyperbolic singularity / holomorphic vector field / transversity |
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| Research Abstract |
We explain principal results.
Let ω be an integrable holomorphic one form defined in a neighborhood ∪ of the disk D^<2n>(1)⊂C^n, n【greater than or equal】2. Assume that the holomorphic foliation F(ω) of codimension one defined by ω is transverse to the boundary S^<2n-1>(1) of D^<2n>(1). By Mobius transformation, we can suppose that the only one singular point of ω inside D^<2n>(1) is the origin 0.
Theorem([6]) If F(ω) has a leaf L such that L has the following properties (i)〜(iii), then n=2.
(i) 0∈L^^-, (ii) L is closed in ∪＼Sing(ω), (iii) L is transverse to each shpere S^<2n-1>(r), 0<r【less than or equal】1.
Let ω be a holomorphic one form defined in a neighborhood ∪ of D^<2n>(1)⊂C^n, n【greater than or equal】3 such that Sing(ω)_∩S^<2n-1>(1)=φ. Let ξ be a holomorphic vector field defined in ∪.
Theorem([7]) If ω(ξ)=0 and ξ is transverse to S^<2n-1>(1), then ω is not integrable.
Theorem([8]) Let X be a polynomial vector field on C^n, n【greater than or equal】2 with isolated singularities. If the holomorphic foliation F(X) defined by solutions of X on CP (n) has singularities of hyperbolic type, the following conditions are equivalent.
(i) F(X) has n separatrices on C^n and is transverse to a sequence of spheres S^<2n-1>(p_j, R_j)⊂C^n where <lim>___<j→∞> R_j=+∞.
(ii) X is linear (of Poincare hyperbolic type) in some affine chart on C^n.
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