Poincare-Bendixson type theoreom for holomorphic foliation of codimension one and its application

本课题的目的是将a . douady和T.Ito证明的“全纯向量场的庞加莱-本迪克森型定理”推广到余维为1的全纯叶理。我们将研究以下问题：问题设F是定义在闭盘的邻域上的余维为1的全纯叶形<B^<2n>(R)>^^^-={Z∈C^n||Z|【小于或等于】R}如果F横向于S^<2n-1>(R)=∂<B^<2n>(R)>^^^-，那么关于F可以说什么？得到以下结果：(1)不存在使F横切到S^<4m+1>(R)的全纯叶化F （C^<2m+1>）。(2)如果F由齐次可积的一种形式ω定义，则F不横向于S^<2n-1>(R)， n【大于或等于】3。(3)设F定义在闭合多盘<Δ^n(1)>^^ - (C^n)的邻域U上。若F横过<Δ^n(1)>^^ -的边界，则F是ψ^*(L)，其中L是C^n上的双曲线性对数叶理，映射ψ:U→C^n是全纯的。(4)设ω是定义在<B^<2n> ^^- (R)>^^ - (C^n)的邻域上的全纯一形式。如果Ker（ω）横向于S^<2n-1>(R)，那么我们就有了ω的庞加莱-霍普夫型定理。(5)若Ker（ω）横于S^<2n-1>(R)，则在B^<2n>(R)内只有一个ω奇点，且该点简单。

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.