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

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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.