A Poincare-Hopf type theorem for holomorphic one forms

我们解释主要结果。设ω是定义在圆盘D^<2n>(1)的邻域上的可积全纯一形式∪C^n， n【大于等于】2。假设余维数为1的全纯叶理F（ω）由ω定义，它横向于D^<2n>(1)的边界S^<2n-1>(1)。通过莫比乌斯变换，我们可以假设ω在D^<2n>(1)内的唯一一个奇点是原点0。定理（[6]）如果F（ω）有一个叶L，使得L具有下列性质(i) ~ (iii)，则n=2。(i) 0∈L^^-, (ii) L闭合于∪\Sing(ω), （iii) L横向于各球面S^<2n-1>(r), 0<r <小于等于）1。设ω是定义在邻域∪D^<2n>(1)∧C^n， n【大于等于】3中的全纯一种形式，使得Sing(ω)_∩S^<2n-1>(1)=φ。设ξ是定义在∪中的全纯向量场。定理（[7]）如果ω(ξ)=0且ξ横于S^<2n-1>(1)，则ω不可积。定理（[8]）设X是C^n上的多项式向量场，n[大于等于]2，具有孤立奇点。如果由X在CP (n)上的解定义的全纯叶理F(X)具有双曲型奇点，则下列条件是等价的。(i) F(X)在C^n上有n个分离矩阵，并与一系列球面S^<2n-1>(p_j, R_j)∧C^n，其中<lim>___<j→∞> R_j=+∞。（ii）在C^n上的仿射图中X是线性的（庞加莱双曲型）。

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.