A Poincare-Hopf type theorem for holomorphic one forms

我们解释了主要结果.设ω是定义在圆盘D^&lt；2n&gt；(1)∪C^n，n[大于或等于]的邻域内的可积全纯形式2.假设由⊂定义的余维1的全纯叶F(ω)横跨于D^&lt；2n&gt；(1)的边界S^&lt；2n-1&gt；(1).通过莫比乌斯变换，我们可以假设ω在D^&lt；2n&gt；(1)内唯一的一个奇点是原点。定理([6])如果F(ω)有叶L，使得L具有下列性质：(I)0∈L^-，(Ii)L在∪\Sing(ω)中闭，(Iii)L横切于每个叶S^&lt；2n-1&gt；(R)，0&lt；R[小于等于]1.设ω是定义在D^&lt；2n&gt；(1)的邻域∪中的全纯One形式，使得⊂C^n，n[大于等于]3使得Sing(ω)_∩S^&lt；2n-1&gt；(1)=φ.设ξ是∪定义的全纯向量场，定理([7])若ω(ξ)=0，且ξ横切于S^&lt；2n-1&gt；(1)，则ω不可积.定理([8])设X是C^n，n[大于或等于]2上具有孤立奇点的多项式向量场.如果由X在Cp(N)上的解所定义的全纯叶层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个分离阵，且横切于球面序列S^&lt；2n-1&gt；(p_j，R_j)⊂C^n其中&lt；Lim&gt；_&lt；j→∞&gt；R_j=+∞。(Ii)X在C^n上的某个仿射图中是线性的(Poincare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.