A Poincaré-Hopf-type theorem for holomorphic one-forms

We prove that if a holomorphic one-form Ω in a neighborhood of a closed euclidian ball B2n⊂Cn, in the n-dimensional complex affine space, defines a distribution transverse to the boundary sphere S2n−1=∂B2n, then n is even and Ω admits a sole singularity q∈B2n. Moreover, this singularity is simple.