In this note we announce some achievements in the study of holomorphic distributions admitting transverse closed real hypersurfaces. We consider a domain with smooth boundary in the complex affine space of dimension two or greater. Assume that the domain satisfies some cohomology triviality hypothesis (for instance, if the domain is a ball). We prove that if a holomorphic one form in a neighborhood of the domain is such that the corresponding holomorphic distribution is transverse to the boundary of the domain then the Euler-Poincaré-Hopf characteristic of the domain is equal to the sum of indexes of the one-form at its singular points inside the domain. This result has several consequences and applies, for instance, to the study of codimension one holomorphic foliations transverse to spheres.
Holomorphic one-form; vector field; Euler-Poincaré characteristic; foliation; distribution
