On the Poincar′e-Hopf Index Theorem for the complex case

In this paper we consider holomorphic vector fields and holomorphic one-forms defining distributions which are transverse to the boundary of a regular domain on a complex manifold or on the complex affine space. In both cases we prove that, under a natural extensibility topological hypothesis, the Euler-Poincar′e-Hopf characteristic of the domain is equal to the sum of indexes of the complex vector field or complex one-form at its singular points inside the domain.