Differentiable equisingularity of holomorphic foliations

Abstract

We prove that a $C^{\infty }$ equivalence between germs holomorphic foliations at $({\mathbb{C}}^2,0)$ establishes a bijection between the sets of formal separatrices preserving equisingularity classes. As a consequence, if one of the foliations is of second type, so is the other and they are equisingular.