Holomorphic foliations desingularized by punctual blow-ups

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Given F be a germ of codimension-one singular holomorphic foliation at the origin 0∈ C 3 . We assume that F can be desingularized by a certain sequence of punctual blow-ups producing only simple singularities (Definition 1). This case is studied in analogy with the case of Kleinian singularities of complex surfaces. It is proved that F is given by a simple poles closed meromorphic 1-form provided that, along the reduction process, the simple singularities exhibit a hyperbolic transverse type (Theorem 3). In the non-hyperbolic case, we prove the existence of a formal integrating factor if we interdict the existence of holomorphic first integrals for the transverse types (Theorem 4). The proof relies strongly on a result of Deligne regarding the fundamental group of the complement of algebraic curves in the complex projective plane.