Holomorphic foliations desingularized by punctual blow-ups
Résumé
Given View the MathML source be a germ of codimension-one singular holomorphic foliation at the origin View the MathML source. We assume that View the MathML source 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 View the MathML source 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.