Topological invariants and equidesingularization for holomorphic vector fields

Moduli spaces of germs of holomorphic foliations in the plane

“…Therefore F is a non dicritical generalized curve (cf. [2]) having ν + 1 smooth transverse separatrices through the origin, and it reduces after one blow-up. Remark 1.5.…”

Classification of absolutely dicritical foliations of cusp type

“…This homeomorphism can be blown down in a neighborhood of C 2 and is a C 0 -conjugacy of the foliations F 0 and F 1 . Now, if F 0 is topologically equivalent to an absolutely dicritical foliation of cusp type, a theorem of C. Camacho and A. Lins Neto and P. Sad [1] ensures that the process of reduction of F 0 is the one of an absolutely dicritical foliation. Since, they also shared the same dicritical components, F 0 is absolutely dicritical of cusp type.…”

“…But the deformation given by λ → f •Λ λ λ is an equisingular unfolding of f 0 since ∆ λ goes to 0 when λ → 0. Using the semi-universality of the family introduced at the beginning of the proof, for λ small enough, there exists some α such that the following conjugacies holds (1). If α = 0, applying some well-chosen homothetie, we can suppose α = 1.…”

Connectedness of the Fibers of a Liouvillian Function

“…Apart from the natural idea to extend the singularity theory of single valued maps to a class of multivalued ones, we may also consider the point of view of ordinary differential equations. In dimension 2, notice that// 1 -• fp p is a first integral for a germ of holomorphic one-form, whose type is called "non-dicritical generalized curve" by C. Camacho, A. Lins Neto and P. Sad in [2]: These forms allow a desingularization without "dicritical components", and without "saddle-node" singularities. One cannot hope to be able to describe the topology of all generalized curves.…”

“…Because the inverse of A, 2 /A>i ls not a number, (z 2 =0)n U is also in the closure of F~l(c). Given an arbitrary index y, different from 1,2, one of the ratios A///^ or Aj/A 2 is not a real number.…”