Rigidity for dicritical germ of foliation in ℂ2

Differentiable equisingularity of holomorphic foliations

“…We say that a saddle-node singularity forF is tangent if its weak separatrix is contained in E. Non-tangent saddle-nodes are also known as well-oriented. The following definition is due to J.-F. Mattei and E. Salem (see [6] and also [2], [3] [4]):…”

Residue-type indices and holomorphic foliations

“…The tangency excess measures the extent that a balanced divisor of separatrices computes the algebraic multiplicity, as expressed in the following result [10]: Proposition 2.4. Let F be a germ of singular foliation at (C 2 , p) with B as a balanced divisor of separatrices.…”

“…Our work is strongly based on the notion of balanced set or balanced equation of separatrices ( [10] and Definition 2.3). This is a geometric objet formed by a finite set of separatrices with weights -possibly negative, corresponding to poles -that, in the nondicritical case, coincides with the whole separatrix set.…”

Local polar invariants and the Poincaré problem in the dicritical case

“…They are characterized by the fact that their desingularization coincide with the reduction of the set of formal separatrices. These foliations satisfy a property of minimization of the algebraic multiplicity [17,13].…”

“…The difficulty now lies in choosing a finite set of separatrices in order to produce such a "reference foliation". The solution is to use a balanced equation of separatrices, a concept introduced by the first author in [13] for the study of the "realization problem" -the existence of foliations with prescribed reduction of singularities and projective holonomy representations. Given a local foliation F at (C 2 , 0) with minimal reduction of singularities E : (M, D) → (C 2 , 0), an irreducible component D ⊂ D is said to be non-dicritical -respectively dicritical -if it is invariant -respectively non-invariant -by the strict transform foliation E * F .…”