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Finite determinacy of dicritical singularities in \((\mathbb C^2 ,0)\). (English) Zbl 1135.32033
The article targets those germs of singularities of holomorphic foliations in \(({\mathbb C}^2,0)\) which are regular after one blow up. For such a foliation, one considers those (finitely many) points \(p\) on the exceptional divisor \(E\) of the blow up where the leaf of the lifted foliation is tangent to \(E\) with contact order \(r(p)+1\). Then by a result of Klughertz the set \(\{r(p)\}_p\) is a complete topological invariant of the above germs of foliations.
The analytic classification, which is the subject of the present article, is more complicated. First, one introduces a new analytic invariant, the transverse structure of the foliation, as a finite subgroup of \(\text{Diff}(E)\) (or its conjugacy class by the Möbius transformations of \(E\)). Although in some particular cases this already has an analytic classification power, in general this is not the case. The main result of the article says that the transverse structure together with a finite number of numerical parameters decide whether two such foliations are analytically equivalent. As a consequence, the formal analytic rigidity of this kind of singular foliations is obtained. (Similar result was obtained by Klughertz in her thesis; the present work improves the order of jets involved.)

32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
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