Monodromy and topological classification of germs of holomorphic foliations

Abstract

We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representation whose topological invariance was conjectured in the eighties by Cerveau and Sad and proved here under mild hypotheses.

Topological Moduli Space for Germs of Holomorphic Foliations

“…Because the dicriticity of a irreducible component D can be characterized by the vanishing of the Camacho-Sad indices along all the adjacent components at their intersection points with D, we have: Proof of Theorem 11.4. This result extends Theorem 5.0.2 of [15] and we will sketch an inductive proof similar to that described in Chapter 8 of [15]. We proceed in four steps: first we extend to our new context the notion of monodromy; then we construct a conjugation Φ 1 between F ♯ and G ♯ on a neighborhood of the union of all non exceptional cut-components of E F ; in a third step we define a conjugation Φ 2 along the exceptional cut-component except at the nodal singularities; finally in the fourth and last step we extend and glue Φ 1 and Φ 2 at the nodal singularities and along the dicritical components.…”

“…(16). With this identification and thanks to the isomorphisms (14), (15) and (17) it can be easily checked, using the proof of Lemma 3.7, that the map δ = ⊕ α δ α given by the connecting maps δ α of the Mayer-Vietoris exact sequences (18) is the surjective morphism δ :…”

“…where with notations of Remark 11.3, Q F ∞ is the inverse system ( Q F Un ) n∈N , An ← − is the category of pro-objects associated to the category of analytic spaces and Top ← − − is the category of pro-objets associated to the category of topological spaces and continuous functions. The monodromy [15], is immediate; the other two follow directly from the proofs in [15]. b) To start the induction, we proceed as in [15, §8.4] but with an F-suitable collection of transversals (∆ C ) C∈C in the meaning that it is obtained in the following way: C is the set of non exceptional cut-components of E F ; for each C ∈ C we choose one separatrix S C whose strict transform meets C and an embedded conformal disc ∆ C transversal to S C at a regular point; these discs are small enough so their pairwise intersections are empty and they are tranversal to the foliation.…”

“…Because the cut-components are non exceptional and thanks to the extended version of the main theorem of [29], Rigidity condition (TR) of the introduction implies the analyticity of the restrictions ψ |∆ C . Then we finish the base case of the induction by the construction of a geometric representation of the conjugation (ψ * , h ψ ) satisfying condition (23) of [15,Extension Lemma 8.3.2], as in [15, §8.4].…”

Nodal separators of holomorphic foliations

“…A key step in the proof of this theorem is to establish a correspondence, after resolution, between the singularities of F and F. When a singularity p in the resolution of F is not a corner, we can use the separatrix issuing from p to define the corresponding singularityp in the resolution of F. Moreover, By Zariski's Theorem [6], the singularities p andp are in "isomorphic positions" in their corresponding exceptional divisors. If the singularity p is a corner, we have no separatrix issuing from p and this is the main difficulty when we deal with corner singularities -recall that F is not necessarily of Generic General Type, so the techniques of [4] does not work -. However, if the corner singularity p is a node, we can overcome this difficulty by using a nodal separator issuing from p and Theorem 3 to define the singularityp corresponding to p in the resolution of F. Moreover, Theorem 2 guarantees that p andp are in "isomorphic positions" in their corresponding exceptional divisors.…”