Topology of singular holomorphic foliations along a compact divisor

Abstract

We consider a singular holomorphic foliation F defined near a compact curve C of a complex surface. Under some hypothesis on (F, C) we prove that there exists a system of tubular neighborhoods U of a curve D containing C such that every leaf L of F |(U \D) is incompressible in U \ D. We also construct a representation of the fundamental group of the complementary of D into a suitable automorphism group, which allows to state the topological classification of the germ of (F, D), under the additional but generic dynamical hypothesis of transverse rigidity. In particular, we show that every topological conjugation between such germs of holomorphic foliations can be deformed to extend to the exceptional divisor of their reductions of singularities.

Topological Moduli Space for Germs of Holomorphic Foliations

“…By construction, Exp ar = Exp sr , cf. (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 :…”

“…The following incompressibility property is proven under some additional assumptions in [13], [16] and an optimal version is obtained by L. Teyssier in [32]:…”

“…It is enough to prove these equalities when s is an intersection point of the strict transform of a separatrix S of F with an irreducible component D of E F . Indeed according to an extension of [23] given in [16,Lemma 1.9] or in [3,Theorem 8], there is such a point on any cut-component C of E F . Thus the induction given in [15, § 7.3] will remain valid and equalities (22) will be satisfied at every singular point of the foliation.…”

“…We distinguish three possibilities. Another proof is given in [16,Theorem 1.12] that remains valid for λ < 0. b) C is not exceptional and s is not a nodal singular point. Then by using (TR) and thanks to an extended version [15] of the rigidity theorem in [29], ψ is transversely holomorphic on the image by E F of a neighborhood of C and specifically at the points of the separatrix S. In this case the proof of (22) given in [15, Chapter 2] remains valid.…”

Germes de feuilletages présentables du plan complexe

“…Dans deux travaux récents [MM08,MM21], D. Marín et J.-F. Mattei ont dégagé des conditions suffisantes sous lesquelles ce résultat se généralise lorsque F n'admet pas d'intégrale première holomorphe non triviale. Précisons cela.…”