Constructing equivalences with some extensions to the divisor and topological invariance of projective holonomy

Abstract

Given topologically equivalent germs of holomorphic foliations \mathcal{F} and \mathcal{F}' , under some hypothesis, we construct topological equivalences extending to some regions of the divisor after resolution of singularities. As an application we study the topological invariance of the projective holonomy representation.

Bilipschitz Invariants for Germs of Holomorphic Foliations

“…Precisely, the results of this section are used in the proofs of Proposition 21 and Proposition 24. Note that Proposition 19 below is a special version of a kind of results previously obtained in [9] (Theorem 6.2.1) and [18] (section 5). Let w ∈ V and consider the map φ w defined by φ w (T ) = φ T (w) and whose domain is the connected component of 0 ∈ C of the set where φ T (w) is defined.…”

Topology of singular holomorphic foliations along a compact divisor

“…Moreover, the topological conjugations considered in [12] are assumed to send nodal separatrices into nodal separatrices preserving its corresponding Camacho-Sad indices. In this paper we have used the following result of R. Rosas [16,Proposition 11] which allows us to eliminate this constraint and to extend our results to general topological conjugations. Theorem 1.12.…”

Nodal separators of holomorphic foliations

“…Proof of Theorem 4. By [5] there exists a topological equivalence h between F and F which, after resolution, extends as a homeomorphism to a neighborhood of each linearizable or resonant singularity which is not a corner. We denote by E and E the exceptional divisors in the resolutions of F and F, respectively.…”

“…Analogously define E 1 and E 2 . We will use the ideas used in [5] to construct the topological equivalence near a nodal non corner point. We will think for a moment that p is not a corner: think that {x = 0} is a separatrix and that the exceptional divisor is reduced to E 1 .…”

“…In a natural way we can think that H 1 (T ) = H 1 ( T ). Then, the fact that {x = 0} is a separatrix is used in [5] to prove that the isomorphism h * is the identity or the inversion isomorphism according to h preserves or reverses the natural orientation of the leaves. In our case we already have this property, by Proposition 21.…”