Topology of singular holomorphic foliations along a compact divisor (Q2879069)

The authors consider a singular holomorphic foliation \(\underline{\mathcal{F}}\) of a complex surface \(\underline{M}\), looking near a compact connected holomorphic curve \(\underline{\mathcal{C}}\) which is a (probably singular) leaf. A particular situation in this context occurs when the pair \((\underline{M}, \;\underline{ \mathcal{C}})\) comes from a resolution of a surface singularity \((S;O)\).NEWLINENEWLINEUnder suitable hypotheses on the pair \((\underline{\mathcal{F}},\;\underline{\mathcal{C}})\), the authors prove that there exists a system of tubular neighborhoods \(U\) of an adapted curve \(\underline{\mathcal{D}}\), containing \(\underline{\mathcal{C}}\), such that every leaf \(\mathcal{L}\) of the foliation \(\underline{\mathcal{F}}\) restricted on \(U\backslash \underline{\mathcal{D}}\) is incompressible in \(U\backslash \underline{\mathcal{D}}\).NEWLINENEWLINEAlso a representation of the fundamental group of the complementary of \(\underline{\mathcal{D}}\) into a suitable automorphism group is constructed. This requires a fine technical work, and is directly equivalent to the classical holonomy representation of \(\pi_1 (\underline{\mathcal{C}})\) into the automorphisms of a transverse section.NEWLINENEWLINEThe second main result allows to state the topological classification of the germ of \((\underline{\mathcal{F}},\;\underline{\mathcal{D}})\), under the additional but generic dynamical hypothesis of transverse rigidity. In particular, it follows that every topological conjugation between such germs of holomorphic foliations can be deformed to extend to the exceptional divisor of their reductions of singularities.