Invariant curves for holomorphic foliations on singular surfaces

TL;DR

This paper extends the existence of invariant curves (separatrices) for holomorphic foliations on complex surfaces to cases with more complex singularities, under specific conditions excluding saddle-nodes.

Contribution

It generalizes the Separatrix Theorem to singular surfaces with non-tree dual graphs, requiring the foliation to lack saddle-nodes in its reduction.

Findings

Existence of separatrices under new singularity conditions

Extension of classical results to more complex surface singularities

Identification of the absence of saddle-nodes as a key condition

Abstract

The Separatrix Theorem of C. Camacho and P. Sad guarantees the existence of invariant curve (separatrix) passing through the singularity of germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.