Invariant curves for holomorphic foliations on singular surfaces

Abstract

The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a 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.In 1988, C. Camacho showed in [3] an extension of this result to include foliations on some singular surfaces:The author would like to thanks J. V. Pereira for incentive, valuable ideas and conversations.