Arkiv för Matematik

Volume 58 (2020)

Number 1

Invariant curves for holomorphic foliations on singular surfaces

Pages: 179 – 195

DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n1.a11

Author

Edileno de Almeida Santos (Instituto de Ciência, Engenharia e Tecnologia, Universidade Federal dos Vales do Jequitinhonha e Mucuri (UFVJM), Teófilo Otoni, MG, Brazil)

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.

Keywords

foliations, invariant curves, birational geometry

2010 Mathematics Subject Classification

37F75

Full Text (PDF format)

Received 7 November 2018

Received revised 10 September 2019

Accepted 27 September 2019

Published 21 July 2022