Sliding invariants and classification of singular holomorphic foliations in the plane (Q5962658)

A germ of a singular foliation \(\mathcal{F}\) in \((\mathbb{C}^2,0)\) is called \textit{reduced} if there exists a coordinate system in which it is defined by a 1-form whose linear part is NEWLINE\[NEWLINE \lambda_1 ydx +\lambda_2 xdy, \quad \frac{\lambda_2}{\lambda_1} \notin \mathbb{Q}_{<0}, NEWLINE\]NEWLINE \(\lambda=-{\lambda_2}/{\lambda_1}\) is called the \textit{Camacho-Sad index} of \(\mathcal{F}\). The problem of classification of germs of singular foliations in the complex plane was stated by R. Thom.NEWLINENEWLINEBy introducing a new invariant called the set of slidings, the author gives a complete strict classification of the class of germs of non-dicritical holomorphic foliations \(\{\mathcal{F}\}\) in the plan whose Camacho-Sad indices are not rational. Moreover, he shows that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomic non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps is proved.NEWLINENEWLINEThe present work enlarges previous results in [\textit{Y. Genzmer}, J. Differential Equations 245, No. 6, 1656--1680 (2008; Zbl 1210.37031); \textit{Y. Genzmer} et al., Mosc. Math. J. 11, No. 1, 41--72 (2011; Zbl 1222.32056)].