About the topology of polar curves of some singular foliations
Let \(\mathcal F\) be a germ of a singular foliation on \((\mathbb C^2,0)\) defined as the kernel of a differential 1-form \(\omega\), that is, by the equation \(\omega =0\). Then, if \(\omega_1\) is a nonsingular 1-form, the polar curve \(\Gamma_{\omega_{1}}\) of \(\mathcal F\) with respect to \(\omega_1\) is the curve defined by \(\omega\wedge\omega_1=0\). The definition of a polar curve does not depend on the choice of the generator \(\omega\). There is a natural notion of a generic polar curve, whose definition requires the use of a coordinate system, but whose equisingularity type does not depend on the choice of the coordinate system. In this paper, the author studies the topological properties (in particular, the equisingularity type) of a generic polar curve in terms of the reduction types of the singularities of the foliation. In particular, the author extends a decomposition theorem for a generic polar foliation, which was obtained in some special cases by \textit{M. Merle} [Invent. Math. 41, 103--111 (1977; Zbl 0371.14003)] and in other special cases by \textit{Lê Dûng Trang, F. Michel} and \textit{C. Weber} [Compos. Math. 72, 87--113 (1989; Zbl 0705.32021)] and by \textit{E. García Barroso} [`Invariants des singularités de courbes planes et courbure des fibres de Milnor', No.~1, Thesis, Ecole Normale Supérieure, LMENS 96-35 (1996)].
scientific article; zbMATH DE number 1940712
scientific article