Holomorphic foliations tangent to Rolle-pfaffian hypersurfaces

TL;DR

This paper investigates holomorphic foliations tangent to Rolle-Pfaffian hypersurfaces in the complex plane, showing they are defined by closed meromorphic 1-forms and classifying their simple models.

Contribution

It establishes that such foliations are characterized by closed meromorphic 1-forms and provides a classification of their simple singularity models.

Findings

Foliations tangent to Rolle-Pfaffian hypersurfaces are defined by closed meromorphic 1-forms.

Classification of simple models in the reduction of singularities.

Extension of Rolle-Khovanskii condition to complex holomorphic foliations.

Abstract

In this paper we study germs of holomorphic foliations, at the origin of the complex plane, tangent to Pfaffian hypersurfaces - integral hypersurfaces of real analytic 1-forms - satisfying the Rolle-Khovanskii condition. This hypothesis leads us to conclude that such a foliation is defined by a closed meromorphic 1-form, also allowing the classification of the simple models in its reduction of singularities.