Singular sets of holonomy maps for algebraic foliations

TL;DR

This paper studies the domains of holonomy maps in singular algebraic foliations, revealing complex singularity structures including large, Cantor, and open sets, influenced by dynamic properties of the foliation.

Contribution

It characterizes the singularity sets of holonomy maps for algebraic foliations, providing explicit examples with diverse and complex singularity structures.

Findings

Germs of holonomy can have large sets of singularities.

Examples include natural boundary and maximal singularity sets.

Presence of rich contracting dynamics influences singularity complexity.

Abstract

In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets of singularities for the analytic continuation. In the Riccati context we provide examples with natural boundary and maximal sets of singularities. In the generic case we provide examples having at least a Cantor set of singularities and even a nonempty open set of singularities. The examples provided are based on the presence of sufficiently rich contracting dynamics in the holonomy pseudogroup of the foliation.