Singularities for analytic continuations of holonomy germs of Riccati foliations

TL;DR

This paper investigates the analytic continuation of holonomy germs in Riccati foliations, revealing that most geodesic rays lead to dense, uncountable singularities, thus addressing a conjecture in the field.

Contribution

It demonstrates the density and uncountability of singularities in holonomy germs for specific Riccati foliations, providing a negative answer to Loray's conjecture.

Findings

Singularities are dense in the limit set.

Almost every geodesic ray leads to singularities.

The set of singularities is uncountable.

Abstract

In this paper we study the problem of analytic extension of germs of holonomy of algebraic foliations. More precisely we prove that for a Riccati foliation associated to a branched projective structure over a finite type surface which is non-elementary and parabolic, all the germs of holonomy between a fiber and a holomorphic section of the bundle are led to singularities by almost every developed geodesic ray. We study in detail the distribution of these singularities and prove in particular that they are included and dense in the limit set and uncountable, giving another negative answer to a conjecture of Loray.