Resolving singularities and monodromy reduction of Fuchsian connections

TL;DR

This paper investigates the monodromy reduction of Fuchsian connections with four singularities, using sheaf theory and bundle modifications to explain classical hypergeometric expansions and discover new ones, linking to Painlevé VI solutions.

Contribution

It introduces a sheaf-theoretic approach with bundle modifications to analyze monodromy reduction, providing new hypergeometric expansions and a geometric proof of eigenvalue inclusion.

Findings

Explains Erdélyi's hypergeometric expansions using geometric methods.

Derives new hypergeometric expansions not previously documented.

Establishes a link between monodromy reduction criteria and Painlevé VI solutions.

Abstract

We study monodromy reduction of Fuchsian connections from a sheave theoretic viewpoint, focusing on the case when a singularity of a special connection with four singularities has been resolved. The main tool of study is {based on} a bundle modification technique due to Drinfeld and Oblezin. This approach via invariant spaces and eigenvalue problems allows us not only to explain Erd\'elyi's classical infinite hypergeometric expansions of solutions to Heun equations, but also to obtain new expansions not found in his papers. As a consequence, a geometric proof of Takemura's eigenvalues inclusion theorem is obtained. Finally, we observe a precise matching between the monodromy reduction criteria giving those special solutions of Heun equations and that giving classical solutions of the Painlev\'e VI equation.