Unfolding of wild character varieties

TL;DR

This paper investigates wild character varieties on Riemann surfaces, constructing Poisson maps to tame varieties via unfolding irregular singularities, and proves these maps induce birational equivalences, confirming a conjecture.

Contribution

It introduces a method to relate wild and tame character varieties through unfolding, establishing Poisson birational equivalences and confirming a conjecture.

Findings

Poisson maps from wild to tame character varieties are constructed.

Unfolding irregular singularities yields Poisson birational equivalences.

The conjecture by Klimes, Paul, and Ramis is affirmed.

Abstract

In this paper, we study wild character varieties on compact Riemann surfaces and construct Poisson maps from wild to tame character varieties by unfolding irregular singularities into regular ones. Furthermore, we show that these unfolding Poisson maps induce Poisson birational equivalences between wild and tame character varieties. This result provides an affirmative answer to a conjecture posed by Klimes, Paul, and Ramis.