Unfolding of wild character varieties
Kazuki Hiroe, Daisuke Yamakawa
TL;DR
The paper addresses the problem of relating wild character varieties $\mathcal{M}_{\mathrm{B}}(\mathbf{\Sigma})$ to tame (unfolded) ones $\mathcal{M}_{\mathrm{B}}(\mathbf{\Sigma}')$ by constructing Poisson maps that unfold irregular singularities. Employing the quasi-Hamiltonian framework with fission spaces and fusion/gluing, the authors build explicit unfolding morphisms $\Upsilon_{(}{}^i t_j{)}$ that connect wild data to a product of conjugacy classes, producing a Poisson morphism between the corresponding character varieties. They prove that, under suitable irreducibility and stability hypotheses, these maps yield Poisson birational equivalences between irreducible components, thereby affirmatively answering a conjecture of Klimeš, Paul, and Ramis in the untwisted wild setting. The work generalizes deformation results for moduli of meromorphic connections (Hirzebruch–Dubrovin type unfoldings) to the wild character variety context and provides a concrete, algebraic-geometry framework for understanding confluences of irregular singularities via unfolding. The results have implications for Painlevé-type phenomena and for the broader study of moduli spaces of meromorphic connections through Poisson geometry.
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.