Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities

Abstract: The unfolding of singular points of linear differential equations is a classical technique to study the properties of irregular singular points from those of regular singular points. In this paper, we propose a general framework for the unfolding of unramified irregular singularities of meromorphic $G$-connections on $\mathbb{P}^{1}$. As one of main results, we give a description for the unfolding of singularities of connections as the deformation of moduli spaces of them, and show that every moduli space of irreducible meromorphic $G$-connections with unramified irregular singularities on $\mathbb{P}^{1}$ has a deformation to a moduli space of irreducible Fuchsian $G$-connections on $\mathbb{P}^{1}$. Furthermore, we can consider the unfolding of additive Delinge-Simpson problems, namely, the unfolding of irregular singularities generates a family of additive Deligne-Simpson problems. Then as an application of our main result, we show that a Deligne-Simpson problem for $G$-connections with unramified irregular singularities has a solution if and only if every other unfolded Deligne-Simpson problem simultaneously has a solution. Also we give a combinatorial and diagrammatic description of the unfolding of irregular singularities via spectral types and unfolding diagrams, and then consider a conjecture proposed by Oshima which asks the existence of irreducible $G$-connections realizing the given spectral types and their unfolding. Our main result gives an affirmative answer of this conjecture.