Deformation of moduli spaces of meromorphic G-connections on P¹ via unfolding of irregular singularities

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 P¹. 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 P¹ has a deformation to a moduli space of irreducible Fuchsian G-connections on P¹. 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.