Versal Unfolding of Irregular Singularities of a Linear Differential Equation on the Riemann Sphere

Abstract

For a linear differential operator $P$ on ${\mathbb{P}}^1$ with unramified irregular singular points, we examine a realization of $P$ as a confluence of singularities of a Fuchsian differential operator $\tilde{P}$ having the same index of rigidity as $P$, which we call an unfolding of $P$. We conjecture that this is always possible. For example, if $P$ is rigid, this is true and the unfolding helps us to study the equation $Pu=0$.