Semilocal Monodromy of Rigid Local Systems

Abstract

The rigid local system on \(\mathbb {P}^1\setminus S\) with a set S of finite points is realized as a rigid Fuchsian differential equation \(\mathscr {M}\) of Schlesinger canonical form. Here “rigid” means that the equation is uniquely determined by the equivalence classes of residue matrices of \(\mathscr {M}\) at the points in S. The semilocal monodromy in this paper is the conjugacy class of the monodromy matrix obtained by the analytic continuation of the solutions of \(\mathscr {M}\) along an oriented simple closed curve \(\gamma \) on \(\mathbb {C}\setminus S\). Since it corresponds to the sum of residue matrices at the singular points surrounded by \(\gamma \) and the equation \(\mathscr {M}\) is obtained by applying additions and middle convolutions to the trivial equation, we study the application of the middle convolution to the sums of residue matrices. In this way we give an algorithm calculating this semilocal monodromy, which also gives the local monodromy at the irregular singular point obtained by the confluence of these points.