On solutions of the Schlesinger equation in the neigborhood of the Malgrange Θ-divisor

Abstract

We say that the family (1) is isomonodromic if, for all a ∈ D(a0), the monodromies χa : π1(C \ {a1, . . . , an}) → G = GL(p,C) of the corresponding system are equal to each other. (Under small variations of the parameter a, there exists a canonical isomorphism of the fundamental groups π1(C \ {a1, . . . , an}) and π1(C \ {a1, . . . , an}) generating the canonical isomorphism Hom(π1(C \ {a1, . . . , an}), G)/G ∼= Hom(π1(C \ {a1, . . . , an}), G)/G of the spaces of classes of the duality representations for these fundamental groups; this allows one to compare χa for various a ∈ D(a0).) For example, if the matrix Bi(a) satisfies the Schlesinger equation dBi(a) = − n ∑