Ramified covers and tame isomonodromic solutions on curves

Abstract

In this paper, we investigate the possibility of constructing isomonodromic deformations by ramified covers. We give new examples and prove a classification result.1991 Mathematics Subject Classification. 34M55, 34M56, 34M03.

A Map Between Moduli Spaces of Connections

“…We begin with the map Φ top : Repν P 1 , D → Repν(C, T ), which is the topological counterpart of our map Φ : Conμ ν P 1 , D → Conμ ν (C, T ) via the Riemann-Hilbert correspondence. It is proved in [5,Theorem 1] that this map is dominant and generically 2 : 1. In Section 4 this result is extended by proving, in Theorem 4.1, that Φ top is surjective and a ramified cover between GIT spaces of representations; the ramification and branch loci of Φ top are described.…”

“…The map Φ is also defined in a more topological setting. This was originally introduced in [5]. We repeat here the construction presented in Section 4 of the cited paper.…”

“…The map Φ was originally defined for monodromy representations in [5]. In Section 4 we further discuss several properties of this map.…”

“…The case studied here, 2 punctures on genus 1 curves, was first considered in [5] for representations. There it was proved that the result of Goldman recalled above, [9,Theorem 10.2], extends as follows.…”

“…The divisor D above contains 5 points: the four branch points of π : C → P 1 and the unique point t ∈ P 1 which satisfies π −1 (t) = T (see Section 3.2 for the explicit construction). This transformation was originally introduced in [5], using the associated monodromy representations. There, it was shown to be dominant and generically 2 : 1.…”