Poisson structure on character varieties. II

TL;DR

This paper establishes a Poisson structure on the moduli space of logarithmic G-connections on a Riemann surface and proves that the monodromy map to character varieties preserves this structure.

Contribution

It introduces a Poisson structure on the moduli space of logarithmic G-connections and demonstrates that the monodromy map is a Poisson morphism.

Findings

Poisson structure on the moduli space of logarithmic G-connections

Monodromy map preserves the Poisson structure

Extension of previous results to new moduli spaces

Abstract

Let G be a complex reductive group and D a finite subset of a compact Riemann surface X. It was shown in [BJ] that the moduli space of G-characters of the complement of D in X has a natural Poisson structure. We show that the moduli space of logarithmic G-connections on X singular over D has a Poisson structure. It is proved that the monodromy map from the moduli space of logarithmic G-connections to the moduli space of G-characters is Poisson structure preserving.