Infinitesimal deformations of parabolic connections and parabolic opers

TL;DR

This paper analyzes the infinitesimal deformations of parabolic connections and opers on Riemann surfaces, revealing the structure of their moduli spaces and the properties of the monodromy map.

Contribution

It provides explicit computations of infinitesimal deformations for parabolic connections and opers, and proves the monodromy map is an immersion.

Findings

Computed infinitesimal deformations of parabolic connections and opers.

Established the monodromy map as an immersion.

Enhanced understanding of moduli space structures.

Abstract

We compute the infinitesimal deformations of quadruples of the form $$(X, S, E_*, D),$$ where $(X, S)$ is a compact Riemann surface with $n$ marked points, $E_*$ is a parabolic vector bundle on $X$ with parabolic structure over $S$, and $D$ is a parabolic connection on $E_*$. Using it we compute the infinitesimal deformations of $(X, S, D)$, where $D$ is a parabolic SL(r, C)-oper on $(X, S)$. It is shown that the monodromy map, from the moduli space of triples $(X, S, D)$, where $D$ is a parabolic SL(r, C)-oper on $(X, S)$, to the SL(r, C)-character variety of X - S, is an immersion.