From $\lambda$-connections to $PSL_2(\mathbb{C})$-opers with apparent singularities

TL;DR

This paper constructs and analyzes a Poisson map from $ ext{SL}_2( ext{C})$ $ ext{lambda}$-connections with sub-line bundles to a space parametrizing apparent singularities on a genus > 1 Riemann surface, revealing geometric and Poisson structure relations.

Contribution

It introduces a method to derive opers with apparent singularities from $ ext{lambda}$-connections and sub-line bundles, establishing the Poisson nature of the associated map.

Findings

The constructed map is Poisson with respect to natural structures.

Relations to wobbly bundles and Lagrangians in Higgs moduli spaces are explored.

The construction links $ ext{lambda}$-connections to apparent singularities and moduli space geometry.

Abstract

On a Riemann surface of genus $> 1$, we discuss how to construct opers with apparent singularities from $SL_2(\mathbb{C})$ $\lambda$-connections $(E, \nabla_\lambda)$ and sub-line bundles $L$ of $E$. This construction defines a rational map from a space which captures important data of triples $(E, L, \nabla_\lambda)$ to a space which parametrises the positions and residue parameters of the induced apparent singularities. We show that this is a Poisson map with respect to natural Poisson structures. The relations to wobbly bundles and Lagrangians in the moduli spaces of Higgs bundles and $\lambda$-connections are discussed.