A compactification of the moduli space of principal Higgs bundles over singular curves

TL;DR

This paper constructs the moduli space of principal Higgs bundles over singular curves using decorated vector bundles and relates it to framed modules, extending the theory to singular algebraic curves.

Contribution

It introduces a new construction of the moduli space for principal Higgs G-bundles over singular curves via decorated vector bundles and parabolic structures.

Findings

Moduli space constructed using decorated vector bundles.

Relation established between the moduli space and framed modules.

Extension of Higgs bundle theory to singular curves.

Abstract

A principal Higgs bundle $(P,\phi)$ over a singular curve $X$ is a pair consisting of a principal bundle $P$ and a morphism $\phi:X\to\text{Ad}P \otimes \Omega^1_X$. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve $X$ using the theory of decorated vector bundles. More precisely, given a faithful representation $\rho:G\to Sl(V)$ of $G$, we consider principal Higgs bundles as triples $(E,q,\phi)$ where $E$ is a vector bundle with $\rk{E}=\dim V$ over the normalization $\xtilde$ of $X$, $q$ is a parabolic structure on $E$ and $\phi:E\ab{}\to L$ is a morphism of bundles, being $L$ a line bundle and $E\ab{}\doteqdot (E^{\otimes a})^{\oplus b}$ a vector bundle depending on the Higgs field $\phi$ and on the principal bundle structure. Moreover we show that this moduli space for suitable integers $a,b$ is related to the space of framed modules.