Compact relative $\mathrm{SO}_0(2,q)$-character varieties of punctured spheres

TL;DR

This paper constructs specific compact, non-hyperbolic relative $ ext{SO}_0(2,q)$-character varieties of punctured spheres, expanding understanding of their geometric properties using Higgs bundles and GIT.

Contribution

It completes the classification of certain relative $ ext{SO}_0(2,q)$-character varieties by proving their compactness and non-hyperbolicity, and shows these moduli spaces are projective varieties.

Findings

Existence of compact, non-hyperbolic character varieties

Dense representations within these varieties

Moduli spaces are projective algebraic varieties

Abstract

We prove that there are relative $\mathrm{SO}_0(2,q)$-character varieties of the punctured sphere which are compact, totally non-hyperbolic and contain a dense representation. This work fills a remaining case of the results of N. Tholozan and J. Toulisse. Our approach relies on the non-abelian Hodge correspondence and we study the moduli space of parabolic $\mathrm{SO}_0(2,q)$-Higgs bundles with some fixed weight. Additionally, we provide a construction based on Geometric Invariant Theory (GIT) to demonstrate that the considered moduli spaces can be viewed as a projective variety over $\mathbb{C}$.