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

Abstract

We prove that there are relative \({\textrm{SO}}_0(2,q)\)-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 \({\textrm{SO}}_0(2,q)\)-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 \(\mathbb {C}\).