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

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 $\mathbb{C}$.

Final version, to appear in Annals of Global Analysis and Geometry. 39 pages, comments are very welcome