Compact connected components in relative character varieties of punctured spheres

We prove that some relative character varieties of the fundamental group a punctured sphere into Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, image any simple closed curve is an elliptic element. These results extend recent work Deroin and first author, which treated case $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on non-Abelian Hodge correspondance between parabolic Higgs bundles. examples we construct rather explicit description as projective obtained via Geometric Invariant Theory.