Compact connected components in relative character varieties of punctured spheres

Abstract

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

References in corpus (2)

• Poisson Geometry of SL(3,C)-Character Varieties Relative to a Surface with Boundary
• Topology of representation spaces of surface groups in PSL(2,R) with assigned boundary monodromy and nonzero Euler number