## Nicolas Tholozan ; Jérémy Toulisse - Compact connected components in relative character varieties of punctured spheres

epiga:5894 - Épijournal de Géométrie Algébrique, April 19, 2021, Volume 5 - https://doi.org/10.46298/epiga.2021.volume5.5894
Compact connected components in relative character varieties of punctured spheres

Authors: Nicolas Tholozan ; Jérémy Toulisse

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 $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. 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.

Volume: Volume 5
Published on: April 19, 2021
Submitted on: November 5, 2019
Keywords: Mathematics - Geometric Topology,Mathematics - Algebraic Geometry