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.

Source: arXiv.org:1811.01603

Volume: Volume 5

Published on: April 19, 2021

Accepted on: February 9, 2021

Submitted on: November 5, 2019

Keywords: Mathematics - Geometric Topology,Mathematics - Algebraic Geometry

Funding:

- Source : OpenAIRE Graph
*RNMS: Geometric structures and representation varieties*; Funder: National Science Foundation; Code: 1107452*RNMS: Geometric Structures and Representation Varieties*; Funder: National Science Foundation; Code: 1107263*RNMS: Geometric Structures and Representation Varieties*; Funder: National Science Foundation; Code: 1107367

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