Geodesic
Compact connected components in relative character varieties of punctured spheres
Épijournal de Géométrie Algébrique, Tome 5 (2021)

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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. 
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     author = {Tholozan, Nicolas and Toulisse, J\'er\'emy},
     title = {Compact connected components in relative character varieties of punctured spheres},
     journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
     publisher = {mathdoc},
     volume = {5},
     year = {2021},
     doi = {10.46298/epiga.2021.volume5.5894},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2021.volume5.5894/}
}
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Tholozan, Nicolas; Toulisse, Jérémy. Compact connected components in relative character varieties of punctured spheres. Épijournal de Géométrie Algébrique, Tome 5 (2021). doi: 10.46298/epiga.2021.volume5.5894

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