Compact connected components in relative character varieties of
punctured spheres
Keyword(s):
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.
Keyword(s):
2017 ◽
Vol 2019
(18)
◽
pp. 5777-5810
◽
Keyword(s):
1960 ◽
Vol 24
(2)
◽
pp. 163-172