Character Varieties for Seifert Groups and Higgs Bundles over Orbifolds

Abstract

This PhD thesis investigates Hodge structures on a particular class of complex algebraic varieties known as character varieties. Our objective is to integrate a powerful result stemming from motivic integration and the proofs of the Weil conjectures, with classical representation-theoretic point-counting techniques. In this work, we develop a hybrid approach combining arithmetic and geometric methods to compute the Deligne-Hodge polynomials of SL2 (C)- character varieties associated with the fundamental group of a circle bundle - referred to as a Seifert group - over an orbifold with a single ramified point. Additionally, using an orbifold version of the non-abelian Hodge correspondence, we derive certain topological properties of the moduli space of trace-free Higgs bundles, with fixed determinant, over the same base orbifold.