# Topology and Arithmetic of GL(n,C)-Character varieties

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A Character variety X_FG is a space of representations of a finitely generated group F into a Lie group G. The most interesting cases are when F is the fundamental group of a Kähler manifold M, and G is a reductive group, since then X_FG is homeomorphic to a space of so-called G-Higgs bundles over M. Typically, X_FG are singular algebraic varieties, defined over the integers, and many of its topological and arithmetic properties can be encoded in a polynomial generalization of the Euler-Poincaré characteristic: the E-polynomial.