Character varieties (Q2841399)

Moduli spaces of representations (also known as character varieties) are ubiquitous in modern mathematics; the extensive bibliography of this paper attests to this fact. The paper under review, both well-written and detailed, gives a thorough treatment of many of their properties.NEWLINENEWLINELet \(\Gamma\) is finitely generated group, and let \(G\) be a reductive affine group scheme acting by conjugation on the algebraic scheme \(\mathcal{H}om(\Gamma, G)\). The Geometric Invariant Theory quotient by this action, denoted \(\mathcal{X}_\Gamma(G)\), is called the \textit{\(G\)-character scheme of \(\Gamma\)}; and is one of the main objects of study in this paper. The subspace of geometric points (maximal ideals) of \(\mathcal{X}_\Gamma(G)\) is called the \textit{\(G\)-character variety of \(\Gamma\)}.NEWLINENEWLINEThis paper contains many theorems and preceding results of practical importance in the theory. Perhaps the main theorem of the paper describes the tangent space at a point \([\rho]\) in \(\mathcal{X}_\Gamma(G)\). Contrary to popular belief, the tangent space is not always isomorphic to a cohomology space, although this theorem shows that the tangent space is the tangent space of a quotient of a cohomology space when the orbit of \(\rho\) is closed, and \(\rho\) is non-singular in \(\mathcal{H}om(\Gamma, G)\). Another main result shown in this paper concerns Lagrangian subspaces of \(G\)-character varieties of surface groups.NEWLINENEWLINEOne of the highlights of the paper is that the author starts with a general discussion of reductive groups and their associated action on \(\mathcal{H}om(\Gamma, G)\), and builds from there with explicit constructions and proofs or concrete references. Many examples are given to show how results are sharp, and definitions are precise and general.NEWLINENEWLINEAlthough many of the results can be found, in part or in special cases, in other earlier works, this treatise is one of the best comprehensive treatments this reviewer knows. I recommend this paper to both beginner and expert alike interested in practically working in the important and fascinating theory of character varieties.