Representation varieties of the fundamental groups of compact orientable surfaces (Q1912773)

Let \(\Gamma_g\) be the fundamental group of a compact orientable surface, \(\mathbb{R}_n (\Gamma_g)\) the variety of \(n\)-dimensional representations of \(\Gamma_g\) (in characteristic zero) and \(\mathbb{X}_n (\Gamma_g)\) the variety of \(n\)-dimensional characters of \(\Gamma_g\) (i.e., the categorical quotient of \(\mathbb{R}_n (\Gamma_g)\) modulo the natural action of \(GL_n)\). The authors prove the following statements: (a) \(\mathbb{R}_n (\Gamma_g)\) is an absolutely irreducible \(\mathbb{Q}\)-rational variety of dimension \((2g-1) n^2+1\) if \(g>1\) and \(n^2 +n\) if \(g=1\); (b) \(\mathbb{X}_n (\Gamma_g)\) is irreducible and \(\mathbb{Q}\)-unirational, of dimension \((2g-2) n^2+2\) (resp., \(2n)\) for \(g>1\) (resp., \(g=1)\). Moreover, \(\mathbb{X}_n (\Gamma_g)\) is \(\mathbb{Q}\)-rational if \(g>1\) and \(n\leq 3\).