When does the zero fiber of the moment map have rational singularities?

Abstract

Let $G$ be a complex reductive group and $V$ a $G$-module. There is a natural moment mapping $\mu\colon V\oplus V^*\to\mathfrak{g}^*$ and we denote $\mu^{-1}(0)$ (the shell) by $N_V$. We use invariant theory and results of Musta\c{t}\u{a} [Mus01] to find criteria for $N_V$ to have rational singularities and for the categorical quotient $N_V /\!\!/ G$ to have symplectic singularities, the latter results improving upon [HSS20]. It turns out that for ``most'' $G$-modules $V$, the shell $N_V$ has rational singularities. For the case of direct sums of classical representations of the classical groups, $N_V$ has rational singularities and $N_V /\!\!/ G$ has symplectic singularities if $N_V$ is a reduced and irreducible complete intersection. Another important special case is $V=p\,\mathfrak{g}$ (the direct sum of $p$ copies of the Lie algebra of $G$) where $p\geq 2$. We show that $N_V$ has rational singularities and that $N_V /\!\!/ G$ has symplectic singularities, improving upon results of [Bud19], [AA16], [Kap19] and [GH20]. Let $\pi=\pi_1(\Sigma)$ where $\Sigma$ is a closed Riemann surface of genus $p\geq 2$. Let $G$ be semisimple and let $\operatorname{Hom}(\pi,G)$ and $\mathscr X\!(\pi,G)$ be the corresponding representation variety and character variety. We show that $\operatorname{Hom}(\pi,G)$ is a complete intersection with rational singularities and that $\mathscr X\!(\pi,G)$ has symplectic singularities. If $p>2$ or $G$ contains no simple factor of rank $1$, then the singularities of $\operatorname{Hom}(\pi,G)$ and $\mathscr X\!(\pi,G)$ are in codimension at least four and $\operatorname{Hom}(\pi,G)$ is locally factorial. If, in addition, $G$ is simply connected, then $\mathscr X\!(\pi,G)$ is locally factorial.