When does the zero fiber of the moment map have rational singularities?
Hans-Christian Herbig, Gerald W. Schwarz, Christopher Seaton
TL;DR
The paper develops a general framework to determine when the zero fiber N_V of the moment map μ: V ⊕ V^* → 𝔤^* has rational singularities and when the invariant quotient N_V // G has symplectic singularities. Building on Mustaţă's jet-scheme criterion and the symplectic-slice theorem, the authors establish CIFR (complete intersection with FPIG and rational singularities) for broad families, including V = p 𝔤, direct sums of classical representations, and several cases related to representation and character varieties. They show that Hom(π,G) is CIFR and X(π,G) has symplectic singularities for surface groups π, with detailed codimension and factoriality statements, and extend these results to reductive G. The approach hinges on jet-scheme analysis, dimension bounds for linear subspaces of null cones, and careful analysis of symplectic slice representations to control singularities, yielding new factorial and symplectic-structure results that improve prior work. Overall, the paper provides a unifying mechanism to verify rational and symplectic singularities across shells, adjoint-copies, classical representations, and moduli spaces arising in geometric representation theory.
Abstract
Let $G$ be a complex reductive group and $V$ a $G$-module. There is a natural moment mapping $μ\colon V\oplus V^*\to\mathfrak{g}^*$ and we denote $μ^{-1}(0)$ (the shell) by $N_V$. We use invariant theory and results of Mustaţă [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 $π=π_1(Σ)$ where $Σ$ is a closed Riemann surface of genus $p\geq 2$. Let $G$ be semisimple and let $\operatorname{Hom}(π,G)$ and $\mathscr X\!(π,G)$ be the corresponding representation variety and character variety. We show that $\operatorname{Hom}(π,G)$ is a complete intersection with rational singularities and that $\mathscr X\!(π,G)$ has symplectic singularities. If $p>2$ or $G$ contains no simple factor of rank $1$, then the singularities of $\operatorname{Hom}(π,G)$ and $\mathscr X\!(π,G)$ are in codimension at least four and $\operatorname{Hom}(π,G)$ is locally factorial. If, in addition, $G$ is simply connected, then $\mathscr X\!(π,G)$ is locally factorial.