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 $\mu :V\oplus V^{\ast }\to {\mathfrak{g}}^{\ast }$ and we denote ${\mu }^{-1}(0)$ (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 $p\ge 2$. 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 $p\ge 2$. Let $G$ be semisimple and let $\mathrm{Hom}(\pi ,G)$ and $\mathcal{X}(\pi ,G)$ be the corresponding representation variety and character variety. We show that $\mathrm{Hom}(\pi ,G)$ is a complete intersection with rational singularities and that $\mathcal{X}(\pi ,G)$ has symplectic singularities. If $p>2$ or $G$ contains no simple factor of rank $1$, then the singularities of $\mathrm{Hom}(\pi ,G)$ and $\mathcal{X}(\pi ,G)$ are in codimension at least four and $\mathrm{Hom}(\pi ,G)$ is locally factorial. If, in addition, $G$ is simply connected, then $\mathcal{X}(\pi ,G)$ is locally factorial.

21 pages. v2: 22 pages, clarified and improved arguments in Section 3, improved exposition and minor corrections. v3: 30 pages, reformatted, minor corrections. v4, 31 pages, improved exposition and minor corrections following referee's remarks

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