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

Herbig, Hans-Christian¹ ; Schwarz, Gerald W² ; Seaton, Christopher³

¹ Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
² Department of Mathematics, Brandeis University, Waltham, MA, United States
³ Department of Mathematics and Computer Science, Rhodes College, Memphis, TN, United States, Department of Mathematics and Statistics, Skidmore College, Saratoga Springs, NY, United States

Geometry & topology, Tome 28 (2024) no. 7, pp. 3475-3510.

Résumé

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 (Invent. Math. 145 (2001) 397–424) 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 our earlier work (Herbig et. al., Compos. Math. 156 (2020) 613–646). 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 previous results of Aizenbud-Avni, Budur, Glazer–Hendel, and Kapon. Let $\pi ={\pi }_1(\mathrm{\Sigma })$, where $\mathrm{\Sigma }$ 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.

Détail

DOI : 10.2140/gt.2024.28.3475

Keywords: singular symplectic reduction, moment map, rational singularities, symplectic singularities, representation variety, character variety, representation growth of linear groups

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