When does the zero fiber of the moment map have rational singularities?
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2021
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| author | Herbig, Hans-Christian Schwarz, Gerald W. Seaton, Christopher |
| author_facet | Herbig, Hans-Christian Schwarz, Gerald W. Seaton, Christopher |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2108_07306 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | When does the zero fiber of the moment map have rational singularities? Herbig, Hans-Christian Schwarz, Gerald W. Seaton, Christopher Algebraic Geometry Symplectic Geometry Primary 53D20, 14B05, Secondary 13A50, 13H10, 14M35, 20G20 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. |
| title | When does the zero fiber of the moment map have rational singularities? |
| topic | Algebraic Geometry Symplectic Geometry Primary 53D20, 14B05, Secondary 13A50, 13H10, 14M35, 20G20 |
| url | https://arxiv.org/abs/2108.07306 |