Geometry & Topology Volume 28, issue 7 (2024)

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

Hans-Christian Herbig, Gerald W Schwarz and Christopher Seaton

Geometry & Topology 28 (2024) 3475–3510

DOI: 10.2140/gt.2024.28.3475

Abstract

Let $G$ be a complex reductive group and $V$ a $G$ –module. There is a natural moment mapping $μ : V \oplus V^{*} \to 𝔤^{*}$ and we denote $μ^{- 1} (0)$ (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 $V = p 𝔤$ (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 previous results of Aizenbud-Avni, Budur, Glazer–Hendel, and Kapon. Let $π = π_{1} (Σ)$ , where $Σ$ is a closed Riemann surface of genus $p \geq 2$ . Let $G$ be semisimple and let $Hom (π, G)$ and $𝒳 (π, G)$ be the corresponding representation variety and character variety. We show that $Hom (π, G)$ is a complete intersection with rational singularities and that $𝒳 (π, G)$ has symplectic singularities. If $p > 2$ or $G$ contains no simple factor of rank $1$ , then the singularities of $Hom (π, G)$ and $𝒳 (π, G)$ are in codimension at least four and $Hom (π, G)$ is locally factorial. If, in addition, $G$ is simply connected, then $𝒳 (π, G)$ is locally factorial.

Keywords

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

Mathematical Subject Classification

Primary: 14B05, 53D20

Secondary: 13A50, 13H10, 14M35, 20G20

References

Publication

Received: 28 November 2022

Revised: 13 May 2023

Accepted: 15 June 2023

Published: 25 November 2024

Proposed: Mark Gross

Seconded: Jim Bryan, Dan Abramovich

Authors

Hans-Christian Herbig
Departamento de Matemática Aplicada Universidade Federal do Rio de Janeiro Rio de Janeiro Brazil

Gerald W Schwarz
Department of Mathematics Brandeis University Waltham, MA United States

Christopher Seaton
Department of Mathematics and Computer Science Rhodes College Memphis, TN United States	Department of Mathematics and Statistics Skidmore College Saratoga Springs, NY United States

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