Isomorphisms of Symplectic Torus Quotients

TL;DR

This paper investigates the structure of symplectic quotients arising from quasi-toral complex groups acting on modules, showing how the quotient determines the module and group up to isomorphism under certain codimension conditions.

Contribution

It proves that the symplectic quotient uniquely determines the module and group when the singular set has codimension at least four, and provides a reduction process for lower codimension cases.

Findings

The quotient $M$ determines $Vigoplus V^*$ and $G$ if $ ext{codim}_N N_ ext{sing}\u2265 4.

A process exists to produce a subgroup $G'$ and submodule $V'$ with higher codimension singularities.

Results extend to real symplectic quotients for compact Lie groups.

Abstract

We call a reductive complex group $G$ quasi-toral if $G^0$ is a torus. Let $G$ be quasi-toral and let $V$ be a faithful $1$-modular $G$-module. Let $N$ (the shell) be the zero fiber of the canonical moment mapping $\mu\colon V\oplus V^*\to\mathfrak{g}^*$. Then $N$ is a complete intersection variety with rational singularities. Let $M$ denote the categorical quotient $N/\!\!/ G$. We show that $M$ determines $V\oplus V^*$ and $G$, up to isomorphism, if $\operatorname{codim}_N N_\mathrm{sing}\geq 4$. If $\operatorname{codim}_NN_\mathrm{sing}=3$, the lowest possible, then there is a process to produce an algebraic (hence quasi-toral) subgroup $G'\subset G$ and a faithful $1$-modular $G'$-submodule $V'\subset V$ with shell $N'$ such that $\operatorname{codim}_{N'}(N')_\mathrm{sing}\geq 4$. Moreover, there is a $G'$-equivariant morphism $N'\to N$ inducing an isomorphism $N'/\!\!/ G'\xrightarrow{\sim} N/\!\!/ G$. Thus, up to isomorphism, $M$ determines $V'\oplus (V')^*$ and $G'$, hence also $N'$. We establish similar results for real shells and real symplectic quotients associated to unitary modules for compact Lie groups.