The Koszul complex of a moment map

TL;DR

This paper investigates when the Koszul complex associated with a moment map provides a resolution of smooth functions on the zero level set, linking this property to the 1-large condition of complexified representations.

Contribution

It establishes a criterion based on the 1-large condition for the Koszul complex to resolve functions on the zero set of a moment map, extending previous work to symplectic manifolds.

Findings

Koszul complex resolves functions on zero set if and only if the complexified representation is 1-large.

Resolution criterion applies both in linear and symplectic settings.

Provides a link between geometric invariant theory and symplectic geometry.

Abstract

Let $K\to U(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $\rho\colon V\to\mathfrak k^*$. We have the Koszul complex ${\mathcal K}(\rho,\mathcal C^\infty(V))$ of the component functions $\rho_1,...,\rho_k$ of $\rho$. Let $G=K_{\mathbb C}$, the complexification of $K$. We show that the Koszul complex is a resolution of the smooth functions on $\rho^{-1}(0)$ if and only if $G\to\GL(V)$ is 1-large, a concept introduced in earlier work of the second author. Now let $M$ be a symplectic manifold with a Hamiltonian action of $K$. Let $\rho$ be a moment mapping and consider the Koszul complex given by the component functions of $\rho$. We show that the Koszul complex is a resolution of the smooth functions on $Z=\rho^{-1}(0)$ if and only if the complexification of each symplectic slice representation at a point of $Z$ is 1-large.