Journal of Symplectic Geometry

The Koszul complex of a moment map

Hans-Christian Herbig and Gerald W. Schwarz

Full-text: Open access

Abstract

Let $K \to \operatorname{U}(V)$ 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 $\rho_1, \dots, \rho_k$ 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 $G \to \operatorname{GL}(V)$ is $1$-large, a concept introduced in [11,12]. 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.

Article information

Source
J. Symplectic Geom., Volume 11, Number 3 (2013), 497-508.

Dates
First available in Project Euclid: 12 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.jsg/1384282847

Mathematical Reviews number (MathSciNet)
MR3100804

Zentralblatt MATH identifier
1286.53082

Citation

Herbig, Hans-Christian; Schwarz, Gerald W. The Koszul complex of a moment map. J. Symplectic Geom. 11 (2013), no. 3, 497--508. https://projecteuclid.org/euclid.jsg/1384282847


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