Journal of Symplectic Geometry

Volume 11 (2013)

Number 3

The Koszul complex of a moment map

Pages: 497 – 508

DOI: http://dx.doi.org/10.4310/JSG.2013.v11.n3.a9

Authors

Hans-Christian Herbig (Institut for Matematiske Fag, Aarhus Universitet, Aarhus, Denmark)

Gerald W. Schwarz (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

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.

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