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September 2013 The Koszul complex of a moment map
Hans-Christian Herbig, Gerald W. Schwarz
J. Symplectic Geom. 11(3): 497-508 (September 2013).

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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Hans-Christian Herbig. Gerald W. Schwarz. "The Koszul complex of a moment map." J. Symplectic Geom. 11 (3) 497 - 508, September 2013.

Information

Published: September 2013
First available in Project Euclid: 12 November 2013

zbMATH: 1286.53082
MathSciNet: MR3100804

Rights: Copyright © 2013 International Press of Boston

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Vol.11 • No. 3 • September 2013
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