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The Koszul complex of a moment map

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  • Hans-Christian Herbig, Denmark
  • Gerald W. Schwarz, Brandeis, United States
Let $K\to\U(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $\rho\colon V\to\liek^*$. We have the Koszul complex ${\mathcal K}(\rho,\mathcal C^\infty(V))$ of the component functions $\rho_1,\dots,\rho_k$ of $\rho$. Let $G=K_\C$, the complexification of $K$. We show that the Koszul complex is a resolution of the smooth functions on $\rho\inv(0)$ if and only if $G\to\GL(V)$ is $1$-large, a concept introduced in \cite{JAlg94,LiftingDO}. Now let $M$ be a symplectic manifold with a Hamiltonian action of the compact Lie group $K$. 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 $Z=\rho\inv(0)$ if and only if the complexification of each symplectic slice representation at a point of $Z$ is $1$-large.
Original languageEnglish
JournalJournal of Symplectic Geometry
Volume11
Issue3
Pages (from-to)497-508
Number of pages12
ISSN1527-5256
Publication statusPublished - 2013

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