The Koszul complex of a moment map

الأصل · EN

Let K→ U(V) be a unitary representation of the compact Lie group K. Then there is a canonical moment mapping ρ V. We have the Koszul complex K(ρ,C∞(V)) of the component functions ρ₁,...,ρₖ of ρ. Let G=KC, the complexification of K. We show that the Koszul complex is a resolution of the smooth functions on ρ⁻¹(0) if and only if G→(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 ρ be a moment mapping and consider the Koszul complex given by the component functions of ρ. We show that the Koszul complex is a resolution of the smooth functions on Z=ρ⁻¹(0) if and only if the complexification of each symplectic slice representation at a point of Z is 1-large.