The Koszul complex of a moment map

Abstract

Let K → U(V ) be a unitary representation of the compact Lie group K. Then there is a canonical moment mapping ρ :We show that the Koszul complex is a resolution of the smooth functions on ρ −1 (0) if and only if G → 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 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.

When is a symplectic quotient an orbifold?

“…Recall from Section 2 that by [12,Corollary 4.2], M is Zariski dense in N so that tensoring with C yields an isomorphism χ * …”

“…By [12,Theorem 2.2], N is normal so that X 0 is normal by [5, Proposition 5.4.1]. The codimension of (X 0 ) (X 0 ) pr is at least two, and thus φ extends to an isomorphism W ⊕ W * → X 0 .…”

“…As V is stable, this latter assumption amounts to assuming that V has TPIG as a G-module. It follows by [12,Theorem 3.2] that the G-module V is 1-large.…”

“…In fact, when the action of G on V is 2-large (or even 1-large), the real analytic ideal of the zero set of the moment map is generated by the component functions of the moment map, see [12,Corollary 4.3]. The proof of Theorem 1.3 can be used to show that, with the same hypotheses, there can be no germ of a real analytic symplectomorphism from a neighborhood of the origin of M 0 (the image of 0 in V ) to a linear symplectic orbifold.…”

An Impossibility Theorem for Linear Symplectic Circle Quotients

“…The reader is cautioned that the definition of 2-large in [28] is stronger than the definition used in [15]. Specifically, the definition in [28] requires FPIG, while that in [15] does not. This causes no issue in the above argument as FPIG has been established.…”