Invariant Kähler potentials and symplectic reduction

For a proper Hamiltonian action of a Lie group $G$ on a Kähler manifold $(X,ω)$ with momentum map $μ$ we show that the symplectic reduction $μ^{-1}(0)/G$ is a normal complex space. Every point in $μ^{-1}(0)$ has a $G$-stable open neighborhood on which $ω$ and $μ$ are given by a $G$-invariant Kähler potential. This is used to show that $μ^{-1}(0)/G$ is a Kähler space. Furthermore we examine the existence of potentials away from $μ^{-1}(0)$ with both positive and negative results.

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