Invariant Kähler potentials and symplectic reduction

Abstract

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

43 pages