Homogeneous K\"ahler and Hamiltonian manifolds

TL;DR

This paper studies the actions of reductive complex Lie groups on K"ahler manifolds, showing orbit closures are complex-analytic and characterizing homogeneous K"ahler manifolds via isotropy subgroups, with conditions for the existence of moment maps.

Contribution

It establishes a link between orbit closures and complex-analyticity, and characterizes homogeneous K"ahler manifolds based on isotropy subgroup properties.

Findings

Orbit closures are complex-analytic in K"ahler manifolds.

Homogeneous K"ahler manifolds are characterized by their isotropy subgroups.

Existence of K-moment maps depends on isotropy groups being algebraic.

Abstract

We consider actions of reductive complex Lie groups $G=K^C$ on K\"ahler manifolds $X$ such that the $K$--action is Hamiltonian and prove then that the closures of the $G$--orbits are complex-analytic in $X$. This is used to characterize reductive homogeneous K\"ahler manifolds in terms of their isotropy subgroups. Moreover we show that such manifolds admit $K$--moment maps if and only if their isotropy groups are algebraic.