A structure theorem along fibers of extreme points of the momentum polytope

Abstract: Let G be a complex reductive Lie group acting on a compact K\"ahler manifold X and assume that the action of a maximal compact subgroup K of G is Hamiltonian. For each extreme point of the convex hull of the momentum map image, there is an associated open dense subset of X, which is invariant under a parabolic subgroup Q of G. We prove a Q-equivariant product decomposition for the Q-action on this subset and discuss some applications of the result. We show a similar statement for real reductive subgroups of G for the restricted momentum map.