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

TL;DR

This paper establishes a structure theorem for the fibers over extreme points of the momentum polytope in Hamiltonian G-actions on Kähler manifolds, revealing a product decomposition related to parabolic subgroups.

Contribution

It introduces a novel Q-equivariant product decomposition for the action on dense subsets associated with extreme momentum map points, extending to real reductive subgroups.

Findings

Product decomposition for the Q-action on dense subsets

Extension of results to real reductive subgroups

Applications to the structure of momentum map fibers

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.