Spherical gradient manifolds

TL;DR

This paper investigates the properties of gradient maps induced by group actions on Kähler manifolds, establishing a criterion for orbit separation related to the existence of open orbits of minimal parabolic subgroups.

Contribution

It generalizes Brion's characterization by linking the orbit separation property of the gradient map to the presence of open orbits of minimal parabolic subgroups in the setting of real-reductive group actions.

Findings

Gradient map almost separates $K$-orbits under certain conditions.

Minimal parabolic subgroup has an open orbit if and only if the gradient map separates orbits.

Generalization of Brion's characterization to real-reductive group actions.

Abstract

We study the action of a real-reductive group $G=K\exp(\lie{p})$ on real-analytic submanifold $X$ of a K\"ahler manifold $Z$. We suppose that the action of $G$ extends holomorphically to an action of the complexified group $G^\mbb{C}$ such that the action of a maximal Hamiltonian subgroup is Hamiltonian. The moment map $\mu$ induces a gradient map $\mu_\lie{p}\colon X\to\lie{p}$. We show that $\mu_\lie{p}$ almost separates the $K$--orbits if and only if a minimal parabolic subgroup of $G$ has an open orbit. This generalizes Brion's characterization of spherical K\"ahler manifolds with moment maps.