Properties of gradient maps associated with action of real reductive group

Abstract

Let [Formula: see text] be a Kähler manifold and let [Formula: see text] be a compact connected Lie group with Lie algebra [Formula: see text] acting on [Formula: see text] and preserving [Formula: see text]. We assume that the [Formula: see text]-action extends holomorphically to an action of the complexified group [Formula: see text] and the [Formula: see text]-action on [Formula: see text] is Hamiltonian. Then there exists a [Formula: see text]-equivariant momentum map [Formula: see text]. If [Formula: see text] is a closed subgroup such that the Cartan decomposition [Formula: see text] induces a Cartan decomposition [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the Lie algebra of [Formula: see text], there is a corresponding gradient map [Formula: see text]. If [Formula: see text] is a [Formula: see text]-invariant compact and connected real submanifold of [Formula: see text] we may consider [Formula: see text] as a mapping [Formula: see text] Given an [Formula: see text]-invariant scalar product on [Formula: see text], we obtain a Morse like function [Formula: see text] on [Formula: see text]. We point out that, without the assumption that [Formula: see text] is a real analytic manifold, the Lojasiewicz gradient inequality holds for [Formula: see text]. Therefore, the limit of the negative gradient flow of [Formula: see text] exists and it is unique. Moreover, we prove that any [Formula: see text]-orbit collapses to a single [Formula: see text]-orbit and two critical points of [Formula: see text] which are in the same [Formula: see text]-orbit belong to the same [Formula: see text]-orbit. We also investigate convexity properties of the gradient map [Formula: see text] in the Abelian case. In particular, we study two-orbit variety [Formula: see text] and we investigate topological and cohomological properties of [Formula: see text].