| Abstrakt: |
Let (Z , ω) be a Kähler manifold and let U be a compact connected Lie group with Lie algebra acting on Z and preserving ω. We assume that the U -action extends holomorphically to an action of the complexified group U ℂ and the U -action on Z is Hamiltonian. Then there exists a U -equivariant momentum map μ : Z → . If G ⊂ U ℂ is a closed subgroup such that the Cartan decomposition U ℂ = U exp (i) induces a Cartan decomposition G = K exp () , where K = U ∩ G , = ∩ i and = ⊕ is the Lie algebra of G , there is a corresponding gradient map μ : Z → . If X is a G -invariant compact and connected real submanifold of Z , we may consider μ as a mapping μ : X →. Given an Ad (K) -invariant scalar product on , we obtain a Morse like function f = 1 2 ∥ μ ∥ 2 on X. We point out that, without the assumption that X is a real analytic manifold, the Lojasiewicz gradient inequality holds for f. Therefore, the limit of the negative gradient flow of f exists and it is unique. Moreover, we prove that any G -orbit collapses to a single K -orbit and two critical points of f which are in the same G -orbit belong to the same K -orbit. We also investigate convexity properties of the gradient map μ in the Abelian case. In particular, we study two-orbit variety X and we investigate topological and cohomological properties of X. [ABSTRACT FROM AUTHOR] |
