2021/06/24 by Leonardo Biliotti, Biliotti, Leonardo, Oluwagbenga Joshua Windare +1 · 1 citation
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2106.13074
Let (Z,ω) be a \Keler manifold and let U be a compact connected Lie group with Lie algebra \mathfraku acting on Z and preserving ω. We assume that the U-action extends holomorphically to an action of the complexified group U\mathbb C and the U-action on Z is Hamiltonian. Then there exists a U-equivariant momentum map μ: Z→ \mathfraku. If G⊂ U\mathbb C is a closed subgroup such that the Cartan decomposition U\mathbb C = Uexp(i\mathfraku) induces a Cartan decomposition G = Kexp(\mathfrakp), where K = U∩ G, \mathfrakp = \mathfrakg∩ i\mathfraku and \mathfrakg=\mathfrak k ⊕ \mathfrak p is the Lie algebra of G, there is a corresponding gradient map μ_\mathfrakp : Z→ \mathfrakp. If X is a G-invariant compact and connected real submanifold of Z, we may consider μ\mathfrak p as a mapping μ_\mathfrakp : X→ \mathfrakp. Given an Ad(K)-invariant scalar product on \mathfrak p, we obtain a Morse like function f=(1)/(2)∥ μ\mathfrak p ∥2 on X. We point out that, without the assumption that X is 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_\mathfrakp in the Abelian cases. In particular, we study two orbits variety X and we investigate topological and cohomological properties of X.