Properties of Gradient maps associated with Action of Real reductive Group

Abstract

Let $(Z,\omega)$ be a \Keler manifold and let $U$ be a compact connected Lie group with Lie algebra $\mathfrak{u}$ acting on $Z$ and preserving $\omega$. 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 $\mu : Z\to \mathfrak{u}$. If $G\subset U^{\mathbb C}$ is a closed subgroup such that the Cartan decomposition $U^{\mathbb C} = U\text{exp}(i\mathfrak{u})$ induces a Cartan decomposition $G = K\text{exp}(\mathfrak{p}),$ where $K = U\cap G$, $\mathfrak{p} = \mathfrak{g}\cap i\mathfrak{u}$ and $\mathfrak{g}=\mathfrak k \oplus \mathfrak p$ is the Lie algebra of $G$, there is a corresponding gradient map $\mu_\mathfrak{p} : Z\to \mathfrak{p}$. If $X$ is a $G$-invariant compact and connected real submanifold of $Z,$ we may consider $\mu_{\mathfrak p}$ as a mapping $\mu_\mathfrak{p} : X\to \mathfrak{p}.$ Given an $\mathrm{Ad}(K)$-invariant scalar product on $\mathfrak p$, we obtain a Morse like function $f=\frac{1}{2}\parallel \mu_{\mathfrak p} \parallel^2$ 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 $\mu_\mathfrak{p}$ in the Abelian cases. In particular, we study two orbits variety $X$ and we investigate topological and cohomological properties of $X$.