Remarks on Semistable Points and Nonabelian Convexity of Gradient Maps
Oluwagbenga Joshua Windare
TL;DR
The paper develops a gradient-map framework for real reductive group actions on Kähler manifolds, proving that the semistable set for the gradient map $\mu_{\mathfrak{p}}$ is open, dense, and connected when semistability holds for the complexified action. It establishes a nonabelian convexity theorem on $G$-invariant compact Lagrangian submanifolds $X$ with $X\subset \mu_{\mathfrak{k}}^{-1}(0)$, showing $\mu_{\mathfrak{p}}(X)\cap \mathfrak{a}_+$ is a convex polytope, and proves a convexity result for two-orbit varieties via Morse-Bott structure of $f_{\mathfrak{p}}$. The work develops the norm-square gradient map $f_{\mathfrak{p}}$ and its stratification, introduces the energy-complete condition and the maximal-weight criterion, and connects these to a topological Hilbert quotient framework. Overall, it extends nonabelian convexity phenomena to gradient-map settings, with implications for quotient geometry and orbit structure in Kähler settings.
Abstract
We study the action of a real reductive group $G$ on a Kahler manifold $Z$ which is the restriction of a holomorphic action of a complex reductive Lie group $U^\mathbb{C}.$ We assume that the action of $U$, a maximal compact connected subgroup of $U^\mathbb{C}$ on $Z$ is Hamiltonian. If $G\subset U^\mathbb{C}$ is compatible, there is a corresponding gradient map $μ_\mathfrak{p}: Z\to \mathfrak{p}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Our main results are the openness and connectedness of the set of semistable points associated with $G$-action on $Z$, a convexity theorem for the $G$-action on a $G$-invariant compact Lagrangian submanifold of $Z$, and a convexity result for two-orbit variety.