A Hilbert-Mumford criterion for polystability in Kahler geometry

Abstract

Consider a Hamiltonian action of a compact Lie group $ K$ on a Kaehler manifold $ X$ with moment map $ \mu:X\to\mathfrak{k}^*$. Assume that the action of $ K$ extends to a holomorphic action of the complexification $ G$ of $ K$. We characterize which $ G$-orbits in $ X$ intersect $ \mu^{-1}(0)$ in terms of the maximal weights $ \lim_{t\to\infty}\langle\mu(e^{\mathbf{i} ts}\cdot x),s\rangle$, where $ s\in\mathfrak{k}$. We do not impose any a priori restriction on the stabilizer of $ x$. Under some mild restrictions on the action $ K\circlearrowright X$, we view the maximal weights as defining a collection of maps: for each $ x\in X$, $\displaystyle \lambda_x:\partial_{\infty}(K\backslash G)\to\mathbb{R}\cup\{\infty\},$ where $ \partial_{\infty}(K\backslash G)$ is the boundary at infinity of the symmetric space $ K\backslash G$. We prove that $ G\cdot x\cap\mu^{-1}(0)\neq\emptyset$ if: (1) $ \lambda_x$ is everywhere nonnegative, (2) any boundary point $ y$ such that $ \lambda_x(y)=0$ can be connected with a geodesic in $ K\backslash G$ to another boundary point $ y'$ satisfying $ \lambda_x(y')=0$. We also prove that the maximal weight functions are $ G$-equivariant: for any $ g\in G$ and any $ y\in \partial_{\infty}(K\backslash G)$ we have $ \lambda_{g\cdot x}(y)=\lambda_x(y\cdot g)$.