A Hilbert–Mumford criterion for polystability in Kaehler 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 $x\in X$, \[ \lambda _x:\partial _{\infty }(K\backslash G)\to \mathbb {R}\cup \{\infty \},\] 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 $\lambda _{g\cdot x}(y)=\lambda _x(y\cdot g)$.