A Hilbert--Mumford criterion for polystability in Kaehler geometry

TL;DR

This paper establishes a new criterion based on maximal weights for determining polystability of orbits in Kaehler geometry under Hamiltonian group actions, extending classical stability notions without stabilizer restrictions.

Contribution

It introduces a Hilbert--Mumford type criterion for polystability in Kaehler geometry using boundary at infinity maps and maximal weights, generalizing previous stability conditions.

Findings

Characterizes orbit intersection with zero level set of moment map.

Defines a boundary map mbda_x for maximal weights.

Provides conditions involving nonnegativity and boundary connectivity for orbit stability.

Abstract

Consider a Hamiltonian action by biholomorphisms of a compact Lie group $K$ on a Kaehler manifold $X$, with moment map $\mu:X\to\klie^*$. We characterize which orbits of the complexified action of $G=K^{\CC}$ in $X$ intersect $\mu^{-1}(0)$ in terms of the maximal weights $\lim_{t\to\infty}\la\mu(e^{\imag ts}\cdot x),s\ra$, where $s$ belongs to the Lie algebra of $K$. We do not impose any a priori restriction on the stabilizer of $x$. Assuming some mild growth conditions on the action of $K$ on $X$, we view the maximal weights as defining a maps $\lambda_x$ from the boundary at infinity of the symmetric space $K\backslash G$ to $\RR\cup\{\infty\}$. We prove that $G\cdot x$ meets $\mu^{-1}(0)$ 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 $\lambda_{g\cdot x}(y)=\lambda_x(y\cdot g)$ for any $g\in G$ and $y\in \partial_{\infty}(K\backslash G)$.