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 μ: X → * . 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 μ ―1 (0) in terms of the maximal weights lim t→∞ 〈μ(e its · x), s), where s ∈ . We do not impose any a priori restriction on the stabilizer of x. Under some mild restrictions on the action K X, we view the maximal weights as defining a collection of maps: for each x ∈ X, λ x : ∂ ∞ (K\G) → ℝ ∪ {∞}, where ∂ ∞ (K\G) is the boundary at infinity of the symmetric space K\G. We prove that G · x ∩ μ ―1 (0) ≠ 0 if: (1) λ x is everywhere nonnegative, (2) any boundary point y such that λ x (y) = 0 can be connected with a geodesic in K\G to another boundary point y' satisfying λ x (y') = 0. We also prove that the maximal weight functions are G-equivariant: for any g ∈ G and any y ∈ ∂ ∞ (K\G) we have λ g·x (y) = λ x (y · g).