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A HILBERT–MUMFORD CRITERION FOR POLYSTABILITY IN KAEHLER GEOMETRY

By I. Mundet and I Riera

Abstract

Abstract. Consider a Hamiltonian action of a compact Lie group K on a Kaehler manifold X with moment map μ: X → 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 μ−1 (0) in terms of the maximal weights limt→∞〈μ(eits · x),s〉, wheres∈k. 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, λx: ∂∞(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) = ∅ if: (1) λx is everywhere nonnegative, (2) any boundary point y such that λx(y) = 0 can be connected with a geodesic in K\G to another boundary point y ′ satisfying λx(y ′) = 0. We also prove that the maximal weight functions are G-equivariant: for any g ∈ G and any y ∈ ∂∞(K\G) wehaveλg·x(y) =λx(y · g). 1

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.352.3692
Provided by: CiteSeerX
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