Hamiltonian circle action with self-indexing moment map

Abstract

Let (M,ω) be a 2n-dimensional closed symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, and let μ : M → ℝ be a moment map. Then it is well-known that μ is a Morse function whose critical point set coincides with the fixed point set MS1. Let ∧2k be the set of all fixed points of Morse index 2k. In this paper, we will show that if μ is constant on ∧2k for each k ≤ n, then (M,ω) satisfies the hard Lefschetz property. In particular, if (M,ω) admits a self-indexing moment map, i.e. μ(z) = 2k for every k ≤ n and z ∈∧2k, then (M,ω) satisfies the hard Lefschetz property