Hamiltonian circle action with self-indexing moment map

الأصل · EN

Let (M,ω) be a 2n-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points and let μ: M → be a corresponding moment map. Let Λ₂ₖ be the set of all fixed points of index 2k. In this paper, we will show that if μ is constant on Λ₂ₖ for each k, then (M,ω) satisfies the hard Lefschetz property. In particular, if (M,ω) admits a self-indexing moment map, i.e. μ(p) = 2k for every p ∈ Λ₂ₖ and k=0,1,,n, then (M,ω) satisfies the hard Lefschetz property.