Hamiltonian circle action with self-indexing moment map

Abstract

Let $(M,\omega )$ be a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points and let $\mu :M\to \mathbb{R}$ be a corresponding moment map. Let ${\Lambda }_{2k}$ be the set of all fixed points of index $2k$. In this paper, we will show that if $\mu$ is constant on ${\Lambda }_{2k}$ for each $k$, then $(M,\omega )$ satisfies the hard Lefschetz property. In particular, if $(M,\omega )$ admits a self-indexing moment map, i.e. $\mu (p)=2k$ for every $p\in {\Lambda }_{2k}$ and $k=0,1,\cdots ,n,$ then $(M,\omega )$ satisfies the hard Lefschetz property.

7 pages, 2 figures

References in corpus (1)

• Hard Lefschetz Property for Hamiltonian torus actions on 6-dimensional GKM manifolds