Mathematical Research Letters

Volume 23 (2016)

Number 3

Hamiltonian circle action with self-indexing moment map

Pages: 719 – 732



Yunhyung Cho (School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea; and Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, Korea)

Min Kyu Kim (Department of Mathematics Education, Gyeongin National University of Education, Incheon, Korea)


Let $(M,\omega)$ be a $2n$-dimensional closed symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, and let $\mu : M \to \mathbb{R}$ be a moment map. Then it is well-known that $\mu$ is a Morse function whose critical point set coincides with the fixed point set $M^{S^1}$. Let $\Lambda_{2k}$ be the set of all fixed points of Morse index $2k$. In this paper, we will show that if $\mu$ is constant on $\Lambda_{2k}$ for each $k \leq n$, then $(M,\omega)$ satisfies the hard Lefschetz property. In particular, if $(M,\omega)$ admits a self-indexing moment map, i.e. $\mu (z) = 2k$ for every $k \leq n$ and $z \in \Lambda_{2k}$, then $(M,\omega)$ satisfies the hard Lefschetz property.

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