Journal of Symplectic Geometry

Volume 13 (2015)

Number 4

Semifree Hamiltonian circle actions on 6-dimensional symplectic manifolds with non-isolated fixed point set

Pages: 963 – 1000



Yunhyung Cho (Center for Geometry and Physics, Institute for Basic Science, Pohang, Korea)

Taekgyu Hwang (Department of Mathematical Sciences, KAIST, Daejeon, Korea)

Dong Youp Suh (Department of Mathematical Sciences, KAIST, Daejeon, Korea)


Let $(M, \omega)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\mathrm{dim} \, M^{S^1} \leq 2$. Assume that $\omega$ is integral with a generalized moment map $\mu$. We first prove that the action is Hamiltonian if and only if $b^+_2 (M_{\mathrm{red}}) = 1$, where $ M_{\mathrm{red}}$ is any reduced space with respect to $\mu$. It means that if the action is non-Hamiltonian, then $b^+_2 (M_{\mathrm{red}}) \geq 2$. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if $ M^{S^1}$ consists of surfaces, then the number $k$ of fixed surfaces with positive genera is at most four. In particular, if the extremal fixed surfaces are spheres, then k is at most one. Finally, we prove that $k \neq 2$ and we construct some examples of 6-dimensional semifree Hamiltonian S1-manifolds such that $ M^{S^1}$ contains $k$ surfaces of positive genera for $k = 0$ and $4$. Examples with $k = 1$ and $3$ were given in [L2].

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Published 17 March 2016