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

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