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

Let $(M, ω)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\dim M^{S^1} \leq 2$. Assume that $ω$ is integral with a generalized moment map $μ$. We first prove that the action is Hamiltonian if and only if $b_2^+(M_{\red})=1$, where $M_{\red}$ is any reduced space with respect to $μ$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{\red}) \geq 2$. 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 $S^1$-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 \cite{L2}.

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