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

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Let (M, ω) be a 6-dimensional closed symplectic manifold with a symplectic S¹-action with Mˢ¹ ≠ and Mˢ¹ ≤ 2. Assume that ω is integral with a generalized moment map μ. We first prove that the action is Hamiltonian if and only if b₂+(M)=1, where M is any reduced space with respect to μ. It means that if the action is non-Hamiltonian, then b₂+(M) ≥ 2. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if Mˢ¹ 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 ≠ 2 and we construct some examples of 6-dimensional semifree Hamiltonian S¹-manifolds such that Mˢ¹ contains k surfaces of positive genera for k = 0 and 4. Examples with k=1 and 3 were given in L2.