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

Abstract

Let (M; ω) be a 6-dimensional closed symplectic manifold with a symplectic S1-action with MS1 ≠0 ; and dim 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 b+ 2 (Mred) = 1, where Mred is any reduced space with respect to μ. It means that if the action is non-Hamiltonian, then b+ 2 (Mred) ≥ 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]. © 2015, International Press of Boston, Inc. All rights reserved