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

Abstract

Let (M, w) be a 6-dimensional closed symplectic manifold with a symplectic S-1-action with M-S1 not equal empty set and dim M-S1 <= 2. Assume that w is integral with a generalized moment map mu. 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 mu. It means that if the action is non-Hamiltonian, then b(2)(+)(M-red) >= 2. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if M-S1 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 not equal 2 and we construct some examples of 6-dimensional semifree Hamiltonian S-1-manifolds such that M-S1 contains k surfaces of positive genera for k = 0 and 4. Examples with k = 1 and 3 were given in [L2].