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The question of what conditions guarantee that a symplectic $S^1$ action isHamiltonian has been studied for many years. In a 1998 paper, Sue Tolman andJonathon Weitsman proved that if the action is semifree and has a non-empty setof isolated fixed points then the action is Hamiltonian. Furthermore, in a 2010paper Cho, Hwang, and Suh proved in the 6-dimensional case that if we have$b_2^+=1$ at a reduced space at a regular level $\lambda$ of the circle valuedmoment map, then the action is Hamiltonian. In this paper, we will use this toprove that certain 6-dimensional symplectic actions which are not semifree andhave a non-empty set of isolated fixed points are Hamiltonian. In this case,the reduced spaces are 4-dimensional symplectic orbifolds, and we will resolvethe orbifold singularities and use J-holomorphic curve techniques on theresolutions.

Hamiltonian $S^1$ actions with Isolated Fixed Points on 6-Dimensional Symplectic Manifolds