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

TL;DR

This paper investigates conditions under which symplectic $S^1$ actions with isolated fixed points on 6-dimensional manifolds are Hamiltonian, extending previous results to non-semifree actions using orbifold resolution and J-holomorphic curves.

Contribution

It proves that certain non-semifree symplectic $S^1$ actions with isolated fixed points on 6-manifolds are Hamiltonian, utilizing orbifold resolution and J-holomorphic techniques.

Findings

Non-semifree actions with isolated fixed points are Hamiltonian under certain conditions.

Resolution of orbifold singularities is effective in analyzing symplectic actions.

J-holomorphic curve methods can be applied to orbifold resolutions in this context.

Abstract

The question of what conditions guarantee that a symplectic $S^1$ action is Hamiltonian has been studied for many years. In a 1998 paper, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of isolated fixed points then the action is Hamiltonian. Furthermore, in a 2010 paper 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 valued moment map, then the action is Hamiltonian. In this paper, we will use this to prove that certain 6-dimensional symplectic actions which are not semifree and have a non-empty set of isolated fixed points are Hamiltonian. In this case, the reduced spaces are 4-dimensional symplectic orbifolds, and we will resolve the orbifold singularities and use J-holomorphic curve techniques on the resolutions.