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Isolated fixed points and moment maps of symplectic manifolds

Clearly every Hamiltonian circle action on a compact symplectic manifold must have fixed points, due to the existence of the maximum and minimum points of the moment map. The goal of this paper is, conversely, to investigate when a symplectic circle action on a compact symplectic manifold becomes Hamiltonian in terms of the fixed point data. As a consequence, we show that if the fixed point set is non-empty and isolated then the symplectic circle action becomes Hamiltonian. The main strategy for showing this is to use a variant of the Euler number at the regular level and its change after passing a critical level.