Symplectic circle actions with isolated fixed points

Abstract

Consider a symplectic circle action on a closed symplectic manifold with\nnon-empty isolated fixed points. Associated to each fixed point, there are\nwell-defined non-zero integers, called weights. We prove that the action is\nHamiltonian if the sum of an odd number of weights is never equal to zero (the\nweights may be taken at different fixed points). Moreover, we show that if\n$\\dim M=6$, or if $\\dim M=2n \\leq 10$ and each fixed point has weights $\\{\\pm\na_1, \\cdots, \\pm a_n\\}$ for some positive integers $a_i$, it is enough to\nconsider the sum of three weights. As applications, we recover the results for\nsemi-free actions, and for certain circle actions on six-dimensional manifolds.\n