Frankel's theorem in the symplectic category

Abstract

We prove that if an (n - I)-dimensional torus acts symplectically on a 2n-dimensional symplectic manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel's theorem which says that a Kahler circle action has a fixed point if and only if it is Hamiltonian. The case of n = 2 is the well-known theorem by McDuff.