Frankel's theorem in the symplectic category

Abstract

We prove that if an (n-1)-dimensional torus acts symplectically on a 2n-dimensional 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 theorem. The case of n=2 is the well known theorem of McDuff. From the well known example of McDuff (a six dimensional symplectic non-Hamiltonian circle action with fixed tori) and its products with copies of a two dimensional sphere with the usual rotation, the condition on the dimension of the acting torus is optimal to obtain the result.