Symplectic actions on compact manifolds

Abstract:

Let $M^{2n}$ be a compact connected Kähler manifold. We prove two theorems that in this case imply the following:

1. If we have a symplectic action of a compact connected Lie group $G$ on $M^{2n}$ and there exists an orbit under the action of $G$ that is contained in a subset $A$ of $M^{2n}$ whose first Betti number equals zero, then our action is Hamiltonian.

2. If we have a continuous symplectic flow on $M^{2n}$ that preserves some invariant metric, then exactly one of the following statements is true:

   1. Our flow is Hamiltonian.

   2. Our flow has a continuous eigenfunction which is not invariant under the flow.