Paper detail

Applying Hodge theory to detect Hamiltonian flows

We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of harmonic one-forms. For example, this is the case for complete Kähler manifolds for which the symplectic form has an appropriate decay at infinity. This extends a classical theorem of Frankel for compact Kähler manifolds to complete non-compact Kähler manifolds.