L2-cohomology and complete Hamiltonian manifolds

A classical theorem of Frankel for compact Kähler manifolds states that a Kähler S1S1-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when the Hodge theory holds on non-compact manifolds, Frankel’s theorem still holds. Finally, we present several concrete situations in which the assumptions of the metatheorem hold.