Applying Hodge theory to detect Hamiltonian flows

TL;DR

This paper extends Frankel's theorem to certain non-compact Kähler manifolds by using Hodge theory to establish conditions under which symplectic group actions are Hamiltonian.

Contribution

It introduces a new approach leveraging Hodge theory on non-compact manifolds to identify Hamiltonian actions, extending classical results to broader settings.

Findings

Hodge theory can be applied to non-compact symplectic manifolds with decay conditions.

Fixed point symplectic actions are Hamiltonian under certain Hodge-theoretic conditions.

Extension of Frankel's theorem to complete non-compact Kähler manifolds.

Abstract

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 K\"ahler manifolds for which the symplectic form has an appropriate decay at infinity. This extends a classical theorem of Frankel for compact K\"ahler manifolds to complete non-compact K\"ahler manifolds.