Applying Hodge theory to detect Hamiltonian flows

Details Applying Hodge theory to detect Hamiltonian flows Applying Hodge theory to detect Hamiltonian flows

2010-05-12

arxiv.org/abs/1005.2163v1

Mathematics

Symplectic Geometry

12 pages

Scientific paper

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.

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