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\(L^2\)-cohomology and complete Hamiltonian manifolds. (English) Zbl 1304.53086

Summary: A classical theorem of Frankel for compact Kähler manifolds states that a Kähler \(S^1\)-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.

MSC:

53D25 Geodesic flows in symplectic geometry and contact geometry
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
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