On irrational ruled symplectic 4-manifolds, homologically trivial cyclic actions of order k>2 extend to Hamiltonian circle actions, but explicit involutions with connected fixed point sets do not.
Chen,Group actions on4-manifolds: some recent results and open questions, Proceedings of the G¨ okova Geometry-Topology Conference 2009, 1–21, Int
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Periodic symplectic and Hamiltonian diffeomorphisms on irrational ruled surfaces
On irrational ruled symplectic 4-manifolds, homologically trivial cyclic actions of order k>2 extend to Hamiltonian circle actions, but explicit involutions with connected fixed point sets do not.