Periodic symplectic and Hamiltonian diffeomorphisms on irrational ruled surfaces

Abstract

We investigate when finite-order Hamiltonian diffeomorphisms extend to Hamiltonian circle actions, probing the transition from discrete to continuous symmetry in symplectic topology. Focusing on irrational ruled symplectic $4$-manifolds, we show that homologically trivial symplectic cyclic actions of order $k>2$ always extend to Hamiltonian $S^1$-actions, possibly after modifying the symplectic form. In contrast, we construct explicit symplectic involutions that cannot be so extended, even on minimal irrational ruled surfaces. These examples reveal geometric obstructions to extending discrete symmetries and highlight new exotic symplectic actions not equivalent to holomorphic ones. Our results also apply to higher-dimensional and non-cyclic group actions, and we establish several structural results on the isomorphism types of finite groups that can act on irrational ruled symplectic $4$-manifolds.