Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let $M=(M,\omega)$ be either the product $S^2\times S^2$ or the non-trivial $S^2$ bundle over $S^2$ endowed with any symplectic form $\omega$. Suppose a finite cyclic group $\mathbf{Z}_n$ is acting effectively on $(M,\omega)$ through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism $\mathbf{Z}_n\hookrightarrow \mathrm{Ham}(M,\omega)$. In this paper, we investigate the homotopy type of the group $\mathrm{Symp}^{\mathbf{Z}_n}(M,\omega)$ of equivariant symplectomorphisms. We prove that for some infinite families of $\mathbf{Z}_n$ actions satisfying certain inequalities involving the order $n$ and the symplectic cohomology class $[\omega]$, the actions extends to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on $J$-holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczyński classification of smooth $\mathbf{Z}_n$-actions on Hirzebruch surfaces.

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