Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let M = (M, ω) M= (M, ) be either S 2 × S 2 S² S² or C P 2 # C P 2 ¯ CP²\# {CP²} endowed with any symplectic form ω. Suppose a finite cyclic group Z n Zₙ is acting effectively on (M, ω) (M, ) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Z n ↪ H a m (M, ω) Zₙ Ham (M, ). In this paper, we investigate the homotopy type of the group S y m p Z n (M, ω) Symp^ {Zₙ} (M, ) of equivariant symplectomorphisms. We prove that for some infinite families of Z n Zₙ actions satisfying certain inequalities involving the order n n and the symplectic cohomology class ω, the actions extend 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 J -holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on 4 4 -manifolds, and on the Chen-Wilczyński classification of smooth Z n Zₙ -actions on Hirzebruch surfaces.