Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

TL;DR

This paper studies the structure of the centralizers of finite cyclic group actions on certain symplectic 4-manifolds, revealing their homotopy types and extending some actions to toric actions using advanced geometric techniques.

Contribution

It characterizes the homotopy type of equivariant symplectomorphism groups for cyclic actions on rational ruled surfaces, extending some actions to toric actions and classifying their centralizers.

Findings

Centralizers are homotopically equivalent to finite-dimensional Lie groups or homotopy pushouts of tori.

Certain cyclic actions extend to toric actions under specific inequalities involving the order and symplectic class.

The results utilize $J$-holomorphic techniques, Delzant's, Karshon's, and Chen-Wilczyński's classifications.

Abstract

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 $Z_n$ is acting effectively on $(M,\omega)$ through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism $Z_n\hookrightarrow Ham(M,\omega)$. In this paper, we investigate the homotopy type of the group $Symp^{Z_n}(M,\omega)$ of equivariant symplectomorphisms. We prove that for some infinite families of $Z_n$ 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-Wilczy\'nski classification of smooth $Z_n$-actions on Hirzebruch surfaces.