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Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Pranav V. Chakravarthy, Martin Pinsonnault

TL;DR

The work investigates the homotopy type of centralizers of Hamiltonian finite cyclic actions on rational ruled surfaces by linking Z_n-actions to toric extensions via Delzant-type and Chen–Wilczynski classifications. It develops a robust J-holomorphic framework that reduces the problem to stratifications of invariant almost complex structures, enabling homotopy decompositions into pushouts of toric isometry groups when actions are tractable. For actions admitting a single toric extension, the centralizer is typically a finite-dimensional Lie group, while actions with two toric extensions along a common circle yield a pushout description that produces a loop-space–heavy factor ΩS^3 in the homotopy type. The results cover S^2×S^2 and CP^2#\overline{CP}^2 with a broad family of Z_n-actions, and analogous statements hold for the odd Hirzebruch surfaces in CP^2#\overline{CP}^2, thereby extending the understanding of equivariant symplectomorphism groups beyond circle actions. Open cases remain where a complete equivariant symplectic classification of Z_n-actions is not yet available, highlighting directions for future work in toric-extension analysis and Chen–Wilczynski-type classifications.

Abstract

Let $M=(M,ω)$ be either the product $S^2\times S^2$ or the non-trivial $S^2$ bundle over $S^2$ endowed with any symplectic form $ω$. Suppose a finite cyclic group $Z_n$ is acting effectively on $(M,ω)$ through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism $Z_n\hookrightarrow Ham(M,ω)$. In this paper, we investigate the homotopy type of the group $Symp^{Z_n}(M,ω)$ 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 $[ω]$, 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 $Z_n$-actions on Hirzebruch surfaces.

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

TL;DR

The work investigates the homotopy type of centralizers of Hamiltonian finite cyclic actions on rational ruled surfaces by linking Z_n-actions to toric extensions via Delzant-type and Chen–Wilczynski classifications. It develops a robust J-holomorphic framework that reduces the problem to stratifications of invariant almost complex structures, enabling homotopy decompositions into pushouts of toric isometry groups when actions are tractable. For actions admitting a single toric extension, the centralizer is typically a finite-dimensional Lie group, while actions with two toric extensions along a common circle yield a pushout description that produces a loop-space–heavy factor ΩS^3 in the homotopy type. The results cover S^2×S^2 and CP^2#\overline{CP}^2 with a broad family of Z_n-actions, and analogous statements hold for the odd Hirzebruch surfaces in CP^2#\overline{CP}^2, thereby extending the understanding of equivariant symplectomorphism groups beyond circle actions. Open cases remain where a complete equivariant symplectic classification of Z_n-actions is not yet available, highlighting directions for future work in toric-extension analysis and Chen–Wilczynski-type classifications.

Abstract

Let be either the product or the non-trivial bundle over endowed with any symplectic form . Suppose a finite cyclic group is acting effectively on through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism . In this paper, we investigate the homotopy type of the group of equivariant symplectomorphisms. We prove that for some infinite families of actions satisfying certain inequalities involving the order and the symplectic cohomology class , 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 -holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on -manifolds, and on the Chen-Wilczyński classification of smooth -actions on Hirzebruch surfaces.
Paper Structure (37 sections, 79 theorems, 91 equations, 3 figures, 4 tables)

This paper contains 37 sections, 79 theorems, 91 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Overview of the argument
  3. Abelian actions on $(S^2 \times S^2,\omega_\lambda)$
  4. Hamiltonian actions and almost-complex structures
  5. Case of a unique toric extension
  6. Case of exactly two toric extensions along a common circle
  7. Tractable Hamiltonian $\mathbb{Z}_{n}$ actions
  8. $\mathbb{Z}_n(a,b;r)$ actions with $\gcd(a,n)\neq 1\pmod{n}$
  9. $\mathbb{Z}_n(a,b;r)$ actions with $\gcd(a,n)= 1\pmod{n}$
  10. Remaining cases
  11. Toric extensions of finite cyclic groups actions
  12. Hamiltonian actions on $S^2 \times S^2$ and $\mathbb{C}P^2\# \overline{\mathbb{C}P^2}$
  13. Proof of Proposition \ref{['prop:gcdConditionSingleToricExtension']}
  14. The Chen--Wilczyński classification
  15. Consequences of the smooth classification for toric extensions
  16. ...and 22 more sections

Key Result

Proposition 2.2

Let $M_{\lambda}$ be the product $S^2 \times S^2$ endowed with the split symplectic form $\omega_\lambda=\lambda\sigma\times\sigma$, where $1\leq\lambda =\ell+\delta$ with $\ell\geq0$ an integer and $0<\delta\leq 1$.

Figures (3)

  • Figure 1: Even Hirzebruch polygon
  • Figure 2: Odd Hirzebruch polygon
  • Figure 3: Moment polytope for $\mathbb{T}^2_0$ action.

Theorems & Definitions (159)

  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4: Proposition 4.7 in Liat
  • Corollary 2.5
  • proof
  • Proposition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Proposition 2.9
  • ...and 149 more