Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911227088732160 |
|---|---|
| author | Chakravarthy, Pranav V. Pinsonnault, Martin |
| author_facet | Chakravarthy, Pranav V. Pinsonnault, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15046 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces Chakravarthy, Pranav V. Pinsonnault, Martin Symplectic Geometry Primary 53D35, Secondary 57R17, 57S05, 57T20 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. |
| title | Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces |
| topic | Symplectic Geometry Primary 53D35, Secondary 57R17, 57S05, 57T20 |
| url | https://arxiv.org/abs/2306.15046 |