Centralizers of Hamiltonian circle actions on rational ruled surfaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866911222092267520 |
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| author | Chakravarthy, Pranav Pinsonnault, Martin |
| author_facet | Chakravarthy, Pranav Pinsonnault, Martin |
| contents | In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$ under the presence of Hamiltonian group actions of the circle $S^1$. We prove that the group of equivariant symplectomorphisms are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. This follows from the analysis of the action of equivariant symplectomorphisms on the space of compatible and invariant almost complex structures $\mathcal{J}^{S^1}_ω$. In particular, we show that this action preserves a decomposition of $\mathcal{J}^{S^1}_ω$ into strata which are in bijection with toric extensions of the circle action. Our results rely on $J$-holomorphic techniques, on Delzant's classification of toric actions and on Karshon's classification of Hamiltonian circle actions on $4$-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_08255 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Centralizers of Hamiltonian circle actions on rational ruled surfaces Chakravarthy, Pranav Pinsonnault, Martin Symplectic Geometry Differential Geometry 53D35, 57S05 In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$ under the presence of Hamiltonian group actions of the circle $S^1$. We prove that the group of equivariant symplectomorphisms are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. This follows from the analysis of the action of equivariant symplectomorphisms on the space of compatible and invariant almost complex structures $\mathcal{J}^{S^1}_ω$. In particular, we show that this action preserves a decomposition of $\mathcal{J}^{S^1}_ω$ into strata which are in bijection with toric extensions of the circle action. Our results rely on $J$-holomorphic techniques, on Delzant's classification of toric actions and on Karshon's classification of Hamiltonian circle actions on $4$-manifolds. |
| title | Centralizers of Hamiltonian circle actions on rational ruled surfaces |
| topic | Symplectic Geometry Differential Geometry 53D35, 57S05 |
| url | https://arxiv.org/abs/2202.08255 |