Centralizers of Hamiltonian circle actions on rational ruled surfaces

Published in arxiv.org, 2022

Recommended citation: Pranav V. Chakravarthy, Martin Pinsonnault. (2023). "Centralizers of Hamiltonian circle actions on rational ruled surfaces." arxiv:2202.08255 . https://arxiv.org/abs/2202.08255

In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of S2×S2 and CP2¯CP2 under the presence of Hamiltonian group actions of the circle S1. 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 JS1(ω). In particular, we show that this action preserves a decomposition of JS1(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.

To appear in Memoirs of the AMS.

arXiv version