### Maximal tori in the symplectomorphism groups of Hirzebruch surfaces

Yael Karshon
2003 Mathematical Research Letters
We count the conjugacy classes of maximal tori in the groups of symplectomorphisms of S 2 × S 2 and of the blow-up of C P 2 at a point. Consider the group Ham(M, ω) of Hamiltonian symplectomorphisms of a symplectic manifold. 1 A k-dimensional torus in Ham(M, ω) is a subgroup which is isomorphic to (S 1 ) k . A maximal torus is one which is not contained in any strictly larger torus. An action of (S 1 ) k on (M, ω) is called Hamiltonian if it admits a moment map, i.e., a map Φ : ξ k are the
more » ... : ξ k are the vector fields on M that generate the action. A Hamiltonian (S 1 ) k -action defines a homomorphism from (S 1 ) k to Ham(M, ω). The action is effective if and only if this homomorphism is one to one. Its image is then a k-dimensional torus in Ham(M, ω). Every k-dimensional torus in Ham(M, ω) is obtained in this way, and two Hamiltonian actions give the same torus if and only if they differ by a reparametrization of (S 1 ) k . Tori in Ham(M, ω) have dimension at most 1 2 dim M . A Hamiltonian action of a ( 1 2 dim M )-dimensional torus is called toric. Theorem 1. Let (M, ω) be a compact symplectic four-manifold. Suppose that dim H 2 (M, R) ≤ 3 and dim H 1 (M, R) = 0. Then every Hamiltonian circle action on (M, ω) extends to a toric action. Recall, a Hamiltonian symplectomorphism is one which can be connected to the identity by a path ψ t such that d dt ψ t = X t • ψ t and ι(X t )ω = dH t for a smooth H t : M → R . 125