A compact symplectic four-manifold admits only finitely many inequivalent toric actions

Abstract

Let (M, ) be a four-dimensional compact connected symplectic manifold.We prove that (M, ) admits only finitely many inequivalent Hamiltonian effective 2-torus actions.Consequently, if M is simply connected, the number of conjugacy classes of 2-tori in the symplectomorphism group Sympl(M, ) is finite.Our proof is "soft".The proof uses the fact that if a symplectic four-manifold admits a toric action, then the restriction of the period map to the set of exceptional homology classes is proper.In an appendix, we present the Gromov-McDuff properness result for a general compact symplectic four-manifold.