Journal of Symplectic Geometry
Volume 5 (2007)
A compact symplectic four-manifold admits only finitely many inequivalent toric actions
Pages: 139 – 166
Let ($M$, \omega) be a four-dimensional compact connected symplectic manifold. We prove that ($M$, \omega) 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($M$, \omega) 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.