A compact symplectic four-manifold admits only finitely many inequivalent toric actions
Published in The Journal of Symplectic Geometry, 2006
Recommended citation: Karshon, Y., Kessler, L., Pinsonnault, M., A compact symplectic four-manifold admits only finitely many inequivalent toric actions. J. Symplectic Geom. 5 (2007), no.2, 139–166. http://projecteuclid.org/euclid.jsg/1202004454
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 $\mathrm{Sympl}(M,\omega)$ is finite. Our proof is “soft”. The proof uses the fact that for symplectic blow-ups of $\mathbf{C}P^2$ the restriction of the period map to the set of exceptional homology classes is proper. In an appendix, we describe results of McDuff that give a properness result for a general compact symplectic four-manifold, using the theory of $J$-holomorphic curves.