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

Abstract

Let (M,ω) be a four dimensional compact connected symplectic manifold. We prove that (M,ω) 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,ω) is finite. Our proof is "soft". The proof uses the fact that for symplectic blow-ups of \CP^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.

25 pages, 8 figures. Version 2 contains a completely "soft" proof of the main result. The proof in Version 1 used a (soft) lemma about the existence of certain symplectically embedded spheres in toric manifolds. The proof of this lemma has been removed from the published version but it is included in this Eprint as a second appendix