  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 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.
