Maximal compact tori in the Hamiltonian groups of 4-dimensional symplectic manifolds

We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply connected symplectic 4-manifold with $b_{2}\geq 3$, and if $\widetilde{M}_{\delta}$ denotes a blow-up of $M$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group of $\widetilde{M}_{\delta})$ is not finitely generated. Both results are based on the fact that in a symplectic 4-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable. Some applications and examples are given.