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

Abstract

We prove that the group of Hamiltonian automorphisms of a symplectic$4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely manyconjugacy classes of maximal compact tori with respect to the actionof the full symplectomorphism group Symp$(M,\omega)$. We also considerthe set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ withrespect to Hamiltonian conjugation and show that its finiteness isequivalent to the finiteness of the symplectic mapping class group$\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rationaland ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with$b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity$\delta$, then the rational cohomology algebra of the Hamiltoniangroup Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitelygenerated. Our 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.