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

Published in Journal of Modern Dynamics, 2006

Recommended citation: Pinsonnault, M., Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. J. Mod. Dyn. 2 (2008), no.3, 431–455. https://doi.org/10.3934/jmd.2008.2.431

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 b23, and if ˜Mδ denotes a blow-up of M of small enough capacity δ, then the rational cohomology algebra of the Hamiltonian group of ˜Mδ 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.

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