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July  2008, 2(3): 431-455. doi: 10.3934/jmd.2008.2.431

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

Martin Pinsonnault 1,

1. 

Fields Institute, Toronto, M5T 3J1, Canada

Received  October 2007 Revised  February 2008 Published  April 2008

We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega)$. We also consider the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with respect to Hamiltonian conjugation and show that its finiteness is equivalent to the finiteness of the symplectic mapping class group $\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational and 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 Hamiltonian group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely generated. 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.
Keywords: J - holomorphic curves., Symplectomorphism groups, 4-manifolds, Hamiltonian actions.
Mathematics Subject Classification: Primary: 57R17, Secondary: 53D2.
Citation: Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431
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