Journal of Modern Dynamics (JMD)

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

Pages: 431 - 455, Issue 3, July 2008      doi:10.3934/jmd.2008.2.431

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Martin Pinsonnault - Fields Institute, Toronto, M5T 3J1, Canada (email)

Abstract: 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:  Symplectomorphism groups, 4-manifolds, Hamiltonian actions, J - holomorphic curves.
Mathematics Subject Classification:  Primary: 57R17, Secondary: 53D20.

Received: October 2007;      Revised: February 2008;      Available Online: April 2008.