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

2006/12/19 by Martin Pinsonnault · 2 citations
Mathematics · #math.SG #math.DG #msc:57R17 #msc:57S15 #msc:53D20

paper · pdf

published as J. Modern Dynamics, Vol. 2, NO. 3, 2008, 431-455 · 22 pages

arxiv created 2006/12/19 · arxiv updated 2011/04/26

Abstract

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 b₂≥ 3, and if \widetildeM_δ denotes a blow-up of M of small enough capacity δ, then the rational cohomology algebra of the Hamiltonian group of \widetildeM_δ) 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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