A compact symplectic four-manifold admits only finitely many inequivalent toric actions

Details A compact symplectic four-manifold admits only finitely many inequivalent toric actions A compact symplectic four-manifold admits only finitely many inequivalent toric actions

2006-09-01

arxiv.org/abs/math/0609043v2

J. Symplectic Geom. Volume 5, Number 2 (2007), 139-166

Mathematics

Symplectic Geometry

25 pages, 8 figures. Version 2 contains a completely "soft" proof of the main result. The proof in Version 1 used a (soft) lem

Scientific paper

Let (M,\omega) be a four dimensional compact connected symplectic manifold. We prove that (M,\omega) admits only finitely many inequivalent Hamiltonian effective 2-torus actions. Consequently, if M is simply connected, the number of conjugacy classes of 2-tori in the symplectomorphism group Sympl(M,\omega) is finite. Our proof is "soft". The proof uses the fact that for symplectic blow-ups of \CP^2 the restriction of the period map to the set of exceptional homology classes is proper. In an appendix, we describe results of McDuff that give a properness result for a general compact symplectic four-manifold, using the theory of J-holomorphic curves.

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