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

Yael Karshon; Liat Kessler; Martin Pinsonnault

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

Yael Karshon·Liat Kessler·Martin Pinsonnault

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Abstract

Let (M,ω) be a four dimensional compact connected symplectic manifold. We prove that (M,ω) 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,ω) 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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BibTeX RIS

Yael Karshon, Liat Kessler, Martin Pinsonnault. 2006-11-14. A compact symplectic four-manifold admits only finitely many inequivalent toric actions. https://arxiv.org/abs/math/0609043

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