Circle actions on symplectic four-manifolds

We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and $2$-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of $S^2$-bundles over a Riemann surface of positive genus. These do not admit $2$-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of these actions. As a by-product, we provide an algorithm that determines the $g$-reduced form of a blowup form. Our work is a combination of “soft” equivariant and combinatorial techniques, using the momentum map and related data, with “hard” holomorphic techniques, including Gromov–Witten invariants.

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with an Appendix by Tair Pnini

Received 22 July 2015

Accepted 12 January 2017

Published 23 August 2019