Communications in Analysis and Geometry

Volume 27 (2019)

Number 2

Circle actions on symplectic four-manifolds

Pages: 421 – 464



Tara S. Holm (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.)

Liat Kessler (Department of Mathematics, Physics, and Computer Science, University of Haifa at Oranim, Tivon, Israel)


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.

with an Appendix by Tair Pnini

Received 22 July 2015

Accepted 12 January 2017

Published 23 August 2019