Circle actions on symplectic four-manifolds

TL;DR

This paper completes the classification of Hamiltonian circle actions on certain symplectic four-manifolds, especially blowups of S^2-bundles over Riemann surfaces, using a mix of combinatorial and holomorphic techniques.

Contribution

It characterizes Hamiltonian circle actions on blowups of S^2-bundles over Riemann surfaces and provides an algorithm for determining the g-reduced form of a blowup form.

Findings

Characterization of Hamiltonian circle actions on blowups of S^2-bundles over Riemann surfaces.

Development of an algorithm to determine the g-reduced form of a blowup form.

Combination of equivariant, combinatorial, and holomorphic techniques in the analysis.

Abstract

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.