Abstract. We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian \(S^1\)-spaces. Additionally, we show that all these spaces are Kähler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are isolated then the space is a toric variety.

Readership

Graduate students and research mathematicians interested in global analysis, analysis on manifolds, and symplectic geometry.

Table of Contents

• Introduction
• Graphs
• Metrics
• Uniqueness: Graph determines space
• Isolated fixed points implies toric variety
• Blowing-up
• Completing the classification; our spaces are Kähler
• Appendices
• References