Special Hamiltonian $S^1$-actions on symplectic 4-manifolds

TL;DR

This paper classifies exact special Hamiltonian $S^1$-spaces on symplectic 4-manifolds with vanishing first Chern class, showing they are Stein surfaces, and proves no compact examples exist.

Contribution

It introduces the concept of special Hamiltonian $S^1$-spaces with an equivariant Maslov condition and classifies all exact cases as Stein surfaces.

Findings

No compact special Hamiltonian $S^1$-spaces exist.

All exact special Hamiltonian $S^1$-spaces are Stein surfaces.

Abstract

In this paper we consider symplectic 4-manifolds $(M,\omega)$ with $c_1(M,\omega)=0$ which admit a Hamiltonian $S^1$-action together with an equivariant Maslov condition on orbits of the group action. We call such spaces {\em special Hamiltonian $S^1$-spaces}. It turns out that there are no compact special Hamiltonian $S^1$-spaces. We classify all exact special Hamiltonian $S^1$-spaces and show that all of them admit the structure of a Stein surface.