From compact semi-toric systems to Hamiltonian $S^1$-spaces

Abstract

We show how any labeled convex polygon associated to a compact semi-toric system, as defined by Vũ ngọc, determinesKarshon's labeled directed graph which classifies the underlyingHamiltonian $S^1$-space up to isomorphism. Then we characterizeadaptable compact semi-toric systems, i.e. those whoseunderlying Hamiltonian $S^1$-action can be extended to an effectiveHamiltonian $\mathbb{T}^2$-action, as those which have at least oneassociated convex polygon which satisfies the Delzant condition.