Moment polytopes for symplectic manifolds with monodromy

A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such “almost-toric 4-manifolds” which admits a Hamiltonian S1-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin–Sternberg. As an application, we derive a Duistermaat–Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system.