A limit of toric symplectic forms that has no periodic Hamiltonians

Abstract

We calculate the Riemann-Roch number of some of the pentagon spaces defined in [Kl], [KM], [HK1]. Using this, we show that while the regular pentagon space is diffeomorphic to a toric variety, even symplectomorphic to one under arbitrarily small perturbations of its symplectic structure, it does not admit a symplectic circle action. In particular, within the cohomology classes of symplectic structures, the subset admitting a circle action is not closed.