Liouville domains from Okounkov bodies

Abstract

Given a strictly concave rational PL function $\phi$ on a complete $n$-dimensional fan $\Sigma$, we construct an exact symplectic structure of finite volume on $(\mathbb{C}^\times)^n$ and a family of functions $H_{\phi,\epsilon}$ called polyhedral Hamiltonians. We prove that for each $\epsilon$ the one-periodic orbits of $H_{\phi,\epsilon}$ come in families corresponding to finitely many primitive lattice points of $\Sigma$ and determine their topology. When $\phi$ is negative on the rays of $\Sigma$, we show that the level sets of polyhedral Hamiltonians are hypersurfaces of contact type. As a byproduct, this construction provides a dynamical model for the singularities of toric varieties obtained as degenerations of Fano manifolds in any dimension via Okounkov bodies.