Liouville domains from Okounkov bodies

TL;DR

This paper constructs symplectic structures and Hamiltonians on complex tori from Okounkov bodies, linking algebraic geometry with symplectic topology and providing models for toric degenerations of Fano manifolds.

Contribution

It introduces a novel method to build symplectic forms and Hamiltonians from rational PL functions on fans, connecting Okounkov bodies with symplectic and contact geometry.

Findings

Families of periodic orbits correspond to lattice points of the fan.

Level sets of Hamiltonians are contact hypersurfaces under certain conditions.

Provides a dynamical model for toric degenerations of Fano manifolds.

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.