论文标题
Okounkov尸体的Liouville域
Liouville domains from Okounkov bodies
论文作者
论文摘要
给定一个严格的凹入合理的PL功能$ ϕ $在完整的$ n $二维风扇上$σ$，我们在$（\ Mathbb {c}^\ times）^n $上构建了有限体积的确切符号结构，并且函数$ h_ {ϕ，ε} $称为Polyhedral Hamilton。我们证明，对于$ h_ {ϕ，ε} $的每个$ε$的单周期轨道都来自与有限的许多原始晶格点相对应的$σ$，并确定其拓扑。当$ ϕ $在$σ$的光线上为负数时，我们表明多面体汉密尔顿人的水平集是接触类型的超出表面。作为副产品，这种结构为通过Okounkov体在任何维度中的Fano歧管变性而获得的曲折品种的奇异性提供了动力学模型。
Given a strictly concave rational PL function $ϕ$ on a complete $n$-dimensional fan $Σ$, we construct an exact symplectic structure of finite volume on $(\mathbb{C}^\times)^n$ and a family of functions $H_{ϕ,ε}$ called polyhedral Hamiltonians. We prove that for each $ε$ the one-periodic orbits of $H_{ϕ,ε}$ come in families corresponding to finitely many primitive lattice points of $Σ$ and determine their topology. When $ϕ$ is negative on the rays of $Σ$, 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.
