Lagrangian Skeleta of Hypersurfaces in $(\mathbb{C}^*)^n$

Abstract

Let $W(z_1, \cdots, z_n): (\mathbb{C}^*)^n \to \mathbb{C}$ be a Laurent polynomial in $n$ variables, and let $\mathcal{H}$ be a generic smooth fiber of $W$. In \cite{RSTZ} Ruddat-Sibilla-Treumann-Zaslow give a combinatorial recipe for a skeleton for $\mathcal{H}$. In this paper, we show that for a suitable exact symplectic structure on $\mathcal{H}$, the RSTZ-skeleton can be realized as the Liouville Lagrangian skeleton.