Tailoring a pair of pants

Abstract

We show how to deform the map $\mathrm{Log}:({\mathbb{C}}^{\ast }{)}^n\to {\mathbb{R}}^n$ such that the image of the complex pair of pants $P^{\circ }\subset ({\mathbb{C}}^{\ast }{)}^n$ is the tropical hyperplane by showing an (ambient) isotopy between $P^{\circ }\subset ({\mathbb{C}}^{\ast }{)}^n$ and a natural polyhedral subcomplex of the product of the two skeleta $S\times \Sigma \subset \mathcal{A}\times \mathcal{C}$ of the amoeba $\mathcal{A}$ and the coamoeba $\mathcal{C}$ of $P^{\circ }$. This lays the groundwork for having the discriminant to be of codimension 2 in topological Strominger-Yau-Zaslow torus fibrations.

final version, to appear in Adv. Math