Geometric Monodromy around the Tropical Limit

TL;DR

This paper describes how the monodromy transformation of a family of smooth hypersurfaces in a toric variety around infinity can be explicitly understood using tropical geometry and Mikhalkin's tropical localization.

Contribution

It provides a concrete tropical geometric description of monodromy transformations for hypersurface families near the tropical limit, advancing the understanding of their geometric structure.

Findings

Explicit description of monodromy in tropical terms

Application of tropical localization to monodromy analysis

Enhanced understanding of hypersurface behavior near the tropical limit

Abstract

Let $\{V_q\}_{q}$ be a complex one-parameter family of smooth hypersurfaces in a toric variety. In this paper, we give a concrete description of the monodromy transformation of $\{V_q\}_q$ around $q=\infty$ in terms of tropical geometry. The main tool is the tropical localization introduced by Mikhalkin.