Abelian tropical covers

TL;DR

This paper classifies harmonic abelian covers of tropical curves using cohomology, provides realizability criteria, and connects the theory to graph flow problems, advancing understanding of tropical geometry and combinatorics.

Contribution

It introduces a cohomological classification of harmonic abelian covers of tropical curves and establishes realizability criteria linking local data to global structures.

Findings

Classification of harmonic e6-covers via sheaf cohomology

Realizability criterion based on local monodromy data

Connection between realizability and nowhere-zero flow problem

Abstract

Let $\mathfrak{A}$ be a finite abelian group. In this article, we classify harmonic $\mathfrak{A}$-covers of a tropical curve $\Gamma$ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on $\Gamma$. We give a realizability criterion for harmonic $\mathfrak{A}$-covers by patching local monodromy data in an extended homology group on $\Gamma$. As an explicit example, we work out the case $\mathfrak{A}=\mathbb{Z}/p\mathbb{Z}$ and explain how realizability for such covers is related to the nowhere-zero flow problem from graph theory.