Higher convexity for complements of tropical varieties

TL;DR

This paper investigates higher convexity properties of complements of tropical varieties, proving new results for hypersurfaces, curves, and nonarchimedean amoebas, and proposing a conjecture supported by partial proofs.

Contribution

It establishes higher convexity for complements of certain tropical varieties and introduces a conjecture generalizing these results.

Findings

Proved higher convexity for complements of tropical hypersurfaces and curves.

Established higher convexity for nonarchimedean amoebas of complete intersection varieties.

Formulated and partially proved a conjecture on the convexity of tropical variety complements.

Abstract

We consider Gromov's homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedean amoebas of varieties that are complete intersections over the field of complex Puiseaux series. Based on these results, we conjecture that the complement of a tropical variety has this higher convexity, and we prove a weak form of our conjecture for the nonarchimedean amoeba of a variety over the complex Puiseaux field. One of our main tools is Jonsson's limit theorem for tropical varieties.