Convexity of complements of tropical varieties, and approximations of currents

TL;DR

This paper investigates the convexity properties of complements of tropical varieties, confirming a local conjecture, and presents counter-examples to a global conjecture and the generalized Hodge conjecture, revealing new obstructions.

Contribution

It affirms a local convexity conjecture for tropical varieties and introduces counter-examples to the global conjecture and the generalized Hodge conjecture in higher dimensions.

Findings

Confirmed local convexity of complements of tropical varieties.

Constructed counter-examples to the global Nisse--Sottile conjecture.

Provided counter-examples to the generalized Hodge conjecture for positive currents.

Abstract

The goal of this note is to affirm a local version of the conjecture of Nisse--Sottile [NS16] on higher convexity of complements of tropical varieties, while providing a family of counter-examples for the global Nisse--Sottle conjecture in any codimension and dimension higher than 1. Moreover, it will be shown that, surprisingly, this family also provides a family of counter-examples for the generalized Hodge conjecture for positive currents in these dimensions, and gives rise to further approximability obstruction.