Dimensionality of tropical Chow groups

TL;DR

This paper investigates the structure of tropical Chow groups, showing that higher-degree forms imply infinite dimensionality, while certain 1-forms do not, drawing parallels with classical algebraic geometry results.

Contribution

It establishes a tropical analog of classical Chow group results, linking the existence of tropical forms to the dimensionality of tropical Chow groups.

Findings

Non-zero tropical forms of degree ≥ 2 imply infinite-dimensional tropical Chow groups.

Existence of tropical 1-forms on surfaces does not necessarily imply infinite dimensionality.

Tropical Klein bottle serves as a counterexample for 1-forms and dimensionality.

Abstract

We show that the existence of non-zero tropical forms of degree at least two implies that the tropical Chow group of points of a tropical affine manifold is infinite-dimensional. This can be seen as a tropical analog of classical results of Mumford and Roitman for Chow groups of smooth (complex) projective algebraic varieties. We also show that the existence of tropical 1-forms on tropical surfaces does not imply infinite dimensionality by considering the case of a tropical Klein bottle.