Dimensionality of tropical Chow groups
arXiv:2408.12417 · doi:10.1017/prm.2026.10165
The paper proves that if a tropical affine manifold admits non‑zero tropical forms of degree ≥2, then its tropical Chow group of points is infinite‑dimensional, and shows that tropical 1‑forms on surfaces do not necessarily give infinite dimensionality, illustrated by a tropical Klein bottle.
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.
Minor changes to the exposition. New figures. This version was accepted for publication in Proceedings of the Royal Society of Edinburgh, Section A: Mathematics