Cycles, cocycles, and duality on tropical manifolds

TL;DR

This paper establishes Poincaré duality for Chow rings of smooth fans in tropical geometry, enabling the definition of pull-backs of tropical cycles along arbitrary morphisms with smooth targets.

Contribution

It proves a Poincaré duality for Chow rings of tropical linear spaces and introduces a framework for pull-backs of tropical cycles on manifolds.

Findings

Poincaré duality holds for Chow rings of smooth tropical fans.

Cycles and cocycles are dual on tropical manifolds.

Pull-backs of tropical cycles are well-defined along arbitrary morphisms.

Abstract

We prove a Poincar\'e duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincar\'e dual to each other. This allows us to define pull-backs of tropical cycles along arbitrary morphisms with smooth target.