Cocycles on tropical varieties via piecewise polynomials

TL;DR

This paper introduces a new framework for tropical cocycles using piecewise polynomials, extending Cartier divisors to higher codimensions, and establishes a tropical Poincaré duality through an intersection product.

Contribution

It generalizes tropical Cartier divisors to higher codimensions and develops a tropical intersection theory with Poincaré duality.

Findings

Defined tropical cocycles via piecewise polynomials

Established an intersection product analogous to cap product

Proved cases of tropical Poincaré duality

Abstract

We use piecewise polynomials to define tropical cocycles generalising the well-known notion of tropical Cartier divisors to higher codimensions. Groups of cocycles are tropical analogues of Chow cohomology groups. We also introduce an intersection product of cocycles with tropical cycles - the counterpart of the classical cap product - and prove that this gives rise to a Poincar\'e duality in some cases.