On Tropical Intersection Theory

TL;DR

This paper extends tropical intersection theory to include arbitrary polytopes and smooth differential forms, providing a more flexible and comprehensive formalism for tropical geometry.

Contribution

It introduces a new tropical intersection formalism that generalizes classical theory to non-rational polytopes and smooth forms, unifying different intersection definitions.

Findings

Formalism works with arbitrary polytopes, including non-rational ones

Defines intersection product via diagonal intersection and fan displacement methods

Proves equivalence of different intersection definitions

Abstract

We develop a tropical intersection formalism of forms and currents that extends classical tropical intersection theory in two ways. First, it allows to work with arbitrary polytopes, also non-rational ones. Second, it allows for smooth differential forms as coefficients. The intersection product in our formalism can be defined through the diagonal intersection method of Allermann--Rau or the fan displacement rule. We prove with a limiting argument that both definitions agree.