A tropical formula for non-Archimedean local heights

TL;DR

This paper develops a new framework using delta-forms on tropical toric varieties to compute non-Archimedean local heights of projective varieties, connecting tropical geometry with arithmetic geometry.

Contribution

It introduces delta-forms on tropical toric varieties and demonstrates their use in calculating non-Archimedean local heights via a star-product with Green functions.

Findings

Delta-forms generalize previous constructions to tropical toric varieties.

Non-Archimedean local heights can be computed using the star-product on tropical toric varieties.

Open subsets of tropical toric varieties admit locally finite simplicial decompositions.

Abstract

We introduce delta-forms on tropical toric varieties generalizing the construction of Mihatsch for $R^n$. These delta-forms will be used to define the star-product with Green functions of piecewise smooth type on a tropical toric variety. As an application, we show that non-archimedean local heights of projective varieties can be computed using the star-product on a suitable complete tropical toric variety. On the way, we show that open subsets of a simplicial tropical toric variety have a locally finite simplicial decomposition which is constant towards the boundary.