Global Spherical Tropicalization via Toric Embeddings

TL;DR

This paper introduces a new approach to spherical tropicalization by embedding spherical varieties into toric varieties, enabling a simplified description and proving that closure operations commute with tropicalization.

Contribution

It provides an equivalent description of spherical tropicalization using toric embeddings and demonstrates that taking closures commutes with the tropicalization process.

Findings

New description of spherical tropicalization via toric embeddings

Proof that closure operations commute with tropicalization

Simplification of the spherical tropicalization process

Abstract

The first steps in defining tropicalization for spherical varieties have been taken in the last few years. There are two parts to this theory: tropicalizing subvarieties of homogeneous spaces and tropicalizing their closures in spherical embeddings. In this paper, we obtain a new description of spherical tropicalization that is equivalent to the other theories. This works by embedding in a toric variety, tropicalizing there, and then applying a particular piecewise projection map. We use this theory to prove that taking closures commutes with the spherical tropicalization operation.