Spherical tropicalization and Berkovich analytification

TL;DR

This paper extends the concept of tropicalization to spherical varieties, demonstrating that the tropicalization map factors through Berkovich analytification and acts as a strong deformation retraction, generalizing known results from toric varieties.

Contribution

It generalizes tropicalization and deformation retraction results from toric to spherical varieties, connecting tropical geometry with Berkovich analytification in this broader context.

Findings

Tropicalization factors through Berkovich analytification for spherical varieties.

Tropicalization acts as a strong deformation retraction of the analytification.

Provides a deformation retraction of Thuillier's analytification onto a subspace.

Abstract

Let $X$ be a spherical variety. We show that Tevelev and Vogiannou's tropicalization map from $X$ to its tropicalization factors through the Berkovich analytification $X^{\text{an}}$, as in the case for toric varieties. Furthermore we show that the tropicalization is a strong deformation retraction of $X^{\text{an}}$. We also give a strong deformation retraction of Thuillier's analytification $X^{\beth}$ onto a subspace described using the colored fan of $X$.