Spherical Tropicalization

TL;DR

This paper extends tropicalization techniques from algebraic tori to spherical homogeneous spaces, establishing a correspondence between tropicalizations and colored fans, supported by an equivariant blow-up theorem and numerous examples.

Contribution

It introduces a new framework for tropicalization in spherical varieties, generalizing existing methods and proving a key compatibility result with colored fans.

Findings

Support of the colored fan matches tropicalization support

Equivariant blow-up theorem underpins the results

Numerous illustrative examples provided

Abstract

We extend tropicalization and tropical compactification of subvarieties of algebraic tori to subvarieties of spherical homogeneous spaces. Given a tropical compactification of a subvariety, we show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the subvariety. The proof is based on our equivariant version of the attening by blow-up theorem. We provide many examples.