Spherical tropicalization and Berkovich analytification

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.