Analytification and Tropicalization

Let $K$ be an algebraically closed field, complete with respect to a nonarchimedean valuation. Analytification and Tropicalization are procedures that let us pass from certain algebraic $K$-varieties to associated Hausdorff topological spaces. In 2009, S. Payne showed that if $X$ is a quasiprojective $K$-variety, then one can construct a homeomorphism between the Berkovich analytification of $X$ and the inverse limit of all quasiprojective tropicalizations of $X$. In this talk, I will discuss results obtained with S. Payne and P. Gross, in which we extend Payne’s 2009 result to any $K$-scheme $X$ that can be realized as a closed subscheme of at least one toric variety.