On Picard Groups of Perfectoid Covers of Toric Varieties

Abstract

Let $X$ be a proper smooth toric variety over a perfectoid field of prime residue characteristic $p$. We study the perfectoid space ${\mathcal{X}}^{perf}$ which covers $X$ constructed by Scholze, showing that $\text{Pic}({\mathcal{X}}^{perf})$ is canonically isomorphic to $\text{Pic}(X)[p^{-1}]$. We also compute the cohomology of line bundles on ${\mathcal{X}}^{perf}$ and establish analogs of Demazure and Batyrev-Borisov vanishing. This generalizes the first author's analogous results for "projectivoid space".

29 pages, accepted for publication in the European J. of Mathematics