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 X perf which covers X constructed by Scholze, showing that Pic(X perf ) is canonically isomorphic to Pic(X)[p −1 ]. We also compute the cohomology of line bundles on X perf and establish analogs of Demazure and Batyrev-Borisov vanishing. This generalizes the first author's analogous results for projectivoid space.

Line bundles on perfectoid covers: case of good reduction

“…The case of X = P n has been studied by Dorfsman-Hopkins, who showed in his thesis [16] that ( 2) is an isomorphism for X = P n , extending the case of n = 1 previously treated by Das [14]. Dorfsman-Hopkins-Ray-Wear have extended this to smooth projective toric varieties [18], where the map (2) is also an isomorphism.…”