Specializing varieties and their cohomology from characteristic $0$ to characteristic $p$

TL;DR

This paper establishes a semicontinuity relation between mod p singular cohomology in characteristic zero and de Rham cohomology in characteristic p, using p-adic Hodge theory and perfectoid geometry.

Contribution

It proves a new semicontinuity theorem connecting cohomology in different characteristics, advancing understanding of algebraic varieties via p-adic methods.

Findings

Bound on mod p singular cohomology rank by de Rham cohomology rank

Application of p-adic Hodge theory and perfectoid geometry

Survey of relevant p-adic and perfectoid concepts

Abstract

We present a semicontinuity result, proven in recent joint work with Morrow and Scholze, relating the mod $p$ singular cohomology of a smooth projective complex algebraic variety X to the de Rham cohomology of a smooth characteristic $p$ specialization of X: the rank of the former is bounded above by that of the latter. The path to this result passes through $p$-adic Hodge theory and perfectoid geometry, so we survey the relevant aspects of those subjects as well.