Specializing varieties and their cohomology from characteristic 0 to characteristic p

2016/06/05 by Bhargav Bhatt, Bhatt, Bhargav · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1606.01463

openalex publication_date 2016/06/05 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

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.

Cited by

• Bounding crystalline torsion from étale torsion

Related

• Specialization of Néron-Severi groups in positive characteristic
• p-adic cohomology
• Arbitrarily large jumps in the de Rham and Hodge cohomology of families in characteristic p
• Dwork cohomology, de Rham cohomology, and hypergeometric functions
• Degeneration of families of projective hypersurfaces and Hodge conjecture