Hodge Theory of $p$-adic analytic varieties: a survey

TL;DR

This survey reviews the development of Hodge Theory for $p$-adic analytic varieties, highlighting key conjectures, proofs, and new phenomena, especially in the nonproper case, with a focus on recent advances since 2011.

Contribution

It summarizes recent progress and conjectures in $p$-adic Hodge Theory, emphasizing nonproper varieties and new mathematical objects introduced.

Findings

Scholze's proof of Tate's conjecture using perfectoid methods

Comparison theorems and dualities for $p$-adic Hodge Theory

Emergence of new objects in the study of nonproper varieties

Abstract

Hodge Theory of $p$-adic analytic varieties was initiated by Tate in his 1967 paper on $p$-divisible groups, where he conjectured the existence of a Hodge-like decomposition for the $p$-adic \'etale cohomology of proper analytic varieties. Tate's conjecture was refined by Fontaine who gave the theory its definite shape. A lot of work has been done for algebraic varieties and a number of proofs of Fontaine's conjectures have been obtained between years 1985 and 2011. But the study of Hodge Theory of $p$-adic analytic varieties started really only in 2011 with Scholze's proof of Tate's conjecture using perfectoid methods. Methods that opened the way to an avalanche of results. In this paper, we survey our results and conjectures (comparison theorems and their geometrization, dualities, etc.), focusing on the case of nonproper analytic varieties, where a number of new phenomena occur. We also describe the new objects that appeared along the way.