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

Pierre Colmez, Wiesława Nizioł

TL;DR

The paper surveys the development of Hodge theory for $p$-adic analytic varieties, tracing from Tate and Fontaine to Scholze’s perfectoid methods and focusing on nonproper spaces. It presents a unified framework in which pro-étale cohomology, Hyodo–Kato cohomology, and de Rham data are linked by refined comparison theorems, syntomic regulators, and period sheaves, culminating in a geometrization via Banach–Colmez spaces and TVS-like categories. Key contributions include the basic comparison theorem (and its arithmetic analog), the ${ m C}_{ m st}$ conjecture and its Stein/compact-support variants, and the demonstration of Poincaré duality in the geometric and arithmetic $p$-adic pro-étale setting. These results provide a robust structural picture for $p$-adic Hodge theory in analytic geometry and lay groundwork for applications to $p$-adic Langlands and related geometric programs. Overall, the work advances a cohesive, duality-rich, and geometrized perspective on $p$-adic Hodge theory for analytic varieties beyond the proper case, highlighting new objects such as Banach–Colmez spaces and their role in organizing $p$-adic cohomological information.

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 étale 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.

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

TL;DR

The paper surveys the development of Hodge theory for -adic analytic varieties, tracing from Tate and Fontaine to Scholze’s perfectoid methods and focusing on nonproper spaces. It presents a unified framework in which pro-étale cohomology, Hyodo–Kato cohomology, and de Rham data are linked by refined comparison theorems, syntomic regulators, and period sheaves, culminating in a geometrization via Banach–Colmez spaces and TVS-like categories. Key contributions include the basic comparison theorem (and its arithmetic analog), the conjecture and its Stein/compact-support variants, and the demonstration of Poincaré duality in the geometric and arithmetic -adic pro-étale setting. These results provide a robust structural picture for -adic Hodge theory in analytic geometry and lay groundwork for applications to -adic Langlands and related geometric programs. Overall, the work advances a cohesive, duality-rich, and geometrized perspective on -adic Hodge theory for analytic varieties beyond the proper case, highlighting new objects such as Banach–Colmez spaces and their role in organizing -adic cohomological information.

Abstract

Hodge Theory of -adic analytic varieties was initiated by Tate in his 1967 paper on -divisible groups, where he conjectured the existence of a Hodge-like decomposition for the -adic étale 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 -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.
Paper Structure (49 sections, 20 theorems, 79 equations)

This paper contains 49 sections, 20 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Complex periods
  3. ${\mathcal{C}}^\infty$-manifolds
  4. Complex algebraic and analytic varieties
  5. $p$-adic periods
  6. $p$-adic analytic varieties
  7. The basic comparison theorem
  8. An example
  9. The complex unit disk
  10. The $p$-adic unit disk
  11. Slogans
  12. Comparison theorems as refined Lazard isomorphisms
  13. Hyodo-Kato cohomology as an avatar of $\ell$-adic cohomology
  14. Étale versus pro-étale cohomology
  15. The basic comparison theorem
  16. ...and 34 more sections

Key Result

Theorem 2.4

(Basic comparison theorem CN4) Let $n\geq r$ be positive integers. We have a long exact sequence in ${\mathcal{D}}({\bf Q}_{p,\Box})$ where

Theorems & Definitions (43)

  • Conjecture 1.4
  • Remark 1.5
  • Remark 2.1
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.6
  • Remark 2.7
  • Remark 2.9
  • Theorem 2.11
  • Remark 2.13
  • ...and 33 more