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Hyodo-Kato cohomology in rigid geometry: some foundational results

Xinyu Shao

TL;DR

This work develops Hyodo-Kato cohomology in rigid geometry, establishing fundamental geometric properties (Mayer–Vietoris, Poincaré duality, Gysin) for both algebraic and analytic HK, including compact support. Leveraging these properties, the authors prove a GAGA-type comparison between algebraic and analytic HK cohomology and establish a semistable conjecture for étale cohomology of almost proper rigid analytic varieties via a period morphism and descent arguments. They formulate and analyze the compactly supported HK theory, relate it to Tsuji’s log-crystalline cohomology, and show compatibility across daggers, rigid spaces, and their pro-étale and de Rham realizations. The results provide a coherent framework unifying HK theory across algebraic and analytic worlds, with explicit comparisons to log-crystalline cohomology and a robust descent apparatus built from éh/h-topologies and Beilinson bases.

Abstract

By exploring the geometric properties of Hyodo-Kato cohomology in rigid geometry, we establish several foundational results, including the semistable conjecture for étale cohomology of almost proper rigid analytic varieties, and GAGA (comparison between algebraic and analytic) for Hyodo-Kato cohomology. A central component of our approach is the Gysin sequence for Hyodo-Kato cohomology, which we construct using the open-closed exact sequence for compactly supported Hyodo-Kato cohomology and Poincaré duality.

Hyodo-Kato cohomology in rigid geometry: some foundational results

TL;DR

This work develops Hyodo-Kato cohomology in rigid geometry, establishing fundamental geometric properties (Mayer–Vietoris, Poincaré duality, Gysin) for both algebraic and analytic HK, including compact support. Leveraging these properties, the authors prove a GAGA-type comparison between algebraic and analytic HK cohomology and establish a semistable conjecture for étale cohomology of almost proper rigid analytic varieties via a period morphism and descent arguments. They formulate and analyze the compactly supported HK theory, relate it to Tsuji’s log-crystalline cohomology, and show compatibility across daggers, rigid spaces, and their pro-étale and de Rham realizations. The results provide a coherent framework unifying HK theory across algebraic and analytic worlds, with explicit comparisons to log-crystalline cohomology and a robust descent apparatus built from éh/h-topologies and Beilinson bases.

Abstract

By exploring the geometric properties of Hyodo-Kato cohomology in rigid geometry, we establish several foundational results, including the semistable conjecture for étale cohomology of almost proper rigid analytic varieties, and GAGA (comparison between algebraic and analytic) for Hyodo-Kato cohomology. A central component of our approach is the Gysin sequence for Hyodo-Kato cohomology, which we construct using the open-closed exact sequence for compactly supported Hyodo-Kato cohomology and Poincaré duality.

Paper Structure

This paper contains 38 sections, 57 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Semistable conjecture for open varieties
  3. Algebraic and analytic Hyodo-Kato cohomology
  4. Cohomology with compact support
  5. The proof of semistable conjecture
  6. Structure of the article
  7. Preliminaries
  8. The éh-topology and h-topology
  9. Beilinson basis
  10. h-topology for algebraic varieties
  11. éh-topology for rigid analytic varieties
  12. de Rham cohomology and $B^+_{\operatorname{dR}}$-cohomology
  13. Cohomology for dagger varieties
  14. Hyodo-Kato cohomology (with compact support)
  15. Hyodo-Kato cohomology for algebraic varieties
  16. ...and 23 more sections

Key Result

Theorem 1.2

(Theorem ss) Suppose $X$ is a proper smooth rigid analytic variety over $C$, $Z \subset X$ is a strictly normal crossing divisor, and $U=X-Z$. (1) We have a $B_{\mathrm{st}}$-linear functorial isomorphism commuting with $\varphi$ and $N$ that induces a $B_{\mathrm{dR}}$-linear filtered isomorphism Here, $H^i_{B_{\operatorname{dR}}^+}(X)$ is the logarithmic $B_{\operatorname{dR}}^+$-cohomology int

Theorems & Definitions (116)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6: open-closed exact sequence
  • Theorem 1.7
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • ...and 106 more