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The zariskian p-adic bifiltered El Zein-Steenbrink-Zucker complex of a proper SNCL scheme with a relative SNCD

Yukiyoshi Nakkajima

TL;DR

This work advances log $p$-adic Hodge theory for SNCL schemes with relative SNCDs by constructing a zariskian bifiltered $p$-adic complex $(A_{ m zar}((X,D)/S),P^D,P)$ that computes log crystalline cohomology and carries natural monodromy and weight filtrations. It formulates the log $p$-adic relative monodromy-weight conjecture and proves it in several nontrivial cases, notably when the relative dimension is small or when fibers arise from semistable degenerations, while establishing key structural results: $E_2$-degeneration of the weight spectral sequence, base change, infinitesimal deformation invariance, and strict compatibility. The paper also develops contravariant functoriality and monodromy theory at the level of zariskian complexes, linking the relative theory to the existing $p$-adic monodromy framework and providing a robust toolkit for studying weight filtrations in the log-crystalline setting. The results have significant implications for understanding weight filtrations, monodromy, and their interactions in relative $p$-adic geometry, with potential applications to arithmetic and motivic aspects of SNCL degenerations.

Abstract

We give the log $p$-adic relative monodromy-weight conjecture and prove it in certain cases.

The zariskian p-adic bifiltered El Zein-Steenbrink-Zucker complex of a proper SNCL scheme with a relative SNCD

TL;DR

This work advances log -adic Hodge theory for SNCL schemes with relative SNCDs by constructing a zariskian bifiltered -adic complex that computes log crystalline cohomology and carries natural monodromy and weight filtrations. It formulates the log -adic relative monodromy-weight conjecture and proves it in several nontrivial cases, notably when the relative dimension is small or when fibers arise from semistable degenerations, while establishing key structural results: -degeneration of the weight spectral sequence, base change, infinitesimal deformation invariance, and strict compatibility. The paper also develops contravariant functoriality and monodromy theory at the level of zariskian complexes, linking the relative theory to the existing -adic monodromy framework and providing a robust toolkit for studying weight filtrations in the log-crystalline setting. The results have significant implications for understanding weight filtrations, monodromy, and their interactions in relative -adic geometry, with potential applications to arithmetic and motivic aspects of SNCL degenerations.

Abstract

We give the log -adic relative monodromy-weight conjecture and prove it in certain cases.

Paper Structure

This paper contains 13 sections, 63 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. SNCL schemes
  3. SNCL schemes with relative SNCD's
  4. Preweight filtrations
  5. Zariskian $p$-adic bifiltered El Zein-Steenbrink-Zucker complexes
  6. Contravariant functoriality of zariskian $p$-adic bifiltered El Zein-Steenbrink-Zucker complexes
  7. Monodromy operators
  8. Bifiltered base change theorem
  9. Infinitesimal deformation invariance
  10. The $E_2$-degeneration of the $p$-adic weight spectral sequence
  11. Log convergences of the weight filtrations
  12. Strict compatibility
  13. Log $p$-adic relative monodromy-weight conjecture

Key Result

Theorem 1.1

Let $\varpi_{\rm crys}^{(m)}(\overset{\circ}{X}/\overset{\circ}{S})$$(m\in {\mathbb N})$ be the crystalline orientation sheaf associated to the set $\{\overset{\circ}{X}_{\lambda}\}_{\lambda \in \Lambda}$ for $m$. That is, $\varpi_{\rm crys}^{(m)}(\overset{\circ}{X}/\overset{\circ}{S})$ is the exten with a canonical isomorphism in $D^+(f^{-1}({\cal O}_S))$ such that in $D^+(f^{-1}({\cal O}_S))$.

Theorems & Definitions (124)

  • Theorem 1.1: nb
  • Conjecture 1.2: $p$--adic monodromy-weight conjecture
  • Theorem 1.3
  • Conjecture 1.4: Relative $p$--adic monodromy conjecture
  • Theorem 1.5
  • Definition 2.1: nb
  • Definition 2.2
  • Definition 2.3: A special case of nb
  • Definition 3.1: ny
  • Proposition 3.2
  • ...and 114 more