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

Yukiyoshi Nakkajima

TL;DR

The paper develops a comprehensive $l$-adic bifiltered framework for SNCL schemes with relative SNCDs, introducing the $l$-adic El Zein–Steenbrink–Zucker complex $(A_{l^{ exists}}((X,D)/S),P^D,P)$ and proving its cohomological realization as $R^qf_{(X_{l^{ exists}},D)/S*}({f Z}_l)$. It establishes two $l$-adic weight spectral sequences degenerating modulo torsion in key cases and formulates the log $l$-adic relative monodromy-weight conjecture, linking the relative monodromy filtration to the weight filtration. The core methodological advance is a robust derived-category theory of bifiltered complexes, including strictly injective/flat resolutions, ${ m RHom}$ and $igotimes^L$ functors, enabling precise manipulation of filtrations $P$ and $P^D$, edge morphisms, and functoriality. The results recover known equal-characteristic and low-dimensional cases, extend Nakayama–Rapoport–Zink–Steenbrink–Zucker-type weight spectral sequences to the log $l$-adic context, and provide a pathway to $p$-adic and $ ty$-adic analogues via compatible bifiltered constructions and semistable reduction arguments.

Abstract

For a family of log points with constant log structure and for a proper SNCL scheme with an SNCD over the family, we construct a fundamental l-adic bifiltered complex as a geometric application of the theory of the derived category of (bi)filtered complexes in our papers. By using this bifiltered complex, we give the formulation of the log l-adic relative monodromy-weight conjecture with respect to the filtration arising from the SNCD. That is, we state that the relative l-adic monodromy filtration should exist for the Kummer log etale cohomological sheaf of the proper SNCL scheme with an SNCD and it should be equal to the l-adic weight filtration.

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

TL;DR

The paper develops a comprehensive -adic bifiltered framework for SNCL schemes with relative SNCDs, introducing the -adic El Zein–Steenbrink–Zucker complex and proving its cohomological realization as . It establishes two -adic weight spectral sequences degenerating modulo torsion in key cases and formulates the log -adic relative monodromy-weight conjecture, linking the relative monodromy filtration to the weight filtration. The core methodological advance is a robust derived-category theory of bifiltered complexes, including strictly injective/flat resolutions, and functors, enabling precise manipulation of filtrations and , edge morphisms, and functoriality. The results recover known equal-characteristic and low-dimensional cases, extend Nakayama–Rapoport–Zink–Steenbrink–Zucker-type weight spectral sequences to the log -adic context, and provide a pathway to -adic and -adic analogues via compatible bifiltered constructions and semistable reduction arguments.

Abstract

For a family of log points with constant log structure and for a proper SNCL scheme with an SNCD over the family, we construct a fundamental l-adic bifiltered complex as a geometric application of the theory of the derived category of (bi)filtered complexes in our papers. By using this bifiltered complex, we give the formulation of the log l-adic relative monodromy-weight conjecture with respect to the filtration arising from the SNCD. That is, we state that the relative l-adic monodromy filtration should exist for the Kummer log etale cohomological sheaf of the proper SNCL scheme with an SNCD and it should be equal to the l-adic weight filtration.

Paper Structure

This paper contains 13 sections, 80 theorems, 356 equations.

Table of Contents

  1. Introduction
  2. The definition of the derived category of bifiltered complexes
  3. Strictly injective resolutions
  4. Strictly flat resolutions
  5. ${\rm RHom}$
  6. $\otimes^L_{\cal A}$
  7. Complements
  8. A remark on bifiltered derived categories
  9. $l$-adic bifiltered El Zein-Steenbrink-Zucker complex
  10. $l$-adic weight spectral sequences
  11. Fundamental properties of the $l$-adic bifiltered complex and the $l$-adic weight spectral sequences
  12. Strict semistable family
  13. Edge morphisms between the $E_1$-terms of $l$-adic weight spectral sequences

Key Result

Proposition 2.2

Let $f \colon (E,\{E^{(i)}\}_{i=1}^n) \longrightarrow (F,\{F^{(i)}\}_{i=1}^n)$ be a morphism in ${\rm MF}^n({\cal A})$. For $1\leq i \leq n$, let $f^{(i)}\colon (E,E^{(i)}) \longrightarrow (F,F^{(i)})$ be the filtered morphism. If $f$ is injective, then $f$ is strict if and only if $f^{(i)}\colon (E

Theorems & Definitions (184)

  • Conjecture 1.1: Log $l$-adic monodromy-weight conjecture
  • Conjecture 1.2: Log $l$-adic relative monodromy-weight conjecture
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • proof
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 174 more