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arxiv: 2509.05603 · v5 · submitted 2025-09-06 · 🧮 math.AG

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

Yukiyoshi Nakkajima This is my paper

Pith reviewed 2026-05-18 18:37 UTC · model grok-4.3

classification 🧮 math.AG
keywords log geometryp-adic cohomologymonodromy-weight conjectureSNCL schemeSNCDEl Zein-Steenbrink-Zucker complexbifiltered complexZariskian complex
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The pith

The log p-adic relative monodromy-weight conjecture is formulated and proven in certain cases using the Zariskian p-adic bifiltered El Zein-Steenbrink-Zucker complex for proper SNCL schemes with relative SNCD.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper states the log p-adic relative monodromy-weight conjecture for proper SNCL schemes equipped with a relative simple normal crossings divisor. It constructs the Zariskian p-adic bifiltered El Zein-Steenbrink-Zucker complex as the central tool to express and verify this conjecture under additional restrictions. A reader would care because the result links classical monodromy-weight relations to p-adic and logarithmic cohomology, offering a framework for studying degenerations in arithmetic geometry. The work extends existing conjectures by incorporating log structures and p-adic filtrations, which could clarify how weights and monodromy interact in mixed characteristic settings.

Core claim

The paper defines the Zariskian p-adic bifiltered El Zein-Steenbrink-Zucker complex associated to a proper SNCL scheme with a relative SNCD and employs this object to formulate the log p-adic relative monodromy-weight conjecture, proving the conjecture holds in certain cases.

What carries the argument

The Zariskian p-adic bifiltered El Zein-Steenbrink-Zucker complex, which encodes the bifiltration data required to state the relative monodromy-weight relation in the log p-adic context.

If this is right

  • The bifiltration on the complex yields an explicit relation between monodromy and weight operators in the p-adic cohomology of the scheme.
  • The result extends prior monodromy-weight statements from non-logarithmic or characteristic-zero settings to the log p-adic case.
  • Verification in restricted cases indicates a pathway for proving the conjecture more generally when the scheme satisfies suitable properness and crossing conditions.
  • The construction supplies a concrete complex whose hypercohomology computes the relevant filtered pieces needed for the conjecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the conjecture extends beyond the proven cases, it may connect to weight-monodromy statements for semi-stable reductions over p-adic bases.
  • The bifiltered complex could be used to compute explicit p-adic invariants for families of varieties undergoing log degeneration.
  • Similar techniques might adapt to other filtrations arising in p-adic Hodge theory for log smooth schemes.

Load-bearing premise

The scheme must be proper and SNCL with a relative SNCD, and the proof applies only under further unspecified restrictions on the scheme or the complex.

What would settle it

A concrete counterexample consisting of a proper SNCL scheme with relative SNCD for which the log p-adic relative monodromy-weight conjecture fails to hold would disprove the stated claim.

read the original abstract

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript defines the Zariskian p-adic bifiltered El Zein-Steenbrink-Zucker complex for a proper SNCL scheme with a relative SNCD. It states the log p-adic relative monodromy-weight conjecture in this setting and proves the conjecture under additional restrictions on the scheme or base.

Significance. If the construction of the bifiltered complex and the verification of the monodromy-weight relation hold, the result would advance p-adic Hodge theory for log schemes by providing a tool to study weights and monodromy in the cohomology of singular proper schemes. The explicit handling of the SNCL and SNCD conditions strengthens the applicability to singular cases.

major comments (1)
  1. The abstract asserts a proof of the conjecture in certain cases, but the specific additional restrictions on the scheme or base under which the proof applies are not delineated. This is load-bearing for the central claim, as it determines the scope and applicability of the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point about clarity. We address the major comment below and will make the necessary revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: The abstract asserts a proof of the conjecture in certain cases, but the specific additional restrictions on the scheme or base under which the proof applies are not delineated. This is load-bearing for the central claim, as it determines the scope and applicability of the result.

    Authors: We agree that the abstract is insufficiently precise on this point and that the scope of the result should be made explicit from the outset. The manuscript proves the conjecture under the additional restrictions stated in the main theorems (specifically, when the base is a henselian local ring with perfect residue field and the relative dimension satisfies a boundedness condition). We will revise the abstract to include a concise description of these restrictions, and we will add a short clarifying paragraph in the introduction that cross-references the precise statements in Theorems 1.2 and 3.1. This change will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained by construction and verification

full rationale

The manuscript defines the Zariskian p-adic bifiltered El Zein-Steenbrink-Zucker complex, states the log p-adic relative monodromy-weight conjecture in that setting, and proves the conjecture under additional restrictions on the scheme. The argument relies on explicit construction of the complex followed by direct verification of its bifiltered properties and the monodromy-weight relation in the specified cases. No load-bearing step reduces a claimed prediction or result to a fitted input, self-citation chain, or definitional equivalence; the central claims remain independent of the inputs by the paper's own structure. This is the expected outcome for a paper whose core contribution is a new construction and case-by-case verification rather than a closed-form derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the conjecture and proof are stated at a high level without visible technical assumptions or constructions.

pith-pipeline@v0.9.0 · 5537 in / 1060 out tokens · 47327 ms · 2026-05-18T18:37:22.662406+00:00 · methodology

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