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Arithmetic geometry · ℓ-adic cohomology · monodromy filtrations · Frobenius weights

Weight–Monodromy Conjecture

Collaboration beta

When a variety degenerates over a local field, monodromy organizes its cohomology into layers. The conjecture says each layer has exactly the Frobenius weight predicted by its position. The general mixed-characteristic case remains open.

grrMVmis pure of Frobenius weightm+r
Known results and sources
A dark algebraic-geometry visualization of a degenerating variety, a symmetric monodromy filtration crossed by N, and nested Frobenius-weight circles with an unresolved seam.
The conjecture predicts that monodromy layers align with exact Frobenius weights; the general mixed-characteristic comparison remains open.

Research problem

Exact mathematical statement

Let K be a non-Archimedean local field with residue characteristic p, let ℓ≠p, and let X/K be smooth and proper. After finite extension, let N be the nilpotent logarithm of tame inertia on

Vm=He´tm(XK¯,E),V_m=H^m_{\acute{e}t}(X_{\overline K},E),

and let M be its monodromy filtration centered at zero. Prove for every m and r that

grrMVmis pure of Frobenius weightm+r.\operatorname{gr}^{M}_{r}V_m \text{ is pure of Frobenius weight }m+r.

the source's one-middle-square and canonical-quotient formulations are reductions for a fixed smooth projective X, not a completed universal comparison theorem.

Problem infographic

Problem at a glance

Three-panel Weight–Monodromy explainer defining K, X, cohomology and N; stating the exact graded-piece weight; illustrating a length-two string; and separating known special cases from the open general mixed-characteristic case.
The exact weight prediction is simple to state. The packet reduces a fixed projective variety to purity of a canonical quotient, while the universal geometric comparison remains open.

Current mathematical picture

Where work on Weight–Monodromy Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureUnrestricted finite-level congruence transfer

A source-proved two-dimensional broken pure string has nonzero N_L at every finite level, N_L≡0 mod π^L, and a split limit that violates weight–monodromy. Rank-controlled comparison, bounded Smith exponents, exact invariant-space or reductive-quotient injection, and closed determinantal conditions remain viable interfaces.

Route status · Narrowed route
Main reductionNative Picard–Lefschetz rank gap

Within the reported P¹ setup, all Picard–Lefschetz defects vanish exactly when the two native global block ranks agree.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve a semisimple local-invariant-cycle support or surjection onto the canonical quotient R_X for the relevant middle square.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Weight–Monodromy Conjecture in numbers

1.4kretained lines of mathematical investigation1,379 in the current working snapshot
Argument development
1,192 · 86%
Explored or eliminated routes
39 · 3%
Computational analysis
1 · 0%
Open obligations
40 · 3%
Definitions and setup
107 · 8%
8selected mapped statements1routes investigated4open questions4contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Weight–Monodromy ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Monodromy depth should determine the exact Frobenius weight. — Depends on missing premiseMonodromy depth shoulddetermine the exactFrobenius…Canonical quotient test — Depends on missing premiseCanonical quotient testCurrent reduction — Depends on missing premiseCurrent reductionExact weight prediction — Depends on missing premiseExact weight predictionMiddle-square test — Depends on missing premiseMiddle-square testNative Picard–Lefschetz rank gap — Depends on missing premiseNative Picard–Lefschetz rankgapClosing target — Depends on missing premiseClosing targetFinite-level transfer obstruction — Depends on missing premiseFinite-level transferobstructionUnrestricted finite-level congruence transfer — stoppedUnrestricted finite-levelcongruence transferProve a semisimple local-invariant-cycle support or surjection onto the canonical quotient R_X for the relevant middle square. — OpenProve a semisimplelocal-invariant-cyclesupport…Construct an arithmetic-Frobenius-equivariant tame comparison with one-conductor or two-kernel rank control at arbitrarily deep precision. — OpenConstruct anarithmetic-Frobenius-equiva‑riant…Audit the external P¹ arithmetic-Kashiwara reduction and prove the native Picard–Lefschetz rank equality. — OpenAudit the external P¹arithmetic-Kashiwarareduction…Universal geometric comparison — OpenUniversal geometriccomparison
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeUnrestricted finite-level congruence transfer

A source-proved two-dimensional broken pure string has nonzero N_L at every finite level, N_L≡0 mod π^L, and a split limit that violates weight–monodromy. Rank-controlled comparison, bounded Smith exponents, exact invariant-space or reductive-quotient injection, and closed determinantal conditions remain viable interfaces.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Prove a semisimple local-invariant-cycle support or surjection onto the canonical quotient R_X for the relevant middle square.Suggested move: State the source and target with twists and Frobenius conventions fixed, then verify eigenvalue support on R_X rather than only abutment-level cohomology.
Ready to work on
02
Construct an arithmetic-Frobenius-equivariant tame comparison with one-conductor or two-kernel rank control at arbitrarily deep precision.Suggested move: Track a common tame generator, integral lattices, Smith exponents, invariant dimensions, and a fixed-field or Northcott finiteness package for the local factor.
Ready to work on
03
Universal geometric comparison

Produce a universal comparison retaining arithmetic Frobenius, rank or eigenvalue support, and global arithmetic boundedness.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Audit the external P¹ arithmetic-Kashiwara reduction and prove the native Picard–Lefschetz rank equality.Suggested move: Work directly with the native negative and positive boundary blocks, preserve the closed upper-rank condition, and avoid reintroducing the superseded matching matrix.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The general mixed-characteristic weight-monodromy conjecture remains open in the checked source scope. Equal characteristic and specified complete-intersection settings are solved special cases and do not establish the unrestricted statement.

[1][2][3]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedBinda, Kato, and Vezzani prove the p-adic weight-monodromy conjecture for scheme-theoretic complete intersections in projective smooth toric varieties.[2]
  2. PreprintWear reports a proof for complete intersections in abelian varieties.[3]
  3. Peer reviewedIto proves coincidence of the weight and shifted monodromy filtrations for proper smooth varieties over equal-characteristic local fields.[1]
3 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusWeight-monodromy conjecture
Solved special caseequal-characteristic case

The equal-characteristic local-field case is proved for proper smooth varieties.

[1]
Solved special casetoric complete intersections

A peer-reviewed theorem proves the p-adic conjecture for scheme-theoretic complete intersections in projective smooth toric varieties.

[2]
Solved special caseabelian-variety complete intersections

A preprint reports the conjecture for complete intersections in abelian varieties.

[3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetNo checked formal statement alignment for the general mixed-characteristic filtration identity was located.
  • Formalization targetThe required étale-cohomology, monodromy, and weight-filtration infrastructure was not assessed in a formal library.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction2 of 82
  • lemma1 of 81
  • equivalence3 of 83
  • counterexample1 of 81
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementMonodromy depth should determine the exact Frobenius weight.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact weight predictionintermediate
  • retained route statementMiddle-square testintermediate
  • retained route statementCanonical quotient testintermediate
  • retained route statementFinite-level transfer obstructionintermediate
  • retained route statementNative Picard–Lefschetz rank gapintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureUnrestricted finite-level congruence transferreported failure
  • Research targetProve a semisimple local-invariant-cycle support or surjection onto the canonical quotient R_X for the relevant middle square.open
  • Research targetConstruct an arithmetic-Frobenius-equivariant tame comparison with one-conductor or two-kernel rank control at arbitrarily deep precision.open
  • Research targetAudit the external P¹ arithmetic-Kashiwara reduction and prove the native Picard–Lefschetz rank equality.open
  • Research targetUniversal geometric comparisonopen
  • Narrowed routeUnrestricted finite-level congruence transferA source-proved two-dimensional broken pure string has nonzero N_L at every finite level, N_L≡0 mod π^L, and a split limit that violates weight–monodromy. Rank-controlled comparison, bounded Smith exponents, exact invariant-space or reductive-quotient injection, and closed determinantal conditions remain viable interfaces.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve a semisimple local-invariant-cycle support or surjection onto the canonical quotient R_X for the relevant middle square.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve a semisimple local-invariant-cycle support or surjection onto the canonical quotient R_X for the relevant middle square.

Weight–Monodromy Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

When a variety degenerates over a local field, monodromy organizes its cohomology into layers. The conjecture says each layer has exactly the Frobenius weight predicted by its position. The general mixed-characteristic case remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Weight-monodromy conjecture over equal characteristic local fieldspeer reviewed result · Tetsushi Ito · American Journal of Mathematics · 2005 · ARXIV math/0308141 · accessed Aug 14, 2026
  2. 2
    On the p-adic weight-monodromy conjecture for complete intersections in toric varietiespeer reviewed result · Federico Binda, Hiroki Kato, Alberto Vezzani · Inventiones Mathematicae · 2025-06-17 · DOI 10.1007/s00222-025-01344-x · accessed Aug 14, 2026
  3. 3
    Perfectoid covers of abelian varieties and the weight-monodromy conjecturepreprint · Peter Wear · arXiv · 2023-03-09 · ARXIV 2303.05610 · accessed Aug 14, 2026

Important qualifications

  • This was a bounded source check, not a systematic literature review.
  • The general-open conclusion is an inference from primary sources that prove restricted cases; no single maintained registry was used as a universal status authority.
  • Wear's abelian-variety result is a preprint and is represented with that evidence posture.
  • No formalization, proof artifact, software, or independently reproduced computation was identified or executed.

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