ProofAtlas

Singularity theory, zeta functions, and monodromy

Igusa–Denef–Loeser Monodromy Conjecture

Collaboration beta

Must every pole predicted by resolution data appear as monodromy somewhere arbitrarily near the singular point?

s0pole ofZtop,x(f,s)⇒e2πis0is a nearby local monodromy eigenvalue
Known results and sources
A resolution divisor graph feeding a zeta-function pole toward a nearby monodromy loop, with the final support connection left open.
The monodromy conjecture asks whether each topological-zeta pole has a nearby monodromy eigenvalue.

Research problem

Exact mathematical statement

Let f:(X,x)→(C,0)f:(X,x)\to(\mathbf C,0) be a nonconstant holomorphic germ on a smooth complex variety. For an embedded log resolution with total-transform components EiE_i, write

Ni=ordEi(f∘π),νi-1=ordEiJacπ,N_i=\operatorname{ord}_{E_i}(f\circ\pi),\qquad \nu_i-1=\operatorname{ord}_{E_i}\operatorname{Jac}\pi,

and

Ztop,x(f,s)=∑∅≠Iχc(EI∘∩π-1(x))∏i∈I1νi+Nis.Z_{\mathrm{top},x}(f,s)=\sum_{\varnothing\ne I}\chi_c(E_I^\circ\cap\pi^{-1}(x))\prod_{i\in I}\frac1{\nu_i+N_i s}.

If s0s_0 is a pole of Ztop,x(f,s)Z_{\mathrm{top},x}(f,s), then e2πis0e^{2\pi i s_0} is a local monodromy eigenvalue at some point of f-1(0)f^{-1}(0) arbitrarily near xx.

Problem infographic

Problem at a glance

A problem-first explainer linking a holomorphic singularity, its resolution data and topological zeta pole, to the conjectured nearby monodromy eigenvalue.
Resolution data produces candidate zeta poles; the conjecture demands that each actual pole be carried by nearby monodromy.

Current mathematical picture

Where work on Igusa–Denef–Loeser Monodromy Conjecture stands

Partially resolved

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

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: A perturbation lying above only the compact Newton facet is automatically covered by the one-edge theorem. No functorial filtered complex currently turns a nonzero complete Laurent principal part into nonempty completed nearby-cycle support.

Route status · Narrowed route
Main reductionGeneric divisorial reduction

The current work labels a generic divisorial marked-quadratic reduction proved in chat and places new obstructions in codimension at least two.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeConstruct the SNC translated-layer complex and prove that its Euler or Lefschetz characteristic equals the complete Laurent principal part.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

Igusa–Denef–Loeser Monodromy Conjecture in numbers

3.5kretained lines of mathematical investigation3,462 in the current working snapshot
Argument development
2,928 · 85%
Explored or eliminated routes
63 · 2%
Computational analysis
47 · 1%
Open obligations
213 · 6%
Definitions and setup
211 · 6%
7selected 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

12 selected steps

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

12 selected steps

Scroll horizontally to explore the route

Working route overview for Igusa–Denef–Loeser 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.Do zeta-function poles always have nearby monodromy carriers? — Depends on missing premiseDo zeta-function polesalways have nearby monodromycarriers?Current reduction — Depends on missing premiseCurrent reductionGeneric divisorial reduction — Depends on missing premiseGeneric divisorial reductionClosing target — Depends on missing premiseClosing targetComplete principalization chart cover — Depends on missing premiseComplete principalizationchart coverCorrected one-edge family — Depends on missing premiseCorrected one-edge familyNeighborhood pole-to-eigenvalue statement — Depends on missing premiseNeighborhoodpole-to-eigenvalue statementSource-reported limitation — stoppedSource-reported limitationConstruct the SNC translated-layer complex and prove that its Euler or Lefschetz characteristic equals the complete Laurent principal part. — OpenConstruct the SNCtranslated-layer complex andprove…Prove blowup invariance and strict proper pushforward for the filtered object while retaining finite-cover action. — OpenProve blowup invariance andstrict proper pushforwardfor…Close the several-rupture nonsplit tree and the extra-facet three-dimensional marked-quadratic model. — OpenClose the several-rupturenonsplit tree and theextra-facet…Filtered Laurent-to-carrier bridge — OpenFiltered Laurent-to-carrierbridge
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 routeSource-reported limitation

The following claim is rejected or insufficient in the recorded route: A perturbation lying above only the compact Newton facet is automatically covered by the one-edge theorem. No functorial filtered complex currently turns a nonzero complete Laurent principal part into nonempty completed nearby-cycle support.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Construct the SNC translated-layer complex and prove that its Euler or Lefschetz characteristic equals the complete Laurent principal part.Suggested move: Define the complex on an SNC chart and reproduce every mandatory translated-layer, cusp, one-edge, and principalization identity before any global claim.
Ready to work on
02
Prove blowup invariance and strict proper pushforward for the filtered object while retaining finite-cover action.Suggested move: Start with elementary blowup contraction and compare its pushed-forward support to the canonical specialization carrier.
Ready to work on
03
Filtered Laurent-to-carrier bridge

A functorial filtered complex connecting the complete Laurent principal part to nearby-cycle support has not been constructed.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Close the several-rupture nonsplit tree and the extra-facet three-dimensional marked-quadratic model.Suggested move: Use principalized interaction charts and the corrected two-facet geometry, keeping the new t-squared vertex outside the one-edge theorem.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusPartially resolved

The broad monodromy conjecture remains open; the two-variable case and several higher-dimensional classes are proved.

[1]
External progress

What the literature has established

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

  1. Authoritative summaryA current survey states the general conjecture as open, explains a proof in two variables, and reviews partial higher-dimensional results.[1]
  2. Peer reviewedA strong monodromy result is proved for a semi-quasihomogeneous hypersurface class; this is a special class, not the general conjecture.[3]
  3. PreprintThe local motivic monodromy conjecture is proved for nondegenerate hypersurfaces satisfying a simplicial condition.[2]
3 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusIgusa–Denef–Loeser Monodromy Conjecture
Solved special casePlane-curve monodromy conjecture

The two-variable case is proved.

[1]
Solved special caseSimplicial nondegenerate hypersurfaces

A simplicial nondegenerate class satisfies a local motivic form.

[2]
Stronger or generalized formMotivic and strong monodromy conjectures

Motivic and strong monodromy formulations strengthen or refine the topological pole-to-eigenvalue statement.

[1]

Formalization opportunities

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

  • Formalization targetNo current statement-aligned formal proof of the general neighborhood theorem was found.
  • Formalization targetResolution, topological zeta, and nearby-cycle infrastructure would need exact formal scope before any artifact claim.

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 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.

5 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma4 of 74
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementDo zeta-function poles always have nearby monodromy carriers?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementNeighborhood pole-to-eigenvalue statementintermediate
  • retained route statementCorrected one-edge familyintermediate
  • retained route statementGeneric divisorial reductionintermediate
  • retained route statementComplete principalization chart coverintermediate
  • 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 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 failureSource-reported limitationreported failure
  • Research targetConstruct the SNC translated-layer complex and prove that its Euler or Lefschetz characteristic equals the complete Laurent principal part.open
  • Research targetProve blowup invariance and strict proper pushforward for the filtered object while retaining finite-cover action.open
  • Research targetClose the several-rupture nonsplit tree and the extra-facet three-dimensional marked-quadratic model.open
  • Research targetFiltered Laurent-to-carrier bridgeopen
  • Research targetNonsplit and extra-facet frontiersuperseded
  • Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: A perturbation lying above only the compact Newton facet is automatically covered by the one-edge theorem. No functorial filtered complex currently turns a nonzero complete Laurent principal part into nonempty completed nearby-cycle support.
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 bridgeConstruct the SNC translated-layer complex and prove that its Euler or Lefschetz characteristic equals the complete Laurent principal part.

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.

Continue the mathematics

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Prepared starting pointConstruct the SNC translated-layer complex and prove that its Euler or Lefschetz characteristic equals the complete Laurent principal part.

Igusa–Denef–Loeser Monodromy Conjecture · ready to start

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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

Must every pole predicted by resolution data appear as monodromy somewhere arbitrarily near the singular point?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

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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
    Introduction to the monodromy conjecturesurvey or monograph · Willem Veys · arXiv · 2024 · ARXIV 2403.03343 · accessed Aug 14, 2026
  2. 2
    The local motivic monodromy conjecture for simplicial nondegenerate singularitiespreprint · Matt Larson, Sam Payne, Alan Stapledon · arXiv · 2022 · ARXIV 2209.03553 · accessed Aug 14, 2026
  3. 3
    The strong monodromy conjecture for a class of semi-quasihomogeneous hypersurface singularitiespeer reviewed result · Guillem Blanco · Mathematische Nachrichten · 2023 · DOI 10.1002/mana.202100376 · accessed Aug 14, 2026

Important qualifications

  • Scoped to the neighborhood/topological statement and selected statement-aligned special cases.
  • Motivic, Igusa, strong, and local/global variants were not exhaustively reconciled.

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