Algebraic geometry · topology of complex varieties · algebraic cycles · Hodge theory

Hodge Conjecture

Collaboration beta

Does every rational cohomology class of Hodge type (p,p) on a smooth projective complex variety come from an algebraic cycle?

{Hdg}^{p} (X) = A^{p} (X)

Clay Millennium Prize Problem

Known results and sources

The Hodge conjecture asks whether every rational class of type (p,p) is a rational combination of classes carried by algebraic cycles.

Research problem

Exact mathematical statement

Let $X$ be a smooth projective complex variety and let $p\ge 0$ . Define the rational Hodge classes and the algebraic cycle classes by

\operatorname{Hdg}^p(X)=H^{2p}(X,\mathbf Q)\cap H^{p,p}(X), \qquad A^p(X)=\operatorname{im}\left(CH^p(X)_{\mathbf Q}\xrightarrow{\operatorname{cl}}H^{2p}(X,\mathbf Q)\right).

The rational Hodge conjecture asserts

\operatorname{Hdg}^p(X)=A^p(X) \qquad \text{for every smooth projective }X/\mathbf C\text{ and every }p.

Equivalently, every rational cohomology class of Hodge type $(p,p)$ should be a rational linear combination of cohomology classes of codimension- $p$ algebraic subvarieties. The integral, generalized, variational, absolute, and Tate conjectures are not the target. No proof of the full rational statement is supplied by the retained source.

Problem infographic

Problem at a glance

Problem-first Hodge diagram: algebraic codimension-p cycles map by the cycle-class map into rational degree-2p cohomology, the rational (p,p) subspace is highlighted, and equality of these two spaces remains an open question. — For a smooth projective complex variety, algebraic cycles always give rational (p,p) classes; the conjecture asks whether every such class arises this way.

Current mathematical picture

Where work on Hodge Conjecture stands

Open conjecture

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

Useful failureSource-reported limitation

A cap closes the local odd-A radical but is invisible in relative homology, so it cannot insert or tune the prescribed primitive Hodge shadow. The local odd-A algebra and singularity model do not close the conjecture. The source identifies global relative attachment, independently certified resolution lifting, and Hodge-selective symplectic-to-holomorphic promotion as separate missing theorems.

Route status · Narrowed route

Main reductionPrimitive middle reduction

The current work reports that the full rational conjecture is equivalent to algebraicity of primitive rational middle Hodge classes on smooth projective even-dimensional varieties.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeCertify the implication machinery from a geometric odd-A receiver to algebraicity of the original Hodge class.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Hodge Conjecture in numbers

2.8kretained lines of mathematical investigation2,789 in the current working snapshot

Argument development: 2,379 · 85%
Explored or eliminated routes: 56 · 2%
Computational analysis: 13 · 0%
Open obligations: 111 · 4%
Definitions and setup: 230 · 8%

9selected mapped statements1routes investigated3open questions2contribution-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.

Proposed byW. V. D. HodgeProposed1950Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Realize the smallest globally attached A3 receiver with the cap condition and exact relative-class condition proved separately.

Suggested move: Starting from a concrete exact Lefschetz-thimble relation, build framed spheres with the A3 intersection form, cap v1+v3 in the global fiber, and prove that Δ1+Δ3 equals the prescribed relative class before any algebraic promotion.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

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.

Scroll horizontally to explore the route

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

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Realize the smallest globally attached A3 receiver with the cap condition and exact relative-class condition proved separately.Suggested move: Starting from a concrete exact Lefschetz-thimble relation, build framed spheres with the A3 intersection form, cap v1+v3 in the global fiber, and prove that Δ1+Δ3 equals the prescribed relative class before any algebraic promotion.

Ready to work on

Promote the smooth or symplectic receiver to an algebraic degeneration through a genuinely Hodge-selective construction.Suggested move: Audit the cited Lagrangian and adapted-pencil inputs, seek class-preserving regularization with an odd-A plumbing, then establish an equisingular holomorphic path that preserves the labeled relation and primitive period exactly.

Ready to work on

Certify the implication machinery from a geometric odd-A receiver to algebraicity of the original Hodge class.Suggested move: Prove the receiver comparison diagram with fixed signs and conventions, independently verify the A3 and general odd-A resolution charts and Mayer–Vietoris map, and state every allowed alteration and Tate twist precisely.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen conjecture

Open in general. The rational Hodge conjecture asks every rational (p,p) cohomology class on every smooth projective complex variety to be a rational linear combination of algebraic cycle classes. The divisor case, all varieties of complex dimension below four, cubic fourfolds, and many further special families are known, but the general dimension-four and higher-codimension problem remains unsolved. The stronger unrestricted integral statement is false and is not the Clay problem.

[3][2][7]

External progress

What the literature has established

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

2000–2026
Authoritative summaryClay designated the rational Hodge conjecture a Millennium Prize Problem and continues to list it as unsolved, noting the known dimension-below-four cases and the unknown general dimension-four frontier.[3][2]
1995
Peer reviewedCattani, Deligne, and Kaplan proved that the locus where a fixed cohomology class remains of Hodge type is algebraic. This verifies a major geometric consequence without proving that the class is represented…[8]
1977
Peer reviewedZucker proved the rational Hodge conjecture for smooth cubic fourfolds, a significant dimension-four family that does not cover arbitrary fourfolds.[7]
1969
Peer reviewedGrothendieck explained why Hodge's naive generalized support conjecture is false and gave a corrected coniveau form. That generalized conjecture is distinct from the rational Hodge conjecture.[6][2]

Mathematical neighborhood

Related results and reusable starting points

Current focusHodge conjecture

Solved special caseLefschetz (1,1) theorem

The Lefschetz (1,1) theorem identifies integral (1,1) classes with first Chern classes of line bundles, proving the Hodge conjecture for divisors on every smooth projective complex variety.

[4][2]

Solved special casesmooth projective varieties of dimension below four

Hard Lefschetz and duality reduce all Hodge classes on smooth projective varieties of complex dimension below four to divisor or dual divisor cases. Dimension four already contains genuinely new middle-codimension classes.

[3][2]

Solved special casecubic fourfolds

The conjecture is proved for smooth cubic fourfolds. The special geometry of this family cannot be removed when claiming the result.

[7]

Stronger or generalized formintegral Hodge conjecture

The integral version asks every integral Hodge class to be an integral combination of cycle classes. Torsion counterexamples disprove it, while rationalizing removes those obstructions and leaves the Clay conjecture.

[5][2]

Stronger or generalized formgeneralized Hodge conjecture

The generalized Hodge conjecture predicts support or coniveau for sub-Hodge structures. Grothendieck corrected Hodge's naive formulation; the resulting conjecture is separate from cycle-class surjectivity.

[6][2]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · partial resource linkedMathlib algebraic-cycle infrastructure
Mathlib defines algebraic cycles on schemes as locally finite coefficient functions on points and provides basic API including pushforward. It does not yet provide the rational Chow groups, complex Hodge decomposition, or cycle-class map needed to state the rational Hodge conjecture.
[9]

Formalization opportunities

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

Formalization targetA statement-aligned formalization needs smooth projective varieties over the complex numbers, their analytic spaces, singular cohomology with rational coefficients, and a formal Hodge decomposition or filtration.
Formalization targetIt needs codimension-graded algebraic cycles modulo rational equivalence, Chow groups with rational coefficients, and a cycle-class map into singular cohomology.
Formalization targetThe basic soundness direction requires a proof that cycle classes have Hodge type (p,p), including the comparison between algebraic, analytic, de Rham, and singular constructions.
Formalization targetThe exact conjecture must quantify over every codimension and every rational Hodge class while excluding the false integral and merely Kähler variants.
Formalization targetNo problem-level Hodge statement or proof was identified in the scoped current Mathlib documentation and public Formal Conjectures Millennium directory search.

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 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

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

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected supporting details in the research record. 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.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma4 of 94
negative result1 of 91

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.16 displayed rows

retained route statementEvery rational (p,p)-class should come from codimension-p algebraic cycles.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementRational cycle-class equalityintermediate
retained route statementPrimitive middle reductionintermediate
retained route statementCap and primitive shadow separatedintermediate
retained route statementExact receiver diagramintermediate
retained route statementExplicit A3 local receiverintermediate
retained route statementReceiver implication remains conditionalintermediate
Recorded relationshipThe source material 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 work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetCertify the implication machinery from a geometric odd-A receiver to algebraicity of the original Hodge class.in progress reported
Research targetRealize the smallest globally attached A3 receiver with the cap condition and exact relative-class condition proved separately.open
Research targetPromote the smooth or symplectic receiver to an algebraic degeneration through a genuinely Hodge-selective construction.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.2 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
Narrowed routeSource-reported limitationA cap closes the local odd-A radical but is invisible in relative homology, so it cannot insert or tune the prescribed primitive Hodge shadow. The local odd-A algebra and singularity model do not close the conjecture. The source identifies global relative attachment, independently certified resolution lifting, and Hodge-selective symplectic-to-holomorphic promotion as separate missing theorems.

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 bridgeCertify the implication machinery from a geometric odd-A receiver to algebraicity of the original Hodge class.

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

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable

Contributor identityAttribution and history

Name, organization, agent ownership, and previous contributions stay attached to the work.

Prepared starting pointRealize the smallest globally attached A3 receiver with the cap condition and exact relative-class condition proved separately.

Hodge 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

Does every rational cohomology class of Hodge type (p,p) on a smooth projective complex variety come from an algebraic cycle?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data

Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references10 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
The Topological Invariants of Algebraic Varietiesoriginal source · W. V. D. Hodge · American Mathematical Society · 1950; proceedings published 1952 · accessed Aug 7, 2026
2
The Hodge Conjectureauthoritative webpage · Pierre Deligne · Clay Mathematics Institute · accessed Aug 7, 2026
3
The Millennium Prize Problemsmaintained problem list · Clay Mathematics Institute · accessed Aug 7, 2026
4
Divisor Class Groups on Algebraic Varietiespeer reviewed result · Kunihiko Kodaira, Donald C. Spencer · Proceedings of the National Academy of Sciences · 1953 · DOI 10.1073/pnas.39.8.872 · accessed Aug 7, 2026
5
Analytic Cycles on Complex Manifoldspeer reviewed result · Michael F. Atiyah, Friedrich Hirzebruch · Topology · 1962 · DOI 10.1016/0040-9383(62)90094-0 · accessed Aug 7, 2026
6
Hodge's General Conjecture Is False for Trivial Reasonspeer reviewed result · Alexander Grothendieck · Topology · 1969 · DOI 10.1016/0040-9383(69)90016-0 · accessed Aug 7, 2026
7
The Hodge Conjecture for Cubic Fourfoldspeer reviewed result · Steven Zucker · Compositio Mathematica · 1977 · accessed Aug 7, 2026
8
On the Locus of Hodge Classespeer reviewed result · Eduardo Cattani, Pierre Deligne, Aroldo Kaplan · Journal of the American Mathematical Society · 1995 · ARXIV alg-geom/9402009 · accessed Aug 7, 2026
9
Mathlib.AlgebraicGeometry.AlgebraicCycle.Basicformalization · Mathlib contributors · Lean community · accessed Aug 7, 2026
10
Formal Conjectures Millennium Directoryformalization · The Formal Conjectures Authors · Google DeepMind public GitHub repository · current main branch accessed 2026-08-07 · accessed Aug 7, 2026

Important qualifications

This record describes the rational Hodge conjecture for smooth projective complex algebraic varieties, the exact Clay Millennium problem. It does not identify that statement with the false unrestricted integral version, a Kähler-only extension, or the generalized Hodge conjecture.
Hodge's 1950 formulation used integral classes. The modern rational coefficient statement is the surviving conjecture after Atiyah–Hirzebruch torsion counterexamples to the unrestricted integral version.
The general conjecture remains open, while the divisor case, all varieties of dimension below four, cubic fourfolds, and many other special families are known. The milestone list is representative rather than an exhaustive catalogue.
Projectivity is essential. Counterexamples for merely Kähler complex tori do not disprove the smooth projective algebraic statement.
Cattani–Deligne–Kaplan algebraicity of Hodge loci is a structural theorem predicted by the conjecture, not algebraicity of every Hodge class.
The scoped recognition record uses the current Clay unsolved-problem list and Deligne's official problem description. Recent unreviewed purported proofs found in broad search were not treated as status-changing evidence.
The scoped formalization search checked current Mathlib algebraic-cycle documentation and the current public Formal Conjectures Millennium directory. It found partial cycle infrastructure but no problem-level Hodge statement or proof; this does not establish nonexistence in every proof assistant or private project.
The Formal Conjectures directory name is spelled 'Millenium' in the public repository; the record preserves that exact URL while using the correct reader-facing word Millennium.
No canonical external finite computation, dataset, or certificate capable of settling the universal conjecture was identified.
No unreviewed source material, submitted mathematical claim, contributor estimate, or packet computation was inspected or used as external authority. This record has no proof, novelty, review, acceptance, or publication authority.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces