Algebraic geometry · algebraic cycles · Weil cohomology · motives

Grothendieck’s Standard Conjectures on Algebraic Cycles

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Do algebraic cycles obey the same Lefschetz decomposition, projector, equivalence, and positivity structures that cohomology predicts?

A (X), B (X), C (X), D (X), I (X)

Known results and sources

The standard conjectures ask whether the Lefschetz structures visible in cohomology are realized by algebraic cycles and correspondences.

Research problem

Exact mathematical statement

Let $X$ be a smooth projective variety of dimension $d$ over a field $k$ , let $H^*$ be a characteristic-zero Weil cohomology theory, let $L$ denote cup product with an ample divisor class, and write

A_H^r(X)=\operatorname{im}\left(\operatorname{CH}^r(X)_{\mathbf Q}\to H^{2r}(X)(r)\right).

The principal standard conjectures assert:

Algebraic hard Lefschetz $A(X)$ : for $0\le r\le d/2$ ,

L^{d-2r}:A_H^r(X)\xrightarrow{\sim}A_H^{d-r}(X).

Lefschetz type $B(X)$ : the inverse Lefschetz maps, equivalently the lowering operator $\Lambda$ , are induced by algebraic correspondences.
Künneth type $C(X)$ : the cohomological Künneth projectors $\pi_i$ in the decomposition of the diagonal are algebraic.
Numerical equals homological $D(X)$ : homological and numerical equivalence agree for algebraic cycles with rational coefficients.
Hodge type $I(X)$ : after the standard sign correction, the Lefschetz intersection form is positive definite on primitive algebraic classes.

These are related conjectures rather than a theorem established by the source packet. Over $\mathbf C$ , analytic Hodge–Riemann supplies cohomological positivity, but the required algebraicity remains the issue; in arbitrary characteristic, $I$ remains an independent central input.

Problem infographic

Problem at a glance

Problem-first explainer for Grothendieck’s standard conjectures: a divisor, a codimension-two cycle, and points on a smooth projective variety of dimension d map into Weil cohomology; an ample class raises degrees by the Lefschetz operator; conjectures A, B, C, D, and I ask for algebraic Lefschetz structure, algebraic projectors, numerical-homological equality, and primitive positivity; the package remains open. — Cohomology has a rigid Lefschetz architecture. Grothendieck’s standard conjectures ask whether algebraic cycles and correspondences realize that architecture, and whether numerical and homological equivalence coincide.

Current mathematical picture

Where work on Grothendieck’s Standard Conjectures on Algebraic Cycles stands

Open problem

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

Main reductionA failure becomes a primitive ghost

The critical obstruction is a nonzero primitive rational Hodge class $p$ that is not algebraic even though $z=Lp$ is algebraic, numerically trivial, and annihilated by every algebraic divisor.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeTurn smoothability into support mobility.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

Grothendieck’s Standard Conjectures on Algebraic Cycles in numbers

1.1kretained lines of mathematical investigation1,104 in the current working snapshot

Argument development: 862 · 78%
Explored or eliminated routes: 45 · 4%
Computational analysis: 48 · 4%
Open obligations: 54 · 5%
Definitions and setup: 95 · 9%

9selected mapped statements3open questions3contribution-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 byAlexander Grothendieck, Enrico BombieriProposed1968Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Turn smoothability into support mobility.

Suggested move: Use the Kollár–Voisin flat-pushforward normal form for the critical cycle, vary its divisor factors, and test whether the resulting rationally equivalent smooth representatives yield a dominant family of algebraic carrier preimages; record an exact incidence obstruction if not.

What would count as progress

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

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.

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.

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Turn smoothability into support mobility.Suggested move: Use the Kollár–Voisin flat-pushforward normal form for the critical cycle, vary its divisor factors, and test whether the resulting rationally equivalent smooth representatives yield a dominant family of algebraic carrier preimages; record an exact incidence obstruction if not.

Ready to work on

Resolve the stage-2 nodal carrier defect globally.Suggested move: For a fivefold ghost carried by a nodal threefold, compute the local class groups, ruling-difference intersection table, local-to-global class-group sequence, global location of

\kappa

, and the change induced by a genuinely support-changing residual link.

Ready to work on

Extract the rank-one external tensor.Suggested move: Spread

\kappa_t\boxtimes\kappa_t

over the uniform resolved-carrier family, decompose it with blowup and projective-bundle formulas, and construct an algebraic relative projector isolating the ambient coefficient

p\boxtimes p

while eliminating mixed and vanishing terms.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen problem

Open in general. The full package predicts algebraic correspondences for Lefschetz inverses and Künneth projectors, positivity on primitive algebraic cycles, and compatibility of homological and numerical equivalence. Classical and modern theorems cover important varieties and individual components, including curves, surfaces, abelian varieties in specified settings, K3-Hilbert deformation types, and certain Lagrangian fibrations, but they do not establish every conjecture for all smooth projective varieties and Weil cohomology theories.

[1][3][7]

External progress

What the literature has established

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

2025
Peer reviewedAncona established relative Lefschetz standard results and deduced absolute cases for specified Lagrangian fibrations satisfying support-theorem hypotheses, enlarging the hyperkähler frontier without proving the universal conjecture.[8]
2021
Peer reviewedAncona proved the standard conjecture of Hodge type for abelian fourfolds in positive characteristic. Combined with Clozel's theorem, this gives numerical equals l-adic homological equivalence for infinitely many l, while l-independence remains missing.[7]
2013
Peer reviewedCharles and Markman proved the standard conjectures for complex projective varieties deformation equivalent to Hilbert schemes of points on K3 surfaces.[6]
2002
Peer reviewedMilne proved the Hodge standard conjecture for abelian varieties in arbitrary characteristic conditional on the Hodge conjecture for complex CM-type abelian varieties, and unconditionally for abelian varieties having no exotic algebraic classes.[5]

Mathematical neighborhood

Related results and reusable starting points

Current focusGrothendieck standard conjectures

Related problemLefschetz standard conjecture B

The Lefschetz standard conjecture asks that the inverse to hard Lefschetz be induced by an algebraic correspondence on the self-product. Hard Lefschetz alone gives a cohomological inverse, not its algebraicity.

[1][2]

Related problemKünneth standard conjecture C

The Künneth form asks that the projectors onto cohomological degrees be algebraic cycles on the self-product. It is part of the standard-conjecture package and supports weight decompositions of pure motives.

[2][3]

Related problemHodge standard conjecture

The Hodge-type standard conjecture is a positivity statement for primitive algebraic cycles. It follows from classical Hodge theory in characteristic zero, while positive-characteristic cases such as Ancona's abelian fourfold theorem require different arguments; none should be read as algebraicity of all Hodge or Tate classes.

[1][5]

Logical consequencestandard conjecture D on homological and numerical equivalence

Under the precise standard-conjecture hypotheses, homological and numerical equivalence coincide. Ancona's fourfold result obtains this for infinitely many l, while the remaining l-independence qualification shows why it is not the general statement.

[3][7]

Logical consequenceWeil conjectures

Grothendieck designed the package to yield the Riemann-hypothesis part of the Weil conjectures by algebraic-cycle methods. Deligne's later independent proof establishes the Weil theorem but does not reverse the implication to prove the standard conjectures.

[1][2]

Logical consequencesemisimple category of pure motives

Algebraic Künneth projectors, equality of equivalence relations, and positivity provide the semisimple pure-motive framework envisioned by Grothendieck. Jannsen's semisimplicity for numerical motives does not itself prove that numerical and homological equivalence coincide.

[3][9]

Weaker or relaxed formAndré motivated cycles

Motivated cycles enlarge algebraic cycles so that the desired characteristic-zero motivic formalism becomes unconditional. This gives a powerful substitute but weakens the algebraicity requirement of the original conjectures.

[4]

Solved special casemodern hyperkähler and abelian special cases

K3-Hilbert deformation types, the Hodge-type statement for abelian fourfolds, and specified Lagrangian-fibration cases are rigorous modern footholds. Each retains its stated variety, characteristic, component, or support hypotheses.

[6][7]

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. This is preliminary infrastructure only; it does not formalize Chow groups, Weil cohomology, the Lefschetz inverse, Künneth projectors, primitive positivity, or the standard conjectures.
[10]

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 rational equivalence and Chow groups, intersection products, proper pushforward and pullback, exterior products, and algebraic correspondences with composition.
Formalization targetIt needs one or more precise Weil cohomology theories with cycle-class maps, Künneth isomorphisms, Poincaré duality, hard Lefschetz, primitive decompositions, and comparison of induced correspondence actions.
Formalization targetThe package requires separate formal statements of the Lefschetz inverse algebraicity, Künneth-projector algebraicity, Hodge-type positivity, and numerical-versus-homological equivalence, together with only the valid conditional implications between them.
Formalization targetThe motivic consequences require pseudo-abelian tensor categories of correspondences, adequate equivalence relations, pure motives, polarizations, and semisimplicity arguments.
Formalization targetNo problem-level formal statement or proof was identified in the scoped current Mathlib documentation and public Formal Conjectures repository 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 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

Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected

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 questions

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction4 of 94
lemma3 of 93
equivalence1 of 91

Selected mathematical clusters2 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementMake the cohomological Lefschetz package algebraic and identify numerical with homological equivalence.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementGlobal D yields Lefschetz correspondencesintermediate
retained route statementLower stages algebraize primitivesintermediate
retained route statementThe new stage is radical vanishingintermediate
retained route statementFirst failure occurs in dimension 2r+1intermediate
retained route statementA failure becomes a primitive ghostintermediate
retained route statementThe ghost becomes an algebraic defect pairintermediate
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
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 targetTurn smoothability into support mobility.open
Research targetResolve the stage-2 nodal carrier defect globally.open
Research targetExtract the rank-one external tensor.open

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 bridgeTurn smoothability into support mobility.

The current research map records this as an open mathematical step.

Evidence needed nextConcrete conditions for progress

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Prepared starting pointTurn smoothability into support mobility.

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Research contextPrepared context for any AI agent

Do algebraic cycles obey the same Lefschetz decomposition, projector, equivalence, and positivity structures that cohomology predicts?

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Sources and references11 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
Standard Conjectures on Algebraic Cyclesoriginal source · Alexander Grothendieck · Oxford University Press · Bombay colloquium 1968; published 1969 · accessed Aug 7, 2026
2
Algebraic Cycles and the Weil Conjecturessurvey or monograph · Steven L. Kleiman · North-Holland · 1968 · accessed Aug 7, 2026
3
The Standard Conjecturessurvey or monograph · Steven L. Kleiman · American Mathematical Society · 1994 · accessed Aug 7, 2026
4
Pour une théorie inconditionnelle des motifspeer reviewed result · Yves André · Publications Mathématiques de l'IHÉS · 1996 · DOI 10.1007/BF02698643 · MR MR1423019 · accessed Aug 7, 2026
5
Polarizations and Grothendieck's Standard Conjecturespeer reviewed result · James S. Milne · Annals of Mathematics · 2002 · DOI 10.2307/3062126 · MR MR1906596 · accessed Aug 7, 2026
6
The Standard Conjectures for Holomorphic Symplectic Varieties Deformation Equivalent to Hilbert Schemes of K3 Surfacespeer reviewed result · François Charles, Eyal Markman · Compositio Mathematica · 2013-02-07 · ARXIV 1009.0413 · DOI 10.1112/S0010437X12000607 · accessed Aug 7, 2026
7
Standard Conjectures for Abelian Fourfoldspeer reviewed result · Giuseppe Ancona · Inventiones Mathematicae · 2021 · DOI 10.1007/s00222-020-00990-7 · accessed Aug 7, 2026
8
Relative and Absolute Lefschetz Standard Conjectures for Some Lagrangian Fibrationspeer reviewed result · Giuseppe Ancona · Journal of the London Mathematical Society · 2025 · DOI 10.1112/jlms.70133 · accessed Aug 7, 2026
9
Motives — Grothendieck's Dreamsurvey or monograph · James S. Milne · James S. Milne · accessed Aug 7, 2026
10
Mathlib.AlgebraicGeometry.AlgebraicCycle.Basicformalization · Mathlib contributors · Lean community · accessed Aug 7, 2026
11
Formal Conjecturesformalization · The Formal Conjectures Authors · Google DeepMind public GitHub repository · current main branch accessed 2026-08-07 · accessed Aug 7, 2026

Important qualifications

The Grothendieck standard conjectures are a package with several named formulations and conditional implications. This record centers the Lefschetz and Hodge-type conjectures and records the Künneth and numerical-versus-homological forms without treating every formulation as definitionally identical.
The 1968 proposed year records Grothendieck's Bombay presentation, published in 1969. The paper says the core conjectures arose earlier and credits Bombieri with an independent formulation; the record does not claim a single-day priority chronology.
The general package remains open, while components are known for many special varieties. Representative milestones are not an exhaustive catalogue of every stable construction, degree, cohomology theory, or variety class.
Deligne's proof of the Weil conjectures did not prove the standard conjectures; it supplied the sought Riemann-hypothesis conclusion by another route.
Ancona's abelian-fourfold theorem proves the Hodge-type standard conjecture and obtains numerical-homological coincidence for infinitely many l, while explicitly retaining l-independence as missing from the full package.
André's motivated cycles provide an unconditional characteristic-zero substitute by enlarging the class of cycles; this is not algebraicity of the correspondences demanded by the original conjectures.
The scoped recognition search found no current Clay, Hilbert, Smale, Erdős, Epoch/FrontierMath, named-prize, or comparable selective-list designation. Absence from this bounded record is not a global nonexistence claim.
The scoped formalization search checked current Mathlib algebraic-cycle documentation and the public Formal Conjectures repository. It found partial cycle infrastructure but no problem-level statement or proof of the standard-conjecture package; this does not establish nonexistence in every proof assistant or private project.
No canonical external computation, dataset, or certificate for the general existence and positivity package 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.

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