Deligne · official Hodge description
Official Clay problem description; historical exposition, not a current literature survey.
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
Hodge classes and their supports can be studied cohomologically, but constructing the required cycles remains the general obstacle.
Passage ID: hodge-conjecture:hodge-cycle-obstruction:v1
Source item: §3–§5, PDF pages 3–4: intermediate Jacobians and detecting Hodge classes
Source record: clay-deligne-hodge@official-description
For a smooth projective complex variety X, every rational class in H²(X,Q) of type (1,1) is a rational combination of divisor classes.
Passage ID: hodge-conjecture:hodge-divisor-classes:v1
Source item: Section 2(iii), PDF page 2
Source record: clay-deligne-hodge@official-description
Deligne cites the 1953 divisor-class account for the Lefschetz (1,1) criterion.
Passage ID: hodge-conjecture:hodge-divisor-history:v1
Source item: §2(iii) and reference [7], Kodaira–Spencer (1953)
Source record: clay-deligne-hodge@official-description
Replacing smooth projective varieties by all compact Kähler manifolds makes the cycle-algebraicity assertion false.
Passage ID: hodge-conjecture:hodge-kahler-boundary:v1
Source item: §2(v), PDF page 3
Source record: clay-deligne-hodge@official-description
Under rational Hodge on X×X, the inverse hard-Lefschetz maps are induced by algebraic correspondences.
Passage ID: hodge-conjecture:hodge-lefschetz-consequence:v1
Source item: §4 Example 2, PDF page 4; implication from §1
Source record: clay-deligne-hodge@official-description
Hodge’s broader conjecture appeared in the ICM 1950 proceedings published in 1952; later coefficient and scope corrections are essential.
Passage ID: hodge-conjecture:hodge-origin-history:v1
Source item: §2(vi) and reference [6], ICM 1950 proceedings published 1952
Source record: clay-deligne-hodge@official-description
Under rational Hodge on X×X, the rational Künneth components of the diagonal have algebraic representatives.
Passage ID: hodge-conjecture:hodge-projector-consequence:v1
Source item: §4 Example 1, PDF page 4; implication from §1
Source record: clay-deligne-hodge@official-description
For every smooth projective complex variety X and every codimension p, CH^p(X)⊗Q maps onto H^{2p}(X,Q) ∩ H^{p,p}(X).
Passage ID: hodge-conjecture:hodge-rational-target:v1
Source item: §1, PDF page 1: rational cycle-class formulation
Source record: clay-deligne-hodge@official-description
The integral analogue of the Hodge conjecture is false: there exist integral Hodge classes on smooth projective complex varieties not represented by integral algebraic cycles.
Passage ID: hodge-conjecture:integral-hodge-counterexamples:v1
Source item: Section 2(iv), PDF page 2
Source record: clay-deligne-hodge@official-description
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v15
Source assessment
Proved
Divisor classes · Lefschetz (1,1)
Codimension one in any dimension; rational coefficients.
Observed Oct 10, 2026
Source assessment
Proved
Counterexamples to the unrestricted compact Kähler extension
The stated failure of the unrestricted compact Kähler extension is established by counterexamples in the cited exposition. Dropping projectivity changes the domain; this does not refute rational Hodge for smooth projective varieties.
Observed Oct 12, 2026
Source assessment · superseded assessment
Disproved
Exact scoped statement in cited source
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Source assessment
Proved
Exact scoped statement in cited source
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Source assessment
Proved
Exact scoped statement in cited source
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Source assessment
Open
Exact scoped statement in cited source
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Source assessment
Proved
Existence of counterexamples to integral Hodge
The stated existential counterexample theorem is proved in the cited exposition. It refutes the positive integral variant, not rational Hodge.
Observed Oct 12, 2026
Source assessment · superseded assessment
Disproved
Integral Hodge variant · counterexamples
Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.
Observed Oct 10, 2026
Question
Hodge conjecture
History
The divisor-class criterion in the cited exposition
Connection
Dropping projectivity changes the class of spaces and yields a false extension.
History
Hodge’s ICM conjecture and its corrected scope
Connection
Applying the global conjecture to X×X would algebraize its diagonal projectors.
Connection
Integral coefficients ask for a stronger assertion and admit counterexamples, whereas the canonical question permits rational combinations.
Further reading
Read the setting and notation before comparing the results.
History
Integral Hodge counterexamples fix the coefficient boundary
Linked manuscripts, code and data. External proof reports, statement correspondence and OpenMath execution are separate checks.
Manuscript / exposition · external
External evidence linked; no OpenMath execution or independent reproduction is asserted.
Resource revision: official-description
Recorded authorship
Pierre Deligne
The cited work does not report AI involvement; this is not an assertion that no AI was used.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
Deligne · official Hodge description
No upstream full text or code is republished by this record.
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