Conjecture · version 2
For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.
All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.
Open
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Deligne · official Hodge description · §1, PDF page 1: rational cycle-class formulation →
Observed Oct 10, 2026
In this problem · publication v13 · Selected context
Primary formulation · Open
Integral coefficients ask for a stronger assertion and admit counterexamples, whereas the canonical question permits rational combinations.
An integral obstruction can disappear after tensoring with Q and does not refute rational Hodge.
Open · version 2
For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.
Scope: All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.
Proved · version 3
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.
Scope: Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.
Relationship not independently reviewed.
Connection in Hodge conjecture →Dropping projectivity changes the class of spaces and yields a false extension.
A proof of rational Hodge must use the smooth projective setting; these counterexamples have a different domain.
Open · version 2
For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.
Scope: All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.
Proved · version 3
The assertion that every rational (p,p) class on every compact Kähler manifold is a rational combination of analytic cycle classes is false; complex-torus counterexamples are recorded in the cited description.
Scope: A refuted different-domain extension; it is not a counterexample among smooth projective varieties.
Relationship not independently reviewed.
Connection in Hodge conjecture →Applying the global conjecture to X×X would algebraize its diagonal projectors.
The linked Claim proves the implication under Hodge on the product; these projectors are not constructed unconditionally.
Open · version 2
For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.
Scope: All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.
Proved · version 2
If the rational Hodge conjecture holds on X×X for a smooth projective complex variety X of dimension d, each rational Künneth component of [Δ_X] in H^{2d}(X×X,Q) is represented by a rational codimension-d algebraic cycle.
Scope: A proved conditional implication using the Hodge type of diagonal components, not an unconditional construction of those cycles.
Relationship not independently reviewed.
Connection in Hodge conjecture →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v15
Primary formulation · Open
Integral coefficients ask for a stronger assertion and admit counterexamples, whereas the canonical question permits rational combinations.
An integral obstruction can disappear after tensoring with Q and does not refute rational Hodge.
Open · version 2
For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.
Scope: All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.
Proved · version 3
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.
Scope: Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.
Relationship not independently reviewed.
Connection in Hodge conjecture →Dropping projectivity changes the class of spaces and yields a false extension.
A proof of rational Hodge must use the smooth projective setting; these counterexamples have a different domain.
Open · version 2
For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.
Scope: All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.
Proved · version 3
The assertion that every rational (p,p) class on every compact Kähler manifold is a rational combination of analytic cycle classes is false; complex-torus counterexamples are recorded in the cited description.
Scope: A refuted different-domain extension; it is not a counterexample among smooth projective varieties.
Relationship not independently reviewed.
Connection in Hodge conjecture →Applying the global conjecture to X×X would algebraize its diagonal projectors.
The linked Claim proves the implication under Hodge on the product; these projectors are not constructed unconditionally.
Open · version 2
For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.
Scope: All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.
Proved · version 2
If the rational Hodge conjecture holds on X×X for a smooth projective complex variety X of dimension d, each rational Künneth component of [Δ_X] in H^{2d}(X×X,Q) is represented by a rational codimension-d algebraic cycle.
Scope: A proved conditional implication using the Hodge type of diagonal components, not an unconditional construction of those cycles.
Relationship not independently reviewed.
Connection in Hodge conjecture →Relationship not independently reviewed. This concerns its use in this problem.
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).
paraphrase · No upstream full text or code is republished by this record.
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.
paraphrase · No upstream full text or code is republished by this record.
Under rational Hodge on X×X, the rational Künneth components of the diagonal have algebraic representatives.
paraphrase · No upstream full text or code is republished by this record.
Replacing smooth projective varieties by all compact Kähler manifolds makes the cycle-algebraicity assertion false.
paraphrase · No upstream full text or code is republished by this record.
Pierre Deligne · Published
Official Clay problem description; historical exposition, not a current literature survey.
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Official description PDF: official-description. External evidence linked; no OpenMath execution or independent reproduction is asserted.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Open
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 passage →