Theorem · version 1
Earlier version 1. View current 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.
A proved conditional implication using the Hodge type of diagonal components, not an unconditional construction of those cycles.
Proved
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 · §4 Example 1, PDF page 4; implication from §1 →
Observed Oct 10, 2026
In this problem · publication v7 · Selected context
Conditional implication · Proved
Source-grounded deduction: Deligne §4 identifies these components as Hodge classes; §1 then supplies their algebraicity under the stated premise.
Scope: Algebraic representatives for the Künneth components of the diagonal, if Hodge holds on X×X.
Limit: Algebraicity of these particular classes remains open in general; the implication proves no new unconditional case.
A consequence under an open premise, not an additional resolved part of that premise.
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 1
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 1
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 v8
Conditional implication · Proved
Source-grounded deduction: Deligne §4 identifies these components as Hodge classes; §1 then supplies their algebraicity under the stated premise.
Scope: Algebraic representatives for the Künneth components of the diagonal, if Hodge holds on X×X.
Limit: Algebraicity of these particular classes remains open in general; the implication proves no new unconditional case.
A consequence under an open premise, not an additional resolved part of that premise.
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 1
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 1
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.
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.
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.
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.
Proved
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 →