Theorem · 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.
A refuted different-domain extension; it is not a counterexample among smooth projective varieties.
Proved
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.
Deligne · official Hodge description · §2(v), PDF page 3 →
Observed Oct 12, 2026
Replaces an earlier source assessment. Earlier assessment ↓
In this problem · publication v14 · Selected context
Different statement · related variant · Proved
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 →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v15
Different statement · related variant · Proved
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 →Relationship not independently reviewed. This concerns its use in this problem.
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.
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
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 passage →
Disproved · Superseded in an indexed context
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 →