Theorem · 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.
Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.
Proved
The stated existential counterexample theorem is proved in the cited exposition. It refutes the positive integral variant, not rational Hodge.
Deligne · official Hodge description · Section 2(iv), PDF page 2 →
Observed Oct 12, 2026
Replaces an earlier source assessment. Earlier assessment ↓
In this problem · publication v10 · Selected context
Different statement · related variant · Proved
Changing the coefficient ring changes the statement. Rational Hodge stays open; this refutation must never label the canonical Problem refuted.
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 →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v15
Different statement · related variant · Proved
Changing the coefficient ring changes the statement. Rational Hodge stays open; this refutation must never label the canonical Problem refuted.
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 →Relationship not independently reviewed. This concerns its use in this problem.
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.
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 existential counterexample theorem is proved in the cited exposition. It refutes the positive integral variant, not rational Hodge.
Observed Oct 12, 2026 · Source passage →
Disproved · Superseded in an indexed context
Observed Oct 10, 2026 · Source passage →