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Theorem · version 3

All codimensions in dimension ≤ 3

Statement

The rational Hodge conjecture holds for every smooth projective complex variety of complex dimension at most three.

Assumptions

  • X is smooth and projective over C.
  • Complex dimension of X is at most 3.
  • Hodge classes and cycle combinations have rational coefficients.

Scope

Every codimension, restricted to complex dimension ≤ 3.

Evidence

Connections

In this problem · publication v13 · Selected context

Hodge conjecture →

Special case · Proved

Includes the divisor case. For threefolds the curve-class case follows from hard Lefschetz and the divisor case; arbitrary fourfolds are excluded.

Scope: All rational Hodge classes on curves, surfaces and threefolds.

Limit: General fourfolds and higher-dimensional varieties are not covered.

Related results · 1

Relationship not independently reviewed. This concerns its use in this problem.

In this problem · publication v15

Hodge conjecture →

Special case · Proved

Includes the divisor case. For threefolds the curve-class case follows from hard Lefschetz and the divisor case; arbitrary fourfolds are excluded.

Scope: All rational Hodge classes on curves, surfaces and threefolds.

Limit: General fourfolds and higher-dimensional varieties are not covered.

Related results · 1

Relationship not independently reviewed. This concerns its use in this problem.

Sources

Deligne · official Hodge description →

  • Section 2(iii), PDF page 2 → · Original passage ↗
    Recorded wording

    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.

    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.

Revision & attribution

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.

History

Statement versions

Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.

Assessment history · 1

Assessments retained in these publication records. Earlier statement versions retain their own source histories.

Record details
Claim ID
claim-hodge-dimension-at-most-three
Statement version
3
SHA-256
b20d7d5f3518582cadee9b3e334a90b5fb1fe3a39ed2e4df6129c47197f64093

Publication records