All codimensions in dimension ≤ 3

Statement

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

Every codimension, restricted to complex dimension ≤ 3.

Proved

Observed Oct 10, 2026

In this problem · publication v8 · Selected context

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.

Clay Hodge page, overview: dimension less than four → · Original passage ↗
Recorded wording
The rational Hodge conjecture holds for every smooth projective complex variety of complex dimension at most three.
paraphrase · Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.

Clay Mathematics Institute · Revision observed-2026-10-10

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.

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.

Proved

Observed Oct 10, 2026 · Source passage →

Record details