Divisor classes · Lefschetz (1,1)

Statement

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.

Codimension one in any dimension; rational coefficients.

Proved

Observed Oct 10, 2026

In this problem · publication v15 · Selected context

Special case · Proved

This covers the divisor case. It overlaps the low-dimensional theorem and every other indexed family in codimension one.

Scope: All divisor Hodge classes, in any dimension.

Limit: Higher codimension is not covered by this theorem.

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

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.

Official description PDF ↗Manuscript / exposition · external

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.

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.

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