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Theorem · version 2

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.

Assumptions

  • X is smooth and projective over C.
  • The class has cohomological degree 2 and Hodge type (1,1).

Scope

Codimension one in any dimension; rational coefficients.

Evidence

Connections

In this problem · publication v10 · Selected context

Hodge conjecture →

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.

In this problem · publication v15

Hodge conjecture →

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.

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.

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.

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-divisor-classes
Statement version
2
SHA-256
0308de2ca367ee82f430fce23c5a2fe7ea81d0ed0b2eccafc7797d117d599052

Publication records