Theorem · version 1
Earlier version 1. View current version 2 →
For smooth projective X/C of dimension d, an ample hyperplane class η, and 0 ≤ p ≤ d, assume rational Hodge on X×X in codimension d−p. Then the inverse of η^p: H^{d−p}(X,Q) → H^{d+p}(X,Q) is induced by a rational algebraic correspondence of codimension d−p on X×X.
A proved implication for the Lefschetz standard-conjecture classes in characteristic zero; no assertion in arbitrary positive characteristic follows.
Proved
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Deligne · official Hodge description · §4 Example 2, PDF page 4; implication from §1 →
Observed Oct 10, 2026
In this problem · publication v6 · Selected context
Conditional implication · Proved
Source-grounded deduction from Deligne §4 Example 2 and the conjecture’s cycle-class surjectivity. Hard Lefschetz itself is already a theorem.
Scope: Inverse Lefschetz isomorphisms would be induced by cycles, under Hodge on the product.
Limit: The premise must cover X×X in the specified codimension. General cycle construction is still missing.
A consequence under an open premise, not an additional resolved part of that premise.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v8
Conditional implication · Proved
Source-grounded deduction from Deligne §4 Example 2 and the conjecture’s cycle-class surjectivity. Hard Lefschetz itself is already a theorem.
Scope: Inverse Lefschetz isomorphisms would be induced by cycles, under Hodge on the product.
Limit: The premise must cover X×X in the specified codimension. General cycle construction is still missing.
A consequence under an open premise, not an additional resolved part of that premise.
Relationship not independently reviewed. This concerns its use in this problem.
Under rational Hodge on X×X, the inverse hard-Lefschetz maps are induced by algebraic correspondences.
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
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →