Domain 03 · Hodge Conjecture

Campaign ID: HC-001
Mathematical status: open conjecture
Geometric category: smooth projective varieties over C
Coefficient field: rational
Governance: ADR-0004

Canonical challenge

For every smooth projective complex variety X and codimension p, determine whether every rational Hodge class of type (p,p) lies in the rational span of algebraic cycle classes.

The campaign preserves the rational Hodge conjecture exactly. Integral variants, generalized Hodge conjectures, analytic substitutes, and numerical evidence are separate statements.

Programme posture

The authorized work is source, normalization, and equivalence control. A reformulation is accepted only when the ambient category, coefficient field, cycle-class map, and quantifiers are preserved. No theorem-strengthening claim is made.

Canonical artifacts

• Domain master plan
• WP00 campaign directory
• Review record
• Governing decision ADR-0004

Claim boundary

The programme has not proved new algebraicity results, resolved a nontrivial special case, or produced a formal certificate of the conjecture. The current achievement is a governed source-normalized campaign boundary.