All open problems
Open problemChecked July 24, 2026
Hodge Conjecture
A bridge between the topology of complex varieties and their algebraic subvarieties.
Mathematical statement
On a smooth projective complex variety, every rational Hodge class of type is a rational linear combination of classes of algebraic cycles.
Statement source: Therefore editorial statement
Statement terms: Source-specific
Editorial wording by Therefore. No statement reuse license is granted.
Statement artifacts, not proofs
These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.
Pinned Lean formulation 1
MillenniumHodge.clay_prize_hodge_conjecture
Complete statement target, proof intentionally absentLean 4
theorem clay_prize_hodge_conjecture : ClayHodge.{u₁, u₂, u₃} := by sorry- Statement source
- Lean Millennium Prize Problems
- Lean version
- v4.31.0
- Placeholder
- Present; no proof artifact
- Source evidence
- Pinned source index
- Fidelity review
- Community formulation
References