[#P7] Hodge conjecture
Problem. Let \(X\) be a smooth projective complex variety and let \(p\ge 0\). Every class in \(H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)\) is a rational linear combination of cohomology classes of algebraic cycles of codimension \(p\).
1Context
The conjecture asks which topological classes on projective varieties arise from algebraic subvarieties.
2Problem setup
Definition 1 (A rational Hodge class of degree 2p). A rational Hodge class of degree 2p is a class in rational cohomology whose complexification lies in the (p,p) summand of the Hodge decomposition.
Definition 2 (An algebraic cycle of codimension p). An algebraic cycle of codimension p is a finite integer linear combination of codimension-p algebraic subvarieties; its cycle class lies in degree-2p cohomology.
Remark 1. The conjecture asks which topological classes on projective varieties arise from algebraic subvarieties.
3What counts as a solution
- Prove the stated rational cycle-class assertion for every smooth projective complex variety and every p, or give a smooth projective complex variety with a rigorously verified rational (p,p) class outside the rational span of algebraic cycle classes.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Clay Mathematics Institute continues to list the rational Hodge conjecture as unsolved and notes known special cases, including varieties of complex dimension below four. Exact unresolved remainder: Prove the rational cycle-class assertion for every smooth projective complex variety and codimension, or give a rigorously verified rational Hodge class outside the rational span of algebraic cycle classes.[1]
1Records
Notes and companion material
Original intake status. The cited authoritative source listed this problem as unsolved when checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Consult the cited official problem description for the integral, rational, and primitive formulations and known cases.
2See also
- Set-theoretic complete intersections for complex space curvesalgebraic geometry
- A polynomial bijection from the rational plane to the rational linealgebraic geometry
- Plane Jacobian conjecturealgebraic geometry
How to cite
TheoremDB contributors, “Hodge conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/hodge-conjectureThis page as plain text: hodge-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from claymath.org[1], current as of July 31, 2026.
1References
- Packet source. Clay Mathematics Institute, Hodge Conjecture, official Millennium Prize Problem page, checked 2026-08-01. Official Problem Description by Pierre Deligne; listed under Unsolved Millennium Prize Problems. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at official problem description, known special cases, and unsolved classification.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides the authoritative rational formulation and current status boundary.Source named by the research packet.
An original CC0 restatement prepared by TheoremDB maintainers.