[#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 \(p\).

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Definitions and notation

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

What counts as a solution

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]

1Packet records

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Hodge conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
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.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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.^[1] Relevance to this problem For Hodge conjecture, record hodge-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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

A polynomial bijection from the rational plane to the rational linealgebraic geometry
Set-theoretic complete intersections for complex space curvesalgebraic geometry
Plane Jacobian conjecturealgebraic geometry

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Plain text

“Hodge conjecture.” TheoremDB. P7. Problem statement; statement text SHA-256 0b5de7b28c815999a167bcfea712400d0416b3b0a0bf4ae16f1a3daaf2f521f7. https://theoremdb.org/statement/?ref=P7

BibTeX

@misc{theoremdb-problem-0b5de7b28c815999a167bcfea712400d0416b3b0a0bf4ae16f1a3daaf2f521f7,
  title = {{Hodge conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 0b5de7b28c815999a167bcfea712400d0416b3b0a0bf4ae16f1a3daaf2f521f7},
  url = {https://theoremdb.org/statement/?ref=P7}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from claymath.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P7] Hodge conjecture

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