Conjecture Fieldbook

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Hodge conjecture

Canonical problem dossier

Certain cohomology classes on smooth projective complex varieties should come from algebraic cycles.

Status
Open
Field
Algebraic geometry
Verification
Reviewed dossier
Source grade
mixed
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

H2p(X,Q)∩Hp,p(X)=?span⁡Q{[Z]}H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)\stackrel{?}{=}\operatorname{span}_{\mathbb{Q}}\{[Z]\}

Why it matters

The conjecture asks how much topological information about algebraic varieties is itself algebraic.

Context and known approaches

It concerns rational Hodge classes of type (p,p) and whether they are rational linear combinations of classes of algebraic cycles.

Formal boundary

Every rational Hodge class on a non-singular projective complex variety is a rational linear combination of cohomology classes of algebraic cycles.

Known partial results

Pending review. No structured partial-result note has been added to this dossier.

Source trail

  1. Clay Mathematics Institute — Official Millennium problem overview.
  2. Wikipedia — Definitions, history, and references.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.