Math Conjectures

Generalized Hodge Conjecture

OPENMajorConjectureProposed 1969 · Full conjecture

Canonical statement

Let k,c≥0k,c\ge0 be integers and let XX be a smooth projective complex variety. If V⊆Hk(X,Q)V\subseteq H^k(X,\mathbb Q) is a rational Hodge substructure such that VCp,q=0V_\mathbb C^{p,q}=0 whenever p<cp<c or q<cq<c, where VC=V⊗QC=⨁p+q=kVCp,qV_\mathbb C=V\otimes_{\mathbb Q}\mathbb C =\bigoplus_{p+q=k}V_\mathbb C^{p,q}, then there exists a closed algebraic subset Z⊆XZ\subseteq X of codimension at least cc such that
V⊆ker⁡ ⁣(Hk(X,Q)⟶Hk(X∖Z,Q)). V\subseteq\ker\!\left( H^k(X,\mathbb Q)\longrightarrow H^k(X\setminus Z,\mathbb Q) \right).
View source LaTeX
Let \(k,c\ge0\) be integers and let \(X\) be a smooth projective complex variety. If \(V\subseteq H^k(X,\mathbb Q)\) is a rational Hodge substructure such that \(V_\mathbb C^{p,q}=0\) whenever \(p<c\) or \(q<c\), where \(V_\mathbb C=V\otimes_{\mathbb Q}\mathbb C =\bigoplus_{p+q=k}V_\mathbb C^{p,q}\), then there exists a closed algebraic subset \(Z\subseteq X\) of codimension at least \(c\) such that
\[
  V\subseteq\ker\!\left(
    H^k(X,\mathbb Q)\longrightarrow H^k(X\setminus Z,\mathbb Q)
  \right).
\]
AG-CYCLE-002:[Grothendieck1969Hodge][Lewis1999Hodge][Voisin2025GeneralizedHodge]
Algebraic geometryGeometry and topology

Notes

For a smooth projective complex variety XX, the generalized Hodge conjecture predicts that Hodge-theoretic coniveau is detected by geometry: if a rational Hodge substructure V⊆Hk(X,Q)V\subseteq H^k(X,\mathbb Q) has all its Hodge components concentrated in bidegrees (p,q)(p,q) with p,q≥cp,q\ge c, then VV should die upon restriction to the complement of some closed algebraic subset of codimension at least cc. The statement in this form is due to Grothendieck, who observed in 1969 that Hodge's original formulation fails "for trivial reasons" and proposed the corrected version recorded here [Grothendieck1969Hodge].

Equivalently, the conjecture asserts that the Hodge coniveau filtration on cohomology coincides with the geometric coniveau filtration. It contains the ordinary Hodge conjecture as a special case but is genuinely stronger in general. It is known in coniveau one in important settings and for various special varieties; the surrounding circle of ideas is surveyed by Lewis [Lewis1999Hodge] and, with emphasis on coniveau and algebraic cycles, by Voisin [Voisin2025GeneralizedHodge].

The conjecture remains open: no current method produces the asserted geometric support for an arbitrary Hodge substructure of higher coniveau, and a resolution would require constructing such supporting subvarieties in complete generality.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.