Math Conjectures

Hodge Conjecture

OPENIconicConjectureProposed 1950 · Full conjecture

Canonical statement

Let XX be a smooth projective variety over C\mathbb C. For every integer p≥0p\ge0,
H2p(X,Q)∩Hp,p(X)=span⁡Q{cl⁡(Z):Z⊆X is a closed algebraic subvariety of codimension p}, H^{2p}(X,\mathbb Q)\cap H^{p,p}(X) =\operatorname{span}_{\mathbb Q} \bigl\{\operatorname{cl}(Z): Z\subseteq X\text{ is a closed algebraic subvariety of codimension }p\bigr\},
where cl⁡(Z)\operatorname{cl}(Z) is the singular-cohomology cycle class and Hp,p(X)H^{p,p}(X) is the (p,p)(p,p)-summand of the Hodge decomposition of H2p(X,C)H^{2p}(X,\mathbb C).
View source LaTeX
Let \(X\) be a smooth projective variety over \(\mathbb C\). For every integer \(p\ge0\),
\[
  H^{2p}(X,\mathbb Q)\cap H^{p,p}(X)
    =\operatorname{span}_{\mathbb Q}
      \bigl\{\operatorname{cl}(Z):
        Z\subseteq X\text{ is a closed algebraic subvariety of
        codimension }p\bigr\},
\] where \(\operatorname{cl}(Z)\) is the singular-cohomology cycle class and \(H^{p,p}(X)\) is the \((p,p)\)-summand of the Hodge decomposition of \(H^{2p}(X,\mathbb C)\).
AG-CYCLE-001:[Hodge1950][Deligne2006Hodge][Voisin2003Hodge]
Algebraic geometryGeometry and topology

Notes

For a smooth projective variety XX over C\mathbb C, every algebraic subvariety of codimension pp has a cohomology class which is rational and of Hodge type (p,p)(p,p). The Hodge conjecture asserts the converse: every class in H2p(X,Q)∩Hp,p(X)H^{2p}(X,\mathbb Q)\cap H^{p,p}(X) is a Q\mathbb Q-linear combination of classes of algebraic subvarieties. The question goes back to Hodge's study of the topological invariants of algebraic varieties around 1950 [Hodge1950].

The case p=1p=1 is settled by the classical Lefschetz (1,1)(1,1) theorem, so the conjecture holds for divisor classes on every smooth projective variety, and it is also known for various special classes of varieties in higher codimension; see [Deligne2006Hodge] and [Voisin2003Hodge] for careful accounts of what is and is not known. The naive integral refinement of the statement is false, which is one reason the rational formulation above is the accepted one.

For arbitrary smooth projective varieties and codimension p≥2p\ge2 the conjecture remains open; a proof would require genuinely new methods for constructing algebraic cycles from Hodge-theoretic data.

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.