1.1. The construction in outline🔗

1. Form the space X(\mathbb C) of complex points with its analytic topology, and the holomorphic de Rham complex \Omega_X^\bullet on it. Rational cohomology H^n(X;\mathbb Q) and de Rham cohomology H^n_{\mathrm{dR}}(X) are hypercohomology groups of sheaf complexes on X(\mathbb C), and the maps of complexes \underline{\mathbb Q}_X\to\underline{\mathbb C}_X\to\Omega_X^\bullet induce a comparison map from the first to the second.

2. Define F^pH^n_{\mathrm{dR}}(X) as the image of the hypercohomology of the truncated complex \Omega_X^{\ge p}, transport complex conjugation to de Rham cohomology, and set H^{p,q}=F^p\cap\overline{F^q}. The Hodge classes \operatorname{Hdg}^p(X;\mathbb Q) are the rational classes of degree 2p whose image under the comparison map lies in H^{p,p}; over \mathbb Q this is the same as lying in F^p.

3. Represent an irreducible subvariety Z\subseteq X of codimension p by its generic point, a point of the scheme X.left of coheight p, and form the cohomology of X(\mathbb C) with support in Z.

4. On the smooth locus of Z, the complex orientation of the normal directions singles out a generator of the cohomology with support in Z in degree 2p, locally in charts, and these local generators glue.

5. Extend the resulting class uniquely across the singular locus of Z, which has codimension at least p+1, so that its cohomology with support vanishes in degrees 2p and 2p+1. Forgetting the support gives \operatorname{cl}_X(Z)\in H^{2p}(X;\mathbb Q).

6. Extend \operatorname{cl}_X additively to cycles and \mathbb Q-linearly to rational cycles. The statement itself uses only the span of the classes of individual subvarieties.

The guide follows this order: steps 1 and 2 are the subject of Hodge classes, step 3 of Cycles and cohomology with support, steps 4 and 5 of The class of a subvariety, and step 6 of The statement.

Everything happens in cohomology with support. Other/ also builds a Borel–Moore homology of the pair Z\subset X and the duality that identifies the class above with a fundamental class, but nothing in the statement passes through it, so this guide leaves it aside.

1.2. Scope and status🔗

Everything that enters the statement is constructed. The class of a subvariety of any codimension, singular or not, is defined outright, with no hypotheses beyond the smoothness, projectivity, and integrality of the ambient variety that the statement itself assumes.

Two cases of the conjecture itself are proved, both unconditionally, and both in Other/AlgebraicGeometry/.

• Codimension zero. Every degree-zero class is a Hodge class, and the class of X itself spans H^0(X;\mathbb Q), so \operatorname{Hdg}^0(X;\mathbb Q)=A^0(X). The two inputs are that the analytification of a smooth integral complex scheme is connected and that the constructed class of the whole variety is nonzero; both are proved here. The second is the useful one: it shows that the construction of the class of a subvariety does not return zero, which is the way a statement of this shape can hold for the wrong reason.

• Above the dimension. For p>\dim X both sides are \bot: there is no point of coheight p, and the Hodge classes vanish because F^p does. The inclusion is then vacuous, but it does check that the two sides degenerate in the same place.

See What the repository proves about the statement for the declarations.

Three classical facts about the construction are not formalized. None is needed to state the conjecture, but they are needed for the usual equivalent formulations, and the first is what stands between the two cases above and the general one.

• Cohomological purity: H^{2p}_Z(X;\mathbb Q) is the line on the fundamental class, so that forgetting support is injective on it and \operatorname{cl}_X(Z)\ne0 for every subvariety Z. The normalized local section the class is built from is proved nonzero for every component; purity is what would carry that through to ordinary cohomology. Without it the span A^p(X) is not known to be as large as the classes generating it suggest.

• The class of a subvariety is a Hodge class: \operatorname{cl}_X(Z)\in\operatorname{Hdg}^p(X;\mathbb Q). This is the reverse inclusion, and it is what would turn the statement into the usual equality.

• The comparison between constant-sheaf and singular cohomology is not known to commute with forgetting support.

1.3. Degree and support conventions🔗

A subvariety Z of codimension p in a smooth variety X of dimension d has p complex normal directions, so its class sits in the cohomological degree 2p that the complex orientation of those directions can generate. This is the origin of the index 2 * (p : ℤ) throughout the code; the complementary index 2 * ((d - p : ℕ) : ℤ), the real dimension of Z, appears wherever a construction is indexed by Z rather than by its codimension. A class with support in Z lies in H_Z^{2p}(X;\mathbb Q); the map forgetSupport sends it to the ordinary group H^{2p}(X;\mathbb Q), in which the conjecture is stated.