Reformulation in terms of algebraic cycles

Another way of phrasing the Hodge conjecture involves the idea of an algebraic cycle. An algebraic cycle on X is a formal combination of subvarieties of X; that is, it is something of the form

${\displaystyle \sum _{i}c_{i}Z_{i}.}$

The coefficients are usually taken to be integral or rational. We define the cohomology class of an algebraic cycle to be the sum of the cohomology classes of its components. This is an example of the cycle class map of de Rham cohomology, see Weil cohomology. For example, the cohomology class of the above cycle would be

${\displaystyle \sum _{i}c_{i}[Z_{i}].}$

Such a cohomology class is called algebraic. With this notation, the Hodge conjecture becomes

Let X be a projective complex manifold. Then every Hodge class on X is algebraic.

The assumption in the Hodge conjecture that X be algebraic (projective complex manifold) cannot be weakened. In 1977, Steven Zucker showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology of type ${\displaystyle (p,p)}$, which is not projective algebraic. (see appendix B of Zucker (1977))

Low dimension and codimension

The first result on the Hodge conjecture is due to Lefschetz (1924). In fact, it predates the conjecture and provided some of Hodge's motivation.

Theorem (Lefschetz theorem on (1,1)-classes) Any element of ${\displaystyle H^{2}(X,\mathbb {Z} )\cap H^{1,1}(X)}$ is the cohomology class of a divisor on ${\displaystyle X}$. In particular, the Hodge conjecture is true for ${\displaystyle H^{2}}$.

A very quick proof can be given using sheaf cohomology and the exponential exact sequence. (The cohomology class of a divisor turns out to equal to its first Chern class.) Lefschetz's original proof proceeded by normal functions, which were introduced by Henri Poincaré. However, the Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.

By the Hard Lefschetz theorem, one can prove:^[1]

Theorem. If for some ${\displaystyle p<{\frac {n}{2}}}$ the Hodge conjecture holds for Hodge classes of degree ${\displaystyle p}$, then the Hodge conjecture holds for Hodge classes of degree ${\displaystyle 2n-p}$.

Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree ${\displaystyle 2n-2}$. This proves the Hodge conjecture when ${\displaystyle X}$ has dimension at most three.

The Lefschetz theorem on (1,1)-classes also implies that if all Hodge classes are generated by the Hodge classes of divisors, then the Hodge conjecture is true:

Corollary. If the algebra ${\displaystyle \operatorname {Hdg} ^{*}(X)=\bigoplus \nolimits _{k}\operatorname {Hdg} ^{k}(X)}$ is generated by ${\displaystyle \operatorname {Hdg} ^{1}(X)}$, then the Hodge conjecture holds for ${\displaystyle X}$.

Hypersurfaces

By the strong and weak Lefschetz theorem, the only non-trivial part of the Hodge conjecture for hypersurfaces is the degree m part (i.e., the middle cohomology) of a 2m-dimensional hypersurface ${\displaystyle X\subset \mathbf {P} ^{2m+1}}$. If the degree d is 2, i.e., X is a quadric, the Hodge conjecture holds for all m. For ${\displaystyle m=2}$, i.e., fourfolds, the Hodge conjecture is known for ${\displaystyle d\leq 5}$.^[2]

Abelian varieties

For most abelian varieties, the algebra Hdg*(X) is generated in degree one, so the Hodge conjecture holds. In particular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, and for simple abelian varieties of prime dimension.^[3][4][5] However, Mumford (1969) constructed an example of an abelian variety where Hdg²(X) is not generated by products of divisor classes. Weil (1977) generalized this example by showing that whenever the variety has complex multiplication by an imaginary quadratic field, then Hdg²(X) is not generated by products of divisor classes. Moonen & Zarhin (1999) proved that in dimension less than 5, either Hdg*(X) is generated in degree one, or the variety has complex multiplication by an imaginary quadratic field. In the latter case, the Hodge conjecture is only known in special cases.

The integral Hodge conjecture

Hodge's original conjecture was:

Integral Hodge conjecture. Let X be a projective complex manifold. Then every cohomology class in ${\displaystyle H^{2k}(X,\mathbb {Z} )\cap H^{k,k}(X)}$ is the cohomology class of an algebraic cycle with integral coefficients on X.

This is now known to be false. The first counterexample was constructed by Atiyah & Hirzebruch (1961). Using K-theory, they constructed an example of a torsion cohomology class—that is, a cohomology class α such that nα = 0 for some positive integer n—which is not the class of an algebraic cycle. Such a class is necessarily a Hodge class. Totaro (1997) reinterpreted their result in the framework of cobordism and found many examples of such classes.

The simplest adjustment of the integral Hodge conjecture is:

Integral Hodge conjecture modulo torsion. Let X be a projective complex manifold. Then every cohomology class in ${\displaystyle H^{2k}(X,\mathbb {Z} )\cap H^{k,k}(X)}$ is the sum of a torsion class and the cohomology class of an algebraic cycle with integral coefficients on X.

Equivalently, after dividing ${\displaystyle H^{2k}(X,\mathbb {Z} )\cap H^{k,k}(X)}$ by torsion classes, every class is the image of the cohomology class of an integral algebraic cycle. This is also false. Kollár (1992) found an example of a Hodge class α which is not algebraic, but which has an integral multiple which is algebraic.

Rosenschon & Srinivas (2016) have shown that in order to obtain a correct integral Hodge conjecture, one needs to replace Chow groups, which can also be expressed as motivic cohomology groups, by a variant known as étale (or Lichtenbaum) motivic cohomology. They show that the rational Hodge conjecture is equivalent to an integral Hodge conjecture for this modified motivic cohomology.

The Hodge conjecture for Kähler varieties

A natural generalization of the Hodge conjecture would ask:

Hodge conjecture for Kähler varieties, naive version. Let X be a complex Kähler manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

This is too optimistic, because there are not enough subvarieties to make this work. A possible substitute is to ask instead one of the two following questions:

Hodge conjecture for Kähler varieties, vector bundle version. Let X be a complex Kähler manifold. Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X.

Hodge conjecture for Kähler varieties, coherent sheaf version. Let X be a complex Kähler manifold. Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of coherent sheaves on X.

Voisin (2002) proved that the Chern classes of coherent sheaves give strictly more Hodge classes than the Chern classes of vector bundles and that the Chern classes of coherent sheaves are insufficient to generate all the Hodge classes. Consequently, the only known formulations of the Hodge conjecture for Kähler varieties are false.

The generalized Hodge conjecture

Hodge made an additional, stronger conjecture than the integral Hodge conjecture. Say that a cohomology class on X is of co-level c (coniveau c) if it is the pushforward of a cohomology class on a c-codimensional subvariety of X. The cohomology classes of co-level at least c filter the cohomology of X, and it is easy to see that the cth step of the filtration N^cH^k(X, Z) satisfies

${\displaystyle N^{c}H^{k}(X,\mathbf {Z} )\subseteq H^{k}(X,\mathbf {Z} )\cap (H^{k-c,c}(X)\oplus \cdots \oplus H^{c,k-c}(X)).}$

Hodge's original statement was:

Generalized Hodge conjecture, Hodge's version. ${\displaystyle N^{c}H^{k}(X,\mathbf {Z} )=H^{k}(X,\mathbf {Z} )\cap (H^{k-c,c}(X)\oplus \cdots \oplus H^{c,k-c}(X)).}$

Grothendieck (1969) observed that this cannot be true, even with rational coefficients, because the right-hand side is not always a Hodge structure. His corrected form of the Hodge conjecture is:

Generalized Hodge conjecture. N^cH^k(X, Q) is the largest sub-Hodge structure of H^k(X, Z) contained in ${\displaystyle H^{k-c,c}(X)\oplus \cdots \oplus H^{c,k-c}(X).}$

This version is open.