Hodge conjecture

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000 for whoever can prove or disprove the Hodge conjecture.
Motivation.
Let "X" be a compact complex manifold of complex dimension "n". Then "X" is an orientable smooth manifold of real dimension formula_1, so its cohomology groups lie in degrees zero through formula_1. Assume "X" is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients
where formula_4 is the subgroup of cohomology classes which are represented by harmonic forms of type formula_5. That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates formula_6, can be written as a harmonic function times
(See Hodge theory for more details.) Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product is compatible with the Hodge decomposition:
Since "X" is a compact oriented manifold, "X" has a fundamental class.
Let "Z" be a complex submanifold of "X" of dimension "k", and let formula_9 be the inclusion map. Choose a differential form formula_10 of type formula_5. We can integrate formula_10 over "Z":
To evaluate this integral, choose a point of "Z" and call it 0. Around 0, we can choose local coordinates formula_14 on "X" such that "Z" is just formula_15. If formula_16, then formula_10 must contain some formula_18 where formula_19 pulls back to zero on "Z". The same is true if formula_20. Consequently, this integral is zero if formula_21.
More abstractly, the integral can be written as the cap product of the homology class of "Z" and the cohomology class represented by formula_10. By Poincaré duality, the homology class of "Z" is dual to a cohomology class which we will call ["Z"], and the cap product can be computed by taking the cup product of ["Z"] and α and capping with the fundamental class of "X". Because ["Z"] is a cohomology class, it has a Hodge decomposition. By the computation we did above, if we cup this class with any class of type formula_21, then we get zero. Because formula_24, we conclude that ["Z"] must lie in formula_25. Loosely speaking, the Hodge conjecture asks:
Statement of the Hodge conjecture.
Let:
We call this the group of "Hodge classes" of degree 2"k" on "X".
The modern statement of the Hodge conjecture is:
A projective complex manifold is a complex manifold which can be embedded in complex projective space. Because projective space carries a Kähler metric, the Fubini–Study metric, such a manifold is always a Kähler manifold. By Chow's theorem, a projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero set of a collection of homogeneous polynomials.
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:
The coefficient 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:
Such a cohomology class is called "algebraic". With this notation, the Hodge conjecture becomes:
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 formula_30, which is not projective algebraic. (see appendix B of )
Known cases of the Hodge conjecture.
Low dimension and codimension.
The first result on the Hodge conjecture is due to . In fact, it predates the conjecture and provided some of Hodge's motivation.
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:
Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree formula_36. This proves the Hodge conjecture when formula_32 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:
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 2"m"-dimensional hypersurface formula_42. If the degree "d" is 2, i.e., "X" is a quadric, the Hodge conjecture holds for all "m". For formula_43, i.e., fourfolds, the Hodge conjecture is known for formula_44.
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. However, constructed an example of an abelian variety where Hdg2("X") is not generated by products of divisor classes. 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. 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.
Generalizations.
The integral Hodge conjecture.
Hodge's original conjecture was:
This is now known to be false. The first counterexample was constructed by . Using K-theory, they constructed an example of a torsion cohomology class—that is, a cohomology class such that for some positive integer —which is not the class of an algebraic cycle. Such a class is necessarily a Hodge class. reinterpreted their result in the framework of cobordism and found many examples of such classes.
The simplest adjustment of the integral Hodge conjecture is:
Equivalently, after dividing formula_45 by torsion classes, every class is the image of the cohomology class of an integral algebraic cycle. This is also false. found an example of a Hodge class which is not algebraic, but which has an integral multiple which is algebraic.
 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:
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:
 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 "c"th step of the filtration "N'H"("X", Z) satisfies
Hodge's original statement was:
 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:
This version is open.
Algebraicity of Hodge loci.
The strongest evidence in favor of the Hodge conjecture is the algebraicity result of . Suppose that we vary the complex structure of "X" over a simply connected base. Then the topological cohomology of "X" does not change, but the Hodge decomposition does change. It is known that if the Hodge conjecture is true, then the locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, that is, it is cut out by polynomial equations. Cattani, Deligne & Kaplan (1995) proved that this is always true, without assuming the Hodge conjecture.