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Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry that relates the algebraic topology of a nonsingular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says 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, U.S. 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 using "some argument".
Motivation
Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2n, so its cohomology groups lie in degrees zero through 2n. Assume X is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients:
\( H^k(X, \mathbf{C}) = \bigoplus_{p+q=k} H^{p,q}(X),\, \)
where H^{p, q}(X) is the subgroup of cohomology classes which are represented by harmonic forms of type (p, q). That is, these are the cohomology classes represented by differential forms which, in some choice of local coordinates z_{1}, ..., z_{n}, can be written as a harmonic function times
\( dz_{i_1} \wedge \cdots \wedge dz_{i_p} \wedge d\bar z_{j_1} \wedge \cdots \wedge d\bar z_{j_q}.\)
(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:
\( \cup : H^{p,q}(X) \times H^{p',q'}(X) \rightarrow H^{p+p',q+q'}(X).\,\)
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 i : Z → X be the inclusion map. Choose a differential form α of type (p, q). We can integrate α over Z:
\( \int_Z i^*\alpha.\!\,\)
To evaluate this integral, choose a point of Z and call it 0. Around 0, we can choose local coordinates z_{1}, ..., z_{n} on X such that Z is just z_{k + 1} = ... = z_{n} = 0. If p > k, then α must contain some dz_{i} where z_{i} pulls back to zero on Z. The same is true if q > k. Consequently, this integral is zero if (p, q) ≠ (k, k).
More abstractly, the integral can be written as the cap product of the homology class of Z and the cohomology class represented by α. 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 (p, q) ≠ (k, k), then we get zero. Because H^{2n }(X, C) = H^{n, n}(X), we conclude that [Z] must lie in H^{nk, nk}(X, C). Loosely speaking, the Hodge conjecture asks:
 Which cohomology classes in H^{k, k}(X) come from complex subvarieties Z?
Statement of the Hodge conjecture
Let:
\(\operatorname{Hdg}^k(X) = H^{2k}(X, \mathbf{Q}) \cap H^{k,k}(X).\)
We call this the group of Hodge classes of degree 2k on X.
The modern statement of the Hodge conjecture is:
Hodge conjecture. Let X be a nonsingular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.
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:
\( \sum_i c_iZ_i.\,\)
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:
\( \sum_i c_i[Z_i].\,\)
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 Zucker showed that it is possible to construct a counterexample to the Hodge conjecture as complex tori with analytic rational cohomology of type (p,p), which is not projective algebraic. (see the appendix B: in Zucker (1977))
Known cases of the Hodge conjecture
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 H2(X, Z) ∩ H1,1(X) is the cohomology class of a divisor on X. In particular, the Hodge conjecture is true for H2.
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, Griffiths transversality theorem shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.
By the Hard Lefschetz theorem, one can prove:
Theorem. If the Hodge conjecture holds for Hodge classes of degree p, p < n, then the Hodge conjecture holds for Hodge classes of degree 2n − p.
Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree 2n − 2. This proves the Hodge conjecture when 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
\( \operatorname{Hdg}^*(X) = \sum_k \operatorname{Hdg}^k(X)\,\)
is generated by Hdg1(X), then the Hodge conjecture holds for X.
Hypersurfaces
By the strong and weak Lefschetz theorem, the only nontrivial part of the Hodge conjecture for hypersurfaces is the degree m part (i.e., the middle cohomology) of a 2mdimensional hypersurface \(X \subset \mathbf P^{2m+1}\). If the degree d is 2, i.e., X is a quadric, the Hodge conjecture holds for all m. For m=2, i.e., fourfolds, the Hodge conjecture is known for d \le 5.[1]
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^{[citation needed]}. However, Mumford (1969) constructed an example of an abelian variety where Hdg^{2}(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 <Hdg^{2}(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.
Generalizations
The integral Hodge conjecture
Hodge's original conjecture was:

 Integral Hodge conjecture. Let X be a projective complex manifold. Then every cohomology class in H^{2k}(X, Z) ∩ 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 Ktheory, they constructed an example of a torsion Hodge class, that is, a Hodge class α such that for some positive integer n, n α = 0. Such a cohomology class cannot be the class of a cycle. Totaro (1997) reinterpreted their result in the framework of cobordism and found many examples of torsion 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 H^{2k}(X, Z) ∩ 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 H^{2k}(X, Z) ∩ 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.
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 level c if it is the pushforward of a cohomology class on a ccodimensional subvariety of X. The cohomology classes of level at least c filter the cohomology of X, and it is easy to see that the cth step of the filtration N^{c} H^{k}(X, Z) satisfies
\( N^cH^k(X, \mathbf{Z}) \subseteq H^k(X, \mathbf{Z}) \cap (H^{kc,c}(X) \oplus\cdots\oplus H^{c,kc}(X)).\)
Hodge's original statement was:
Generalized Hodge conjecture, Hodge's version. \(N^cH^k(X, \mathbf{Z}) = H^k(X, \mathbf{Z}) \cap (H^{kc,c}(X) \oplus\cdots\oplus H^{c,kc}(X))\).
Grothendieck (1969) observed that this cannot be true, even with rational coefficients, because the righthand side is not always a Hodge structure. His corrected form of the Hodge conjecture is:
Generalized Hodge conjecture. Nc Hk(X, Q) is the largest subHodge structure of Hk(X, Z) contained in \( H^{kc,c}(X) \oplus\cdots\oplus H^{c,kc}(X)\).
This version is open.
Algebraicity of Hodge loci
The strongest evidence in favor of the Hodge conjecture is the algebraicity result of Cattani, Deligne & Kaplan (1995). 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.
See also
Tate conjecture
Hodge theory
Hodge structure
period mapping
References
James Lewis: A Survey of the Hodge Conjecture, 1991, Example 7.21
Atiyah, M. F.; Hirzebruch, F. (1961), "Vector bundles and homogeneous spaces", Proc. Sympos. Pure Math. 3: 7–38
Cattani, Eduardo; Deligne, Pierre; Kaplan, Aroldo (1995), "On the locus of Hodge classes", Journal of the American Mathematical Society 8 (2): 483–506, doi:10.2307/2152824, JSTOR 2152824, MR 1273413.
Grothendieck, A. (1969), "Hodge's general conjecture is false for trivial reasons", Topology 8 (3): 299–303, doi:10.1016/00409383(69)900160.
Hodge, W. V. D. (1950), "The topological invariants of algebraic varieties", Proceedings of the International Congress of Mathematicians (Cambridge, MA) 1: 181–192.
Kollár, János (1992), "Trento examples", in Ballico, E.; Catanese, F.; Ciliberto, C., Classification of irregular varieties, Lecture Notes in Math. 1515, Springer, p. 134, ISBN 3540552952.
Lefschetz, Solomon (1924), L'Analysis situs et la géométrie algébrique, Collection de Monographies publiée sous la Direction de M. Emile Borel (in French), Paris: GauthierVillars Reprinted in Lefschetz, Solomon (1971), Selected papers, New York: Chelsea Publishing Co., ISBN 9780828402347, MR 0299447.
Moonen, B. J. J.; Zarhin, Yu. G. (1999), "Hodge classes on abelian varieties of low dimension", Mathematische Annalen 315 (4): 711–733, arXiv:math/9901113, doi:10.1007/s002080050333.
Mumford, D. (1969), "A Note of Shimura's paper "Discontinuous groups and abelian varieties"", Math. Ann. 181 (4): 345–351, doi:10.1007/BF01350672.
Totaro, B. (1997), "Torsion algebraic cycles and complex cobordism", Journal of the American Mathematical Society 10 (2): 467–493, arXiv:alggeom/9609016, doi:10.1090/S0894034797002324, JSTOR 2152859.
Voisin, Claire (2002), "A counterexample to the Hodge conjecture extended to Kähler varieties", Int Math Res Notices 2002 (20): 1057–1075, doi:10.1155/S1073792802111135.
Weil, A. (1977), "Abelian varieties and the Hodge ring", Collected papers III: 421–429
Zucker, S. (1977), "The Hodge conjecture for cubic fourfolds", Comp. Math 34: 199–209 http://archive.numdam.org/ARCHIVE/CM/CM_1977__34_2/CM_1977__34_2_199_0/CM_1977__34_2_199_0.pdf
External links
The Clay Math Institute Official Problem Description by P. Deligne (pdf)
Popular lecture on Hodge Conjecture by Dan Freed (University of Texas) (Real Video) (Slides)
Indranil Biswas, Kapil Paranjape. The Hodge Conjecture for general Prym varieties
Burt Totaro, Why believe the Hodge Conjecture?
Claire Voisin, Hodge loci
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