- Hodge conjecture
The Hodge conjecture is a major unsolved problem in
algebraic geometry which relates thealgebraic topology of anon-singular complexalgebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certainde Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties. The Hodge conjecture is one of theClay Mathematics Institute 's Millennium Prize Problems, so there is a US$1,000,000 prize for proving the Hodge conjecture.Motivation
Let "X" be a connected
complex manifold of complex dimension "n". Then "X" is anorientable smooth manifold of dimension "2n", so its cohomology groups lie in degrees zero through "2n". Assume that "X" is aKähler manifold , so that there is a decomposition on its cohomology with complex coefficients::H^k(X, mathbf{C}) = igoplus_{p+q=k} H^{p,q}(X),where H^{p,q}(X) is the subgroup of cohomology classes which are represented byharmonic form s of type ("p", "q"). That is, these are the cohomology classes represented bydifferential form s which, in some choice of local coordinates z_1, ldots, z_n, can be written as aharmonic function times dz_{i_1} wedge cdots wedge dz_{i_p} wedge dar z_{j_1} wedge cdots wedge dar z_{j_q}. (SeeHodge theory for more details.) Taking wedge products of these harmonic representatives corresponds to thecup product in cohomology, so the cup product is compatible with the Hodge decomposition::cup : H^{p,q}(X) imes H^{p',q'}(X) ightarrow H^{p+p',q+q'}(X).Since "X" is a complex manifold, "X" has afundamental 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 alpha of type ("p", "q"). We can integrate alpha 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,ldots,z_n on "X" such that "Z" is just z_{k+1} = cdots = z_n = 0. If "p" > "k", then alpha 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 alpha. 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 alpha 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, mathbf{C}) = H^{n,n}(X), we conclude that ["Z"] must lie in H^{n-k,n-k}(X, mathbf{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 projective complex 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, theFubini-Study metric , such a manifold is always a Kähler manifold. ByChow's theorem , a projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero set of a collection of homogenous 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.
Known cases of the Hodge conjecture
Low dimension and codimension
The first result on the Hodge conjecture is due to
Solomon Lefschetz . In fact, it predates the conjecture and provided some of Hodge's motivation.:Theorem (
Lefschetz theorem on (1,1)-classes ) Any element of H^{2}(X, mathbf{Z}) cap H^{1,1}(X) is the cohomology class of a divisor on "X". In particular, the Hodge conjecture is true for H^2.A very quick proof can be given using
sheaf cohomology and theexponential exact sequence . (The cohomology class of a divisor turns out to equal to its firstChern class .) Lefschetz's original proof proceeded bynormal function s, which were introduced byHenri Poincaré . However,Griffiths's 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 operatorname{Hdg}^1(X), then the Hodge conjecture holds for "X".
Abelian varieties
For most abelian varieties, the algebra operatorname{Hdg}^*(X) is generated in degree one, so the Hodge conjecture holds. In particular, the Hodge conjecture holds for general abelian varieties, for products of elliptic curves, and for simple abelian varieties. However, Mumford constructed an example of an abelian variety where operatorname{Hdg}^2(X) is not generated by products of divisor classes. André Weil generalized this example by showing that whenever the variety has
complex multiplication by an imaginary quadratic field, then operatorname{Hdg}^2(X) is not generated by products of divisor classes. Moonen and Zahren proved that in dimension less than 5, either operatorname{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, mathbf{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 and Hirzebruch. Using
K-theory , they constructed an example of a torsion Hodge class, that is, a Hodge class alpha such that for some positive integer "n", nalpha = 0. Such a cohomology class cannot be the class of a cycle.Burt Totaro reinterpreted their result in the framework ofcobordism 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 non-torsion cohomology class in H^{2k}(X, mathbf{Z}) cap H^{k,k}(X) is the cohomology class of an algebraic cycle with integral coefficients on "X".
This is also false. Kollár found an example of a Hodge class alpha 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".
Claire Voisin 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 "c"-codimensional subvariety of "X". The cohomology classes of level at least "c" filter the cohomology of "X", and it is easy to see that the "c"th step of the filtration N^cH^k(X, mathbf{Z}) satisfies:N^cH^k(X, mathbf{Z}) subseteq H^k(X, mathbf{Z}) cap (H^{k-c,c}(X) opluscdotsoplus H^{c,k-c}(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^{k-c,c}(X) opluscdotsoplus H^{c,k-c}(X)).
Grothendieck 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, mathbf{Q}) is the largest sub-Hodge structure of H^k(X, mathbf{Z}) contained in H^{k-c,c}(X) opluscdotsoplus H^{c,k-c}(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 and Kaplan. 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, and Kaplan proved that this is always true, without assuming the Hodge conjecture.
References
*Hodge, W. V. D. "The topological invariants of algebraic varieties". "Proceedings of the International Congress of Mathematicians", Cambridge, MA, 1950, vol. 1, pp. 181–192.
*Grothendieck, A. "Hodge's general conjecture is false for trivial reasons". "Topology" 8 1969, pp. 299–303.External links
* [http://www.claymath.org/millennium/Hodge_Conjecture/Official_Problem_Description.pdf The Clay Math Institute Official Problem Description (pdf)]
* Popular lecture on Hodge Conjecture by Dan Freed (University of Texas) [http://claymath.msri.org/hodgeconjecture.mov (Real Video)] [http://www.ma.utexas.edu/users/dafr/HodgeConjecture/netscape_noframes.html (Slides)]
* K. H. Kim, F. W. Roush. [http://arxiv.org/abs/math/0608265 Counterexample to the Hodge Conjecture] - recently revised by authors, and the claim withdrawn
* Indranil Biswas. [http://www.imsc.res.in/~kapil/isw/isw.pdf The Hodge Conjecture for general Prym varieties (pdf)]
Wikimedia Foundation. 2010.