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| {{Millennium Problems}}
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| The '''Hodge conjecture''' is a major unsolved problem in [[algebraic geometry]] which relates the [[algebraic topology]] of a [[non-singular]] [[complex number|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 duality|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 which 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".
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| == Motivation ==
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| Let ''X'' be a [[compact space|compact]] [[complex manifold]] of complex dimension ''n''. Then ''X'' is an [[orientable]] [[smooth manifold]] of real dimension 2''n'', so its [[cohomology]] groups lie in degrees zero through 2''n''. Assume ''X'' is a [[Kähler manifold]], so that there is a decomposition on its cohomology with complex coefficients:
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| :<math>H^k(X, \mathbf{C}) = \bigoplus_{p+q=k} H^{p,q}(X),\,</math>
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| where ''H<sup>p, q</sup>''(''X'') is the subgroup of cohomology classes which are represented by [[harmonic form]]s of type (''p'', ''q''). That is, these are the cohomology classes represented by [[differential form]]s which, in some choice of local coordinates ''z''<sub>1</sub>, ..., ''z<sub>n</sub>'', can be written as a [[harmonic function]] times <math>dz_{i_1} \wedge \cdots \wedge dz_{i_p} \wedge d\bar z_{j_1} \wedge \cdots \wedge d\bar z_{j_q}</math>. (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:
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| :<math>\cup : H^{p,q}(X) \times H^{p',q'}(X) \rightarrow H^{p+p',q+q'}(X).\,</math>
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| Since ''X'' is a compact oriented manifold, ''X'' has a [[fundamental class]].
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| 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'':
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| :<math>\int_Z i^*\alpha.\!\,</math>
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| To evaluate this integral, choose a point of ''Z'' and call it 0. Around 0, we can choose local coordinates ''z''<sub>1</sub>, ..., ''z<sub>n</sub>'' on ''X'' such that ''Z'' is just ''z''<sub>''k'' + 1</sub> = ... = ''z<sub>n</sub>'' = 0. If ''p'' > ''k'', then α must contain some ''dz<sub>i</sub>'' where ''z<sub>i</sub>'' pulls back to zero on ''Z''. The same is true if ''q'' > ''k''. Consequently, this integral is zero if (''p'', ''q'') ≠ (''k'', ''k'').
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| 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''<sup>2''n''</sup>(''X'', '''C''') = ''H<sup>n, n</sup>''(''X''), we conclude that [''Z''] must lie in ''H<sup>n-k, n-k</sup>''(''X'', '''C'''). Loosely speaking, the Hodge conjecture asks:
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| :''Which cohomology classes in ''H<sup>k, k</sup>''(''X'') come from complex subvarieties ''Z''?''
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| == Statement of the Hodge conjecture ==
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| Let:
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| :<math>\operatorname{Hdg}^k(X) = H^{2k}(X, \mathbf{Q}) \cap H^{k,k}(X).</math>
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| We call this the group of ''Hodge classes'' of degree 2''k'' on ''X''.
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| The modern statement of the Hodge conjecture is:
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| ::'''Hodge conjecture.''' Let ''X'' be a non-singular 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''.
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| 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 [[Algebraic geometry and analytic geometry#Chow.27s theorem|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 homogenous polynomials.
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| === Reformulation in terms of algebraic cycles ===
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| 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:
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| : <math>\sum_i c_iZ_i.\,</math>
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| 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:
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| :<math>\sum_i c_i[Z_i].\,</math>
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| Such a cohomology class is called ''algebraic''. With this notation, the Hodge conjecture becomes:
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| ::Let ''X'' be a projective complex manifold. Then every Hodge class on ''X'' is algebraic.
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| 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 {{Harvtxt|Zucker|1977}})
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| == Known cases of the Hodge conjecture ==
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| === Low dimension and codimension ===
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| The first result on the Hodge conjecture is due to {{Harvtxt|Lefschetz|1924}}. In fact, it predates the conjecture and provided some of Hodge's motivation.
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| ::'''Theorem''' ([[Lefschetz theorem on (1,1)-classes]]) Any element of ''H''<sup>2</sup>(''X'', '''Z''') ∩ ''H''<sup>1,1</sup>(''X'') is the cohomology class of a [[divisor (algebraic geometry)|divisor]] on ''X''. In particular, the Hodge conjecture is true for ''H''<sup>2</sup>.
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| 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 function (geometry)|normal function]]s, which were introduced by [[Henri Poincaré]]. However, [[Griffiths transversality theorem]] shows that this approach cannot prove the Hodge conjecture for higher codimensional subvarieties.
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| By the [[Hard Lefschetz theorem]], one can prove:
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| ::'''Theorem.''' If the Hodge conjecture holds for Hodge classes of degree ''p'', ''p'' < ''n'', then the Hodge conjecture holds for Hodge classes of degree 2''n'' − ''p''.
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| Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree 2''n'' − 2. This proves the Hodge conjecture when ''X'' has dimension at most three.
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| 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:
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| ::'''Corollary.''' If the algebra
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| :: <math>\operatorname{Hdg}^*(X) = \sum_k \operatorname{Hdg}^k(X)\,</math>
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| : is generated by Hdg<sup>1</sup>(''X''), then the Hodge conjecture holds for ''X''.
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| === Abelian varieties ===
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| For most [[abelian variety|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|date=April 2012}}. However, {{Harvtxt|Mumford|1969}} constructed an example of an abelian variety where Hdg<sup>2</sup>(''X'') is not generated by products of divisor classes. {{Harvtxt|Weil|1977}} generalized this example by showing that whenever the variety has [[complex multiplication]] by an [[imaginary quadratic field]], then <Hdg<sup>2</sup>(''X'') is not generated by products of divisor classes. {{Harvtxt|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.
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| == Generalizations ==
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| === The integral Hodge conjecture ===
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| Hodge's original conjecture was:
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| ::'''Integral Hodge conjecture.''' Let ''X'' be a projective complex manifold. Then every cohomology class in ''H''<sup>2''k''</sup>(''X'', '''Z''') ∩ ''H<sup>k, k</sup>''(''X'') is the cohomology class of an algebraic cycle with integral coefficients on ''X''.
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| This is now known to be false. The first counterexample was constructed by {{Harvtxt|Atiyah|Hirzebruch|1961}}. Using [[K-theory]], 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. {{Harvtxt|Totaro|1997}} reinterpreted their result in the framework of [[cobordism]] and found many examples of torsion classes.
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| The simplest adjustment of the integral Hodge conjecture is:
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| ::'''Integral Hodge conjecture modulo torsion.''' Let ''X'' be a projective complex manifold. Then every cohomology class in ''H''<sup>2''k''</sup>(''X'', '''Z''') ∩ ''H<sup>k,k</sup>''(''X'') is the sum of a torsion class and the cohomology class of an algebraic cycle with integral coefficients on ''X''.
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| Equivalently, after dividing ''H''<sup>2''k''</sup>(''X'', '''Z''') ∩ ''H<sup>k,k</sup>''(''X'') by torsion classes, every class is the image of the cohomology class of an integral algebraic cycle. This is also false. {{Harvtxt|Kollár|1992}} found an example of a Hodge class α which is not algebraic, but which has an integral multiple which is algebraic.
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| === The Hodge conjecture for Kähler varieties === | |
| A natural generalization of the Hodge conjecture would ask:
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| ::'''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''.
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| 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:
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| ::'''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''.
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| ::'''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''.
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| {{Harvtxt|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.
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| === The generalized Hodge conjecture ===
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| 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<sup>c</sup>'' ''H<sup>k</sup>''(''X'', '''Z''') satisfies
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| :<math>N^cH^k(X, \mathbf{Z}) \subseteq H^k(X, \mathbf{Z}) \cap (H^{k-c,c}(X) \oplus\cdots\oplus H^{c,k-c}(X)).</math>
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| Hodge's original statement was:
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| ::'''Generalized Hodge conjecture, Hodge's version.''' <math>N^cH^k(X, \mathbf{Z}) = H^k(X, \mathbf{Z}) \cap (H^{k-c,c}(X) \oplus\cdots\oplus H^{c,k-c}(X)).</math>
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| {{harvtxt|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:
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| ::'''Generalized Hodge conjecture.''' ''N<sup>c</sup>'' ''H<sup>k</sup>''(''X'', '''Q''') is the largest sub-Hodge structure of ''H<sup>k</sup>''(''X'', '''Z''') contained in <math>H^{k-c,c}(X) \oplus\cdots\oplus H^{c,k-c}(X).</math> | |
| This version is open.
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| == Algebraicity of Hodge loci ==
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| The strongest evidence in favor of the Hodge conjecture is the algebraicity result of {{Harvtxt|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.
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| ==See also==
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| *[[Tate conjecture]]
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| *[[Hodge theory]]
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| *[[Hodge structure]]
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| *[[period mapping]]
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| ==References==
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| *{{citation |last1=Atiyah |first1=M. F. |author1-link=Michael Atiyah |last2=Hirzebruch |first2=F. |author2-link=Friedrich Hirzebruch |year=1961 |title=Vector bundles and homogeneous spaces |journal=Proc. Sympos. Pure Math. |volume=3 |issue= |pages=7–38 |doi= }}
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| *{{Citation | last1=Cattani | first1=Eduardo |author1-link=Eduardo Cattani | last2=Deligne | first2=Pierre | author2-link=Pierre Deligne | last3=Kaplan | first3=Aroldo |author3-link=Aroldo Kaplan | title=On the locus of Hodge classes | mr=1273413 | year=1995 | journal=[[Journal of the American Mathematical Society]] | volume=8 | issue=2 | pages=483–506 | doi=10.2307/2152824 | jstor=2152824 }}.
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| *{{citation|last=Grothendieck|first=A.|authorlink=Alexander Grothendieck|title=Hodge's general conjecture is false for trivial reasons|journal=[[Topology (journal)|Topology]]|volume=8|year=1969|pages=299–303|doi=10.1016/0040-9383(69)90016-0|issue=3}}.
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| *{{citation|last=Hodge|first=W. V. D.|authorlink=W. V. D. Hodge|title=The topological invariants of algebraic varieties|journal=Proceedings of the International Congress of Mathematicians|publication-place=Cambridge, MA|year=1950|volume=1|pages=181–192}}.
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| *{{Citation |last=Kollár |first=János |authorlink=János Kollár |chapter=Trento examples |title=Classification of irregular varieties |page=134 |editor1-last=Ballico |editor1-first=E. |editor2-first=F. |editor2-last=Catanese |editor3-first=C. |editor3-last=Ciliberto |series=Lecture Notes in Math. |volume=1515 |location= |publisher=Springer |year=1992 |isbn=3-540-55295-2 }}.
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| *{{Citation | last1=Lefschetz | first1=Solomon |authorlink=Solomon Lefschetz | title=L'Analysis situs et la géométrie algébrique | publisher=Gauthier-Villars | language=French | series=Collection de Monographies publiée sous la Direction de M. Emile Borel | location=Paris | year=1924}} Reprinted in {{Citation | last1=Lefschetz | first1=Solomon | title=Selected papers | publisher=Chelsea Publishing Co. | location=New York | isbn=978-0-8284-0234-7 | mr=0299447 | year=1971}}.
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| *{{citation |last=Moonen |first=B. J. J. |author1-link=Ben Moonen |last2=Zarhin |first2=Yu. G. |author2-link=Yuri Zarhin |year=1999 |title=Hodge classes on abelian varieties of low dimension |journal=[[Mathematische Annalen]] |volume=315 |issue=4 |pages=711–733 |doi=10.1007/s002080050333 |arxiv=math/9901113 }}.
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| *{{citation |last=Mumford |first=D. |authorlink=David Mumford |title=A Note of Shimura's paper "Discontinuous groups and abelian varieties" |journal=[[Mathematische Annalen|Math. Ann.]] |volume=181 |issue=4 |year=1969 |pages=345–351 |doi=10.1007/BF01350672 }}.
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| *{{citation |last=Totaro |first=B. |authorlink=Burt Totaro |title=Torsion algebraic cycles and complex cobordism |journal=Journal of the American Mathematical Society |volume=10 |issue=2 |pages=467–493 |year=1997 |jstor=2152859 |arxiv=alg-geom/9609016 |doi=10.1090/S0894-0347-97-00232-4 }}.
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| *{{citation |last=Voisin |first=Claire |authorlink=Claire Voisin |year=2002 |title=A counterexample to the Hodge conjecture extended to Kähler varieties |journal=Int Math Res Notices |volume=2002 |issue=20 |pages=1057–1075 |doi=10.1155/S1073792802111135 }}.
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| *{{citation |last=Weil |first=A. |authorlink=André Weil |title=Abelian varieties and the Hodge ring |work=Collected papers |year=1977 |pages=421–429 |volume=III }}
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| *{{citation |last=Zucker |first=S. |title=The Hodge conjecture for cubic fourfolds |journal=Comp. Math |volume=34 |pages=199–209 |year=1977}} http://archive.numdam.org/ARCHIVE/CM/CM_1977__34_2/CM_1977__34_2_199_0/CM_1977__34_2_199_0.pdf
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| == External links ==
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| * [http://www.claymath.org/sites/default/files/hodge.pdf The Clay Math Institute Official Problem Description by P. Deligne (pdf)]
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| * 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)]
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| * [[Indranil Biswas]], Kapil Paranjape. [http://arxiv.org/abs/math/0007192v1 The Hodge Conjecture for general Prym varieties]
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| * [[Burt Totaro]], [http://burttotaro.wordpress.com/2012/03/18/why-believe-the-hodge-conjecture/ Why believe the Hodge Conjecture?]
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| * [[Claire Voisin]], [http://www.math.polytechnique.fr/~voisin/Articlesweb/hodgeloci.pdf Hodge loci]
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| [[Category:Homology theory]]
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| [[Category:Hodge theory]]
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| [[Category:Conjectures]]
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| [[Category:Millennium Prize Problems]]
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| [[Category:Algebraic geometry]]
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