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The Hodge conjecture is a major open mathematical problem that states that on a complex manifold $X$ and its respective Hodge classes, defined as

$Hdg^k(X)= H^{2k}(X,\mathbb{Q})\cap H^{k,k}(X)$

that the Hodge classes are linear combinations with rational coefficients of the cohomology classes of complex subvarieties of $X$.

What are the prequisities for understanding this conjecture?

All answers are greatly appreciated.

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What do you mean by understanding the conjecture? You've stated it. Do you know what the words you've used mean? I think taking a look at the Lefschetz theorem on $(1, 1)$ classes would be a good move (it is the case $k = 1$). –  Michael Albanese Oct 6 '12 at 14:20
    
Pretty much but I wanted to get a more detailed understanding of it. They are rational cohomology classes of those varieties. A deeper understanding always helps, although I should have stated my question differently. –  Jaivir Baweja Oct 6 '12 at 22:24
    
@Jaivir Baweja: Can you specify your question. If you are a specialist, then your question has no sense. If you are a beginer, then I am not sure that narraw time is suffucient to understood this matter. The conjecture is very complicated if you want to make an approach even for understanding it. –  ZE1 Dec 26 '12 at 10:04
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2 Answers 2

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+200

Your question is very large. I am not aware about your background, but you need to begin with the following items:

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:10.1016/0040-9383(69)90016-0.

Hodge, W. V. D. (1950), "The topological invariants of algebraic varieties", Proceedings of the International Congress of Mathematicians (Cambridge, MA) 1: 181–192.

Moonen, B. J. J.; Zarhin, Yu. G. (1999), "Hodge classes on abelian varieties of low dimension", Mathematische Annalen 315 (4): 711–733.

Voisin, Claire (2002), "A counterexample to the Hodge conjecture extended to Kähler varieties", Int Math Res Notices 2002 (20): 1057–1075.

See also these links for more details about the references:

http://www.math.jussieu.fr/~voisin/Articlesweb/takagifinal.pdf

http://www.claymath.org/millennium/Hodge_Conjecture/hodge.pdf

This link is also very important:

http://mathoverflow.net/questions/54197/why-is-the-hodge-conjecture-so-important

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Although Dan Freed's lecture remains a classic introduction, one cannot go wrong with the video series here too.

One needs to familiarize wit basic geometry, calculus, algebraic topology, differential topology, algebraic geometry, Bezout's theorem and Hodge theory in general.

Here is also a glimpse from Kevin Devlin's introduction to the Millenium problems.

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