Prerequisites for understanding the Hodge conjecture

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.

Solution 1:

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:

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

Solution 2:

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.