TITLE: The prerequisites for Deligne's Théorie de Hodge I, II, III
QUESTION [23 upvotes]: I am an undergraduate student. I am not sure if it's OK to ask this question here.  
I want to learn Hodge theory. But I do not know how to start it, and how much mathematics I should need before I read Deligne's paper.
Is there an elementary book or note on Hodge theory for undergraduate students? Is it worthy to read Hodge's book The theory and applications of harmonic integrals?

REPLY [14 votes]: I might be repeating what others have said, but I think the question is whether you want to learn classical Hodge theory or mixed Hodge theory. Deligne's papers are about the latter. In principle it's very possible to learn one of them and not much of the other - a complex analyst can make use of classical Hodge theory without ever caring about weight filtrations and so on, and an algebraic geometer can use mixed Hodge theory as a tool without learning the transcendental parts that go into the construction. (If you are interested in computing the cohomology of algebraic varieties then mixed Hodge theory is an amazingly powerful tool.)
In any case, if it really is mixed Hodge theory you want to learn, then Deligne's papers are not a bad start. There's also the book of Peters and Steenbrink, Deligne's ICM address "Poids dans la cohomologie des variétés algébriques" and Brylinski and Zucker's "An overview of recent advances in Hodge theory". But you should be aware that none of these papers are easy to read! They will assume knowledge of algebraic geometry and a large amount of homological algebra (derived categories, simplicial techniques, et cetera). To make things worse, a large part of the motivation for the theory comes from étale cohomology and the Weil conjectures. 
A better way into mixed Hodge theory might be to learn some applications first and figure out why it's useful. For instance, you could learn about the Hodge-Deligne polynomial of an algebraic variety. In simple cases you can think very concretely about the Hodge-Deligne polynomial as a gadget that counts the numbers of points of your variety over various finite fields. Once you have understood how one can use it as a tool and why it's absolutely amazing that a Hodge-Deligne polynomial even exists, you can read about mixed Hodge theory to understand why it's well defined.