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*?



The prerequisites for Deligne's Théorie de Hodge I, II, III
I would recommend Voisin's "Hodge Theory and Complex Algebraic Geometry" as an introduction to Hodge theory--Volume I should suffice for your purposes. The book also does a bit of the Hodge theory of non-compact varieties, in Section 8.4. The relevant sections in Griffiths-Harris aren't bad either. Deligne's papers also require some familiarity with spectral sequences and homological algebra, but this can be obtained in tons of places.


That said, the actual *proofs* of the Hodge decomposition, etc. are not so relevant for understanding Deligne's work--in my opinion, it would be a much better use of your time to understand the statements and compute some examples if this is really your goal. For example, I found Hodge theory for smooth projective curves (which works over an arbitrary base!) very illuminating--see e.g. Theorem 2 in [these notes](http://math.stanford.edu/%7Edlitt/exposnotes/vhs.pdf) (which have relatively heavy homological algebra preliminaries). Once you have a good understanding of the statements of Hodge theory for compact Kahler manifolds, you can try using the formal functoriality properties of mixed Hodge structures to compute some examples for some open varieties (by finding nice compactifications) and singular varieties (by finding nice resolutions).


From your question, you seem to view Deligne's papers as the apotheosis of Hodge theory--I would argue rather that you should focus your efforts on understanding the case of compact Kahler manifolds, and wait until you have a good feel for them before moving on to the general situations with which Deligne deals.


If you are more algebraically than analytically inclined, the [Deligne-Illusie paper](https://publications.ias.edu/node/403) establishing the degeneration of the Hodge-to-de Rham spectral sequence is quite beautiful and very readable--two more readable accounts are [these notes](http://math.berkeley.edu/%7Eachinger/notes/char-p-5.pdf) of Piotr Achinger, and [this account](http://en.bookfi.org/book/508940) by Illusie. You may also find [the comparison theorem](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1966__29_/PMIHES_1966__29__95_0/PMIHES_1966__29__95_0.pdf) between analytic and algebraic de Rham cohomology interesting.


