22
$\begingroup$

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?

$\endgroup$
29
$\begingroup$

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 (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 establishing the degeneration of the Hodge-to-de Rham spectral sequence is quite beautiful and very readable--two more readable accounts are these notes of Piotr Achinger, and this account by Illusie. You may also find the comparison theorem between analytic and algebraic de Rham cohomology interesting.

$\endgroup$
13
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ +1: This is a great answer. But surprisingly, one doesn't actually need mixed Hodge theory to prove the well-definedness of the Hodge-Deligne polynomial! You can deduce it from resolution of singularities and weak factorization. $\endgroup$ – Daniel Litt Sep 4 '13 at 14:34
10
$\begingroup$

Deligne's papers are probably not the introduction to classical Hodge theory you are looking for.

I would recommend reading "Hodge Theory and Complex Algebraic Geometry I" by Voisin. It covers the algebraic and complex analytic aspects of Hodge Theory, including sketches of parts of Hodge II. However, it skips the main analytic input of Hodge Theory regarding elliptic operators.

Prereqs: a first course in both differential manifolds and complex analysis first, along with the basic ideas of algebraic geometry.

"Principle of Algebraic Geometry" by Griffiths and Harris would be another good reference with similar prerequisites, although it doesn't cover more advanced topics such as mixed Hodge Theory.

"Period Mappings and Period Domains" by Carlson, Muller-Stach, Peters, might be an easier read - it has many exercises, and seems to have a greater emphasis on intuition and examples. It might skip some of the proofs covered in the above. I have not read it, though, so cannot vouch for it.

$\endgroup$
  • $\begingroup$ In principle, the Carlson, Muller-Stach, and Peters books is somewhat self-contained; that said, it basically assumes some familiarity with the basic material, in my opinion. $\endgroup$ – Daniel Litt Sep 4 '13 at 1:12
  • $\begingroup$ I am not sure if Deligne's papers prove anything for general closed Kaehler manifolds (and for most people, Hodge theory is probably a set of some statements about such objects). So "Deligne's papers are probably not the introduction to classical Hodge theory you are looking for" seems to be very true---they do not prove what is being asked for (most likely). $\endgroup$ – user138661 May 13 at 16:15
9
$\begingroup$

One of the best books on this subject is the book of Voisin "Hodge Theory and Complex Algebraic Geometry". But it maynot sometimes be as self contained as you need. You may find the book by Carlson, Müller-Stach, Peters, "Period mappings and Period domains" more readable. A less famous resource is the book by Bertin, Demailley, Illusie and Peters, "Introduction to Hodge theory". It is a good reference and contains many informations from basic to higher levels and from complex and $L^{2}$-methods to characteristic $p$ ones. By the way, now there are many online courses and lecture notes about Hodge theory which you can easily find by googling. For example this one, which is short and elementary or this . The first chapters of Griffthis-Harris book "principles of algebraic geometry" are a good introduction to the complex bases of Hodge theory.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.