Down-to-earth expositions of Hodge theory

Question

What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory?

Of course, since I haven't found a (for me) readable introduction, I don't know what I should learn about Hodge theory, but at least I'm looking for:

• reasons to expect that solutions to the laplacian (rather than other operators) should be studied

• examples where the harmonic forms can be explicitly written down

• how to think of harmonic forms in general

• the hodge diamonds (and how to compute them!) of the standard examples of projective varieties

So far I have read (parts of) the relevant parts of Griffiths-Harris and lecture notes on the web, but still don't how understand how to do computations with Hodge theory, or for which varieties I should expect to be able to use it.

Answer 1

The best explanation of Hodge theory I heard in my student years from my adviser Oleg Viro; I do not know who the author is.

(1) Consider a complex $(C^*,d)$ of Euclidean spaces. Then there is the adjoint differential $d^*$, and elementary linear algebra establishes canonical (given the metric) isomorphisms $H^*=\ker\Delta=\ker d\cap\ker d^*$, where $\Delta:=dd^*+d^*d$. Taking this to the infinite dimensional case of forms on a closed manifold is an analytic technicality.

(2) On any complex manifold there is a canonical $(p,q)$-decomposition of forms. hence a filtration on the cohomology (which does not necessarily induce a grading). However, if the metric is K\"ahler, then $\Delta=2\square=2\bar\square$ (college math); hence, harmonic forms respect the grading and we get a grading in the cohomology (which is determined by the filtration, hence is canonical).

(3) A bit more work ($dd^*+d^*d$, a bit of linear algebra again, and the explicit definition of the higher differentials) shows that the spectral sequence sequence associated with the Hodge filtration collapses. Altogether, this is Hodge theory.

Answer 2

I thought I'd add a few words to answer the questions "reasons to expect that solutions to the laplacian/how to think of harmonic forms". I tend to think of the Hodge theorem as an infinite dimensional version of the least squares. De Rham's theorem tells us that the cohomology of a manifold can be identified with the space of closed forms modulo exact forms. But this raises the question how do we choose a good a representative for cohomology class? Suppose our manifold is compact and orientable, and that we have chosen a Riemanninan metric which allows us to measure size. Then the natural representative would the one that minimizes the norm. This is precisely the harmonic representative. Here is the explanation. If $\Delta\alpha=0$, where $\Delta = dd^*+d^*d$, then $$||d\alpha||^2+||d^\ast\alpha||^2=\langle\Delta\alpha,\alpha\rangle=0$$ so $\alpha$ is closed and co-closed (and conversely such forms are harmonic). Therefore $\alpha$ determines a cohomology class. Any other form in the class is given by $\alpha+d\beta$. We have $$||\alpha+d\beta||^2= ||\alpha||^2+||d\beta||^2+\langle d^*\alpha,\beta\rangle= ||\alpha||^2+||d\beta||^2\ge ||\alpha||^2$$ So indeed, $\alpha$ has the smallest norm in its class.

For applications to algebraic geometry one needs the harmonic theory to play well with the complex geometry (e.g. holomorphic forms should be harmonic). For this, we need a Kähler metric as described in Alex Degtyarev's answer.

Answer 3

You might like Period Mappings and Period Domains, by Carlson, Muller-Stach, and Peters. This book contains a lot of really nice examples, and there are even exercises at the end of many sections.

Answer 4

I am a great fan of Donu Arapura's answers on mathoverflow. He has a very nice book called "algebraic geometry over the complex numbers" which discusses a small fraction of Hodge theory.

Answer 5

If I recall correctly, a fairly readable and straightforward introduction to the basics of Hodge theory (in the differential-geometric setting) can be found in Chapter 6 aptly entitled "Hodge theorem" of the book F. Warner, Foundations of Differentiable Manifolds and Lie Groups.

Answer 6

The book of Kulikov/Kurchanov is very nice (the translation is especially good :))

MR1060327 (91k:14010) Reviewed Kulikov, Vik. S.; Kurchanov, P. V. Complex algebraic manifolds: periods of integrals, Hodge structures. (Russian) Current problems in mathematics. Fundamental directions, Vol. 36 (Russian), 5–231, 280, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989. 14C30 (14D07 32G20 32J25)