Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Hodge decomposition states any $p$ form can be decomposed into three orthogonal $L^2$ components: exact form, co-exact form and hamonic form. But actually we don't know how to decompose a general one. So how to apply the decomposition to other fields? Even if we can decompose one, what does it imply?

share|improve this question
2  
Dear zhangwfjh, What do you mean "other fields"? As for applications of Hodge theory, it is a very basic tool; your question is almost as general as asking for applicaions of de Rham theory, which is to say --- it is pretty broad. Do you know about the Lefschetz decomposition for the cohomology of smooth projecive varieties (which is one basic example)? Regards, –  Matt E Dec 18 '13 at 1:59
2  
@MattE I mean some "fields" that connect to physics, chemistry, biology etc, other than algebraic topology. I think it must be important to hodge theory but before I get into that, I want a more intuitive feeling about that. I don't know Leftschetz decomposition you mentioned. Thank for your comment. I would aprreciate it even if a very tiny list of applications is given. –  zhangwfjh Dec 18 '13 at 2:05

3 Answers 3

If $X$ is a smooth projective variety over $\mathbb C$ (equivalently, a closed complex submanifold of $\mathbb C P^n$), then the cohomology of $X$ (with $\mathbb C$ coeficients) has its Hodge decomposition: $H^n(X, \mathbb C) = \oplus_{p+ q = n} H^{p,q}$; here $H^{p,q}$ consists of class that can be represented by harmonic $(p,q)$-forms.

(The decomposition arises by combining Hodge theory as described in the OP with extra stucture induced by the fact that $X$ is a Kahler manifold.)

One has Hodge symmetry: complex conjugation interchanges $H^{p,q}$ and $H^{q,p}$, and this implies that they have the same dimension.

The Hodge decomposition and Hodge symmery together imply, for example, that if $n$ is odd then the dimension of $H^n(X,\mathbb C)$ is even. This is a major topological constraint on the topology of complex projective varieties. E.g it implies that the Hopf surface $(\mathbb C^2 \setminus \{0\})/ 2^{\mathbb Z}$ (here $2^{\mathbb Z}$ acts by scalar multiplication), which is a compact complex manifold, can't be embedded into projective space. (Its $H^1$ is one-dimensional.)


Some other example applications:

share|improve this answer
    
I wish I could accept this. But I don't touch anything on algebraic geometry now. –  zhangwfjh Dec 18 '13 at 14:23

I'm not sure this is the sort of thing you're looking for but a nice and quick application is a proof of Poincare duality: On an $n$-dimensional manifold $M$, the Hodge star operator, *, gives an isomorphism between harmonic $k$ forms and $n-k$ forms. This is because a form $\alpha$ is harmonic if and only if $d \alpha = 0$ and $*d*\alpha = 0$. Thus $H^k(M;\mathbb R) \simeq H^{n-k}(M;\mathbb R)$.

share|improve this answer

For the interplay of Hodge theory and physics, see various papers of Cantarella, deTurck, and Gluck.

share|improve this answer
    
Oh, I love those papers. Thank you very much. Glad to see relevant ones if you know more. –  zhangwfjh Dec 18 '13 at 14:28

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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