Application of Hodge decomposition

Question

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?

Answer 1

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.)

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Some other example applications:

• Lefschetz decomposition

• Torelli theorems

• Many more, but I'm going to post this now

Answer 2

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 $H^k(M;\mathbb R) \simeq H^{n-k}(M;\mathbb R)$.

Answer 3

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