Applications of Hodge theory to topology and analysis

Question

I am going to give a talk for the PhD students' seminar at my university. The audience is composed mainly by algebraic topologists, algebraic geometers and analysts. I have decided that I'm going to talk about Hodge theory as it has links with all of these fields, starting from the basics and going up to use it to show that deRham cohomology on compact manifolds is finite dimensional. If possible, I would like to give other interesting (and, if possible, not too hard) examples of applications. Does any of you know some?

Answer 1

Here's one of my favorites.

First, remember (part of) the more general version of the Hodge theorem: if $E_i$ are a sequence of vector bundles on a compact smooth manifold, and $0 \to \Gamma(E_0) \xrightarrow{d_0} \Gamma(E_1) \cdots \xrightarrow{d_n} \Gamma(E_n) \to 0$ is an elliptic complex ($d^2 = 0$ and taking symbols gives an exact sequence), then the homology groups of this complex are finite-dimensional. The standard example of an elliptic complex is $\Omega^*(M)$ with $d$.

But take $M$ to be a compact complex manifold and $E_n = \Omega^{0,n}(TM)$ with $\overline \partial$ to be the Dolbeaut complex with coefficients in $TM$. This is indeed an elliptic complex, and the theorem above implies that $\mathcal H_0$ - the space of holomorphic vector fields - is finite dimensional.

Now note that the Lie algebra of $\text{Aut}(M)$, the group of biholomorphisms of $M$, is the space of holomorphic vector fields. We've just proved that $\text{Aut}(M)$ is a finite-dimensional Lie group. This is in stark contrast to the case of $M$ noncompact; every nonvanishing holomorphic function $f$ defines a biholomorphism of $\Bbb C^2$ by $(z,w) \mapsto (z,f(z)w)$, so $\text{Aut}(\Bbb C^2)$ is very infinite-dimensional.

Answer 2

Connect Hodge theory to Algebraic geometry, I think Lefschetz theorem on (1,1)-classes would be interesting, and a generalized one Hodge conjecture.

Kodaira imbedding theorem, Kodaira Vanishing theory may also be relevant.