4 Hodge Theory
We now arrive at the main theorem we were working towards.
(Hodge Decomposition Theorem)
Let be an elliptic complex with and as before. Then
, and is an isomorphism.
(1) follows from regularity. (2) follows from the identities
For (3), we can also decompose . Since , they must be equal. Moreover,
but clearly since
So we must have equality throughout, and is clearly orthogonal to . For (4), it is clear that , and since , that must be an equality.
Let be a closed Riemannian manifold and a Hermitian vector bundle. Let be formally self-adjoint of order . Then we have an orthogonal decomposition
Moreover, each is finite-dimensional, and for any , there are only finitely many eigenvalues of magnitude .
The idea is to apply the spectral theorem for compact self-adjoint operators to the inverse of
. Of course,
need not be invertible. So we do the following:
Consider the operator
. It is then clear that
is elliptic and injective. So
is invertible (since the complement of the image is
), with inverse
induces a bijection between the smooth sections, so does
be the composition
. Then this is compact and self-adjoint (can check this for smooth sections, and use that its “inverse”
is formally self adjoint).
By the spectral theorem of compact self–adjoint operators (and positivity of ),
Moreover, each factor is finite-dimensional, and is the only accumulation point of the spectrum.
We will show that decomposes as a sum of eigenspaces for . We first establish that
Since is self-adjoint, the second equality follows by elliptic regularity. The first equality follows from the computation
plus the density of and surjectivity of .
Now since commutes with , we know acts as a self-adjoint operator on the finite-dimensional vector space . Moreover, restricted to this subspace, we have
So by linear algebra, decomposes into eigenspaces of of eigenvalues , and the theorem follows.