4 Hodge Theory
We now arrive at the main theorem we were working towards.
Let be an elliptic complex with and as before. Then
, and is an isomorphism.
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 .
Consider the operator . It is then clear that is elliptic and injective. So is invertible (since the complement of the image is ), with inverse . Since induces a bijection between the smooth sections, so does . Let 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.□