I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the Riemannian case alone.
Could anyone point me in the right direction? I know some basic sheaf theory and sheaf cohomology, if that matters.
I am extremely skeptical that one could do Hodge theory on a semi-Riemannian manifold without extreme changes, because the Laplacian in signature $(p,q)$ is $\partial/\partial x_1^2 + \dots + \partial/\partial x_p^2 - \dots - \partial/\partial x_{p+q}^2$, which is not an elliptic operator, and ellipticity is completely crucial to the Hodge theory. This is probably not the only issue, but it's certainly first that comes to mind.
A great source for Hodge theory on Riemannian manifolds in the appropriate generality (elliptic complexes of differential operators on vector bundles) is Wells, Differential analysis on complex manifolds. A different approach (less general and different in flavor) can be found in both Griffiths and Harris and Warner; both are more or less the same proof.