Hodge decomposition for non-elliptic complexes

Question

It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/Hermitian metrics on the vector bundles of a differential complex, that might be non-elliptic. Is this still true if we discard the ellipticity?

I am interested in particular in knowing:

• If it is false, true or unknown the bijective correspondence for general differential complexes on compact manifolds.
• If it is false, a counterexample would be appreciated.
• If it is true, a reference to the proof would be appreciated.

I have on my mind the following basic example in which this seems to work. Consider any manifold $X$ and any complex of vector bundles where the differential morphisms $d_i$ are all 0. In this case it is trivial that this holds. In this case we do not need compactness either.

For a less trivial example we can consider the torus $T=S^1\times S^1$. Let $d$ be the exterior derivative on $S^1$. Then define a differential operator $d':C^\infty(T)\to \Omega^1(T)$ given by $d'(f)(x,y)={\pi_1}^* d(f|_{S^1\times\{y\}} )$ where ${\pi_1} :T \to S^1$ is the projection on the first coordinate. The adjoint of $d'$ is $d^* (fdx+gdy)=\frac{\partial f}{\partial x}$ where $dx={\pi_1}^* (ds)$, $dy={\pi_2}^* (ds)$, $\frac{\partial}{\partial x}$ is the vector field given by $dx(\frac{\partial}{\partial x})=1$ and $ds$ is any differential form on $S^1$ that does not vanish. In this case we would have Laplacians $\Delta_0=d^* d'$ and $\Delta_1=d'd^* $. It is clear that $\Delta_0(f)=\frac{\partial^2 f}{\partial x^2}$ and $\Delta_1(fdx+gdy)=\frac{\partial^2 f}{\partial x^2}dx$. Thus, the kernel of $\Delta_0$ equals the kernel of $d'$ and the kernel of $\Delta_1$ is also $\Omega^1(T)/ d'(C^\infty(T))$.

Answer 1

This is false. Equip $T^2 = S^1 \times S^1$ with the Lorentzian metric $g = -ds^2 + dt^2$. For concreteness, regard $S^1 = \mathbb{R} / \mathbb{Z}$. Consider the de Rham complex. The Hodge Laplacian on functions is now the wave operator $\Box = -\frac{\partial^2}{\partial s^2} + \frac{\partial^2}{\partial t^2}$. Its kernel is infinite-dimensional (e.g. $\cos(2\pi ks)\cos(2\pi kt)$ is in the kernel for all nonnegative integers $k$), so is very far from being isomorphic to $H^0(T^2) = \mathbb{R}$.

There are situations where one does have a Hodge isomorphism theorem for a non-elliptic complex. For example, this is true when the complex is only subelliptic, such as in the Rumin complex. More strikingly, it is also true for the Kohn—Rossi complex, even though some of the operators in this complex are not even hypoelliptic.