Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative: $$0 \to \mathcal{E} \xrightarrow{d^\nabla} \Omega^1_M \otimes \mathcal{E} \xrightarrow{d^\nabla} \Omega^2_M \otimes \mathcal{E} \xrightarrow{d^\nabla} \cdots$$ Here, I am using $\mathcal{E}$ to denote the sheaf of smooth sections of $E$. We say that the connection $\nabla$ is flat if $(d^\nabla)^2 = 0$, and in this case we get an actual complex. In this situation, the sheaf $$\mathcal{L} := \mathrm{Ker}\left(\mathcal{E} \xrightarrow{d^\nabla} \Omega^1_M \otimes \mathcal{E} \right) $$ of parallel sections is a local system and we can use this de Rham complex (a soft resolution) to compute its sheaf cohomology and cohomology with compact support.
But what if the connection is not flat? We obviously no longer have a complex, but the differential operators are sheaf homomorphisms, so we still have a sequence of sheaves, and their kernels and images are not necessarily acyclic.
Can we use sheaf cohomology or D-module theory to say anything interesting in this case?
