Local Vanishing of Characteristic Cohomology

Introduction. To a smooth manifold M one can associate in a natural way a new smooth manifold, the manifold of k-jets of n-dimensional submanifolds of M , indicated by G n (M), which parametrizes in a smooth way the k-jets of immersed submanifolds ofM . OnG n (M) one can build in a canonical way a differential ideal, denoted . The cohomology associated to the complexG n (M)/ (k) is called characteristic cohomology. These ideas, which in part go back to [C], are explained here. A more detailed introduction to them can be read, for example, in the introduction to [BG1] or in [BGH1] (see also [BG2], [BGH2], [BGH3]). Characteristic cohomology appears in this picture as a cohomological tool to study n-dimensional submanifolds of M . In this context one should think of submanifolds as solutions to systems of PDEs (partial differential equations). For example, they could be integral manifolds of an integrable distribution or of a differential ideal. Characteristic cohomology (or a variation of it) can then be used to provide invariants for the system of PDEs. The notation for the qth characteristic cohomology group over a smooth manifold M is