Formally Integrable Structures I. Resolution of Solution Sheaves

This is the first of a series of papers on the $L^2$-theory for formally integrable structures. It is devoted to constructing a resolution of the solution sheaf for a class of overdetermined systems introduced by L. H{\"o}rmander. A sufficient condition for global exactness is obtained, which leads to gluing techniques for local solutions formulated as Cousin type problems. In addition, we also prove the local solvability of the Treves complex for formally integrable structures with vanishing Levi forms, including Levi flat structures as special cases. To the best of the authors' knowledge, nothing more than the elliptic case is known about the local $L^2$-solvability of the Treves complex in the Levi flat case.