Hodge decompositions for Lie algebroids on manifolds with boundary

We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that the Chevalley-Eilenberg complex of an elliptic, q-convex Lie algebroid admits a Hodge decomposition in degree q. This generalizes the well-known Hodge decompositions for the exterior derivative on real manifolds and the delbar-operator on q-convex complex manifolds. We establish the results in a more general setting, where the differential does not necessarily square to zero and moreover varies in a family, including an analysis of the behaviour on the deformation parameter. As application we give a proof of a classical holomorphic tubular neighbourhood theorem (which implies the Newlander-Nirenberg theorem) based on the Moser trick, and we provide a finite-dimensionality result for certain holomorphic Poisson cohomology groups.