Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

For curved projective manifolds, we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalize the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geometric contact with the model to a level that is optimal, in a suitable sense. In terms of the trivializations arising from the special frames, normal solutions of classes of natural linear partial differential equation (so-called first Bernstein-Gelfand-Gelfand equations) are shown to be necessarily polynomial in the generalized homogeneous coordinates; the polynomial system is the pull-back of a polynomial system that solves the corresponding problem on the model. Thus, questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of the corresponding polynomial systems and algebraic sets. We show that a normal solution determines a canonical manifold stratification that reflects an orbit decomposition of the model. Applications include the construction of structures that are analogues of Poincare-Einstein manifolds.