Complete complexes and spectral sequences

By analogy with the classical (Chasles-Schubert-Semple-Tyrell) spaces of complete quadrics and complete collineations, we introduce the variety of complete complexes. Its points can be seen as equivalence classes of spectral sequences of a certain type. We prove that the set of such equivalence classes has a structure of a smooth projective variety. We show that it provides a desingularization, with normal crossings boundary, of the Buchsbaum-Eisenbud variety of complexes, i.e., a compactification of the union of its maximal strata.