  We show that elliptic complexes of (pseudo)differential operators on smooth
compact manifolds with boundary can always be complemented to a Fredholm
problem by boundary conditions involving global pseudodifferential projections
on the boundary (similarly as the spectral boundary conditions of Atiyah,
Patodi and Singer for a single operator). We prove that boundary conditions
without projections can be chosen if, and only if, the topological Atiyah-Bott
obstruction vanishes. These results make use of a Fredholm theory for complexes
of operators in algebras of generalized pseudodifferential operators of
Toeplitz type which we also develop in the present paper.
