  A generic compact real codimension two submanifold X of C^(n+2) will have a
CR structure at all but a finite number of points (failing at the complex jump
points J). The main theorem of this paper gives a method of extending the CR
structure on the non-jump points X-J to the jump points. We examine a Gauss map
from X-J to an appropriate flag manifold F and take the closure of the graph of
this map in X x F. This is a version of a Nash blow-up. We give a clean
criterion for when this closure is a smooth manifold and see that the local
differential properties at the points X-J can now be naturally extended to this
new smooth manifold, allowing global techniques from differential geometry to
be applied to compact CR manifolds. As an example, we find topological
obstructions for the manifold to be Levi nondegenerate.
