Formal stability of the CR cross-cap

abstract

For m<n, any real analytic m-submanifold of complex n-space with a
non-degenerate CR singularity is shown to be locally equivalent, under
a formal holomorphic coordinate change, to a fixed real algebraic
variety defined by linear and quadratic polynomials.  The situation is
analogous to Whitney's stability theorem for cross-cap singularities
of smooth maps.  The formal coordinate change is obtained from a
sequence of approximations whose norms are controlled only on
polydiscs shrinking to a point, so the analytic classification remains
an open question.