Analytic stability of the CR cross-cap

article abstract

For m<n, any real analytic m-submanifold of complex n-space with a
nondegenerate CR singularity is shown to be locally equivalent, under
a 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 complex analyticity of the normalizing
transformation is proved using a rapid convergence argument.