Formal Stability of the Cr Cross-cap

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 xed real algebraic variety de ned 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 classi cation remains an open question.