Complexification of the CR cross-cap

Adam Coffman

unpublished lecture notes

Some elementary geometric and algebraic properties of the CR cross-cap
are considered.  This is the 4-dimensional real algebraic variety in
complex 5-space that is a formally stable normal form for generic CR
singularities.  It is shown to be totally real except at one point,
its complexification is calculated, and an interesting multi-valued
reflection is constructed.  Some higher-dimensional and degenerate
cases of the normal form are also exhibited.