Complexification of the CR Cross-Cap

These notes are based on research conducted at the University of Chicago. They represent a more self-contained version of a chapter from my dissertation [C1]. [Note added in 2006: Some of these ideas more recently appeared in [C2], [C3], and [C4].] The main result of [C1] established a normal form for a generic complex tangent of a real m-submanifold in a complex n-manifold, in the case m < n. “Generic” refers to a point where the real tangent space contains exactly one complex line, and so that the defining functions satisfy certain non-degeneracy conditions. The “formal stability” theorem says that for a generic complex tangent of a submanifold defined by real analytic functions in a neighborhood of the origin of C, there exists a formal holomorphic coordinate change transforming the submanifold into a fixed real algebraic variety M. This variety depends only on m and n (there are no continuous invariants, unlike the m = n case) and its defining functions are linear and quadratic polynomials. In these notes, the geometry of these real varieties and their complexifications is considered. There is some choice in the algebraic normal form. The following two are formally equivalent equations for M in C; these dimensions are the smallest where the non-degeneracy conditions are satisfied. The non-degenerate normal form