The Equivalence Problem for Higher-codimensional Cr Structures

The equivalence problem for CR structures can be viewed as a special case of the equivalence problem for G-structure. This paper uses Cartan’s methods (in modernized form) to show that a CR manifold of codimension 3 or greater with suitably generic Levi form admits a canonical connection on a reduced structure bundle whose group is isomorphic to the multiplicative group of complex numbers. As corollaries, it follows that the CR manifold admits a canonical affine connection, and consequently that the automorphisms of the CR manifold constitute a Lie group. The most difficult technical step is to construct a smooth moduli space for generic vector-valued hermitian forms, which is tied to the CR manifold via the Levi map. The techniques used to construct this space are drawn from the classical invariant theory of complex projective hypersurfaces.