CR Manifold - Introduction and Motivation

Introduction and Motivation

The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface.

Suppose for instance that M is the hypersurface of C2 given by the equation

where z and w are the usual complex coordinates on C2. The holomorphic tangent bundle of C2 consists of all linear combinations of the vectors

The distribution L on M consists of all combinations of these vectors which are tangent to M. In detail, the tangent vectors must annihilate the defining equation for M, so L consists of complex scalar multiples of

Note that L gives a CR structure on M, for = 0 (since L is one-dimensional) and since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.

More generally, suppose that M is a real hypersurface in Cn, with defining equation F(z₁, ..., z_n) = 0. Then the CR structure L consists of those linear combinations of the basic holomorphic vectors on Cn:

which annihilate the defining function. In this case, for the same reason as before. Moreover, ⊂ L since the commutator of vector fields annihilating F is again a vector field annihilating F.