  A continuous map C^d -> C^N is a complex k-regular embedding if any k
pairwise distinct points in C^d are mapped by f into k complex linearly
independent vectors in C^N. Our central result on complex k-regular embeddings
extends results of Cohen & Handel (1978), Chisholm (1979) and Blagojevic,
L\"uck & Ziegler (2013) on real k-regular embeddings: We give new lower bounds
for the existence of complex k-regular embeddings. These are obtained by
modifying the framework of Cohen & Handel (1978) and a study of Chern classes
of complex regular representations. The main technical result, used for the
study of the Chern classes, is an upper bound for the height of the cohomology
of an unordered configuration space Furthermore, we give similar lower bounds
for the existence of complex l-skew embeddings C^d -> C^N, for which we require
that the images of the tangent spaces at any l distinct points are skew complex
affine subspaces of C^N.
