On complex highly regular embeddings and the extended Vassiliev conjecture

TL;DR

This paper extends the theory of complex k-regular embeddings by providing new lower bounds for their existence, using advanced topological and cohomological methods, and also explores related l-skew embeddings.

Contribution

It introduces new lower bounds for complex k-regular embeddings and extends existing real embedding results to the complex setting using Chern classes and cohomology analysis.

Findings

New lower bounds for complex k-regular embeddings.

Extension of real embedding results to complex case.

Upper bounds for cohomology height of configuration spaces.

Abstract

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.