Obstructions to generic embeddings

TL;DR

This paper generalizes Laufer's theorem by analyzing Whitney cohomology and cohomology with supports, providing a necessary condition for the global embedding of CR manifolds into Stein manifolds based on cohomology dimensions.

Contribution

It extends Laufer's zero or infinity law to CR manifolds using Whitney cohomology, offering new insights into obstructions to generic embeddings.

Findings

Cohomology groups must be zero or infinite dimensional for embedding

Generalization of Laufer's theorem to CR manifolds

Provides a necessary condition for Stein manifold embeddings

Abstract

In Grauert's paper [G] it is noted that finite dimensionality of cohomology groups sometimes implies vanishing of these cohomomogy groups. Later on Laufer formulated a zero or infinity law for the cohomology groups of domains in Stein manifolds. In this paper we generalize Laufer's Theorem in [L] and its version for small domains of CR manifolds, proved in [Br], by considering Whitney cohomology on locally closed subsets and cohomology with supports for currents. With this approach we obtain a global result for CR manifolds generically embedded in a Stein manifold. Namely a necessary condition for global embedding into an open subset of a Stein manifold is that the de-bar-M-cohomology groups must be either zero or infinite dimensional.