On the present state of the Andersen-Lempert theory

TL;DR

This survey reviews the current state of Andersen-Lempert theory focusing on the density property of Stein manifolds, introduces new results on product manifolds and algebraic surfaces, and discusses recent techniques used in the field.

Contribution

The paper provides two new results: the preservation of the volume density property under products of Stein manifolds and an example of an algebraic surface lacking the algebraic density property.

Findings

Product of Stein manifolds with volume density property also has it.

Existence of an algebraic surface without algebraic density property.

Application of Brunella's technique in the proof.

Abstract

In this survey of the Andersen-Lempert theory we present the state of the art in the study of the density property (which means that the Lie algebra generated by completely integrable holomorphic vector fields on a given Stein manifold is dense in the space of all holomorphic vector fields). There are also two new results in the paper one of which is the theorem stating that the product of Stein manifolds with the volume density property possesses such a property as well. The second one is a meaningful example of an algebraic surface without the algebraic density property. The proof of the last fact requires Brunella's technique.