Flexibility properties in Complex Analysis and Affine Algebraic Geometry

TL;DR

This paper reviews the concepts of flexibility, density, and Oka-Forstneri properties in complex analysis and affine algebraic geometry, highlighting their implications and applications in understanding automorphism groups of complex manifolds.

Contribution

It provides a comprehensive overview of the interrelations and significance of these properties, emphasizing recent developments and their impact on solving complex geometric problems.

Findings

Identification of key properties: density, flexibility, Oka-Forstneri.

Explanation of implications between properties.

Application examples solving complex geometric problems.

Abstract

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka-Forstneri\v{c} manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930's, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview article we present 3 classes of properties: 1. density property, 2. flexibility 3. Oka-Forstneri\v{c}. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.