[Paper Review] Manifolds with infinite dimensional group of holomorphic automorphisms and the Linearization Problem

Frank Kutzschebauch|arXiv (Cornell University)|Mar 3, 2019

Holomorphic and Operator Theory48 references4 citations

TL;DR

This paper investigates the structure of holomorphic automorphism groups on complex manifolds, particularly focusing on those with infinite-dimensional automorphism groups, such as $\mathbb{C}^n$. It introduces and analyzes the density and flexibility properties of these groups, applies them to solve geometric problems, and provides new criteria for the holomorphic linearization problem, showing that certain $G$-actions on Stein manifolds are biholomorphic to linear actions under specific conditions on their quotient spaces and isotropy groups.

ABSTRACT

We overview a number of precise notions for a holomorphic automorphism group to be big together with their implications, in particular we give an exposition of the notions of flexibility and of density property. These studies have their origin in the famous result of Andersén and Lempert from 1992 proving that the overshears generate a dense subgroup in the holomorphic automorphism group of $\C^n, n\ge 2$. There are many applications to natural geometric questions in complex geometry, several of which we mention here. Also the Linearization Problem, well known since the 1950 s and considered by many authors, has had a strong influence on those studies. It asks whether a compact subgroup in the holomorphic automorphism group of $\C^n$ is necessarily conjugate to a group of linear automorphisms. Despite many positive results, the answer in this generality is negative as shown by Derksen and the author. We describe various developments around that problem.

Motivation & Objective

• To understand the structure and largeness of holomorphic automorphism groups on complex manifolds, especially $\mathbb{C}^n$.
• To clarify the role of density and flexibility properties in characterizing complex manifolds with rich automorphism groups.
• To address the long-standing holomorphic linearization problem by identifying conditions under which nonlinear group actions on $\mathbb{C}^n$ are conjugate to linear actions.
• To provide new criteria for when a Stein $G$-manifold is $G$-biholomorphic to a linear $G$-module, based on stratified biholomorphisms of quotient spaces.
• To explore the implications of these results for characterizing $\mathbb{C}^n$ among Stein manifolds via automorphism group structure.

Proposed method

• The paper defines and analyzes the density property and flexibility in terms of generating sets of holomorphic vector fields and automorphisms.
• It uses shear and overshear automorphisms—time-1 maps of complete holomorphic vector fields—as fundamental building blocks for the automorphism group of $\mathbb{C}^n$.
• The approach relies on stratified biholomorphisms between categorical quotients $Q_X$ and $Q_V$ of $G$-manifolds to compare geometric structures.
• It introduces the concept of 'large' $G$-varieties, requiring codimension conditions on isotropy groups, to ensure local and global $G$-biholomorphisms.
• Theorems are proven by combining Luna's slice theorem with the theory of categorical quotients and the structure of reductive group actions.
• The method applies to $G$-actions on Stein manifolds and establishes conditions under which such actions are linearizable via quotient equivalence.

Experimental results

Research questions

• RQ1 Under what conditions is a holomorphic action of a reductive group $G$ on a Stein manifold $X$ biholomorphic to a linear action on a $G$-module $V$?
• RQ2 Can the holomorphic automorphism group of a Stein manifold $X$ be used to characterize $X$ as biholomorphic to $\mathbb{C}^n$?
• RQ3 What role do the density and flexibility properties of automorphism groups play in the linearization problem and in characterizing affine space?
• RQ4 How do the stratified biholomorphisms of categorical quotients $Q_X \simeq Q_V$ relate to the global $G$-biholomorphism of $X$ and $V$?
• RQ5 What is the minimal dimension in which non-linearizable holomorphic actions of a given reductive group $G$ can occur?

Key findings

• The paper establishes that if a Stein $G$-manifold $X$ and a $G$-module $V$ have isomorphic categorical quotients $Q_X \simeq Q_V$ via a stratified biholomorphism, and $V$ is large, then $X$ and $V$ are $G$-biholomorphic.
• For $SL_2(\mathbb{C})$-actions, the existence of a stratified biholomorphism between $Q_X$ and $Q_V$ implies $X$ is $SL_2(\mathbb{C})$-biholomorphic to $V$, even without the large condition.
• When the categorical quotient $Q_V$ is one-dimensional, a stratified biholomorphism between $Q_X$ and $Q_V$ implies $X$ is $G$-biholomorphic to $V$, providing a new criterion for linearization.
• The results show that the holomorphic linearization problem can be approached via quotient space structure, offering a new framework beyond classical approaches.
• The Koras-Russel cubic threefold $M_{KR}$ cannot be analyzed via these criteria due to its Luna quotient not being isomorphic to that of any linear action, leaving its biholomorphy to $\mathbb{C}^3$ open.
• The paper demonstrates that the automorphism group structure alone—without topology—cannot currently resolve the characterization of $\mathbb{C}^n$ among Stein manifolds.