Non-algebraic Examples of Manifolds with the Volume Density Property

TL;DR

This paper introduces the first known examples of non-algebraic Stein manifolds with the volume density property, expanding understanding of automorphism groups in complex geometry and providing potential counterexamples to longstanding conjectures.

Contribution

It adapts a criterion to the holomorphic case and constructs non-algebraic manifolds with the volume density property using suspensions and modifications.

Findings

First non-algebraic manifolds with volume density property

Examples potentially countering the Zariski Cancellation Problem

Implications for the linearization problem of C*-actions on C^3

Abstract

Some Stein manifolds (with a volume form) have a large group of (volume-preserving) automorphisms: this is formalized by the (volume) density property, which has remarkable consequences. Until now all known manifolds with the volume density property are algebraic, and the tools used to establish this property are algebraic in nature. In this note we adapt a known criterion to the holomorphic case, and give the first known examples of non-algebraic manifolds with the volume density property: they arise as suspensions or pseudo-affine modifications over Stein manifolds satisfying some technical properties. As an application we show that there are such manifolds that are potential counterexamples to the Zariski Cancellation Problem, a variant of the Toth-Varolin conjecture, and the problem of linearization of C*-actions on C^3.