Algebraic volume density property of affine algebraic manifolds

TL;DR

This paper introduces the algebraic volume density property for affine algebraic manifolds, establishes its implications, and provides significant examples including affine modifications of complex Euclidean spaces and linear algebraic groups.

Contribution

It defines the algebraic volume density property and demonstrates its implications, along with constructing key classes of Stein manifolds exhibiting this property.

Findings

Algebraic volume density property implies volume density property.

Affine modifications of ^n have this property.

Linear algebraic groups with invariant volume forms also have this property.

Abstract

We introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it, in particular that it implies the volume density property. The main results of the paper are producing two big classes of examples of Stein manifolds with volume density property. One class consists of certain affine modifications of $\C^n$ equipped with a canonical volume form, the other is the class of all Linear Algebraic Groups equipped with the left invariant volume form.