On algebraic volume density property

TL;DR

This paper introduces an effective criterion to verify the algebraic volume density property (AVDP) for smooth affine algebraic varieties and applies it to homogeneous spaces with invariant volume forms, expanding understanding of their algebraic vector fields.

Contribution

It develops a practical criterion for verifying AVDP and proves it for homogeneous spaces with invariant volume forms under linear algebraic groups.

Findings

Established AVDP for homogeneous spaces $G/R$ with invariant volume forms.

Provided an effective method to verify AVDP in algebraic varieties.

Extended the class of varieties known to have the algebraic volume density property.

Abstract

A smooth affine algebraic variety $X$ equipped with an algebraic volume form $\omega$ has the algebraic volume density property (AVDP) if the Lie algebra generated by completely integrable algebraic vector fields of $\omega$-divergence zero coincides with the space of all algebraic vector fields of $\omega$-divergence zero. We develop an effective criterion of verifying whether a given $X$ has AVDP. As an application of this method we establish AVDP for any homogeneous space $X=G/R$ that admits a $G$-invariant algebraic volume form where $G$ is a linear algebraic group and $R$ is a closed reductive subgroup of $G$.