On 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 $G$.