Lie algebras of zero divergence vector fields on complex affine algebraic varieties

TL;DR

This paper extends Lichnerowicz's results on the abelianization of volume-preserving vector fields from smooth manifolds to certain complex affine algebraic varieties with algebraic volume forms, revealing topological dependencies.

Contribution

It provides the first algebraic analogs of Lichnerowicz's finite-dimensionality and topological dependence results for complex affine algebraic varieties.

Findings

Abelianization of algebraic divergence-free vector fields is finite-dimensional.

Dimension depends only on the topology of the variety.

Results apply to classical non-singular complex affine varieties.

Abstract

For a smooth manifold $X$ equipped with a volume form, let $\dL$ be the Lie algebra of volume preserving smooth vector fields on $X$. A. Lichnerowicz proved that the abelianization of $\dL$ is a finite-dimensional vector space, and that its dimension depends only on the topology of $X$. In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties that admit a nowhere-zero algebraic form of top degree (which plays the role of a volume form).