Abstract For a smooth manifold X equipped with a volume form, let 𝓛0 (X) be the Lie algebra of volume preserving smooth vector fields on X. Lichnerowicz proved that the abelianization of 𝓛0 (X) 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 with trivial canonical bundle, which include certain algebraic surfaces and linear algebraic groups. The proofs are based on a remarkable result of Grothendieck on the cohomology of affine varieties, and some techniques that were introduced with the purpose of extending the Andersén–Lempert theory, which was originally developed for the complex spaces ℂn, to the larger class of Stein manifolds that satisfy the density property.

中文翻译：

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Andersén-Lempert 理论的扩展：复仿射代数簇上零散度矢量场的李代数

摘要 对于带有体积形式的光滑流形 X，令 𝓛0 (X) 是 X 上体积保持光滑向量场的李代数。 Lichnerowicz 证明了 𝓛0 (X) 的阿贝尔化是一个有限维向量空间，并且它的维数仅取决于 X 的拓扑结构。在本文中，我们为具有平凡正则丛的非奇异复数仿射代数簇的一些经典例子提供了类似的结果，其中包括某些代数曲面和线性代数群。证明基于格洛腾迪克关于仿射变体的上同调的一个显着结果，以及为了将最初为复空间 ℂn 开发的 Andersén-Lempert 理论扩展到更大的 Stein 类而引入的一些技术满足密度特性的流形。