Extension of the Andersén–Lempert theory: Lie algebras of zero divergence vector fields on complex affine algebraic varieties
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Bibliographic record
Abstract
Abstract For a smooth manifold X equipped with a volume form, let 𝓛 0 ( X ) be the Lie algebra of volume preserving smooth vector fields on X . Lichnerowicz proved that the abelianization of 𝓛 0 ( 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 ℂ n , to the larger class of Stein manifolds that satisfy the density property.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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