An affineness criterion for algebraic groups and applications

TL;DR

The paper establishes a new criterion for determining when a smooth, connected algebraic group is affine based on the invariance of invertible sheaves on normal G-varieties, and applies this to simplify the Chevalley-Barsotti structure theorem.

Contribution

It introduces an affineness criterion for algebraic groups using invertible sheaves and extends G-actions to projective actions, simplifying existing structural theorems.

Findings

A smooth, connected algebraic group is affine iff invertible sheaves on normal G-varieties are G-invariant.

G-invariant invertible sheaves induce projective actions on linear systems.

Provides a new, simpler proof of the Chevalley-Barsotti Theorem.

Abstract

We prove that a smooth and connected algebraic group $G$ is affine if and only if any invertible sheaf on any normal $G$-variety is $G$-invariant. For the proof, a key ingredient is the following result: if $G$ is a connected and smooth algebraic group and $\mathcal L$ is a $G$-invariant invertible sheaf on a $G$-variety $X$, then the action of $G$ on $X$ extends to a projective action on the complete linear ${\mathbb P}(H^0(X,{\mathcal L})$. As an application of the affineness criterion, we give a new and simple proof of Chevalley-Barsotti Theorem on the structure of algebraic groups.