Affine extensions of principal additive bundles over a punctured surface

TL;DR

This paper explores the classification of complex affine threefolds with a focus on principal additive bundles over punctured surfaces, highlighting the role of the variety SL_2 and its G_a-action.

Contribution

It initiates the classification of certain affine threefolds with G_a-actions, emphasizing the structure over punctured surfaces and analyzing the example of SL_2.

Findings

SL_2 admits a G_a-action with a principal G_a-bundle structure over punctured plane

The restriction of the quotient morphism to the punctured surface is a principal G_a-bundle

Provides foundational steps towards classifying affine G_a-threefolds

Abstract

The aim of this article is to make a first step towards the classification of complex normal affine $\mathbb G_a$-threefolds $X$. We consider the case where the restriction of the quotient morphism $\pi\colon X\to S$ to $\pi^{-1}(S_*)$, where $S_*$ denotes the complement of some regular closed point in $S$, is a principal $\mathbb G_a$-bundle. The variety $\mathrm{SL}_2$ will be of special interest and a source of many examples. It has a natural right $\mathbb G_a$-action such that the quotient morphism $\mathrm{SL}_2\to\mathbb A^2$ restricts to a principal $\mathbb G_a$-bundle over the punctured plane $\mathbb A^2_*$.