Affine extensions of principal additive bundles over a punctured surface

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 $\mathbb A^2_*$.