Geometry of equivariant compactifications of \(\mathbb{G}^n_a\).

*(English)*Zbl 0966.14033In this article, the authors study equivariant compactifications of the additive group \({\mathbb{G}}_{a}^{n}\). Without the condition of equivariance, the classification is very complicated and probably impossible. The condition of equivariance reduces the problem to a more feasible task. In the present article, all possible equivariant compactifications into projective space are given. In contrast to projective toric varieties, in which case one finds a unique structure for each projective variety, for the group \({\mathbb{G}}_{a}^{n}\), one finds several inequivalent actions on projective space. In fact, for \(n\geq 6\), one finds infinite families of actions on projective \(n\)-space. All equivariant compactifications are found for \(n=2\) and for smooth projective threefolds with Picard group of rank 1. The theory of compactification of reductive groups, for example, compact toric varieties, has been studied by many authors. They are classified using combinatorial objects. The theory of non-reductive groups in much less developed, and this article shows several important differences with the reductive case.

Reviewer: L.Moser-Jauslin (Dijon)