Additive actions on toric varieties

TL;DR

This paper characterizes additive actions on complete toric varieties, showing they are normalized by the torus and correspond to Demazure roots, with all such actions being isomorphic.

Contribution

It establishes a bijection between normalized additive actions and complete collections of Demazure roots on toric varieties, and proves their uniqueness up to isomorphism.

Findings

Additive actions on complete toric varieties are normalized by the torus.

Normalized additive actions correspond to complete collections of Demazure roots.

All normalized additive actions on a given toric variety are isomorphic.

Abstract

By an additive action on an algebraic variety $X$ of dimension $n$ we mean a regular action $\mathbb{G}_a^n \times X \to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. We prove that if a complete toric variety $X$ admits an additive action, then it admits an additive action normalized by the acting torus. Normalized additive actions on a toric variety $X$ are in bijection with complete collections of Demazure roots of the fan of $X$. Moreover, any two normalized additive actions on $X$ are isomorphic.