On rational multiplicative group actions

Abstract

We establish a one-to-one correspondence between rational multiplicative group actions on an algebraic variety $X$ and derivations $\partial\colon K_X\to K_X$ of the field of fractions $K_X$ of $X$ satisfying that there exists a generating set $\{a_i\}_{i\in I}$ of $K_X$ as a field such that $\partial(a_i)=\lambda_i a_i$ with $\lambda_i \in \mathbb{Z}$ for all $i\in I$. We call such derivations rational semisimple. Furthermore, we also prove the existence of a rational slice for every rational semisimple derivation, i.e., an element $s\in K_X$ such that $\partial(s)=s$. By analogy with the case of additive group actions case, we prove that $K_X\simeq K_X^{\mathbb{G}_m}(s)$ and that under this isomorphism the derivation $\partial$ is given by $\partial=s\frac{d}{ds}$. Here, $K_X^{\mathbb{G}_m}$ is the field of invariant of the $\mathbb{G}_m$-action.