# Finite abelian groups acting on smooth varieties and “invariant” invertible functions

Let $G$ be a finite abelian group acting on a smooth quasi-projective variety $X$ over $\mathbb C$. Let $E(X) = \mathcal{O}(X)^*/\mathbb C^*$.

Under what conditions is $E(X)^G$ the trivial group?

One simple condition which implies that $E(X)^G$ is trivial is that $\mathcal{O}(X) = \mathbb C^*$. That certainly implies that $E(X)^G$ is trivial, as it implies that $E(X)$ is trivial. Examples of such varieties are dense open subsets of $\mathbb A^n$ whose complement is of codimension at least two.

There are some other criteria appearing in Knop-Kraft-Vust's paper entitled The Picard group of a $G$-variety (MR CiteSeerX).

Are any others known? In other words, are there some nice examples of pairs $(X,G)$ as above with $E(X)^G$ trivial?

Note: I edited the question to make it more and more elementary.

• Note that the second author is "Kraft", and the complete reference is: Friedrich Knop, Hanspeter Kraft, Thierry Vust, The Picard group of a $G$-variety. Algebraische Transformationsgruppen und Invariantentheorie, 77–87, DMV Sem., 13, Birkhäuser, Basel, 1989. – Jim Humphreys Oct 26 '16 at 19:21