# Categorical quotient of open subsets of affine varieties

Let $X$ be a complex affine variety and $G$ be a complex reductive group acting on $X$. Let $X//G=\operatorname{Spec}\mathbb{C}[X]^G$ and $$\pi:X\to X//G$$ be the GIT quotient of $X$ by $G$. Suppose that $U\subseteq X$ is a $G$-invariant Zariski-open subset of $X$.

What is the categorical quotient of $U$ by $G$? Is is the restriction $\pi:U\to \pi(U)$?

I know that if $U=\pi^{-1}(\pi(U))$ is saturated, then the restriction $\pi:U\to\pi(U)$ is a categorical quotient for the action of $G$ on $U$, but I am mostly interested in the case where $U$ is not necessarily saturated. In other words, $G\cdot U\subseteq U$, but it is possible that $\overline{G\cdot p}\not\subseteq U$ for some $p\in U$.

• What about $G=\mathbb C^*$ acting on $\mathbb C^n$, $n\ge2$ by scalars and $U=\mathbb C^n\setminus\{0\}$? – Friedrich Knop Oct 25 '16 at 10:47