  Let $K$ be a locally compact field of characteristic 0. Let $G$ be a linear
algebraic group defined over $K$, acting algebraically on an algebraic variety
$V$. We prove that the action of $G(K)$ (the group of $K$-rational points of
$G$) on $V(K)$ is topologically amenable, if and only if all points stabilizers
in $G(K)$ are solvable-by-compact. This follows by combining a result by
Borel-Serre \cite{BoSe} with the following fact: let $G$ be a second countable
locally compact group acting continuously on a second countable locally compact
space $Y$. If the action $G\curvearrowright Y$ is smooth (i.e. the Borel
structure on $G\backslash Y$ is countably separated), then topological
amenability of $G\curvearrowright Y$ is equivalent to amenability of all point
stabilizers in $G$.
