Equivariantly uniformly rational varieties

International audienceWe introduce equivariant versions of uniform rationality: given an algebraic group $G$, a $G$-variety is called $G$-uniformly rational (resp. $G$-linearly uniformly rational) if every point has a $G$-invariant open neighborhood equivariantly isomorphic to a $G$-invariant open subset of the affine space endowed with a $G$-action (resp. linear $G$-action). We establish a criterion for $\mathbb{G}_m$-uniform rationality of affine variety equipped with hyperbolic $\mathbb{G}_m$-action with a unique fixed point, formulated in term of their Altmann-Hausen presentation. We prove the $\mathbb{G}_m$-uniform rationality of Koras-Russel threefolds of the first kind and we also give example of non $\mathbb{G}_m$-uniformly rational but smooth rational $\mathbb{G}_m$-threefold associated to pairs of plane rational curves birationally non equivalent to a union of lines