Equivariantly Uniformly Rational Varieties

We 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 G(m)-uniform rationality of smooth affine varieties equipped with hyperbolic G(m)-actions with a unique fixed point, formulated in terms of their Altmann-Hausen presentation. We prove the G(m)-uniform rationality of Koras-Russell three folds of the first kind, and we also give an example of a non-Gm uniformly rational but smooth rational G(m)-threefold associated with pairs of plane rational curves birationally nonequivalent to a union of lines.