Varieties covered by affine spaces, uniformly rational varieties and their cones

Abstract

It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)] that the affine cones over flag manifolds and rational smooth projective surfaces are elliptic in the sense of Gromov. The latter remains true after successive blowups of points on these varieties. In the present note we extend this to smooth projective spherical varieties (in particular, toric varieties) successively blown up along smooth subvarieties. The same also holds, more generally, for uniformly rational projective varieties, in particular, for projective varieties covered by affine spaces. It occurs also that stably uniformly rational complete varieties are elliptic.