Algebraic Gromov ellipticity of cones over projective manifolds

We find classes of projective manifolds that are elliptic in the sense of Gromov and such that the affine cones over these manifolds also are elliptic off their vertices. For example, the latter holds for any generalized flag manifold of dimension n >= 3 successively blown up in a finite set of points and infinitesimally near points. This also holds for any smooth projective rational surface. For the affine cones, the Gromov ellipticity is a weaker property than the flexibility. For instance, it is known that the affine cones over pluri-anticanonically embedded del Pezzo surfaces of degree <= 3 are not flexible. However, these cones are elliptic in the sense of Gromov. The flexibility of a smooth quasiaffine variety implies infinite transitivity of the action of its automorphism group. The latter may fail for elliptic smooth quasiaffine varieties. Nevertheless, ellipticity still implies infinite transitivity of the action of the endomorphism monoid.