  We analyze the deformation theory of equivariant vector bundles. In
particular, we provide an effective criterion for verifying whether all
infinitesimal deformations preserve the equivariant structure. As an
application, using rigidity of the Frobenius homomorphism of general linear
groups, we prove that projectivizations of Frobenius pullbacks of tautological
vector bundles on Grassmanians are arithmetically rigid, that is, do not lift
over rings where $p \neq 0$. This gives the same conclusion for Totaro's
examples of Fano varieties violating Kodaira vanishing. We also provide an
alternative purely geometric proof of non-liftability mod $p^2$ and to
characteristic zero of the Frobenius homomorphism of a reductive group of
non-exceptional type.
