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We analyze the deformation theory of equivariant vector bundles. Inparticular, we provide an effective criterion for verifying whether allinfinitesimal deformations preserve the equivariant structure. As anapplication, using rigidity of the Frobenius homomorphism of general lineargroups, we prove that projectivizations of Frobenius pullbacks of tautologicalvector bundles on Grassmanians are arithmetically rigid, that is, do not liftover rings where $p \neq 0$. This gives the same conclusion for Totaro'sexamples of Fano varieties violating Kodaira vanishing. We also provide analternative purely geometric proof of non-liftability mod $p^2$ and tocharacteristic zero of the Frobenius homomorphism of a reductive group ofnon-exceptional type.

Arithmetically rigid schemes via deformation theory of equivariant vector bundles