Arithmetically rigid schemes via deformation theory of equivariant vector bundles

TL;DR

This paper studies the deformation theory of equivariant vector bundles, providing criteria for rigidity and applying it to show certain geometric structures are arithmetically rigid, with implications for Frobenius homomorphisms and Fano varieties.

Contribution

It introduces an effective criterion for the preservation of equivariance under infinitesimal deformations and applies it to prove arithmetical rigidity of specific vector bundles and Frobenius homomorphisms.

Findings

Projectivizations of Frobenius pullbacks are arithmetically rigid.

Fano varieties violating Kodaira vanishing are also arithmetically rigid.

Alternative geometric proof of non-liftability of Frobenius homomorphism.

Abstract

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.