Equivariant deformation theory and deformations of homogeneous varieties

Abstract

We show that, in characteristic $p$, almost all projective homogeneous varieties under a reductive group and with non-reduced stabilizer do not lift to characteristic $p^2$. To this end, first we develop a general framework for equivariant deformation theory, leading to practical criteria ensuring that the deformations of a scheme come with deformations of a group scheme action. Second, we establish a formula to compute the cohomology of equivariant vector bundles over projective homogeneous varieties with non-reduced stabilizer, enabling us to determine the $G$-module structure of the first cohomology groups of their tangent bundles.