Equivariant deformation theory & arithmetic deformations of homogeneous varieties

Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to prove a dichotomous behaviour for almost all varieties that are homogeneous under a reductive group : either they deform to characteristic 0, or they admit no deformation to any ring of characteristic greater than p.