Deformation Theory

In these notes we’ll give an introduction to deformation theory and apply it to the special case of abelian schemes. We’ll start by defining the deformation functor and show how the cohomology groups of the sheaf of derivations of a scheme can be used to both determine if deformations exist and if so, what the set of deformations looks like. After that, we’ll introduce the 800 pound gorilla of the subject, Schlessinger’s criteria, which says that under fairly weak and natural assumptions the deformation functor is representable (or more precisely, pro-representable). This will allow us to produce an object encoding the set of all deformations of the given scheme. We’ll then apply this theory to the special case of abelian varieties and see that the object given to us by Schlessinger’s criteria has a simple geometric interpretation. The main source that I used to learn this material is Oort’s paper [1].