This is the second post in a series on deformations of abelian varieties. In the previous post, I described the basic outline of the goals and strategies, as well as a weak version of Schlessinger’s criterion useful in showing prorepresentability of smooth moduli problems without infinitesimal automorphisms. In this post, we’ll see that the deformation problem for abelian varieties satisfies the second condition above: that abelian varieties are rigid. The material here is very classical; I learned it from Oort’s article (from a summer school in the 1970s) and Katz’s article. Most of the material in this post comes from chapter 6 of Mumford’s GIT book, which is surprisingly readable without knowledge of any other parts of it.

Let {R} be an artin ring, and let {X/\mathrm{Spec} R} be an abelian scheme. Consider a morphism of {R}-group schemes

\displaystyle f: X \rightarrow X

inducing the identity on the special fiber. We would like to show that it is the identity, as in the next proposition:

Proposition 7 Such a morphism {f} is necessarily the identity: that is, an infinitesimal deformation of an abelian variety has no nontrivial infinitesimal automorphisms.

This will imply prorepresentability of the deformation functor, using the general form of Schlessinger’s theorem.