I’ve been trying to learn something about deformations of abelian varieties lately. One of the big results is:

Theorem 1 The “local moduli space” of an abelian variety {X/k} is smooth of dimension {g^2}, if {\dim X = g}.

Why might you care about this result? Let’s say you’re in the case {g = 1}, so then presumably you’re interested in the stack {M_{1, 1}} of elliptic curves. This is a Deligne-Mumford stack: that is, there are enough étale maps {\mathrm{Spec} R \rightarrow M_{1, 1}} from affine schemes, and as a result it makes sense to talk about the strict henselianization at a point (or the completion at a “point” from an algebraically closed field). Then, the point of the above theorem is that you can work out exactly what that is: it’s a one-dimensional thing. This isn’t too surprising, because the isomorphism class of an elliptic curve depends on one parameter (the {j}-invariant). So knowing the deformation theory of elliptic curves lets you say what {M_{1, 1}} looks like, very locally.

Let’s make this a bit more precise. An actual formulation of the theorem would specify what “local moduli” really means. For us, it means deformations. A deformation of an abelian variety {X/k} over an artin local ring {R} with residue field {k} is the data of an abelian scheme (that is, a proper flat group scheme with abelian variety fibers) {X' \rightarrow \mathrm{Spec} R} together with an isomorphism of abelian varieties {X' \times_{\mathrm{Spec} R} \mathrm{Spec} k \simeq X}.

Theorem 2 Let {k} be an algebraically closed field, and let {X/k} be an abelian variety. Then the functor of deformations of {X} is prorepresentable by {W(k)[[t_1, \dots, t_{g^2}]]} for {W(k)} the ring of Witt vectors over {k}.

In other words, to give a deformation of {X} over an artin local {R} with residue field {k} is the same as giving a homomorphism of local rings

\displaystyle W(k)[[t_1, \dots, t_{g^2}]] \rightarrow R .

The relevance of {W(k)} here essentially comes from the fact that every complete (e.g. artin) local ring with residue field {k} is uniquely a continuous {W(k)}-algebra. If we restricted ourselves to {k}-algebras, we could replace {W(k)} by {k}. (more…)