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 be an artin ring, and let be an abelian scheme. Consider a morphism of -group schemes
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 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.
1. A general rigidity lemma
To prove this, consider the difference . This is a morphism of schemes which restricts to a constant over the special fiber. Now we have the following useful rigidity lemma, which is extremely handy in generalizing results for abelian varieties to abelian schemes.
Lemma 8 (Rigidity lemma) Let be the of an artin local ring with closed point . Let be -schemes with proper and flat; suppose that . Suppose is such that is a single point (set-theoretically). Then factors through .
Proof: The observation is that, set-theoretically, the morphism of spaces has to factor
because the image of is a point. To describe , though, we need to do more: we need the map of sheaves of rings, since a morphism of schemes is a morphism of locally ringed spaces.
To avoid confusion, we will write for the underlying space of . Let be the underlying map of topological spaces of . Then, since the image of is one point, we find that there is a (unique) map of topological spaces such that factors . Our goal is to make into a morphism of schemes and make the diagram commute in the category of schemes.
The description of comes from together with a map of sheaves of rings . In order to factor through , and upgrade to a map of schemes, we need to define a map of sheaves such that the composite
is the structure map of . (Here is the push-forward of structure map on sheaves of rings from ; it is already determined.)
The whole point, though, is that is an isomorphism; in fact, let be the map making into an -scheme. Then
by the assumption and cohomology and base change. It follows that the pushforward is an isomorphism too, and there is a unique way of defining a map of sheaves of rings to factor through .
This almost upgrades to a map of schemes; the only last thing to check is that is a map of locally ringed spaces (rather than simply ringed spaces). In other words, we need to show that is a local homomorphism of local rings. But this follows because we have a composite map of rings
(for any ), which is a local homomorphism because is a map of schemes.
Suppose admits a section . Then it follows in the above proposition that is actually equal to the composite , by staring at the commutative diagram
which implies that the map comes from .
This corollary was stated for artinian rings, but it quickly yields a powerful generalization.
Corollary 9 (Rigidity lemma; strengthened version) Let be a morphism of separated schemes over a connected scheme . Suppose is proper and flat over and that there is a section . Suppose there exists a point such that is set-theoretically a point and that for all , . Then factors as where is the map .
In other words, if is constant on one fiber, it is constant on all fibers, and a little more.
Proof: Without loss of generality, we can assume irreducible.
Consider the morphism and the other morphism . We want to show that they are equal. The previous version of the rigidity lemma and the comments following it imply that the two morphisms are equal when restricted to , for any nilpotent thickening of .
Now, there is a maximal closed subscheme such that become equal on : this is the equalizer of , which can be represented by a suitable fibered product . What we have seen is that (as closed subschemes) for any nilpotent thickening of . Using the Krull intersection theorem, it follows that must contain a neighborhood of the fiber , which (by properness) we can assume to be of the form for open containing . From this, we want to conclude that .
So far, we find that when restricted to for an open subset containing . Since is dense in , it follows (by flatness) that is dense in , which means that the subscheme must set-theoretically contain all of . Using the artinian version of the rigidity lemma again, it follows that must scheme-theoretically contain all of .
One of the intriguing things to me about this proof is how fundamentally scheme-theoretic it is; although the result is a nontrivial statement even in the category of varieties, the proof requires the use of nilpotents and infinitesimal neighborhoods to get at the global structure. It reminds me of the proof of the theorem of the cube. This is a statement about varieties and can be proved using only varieties (as in ch. 2 of Mumford’s book on abelian varieties), but the proof using a tiny bit of deformation theory is much simpler.
2. The deformation functor preserves products
In any event, we find from this that deformations of abelian varieties have no nontrivial (infinitesimal) automorphisms: in fact, we find that any endomorphism of an abelian scheme over a connected base which is the identity on one fiber is the identity. It follows that, given an abelian variety , the functor
sending an abelian variety to its groupoid of infinitesimal deformations over an artin ring and infinitesimal isomorphisms, is actually discrete. This is intuitively at least unsurprising: an abelian variety is complex analytically a complex torus and isomorphisms morphisms between these are given by isomorphisms of lattices, which are very discrete.
Now, is product-preserving along surjections: this follows from the fact that if are surjections in , then is obtained by “gluing” along . In other words, let for a ring , be the category of schemes flat over . Then we have an equivalence of categories
In other words, to give a scheme flat over is equivalent to giving a scheme flat over , a scheme flat over , together with an isomorphism between the restrictions to (in other words, this is the 2-categorical fiber product rather than the 1-categorical one).
In particular, to give a deformation of an abelian variety over is the same as giving a deformation over , one over , and an isomorphism on the restrictions to . Since there are no infinitesimal automorphisms, though, we can say a little more: an isomorphism class of deformations over is the same as giving an isomorphism class of deformations over , one over , which restrict to the same isomorphism class on .
Corollary 10 The functor which sends an artinian ring to the set of isomorphism classes of deformations of an abelian variety over is product-preserving.
Using the general version of Schlessinger’s criterion (which we did not prove yesterday), it now follows that this functor is actually prorepresentable. In the next post, we’ll see that the functor above is formally smooth, so we can apply the weaker version of Schlessinger’s criterion proved last post to show that is prorepresentable by .