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 ${k}$.

1. Schlessinger’s criterion

It turns out that the above result is part of a very general phenomenon: many nice formal moduli problems turn out to be prorepresentable by power series rings over Witt vectors.

For instance, let ${k}$ be a separably closed field, and let ${f(x,y)}$ be a formal group law over ${k}$ of height ${n < \infty}$. We might ask for deformations of ${f}$: that is, formal group laws over an artin local ring with residue field ${k}$ which reduce to ${f}$ mod the maximal ideal, modulo isomorphisms which are the identity mod the maximal ideal. A theorem of Lubin and Tate states:

Theorem 3 (Lubin, Tate) The functor classifying deformations of ${f}$ (mod isomorphism) is prorepresentable by ${W(k)[[t_1, \dots, t_{n-1}]]}$.

(For instance, the multiplicative group has no nontrivial deformations.)

This theorem turns out to be connected with the formal moduli problem for abelian varieties—a theorem of Serre and Tate states that, for instance, deformations of a supersingular elliptic curve are the same as deformations of their formal group. A supersingular elliptic curve has a formal group law of height two, so Lubin-Tate theory gives the same answer as we would have expected: the deformation theory of such an elliptic curve is prorepresentable by ${W(k)[[t]]}$.

As another example, we might consider the deformation theory of curves of genus ${g \geq 2}$. Again it turns out that the deformations of such a curve are prorepresentable by ${W(k)[[t_1, \dots, t_{3g-3}]]}$; this reflects the fact that such a curve should depend on ${3g-3}$ parameters.

Anyway, it turns out that the Lubin-Tate theorem, as well as the theorem mentioned at the beginning, all fit into a common framework: roughly speaking, moduli problems which do not admit infinitesimal automorphisms, and which have no obstructions to lifting, will be prorepresentable by power series rings over ${W(k)}$.

Let ${k}$ be a field, and let ${\mathrm{Art}_k}$ be the category of artin local algebras with residue field $k$.

Theorem 4 (Schlessinger) Let ${F: \mathrm{Art}_k \rightarrow \textbf{Sets}}$ be a moduli problem (i.e., a functor with ${F(k) = \ast}$). Suppose ${F}$ satisfies:

1. ${F(A' \times_A A'') \simeq F(A') \times_{F(A)} F(A'')}$ for every triple ${A', A, A'' \in \mathrm{Art}_k}$ and surjections ${A', A'' \twoheadrightarrow A}$.
2. ${F}$ is formally smooth: for a surjection ${A \twoheadrightarrow \overline{A}}$, the map ${F(A) \rightarrow F(\overline{A})}$ is surjective.
3. The tangent space to ${F}$ is of finite dimension ${g}$.

Then ${F}$ is prorepresentable by the ring ${W(k)[[t_1, \dots, t_g]]}$.

This is really a special case of Schlessinger’s more general criterion for when a moduli problem is prorepresentable (or more generally, has a “hull”). The tangent space to ${F}$ is defined as ${F(k[\epsilon]/\epsilon^2)}$; it’s not too hard to show that under the above hypotheses, this acquires the structure of a ${k}$-vector space.

In practice, the condition (1) is often not satisfied: many moduli problems (say, deformations of a scheme) will have instead the property that

$\displaystyle F(A' \times_A A'') \rightarrow F(A) \times_{F(A)} F(A'') \ \ \ \ \ (1)$

is a surjection (rather than an isomorphism). However, in many of these cases there will be a lift of ${F}$ to a groupoid-valued functor

$\displaystyle F_1: \mathrm{Art}_k \rightarrow \mathbf{Gpds}$

such that ${F = \pi_0 F_1}$ (i.e., isomorphism classes of elements of ${F_1}$). A very natural condition to require now is that

$\displaystyle F_1(A' \times_A A'') \simeq F_1(A') \times_{F_1(A)} F_1(A''),$

where one uses the appropriate 2-fibered product of groupoids. When passing to ${\pi_0}$, we get a surjection rather than a bijection in (1).

The reason that the condition on ${F_1}$ is natural is that ${\mathrm{Spec} (A' \times_A A'')}$ is obtained by a gluing process: ${\mathrm{Spec} A', \mathrm{Spec} A''}$ are glued along the common closed subscheme ${\mathrm{Spec} A}$. So, roughly, giving a geometric object over ${\mathrm{Spec} (A' \times_A A'')}$ is the same as giving one over ${\mathrm{Spec} A'}$ and one over ${\mathrm{Spec} A''}$ together with an isomorphism of the two restrictions to ${\mathrm{Spec} A}$. For instance, this is true for flat families of schemes.

Now, if the groupoid-valued functor ${F_1}$ is discrete, then the reduction ${F = \pi_0 F}$ does satisfy (1) above. This is going to be the case for deformations of abelian varieties, which are rigid in this sense, and it’s also the case for deformations of formal group laws and curves of genus ${\geq 2}$. Finally, the smoothness condition is what makes the prorepresenting object be a power series ring: it reflects the formal smoothness of the prorepresenting ring.

2. The proof of Schlessinger’s theorem

The proof of Schlessinger’s criterion in the formally smooth, left-exact case (which is all we need here) is a lot easier than in the general case. So, let’s say we have a left-exact functor ${F: \mathrm{Art}_k \rightarrow \textbf{Sets}}$ satisfying the conditions of the theorem. In particular, we have a collection of elements ${a_i \in F(k[\epsilon]/\epsilon^2), 1 \leq i \leq g}$ which form a basis for this vector space. Since ${F}$ is formally smooth, each ${a_i}$ can be lifted to ${W(k)[[t]]/(p^n, t^n)}$ for each ${n}$, and in the limit we get elements

$\displaystyle \alpha_i \in F( W(k)[[t]]) = \varprojlim F(W(k)[[t]]/(p^n, t^n).$

Since ${F}$ is product-preserving, these together define an element of ${F( W(k)[[t_1, \dots, t_g]])}$, or a morphism of functors

$\displaystyle \hom(W(k)[[t_1, \dots, t_g]], \cdot) \rightarrow F(\cdot): \mathrm{Art}_k \rightarrow \mathbf{Sets}. \ \ \ \ \ (2)$

By construction, this natural transformation of functors is an isomorphism on ${k[\epsilon]/\epsilon^2}$; the goal is to show that it is an isomorphism on all rings in ${\mathrm{Art}_k}$.

This can be done by induction on the length of an element of ${\mathrm{Art}_k}$; the key point is that ${F}$ and the functor prorepresented by ${W(k)[[t_1, \dots, t_g]]}$ are formally smooth—because the Witt vectors are formally smooth, though this isn’t immediate. Let ${A \in \mathrm{Art}_k}$, and suppose the map (2) is an isomorphism on rings of smaller length than ${A}$. Now, choose a nonzero ${x \in A}$ annihilated by the maximal ideal ${\mathfrak{m}_A \subset A}$. We have a map ${A \rightarrow A/x}$, which induces a surjective map

$\displaystyle F(A) \rightarrow F(A/x) ;$

this gives a pull-back diagram

Now ${A \times_{A/x} A \simeq A \times_k k[\epsilon]/\epsilon^2}$ via the map which sends ${\epsilon \mapsto x}$ and ${A}$ to the diagonal. It follows that the action of ${k[\epsilon]/\epsilon^2 }$ on ${A}$ over ${A/x}$ (obtained by adding ${x}$) actually exhibits ${A}$ as a ${k[\epsilon]/\epsilon^2}$-torsor over ${A/x}$ in the category ${\mathrm{Art}_k}$.

Since ${F}$ is product-preserving and ${F(A) \twoheadrightarrow F(A/x)}$ is surjective by smoothness, it follows that ${F(A)}$ is naturally a torsor over ${F(A/x)}$ for the tangent space at ${f}$. The same is true for the smooth, prorepresentable functor ${\hom(W(k)[[t_1, \dots, t_g]], \cdot)}$. By induction, the map in (2)

$\displaystyle \hom(W(k)[[t_1, \dots, t_g]], A/x) \rightarrow F(A/x)$

is an isomorphism, and the map on tangent spaces is an isomorphism. It follows that (2) is an isomorphism for ${A}$ as well. This proves Schlessinger’s theorem.

3. Deformations of abelian varieties: outline

Now let’s specialize to the case of interest in this post: deformations of abelian varieties. Let ${X/k}$ be an abelian variety. We are interested in abelian schemes ${X'/\mathrm{Spec} R}$ for ${R}$ a local artin algebra with residue field ${k}$, together with an isomorphism of ${X'}$ with ${X}$ over the closed point. We want to see that this is a moduli problem prorepresentable by ${W(k)[[t_1, \dots, t_{g^2}]]}$. Most of this will follow from:

Proposition 5 Deformations of abelian varieties are unobstructed and have no infinitesimal automorphisms.

In other words, if ${X'/\mathrm{Spec} R}$ is a deformation of ${X}$ and ${\mathrm{Spec} R \hookrightarrow \mathrm{Spec} R_1}$ is an infinitesimal thickening, then the deformation can be lifted to ${\mathrm{Spec} R_1}$; this is the smoothness condition. Moreover, ${X'}$ has no nontrivial automorphisms reducing to the identity on the special fiber. The latter means that the (more natural) groupoid-valued functor

$\displaystyle \mathrm{Art}_k \stackrel{\mathrm{defos\ of \ } X}{\longrightarrow} \textbf{Gpd} ,$

which is product-preserving, is actually discrete. Consequently, the less fancy set-valued functor

$\displaystyle \mathrm{Art}_k \stackrel{\mathrm{iso\ classes \ of \ defos\ of \ } X}{\longrightarrow} \textbf{Sets} ,$

is also product-preserving. This is something special: a scheme admits infinitesimal automorphisms if ${H^0(T_X) \neq 0}$ (i.e., there are nontrivial vector fields). But for an abelian variety, a vector field is determined by its value at zero, and that must be zero if it is an infinitesimal automorphism of the abelian variety.

So, if we believe these two things, then we find by Schlessinger’s theorem that the deformation problem for an abelian variety is prorepresentable by ${W(k)[[t_1, \dots, t_n]]}$ for some ${n}$—to determine which, we need to analyze deformations over ${k[\epsilon]/\epsilon^2}$. Fortunately, that turns out to be relatively straightforward, in view of the following.

Proposition 6 There is a bijection between isomorphism classes of deformations of ${X}$ as an abelian variety over ${R \in \mathrm{Art}_k}$ and abstract deformations of ${X}$ as a plain variety over ${R}$.

In general, abstract deformations of a plain variety over ${k[\epsilon]/\epsilon^2}$ (“first-order deformations”) are classified by ${H^1(T_X)}$. For an abelian variety, ${T_X \simeq \mathcal{O}_X^g}$ because the tangent bundle is parallelizable, and ${H^1(\mathcal{O}_X)}$ has dimension ${g}$. This gives the ${g^2}$ as claimed.

In the next post, I’ll describe the proof of the above facts, and analyze the situation more carefully.