Let ${X}$ be an abelian variety over the algebraically closed field ${k}$. In the previous post, we studied the Picard scheme ${\mathrm{Pic}_X}$, or rather its connected component ${\mathrm{Pic}^0_X}$ at the identity. The main result was that ${\mathrm{Pic}^0_X}$ was itself an abelian variety (in particular, smooth) of the same dimension as ${X}$, which parametrizes precisely the translation-invariant line bundles on ${X}$.

We also saw how to construct isogenies between ${X}$ and ${\mathrm{Pic}^0_X}$. Given an ample line bundle ${\mathcal{L}}$ on ${X}$, the map

$\displaystyle X \rightarrow \mathrm{Pic}^0_X, \quad x \mapsto t_x^* \mathcal{L} \otimes \mathcal{L}^{-1}$

is an isogeny. Such maps were in fact the basic tool in proving the above result.

The goal of this post is to show that the contravariant functor

$\displaystyle X \mapsto \mathrm{Pic}^0_X$

from abelian varieties over ${k}$ to abelian varieties over ${k}$, is a well-behaved duality theory. In particular, any abelian variety is canonically isomorphic to its bidual. (This explains why the double Picard functor on a general variety is the universal abelian variety generated by that variety, the so-called Albanese variety.) In fact, we won’t quite finish the proof in this post, but we will finish the most important step: the computation of the cohomology of the universal line bundle on $X \times \mathrm{Pic}^0_X$.

Motivated by this, we set the notation:

Definition 11 We write ${\hat{X}}$ for ${\mathrm{Pic}^0_X}$.

The main reference for this post is Mumford’s Abelian varieties. (more…)

Let ${k}$ be an algebraically closed field, and ${X}$ a projective variety over ${k}$. In the previous two posts, we’ve defined the Picard scheme ${\mathrm{Pic}_X}$, stated (without proof) the theorem of Grothendieck giving conditions under which it exists, and discussed the infinitesimal structure of ${\mathrm{Pic}_X}$ (or equivalently of the connected component ${\mathrm{Pic}^0_X}$ at the origin).

We saw in particular that the tangent space to the Picard scheme could be computed via

$\displaystyle T \mathrm{Pic}^0_X = H^1(X, \mathcal{O}_X),$

by studying deformations of a line bundle over the dual numbers. In particular, in characteristic zero, a simply connected smooth variety has trivial ${\mathrm{Pic}^0_X}$. To get interesting ${\mathrm{Pic}_X^0}$‘s, we should be looking for non-simply connected varieties: abelian varieties are a natural example.

Let ${X}$ be an abelian variety over ${k}$. The goal in this post is to describe ${\mathrm{Pic}^0_X}$, which we’ll call the dual abelian variety (we’ll see that it is in fact smooth). We’ll in particular identify the line bundles that it parametrizes. Most of this material is from David Mumford’s Abelian varieties and Alexander Polischuk’s Abelian varieties, theta functions, and the Fourier transform. I also learned some of it from a class that Xinwen Zhu taught last spring; my (fairly incomplete) notes from that class are here(more…)

Let ${X}$ be an abelian variety over an algebraically closed field ${k}$. If ${k = \mathbb{C}}$, then ${X}$ corresponds to a complex torus: that is, ${X}$ can be expressed complex analytically as ${V/\Lambda}$ where ${V}$ is a complex vector space of dimension ${\dim X}$ and ${\Lambda \subset V}$ is a lattice (i.e., a ${\mathbb{Z}}$-free, discrete submodule of rank ${2g}$). In this case, one can form the dual abelian variety

$\displaystyle X^{\vee} = \hom(X, S^1) = \hom_{\mathrm{cont}}(V/\Lambda, \mathbb{R}/\mathbb{Z}) \simeq \hom_{\mathbb{R}}(V, \mathbb{R})/2\pi i \hom(\Lambda, \mathbb{Z}).$

At least, ${X^{\vee}}$ as defined is a complex torus, but it turns out to admit the structure of an abelian variety.

The purpose of the next few posts is to describe an algebraic version of this duality: it turns out that ${X^{\vee}}$ can be constructed as a scheme, purely algebraically. I’d like to start with a couple of posts on Picard schemes. A useful reference here is this article of Kleiman.

1. The Picard scheme analytically

Let ${X }$ be a smooth projective variety over the complex numbers ${\mathbb{C}}$. The collection of line bundles ${\mathrm{Pic}(X)}$ is a very interesting invariant of ${X}$. Usually, it splits into two pieces: the “topological” piece and the “analytic” piece. For instance, there is a first Chern class map

$\displaystyle c_1: \mathrm{Pic}(X) \rightarrow H^2(X; \mathbb{Z}) ,$

which picks out the topological type of a line bundle. (Topologically, line bundles on a space are classified by their first Chern class.) The admissible topological types are precisely the classes in ${H^2(X; \mathbb{Z})}$ which project to ${(1,1)}$-classes in ${H^2(X; \mathbb{C})}$ under the Hodge decomposition. (more…)

Let ${A}$ be an abelian variety of dimension ${g}$ over a field ${k}$ of characteristic ${p}$. In the previous posts, we saw that to give a deformation ${R}$ of ${A}$ over a local artinian ring with residue field ${k}$ was the same as giving a continuous morphism of rings ${W(k)[[t_1, \dots, t_{g^2}]] \rightarrow A}$: that is, the local deformation space is smooth on ${g^2}$ parameters.

There is another description of the deformation space in terms of the ${p}$-divisible group, though. Given ${A}$, we can form the formal scheme

$\displaystyle A[p^\infty] = \varinjlim A[p^n],$

where each ${A[p^n]}$ is a finite group scheme of rank ${p^{2ng}}$ over ${k}$. As a formal scheme, this is smooth: that is, given a small extension ${R' \twoheadrightarrow R}$ in ${\mathrm{Art}_k}$ and a morphism ${\mathrm{Spec} R \rightarrow A[p^\infty]}$, there is an extension

In fact, we start by finding an extension ${\mathrm{Spec} R' \rightarrow A}$ (as ${A}$ is smooth), and then observe that this extension must land in some ${A[p^n]}$ if ${\mathrm{Spec} R}$ landed in ${A[p^\infty]}$.

As a result, we can talk about deformations of this formal (group) scheme.

Definition 16 deformation of a smooth, formal group scheme ${G}$ over ${k}$ over a ring ${R \in \mathrm{Art}_k}$ is a smooth, formal group scheme ${G'}$ over ${R}$ which reduces mod the maximal ideal (i.e., when restricted to ${k}$-algebras) to ${G}$.

Suppose ${G(k) = \ast}$. Using Schlessinger’s criterion, we find that (under appropriate finiteness hypotheses), ${G}$ must be prorepresentable by a power series ring ${R[[t_1, \dots, t_n]]}$ for some ${n}$. In other words, ${G}$ corresponds to a formal group over ${R}$.

Suppose for instance that ${A}$ was a supersingular elliptic curve. By definition, each of the ${A[p^n]}$ is a thickening of the zero point, and consequently ${A[p^\infty]}$ is the formal completion ${\hat{A}}$ of ${A}$ at zero. This data is equivalent to the formal group of ${A}$, and as we just saw a deformation of this formal group over a ring ${R \in \mathrm{Art}_k}$ is a formal group over ${R}$ which reduces to the formal group of ${A}$ mod the maximal ideal.

The main result is:

Theorem 17 (Serre-Tate) Let ${R \in \mathrm{Art}_k}$. There is an equivalence of categories between abelian schemes over ${R}$ and pairs ${(A, G)}$ where ${A }$ is an abelian variety over ${k}$ and ${G}$ a deformation of ${A[p^\infty]}$ over ${R}$.

So, in particular, deformations of ${A}$ over ${R}$ are equivalent to deformations of ${A[p^\infty]}$ over ${R}$.

In this post, I’ll describe an argument due to Drinfeld for this result (which I learned from Katz’s article). (more…)

(This is the third in a series of posts on deformations of abelian varieties; the first two posts are here and here.)

Let ${X/k}$ be an abelian variety. In the previous post, we saw that the functor

$\displaystyle \mathrm{Art}_k \rightarrow \mathbf{Sets}$

sending a local artinian algebra ${R}$ with residue field $k$ to the set of isomorphism classes of deformations of ${X}$ over ${R}$ was product-preserving, by showing that there were no infinitesimal automorphisms. The next step is to show that it is smooth: that is, given a deformation over ${R}$ and a surjection ${R' \twoheadrightarrow R}$, it can be lifted to a deformation over ${R'}$. Together with Schlessinger’s theorem, this will yield a description of the local moduli space. (more…)

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.

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…)