Galois Deformations 1: Schlessinger

Today we’ll begin a short series on deformations of Galois representations. I know very little about this topic in general, but since I brought up Taniyama-Shimura and this is one of the main tools used in proving it I thought it would be interesting to take a look at what these are. It will also be a nice example of why abstraction is often better. Today we’ll review standard deformation theory from a geometric viewpoint, but we’ll frame it in very abstract terms (the way Schlessinger did in his fantastic paper). This abstraction will allow us to apply certain results that come from geometric intuition to any functor that satisfies certain criteria.

I’ve posted a little about deformation theory around here. I won’t recall it because our framework will be slightly different. Instead of thinking about individual deformations, let’s do something more Grothendieckian and see what happens when we consider all deformations at the same time. In other words, fix some smooth varitey ${X/k}$, then given some “deformation ring” ${A}$ (to be precise later), we consider the set of all deformations of our variety ${\tilde{X}/A}$. Here is where our formalism comes in. Our set changes with ${A}$ nicely enough that ${Def_X: _\Lambda Art_k \rightarrow Set}$ actually forms a functor.

Let’s talk about the category of deformation rings now. Let ${\Lambda}$ be a Noetherian ring. The category labelled ${_\Lambda Art_k}$ means the category of local Artinian ${\Lambda}$-algebras together with a choice of augmentation map to ${k}$. This last part is important, because the morphisms between two of these rings must be both ${\Lambda}$-algebra maps and maps that commute with the augmentation to ${k}$. The reason for this is that when I pick a deformation ${\tilde{X}/A}$ I’ve done more than just specified an abstract deformation. I also am saying that the pullback square gives me a choice of isomorphism of the special fiber:

${\begin{matrix} X & \hookrightarrow & \tilde{X} \\ \downarrow & & \downarrow \\ Spec k & \rightarrow & Spec A \end{matrix}}$

where that bottom arrow is the one on spectra induced by the augmentation ${A\rightarrow k}$. Based on this geometric picture, we abstract as little as possible and call any functor ${F: _\Lambda Art_k\rightarrow Set}$ a deformation functor if it satisfies certain properties that formally correspond to having a single deformation over ${k}$ (exercise for the uninitiated: prove that if X has a non-trivial automorphism, then the functor will not satisfy this condition if the seemingly strange augmentation condition is dropped. I’ve never seen this exercise written down, but it seems important to me), being able to glue when you ought to be able to, and this gluing being unique over first order infinitesimal neighborhoods. The exact conditions can be found here (I don’t feel bad linking to that nLab page since I wrote it).

A key theorem due to Schlessinger that will come up next time is that under mild conditions to check on this functor we actually get that it is prorepresentable. This just means if we “complete” the category formally by making a new functor ${\widehat{F}: _\Lambda Noeth_k\rightarrow Set}$ by ${\widehat{F}(R)=\lim F(R/m^n )}$ the functor is actually representable here. We won’t dwell on this because when thinking about Galois representations we will have stronger things going on by continuity of our maps which will make this point less important. All this will be made more precise next time, but it is worth pointing out how amazing the forsight of Schlessinger was to formulate this in terms of functors so that it would apply all over math. His criterion says if we check certain conditions based on geometric intuition the functor will be representable. Or more intuitively, we will have a “space” that universally parametrizes everything. You can read more about this at that nLab page.

Next time we’ll bring this back from this abstraction and talk about what this means for being able to “deform” Galois representations.

Heights of Varieties

Now that we’ve defined the height of a ${p}$-divisible group we’ll define the height of a variety in positive characteristic. There are a few ways we can motivate this definition, but really it just works and turns out to be a very useful concept. We’ll mostly follow the paper of Artin and Mazur.

We could do this in more generality, but to keep things as simple as possible we’ll assume that we have a proper variety ${X}$ over a perfect field ${k}$ of characteristic ${p}$. The first motivation is that we can think about ${\mathrm{Pic}(X)}$. One way to get information about this group is to use deformation theory and look at the formal completion ${\widehat{\mathrm{Pic}}(X)}$.

The way to define this is to define the ${S}$-valued points (${S}$ an Artin local ${k}$-algebra with residue field ${k}$) to be the group fitting into the sequence ${0\rightarrow \widehat{\mathrm{Pic}}(X)(S)\rightarrow H^1(X\times S, \mathbb{G}_m)\rightarrow H^1(X, \mathbb{G}_m)}$.

So ${\widehat{\mathrm{Pic}}(X)}$ is a functor which by Schlessinger’s criterion is prorepresentable by a formal group over ${k}$. Notice that ${\widehat{\mathrm{Pic}}(X)(S)=\mathrm{ker}(\mathrm{Pic}(X\times S)\rightarrow \mathrm{Pic}(X))}$, so there is a pretty concrete way to think about what is going on. We take our scheme and consider some nilpotent thickening. The line bundles on this thickening that are just extensions from the trivial line bundle are what is in this formal Picard group.

There is no reason to stop with just ${H^1}$. We could define ${\Phi^r: Art_k\rightarrow Ab}$ by ${\Phi^r(S)}$ is the kernel of the restriction map ${H^r(X\times S, \mathbb{G}_m)\rightarrow H^r(X, \mathbb{G}_m)}$. In the cases we care about, modulo some technical details, we can apply Schlessinger type arguments to this to get that if the dimension of ${X}$ is ${n}$, then ${\Phi^n}$ is not only pro-representable, but by formal Lie group of dimension ${1}$. We’ll call this ${\Phi_X}$.

When ${n=2}$ this is just the well-known Brauer group, and so for instance the height of a K3 surface is the height of the Brauer group. We also have that if ${\Phi_X}$ is not ${\widehat{\mathbb{G}}_a}$ then it is a ${p}$-divisible group and amazingly the Dieudonne module of ${\Phi_X}$ is related to the Witt sheaf cohomology via ${D(\Phi_X)=H^n(X, \mathcal{W})}$. Recall that ${D(\Phi_X)}$ is a free ${W(k)}$-module of rank the height of ${\Phi_X}$, so in particular ${H^n(X, \mathcal{W})}$ is a finite ${W(k)}$-module!

Remember that we computed an example where that wasn’t finitely generated. So non-finite generatedness of ${H^n(X, \mathcal{W})}$ actually is related to the height in that if the variety is of finite height then ${H^n(X, \mathcal{W})}$ is finitely generated. Since we call a variety of infinite height supersingular, we can rephrase this as saying that ${H^n(X, \mathcal{W})}$ is not finitely generated if and only if ${X}$ is supersingular.

Just as an example of what heights can be, an elliptic curve must have height ${1}$ or ${2}$ and a K3 surface can have height between ${1}$ and ${10}$ (inclusive). As of right now it seems that the higher dimensional analogue of if the finite height range of a Calabi-Yau threefold is bounded is still open. People have proved certain bounds in terms of hodge numbers. For instance ${h(\Phi_X)\leq h^{1, 2}+1}$. For a general CY ${n}$-fold we have ${h\leq h^{1, n-1}+1}$.

This is pretty fascinating because my interpretation of this (which could be completely wrong) is that since for K3 surfaces the moduli space is ${20}$ dimensional, we get that (for non-supersingular) ${h^{1,1}=20}$ since this is just the dimension of the tangent space of the deformations, which for a smooth moduli should match the dimension of the moduli space. Thus we get a uniform bound (not the one I mentioned earlier).

But for CY threefolds the moduli space is much less uniform. They aren’t all deformation equivalent. They lie on different components that have different dimensions (this is a guess, I haven’t actually seen this written anywhere). So this doesn’t allow us to say ${h^{1,2}}$ is some number. It depends on the dimension of the component of the moduli that it is on (since ${h^{1,2}=\dim H^2(X, \Omega)=\dim H^1(X, \mathcal{T})}$ using the CY conditions and Serre duality). So I think it is still an open problem for how big that can be. If it can get unreasonably large, then maybe we can arbitrarily large heights of CY threefolds.

Next time maybe we’ll prove some equivalent ways of computing heights for CY varieties and talk about how height has been used by Van der Geer and Katsura and others in a useful way for K3 surfaces.