# 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.