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height of a variety

Idea

The height of a variety should reflect how close the variety is to being ordinary and other arithmetic properties.

Definition

Let $X$ be a smooth proper $n$ dimensional variety over an algebraically closed field $k$ of characteristic $p$. Then one can define the Artin-Mazur formal group $\Phi$. Since it is a one-dimensional formal group, it is completely determined up to isomorphism by its height?. This is the height of the variety . The height could be infinite if $\Phi \simeq \widehat{{𝔾}_a}$, otherwise $\Phi$ is a p-divisible group.

Examples

For an elliptic curve, the height is either $1$ in which case it is ordinary, or $2$ in which case it is supersingular.

A Calabi-Yau variety of any dimension is ordinary if and only if it has height $1$. For K3 surfaces the height is less than or equal to $10$ or infinite, but for all higher dimensional Calabi-Yau varieties the height has no known bound. Infinite height Calabi-Yau varieties are known as supersingular.

The height of an abelian variety depends on its $p$-rank, but must be $1$, $2$, or infinite.

Relation to Witt Cohomology

Let $𝒲$ be the sheaf of Witt vectors on a variety $X$ satisfying the conditions above. If $X$ has finite height, then the Dieudonne module of the Artin-Mazur formal group is isomorphic to $H^n(X,𝒲)$. By standard Dieudonne theory, $D(\Phi )$ is a free of rank $\mathrm{ht}(X)$ module over $W$, so $\mathrm{ht}(X)={\dim }_K{H}^n(X,𝒲)\otimes K$ where $K$ is the fraction field of $W$.

One consequence of the above is that $X$ is supersingular (of infinite height) if $H^n(X,𝒲)$ is not a finite-type $W$-module. It is possible also that it is a torsion module in which case $H^n(X,𝒲)\otimes K=0$ and again we can conclude that $X$ is of infinite height (since if $X$ were of finite height it would be a free module).

Relation to Crystalline Cohomology

Suppose that $X$ is a variety with the above hypotheses, then the torsion free part of the crystalline cohomology $H_{\mathrm{crys}}^n(X/W)$ is a Cartier module under the action of Frobenius. We can consider the part with slopes less than $1$, i.e. $H_{\mathrm{crys}}^n(X/W){\otimes }_W{K}_{[0,1)}$. The dimension of this is the height of $X$.

Related concepts

• formal group

• Artin-Mazur formal group

References

• Michael Artin, Barry Mazur, Formal Groups Arising from Algebraic Varieties, numdam, MR56:15663