nLab
height of a variety

Contents

Idea

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

Definition

Let XX be a smooth proper nn dimensional variety over an algebraically closed field kk of characteristic pp. Then one can define the Artin-Mazur formal group Φ\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 Φ𝔾 a^\Phi\simeq \widehat{\mathbb{G}_a}, otherwise Φ\Phi is a p-divisible group.

Examples

For an elliptic curve, the height is either 11 in which case it is ordinary, or 22 in which case it is supersingular.

A Calabi-Yau variety of any dimension is ordinary if and only if it has height 11. For K3 surfaces the height is less than or equal to 1010 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 pp-rank, but must be 11, 22, or infinite.

Relation to Witt Cohomology

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

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

Relation to Crystalline Cohomology

Suppose that XX is a variety with the above hypotheses, then the torsion free part of the crystalline cohomology H crys n(X/W)H^n_{crys}(X/W) is a Cartier module under the action of Frobenius. We can consider the part with slopes less than 11, i.e. H crys n(X/W) WK [0,1)H^n_{crys}(X/W)\otimes_W K_{[0,1)}. The dimension of this is the height of XX.

References

Revised on July 31, 2011 00:13:42 by hilbertthm90 (24.16.144.159)