The height of a variety should reflect how close the variety is to being ordinary and other arithmetic properties.
Let be a smooth proper dimensional variety over an algebraically closed field of characteristic . Then one can define the Artin-Mazur formal group . 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 , otherwise is a p-divisible group.
For an elliptic curve, the height is either in which case it is ordinary, or in which case it is supersingular.
A Calabi-Yau variety of any dimension is ordinary if and only if it has height . For K3 surfaces the height is less than or equal to 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 -rank, but must be , , or infinite.
Let be the sheaf of Witt vectors on a variety satisfying the conditions above. If has finite height, then the Dieudonne module of the Artin-Mazur formal group is isomorphic to . By standard Dieudonne theory, is a free of rank module over , so where is the fraction field of .
One consequence of the above is that is supersingular (of infinite height) if is not a finite-type -module. It is possible also that it is a torsion module in which case and again we can conclude that is of infinite height (since if were of finite height it would be a free module).
Suppose that is a variety with the above hypotheses, then the torsion free part of the crystalline cohomology is a Cartier module under the action of Frobenius. We can consider the part with slopes less than , i.e. . The dimension of this is the height of .