Vojta's conjecture

Statement of the conjecture

Let F be a number field, let X / F be a non-singular algebraic variety, let D be an effective divisor on X with at worst normal crossings, let H be an ample divisor on X , and let K X be a canonical divisor on X . Choose Weil height functions h H and h K X and, for each absolute value v on F , a local height function λ D , v . Fix a finite set of absolute values S of F , and let ϵ > 0 . Then there is a constant C and a non-empty Zariski open set U ⊆ X , depending on all of the above choices, such that

Examples:

1. Let X = P N . Then K X ∼ − ( N + 1 ) H , so Vojta's conjecture reads ∑ v ∈ S λ D , v ( P ) ≤ ( N + 1 + ϵ ) h H ( P ) + C for all P ∈ U ( F ) .
2. Let X be a variety with trivial canonical bundle, for example, an abelian variety, a K3 surface or a Calabi-Yau variety. Vojta's conjecture predicts that if D is an effective ample normal crossings divisor, then the S -integral points on the affine variety X ∖ D are not Zariski dense. For abelian varieties, this was conjectured by Lang and proven by Faltings (1991).
3. Let X be a variety of general type, i.e., K X is ample on some non-empty Zariski open subset of X . Then taking S = ∅ , Vojta's conjecture predicts that X ( F ) is not Zariski dense in X . This last statement for varieties of general type is the Bombieri-Lang conjecture.

Generalizations

There are generalizations in which P is allowed to vary over X ( F ¯ ) , and there is an additional term in the upper bound that depends on the discriminant of the field extension F ( P ) / F .

There are generalizations in which the non-archimedean local heights λ D , v are replaced by truncated local heights, which are local heights in which multiplicities are ignored. These versions of Vojta's conjecture provide natural higher-dimensional analogues of the ABC conjecture.