Let be a number field, let be a non-singular algebraic variety, let be an effective divisor on with at worst normal crossings, let be an ample divisor on , and let be a canonical divisor on . Choose Weil height functions and and, for each absolute value on , a local height function . Fix a finite set of absolute values of , and let . Then there is a constant and a non-empty Zariski open set , depending on all of the above choices, such that
Let . Then , so Vojta's conjecture reads for all .
Let 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 is an effective ample normal crossings divisor, then the -integral points on the affine variety are not Zariski dense. For abelian varieties, this was conjectured by Lang and proven by Faltings (1991).
Let be a variety of general type, i.e., is ample on some non-empty Zariski open subset of . Then taking , Vojta's conjecture predicts that is not Zariski dense in . This last statement for varieties of general type is the Bombieri–Lang conjecture.
There are generalizations in which is allowed to vary over , and there is an additional term in the upper bound that depends on the discriminant of the field extension .
There are generalizations in which the non-archimedean local heights 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.