Published online by Cambridge University Press: 01 July 2009
The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of , that with respect to the height are “very close” to a given subgroup Γ of finite rank of . Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form.
In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.