  Let F be a global field. In this work, we show that the Brauer-Manin
condition on adelic points for subvarieties of a torus T over F cuts out
exactly the rational points, if either F is a function field or, if F is the
field of rational numbers and T is split. As an application, we prove a
conjecture of Harari-Voloch over global function fields which states, roughly
speaking, that on any rational hyperbolic curve, the local integral points with
the Brauer-Manin condition are the global integral points. Finally we prove for
tori over number fields a theorem of Stoll on adelic points of zero-dimensional
subvarieties in abelian varieties.
