  We formulate a generalization of Vojta's conjecture in terms of log pairs and
variants of multiplier ideals. In this generalization, a variety is allowed to
have singularities. It turns out that the generalized conjecture for a log pair
is equivalent to the original conjecture applied to a log resolution of the
pair. A special case of the generalized conjecture can be interpreted as
representing a general phenomenon that there tend to exist more rational points
near singular points than near smooth points. The same phenomenon is also
observed in relation between greatest common divisors of integer pairs
satisfying an algebraic equation and plane curve singularities, which is
discussed in Appendix. As an application of the generalization of Vojta's
conjecture, we also derive a generalization of a geometric conjecture of Lang
concering varieties of general type to singular varieties and log pairs.
