Vojta's conjecture for singular varieties

Abstract

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.