PROBLEMS OF VALUE DISTRIBUTION IN DIMENSIONS HIGHER THAN UNITY

Abstract

Consider two -dimensional complex manifolds and , where is assumed to be compact. Suppose that on a form is give, which defines an element of volume, and on a function with isolated critical points and such that the domain is relatively compact for all . For each point we construct on a form of bidegree with certain special properties which allow us to use a more or less standard techniques to prove the following “first main theorem”: if a holomorphic map is non-degenerate for at least one point, then where denotes the integral , and the integral ; here is the number of points (including multiplicities) such that .Under various conditions on the exhaustion and the mapping we obtain various theorems which assert that when these conditions hold, then the quantity grows for almost all (over some subsequence of numbers ) at the same rate as .We also consider the case of real manifolds and smooth maps. Here we obtain analogous results, though by different methods.