Equidistribution of divisors for sequences of holomorphic curves

Abstract

I. M. Dektyarev UDC 515.16We study holomorphic curves in an n-dimensional complex manifold on which a family of divisors parametrized by an m-dimensional compact complex manifold is given. If, for a given sequence of such curves, their areas (in the induced metric) monotonically tend to infinitT, then for every divisor o.ne can define a defec~ characterizing the deviation of the frequency at which this sequence intersects the divisor from the average frequency (over the set of all divisors). It turns out that, as well as in the clwssical multidimensional case, the set of divisors with positive defect is very rare. (We estimate how rare it is.) Moreover, the defect of almost all divisors belonging to a linear subsystem is equal to the mean value of the defect over the subsystem, and for all divisors in the subsystem (without any exception) the defect is not less than this mean value.In a general way, the value distribution theory in higher dimensions can be described as follows. Consider a holomorphic mapping of a noncompact complex manifold equipped with an exhaustion function into a compact manifold. The problem is to study functions relating the growth characteristics of this mapping to the growth characteristics of the preimages of points or divisors. One also studies the structure of the set of exceptional (deficient) values or divisors, i.e., values or divisors for which the behavior of these functions differs from the '~ypicar' behavior. Here "growth" is related to values of the exhaustion function.There are some recent papers, in particular, ones related to complex dynamics, in which similar problems are studied for sequences of mappings (usually polynomial). In this case, the role of the exhaustion function is played by the degree of the polynomials defining the mapping. Apparently, [7] is the first paper in this direction. The present paper belongs in the same direction.We study holomorphic curves f: C -~ Q in an n-dimensional complex manifold Q (not necessarily compact) on which a family of divisors parametrized by an m-dimensional (m > 1) compact complex manifold M is given. By D r C Q we denote the divisor corresponding to y E ~/I. Consider the set W -{(q, y) : q E D~} in the direct product Q • iVI. Throughout the following, we assume that W is a submanifold, the projection 0: 17~ --, Q is regular and surjective, and the projection ,~: W -, Ai r is an open mapping. We treat 0 as a bundle with base Q and fiber Sq = {y : y E Ai r, q E Dr} over q E Q.Moreover, we assume that M is equipped with an Hermitian metric such that the associated (