Equidistribution of Divisors for Sequences of Holomorphic Curves

I. M. Dektyarev

Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 3, pp. 17-25

Résumé

We 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 infinity, then for every divisor one can define a defect 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 classical 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.