RUNCATED COUNTING FUNCTIONS IN VALUE DISTRIBUTION THEORY

摘要

We present here some results related to Nevanlinna theory. Let Y be a Riemann surface with a proper surjective holomorphic map πY : Y → C. We denote by N_(ram) πY(r) the ramification counting function, which counts the number of the ramification points of πY in the domain Y(r) = {z ∈ Y; |πy(z)| ＜ r}. Let X be a smooth projective variety and let D is contained in X be a reduced divisor. Given a holomorphic map f : Y → X, we study the truncated counting function N_f, D(r), which counts the number of the points f(Y(r)) ∩ D without counting multiplicities. By the first main theorem in Nevanlinna theory, we have an upper bound N_f, D(r) ≤ T_f,D(r) + O(l) when r → ∞. Here T_f, D(r) is the order function, which is an analogue of the notion of the degree forrational maps. Our main interest here is to bound the truncated counting function N_ft D(r) from below.