The error term of holomorphic mappings in Nevanlinna theory

Abstract

Abstract. We construct a holomorphic mapping from Cm to P" for any m and n with m > n > 1 which shows Cherry-Lang-Wong's upper bound of the error term of the Second Main Theorem in Nevanlinna theory is essentially the best possible. Thus a question of Serge Lang is answered affirmatively in higher dimensions. IntroductionThe classical one-dimensional Nevanlinna theory studies the number of solutions to the equation f(z) -a in the disc of radius r as r tends to infinity asymptotically. In the higher-dimensional Nevanlinna theory one studies holomorphic mappings from Cm into an «-dimensional complex manifold X. One is interested not only in the preimages of points but also in the inverse images of complex analytic subsets of X of positive dimension. Of all the various theorems in Nevanlinna theory, the Second Main Theorem is the most important.In recent years, there has been considerable interest in finding the precise error term of the Second Main Theorem. Such a question was raised by Serge Lange [4] who was inspired by Vojta's [9] analogy between Nevanlinna theory and Diophantine approximation. In 1990, S. Lang [5] found the best nature of the upper bound of the Second Main Theorem by improving Wong's method in [10] in the equidimensional case. Later, Z. Ye [14] showed all upper bounds of the error terms in C conjectured by Lang are sharp. Recently, based on Lang's method, Cherry [1] obtained an upper bound for the error term when the dimension of the domain is not less than that of the image space under the assumption that / is a non-degenerate holomorphic mapping. On the other hand, Wong and Stoll [11] also obtained an upper bound for the error term when the dimension of the domain is less than that of the image space under the assumption that / is a linearly non-degenerate meromorphic mapping. However their upper bound has not been verified to be sharp yet. In this paper, we show the main term in the upper bound in Cherry-Lang-Wong's theorem is sharp when the dimension of the domain is not less than the dimension of the image space.