On Cartan's theorem and Cartan's conjecture

Abstract

We give a mild generalization of Cartan's theorem on value distribution for a holomorphic curve in projective space relative to hyperplanes. This generalization is used to complete the proof of the following theorem claimed in an earlier paper by the author: Given hyperplanes in projective space in general position, there exists a finite union of proper linear subspaces such that all holomorphic curves not contained in that union (even linearly degenerate curves) satisfy the inequality of Cartan's theorem, except for the ramification term. In addition, it is shown how these methods can lead to a shorter proof of Nochka's theorem on Cartan's conjecture and (in the number field case) how Nochka's theorem gives a short proof of Wirsing's theorem on approximation of algebraic numbers by algebraic numbers of bounded degree.