Holomorphic maps into complex projective space omitting hyperplanes

Abstract:

Using methods akin to those of E. Borel and R. Nevanlinna, a generalization of Picard’s theorem to several variables is proved by reduction to a lemma on linear relations among exponentials of entire functions. More specifically, it is shown that a holomorphic map from ${{\mathbf {C}}^m}$ to ${{\mathbf {P}}_n}$ omitting $n + 2$ distinct hyperplanes has image lying in a hyperplane. If the map omits $n + 2$ or more hyperplanes in general position, the image will lie in a linear subspace of low dimension, being forced to be constant if the map omits $2n + 1$ hyperplanes in general position. The limits found for the dimension of the image are shown to be sharp.