Julia directions for holomorphic curves

A theorem of Picard type is proved for entire holomorphic mappings into projective varieties. This theorem has local nature in the sense that the existence of Julia directions can be proved under natural additional assumptions. An example is given which shows that Borel’s theorem on holomorphic curves omitting hyperplanes has no such local counterpart. Let P be complex projective space of dimension m and M ⊂ P be a projective variety. By a divisor onM we mean an intersection of a hyperplane in P with M . We study holomorphic curves f : C→M . Theorem 1. Every holomorphic map f : C → M, omitting 2n + 1 divisors such that any n+ 1 of them have empty intersection, is constant. Remark. The dimension of M is not mentioned in this formulation. Only the intersection pattern is relevant. Corollary. Every holomorphic map C→ P, omitting 2n+1 hypersurfaces, such that any n+ 1 of them have empty intersection, is constant. This Corollary also follows from the results of M. Green [4] and V. F. Babets [1]. Their proofs were based on Borel’s theorem (which we will state later). We start with a simple proof of Theorem 1, independent of Borel’s theorem. The method of the proof first appeared in [3]. It also provides a new proof of ∗Supported by NSF grant DMS-950036