Julia directions for holomorphic curves

TL;DR

The paper proves a Picard-type theorem for entire holomorphic mappings into complex projective varieties, introducing the concept of Julia directions and highlighting differences from Borel's theorem.

Contribution

It establishes a local version of Picard's theorem involving Julia directions and provides a counterexample related to Borel's theorem.

Findings

Existence of Julia directions under certain conditions

Local Picard-type theorem for holomorphic curves

Counterexample to local Borel's theorem

Abstract

A theorem of Picard's type is proved for entire holomorphic mappings into complex projective varieties. This theorem has local character in the sense that the existence of Julia directions can be proved under a natural additional assumption. An example is given which shows that Borel's theorem on holomorphic curves omitting hyperplanes does not have such a local counterpart.