Defect relations of holomorphic curves and their associated curves in CPm

Abstract

A holomorphic curve f := (f0,...,fm) in an m-dimensional complex projective space CP m is nothing but a system of entire functions {fj}m j=0 in the complex line C, which have no common zeros. It is nondegenerate if and only if there is no linear relations among the fJ's. The defect relation for non-degenerate holomorphic curves in CP m was obtained by Ahlfors [i] and Cartan [2]. This defect relation is in terms of a (generalized) Nevanlinna characteristic function for f. The characteristic function defined by Cartan [2] for a system of entire functions is essentially the same as the order function defined by Weyl [7] for the corresponding holomorphic curve, which was then used in [i]. A modern treatment of the work by Ahlfors and Weyl can be found in Wu [8]. However, a holomorphic curve in CP m might lie in a projective linear subspace of CP m. My first result is