Deficiencies of the Associated Curves of a Holomorphic Curve in the Projective Space

Abstract

Let kx be the nonconstant associated holomorphic curve of rank k (1 k 5 n 1) of a transcendental holomorphic curve x: C -Pn C. Itisproved that if 1 _ k n -2 andAk E P1(k) C,j = 1,... 21(k) 2 (1(k) = (kn+)) are in general position and # 0 for all Ak, then 2-21()-2 dk(Ak) _ 21(k) 3 and that in the case when k = n 1, 'An-I sn(16n-l) _ I(n-1), where {An-1} is a finite subset of PI(n_1)_IC in general position such that > 0 for all An-I . These are sharp. 1. The theory of holomorphic (or meromorphic) curves was initiated by H. and J. Weyl in 1938 [7], [8]. Its main problem, the proof of the so-called defect relations, was solved by Ahlfors [1]. A modern detailed treatment was given by Wu [9]. It is a natural generalization of the Picard-Borel-Nevanlinna theory. On the other hand, a generalization of the Nevanlinna theory to systems of entire functions had been tried by Cartan in 1933 [2]. If a system of entire functions (x0, xl, . . ., xn) is given, then we can see it as a reduced representation of a holomorphic curve x: C -> Pn C. By the reasoning in [2, pp. 7-8] and [8, pp. 81-84] or [9, pp. 104-105], the characteristic function defined by Cartan for the system of entire functions is essentially equal to the order function defined by Weyl for the corresponding holomorphic curve in the projective space. Hence the results on systems of entire functions can be restated by the statements on holomorphic curves in the projective space. The defect relation for nondegenerate holomorphic curves in the projective space was obtained by Ahlfors [1] and Cartan [2]. But it seems that a defect relation for degenerate holomorphic curves in the projective space has not been yet obtained in a sharp form. Concerning it there are a result of Toda [5] and a conjecture of Cartan [2]. Recently, Toda [6] made another contribution to the study of degenerate holomorphic curves which is in turn a generalization of the result of the author and Ozawa [3], [4] on entire algebroid functions. In the language of holomorphic curves (cf. [9]), Toda's result concerns the defects of the original curve. Received by the editors June 2, 1975 and, in revised form, December 22, 1975. AMS (MOS) subject classifications (1970). Primary 30A70.