On the defect of holomorphic curves with maximal deficiency sum
by
Nobushige Toda
Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa-ku, Nagoya(466-8555), Japan

Let f be a non-degenerate transcendental holomorphic curve from C into the n-dimensional complex projective space Pⁿ(C) with a reduced representation

(f1, ... , fn+1):C --> Cn+1-{0},

where n is a positive integer.

We use the following notations as usual:

T(r, f)=  1 2\pi ó õ 2\pi 0  log||f(rei\theta)||d\theta-log||f(0)||,

m(r, a, f)=  1 2\pi ó õ 2\pi 0  log  ||a||||f(rei\theta)|| |(a, f(rei\theta)| d\theta    and    \delta(a, f)= liminf r --> \infty   m(r, a, f) T(r, f) ,

where

||f(z)||=(|f1(z)|2+ ... +|fn+1(z)|2)1/2

and for a vector a=(a₁, ... , a_n+1) in Cⁿ⁺¹-{0}

(a, f(z))=a1f1(z)+ ... +an+1fn+1(z).

Let X be a subset of Cⁿ⁺¹-{ 0} in N-subgeneral position; that is to say, # X >= N+1 and any N+1 elements of X generate Cⁿ⁺¹, where N is an integer satisfying N >= n.

H. Cartan (N=n, 1933) and E. I. Nochka (N > n, 1982) gave the following

Defect relation. For any q elements a_j   (j=1, ... , q) of X (2N-n+1 < q < \infty),

q å j=1  \delta(aj, f) <= 2N-n+1.

We are interested in the holomorphic curve the defect relation of which is extremal:

q å j=1  \delta(aj, f) = 2N-n+1.

(1)

Our main result is as follows:

Theorem . Suppose that

(i) N > n=2m  (m in N) and (ii) there exist a₁, ... , a_q in X satisfying (1),
where 2N-n+1 < q < \infty.

Then, there exist at least

[(2N-n+1)/(n+1)]+1

vectors a_j in {a₁, ... , a_q} satisfying \delta(a_j, f)=1.

Date received: July 2, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-35.