Introduction Suppose M is a connected, complex manifold of dimension m. Suppose that there exists a parabolic exhaustion r on M. Then, (M,T) is called a parabolic manifold. Let V be a vector space of dimension n + 1 1. Let / : M P(V) be a meromorphic map and 0 a finite set of target meromorphic maps g : M P(V*) in general position. Let Tj and T5 be the characteristic functions of / and #, respectively. Assume that each g in (* 5 grows slower than / , i.e. that for 0 s r, r-+oo j y ( r , S) Denote by Nfi9 the valence function of the divisor /x/jP. For f or g nonconstant and 0 s r, the Nevanlinna defect of / for g is given by 0 g(/,g) = 1 - limsup ^ | y } r r , 1- Now, suppose M = C and n = 1. For # in N, the incidence can be considered of the image of / and distinct fixed targets {aj}q-x contained in Pi = C U {oo}. In this case, R. Nevanlinna derived in 1924 his defect relation 7= 1 Notice that this gives a generalization of Picard's Theorem. Then, in 1929, Nevan- linna [5] conjectured that this defect relation remains valid if the fixed targets are replaced by a finite set (3 of meromorphic target functions g : C Pi growing slower than / . He proved this conjecture when 0 consists of three target functions. C. T. Chuang [4] verified the conjecture for entire functions in 1964. In 1986, N. Steinmetz [8] generalized Chuang's method and proved Nevanlinna's conjecture. Next, suppose that n 1. Also, suppose that / : C P(V) is a holomorphic curve and 0 is a finite set of holomorphic curves g : C P(Vr*). Then, assuming each g in (3 grows slower than / , W. Stoll [14] proved in 1986 that J ] «(/,/) n(n + l). gee In 1990, under the same assumptions, M. Ru-W. Stoll [7] proved that gee where n + 1 is a sharp bound. This result can be extended to meromorphic maps / : C m P n . Also in 1990, E. Bardis [2] further extended this result to ramified Received by the editor June 6, 1996, and in revised form January 5, 1998 1
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