Exceptional values of holomorphic functions. Remarks on a Nishino's Theorem

Let F(z,w) epsilon O(Cn+1), where (z,w) epsilon C(n)x C. Let z' epsilon C-n such that F(z', w) is not constant. If F(z', w) is not surjective it takes all the values of C minus one pi(z') (Picard). T. Nishino studied in [8] pi(z) when n = 1, F(z, w) is of finite order in w and pi(z) is defined in a set E subset of C with at least one accumulation point. In this work, we see that his result allows to obtain an explicit expression of such a F(z, w) when n >= 1 and F(z', w) is not a constant for any z' epsilon C-n , and conclude that pi(z) = eta(z)-1/xi(z) for eta(z) and xi(z) epsilon O(C-n) when pi(z) is defined on a nonempty open set U subset of C-n . Moreover, we give several applications of this fact. We show that the complement of the graph of pi(z) in Cn+1 is dominated by Cn+1 via a family of surjective fiber-preserving holomorphic maps with non-vanishing Jacobian determinant, which are described in terms of the flow of a complete vector field of type C*. In particular, Buzzard and Lu's results in [2] applied to pi(z) for n = 1 can be extended for n >= 2. It will allow to define new examples of Oka manifolds. (c) 2017 Elsevier Inc. All rights reserved.