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

Let F(z,w)∈O(Cn+1), where (z,w)∈Cn×C. Let z′∈Cn such that F(z′,w) is not constant. If F(z′,w) is not surjective it takes all the values of C minus one π(z′) (Picard). T. Nishino studied in [8] π(z) when n=1, F(z,w) is of finite order in w and π(z) is defined in a set E⊂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′∈Cn, and conclude that π(z)=η(z)−1/ξ(z) for η(z) and ξ(z)∈O(Cn) when π(z) is defined on a nonempty open set U⊂Cn. Moreover, we give several applications of this fact. We show that the complement of the graph of π(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 π(z) for n=1 can be extended for n≥2. It will allow to define new examples of Oka manifolds.