A Unicity Theorem for Meromorphic Mappings

We prove a unicity theorem of Nevanlinna for meromorphic mappings of P into Pm. 1. INTR~DuOTI~N As an application of Nevanlinna’s second main theorem and Borel’s lemma, R. Nevanlinna proved that for any two meromorphic functions in the complex plane @ on which they share four distinct values, then, these two meromorphic functions are the same up to a Mijbius transformation. Since then, there have been a number of papers (e.g. [4], [2], and [8]) working towards this kind of problems. Recently, motivated by the accomplishment of the second main theorem for moving targets (cf. [6]), M. Sh’ lrosaki [9] has proved a unicity theorem of meromorphic functions for moving targets, i.e. replacing four values in the original problem by four ‘small’ functions. However, his result is only dealing with one complex variable. In this paper, we extend this kind of theorem to the case of meromorphic mappings of C? into IF” for moving targets. Broadly speaking, for any two meromorphic mappings of Cc” into F sharing 2(m + 1) ‘small’ mappings in a certain sense, then, there is a non-zero bilinear function vanishing on these two meromorphic mappings. Particularly, when m = 1, these two meromorphic mappings in Cc” are the same up to a Mijbius transformation. Thus, Shirosaki’s result is a special case of ours when n = m = 1. 1991 Mathematics Subject Class$cation. Primary 32A22 and 32H30. The author thanks Manabu Shirosaki for his reprints and the referee for suggestions and comments. 519