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A uniqueness theorem for meromorphic mappings with hypersurfaces and without counting multiplicities

In 1926, R. Nevanlinna [6] showed that for two nonconstant meromorphic functions f and g on the complex plane C , if they have the same inverse images for five distinct values then f=g. In 1975, H. Fujimoto [4] generalized the above result to the case of meromorphic mappings of m C into n. | T¹p chÝ Khoa häc & C«ng nghÖ - Sè 1(45) Tập 1/N¨m 2008 A UNIQUENESS THEOREM FOR MEROMORPHIC MAPPINGS WITH HYPERSURFACES AND WITHOUT COUNTING MULTIPLICITIES Bùi Khánh Trình (Trường Đại học Xây dựng-Hà Nội) 1. Introduction In 1926, R. Nevanlinna [6] showed that for two nonconstant meromorphic functions f and g on the complex plane C , if they have the same inverse images for five distinct values then f=g. In 1975, H. Fujimoto [4] generalized the above result to the case of meromorphic mappings of C m into CP n . Since that time this problem has been studied intensively by H. Fujimoto, W. Stoll, L. Smiley, G. Dethloff, D. D. Thai, T. V. Tan, S. Ji, S. D. Quang, M. Ru and others. We would like to note that their results about uniqueness problem of meromorphic mappings of C m into CP n have been still restricted to the case of hyperplanes. The aim of this paper is to give a uniqueness theorem for the case of hypersurfaces. 2. Preliminaries 1/ 2 2.1. For m 2 , we set z = z = ( z1 ., z m ) ∈ C ∑ zj j =1 m and define: { B (r ) = {z ∈C m : z 〈 r}, S (r ) = z ∈ C m : z = r dc = ( −1 ( ∂ − ∂ ) , υ = dd c z 4π ) 2 m −1 , σ = d c log z ∧ ( dd c log z 2 ) m −1 Let F be a nonzero holomorphic function on C m . For a set α = (α1 ,., α n ) of nonnegative integers, we set α = α1 + α 2 + . + α n and D α F = α ∂ F . We define the map α1 ∂ z1.∂α n zn ν F ( z ) := max{m :D α F ( z ) = 0 for all α with α 1. Second Main Theorem. (Classical version) Let f be a linearly nondegenerate meromorphic mapping of C m into CP n and H1 ,., H q (q ≥ n + 1) be hyperplanes in CP n in general position. Then q (q − n − 1)T f (r ) ≤ ∑ N ([ nf ], H j ) (r ) + ο (T f (r )) j =1 for all r except a subset E of (1, +∞ ) of finite Lebesgue measure. Theorem 2.4 [1] Let f be an algebraically nondegenerate meromorphic mapping of C m into CP n . Let Q1 ,., Qq be hypersurfaces in CP n in general position of common degree d ≥ 1 . Then there exists positive integer L, .