Báo cáo toán học: An Extension of Uniqueness Theorems for Meromorphic Mappings

Trong bài báo này, chúng tôi cung cấp cho một số kết quả về số lượng của ánh xạ meromorphic của Cm vào CP n trong điều kiện là những hình ảnh nghịch đảo của hyperplanes trong CP n. Đồng thời, chúng tôi đưa ra một câu trả lời cho một câu hỏi mở được đặt ra bởi H. Fujimoto vào năm 1998.
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Báo cáo toán học: "An Extension of Uniqueness Theorems for Meromorphic Mappings"Vietnam Journal of Mathematics 34:1 (2006) 71–94 9LHWQD P -RXUQDO RI 0$ 7+ (0$ 7, &6 ‹ 9$67 An Extension of Uniqueness Theorems for Meromorphic Mappings Gerd Dethloff1 and Tran Van Tan2 1 Universit´ de Bretagne Occidentale UFR Sciences et Techniques e D´partement de Math´matiques 6, avenue Le Gorgeu, e e BP 452 29275 Brest Cedex, France 2 Dept. of Math., Hanoi University of Education, 136 Xuan Thuy Road Cau Giay, Hanoi, Vietnam Received February 22, 2005 Revised June 20, 2005Abstract. In this paper, we give some results on the number of meromorphic map-pings of Cm into CP n under a condition on the inverse images of hyperplanes in CP n .At the same time, we give an answer for an open question posed by H. Fujimoto in1998.1. IntroductionIn 1926, Nevanlinna showed that for two nonconstant meromorphic functionsf and g on the complex plane C, if they have the same inverse images for fivedistinct values, then f = g , and that g is a special type of a linear fractional tran-formation of f if they have the same inverse images, counted with multiplicities,for four distinct values. In 1975, Fujimoto [2] generalized Nevanlinna’s result to the case of mero-morphic mappings of Cm into CP n . This problem continued to be studied bySmiley [9], Ji [5] and others. Let f be a meromorphic mapping of Cm into CP n and H be a hyperplanein CP n such that imf H. Denote by v(f,H ) the map of Cm into N0 such thatv(f,H ) (a) (a ∈ Cm ) is the intersection multiplicity of the image of f and H atf (a). Let k be a positive interger or +∞. We set72 Gerd Dethloff and Tran Van Tan 0 if v(f,H ) (a) > k, k) v(f,H ) (a) = v(f,H ) (a) if v(f,H ) (a) k.Let f be a linearly nondegenerate meromorphic mapping of Cm into CP n and{Hj }q=1 be q hyperplanes in general position with j k) k) m − 2 for all 1(a) dim z : v(f,Hi ) (z ) > 0 and v(f,Hj ) (z ) > 0 i 0 . j =1 In [5], Ji proved the followingTheorem J. [5] If q = 3n + 1 and k = +∞, then for three mappings f1 , f2 , f3 ∈Fk {Hj }q=1 , f, 1 , the mapping f1 × f2 × f3 : Cm −→ CP n × CP n × CP n is jalgebraically degenerate, namely, {(f1 (z ), f2 (z ), f3 (z )), z ∈ Cm } is contained ina proper algebraic subset of CP n × CP n × CP n . In 1929, Cartan declared that there are at most two meromorphic functionson C which have the same inverse images (ignoring multiplicities) for four dis-tinct values. However in 1988, Steinmetz [10] gave examples which showed thatCartan’s declaration is false. On the other hand, in 1998, Fujimoto [4] showedthat Cartan’s declaration is true if we assume that meromorphic functions on Cshare four distinct values counted with multiplicities truncated by 2. He gavethe following theoremTheorem F. [4] If q = 3n + 1 and k = +∞ then Fk {Hj }q=1 , f, 2 contains at jmost two mappings. He also proposed an open problem asking if the number q = 3n+1 in TheoremF can be replaced by a smaller one. Inspired by this question, in this paper wewill generalize the above results to the case where the number q = 3n + 1 isin fact replaced by a smaller one. We also obtain an improvement concerningtruncating multiplicities. 2 Denote by Ψ the Segre embedding of CP n × CP n into CP n +2n which isdefined by sending the ordered pair ((w0 , ..., wn ), (v0 , ..., vn )) to (..., wi vj , ...) (inlexicographic order). Let h : Cm −→ CP n × CP n be a meromorphic mapping. Let (h0 : ... :hn2 +2n ) be a representation of Ψ ◦ h . We say that h is linearly degenerate(with the algebraic structure in CP n × CP n given by the Segre embedding) ifh0 , ..., hn2 +2n are linearly dependent over C. Our main results are stated as follows:Theorem 1. There are at most two distinct mappings in Fk {Hj }q=1 , f, p in jeach of the following cases: ...