Second main theorem for holomorphic mappings from the discs into the projective spaces

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In this paper, we establish a second main theorem for holomorphic mappings from a disc (R) into Pn(C) and families of hyperplanes in subgeneral position. Our result is an extension the classical second main theorem of Cartan-Nochka and the second main theorem of Fujimoto.

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Nội dung Text: Second main theorem for holomorphic mappings from the discs into the projective spaces

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0027 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 21-29 This paper is available online at http://stdb.hnue.edu.vn SECOND MAIN THEOREM FOR HOLOMORPHIC MAPPINGS FROM THE DISCS INTO THE PROJECTIVE SPACES Nguyen Van An1 and Nguyen Thi Nhung2 1 Division of Mathematics, Banking Academy 2 Department of Mathematics and Informatics, Thang Long University Abstract. In this paper, we establish a second main theorem for holomorphic mappings from a disc ∆(R) into Pn (C) and families of hyperplanes in subgeneral position. Our result is an extension the classical second main theorem of Cartan-Nochka and the second main theorem of Fujimoto. Keywords: Second main theorem, holomorphic mapping, subgeneral position. 1. Introduction Let f be a holomorphic mapping from C into Pn (C) with a reduced presentation f = (f0 : · · · : fn ) in a fixed homogeneous coordinate system (ω0 : · · · : ωn ) of Pn (C). Let H be a hyperplane in Pn (C) defined by H : {(ω0 : · · · : ωn ) : a0 ω0 + · · · + an ωn = 0}, where ai ∈ C, 0 ≤ i ≤ n. If there is no confusion, we will also denote by H the linear form H(ω0 , . . . , ωn ) = a0 ω0 + · · · + an ωn . We set ||ω|| = (|ω0 |2 + · · · + |ωn |2 )1/2 , ||f || = (|f0 |2 + · · · + |fn |2 )1/2 , ||H|| = (|a0 |2 + · · · + |an |2 )1/2 and H(f ) := a0 f0 + · · · + an fn . Then the function H(f ) depends on the choice of the reduce representation of f and representation of H. However its divisor of the zeros νH(f ) does not depend on these choices. Here, we may consider νH(f ) as a function whose value at a point z0 is the multiple intersection of the image of f and H at the point f (z0 ). The classical second main theorem for holomorphic mappings into projective spaces of H. Cartan, which is the most important second main theorem in Nevanlinna theory, is stated as follows: Theorem A (see [4]). Let f be linearly nondegenerate holomorphic mapping of C into Pn (C) and {Hj }qj=1 be hyperplanes of Pn (C) in general position, where q ≥ n + 2. Then q X (q − n − 1)Tf (r) ≤ Nn (r, νHj (f ) ) + o(Tf (r)). j=1 Received November 16, 2015. Accepted December 10, 2015. Contact Nguyen Thi Nhung, e-mail address: hoangnhung227@gmail.com 21
Nguyen Van An and Nguyen Thi Nhung Here, Tf (r) denotes the characteristic function of f and Nn (r, νHj (f ) ) denotes the counting function of the divisor νHj (f ) with multiplicity truncated to level n. These notions are defined in the next section. In 1994, H. Fujimoto [1] generalized the above result to the case of meromorphic mappings from the balls in Cm into Pn (C). We state here that result for the case of holomorphic mappings from the discs ∆(R) = {z ∈ C : |z| < R} ⊂ C as follows: Theorem B (see [1]). Let Hj (1 ≤ j ≤ q) be q (≥ n + 2) hyperplanes in general position and let f beSa linear meromorphic mapping from ∆(R) ⊂ C into Pn (C) satisfying the condition / qj=1 Hj . Then it holds that, for any r (0 < r < R), f (0) ∈ q X (q − n − 1)Tf (r) ≤ Nn (r, νHj (f ) ) + S(r), j=1 where, for any given positive ε and ρ (r < ρ < R), S(r) is evaluated as follows: 1 1 S(r) ≤K0 + K1 log+ ρ + K2 log+ + K3 log+ ρ−r r q X
|Hj (f )(0)|
q X |Hj (f )(0)|
+K4 log+
log
+ + εTf (ρ) − Wf∗ ||f (0)|| ||f (0)|| j=1 j=1 with constants K0 depending only on ε, Hj (1 ≤ j ≤ q) and with constants Kκ (1 ≤ κ ≤ 4) depending only on n. Here, the quantity Wf∗ is defined by
f0 (0) f1 (0) ··· fn (0)
′
1
f0 (0) f ′ 1 (0) ··· fn′ (0)
Wf∗ = ·
.. .. ..
. ||f (0)||n+1