The second main theorem for meromorphic mappings into a complex projective space
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THE SECOND MAIN THEOREM FOR MEROMORPHIC MAPPINGS INTO A COMPLEX PROJECTIVE SPACE Do Phuong An · Si Duc Quang · Do Duc Thai
Received: 11 October 2012 / Published online: 8 March 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract The main purpose of this article is to show the Second Main Theorem for meromorphic mappings of Cm into Pn (C) intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give two unicity theorems for meromorphic mappings of Cm into Pn (C) sharing few hypersurfaces without counting multiplicity. Keywords Holomorphic curves · Algebraic degeneracy · Defect relation · Nochka weight Mathematics Subject Classification (2000) Primary 32H30 · Secondary 32H04 · 32H25 · 14J70
1 Introduction and main results q
Let {Hj }j =1 be hyperplanes of CP n . Denote by Q the index set {1, 2, . . . , q}. Let N ≥ n q and q ≥ N + 1. We say that the family {Hj }j =1 are in N -subgeneral position if for every subset R ⊂ Q with the cardinality |R| = N + 1
Hj = ∅.
j ∈R
If they are in n-subgeneral position, we simply say that they are in general position. D.P. An · S.D. Quang · D.D. Thai () Department of Mathematics, Hanoi National University of Education, 136 XuanThuy str., Hanoi, Vietnam e-mail: [email protected] D.P. An e-mail: [email protected] S.D. Quang e-mail: [email protected]
188
D.P. AN ET AL. q
Let f : Cm → CP n be a linearly nondegenerate meromorphic mapping and {Hj }j =1 be hyperplanes in N -subgeneral position in CP n . Then Cartan–Nochka’s second main theorem (see [10, 13]) stated that (q − 2N + n − 1)T (r, f ) ≤
q
N [n] r, div(f, Hi ) + o T (r, f ) .
i=1
As usual, by notation “P ” we mean that theassertion P holds for all r ∈ [0, ∞) excluding a Borel subset E of the interval [0, ∞) with E dr < ∞. Cartan–Nochka’s second main theorem plays an extremely important role in Nevanlinna theory, with many applications to Algebraic or Analytic geometry. Over the last few decades, there have been several results generalizing this theorem to abstract objects. Many contributed. We refer readers to the articles [2, 9, 11, 12, 14–19, 21, 22] and the references therein for the development of related subjects. We recall some recent results and which are the best results available at present. Let f : C → Pn (C) be a holomorphic map. Let f˜ = (f0 , . . . , fn ) be a reduced representation of f , where f0 , . . . , fn are entire functions on C and have no common zeros. The Nevanlinna–Cartan characteristic function Tf (r) is defined by 2π 1 Tf (r) = logf˜ reiθ dθ, 2π 0 where
f˜(z) = max f0 (z) , . . . , fn (z) .
The above definition is independent, up to an additive constant, of the choice of a reduced representation of f . Let D be a hypersurface in Pn (C) of degree d. Let Q be the homogeneous polynomial (form) of degree d defining D. The proximity function mf (r, D) is defined as 2π f˜(reiθ )d Q
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