Second Main Theorem of Nevalinna Theory for Nonequidimensional Meromorphic Maps

Wong, Pit-Mann; Stoll, Wilhelm

doi:10.2307/2374940

Cited by 25 publications

(20 citation statements)

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“…Denote by Γ = max 1≤j≤q { n−1 k=0 m f k (s 0 , H j )} and Λ(r) = min k {1/(1 + T f k (r))}. For any µ ∈ T, z ∈ I f , the Product to Sum Estimate (see [27] Lemma 1.12), with λ = Λ(r), reads…”

Section: Theorem 373 (Ahlfors Estimate Ii) Let H Be a Hyperplane Inmentioning

confidence: 99%

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Some generalizations of the second main theorem intersecting hypersurfaces

2014

Methods and Applications of Analysis

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Abstract. In [18], the author established a defect relation for holomorphic mappings from C into the projective variety X ⊂ P N (C) intersecting divisors of X coming from hypersurfaces in P N . The purpose of this paper is to do some further generalizations. It contains two major parts: (I) We extend the defect relation to effective divisors on X with P ic(X) = Z. (II) We extend the defect relation to meromorphic mappings from parabolic manifolds into X ⊂ P N .

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Section: Theorem 373 (Ahlfors Estimate Ii) Let H Be a Hyperplane Inmentioning

confidence: 99%

“…If we take τ = π 2 , Stoll [23] showed that there exists a holomorphic differential form B of degree (m − 1, 0) on M (c.f. [23] or [27]) such that…”

Section: Theorem 373 (Ahlfors Estimate Ii) Let H Be a Hyperplane Inmentioning

confidence: 99%

Some generalizations of the second main theorem intersecting hypersurfaces

2014

Methods and Applications of Analysis

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“…License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use (13) It follows from (7), (10), (11), and (12) that N(E, 0,r)<T(E, r)<logM(E, r) = logE(r)<(l+o(l))N(E,0,r) (r -oo) and so, as r -y oo,…”

Section: Lemmasmentioning

confidence: 99%

“…Recently, based on Lang's method, Cherry [1] obtained an upper bound for the error term when the dimension of the domain is not less than that of the image space under the assumption that / is a non-degenerate holomorphic mapping. On the other hand, Wong and Stoll [11] also obtained an upper bound for the error term when the dimension of the domain is less than that of the image space under the assumption that / is a linearly non-degenerate meromorphic mapping. However their upper bound has not been verified to be sharp yet.…”

Section: Introductionmentioning

confidence: 99%

The error term of holomorphic mappings in Nevanlinna theory

Ye¹

1995

Proc. Amer. Math. Soc.

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Abstract. We construct a holomorphic mapping from Cm to P" for any m and n with m > n > 1 which shows Cherry-Lang-Wong's upper bound of the error term of the Second Main Theorem in Nevanlinna theory is essentially the best possible. Thus a question of Serge Lang is answered affirmatively in higher dimensions. IntroductionThe classical one-dimensional Nevanlinna theory studies the number of solutions to the equation f(z) -a in the disc of radius r as r tends to infinity asymptotically. In the higher-dimensional Nevanlinna theory one studies holomorphic mappings from Cm into an «-dimensional complex manifold X. One is interested not only in the preimages of points but also in the inverse images of complex analytic subsets of X of positive dimension. Of all the various theorems in Nevanlinna theory, the Second Main Theorem is the most important.In recent years, there has been considerable interest in finding the precise error term of the Second Main Theorem. Such a question was raised by Serge Lange [4] who was inspired by Vojta's [9] analogy between Nevanlinna theory and Diophantine approximation. In 1990, S. Lang [5] found the best nature of the upper bound of the Second Main Theorem by improving Wong's method in [10] in the equidimensional case. Later, Z. Ye [14] showed all upper bounds of the error terms in C conjectured by Lang are sharp. Recently, based on Lang's method, Cherry [1] obtained an upper bound for the error term when the dimension of the domain is not less than that of the image space under the assumption that / is a non-degenerate holomorphic mapping. On the other hand, Wong and Stoll [11] also obtained an upper bound for the error term when the dimension of the domain is less than that of the image space under the assumption that / is a linearly non-degenerate meromorphic mapping. However their upper bound has not been verified to be sharp yet. In this paper, we show the main term in the upper bound in Cherry-Lang-Wong's theorem is sharp when the dimension of the domain is not less than the dimension of the image space.

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“…[7], [15], [20], [4] and [22]) from both fields have started to look for more analogues between these two theories. It is known that a famous theorem of Roth in number theory is analogous to a weak form of the second main theorem in Nevanlinna theory, and the Artin-Whaples product formula in number theory can be viewed as an analogue of the first main theorem in Nevanlinna theory.…”

Section: Introductionmentioning

confidence: 99%

An analogue of continued fractions in number theory for Nevanlinna theory

2004

Trans. Amer. Math. Soc.

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Abstract. We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory. IntroductionSince P. Vojta [17] created a dictionary between Nevanlinna theory and Diophantine approximation theory, researchers (e.g. [7], [15], [20], [4] and [22]) from both fields have started to look for more analogues between these two theories. It is known that a famous theorem of Roth in number theory is analogous to a weak form of the second main theorem in Nevanlinna theory, and the Artin-Whaples product formula in number theory can be viewed as an analogue of the first main theorem in Nevanlinna theory. Theoretically speaking, we should be able to find an analogue of any theorem related to the Roth theorem in Diophantine approximation for Nevanlinna theory, and vice versa. An up-to-date account of these matters appears in [2], [18] and [14]. The author [23] has found an analogue of Khinchin's theorem for Nevanlinna theory, which has given an answer to one of S. Lang's questions in [9]. S. Lang also suggested (in a personal conversation) finding an analogue of the continued fractions for Nevanlinna theory. This is an interesting question that has been around for a while. In this paper, we find an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory. Notation and preliminariesFor the convenience of the general readers, we briefly give some definitions and notation in Nevanlinna theory and continued fractions. Standard references are [3] and [12] for Nevanlinna theory, and [5] and [8] for continued fractions.Let f be a meromorphic function on the whole plane and D r = {|z| < r}. Denote the number of poles of f in D r by n(f, ∞, r), and define n(f, a, r) = n(1/(f − a), ∞, r) if a ∈ C. We also let N (f, a, r) = r 0 n(f, a, t) − n(f, a, 0) t dt + n(f, a, 0) log r.

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Second Main Theorem of Nevalinna Theory for Nonequidimensional Meromorphic Maps

Cited by 25 publications

References 20 publications

Some generalizations of the second main theorem intersecting hypersurfaces

Some generalizations of the second main theorem intersecting hypersurfaces

The error term of holomorphic mappings in Nevanlinna theory

An analogue of continued fractions in number theory for Nevanlinna theory

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