Main Theorem For Meromorphic Mappings Research Articles

Meromorphic Mappings into Projective Varieties with Arbitrary Families of Moving Hypersurfaces

In this paper, we prove a general second main theorem for meromorphic mappings into a subvariety $V$ of $\mathbb P^N(\mathbb C)$ with an arbitrary family of moving hypersurfaces. Our second main theorem generalizes and improves all previous results for meromorphic mappings with moving hypersurfaces, in particular for meromorphic mappings and families of moving hypersurfaces in subgeneral position. The method of our proof is different from that of previous authors used for the case of moving hypersurfaces.

Second main theorem for meromorphic mappings intersecting moving targets on parabolic manifolds

In this paper, we establish a new second main theorem for meromorphic mappings from $M$ into $\mathbb{P}(V)$ intersecting moving targets $g_{j}:M\rightarrow\mathbb{P}(V^{\ast}),\ 1\leq j\leq q,$ where $M$ is a parabolic manifold and $V$ is a Hermitian vector space. As an application, we prove the algebraic dependence problem for meromorphic mappings with moving targets in general position.

A general form of the Second Main Theorem for meromorphic mappings from a p-Parabolic manifold to a projective algebraic variety

A general form of the Second Main Theorem for meromorphic mappings from a p-Parabolic manifold to a projective algebraic variety

Degeneracy second main theorem for meromorphic mappings and moving hypersurfaces with truncated counting functions and applications

In this paper, we establish a new second main theorem for meromorphic mappings of [Formula: see text] into [Formula: see text] and moving hypersurfaces with truncated counting functions in the case, where the meromorphic mappings may be algebraically degenerate. A version of the second main theorem with weighted counting functions is also given. Our results improve the recent results on this topic. As an application, an algebraic dependence theorem for meromorphic mappings sharing moving hypersurfaces is given.

Erratum to “Second main theorems for meromorphic mappings and moving hyperplanes with truncated counting functions”

This erratum concerns Lemma 3.1 in the original paper [Proc. Amer. Math. Soc. 147 (2019), pp. 1657–1669]. The statement of that lemma needs to be slightly changed since the statement of Claim 3.5 must be changed. The proof of the lemma is mostly still kept as the original one with some lines modified. This change does not effect the proof of the main results.

A Difference Analogue of Cartan’s Second Main Theorem for Meromorphic Mappings

In this paper, we prove a difference analogue of Cartan’s second main theorem for a meromorphic mapping on ℂm intersecting a finite set of fixed hyperplanes in general position on ℙn(ℂ). As an application, we prove a uniqueness theorem for a class of holomorphic curves by inverse images of n + 4 hyperplanes. This result is so far the best result about the uniqueness problem for holomorphic curves by inverse images of hyperplanes.

Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces

The purpose of this paper has twofold. The first is to establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into an arbitrary projective variety intersecting a family of hypersurfaces in subgeneral position. In our result, the truncation level of the counting functions is estimated explicitly. Our result is an extension of the classical second main theorem of H. Cartan, also is a generalization of the recent second main theorem of M. Ru and improves some recent results. The second purpose of this paper is to give another proof for the second main theorem for the special case where the projective variety is a projective space, by which the truncation level of the counting functions is estimated better than that of the general case.

Second main theorems for meromorphic mappings and moving hyperplanes with truncated counting functions

In this article, we establish some new second main theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ and moving hyperplanes with truncated counting functions. Our results are improvements of the previous second main theorems for moving hyperplanes with truncated (to level $n$) counting functions.

Second main theorem for meromorphic mappings with moving hypersurfaces in subgeneral position

Let Q1,...,Qq be q slowly moving hypersurfaces in Pn(C) of degree di which are located in N-subgeneral position. Let f be a meromorphic mapping from Cm into Pn(C) which is algebraically nondegenerate over the field generated by Qi's. In this paper, we will prove that, for every ϵ>0, there exists a positive integer M such that||(q−(N−n+1)(n+1)−ϵ)Tf(r)≤∑i=1q1diN[M](r,f⁎Qi)+o(Tf(r)). Moreover, an explicit estimate for M is given. Our result is an extension of the previous second main theorems for meromorphic mappings and moving hypersurfaces.

Second Main Theorems for Meromorphic Mappings with Moving Hypersurfaces and a Uniqueness Problem

In this article, we establish some new second main theorems for meromorphic mappings of \({\mathbb {C}}^m\) into \({\mathbb {P}}^{n}({\mathbb {C}})\) and moving hypersurfaces with truncated counting functions in the case where the meromorphic mappings may be algebraically degenerate. Our results are improvements of some recent results on second main theorem in the two cases of moving hyperplanes and of moving hypersurfaces. As an application, a unicity theorem for meromorphic mappings sharing moving hypersurfaces is given.

Second Main Theorem and Unicity of Meromorphic Mappings for Hypersurfaces in Projective Varieties

Let V be a projective subvariety of \(\mathbb P^{n}(\mathbb C)\). A family of hypersurfaces \(\{Q_{i}\}_{i=1}^{q}\) in \(\mathbb P^{n}(\mathbb C)\) is said to be in N-subgeneral position with respect to V if for any 1≤i1<⋯<iN+1≤q, \( V\cap (\bigcap _{j=1}^{N+1}Q_{i_{j}})=\varnothing \). In this paper, we will prove a second main theorem for meromorphic mappings of \(\mathbb C^{m}\) into V intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give a uniqueness theorem for meromorphic mappings of \(\mathbb C^{m}\) into V sharing a few hypersurfaces without counting multiplicity. In particular, we extend the uniqueness theorem for linearly nondegenerate meromorphic mappings of \(\mathbb C^{m}\) into \(\mathbb P^{n}(\mathbb C)\) sharing 2n+3 hyperplanes in general position to the case where the mappings may be linearly degenerated.

A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables

Let c∈Cm, f:Cm→Pn(C) be a linearly nondegenerate meromorphic mapping over the field Pc of c-periodic meromorphic functions in Cm, and let Hj(1≤j≤q) be q(>2N−n+1) hyperplanes in N-subgeneral position of Pn(C). We prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the Casorati determinant of f instead of its Wronskian determinant. As its applications, we obtain a defect relation, a uniqueness theorem and a difference analogue of generalized Picard theorem.

Second main theorems with weighted counting functions and algebraic dependence of meromorphic mappings

The purpose of this article is twofold. The first is to prove a new second main theorem for meromorphic mappings and moving hyperplanes of P n ( C ) \mathbb {P}^n(\mathbb {C}) , where the counting functions are truncated multiplicities and have different weights. Our result is an extension of previous second main theorems for moving hyperplanes with the truncated (to level n n ) counting functions. As its application, the second purpose of this article is to prove a new algebraic dependence theorem for meromorphic mappings having the same inverse images of some moving hyperplanes, where the moving hyperplanes involve the assumption with different roles.

Second main theorems for meromorphic mappings intersecting moving hyperplanes with truncated counting functions and unicity problem

In this article, we show some new second main theorems for the mappings and moving hyperplanes of $\P^n(\C)$ with truncated counting functions. Our results are improvements of recent previous second main theorems for moving hyperplanes with the truncated (to level $n$) counting functions. As their application, we prove a unicity theorem for meromorphic mappings sharing moving hyperplanes.

The second main theorem for meromorphic mappings into a complex projective space

The main purpose of this article is to show the Second Main Theorem for meromorphic mappings of ℂ m into ℙ n (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 ℂ m into ℙ n (ℂ) sharing few hypersurfaces without counting multiplicity.

Second main theorem and unicity theorem for meromorphic mappings sharing moving hypersurfaces regardless of multiplicity

The purpose of this paper is twofold. The first is to establish a new second main theorem for meromorphic mappings of C m into P n ( C ) intersecting moving hypersurfaces with truncated counting functions, where the mappings may be algebraically degenerate. The second is to prove a uniqueness theorem for these mappings which share few moving hypersurfaces without counting multiplicity.

On Nevanlinna's second main theorem in projective space

We first prove a theorem concerning higher order logarithmic partial derivatives for meromorphic functions of several complex variables. Then we show the best nature of the second main theorem in Nevanlinna theory under two different assumptions of non-degeneracy of meromorphic mappingsf : ℂn → ℙn for arbitrary positive integersn andm. Moreover, we derive a upper bound of the error term in the second main theorem for meromorphic mappings of finite order. Finally, we demonstrate the sharpness of all upper bounds in our main theorems.