一个圆环上全纯映射的Nevanlinna理论第二基本定理

为了建立圆环上的全纯映射和处于强l-次一般位置的超曲面交的第二基本定理,采用了Nochka权方法.首先用Nochka权将关于所有超曲面的Weil函数用部分超曲面的Weil函数控制;然后将超曲面嵌入到高维射影空间中变成超平面,并构造线性非退化全纯映射;进而利用圆环上关于超平面推广的Cartan第二基本定理,给出圆环上代数非退化全纯映射和超曲面处于强l-次一般位置的第二基本定理.

A Second Main Theorem of Nevanlinna Theory for Closed Subschemes in Subgeneral Position

In this paper,by using Seshadri constants for subschemes,the author establishes a second main theorem of Nevanlinna theory for holomorphic curves intersecting closed subschemes in(weak)subgeneral position.As an application of his second main theorem,he obtain a Brody hyperbolicity result for the complement of nef effective divisors.He also give the corresponding Schmidt’s subspace theorem and arithmetic hyperbolicity result in Diophantine approximation.

A Cartan’s Second Main Theorem Approach in Nevanlinna Theory

In 2002, in the paper entitled ＂A subspace theorem approach to integral points on curves＂, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt＇s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt＇s subspace in Nevanlinna theory is H. Cartan＇s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan＇s original theorem. We call such method ＂a Cartan＇s Second Main Theorem approach＂. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.

The Degenerated Second Main Theorem and Schmidt's Subspace Theorem

In this paper, we establish a Second Main Theorem for an algebraically degenerate holomorphic curve f : C → Pn(C) intersecting hypersurfaces in general position. The related Diophantine problems are also considered.

Weak Cartan-type Second Main Theorem for Holomorphic Curves

In this paper, a weak Cartan-type second theorem for holomorphic curve f ： C→P^n（C） intersecting hypersurfaces Dj, 1≤j≤q, in P^n（C） in general position with degree dj is given as follows： For every ε〉0, there exists a positive integer M such that ｜｜（q - （n ＋ 1） ε）Tf（r）≤∑j^q=1 1/dj Nf^M（r,Dj）＋o（Tf（r））, where ＂｜｜＂ means the estimate holds for all large r outside a set of finite Lebesgue measure.

Cartan's Second Main Theorem and Mason's Theorem for Jackson Difference Operator

Let f:C→P^(n)be a holomorphic curve of order zero.The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves.In addition,a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial.Furthermore,they extend the Mason’s theorem for m+1 polynomials.Some examples are constructed to show that their results are accurate.

An improvement of Chen-Ru-Yan's degenerated second main theorem

We give an improvement for the second main theorem of algebraically non-degenerate holomorphic curves into a complex projective variety V intersecting hypersurfaces in subgeneral position, obtained by Chen et al.(2012). An explicit estimate for the truncation level is also obtained in the projective normal case.

Riemann Hypothesis and Value Distribution Theory

Riemann Hypothesis was posed by Riemann in early 50’s of the 19th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper” problems of mathematics. The Riemann Hypothesis is closely related to the well-known Prime Number Theorem. The Riemann Hypothesis states that all the nontrivial zeros of the zeta-function lie on the “critical line” . In this paper, we use Nevanlinna’s Second Main Theorem in the value distribution theory, refute the Riemann Hypothesis. In reference [7], we have already given a proof of refute the Riemann Hypothesis. In this paper, we gave out the second proof, please read the reference.

Nevanlinna理论的最新进展（英文）

R.Nevanlinna在Picard定理和Borel定理基础上,发表了他的论文,并建立了一个以其名字命名的理论.此后,Nevanlinna理论已经成为在复分析、复几何和多复变函数的一个重要研究领域.该文旨在回顾以往研究中的一些重要进展,并对Nevanlinna理论研究中最新进展进行了部分综述.

一个关于处于一般位置的nef除子补的Brody双曲性结果

为了建立关于nef除子补的Brody双曲性的定量结果,采用了Nevanlinna理论的方法.首先给出一个关于非常值的全纯曲线与处于一般位置的除子交的Nevanlinna理论第二基本定理型结果,然后通过构造适当的nef和ample除子,给出了全纯曲线的像在不取某些nef除子时必为常值的结果.

从圆环到紧黎曼曲面上全纯映射的第二基本定理

通过定义一个在圆环A(r)={z∈C:1/r<|z|<r}上的次调和函数的一个特殊的流,全纯映射的特征函数和Green-Jensen公式,进而得到了圆环到紧黎曼曲面上全纯映射的第二基本定理。

Degeneracy of holomorphic curves in surfaces

Let X be a complex projective algebraic manifold of dimension 2 and let D1,..., Du be distinct irreducible divisors on X such that no three of them share a common point. Let f: C → X\(U1≤i≤uDi) be a holomorphic map. Assume that u ≥ 4 and that there exist positive integers n1,...,nu, c such that ninj(Di.Dj) = c for all pairs i, j. Then f is algebraically degenerate, i.e. its image is contained in an algebraic curve on X.

Nevanlinna theory through the Brownian motion

In this paper, we introduce the Nevanlinna theory using stochastic calculus, following the works of Davis(1975), Carne(1986) and Atsuji(1995, 2005, 2008 and 2017), etc. In particular, we give(another) proofs of the classical result of Nevanlinna for meromorphic functions and the result of Cartan-Ahlfors for holomorphic curves by using the probabilistic method.

Holomorphic Curves into Projective Varieties Intersecting Closed Subschemes in Subgeneral Position

In this paper,the authors introduce the index of subgeneral position for closed subschemes and obtain a second main theorems based on this notion.They also give the corresponding Schmidt’s subspace type theorem via the analogue between Nevanlinna theory and Diophantine approximation.